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Temperature-driven spectral weight transfer in doped magnetic insulators Möller, Mirko 2016

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Temperature-driven spectral weighttransfer in doped magnetic insulatorsbyMirko Mo¨llerB.Sc., Freie Universita¨t Berlin, 2010M.Sc., Freie Universita¨t Berlin, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c© Mirko Mo¨ller 2016AbstractIn this thesis we study the effects of finite temperature (T ) on the single-electron spectral function of doped magnetic insulators.First, we derive the low-temperature correction to the self-energy of acharge carrier injected with parallel spin into a ferromagnetic backgroundwhich is modeled with both Heisenberg and Ising Hamiltonians so that dif-ferences due to gapless versus gapped magnons can be understood. Besidethe expected thermal broadening of the T = 0 quasiparticle peak whichbecomes a resonance inside a continuum, we find that spectral weight istransferred to regions lying outside this continuum, because the carrier anda thermal magnon can bind into a spin-polaron. This work is valid in di-mensions d ≥ 2, because it does not include the role of magnetic domainswhich are important in 1d.We then consider the role of these magnetic domains in 1d systems,for models where spin-polaron formation is impossible. We present MonteCarlo simulations for the spectral function of three related models of a chargecarrier that is injected into an Ising chain. Both ferromagnetic and antifer-romagnetic coupling between the Ising spins are considered. The interactionbetween the carrier and the Ising spins is also of Ising type. In two of themodels the charge carrier is hosted by a different band, while in the thirdmodel it is hosted by the same band as the Ising spins. We find that thecarrier’s spectral function exhibits a distinctive fine structure due to its tem-porary entrapping inside small magnetic domains, and use these results toconstruct an accurate (quasi)analytic approximation for low and mediumT . While at T = 0, for ferromagnetic order all three models have identical,low-energy quasiparticles, at finite T the low-energy behavior of the firsttwo models remains equivalent, but that of the third model is controlled byrare events due to thermal fluctuations, which transfer spectral weight belowthe T = 0 quasiparticle peak, generating a pseudogaplike phenomenology.Taken together, our results show that the temperature evolution of the spec-tral weight of weakly doped magnetic insulators can be very complex.iiPreface• A version of the work discussed in Chapter 2 is published as M. Mo¨llerand M. Berciu, Physical Review B 88, 195111 (2013). It is an exten-sion of the work on spin-polarons by Shastry and Mattis [95] and makesuse of the formalism developed by Berciu and Sawatzky in Ref. [7].• A version of the work discussed in Chapter 3 is published as M. Mo¨llerand M. Berciu, Physical Review B 90, 075145 (2014).• A version of the work discussed in Chapter 4 is published as M. Mo¨llerand M. Berciu, Physical Review B 92, 214422 (2015). It makes useof the formalism that was developed in the publication above.• During the final two years of my PhD I also worked on polarons inmodels for which the electron-phonon coupling modulates the hop-ping integrals. A part of this work was published as M. M. Mo¨llerand M. Berciu Physical Review B 93, 035130 (2016), while anothermanuscript is currently in preparation. I chose not to discuss this workin the thesis, because the topic is rather different from my previouswork.I carried out all the analytical and numerical work for these publicationsand wrote the first draft of the manuscripts. The preparation of the finaldrafts was assisted by M. Berciu. I also benefited from discussions with G.A. Sawatzky, especially for the work on electron-phonon coupling.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Anomalous temperature-dependence in the parent cuprates . 31.2 Exchange interactions . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Hund’s coupling . . . . . . . . . . . . . . . . . . . . . 61.2.2 Kinetic exchange and super exchange: . . . . . . . . . 71.2.3 Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction 91.2.4 Double exchange . . . . . . . . . . . . . . . . . . . . . 91.2.5 Coupling of conduction electrons to a magnetic impu-rity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Model Hamiltonian for magnetic semiconductors . . . . . . . 131.4 Single particle Green’s functions . . . . . . . . . . . . . . . . 161.4.1 Zero temperature . . . . . . . . . . . . . . . . . . . . 161.4.2 Finite temperature . . . . . . . . . . . . . . . . . . . 191.5 The spin-polaron . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Markov-chain Monte Carlo simulations . . . . . . . . . . . . 261.6.1 The effect of correlations . . . . . . . . . . . . . . . . 291.6.2 The blocking method . . . . . . . . . . . . . . . . . . 311.7 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . 33ivTable of Contents2 Signatures of spin-polaron states at low temperatures . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 The low-temperature expansion . . . . . . . . . . . . . . . . 372.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Local environment effects on a charge carrier injected intoan Ising chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Simplified model of a 1d magnetic semiconductor . . . . . . . 563.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.1 Analytic approximation . . . . . . . . . . . . . . . . . 693.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 On the equivalence of models with similar low-energy quasi-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.1 FM results . . . . . . . . . . . . . . . . . . . . . . . . 824.4.2 AFM results . . . . . . . . . . . . . . . . . . . . . . . 914.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . 945 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101AppendicesA Derivation of the lowest T 6= 0 self-energy term . . . . . . . 111B Exact solution of the undoped Ising chain . . . . . . . . . . 113C Implementation of the Metropolis algorithm . . . . . . . . 114vList of Tables3.1 List of the shortest domains (underlined spins) that form inotherwise ordered backgrounds. The energies for trapping thecarrier in these domains are shown in Figs 3.5 and 3.6. . . . . 66viList of Figures1.1 Superexchange in a two-orbital Hubbard model. After Ref.[49]. Note that panel (a) and (c) also show the electron con-figurations that lead to AFM superexchange in a one-orbitalHubbard model. . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 An energy level diagram of the Anderson-impurity model withthe impurity level singly occupied. The shaded area is occu-pied by the conduction electrons. . . . . . . . . . . . . . . . . 101.3 A sketch of the second order processes from the first channelof Eq. (1.6) that lead to the exchange interaction betweenthe conduction electrons and the magnetic impurity. . . . . . 111.4 The same as Fig. 1.3, but for the second channel. . . . . . . . 121.5 Energy E↑(k) (thick full green line) and spectral weightA(0)↓ (k, ω)(contour map) vs. kx at ky = 0, for the Heisenberg model(left) and the Ising model (right) in 2d, for AFM couplingJ0/t = 3. The dashed red lines mark the expected continuumboundaries in the m = 1 subspace. Other parameters areJ/t = 0.5, S = 0.5, η/t = 0.01. . . . . . . . . . . . . . . . . . . 221.6 Left: A(0)↓ (k = 0, ω) for FM J0/t = −2 in 2d. The lowerc+m continuum edge is marked with dashed red lines. TheIsing model has a discrete state (sp2) below the continuum,but the Heisenberg model does not. Right: Spectral weightA(0)↓ (k, ω) for the Ising model in 2d for ky = 0, kx < 0.3pi.The dashed red line marks the lower c+m continuum edge.The sp2 state appears for small k and then merges into thecontinuum. Other parameters are J/t = 0.5, S = 0.5, η/t =0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.7 Ground-state energy of the Ising sp2 polaron as a function ofJ0/t for J/t = 0.5 (top) and as a function of J/t for J0/t = −2(bottom), for S = 0.5. . . . . . . . . . . . . . . . . . . . . . . 24viiList of Figures1.8 Spectral weight A(0)↓ (k, ω) vs kx for the 2d Heisenberg modelat ky = 0 (left) and ky = pi (right) and FM J0/t = −2.Sp1 appears above the continuum only near the Brillouinzone edge. No sp2 peak is seen below the continuum. Thedashed red lines mark the c+m continuum boundaries and thegreen line marks E↑(k). Other parameters are J/t = 0.5, S =0.5, η/t = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Spectral weight A↑(k, ω) and the real (solid line) and imag-inary (dashed line) part of the self-energy Σ(k, ω) for the2d Heisenberg model with AFM J0/t = 10 and βt = 1, atk = (0, 0) (top) and k = (pi, pi) (bottom). The expected sp1continuum boundaries are marked with dash-dotted blue linesand the expected c+m continuum boundaries with dashed redlines. The E↑(k) energy of the T = 0 δ-peak is marked with athick green line. Other parameters are J/t = 0.5, S = 0.5, η =0.02 (top) and η = 0.05 (bottom). . . . . . . . . . . . . . . . . 412.2 Same as Fig. 2.1 but for the Ising model. All parameters arethe same except βt = 0.5 and η = 0.01 in both panels. Notethat for the Ising model Σ(ω) is independent of k. The insetshows a zoom on Σ(ω) at high energies. . . . . . . . . . . . . 422.3 Spectral weight A↑(0, 0, ω) for the 2d Ising model (panels (a)and (b)) and 2d Heisenberg model (panel (c)) for FM J0/t =−2 at βt = 0.5, η/t = 0.01 (Ising) and βt = 1, η/t = 0.02(Heisenberg). The expected location of various features arealso indicated (see text for more details). Other parametersare J/t = 0.5, S = 0.5. . . . . . . . . . . . . . . . . . . . . . . 442.4 Spectral weight A↑(0, 0, ω) for the 2d Ising (dashed lines) andHeisenberg (full lines) models for J0/t = 10, 5, 3 in the top,middle and bottom panels, respectively. Other parametersare J/t = 0.5, S = 0.5 and βt = 0.5, η/t = 0.01 (Ising),and βt = 1, η/t = 0.02 (Heisenberg). The oscillations visibleespecially in the sp1 continuum are due to finite-size effects(we used N = 1002 and N = 5002 for Heisenberg and Isingmodels, respectively). . . . . . . . . . . . . . . . . . . . . . . 45viiiList of Figures2.5 Spectral weight A↑(k = 0, ω) for the 2d Heisenberg (left)and Ising (right) models with AFM J0/t = 7, at differenttemperatures. Only the sp1 continuum is shown. Its edgesare indicated with dot-dashed blue lines. Other parametersare J/t = 0.5, S = 0.5 and η/t = 0.01 and 0.02 for Ising andHeisenberg, respectively. . . . . . . . . . . . . . . . . . . . . . 462.6 Integrated spectral weight in the c+m continuum as a func-tion of β. Lines are fits described in the text. Parameters are:J0/t = 10, J/t = 0.5, S = 1/2, η/t = 0.01 (Ising), η/t = 0.05(Heisenberg). . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.7 Spectral weight A↑(k = 0, ω) for the 3d Heisenberg model atβt = 1 for FM J0/t = −3 (top) and AFM J0/t = 10 (bot-tom) couplings. The edges of the c+m continuum (dashedred lines) and sp1/sp2 continuum (dot-dashed blue lines) areindicated, as is E↑(0) (thick green line). Other parametersare J/t = 0.5, S = 0.5, η = 0.1. . . . . . . . . . . . . . . . . . 492.8 Spectral weight A↑(k = 0, ω) for the 3d Ising model at βt =0.5 for FM J0/t = −3 (top) and AFM J0/t = 10 (bottom)couplings. The edges of the c+m continuum (dashed redlines) and sp1/sp2 continuum (dot-dashed blue lines) are in-dicated, as is E↑(0) (thick green line). Other parameters areJ/t = 0.5, S = 0.5, η = 0.01. . . . . . . . . . . . . . . . . . . 503.1 Contour plot of the T = 0 spectral functions for FM coupling,|J |/t = 0.5. Other parameters are J0/t = 2.5, h = 0, η/t =0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 The same as in Fig. 3.1, but for AFM coupling. . . . . . . . . 613.3 Spectral function A(0, ω) for different temperatures and FMcoupling. Parameters are |J |/t = 0.5, J0/t = 2.5, h =0, η/t = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 The same as in Fig. 3.3, but for AFM coupling. . . . . . . . . 633.5 Spectral functions A(k, ω) for βt = 1 and k = 0, pi/2, pi, forFM coupling. Solid vertical lines mark the trapping energiesof the carrier in small domains. The corresponding numbersshow the length of the domain (see Table 3.1). The dashedvertical lines mark the band-edges of the T = 0 low-energyband. Other parameters are |J |/t = 0.5, J0/t = 2.5, h =0, η/t = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6 The same as in Fig. 3.5, but for AFM coupling. . . . . . . . . 65ixList of Figures3.7 A(k, ω) for different values of k and FM coupling, at |J |/t =0.5, J0/t = 2.5, βt = 1.0, h = 0, η/t = 0.04. . . . . . . . . . 673.8 The same as in Fig. 3.7, but for AFM coupling. . . . . . . . . 683.9 Spectral functions A(0, ω) for different values of h, for FMcoupling. Parameters are |J |/t = 0.5, J0/t = 2.5, βt =1.0, η/t = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . 693.10 The same as in Fig. 3.9, but for AFM coupling. . . . . . . . . 703.11 Comparison between the Monte Carlo (MC) results (shadedarea) and the analytic approximation of Eq. (3.31) for do-mains with a maximal length of L, for FM coupling |J |/t =0.5, at J0/t = 2.5, h = 0, η/t = 0.04, k = 0 and βt = 3 (top)and βt = 2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . 723.12 The same as in Fig. 3.11 but for AFM coupling. . . . . . . . 743.13 The correlation length ξ for |J | = 0.5. Note that the correla-tion length is the same for FM and AFM J , but for AFM Jthe spin-spin correlation function 〈σnσm〉T contains an addi-tional factor (−1)|m−n|. . . . . . . . . . . . . . . . . . . . . . 754.1 Sketch of the three models. Large, red arrows represent thelocal magnetic moments, empty (filled) blue circles representempty (filled) carrier sites. For Models I and II the carrierspin is represented by a blue arrow, for Model III the carrieris a spinless “hole” in the Ising chain. . . . . . . . . . . . . . 784.2 Spectral weight at k = 0 for a FM background and threedifferent temperatures for (a) Model I with J0/t = 5, J/t =0.5; (b) Model II with J0/t = 5, J/t = 0.5; (c) Model IIIwith J/t = 2.5. Insets in panels (a) and (b) show the spectralweight in the presence of a magnetic field, while in (c) it showsthe two continua appearing at low energies, for βJ = 0.5. Inall cases, the broadening is η/t = 0.04. The vertical linesshow the energy of the T = 0 QP peak. . . . . . . . . . . . . 844.3 A(k, ω) for the three models with FM background at βJ =0.5. Other parameters are as in Fig. 4.2. The dispersionlesslow energy, low weight part of the spectrum of Model III isnot shown. Red, vertical lines indicate the location of theT = 0 QP peaks. . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 When doping “removes” a thermally excited spin-down, theenergy variation upon doping is Eb − Ea = −2dJ + (k) andlies (at least partially) below the T = 0 QP ground-stateenergy of 2dJ − 2dt. . . . . . . . . . . . . . . . . . . . . . . . 87xList of Figures4.5 Rescaled spectral weight eβ4J [A(0, ω) − A(0)(0, ω)] in the re-gion of the continuum centered at −2J , for Model III withFM background and different values of β. For comparisonthe dashed, black line shows −Im[g0,0(ω, {↑, σ0 =↓})]/pi cal-culated with Eq. (4.20). Other parameters are J/t = 2.5 andη/t = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6 A(k = 0, ω) for Model III with FM background, for differentvalues of J at a temperature βJ = 0.5. The dashed red linesshow the location of the T = 0 QP peak. Full blue lines markthe energies −2J±2t. Parts of the spectra have been rescaledfor better visibility. . . . . . . . . . . . . . . . . . . . . . . . . 904.7 T = 0, AFM solutions. Top panels: contour plots of A(k, ω).Bottom panels: Cross sections at k = 0. (a) and (d) ModelI with J0/t = 5, |J |/t = 0.5; (b) and (e) Model II withJ0/t = 5, |J |/t = 0.5; (c) and (f) Model III with |J |/t =2.5. To improve visibility of the continuum a hard cutoff atA(k, ω) = 0.1 was used for the Model III contour plot. In allcases η/t = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . 914.8 AFM Spectral weight at k = 0 for three different tempera-tures for (a) Model I with J0/t = 5, |J |/t = 0.5; (b) ModelII with J0/t = 5, |J |/t = 0.5; (c) Model III with |J |/t = 2.5,the inset shows spectral weight below the T = 0 QP peak forβ|J | = 2.5. In all cases,the broadening is η/t = 0.04. Thevertical lines show the energy of the T = 0 QP peak. . . . . . 934.9 Spiral spin configuration in a 1d XY model. . . . . . . . . . . 944.10 (a) energy difference between the spiral state and the FMstate as a function of θ for different values of J . (b) thedependence of θc on J . In both panels t = 1. . . . . . . . . . 95C.1 The autocorrelation of the magnetization for different valuesof β and FM coupling. The dashed lines indicate when it hasfallen off to 0.1. Parameters are |J |/t = 0.5, J0 = 2.5 andh = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115C.2 Blocking analysis of the relative error of A(k = 0, ω = −4.5t)for FM |J | = 0.5 (cf. Fig. 3.3). Other parameters are k = 0,J0/t = 2.5, h = 0 and η/t = 0.04. . . . . . . . . . . . . . . . . 116C.3 The FM spectral function A(0, ω) for |J0|/t = 2.5, |J |/t = 0.5,h = 0 and η/t = 0.04. The error bars correspond to thelargest error from the blocking method. . . . . . . . . . . . . 116xiAcknowledgmentsFirst and foremost I would like to thank my supervisor Mona Berciu. Besidesbenefiting greatly from her expertise and clear way of explaining things, Iam grateful for her exceptional kindness and for always making time forme when I needed her advice. Discussions with her always left me feelinguplifted and motivated and the knowledge that she would always do her bestto help me was immensely reassuring.I am grateful to the members of my committee, Gordon Semenoff, Ian Af-fleck and George Sawatzky for their feedback and time, especially to GeorgeSawatzky who was very involved with my research and provided me withmany new ideas and opportunities to learn from his vast knowledge.On various occasions I benefited from discussions with the current andformer members of my group. In particular I want to thank ClemensAdolphs, Alfred Cheung, Dominic Marchand, Krzysztof Bieniasz, StevenJohnston, and also Arash Khazraie from George Sawatzky’s group. I amalso grateful to my flatmate and fellow graduate student Mohammad Samaniwhose advice on general coding issues helped me more than he is probablyaware of.I would also like to thank all of my friends, those who helped me makethe past years in Vancouver memorable and those with whom I shared myyears as an undergraduate student in Berlin. The Varsity Outdoor Clubwas and still is an important part of my time at UBC for which I am verygrateful.Finally I would like to thank my parents, Thomas and Birgit, and mybrother, So¨ren, for their continuing love and support and for their under-standing. I know that my decision to pursue my studies so far away fromhome was not always easy for them.xiiChapter 1IntroductionMagnetic materials and their fascinating properties have been studied forcenturies. Their applications range from the use as compass needles andrefrigerator magnets to the use in hard drives and magnetoresistive random-access-memory (MRAM) in computers [106]. While some of the propertiesof magnetic materials can be explained with classical electrodynamics, it isnow well understood that the theory of magnetism is inherently a quantummechanical theory. The exchange interactions which lead to the formationand interaction of magnetic moments are linked to the indistinguishabilityof particles and the antisymmetry of fermionic many-body wavefunctionsunder particle exchange.In many cases the magnetic moments which collectively give a magnet itsproperties are constituted by the partially filled 3d or 4f shells of transitionmetal (TM) or rare earth (RE) ions, respectively. These orbitals have avery small radial extent which causes strong Coulomb repulsion among theelectrons which fill them. In order to minimize the Coulomb energy theseelectrons tend to align their spins, a rule known in atomic physics as Hund’sfirst rule [32, 41]. By maximizing the total spin it is ensured that the spatialpart of the many-electron wavefunction is antisymmetric and the Coulombrepulsion is minimized.When TM or RE ions are incorporated into crystals the 3d or 4f shellsonly hybridize weakly with the surrounding atoms. Compared to the ioniccase the Coulomb repulsion of these orbitals may be screened substantially,but it is generally still much larger than the hybridization with the sur-rounding atoms. As a consequence the electrons in the 3d and 4f shellsoften remain localized in the solid and give rise to local magnetic moments.It is also possible that the 3d/4f orbitals form a narrow conduction band inwhich case one speaks of itinerant moment magnetism, but this scenario isnot the focus of this thesis.The interactions between local magnetic moments give rise to magneticorder. There are many possibilities, including antiferromagnetic (AFM)order which is observed for example in the parent cuprates [88] and ferro-magnetic (FM) order, for instance in FM chalcogenides like EuO and EuS1Chapter 1. Introduction[64, 82]. More complicated types of magnetic order include zig-zag order iniridates [16, 61, 99] and FM layers that are stacked antiferromagneticallyin manganite perovskites [81, 87] and KCuF3 [50, 54]. For the latter twomaterials the magnetic order is furthermore inseparably linked to orbitalorder resulting in a highly complicated ground state.When conduction electrons (or holes) are present, their interaction withthe local magnetic moments can also affect the magnetic order. This isespecially true in dilute magnetic semiconductors such as Ga1−xMnxAs [22,83] where the interaction between magnetic moments is mediated by theconduction electrons. In heavy fermion materials like CeSix.[58, 81], theconduction electrons have a tendency to screen the local magnetic moments,and in the manganite perovskites a rich, doping dependent phase diagramexists [38].Likewise, the presence of magnetic order can hinder the motion of chargecarriers in a conduction band and lead to effective interactions betweenthem. When the interactions are strong enough they renormalize the effec-tive mass and other properties of the charge carriers and it becomes con-venient to use a quasiparticle (QP) picture, where QP refers to a chargecarrier that is “dressed” by local excitations of the magnetic environment.The extent to which this happens depends on the type of magnetic order(generally FM order is preferred over AFM order by itinerant charge car-riers) and on whether the mobile charge carriers are hosted by a separateband or the same band which hosts the magnetic moments. Another im-portant factor are finite temperature effects which lead to the formationof magnetic defects and domain walls off which the carriers are scattered.In certain cases the interaction with magnons can lead to the formation ofspin-polarons which are QP states in which the carrier continuously absorbsand reemits a magnon. To discuss all these interesting properties one needsa model that incorporates the charge carrier degrees of freedom, the spindegrees of freedom and the interactions between the two subsystems. Sucha model and its zero temperature properties are presented below. However,before discussing them we first present a brief review of experimental andtheoretical work on the cuprates which motivates some of the work in thisthesis.21.1. Anomalous temperature-dependence in the parent cuprates1.1 Anomalous temperature-dependence in theparent cupratesA class of materials with unusual T -dependence of the spectral weight arethe cuprates. While the models studied in this thesis are simpler than thoseneeded to study the complex many-body physics of the cuprates, this classof materials serves as a motivation for our work. Even the simpler modelsstudied in this thesis show a variety of interesting and non-trivial propertieswhich could help understand some of the puzzling behavior observed in thecuprates.Upon doping the cuprates become high-temperature superconductorswhich is why, over the past decades, they have been studied in great de-tail, one of the tools of choice is angle resolved photoemission spectroscopy(ARPES) (see Ref. [19] for a review). It is widely believed that the elec-tronic properties of the cuprates are captured by CuO2 layers. The Cu is ina 3d9 configuration corresponding to a single spin 1/2 hole per Cu atom. Inthe undoped, parent compounds these holes tend to align antiferromagnet-ically due to a superexchange interaction mediated by the oxygens. Dopingof these materials introduces holes which tend to occupy the oxygens. Thelow-energy state of the hole-doped compound is believed to be the famousZhang-Rice singlet [114] in which a doped hole which is hosted by the fouroxygens surrounding a Cu atom forms a singlet with the Cu-spin at thecenter. Even though its validity is still being debated [26, 27, 56], theoret-ical studies of these materials are often carried out within the t − J-modelwhere it is assumed that the Zhang-Rice singlets form effective holes inan otherwise AFM ordered background. Often longer range hopping is in-cluded to match the experimental dispersion, in that case one speaks of thet− t′ − t′′ − J-model.Instead of a sharp QP peak, the low-energy part of the ARPES spectra ofthe parent cuprates has a very broad feature which has the dx2−y2-symmetryof the Zhang-Rice singlet, and is often referred to as the QP peak [51]. Aswill be discussed below this terminology is actually not quite correct andtherefore we will refer to this feature simply as the lower Hubbard band(LHB) [98]. What is puzzling about the LHB, are both its width whichis much larger than what one would expect for a coherent QP peak andits strong T -dependence. ARPES data acquired by Kim et al. [51] showsthat both the width and the center of the LHB change with T and thatthe energy scale of this change is incompatible with conventional electron-phonon coupling or a mere broadening of the Fermi-Dirac distribution. Fur-31.1. Anomalous temperature-dependence in the parent cupratesthermore it was found that the broadening of the LHB is linear in T for200K < T < 400K and then levels off [98] and that the T -dependence ismuch weaker for doped materials [51].To explain the anomalous T -dependence Kim et al. propose a heuristicmodel in which an ARPES measurement of initial states which are closein energy leads to final states which are further apart in energy. When Tis increased the distribution of initial states changes which can lead to thetransfer of spectral weight to different final states which are energeticallywell-separated from the final states that are observed at low T . This idea isvery similar to the ideas which lead to spectral weight transfer in the modelsinvestigated in this thesis.The heuristic model proposed by Kim et al. has the potential to ex-plain the anomalous T -dependence of the LHB, but it does not explain thebroadness of this feature at low-T . Here a breakthrough was achieved byShen et al. [97] who showed that for the parent cuprates the QP peak hasvanishingly small spectral weight and is therefore not seen in ARPES mea-surements. Instead one observes the various “shakeup” peaks that occurwhen an electron is removed and since these are very close to each otherin energy the observed feature appears as a broad Gaussian peak. This ef-fect is similar to the Franck-Condon principle from molecular spectroscopy(see also Refs [92] and [19]), but the excitations leading to the Gaussian-likefeature need not be vibrational in nature.Finite T calculations for a single hole in the t − t′ − t′′ − J-model werecarried out by van den Brink and Sushkov with the Self Consistent BornApproximation [105]. They find a broadening of the QP peak, but not suf-ficient to explain the broadening observed in ARPES spectra. Furthermorethey find a uniform, momentum-independent shift of the QP peak to lowerenergy as T increases. They attribute this to the fact that at finite T themagnetic background no longer has long range order making it easier forthe hole to propagate. The shift of the QP peak could be related to theanomalous T -dependence observed in optical measurements by Choi et al.[17]. Their data shows a large redshift of a peak in the optical conductivitywhich they associate with an excitonic process creating an electron in a Cu3d and a hole in an O 2p orbital. They argue that the hole on the O 2porbital will be in a state corresponding to the QP peak in the t− t′− t′′−J-model and that the energy shift of this peak predicted by van den Brink andSushkov is consistent with the observed redshift.Other theoretical studies of the T -dependence of the LHB have focusedon the effects of electron phonon coupling which are not included in thecalculation by van den Brink and Sushkov. Mishchenko and Nagaosa used41.1. Anomalous temperature-dependence in the parent cupratesDiagrammatic Monte Carlo to study the spectral function of the t−J-modelat T = 0 [67]. They found that interactions with the spin-background leadto an effective enhancement of the electron-phonon coupling and argued thatthe parent cuprates are in the strong-coupling regime where the formation ofa small polaron occurs. Their calculated spectra at strong electron-phononcoupling show a broad peak corresponding to the Franck-Condon peak and aQP peak with vanishing spectral weight, in agreement with the experimentalARPES data. The broad peak follows the dispersion of the regular t − J-model without electron-phonon coupling. An explanation for this behaviorwas given by Ro¨sch et al. based on an adiabatic approximation [89, 90]which also allows them to include coupling to different phonon branches.An extension of the results by Mishchenko and Nagaosa was carried outby Cataudella et al for the t − J-model [14] and t − t′ − t′′ − J-model [15].They use a method combining the Momentum Average approximation [8]for electron-phonon coupling and the Self Consistent Born Approximationfor electron-magnon coupling. Their results for the thermal broadening ofthe LHB are in qualitative agreement with ARPES data and show the linearT behavior mentioned above.Experimental data supporting strong electron-phonon coupling in thecuprates was provided by Shen et al. who studied the T -dependence ofpi-bonding O 2ppi bands and the Ca 3p bands in Ca2CuO2Cl2 [98]. Thesebands do not couple to the spin-background and therefore can be used as a“benchmark” for the effect of electron-phonon coupling. Indeed Shen et al.find a substantial T -dependence of these bands, but not as large as that ofthe LHB. This suggests that electron-phonon coupling does indeed play animportant role, but that magnons (or other excitations) also contribute tothe T -dependence of the LHB. This is supported by the findings of Lau et al.who carried out exact diagonalization for a CuO2 layer [56]. In contrast tothe t−J-like model they explicitly include the σ-bonding O 2p orbitals andfind that besides the Zhang-Rice-like states a low-energy spin-polaron withspin 3/2 exists. This spin-polaron is invisible to ARPES at T = 0, but couldbe thermally activated, suggesting a T -dependent broadening mechanismwith the same energy scale as the phononic mechanism discussed above.In summary, while a lot of progress in understanding the T -dependenceof ARPES spectra of the parent cuprates has been made in recent years,the microscopic mechanism behind it is not yet fully understood. From ex-perimental data and theory it is clear that phonons play an important roleand that the electron-phonon coupling is increased by the presence of themagnetic background. However, there is evidence which suggests that apartfrom this indirect role magnons also play a more direct role. It is there-51.2. Exchange interactionsfore desirable to obtain a better understanding of the T -dependence causedby electron-magnon interactions. To achieve this one must first study sim-pler models without the added complexity of coupling to both magnons andphonons. The T -dependent spectral weight transfer of such models is inves-tigated in this thesis. In the following sections we discuss the Hamiltonianneeded to model these simpler problems and the origin of its various terms.1.2 Exchange interactionsThe exchange interactions which lead to all the interesting behavior men-tioned above were first discovered by Heisenberg [37] and Dirac [23] andhave the formHexc =∑i,jJi,jSiSj , (1.1)where Si is the local magnetic moment at site Ri. Often the sum is restrictedto nearest neighbors. In the initial formulation of this model the couplingconstant Ji,j was given by exchange integrals of the Coulomb energyJi,j =∫dr∫dr′ φ∗i (r)φ∗j (r′)e2|r− r′|φi(r′)φj(r), (1.2)where φi and φj are wavefunctions centered at site Ri and Rj , respectively.In most cases these direct exchange integrals between neighboring atoms arenot large enough to explain the magnetic order observed in many materi-als. Instead, the underlying physics which governs the sign and size of thecoupling J is often intricately related to the crystal structure and orbitalstructure of the material under study and can be deduced with the so calledGoodenough-Kanamori-Anderson rules (for a review see Refs [50] and [49]).In the following we sketch a few examples of exchange mechanisms that leadto either AFM or FM coupling. These sketches are not intended as rigoroustreatments, but instead are supposed to give a qualitative idea of the physicsbehind these mechanisms.1.2.1 Hund’s couplingHund’s coupling is an intraatomic coupling and given by the case whereφi and φj in Eq. (1.2) belong to different orbitals centered on the sameatom. This coupling is responsible for Hund’s first rule (originally formu-lated by Hund in 1925 [41], see also Refs [50] and [32]) from atomic physics61.2. Exchange interactionswhich states that when occupying orbitals of a given subshell the electronsmaximize the total spin. The underlying physics is that when the spin ismaximized the spatial part of the wavefunction must be antisymmetric andtherefore minimizes the Coulomb repulsion. This is especially importantfor the highly localized 3d and 4f orbitals. Consequently Hund’s couplingis responsible for the existence of magnetic moments. For example in theEu chalcogenides the magnetic moments are constituted by the half filled 4forbitals.However, since it is mainly an intraatomic effect, Hund’s coupling is notdirectly responsible for the interaction of adjacent magnetic moments. Notethat Hund’s coupling also exists for orbitals of different subshells, but stillon the same atom. In the Eu chalcogenides this leads to the coupling ofconduction electrons, partially hosted by the Eu 5d orbitals, to the localmagnetic moments which are constituted by the half filled Eu 4f orbitals.1.2.2 Kinetic exchange and super exchange:The simplest model that leads to kinetic exchange is the single-band Hub-bard model at half filling. The Hamiltonian is given byHˆU = −t∑〈i,j〉∑σc†i,σcj,σ + U∑ini,↑ni,↓. (1.3)The operator c†i,σ (ci,σ) creates (annihilates) an electron with spin σ at sitei and ni,σ = c†i,σci,σ is the occupation operator for site i.We start from a configuration where every site is occupied by one elec-tron, minimizing the Hubbard repulsion U . When neighboring electronshave antiparallel spins they can tunnel into states with double occupancyand back. To second order in perturbation theory this results in an en-ergy gain ∆E↑↓ = −t2/U (see Fig. 1.1)(a)). Due to the Pauli exclusionprinciple this process is forbidden for neighboring electrons with parallelspins. When one projects onto the subspace with single occupancy thissituation is captured by an effective Heisenberg Hamiltonian with AFMcoupling |J | = 4t2/U .A simple example that leads to FM kinetic exchange is given by a two-orbital Hubbard model at half filling. We assume that the two orbital typesare degenerate, but hopping is only possible between nearest neighbor or-bitals of the same type. The possible spin configurations for two electronsare depicted in Fig. 1.1. Note that in this case an FM configuration isfavored because of the intraatomic Hund’s coupling JH . The superexchange71.2. Exchange interactions(a) ΔE↑↓ = -t2/U(c) ΔE↑↑ = 0(b) ΔE↑↓ = -t2/U(d) ΔE↑↑ = -t2/(U-JH)Figure 1.1: Superexchange in a two-orbital Hubbard model. After Ref. [49].Note that panel (a) and (c) also show the electron configurations that leadto AFM superexchange in a one-orbital Hubbard model.coupling is proportional to the energy difference between the AFM configu-ration and the FM configuration. Assuming JH  U this gives |J | = 4t2U JHU .This type of exchange interaction is also called kinetic exchange.These ideas can be generalized to more complex models where the localmagnetic moments are hosted by TM or RE ions that do not hybridizedirectly, but through their ligands, often oxygens [2, 3, 52]. In this case theexchange interaction is referred to as superexchange. Hopping from one TMion to another is a second order process and J is therefore obtained by fourthorder perturbation theory. Furthermore one needs to consider not just theCoulomb repulsion Udd on the TM ions, but also the energy difference ∆pdbetween the ligand orbitals and the TM orbitals. In addition more than oneorbital per TM ion can be involved in the exchange mechanism and the bond-angles may also play a role. This complexity can lead to both AFM and FMcoupling constants and has been studied extensively by Goodenough andKanamori who developed a set of rules to determine the type (FM or AFM)and strength of the coupling [33–35, 43–45]. An excellent review of theserules can be found in Ref. [50]. Note that in most cases superexchange leadsto AFM order, very specific conditions are needed for FM order [49, 50].Note that higher order terms in perturbation theory can also be consid-ered and lead to interactions which are for example proportional to (S1S2)2and can even involve more than two spin operators [31, 102, 110].81.2. Exchange interactions1.2.3 Ruderman-Kittel-Kasuya-Yosida (RKKY) interactionLocalized magnetic moments in a metal can interact with each other bypolarizing the conduction electrons. This type of interactions leads to acoupling constant that oscillates as a function of the distance r between thelocal magnetic moments. To leading order one obtainsJ ∼ cos(2kF r)r3. (1.4)This interaction was first derived by Ruderman and Kittel to explainthe interaction of nuclear spins via conduction electrons [91]. The extensionto materials with magnetic moments due to localized d or f electrons wascarried out by Yosida [109] and Kasuya [47, 47].Since we are restricting ourselves to models with a single conductionelectron, the RKKY interaction as described above is not relevant for thephysics described in this thesis. However a variant of it, sometimes re-ferred to as the Bloembergen-Rowland-mechanism [10], is responsible forthe magnetic ordering in the Eu chalcogenides. In these materials virtualexcitations of electrons across the semiconducting gap lead to an effectivecoupling between neighboring 4f electrons. For instance it is possible thatan f electron tunnels into the conduction band composed primarily of Eu 5dorbitals. Hund’s rule exchange with the neighboring Eu 4f orbitals can thenlead to a FM coupling. On the other hand, if two 4f electrons tunnel intothe conduction band the Pauli exclusion principle favors a singlet, leadingto AFM exchange. In both cases the existence of the energy gap leads toan exponential decay of the exchange interaction with the distance betweenlocal magnetic moments [10, 49, 62]. The competition between AFM andFM coupling leads to the different types of magnetic order observed in theEu chalcogenide series [48, 49, 57, 64]. EuO and EuS are ferromagnets withCurie temperatures of 70 K and 17 K, respectively. EuSe has a complicatedmagnetic structure and EuTe is an antiferromagnet with a Ne´el temperatureof 10 K [82].1.2.4 Double exchangeThis type of exchange interaction also requires a finite concentration ofcharge carriers. It arises in doped perovskite manganites [49, 50, 111, 112].In these materials the crystal field due to the oxygen ligands causes theMn 3d orbitals to split into two eg and three t2g orbitals. The conductionelectrons which are hosted by Mn eg orbitals experience a strong Hund’scoupling with the localized magnetic moments, constituted by the half filled91.2. Exchange interactionsεfεf+UEFFigure 1.2: An energy level diagram of the Anderson-impurity model withthe impurity level singly occupied. The shaded area is occupied by theconduction electrons.Mn t2g orbitals. This Hund’s coupling is of the same order as the conduc-tion band width which makes the RKKY theory inapplicable [50]. In thissituation, even when the superexchange between neighboring local magneticmoments is AFM, a finite concentration of conduction electrons can lead toFM order. The reason is that an AFM magnetic background suppressesthe hopping of the conduction electrons since hopping to an adjacent sitewould cost an energy of the order of the Hund’s coupling. It can thereforebe energetically preferable for the system to be in a FM state, or at least acanted FM state [20], which allows the conduction electrons to lower theirkinetic energy by delocalizing.A somewhat similar situation occurs in the half-filled, single band Hub-bard model for U →∞. In this case the AFM superexchange coupling goesto zero. If one now dopes holes into the system their motion reshuffles thespins of the magnetic background breaking AFM bonds. This hinders themotion of the hole and in the limit U → ∞ the ground state of the dopedsystem is FM. This was first described and proved rigorously by Nagaoka [72]for the case of a single hole. An extension and short, yet rigorous proof canbe found in Ref. [104]. The situation for more than one hole is significantlymore complicated [24, 30, 96].1.2.5 Coupling of conduction electrons to a magneticimpurityThe coupling of conduction electrons to magnetic impurities is importantfor dilute magnetic semiconductors. For simplicity we consider a single spin1/2 impurity, but the ideas presented here can be generalized to multipleimpurities and larger spin values. The coupling between the conduction101.2. Exchange interactionsεfεf+UEFεfεf+UEFVk'εfεf+UEFVk'εfεf+UEFV*kΔE = ε(k) - εf Figure 1.3: A sketch of the second order processes from the first channelof Eq. (1.6) that lead to the exchange interaction between the conductionelectrons and the magnetic impurity.electrons and the impurity is described by the Anderson-impurity modelHˆA−I =∑k,σ(k)c†k.σck.σ +∑σff†σfσ + Uf†↑f↑f†↓f↓+∑k,σ[Vkf†σck,σ + V∗k c†k,σfσ](1.5)Here (k) is the dispersion of the conduction electrons and c†k.σ, ck.σ aretheir creation and annihilation operators, respectively, while k denotes themomentum and σ = ±1/2 is the spin. The impurity with creation operatorf †σ is assumed to be located at the origin and has an on-site energy f . Itinteracts with the conduction electrons via a hybridization Vk. Double occu-pation of the impurity orbital costs an additional energy U due to Coulombrepulsion.We are interested in the situation where the Fermi energy, EF , lies be-tween f and f+U . In that case the impurity level is on average singly occu-pied, as shown in Fig. 1.2, and acts as a local magnetic moment. States withthe impurity level unoccupied or doubly occupied are energetically unfavor-able, yet, as we will see, they are still important for the interaction with theconduction electrons. One can derive an effective, low-energy Hamiltonianby projecting on the states where the impurity is singly occupied. Start-ing from this subspace, first order processes in Vk lead to either a doublyoccupied or an unoccupied impurity level. The first correction is therefore111.2. Exchange interactionsεfεf+UEFεfεf+UEFV*kεfεf+UEFV*kεfεf+UEFVk'ΔE = εf + U - ε(k') Figure 1.4: The same as Fig. 1.3, but for the second channel.of second order in Vk. The derivation of the effective Hamiltonian can becarried out by using projection operators as described in Ref. [40] or byusing a canonical transformation as was first shown by Schrieffer and Wolff[94]. The result is an exchange interaction between the conduction electronsand the impurity and a scattering term which is not of interest to us. Theexchange coupling is given by [40]Jk,k′ = 2V∗k Vk′[1(k)− f +1f + U − (k′)]. (1.6)It is instructive to qualitatively discuss the origin of this exchange inter-action. There are two channels corresponding to the first and second termin Eq. (1.6). All processes which contribute to the channels start and endwith a singly occupied impurity level. The first channel corresponds to pro-cesses where the electron that occupies the impurity level “jumps” into theconduction band (see Fig. 1.3). The energy difference between this interme-diate state and the original state is (k) − f . The empty impurity level isthen occupied by another conduction electron. Since this electron does notnecessarily have the same spin as the electron which previously occupied theimpurity, this process can result in a flip of the impurity spin.The processes belonging to the second channel are sketched in Fig. 1.4.Here a conduction electron first “jumps” into the impurity level which is thendoubly occupied. The energy difference between the intermediate state andthe original state is therefore given by f +U − (k′). Note that this processis only possible for a conduction electron whose spin is antiparallel to that ofthe singly occupied impurity. In the second step one of the electrons hosted121.3. Model Hamiltonian for magnetic semiconductorsby the impurity “jumps” into the conduction band. Since this can be eitherof the two electrons the final impurity state can have its spin flipped withrespect to the original state.The example of the Anderson-impurity model illustrates two very impor-tant points. First it shows that AFM coupling between the conduction elec-trons and the local magnetic moments can occur. This becomes very clearin the second channel where only conduction electrons with antiparallel spincan hop into the impurity level. Second, it provides a nice illustration of thefact that exchange interactions usually occur in effective Hamiltonians as aresult of projecting out charge fluctuations, in this cases charge fluctuationsof the impurity level. This point also becomes very clear in our treatment ofkinetic exchange in the Hubbard model, where double occupancy of any siteis projected out. As mentioned above the results on the Anderson-impuritymodel can be generalized to multiple impurities and impurities which canhost more than two electrons, i.e. have a spin that is larger than 1/2.1.3 Model Hamiltonian for magneticsemiconductorsA minimal model for magnetic semiconductors must consist of three parts:(i) A part describing the conduction electrons which are assumed to behosted by a broad band; (ii) a part describing the local magnetic momentsand their tendency to order and (iii) a part describing the interaction be-tween the local magnetic moments and the conduction electrons. This isaccomplished by the Kondo-lattice model which in the context of magneticsemiconductors is often referred to as the s-f model. This terminology origi-nates with the Eu chalcogenides, where the conduction electrons are hostedby the Eu 5d and 6s orbitals while the local magnetic moments are hosted bythe half-filled Eu 4f orbitals [64, 82]. In the case of materials with transitionmetal ions such as the manganites this model is also referred to as the s-dmodel [49, 111–113].We consider the limit where the conduction band hosts a single spin-12charge carrier. This charge carrier propagates on a hypercubic lattice withperiodic boundary conditions after Ni sites in the direction i = 1, d; thetotal number of sites is N =∏di=1Ni. In Chapter 2 we present resultsfor d = 2 and d = 3, while in Chapters 3 and 4 we present results for asimplified version of this model for d = 1. Of course, long-range FM orderat finite T only exists in d = 3. However, we also consider anisotropic layeredcompounds, like the manganites, which have 2d FM layers whose finite-T131.3. Model Hamiltonian for magnetic semiconductorslong-range order is stabilized by weak interlayer coupling, but where onecan assume that at very low T the intralayer carrier dynamics determineits properties. In principle, similar arguments can be employed to studyd = 1 chains with FM order at finite T maintained by their immersion in3d lattices or by other mechanisms. For instance, the existence of FM edgestates in graphene nanoribbons is a current topic of investigations [13, 46].An important point to be aware of is that in contrast to higher dimensions,in 1d the energy cost of magnetic domains does not necessarily scale withtheir size. Consequently approximations which work well in 2d and 3d mayperform very poorly in 1d. This is part of the reason why in Chapter 2 weonly present results for d > 1.The carrier is an electron in an otherwise empty band or a hole in an oth-erwise full band, described by a tight binding model with nearest neighbor(nn) hopping:Tˆ =∑k,σ(k)c†k,σck,σ, (1.7)with (k) = −2t∑di=1 cos ki for lattice constant a = 1. c†k,σ creates a carrierwith momentum k and spin σ.The local magnetic moments are described by either a Heisenberg [37]interaction:HˆS = −J∑〈i,j〉(Si · Sj − S2)(1.8)or an Ising interaction [42, 60]:HˆI = −J∑〈i,j〉(Szi Szj − S2), (1.9)where Si is the spin-S moment located at site Ri and only nn exchange isincluded in both models.When J is FM (i.e. J > 0 in Eqs (1.8) and (1.9)) the undoped groundstate for both HˆS and HˆI is |FM〉 = |S, S, . . . 〉 and has zero energy. Thesimplest excited states of interest are the single magnon states:|Φ(q)〉 = S−q√2S|FM〉 =∑jeiqRj√2SNS−j |FM〉. (1.10)Here S±i = Sxi ± iSyi are the raising (+) and lowering (−) operators. Thekey difference between the Heisenberg and Ising interactions is the dis-persion of the single magnon states. For the Heisenberg model this is141.3. Model Hamiltonian for magnetic semiconductorsΩq = 4JS∑di=1 sin2(qi/2), whereas for the Ising model the magnons aredispersionless, Ωq = Ω = 2dJS. Thus, studying both models allows us toevaluate the relevance of having gapped or gapless magnons.The case of AFM coupling, J < 0 is significantly more difficult to treat.For the Ising spins the GS is the Ne´el state in which nearest neighbor spinspoint into opposite directions. For the Heisenberg case, on the other, handthe true AFM GS is only known for d = 1 and S = 1/2, and even withthose restrictions its form is rather complicated [110]. Consequently AFMcoupling will only be treated in Chapters 3 and 4 for simplified versions ofthe s-f(d) model.The interaction between the carrier and the local moments is also aHeisenberg exchange:Hˆexc = J0∑jsj · Sj , (1.11)where si =∑α,β c†i,ασα,β2 ci,β is the carrier spin operator and σ are the Paulimatrices. The coupling J0 can be either FM or AFM; we will consider bothcases. As mentioned above the FM case occurs for example when the con-duction electrons experience Hund’s coupling with local magnetic moments.This is the case for the Eu chalcogenides. AFM coupling can occur when ahybridization between the conduction electrons and the orbitals hosting thelocal magnetic moments is present and the local magnetic moments havemaximum spin. In that case only conduction electrons with opposite spincan make use of the hybridization term to reduce their energy. In this caseone can use a Schrieffer-Wolff transformation [94] to rewrite the hybridiza-tion term as an exchange interaction.It is convenient to split Hˆexc = Hˆzexc + Hˆx,yexc , whereHˆzexc =J02∑j(c†j,↑cj,↑ − c†j,↓cj,↓)Szj (1.12)Hˆx,yexc =J02∑j(c†j,↑cj,↓S−j + c†j,↓cj,↑S+j). (1.13)The first term causes an energy shift ±J0S/2. The second term is respon-sible for spin-flip processes, where the carrier flips its spin by absorbing oremitting a magnon.The total Hamiltonian is:Hˆ = Tˆ + HˆS/I + Hˆexc. (1.14)151.4. Single particle Green’s functionsDue to translational invariance, the total momentum is conserved. Further-more, the z−component Sztot of the total spin (the sum of the carrier spinand lattice spins), is also conserved. Therefore, eigenstates Hˆ|ψ(m)α (k)〉 =E(m)α (k)|ψ(m)α (k)〉 are indexed by the total momentum of the system, k,by the number m of magnons when the carrier has spin-up so that Sztot =NS + 12 −m, and by α which comprises all the other quantum numbers.The s-f model poses a difficult many-body problem and can only besolved in limiting cases. One of them is the zero bandwidth limit [76], whilethe other cases are that of a single carrier injected into completely orderedFM background at T = 0 [95] and that of two carriers with opposite spin[68, 69] at T = 0. In the single carrier case the solution depends dramaticallyon the spin orientation of the carrier. If the carrier has its spin aligned withthe magnetic background (spin up) the system is in the NS + 1/2 subspaceand no spin-flip processes are possible. The solution is therefore trivial. If onthe other hand the carrier is injected with its spin opposite to the magneticmoments, the system is in the NS−1/2 subspace and for sufficiently strongAFM J0 a bound state consisting of a magnon and the carrier, a so-calledspin-polaron can form. We will review this solution below, after introducingthe formalism of single particle Green’s functions.1.4 Single particle Green’s functions1.4.1 Zero temperatureThe single particle GF is a powerful tool in condensed matter physics. As weshow below it gives us access to the eigenenergies and QP weights, propertieswhich can be measured in photoemission experiments [19]. In this thesis weare restricting ourselves to models of a single fermion which interacts withthe degrees of freedom of a lattice of local magnetic moments. Consequentlyonly the single particle, retarded GF is of interest to us. At T = 0 it isdefined as [28, 63, 115]Gα,α′(k, τ) = −iΘ(τ)〈0|ck,α(τ)c†k,α′(0)|0〉 (1.15)Here ck,α(τ) = eiHˆτ ck,α(0)e−iHˆτ is a fermionic annihilation operator in theHeisenberg picture (we set h¯ = 1). The index k is the momentum and αis to be considered a place holder for spin and band indices. The state |0〉is the ground state of the local magnetic moment part of Hˆ whose energyES/I0 we choose to be zero. Finally the Heaviside function, Θ(τ), is defined161.4. Single particle Green’s functionsasΘ(τ) ={1 τ > 00 τ < 0(1.16)Physically Gα,α′(k, τ) corresponds to the probability amplitude that afermion which is injected into the system in state (k, α′) will be in state(k, α) after a time τ while the rest of the system remains unchanged. Notethat we are restricting ourselves to systems where the total momentum isconserved and consequently k cannot change during such a process. If, as isusually the case, the state |0〉 has total momentum zero, then k is the totalmomentum of the system.In many cases it is convenient to work in the frequency domain ratherthan the time domain. This is achieved by Fourier transformingGα,α′(k, ω) =∫ ∞−∞dτ eiωτGα,α′(k, τ). (1.17)To carry out the integral we first insert the identity operator in the form1 =∑n |n〉〈n|, where |n〉 are the eigenvectors of Hˆ with eigenvalues En.This yieldsGα,α′(k, ω) = −i∑n∫ ∞−∞dτ ei(ω+ES/I0 −En)τΘ(τ)〈0|ck,α|n〉〈n|c†k,α′ |0〉 (1.18)Note that since the GS energy of the local magnetic moments ES/I0 = 0 theenergy difference ES/I0 −En simplifies to −En. However, we will see that forfinite temperatures the appearance of energy differences in the exponentialis unavoidable and has physical consequences.We proceed by using the following integral representation for the Heav-iside function [115]Θ(τ) = limη→0+∫ ∞−∞dx2piie−ixτx+ iη. (1.19)Inserting this into Eq. (1.18) we first perform the integration over x as acontour integral and subsequently the integration over τ to obtainGα,α′(k, ω) = limη→0+∑n〈0|ck,α|n〉〈n|c†k,α′ |0〉ω − En + iη . (1.20)171.4. Single particle Green’s functionsThis representation of the GF is known as the Lehmann representation. Inthis form it becomes clear that Gα,α′(k, ω) has poles at the eigenenergies Enof Hˆ, and that their weights correspond to the overlap of the free carrierstates c†k,α/α′ |0〉 with the eigenstates |n〉 of Hˆ. When α = α′ these weightsare referred to as the QP weights Zαn (k) = |〈n|ck,α|0〉|2.The easiest way to extract the eigenenergies and QP weights from the GFis to calculate the spectral function Aα,α(k, ω) = − 1pi ImGα,α(k, ω). Usingthe well-known identity limη→0+ 1x+iη = P 1x + ipiδ(x), where P denotes theCauchy principal value [28], this becomesAα,α(k, ω) = − 1piImGα,α(k, ω) =∑nZαn δ(ω − En). (1.21)When carrying out numerical calculations we will usually use a small,but finite value for η. This results in a broadening of the delta-peaks fromEq. (1.21) into Lorentzians. By reverse Fourier transforming Eq. (1.20) onecan show that choosing a finite value for η introduces a finite lifetime ∼ 1/ηfor the free carrier.From Eq. (1.21) it is clear that the spectral function obeys the sum rule∫ ∞−∞dω A(k, ω) =∑nZαn = 1, (1.22)where we used the completeness relation∑n |n〉〈n| = 1. Furthermore thedensity of states (DOS) ρ(ω) is related to A(k, ω) viaρ(ω) =1N∑qA(q, ω). (1.23)Eq. (1.20) can be recast in the form of the expectation value of anoperator by introducing the resolvent of HˆGˆ(ω) =1ω − Hˆ + iη . (1.24)The GF is then given byGα,α′(k, ω) = 〈0|ck,αGˆ(ω)c†k,α′ |0〉. (1.25)The advantage of working with the resolvent Gˆ(ω) is that it obeysDyson’s identity. If we write the Hamiltonian as the sum of two parts Hˆ0and Vˆ it is easy to verify thatGˆ(ω) = Gˆ0(ω) + Gˆ(ω)Vˆ Gˆ0(ω) (1.26)181.4. Single particle Green’s functionsThis holds independent of the choice of Hˆ0 and Vˆ , but generally one choosesHˆ0 in such a way that it is easy to calculate the expectation values of Gˆ0. Byrepeatedly using Dyson’s identity, one can construct a hierarchy of equationsof motion (EOM) of the GF. In the subsequent Chapters we will make useof this approach to calculate the GFs for various Hamiltonians.1.4.2 Finite temperatureThe finite temperature GF is usually defined in the grand canonical ensemble[28, 63, 115]. However, for the models investigated in this thesis there isalways exactly one fermion in the system and only the number of bosonicexcitations varies. We therefore need to formulate the finite temperaturetheory in the canonical ensemble. The natural extension of Eq. (1.15) tofinite temperatures is thenGα,α′(k, τ) =−iΘ(τ)ZTr[e−βHˆS/Ick,α(τ)ck,α′(0)], (1.27)where β = T−1 is the inverse temperature (we set the Boltzmann constantkB = 1). The trace runs over all the eigenstates of the lattice part, HˆS/I, ofHˆ, i.e. all possible configurations of the system before injection of the extracharge carrier. As usual the partition function Z is given byZ = Tr[e−βHˆS/I](1.28)We can use the same procedure as for the zero temperature GF to trans-form to the frequency domain. In doing so we obtainGα,α′(k, ω) =∑ne−βES/InZ〈S/I, n|ck,αGˆ(ω + ES/In )c†k,α′ |S/I, n〉. (1.29)Here |S/I, n〉 are the eigenstates of HˆS/I with eigenenergies ES/In . It is crucialto realize that ω is shifted by ES/In . This shift comes about because〈S/I, n|eiHˆτ ck,α · · · = eiES/In τ 〈S/I, n|ck,α . . . (1.30)Physically it means that all energies are measured with respect to the energyof the system before injection of the charge carrier. For the zero temperatureGF this is also true, but when the carrier is injected the lattice is in itsground state, which we assume to have zero energy.191.5. The spin-polaronBefore concluding this section, we emphasize that all the calculations inthis thesis are in a canonical ensemble. The chemical potential is not fixedat ω = 0, as customary in grand canonical formulations, instead it can becalculated as µ =(∂F∂N)T→ minα,β[EN+1,α − EN,β] as T → 0, where F isthe free energy and N is the particle number. As pointed out above, hereω = 0 marks the energy of the undoped system.1.5 The spin-polaronIn this section we apply the formalism of zero temperature GFs to theHamiltonian of Eq. (1.14) to study the spin-polaron. This problem wasfirst solved by Shastry and Mattis [95] and later generalized to complex lat-tices by Berciu and Sawatzky [7]. An approach that does not rely on GFs,but yields all the eigenfunctions was presented by Henning et al. [39] andNakano et al. [73].To distinguish the T = 0 GFs we denote them by G(0)↑ (k, ω) for a carrierinjected with spin up (m = 0 sector) and G(0)↓ (k, ω) for a carrier injectedwith spin down (m = 1 sector). Let us start with the solution in the m = 0sector, where the total magnetization is Sztot = NS + 1/2. In that case nospin flips are possible and the eigenenergies are simply given by E↑(k) =(k) + J0S/2. The spectral function A(0)↑ (k) consists of a single δ-peak atE↑(k) and G(0)↑ (k, ω) = [ω−E↑(k)+iη]−1 is just the free particle propagatorup to a shift J0S/2.For the m = 1 sector we need to work a little harder. The GF is definedasG(0)↓ (k, ω) = 〈FM |ck,↓Gˆ(ω)c†k,↓|FM〉. (1.31)Splitting Hˆ into Hˆ0 = Tˆ + HˆS/I and Vˆ = Hˆexc and using Dyson’s identitywe obtainG(0)↓ (k, ω) =[1 + J0√S2N∑qF (0)(k,q, ω)]G(0)↑ (k, ω + J0S), (1.32)where we defined F (0)(k,q, ω) = 〈FM |ck,↓Gˆ(ω)c†k−q,↑ S−q√2S|FM〉.We proceed by using Dyson’s identity again to obtain the EOM for201.5. The spin-polaronF (0)(k,q, ω). This yieldsF (0)(k,q, ω) =J0√S√2NG(0)↓ (k, ω)−J02N∑QF (0)(k,Q, ω)G(0)↑ (k− q, ω − Ωq)(1.33)Note that on the right hand side of both EOM F (0)(k,q, ω) only appearsas a sum. We therefore define f (0)(k, ω) = N−1∑q F(0)(k,q, ω). InsertingEq. (1.33) into this definition yieldsf(k, ω) =J0√S2NG(k, ω)g(0)(k, ω)1 + J02 g(0)(k, ω), (1.34)where g(0)(k, ω) = N−1∑qG(0)↑ (k− q, ω − Ωq) can be calculated sinceG(0)↑ (k, ω) is known. This can be done numerically or by bringing it into theform of an elliptical integral for which a variety of methods of calculationexist [9, 70]. Note that g(0)(k, ω) depends on the magnon dispersion Ωq andis therefore different for HˆS and HˆI.Finally, we can plug f (0)(k, ω) into Eq. (1.32) to obtainG(0)↓ (k, ω) =[ω − (k) + J0S2−J20S2 g(0)(k, ω)1 + J02 g(0)(k, ω)]−1. (1.35)As pointed out above the difference between the Ising and Heisenberg mag-netic background is contained completely in g(0)(k, ω). Note furthermorethat the last term on the right corresponds to the self-energy, up to a con-stant shift J0S/2.We can now investigate the spectral function A(0)↓ (k, ω). Our main focuswill be to understand when a spin-polaron state forms in the m = 1 sector,but we will also verify the presence of the continuum at the expected loca-tion. We assume that |J0| is the largest energy scale, experimentally thisis true for example in the manganite perovskites. While realistically oneexpects J  t, we will set J/t = 0.5 so that its role can be discerned easily.We furthermore specialize to the case S = 1/2 since it is to be expected thateffects of the quantum nature of the spins will be the most pronounced inthis limit.Figure 1.5 shows E↑(kx, ky = 0) (thick full green line) and the spectralweight A(0)↓ (kx, ky = 0, ω) (contour map) for the 2d s-f(d) model, for AFM211.5. The spin-polaron-6-4-2 0 2 4 6 8 0  0.2  0.4  0.6  0.8  1ω/ tkx/pi 0.000 0.005 0.010 0.015 0.020 0.025 0.030-6-4-2 0 2 4 6 8 0  0.2  0.4  0.6  0.8  1ω/ tkx/pi 0.000 0.005 0.010 0.015 0.020 0.025 0.030Figure 1.5: Energy E↑(k) (thick full green line) and spectral weightA(0)↓ (k, ω) (contour map) vs. kx at ky = 0, for the Heisenberg model (left)and the Ising model (right) in 2d, for AFM coupling J0/t = 3. The dashedred lines mark the expected continuum boundaries in the m = 1 subspace.Other parameters are J/t = 0.5, S = 0.5, η/t = 0.01.coupling J0 = 3 and Heisenberg and Ising local magnetic moments. Thespectrum of the m = 1 sector consists of a discrete state at low energies,the spin-polaron, and the up-carrier + magnon (c+m) continuum at higherenergies.We start by discussing the origin of the c+m continuum. Since it isinjected with spin-down the carrier can flip its spin and emit a magnon.In doing so it transfers a momentum q to the magnon. Consequently thecarrier momentum after the emission of a magnon is k− q. Furthermoresince the carrier has flipped its spin it experiences an energy shift +J0S/2.The c+m continuum is caused by states where the carrier and the emittedmagnon are far apart. One also refers to this type of states as scatteringstates. Since any momentum q can be transferred to the magnon, the c+mcontinuum spans the energy range {E↑(k− q) + Ωq}q.Let us now discuss the spin-polaron. Because we will encounter a dif-ferent spin-polaron later on, we will refer to this spin polaron as “sp1”.While in the c+m continuum the carrier and the magnon are far apart,in the sp1 state the carrier continuously emits and reabsorbs a magnonin a coherent fashion. To get a better understanding of this we can useperturbation theory [7]. In the absence of Tˆ and HˆS/I the singlet-likestate 1√2S+1∑jeikRj√N[√2Sc†j,↓ − c†j,↑S−j√2S]|FM〉 is an eigenstates of Hˆexcwith eigenenergy −J0(S + 1)/2. The first order corrections to its energy221.5. The spin-polaron-4.00 -3.75 -3.50 -3.25 -3.00ω/t0.↓(0)(0,0,ω)-5.0 -4.5 -4.0 -3.5 -3.0ω/t0.↓(0)(0,0,ω)IsingHeisenberg -2.0 -2.5 -3.0 -3.5 -4.0  0.1   0.2   0.3ω/ tkx/pi 0.0 0.1 0.2 0.3 0.4 0.5Figure 1.6: Left: A(0)↓ (k = 0, ω) for FM J0/t = −2 in 2d. The lowerc+m continuum edge is marked with dashed red lines. The Ising modelhas a discrete state (sp2) below the continuum, but the Heisenberg modeldoes not. Right: Spectral weight A(0)↓ (k, ω) for the Ising model in 2d forky = 0, kx < 0.3pi. The dashed red line marks the lower c+m continuumedge. The sp2 state appears for small k and then merges into the continuum.Other parameters are J/t = 0.5, S = 0.5, η/t = 0.01.areEsp1(k) ≈ −J02(S + 1) +2S2S + 1(k) +2dJ2S + 1(1.36)For J0  t, J this approximation is indeed very accurate [7, 69] (not shown).Note that to first order the effective mass of the sp1 state is a factor of(2S + 1)/2S larger than the bare carrier mass. The reason for this is thatthe carrier needs to move the magnon along with it, when it propagateson the lattice. For AFM J0 > 0 sp1 is the ground state since states inthe continuum have the carrier with spin up and cost ∼ J0S/2 in exchangeenergy. This suggests that for FM coupling J0 < 0, the sp1 polaron shouldbe located above the c+m continuum. This expectation is confirmed below.Perturbation theory also suggests that the symmetry of the sp1 state issinglet-like.Comparing the two panels of Fig. 1.5, we see that the sp1 dispersion isvery similar for the two models. This is expected because this is a coherentstate where the magnon is locked into a singlet with the carrier, and thisprocess is controlled by J0  J . Whether the magnon is localized (Ising)or has a finite speed ∼ J (Heisenberg) is irrelevant. A difference appearsin the shape of the c+m continuum, however. As mentioned, this mustspan energies {E↑(k− q) + Ωq}q since it consists of up-carrier and magnon231.5. The spin-polaron-4 -3 -2 -1 0J0/t-4.0-3.8-3.6-3.4E sp2/tsp2 polaronlower c+m edge-J0S/2-4t0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0J/t-4.0-3.8-3.6-3.4-3.2E sp2/tsp2 polaronlower c+m edgeFigure 1.7: Ground-state energy of the Ising sp2 polaron as a function ofJ0/t for J/t = 0.5 (top) and as a function of J/t for J0/t = −2 (bottom),for S = 0.5.scattering states. The dashed red lines show the boundaries of this range, inagreement with the data (this is more difficult to see for the upper edge, onthis scale, due to the reduced spectral weight at high energies). Since Isingmagnons are dispersionless the continuum boundaries do not change with k.In contrast, the continuum boundaries for the Heisenberg model vary withk, the continuum being wider at the center of the Brillouin zone than nearits edges.This difference has consequences for a FM coupling J0 < 0. As men-tioned, in this case the c+m continuum is expected to be the low-energyfeature in the m = 1 spectrum, with the sp1 state appearing above it. Thisis indeed the case for the Heisenberg model, however in the Ising model, fora sufficiently large J , a second discrete state emerges below the c+m con-tinuum. We will refer to this state as “sp2” to distinguish it from sp1. Theleft panels of Fig. 1.6 show its presence (absence) for the Ising (Heisenberg)model at k = 0. The right panel shows that even for the Ising model, thesp2 only exists for small k, at least for these parameters.The origin of the sp2 state is suggested by the findings of Henning etal. who showed that for J = 0, polaron-like states exist inside the c+m241.5. The spin-polaron-4-2 0 2 4 6 0  0.2  0.4  0.6  0.8  1ω/ tkx/pi 0.00 0.02 0.04 0.06 0.08 0.10-4-2 0 2 4 6 0  0.2  0.4  0.6  0.8  1ω/ tkx/pi 0.00 0.02 0.04 0.06 0.08 0.10Figure 1.8: Spectral weight A(0)↓ (k, ω) vs kx for the 2d Heisenberg model atky = 0 (left) and ky = pi (right) and FM J0/t = −2. Sp1 appears above thecontinuum only near the Brillouin zone edge. No sp2 peak is seen below thecontinuum. The dashed red lines mark the c+m continuum boundaries andthe green line marks E↑(k). Other parameters are J/t = 0.5, S = 0.5, η/t =0.01.continuum [39]. We believe that the addition of HˆI pushes one of thembelow the continuum. This is possible because for an Ising coupling, thelower continuum edge moves up by Ω = 2dJS, whereas the polaron-likestates experience a smaller energy shift since they include a component withthe carrier having spin-down. For the Heisenberg model, on the other hand,inclusion of HˆS does not change the location of the lower continuum edge atk = 0 since Ωq=0 = 0, so the polaron-like state remains a resonance insidethe continuum.The ground-state energy of the sp2 polaron is explored in Fig. 1.7. Thetop panel shows its dependence on J0/t. The sp2 state has weight on boththe down-carrier and on the up-carrier+magnon components. For J0 = 0 theweight of the latter component must vanish since no spin-flips are possibleand the sp2 state is the same as a free down-carrier, whose energy −J0S/2−4t is also indicated (dashed blue line). These results suggest that as |J0|/tincreases, the sp2 state shifts weight from the down-carrier component to theup-carrier+magnon component until it essentially becomes a continuum-likestate.The bottom panel in Fig. 1.7 shows the sp2 ground-state energy vs. J/tfor fixed J0/t = −2. This value of J0/t was chosen because here the polaroniccharacter of sp2 is especially strong because if we neglect Hx,yexc , the energyof the down-carrier component is equal to that of the up-carrier+magnoncomponent. The distance between sp2 and the continuum increases with251.6. Markov-chain Monte Carlo simulationsJ/t, as expected from our previous discussion.While we have only seen the sp2 polaron for the Ising model, we cannotrule out the possibility that for a very narrow range of momenta and carefullychosen parameters, an sp2 state might also appear in the Heisenberg model.Another important point is that the sp1 state is not guaranteed to exist forall k, either. In Fig. 1.8 we show A(0)↓ (k, ω) for the 2d Heisenberg model.No sp2 state appears below the continuum, and sp1 separates above thecontinuum only near the Brillouin zone edge. This is not a surprise giventhe rather small value of |J0| which controls the separation between sp1and the continuum. For sufficiently large |J0|, the sp1 polaron splits off thecontinuum in the entire Brillouin zone [95].To summarize, the spectrum in the m = 1 (one-magnon) subspace con-tains the expected c+m continuum. For AFM J0 the low-energy featureis the sp1 polaron for both the Heisenberg and the Ising models. For FMJ0, sp1 becomes the high energy feature and may only appear in a smallregion of the Brillouin zone if |J0| is small. For the Ising model and FMJ0, an sp2 polaron is also found to appear below the c+m continuum, ina central region of the Brillouin zone that increases with increasing J . Forthe Heisenberg model and FM J0 we cannot entirely rule out the existenceof sp2, although we provided arguments which suggest that this is unlikely.We focused here more on the sp2 polaron because, to our knowledge, thissolution had not been discussed before, while the sp1 state has been analyzedin great detail [7, 39, 73, 95]. We also note that while we presented only(computationally less costly to generate) 2d results, we find qualitativelysimilar results in 3d. This will become clear from the finite-T results shownin Chapter 2.1.6 Markov-chain Monte Carlo simulationsMonte Carlo simulations can be used to calculate finite-T expectation valuesof the type given in Eq. (1.29). We will use this technique in Chapters 3and 4 for simplified versions of the s-f(d) model. This section gives a shortreview of the technique.Many textbooks have been written on the subject of Markov-chain MonteCarlo simulations. The ideas presented here are loosely based on the text-book by Landau and Binder [55] and a set of lecture notes by N. Prokof’ev[86]. Markov-chain Monte Carlo simulations are a powerful technique to261.6. Markov-chain Monte Carlo simulationscalculate expectation values of the form〈A〉 =∑iρ(si)A(si). (1.37)Here A is some physical quantity of interest, e.g. the magnetization of alattice of local magnetic moments, and the sum runs over all the microstates{s1, s2, . . . } of the many body system under study. The probability ρ(si)for a given microstate is given by ρ(si) = Z−1e−βE(si), where E(si) is theenergy of microstate si and β is the inverse temperature (we set kB = 1).In many cases the sum in Eq. (1.37) cannot be calculated exactly becauseof the sheer number of microstates that exist. For instance for an Ising chainwith L spins the number of microstates is 2L. The simplest approach wouldtherefore be to use a Monte Carlo algorithm which randomly generates Nmicrostates, r1, r2, . . . rN from a uniform distribution and approximate 〈A〉by〈A〉 ≈ A¯(N) =N∑i=1ρ(ri)A(ri) (1.38)When N →∞ the estimate A¯(N) converges to the exact result [55].The problem with the above approach is that one is likely to generatemicrostates which have small weights ρ(ri). Consequently their contributionto the sum is small and the convergence is slow. This can be avoided ifinstead of generating microstates from a uniform distribution one generatesthem from the distribution ρ(si) itself.To do this we start from a microstate of the system and simulate theevolution of the system. If at the beginning of the Monte Carlo simulationthe system is in state x0 it will be in state xn after n steps of the simulation.The number n of Monte Carlo steps therefore plays the role of time. For thisapproach to work it is crucial that the system is ergodic, i.e. that any statesj of the system can be reached from any state si. Furthermore we requirethat the evolution of the system can be described by a Markov process.A Markov process is defined as follows. Let P (xn+1 = sin+1 |xn =sin , xn−1 = sin−1 . . . ) denote the conditional probability that the systemis in state sin+1 after n + 1 Monte Carlo steps if it was in state sin after nsteps and in state sin−1 after n− 1 steps and so on. Processes which satisfyP (xn+1 = sin+1 |xn = sin , xn−1 = sin−1 . . . ) = P (xn+1 = sin+1 |xn = sin), i.e.the evolution of the system only depends on its current state and not on itshistory, are called Markov processes.271.6. Markov-chain Monte Carlo simulationsThe conditional probability P (xn+1 = sj |xn = si) is also called thetransition probability from state si to state sj . For notational conveniencewe will refer to it as wi→j . Our goal is to find a simple expression for wi→jwhich allows us to generate states from the probability distribution ρ(si).Toachieve this we make use of the master equation governing the probabilityP (xn = sj) that the system is in state sj at “time” n [71].∂P (xn = sj)∂n= −∑i 6=jwj→iP (xn−1 = sj) +∑i 6=jwi→jP (xn−1 = si) (1.39)The master equation can be thought of as a flow equation. The first term onthe left hand side describes the probability flow out of the state sj while thesecond term describes the flow into the state sj . In equilibrium P (xn = sj)must be independent of n and identical to ρ(sj). Therefore we have∑i 6=jwj→iρ(sj) =∑i 6=jwi→jρ(si) (1.40)A common way to satisfy Eq. (1.40) is to require thatwi→jwj→i=ρ(sj)ρ(si)= e−β(E(sj)−E(si)), (1.41)a condition which is referred to as detailed balance. Physically, detailedbalance implies that every change in the state of the system is reversible. Inother words if the system can change from state si to state sj the reverseprocess, changing from state sj to state si, must also be allowed if detailedbalance is to be satisfied.When performing a Monte Carlo simulation we split the transition prob-ability into two separate parts: wi→j = Pu(si, sj)Pacc(si, sj). The first part,Pu(si, sj) is the probability that when the system is in state si an updateto state sj is suggested. The second part Pacc(si, sj) is the probability thatthe update to state sj is accepted. If we choose Pu(si, sj) in such a way thatPu(si, sj) = Pu(sj , si), then it drops out from the detailed balance condition.We are therefore left with the task of choosing Pacc(si, sj) in a way whichsatisfies the detailed balance condition. In 1953 Metropolis et al. proposedPacc(si, sj) = min(1, e−β(E(sj)−E(si))) [36, 65]. It is easy to verify that thischoice does indeed obey detailed balance. Furthermore it has the advantagethat states with a lower energy are automatically accepted.Note that the probability Pacc(si, sj) does not necessarily need to benormalized. The normalization is added when calculating the desired ex-pectation value, as shown below. Because it is likely that consecutive states281.6. Markov-chain Monte Carlo simulationsxn and xn+1 are correlated, one often uses only every mth state in the cal-culation of the expectation value. Furthermore one usually discards the firstNth states to ensure that the states are thermalized. The desired expectationvalue is then given byA¯(N) =1NN∑n=1A(xmn+Nth), (1.42)where the states {x1, x2, . . . xmN+Nth} were generated with the Monte Carlosimulation and the factor 1/N ensures normalization.The simplest Metropolis Monte Carlo scheme for calculating the magne-tization of an Ising model consists of the following steps (cf. [55])1. Initialize the system in any state2. Randomly choose a spin (corresponds to Pu).3. Calculate the energy change ∆E which would result if the spin wasflipped.4. If ∆E < 0 accept the spin flip. Otherwise generate a random number0 < r < 1 and accept the spin flip if r < e−β∆E (Corresponds to Pacc).5. Calculate the magnetization and store its value.6. Go to step 2.Generally one refers to the sequence 2-4 as one Monte Carlo step, while step 5is referred to as a Monte Carlo measurement. Note that in this simple schemea measurement is performed after every step which will lead to correlationsbetween the measurements. It is important that the magnetization is alsocalculated when the spin flip in the preceding step was not accepted. Oncethe simulation has ended, the average magnetization and its standard errorcan be calculated. A more sophisticated implementation which circumventsthe problem of correlations is discussed in Appendix C.1.6.1 The effect of correlationsThere are multiple sources of errors which need to be considered when work-ing with Monte Carlo simulations. In this section we discuss statistical errorswhich arise due to a finite sample size and correlations between Monte Carlomeasurements. There are also other sources of error which are not discussedin this section, for instance finite size effects.291.6. Markov-chain Monte Carlo simulationsLet us assume that during a Monte Carlo simulation we made N orderedmeasurements A1, . . . , AN . By ordered we mean that measurement An+1was made immediately after measurement An, but it is possible that a fixednumber of states was discarded in between the two measurements to reducetheir correlation. The measurement index n can then be interpreted as adiscrete time variable with stepsize δt which is directly proportional to thenumber of states discarded in between measurements.The mean value is given by A¯ = N−1∑Ni=1Ai and a measure of itsuncertainty is given by its variance. The variance of the ith measurementis defined as Var(Ai) = 〈(Ai− 〈A〉)2〉. Since all the Ai were drawn from thesame distribution Var(Ai) = Var(A) is independent of i. It is also usefulto define the covariance Cov(Ai, Aj) = 〈(Ai − 〈A〉)(Aj − 〈A〉)〉 which is ameasure of the correlation between Ai and Aj . Note from its definition itfollows that Cov(Ai, Aj) = Cov(Aj , Ai) and since all the Ai were drawnfrom the same distribution and the system is assumed to be in equilibriumCov(Ai, Aj) = Cov(A1, A1+j−i). The variance of the mean is given by:Var(A¯) =1N2 N∑i=1Var(Ai) +N∑i=1N∑j=1,j 6=iCov(Ai, Aj) (1.43)Using the shorthands Var(A) = σ2A and Cov(Ai, Aj) = σ2ACAj−i, this can bebrought into the following formσ2A¯ =σ2AN[1 + 2N−1∑t=1(1− tN)CAt](1.44)The function CAt is known as the autocorrelation function of A. From itsdefinition it is clear that CA0 = 1. Furthermore one expects CAt → 0 forlarge t since Ai and Ai+t are not correlated in that case. If we assume thatCAt is already negligibly small for t  N the second term in the equationabove can be approximated as∑∞t=0CAt ≈ 1δt∫∞0 dt CAt =: τA/δt [71]. Thequantity τA is known as the autocorrelation time of A. Reinserting this weobtain the following expression for the variance σ2A¯of the mean A¯σ2A¯ =σ2AN[1 + 2τAδt]. (1.45)Equation (1.45) confirms the well known result that the variance of themean scales as 1/N . The simplest way to improve the accuracy of the sim-ulation is therefore to increase the sample size. However, often calculating301.6. Markov-chain Monte Carlo simulationsthe observable A for a given state is expensive. In that case the accuracy ofthe simulation can be increased by keeping the sample size fixed but increas-ing the time step δt which corresponds to discarding more states betweenmeasurements, i.e. waiting longer between measurements.1.6.2 The blocking methodThere are various methods to assess the effect of correlations on the out-come of a Monte Carlo simulation. The simplest one is to calculate theautocorrelation function. Another useful method is the so-called blockingmethod. To implement the blocking method we combine M measurementsinto a block average BnBn =1MM∑i=1Ai+nM (1.46)Lets assume that we can form Nb of such blocks and that Nb is a power oftwo. This will prove useful later on.In terms of the block averages the mean A¯ is given byA¯ =1NbNb∑n=1Bn. (1.47)Since all of the blocks comprise an equal number M of measurements weassume that they also have the same variance. To estimate it we calculate.(σ(0)B )2 =1NbNb∑n=1(Bn − A¯)2. (1.48)In a next step we combine adjacent blocks B(1)n = (B2n−1 +B2n)/2. Thisis possible since Nb is a power of two. Note also that the choice of combiningadjacent blocks is merely a choice of convenience, more sophisticated meth-ods where randomly chosen pairs of blocks are combined are also possible.311.6. Markov-chain Monte Carlo simulationsThe variance of the superblocks B(1)n is estimated as(σ(1)B )2 =2NbNb/2∑n=1(B(1)n − A¯)2=12NbNb∑n=1(Bn − A¯)2 + 1NbNb/2∑n=1(B2n−1 − A¯)(B2n − A¯)=(σ(0)B )22+1NbNb/2∑n=1(B2n−1 − A¯)(B2n − A¯) (1.49)The last term in the equation above has the form of a covariance and is ameasure of the correlation between adjacent blocks. If B2n−1 and B2n arecorrelated we expect this term to always be larger than zero and contributeto σ(1)B . If, on the other hand, B2n−1 and B2n are uncorrelated the sign of(B2n−1−A¯)(B2n−A¯) will oscillate and the term will be zero on average. Notethat when there are no correlations we have (σ(1)B )2 = (σ(0)B )2/2, whereascorrelated blocks lead to (σ(1)B )2 > (σ(0)B )2/2.Since we chose Nb to be a power of two the process of combining adjacentblocks can be continued and in the absence of correlations we have (σ(m)B )2 =(σ(m−1)B )2/2 = · · · = (σ(0)B )2/2m. Furthermore we know from the precedingsection that in the absence of correlations the variance of the mean is givenby(σ(m)A¯)2 =(σ(m)B )2Nb/2m=(σ(0)A¯)2Nb. (1.50)This means that in the absence of correlations (σ(m)A¯)2 is independent of m.When correlations are present, on the other hand, we have shown above that(σ(m)B )2 > (σ(m−1)B )2/2 and consequently (σ(m)A¯)2 > (σ(m−1)A¯)2.The effect of correlations can therefore be estimated by plotting σ(m)A¯against the number of blocks N(m)b = Nb/2m. At large values of N(m)b theblocks will be small and likely to be correlated. Consequently σ(m)A¯will beunderestimated in this regime. As we move towards smaller values of N(m)bcorrelations should play less of a role and the size of σ(m)A¯should increaseuntil it becomes roughly constant or at least tapers off to a finite value whichconstitutes a good estimate for the standard error of the mean.321.7. Outline of this thesis1.7 Outline of this thesisAt zero T the spectral function of a spin-up carrier injected into a FMwith all spins up is strikingly different from that of a spin-down carrier.The reason for this is that the spin-up carrier spectrum belongs to theSztot = NS + 1/2 subspace where spin-flips are impossible while the spin-down carrier spectrum belongs to the Sztot = NS − 1/2 subspace. Thisnaturally leads to the question what happens to the spin-up spectrum atfinite T . In that case the carrier is injected into a magnetic backgroundwhere thermal magnons may already be present and consequently the dif-ferent subspace of Sztot mix. Therefore one expects that spectral weight willbe transferred from the T = 0 δ-peak to the spin-polaron states leading toan interesting T dependence of the spin-up spectrum.How this transfer of spectral weight occurs is the topic of Chapter 2,where we develop a low-T expansion for the spin-up GF. The expansionbecomes exact in the limit T → 0. In contrast to previous work we usea canonical ensemble which ensures that exactly one carrier is present inthe system at all times. The small parameter in which we expand is theBoltzmann factor e−βΩq . Essentially the idea for this expansion is that atlow T only states with few magnons contribute to the temperature averageand consequently the trace over the states of the magnetic background canbe truncated. The advantage of this method and the restriction to a singlecarrier is that it allows us to treat the carrier-magnon interaction exactlyrather than using a mean field approach. It furthermore allows us to compareIsing and Heisenberg magnetic backgrounds just as we did for the T = 0spin-polaron above. This work applies for d ≥ 2. In d = 1, a single domainwith opposite FM order costs the same energy as a magnon and restrictionto a single magnon is not a good approximation.To understand the role played by these domains in 1d, we introduce,in Chapter 3, a simplified model where all the exchange interactions are ofIsing type. This simplification makes it possible to obtain exact, numericalresults with a Metropolis algorithm and for any temperature to study bothFM and AFM chains on equal footing. We find that for both AFM and FMmagnetic backgrounds the carrier spectrum shows a distinctive structure ofsmall resonances. These resonances can be attributed to small magneticdomains which entrap the carrier. A similar phenomenon occurs in binaryalloys of the type AxB1−x where charge carriers can become localized indomains of like-atoms. However, there are also some key differences betweenour model and binary alloys. In binary alloys one uses a disorder averagethat keeps the concentration of atoms constant and generally there are no331.7. Outline of this thesiscorrelations. In this cases the carrier can become truly localized. In ourmodel, on the other hand, we use a temperature average which includescorrelations. Furthermore the carrier never becomes truly localized becausethe magnetic background is not static.The numerical results allow us to identify the type of magnetic domainswhich contribute most to the entrapment of the carrier. This informationis then used to construct an analytical approximation that works well atlow to medium T and reproduces most of the resonances found in the exactspectrum.In Chapter 4 we study the differences between multi-band and single-band models. In single-band models the carrier is hosted by the same bandthat gives rise to the magnetic moments, while in multi-band models it ishosted in a separate band. Specifically we compare three different models(one of them being the model from Chapter 3) that for FM order and T = 0have identical QPs. The models can be mapped onto each other in a fashionthat resembles the famous Zhang-Rice mapping from the three band Emerymodel to the t−J model in the cuprates [114]. Surprisingly we find that eventhough the mapping between our models works at T = 0, it does not work atany finite T . The reason for this is again linked to the local environment intowhich the carrier is injected. For a single-band model the carrier effectivelyremoves one of the magnetic moments when it is injected. If this leads tothe removal of a magnetic moment which was previously misaligned withits neighbors the energy of the system is lowered. In two-band models onthe other hand such a process is impossible because the carrier is hosted bya separate band and therefore does not remove magnetic moments upon itsinjection. At T = 0 for FM coupling all the magnetic moments are alignedand these processes do not play any role. However, at any finite T they causethe low-energy states of the single-band to be qualitatively different from themulti-band models. The ramifications of this effect and its generalization tomore complicated models are discussed in detail in Chapter 4Chapter 5 contains a summary and discussion of the work presented inthis thesis and suggestions for further work and extensions.34Chapter 2Signatures of spin-polaronstates at low temperatures2.1 IntroductionIn the previous chapter we derived the zero temperature spectrum for a sin-gle carrier injected into a FM ordered magnetic background. At finite tem-perature, an exact solution is no longer possible since one needs to considerstates with arbitrary numbers of magnons when performing the temperatureaverage. A natural approach for low T is to consider states with a smallnumber of magnons; this is what we do here. As a result, the solution wepropose becomes asymptotically exact in the limit of very low temperatures,where “low” means well-below the Curie critical temperature TC of the FMbackground.As mentioned, a spin-up carrier has a very simple spectrum at T =0, mirroring that of the free carrier, with a single eigenstate for a givenmomentum. At T 6= 0 thermally activated magnons are present in thesystem and the carrier can now flip its spin by absorbing one of them.Interaction with even one such magnon takes the problem in the Hilbertsubspace appropriate for the T = 0 spin-down carrier, which has a verydifferent spectrum. As a result, we expect that spectral weight is transferredfrom the spin-up QP peak to energies in the spectrum of the spin-polaron, asT increases. How exactly does this occur at very low T , and what happensto the infinitely-lived discrete state that was the only feature in the spectrumat T = 0, is the topic of this chapter.Furthermore, we consider two types of exchange between the local mo-ments, namely Heisenberg exchange and Ising exchange (in both cases, thecharacteristic energy scale is J). For the latter the magnon spectrum isgapped, whereas for the former the magnon spectrum is gapless. This al-lows us to contrast the two cases to understand the relevance of the magnon’sspectrum on the evolution with T of the up carrier’s spectral function.Finite temperature studies in the single carrier limit have been previously352.1. Introductioncarried out by Nolting et al. [81], Kubo [53] and Auslender et al. [5]. In thefollowing paragraphs we summarize their approaches and comment on thedifferences to our method.Nolting et al. account for the kinetic energy of the carrier with a tight-binding model with an energy scale t, and for the exchange between the localmoments and the carrier with a Heisenberg exchange with a coupling J0.Unlike the models we consider, Nolting et. al. do not include the exchangeJ between local moments; this is one key difference between our work andtheirs. The second is the approach employed. While, as mentioned, we usea low-T expansion to calculate the propagator, Nolting et al. proposed anansatz for the self-energy chosen so as to reproduce asymptotic limits wherean exact solution is available, specifically the T = 0 solution mentionedabove and the case of finite T but zero bandwidth, t = 0 [76] (This approachwas later generalized to finite carrier concentrations as well [80]). Theiransatz for the self-energy contains several free parameters which are fixedby fitting them to a finite number of exactly calculated spectral moments. Asimilar approach for the spectral weight was previously used by Nolting andOles´ in Refs [77–79]. The temperature dependence is contained implicitlyin the magnetization which enters the self-energy as an external parameter.In the limit of very low T we consider here, the average local moment isessentially unchanged from its T = 0 value, so the effects we uncover arebasically absent in the ansatz of Nolting et al.. In other words, besidesstudying different Hamiltonians by very different means, our studies alsofocus on very different regimes: very low T , in our work, vs. medium andhigh T in Ref. [81]. Needless to say, in the absence of an exact solution it islikely that a collection of approximations valid in different regimes will beneeded in order to fully understand this problem.Kubo [53] used the coherent potential approximation (CPA) to calculatethe carrier LDOS at finite T (see also [103]). The CPA is an approximationthat was introduced by Soven [100] as a means to deal with the disorderaverage in binary alloys through the introduction of an effective medium.In Ref. [53] Kubo extends this procedure to the s-f(d) model by neglect-ing spin-spin correlations. Instead the spin part HS of the Hamiltonian istreated in the molecular field approximation which is essentially a mean fieldapproximation. The approximation is expected to fail at low temperatures,where spin-spin correlations are important, and is therefore not applicablein our case. The advantage of the CPA is that it can be extended to modelsfor dilute magnetic semiconductors where the local magnetic moments aredisordered [101].Auslender et al. [5] derived an expression for the single particle GF of362.2. The low-temperature expansionthe s-f(d) model at temperatures much lower than the Curie temperatureTC . Their result is also obtained by using an EOM approach, but they usea Dyson-Maleev transformation to represent the local spins Si with bosonicoperators. This type of transformation generally works best when S is largewhile our low-T approximation does not impose any requirements on S andalso allows us to study the limiting case S = 1/2. Furthermore Auslender etal. neglect spin-spin interactions which are taken into account in our work.Nevertheless the self energy derived by Auslender et al. is remarkably closeto our result. However, in their analysis they focus on the effect that finitetemperature has on the T = 0 QP peak and not on the transfer of spectralweight to spin-polaron states.2.2 The low-temperature expansionTo calculate the low-T expression of the GF for a spin-up carrier, G↑(k, ω)we start from Eq. (1.29)G↑(k, ω) =∑ne−βES/InZ〈S/I, n|ck,↑Gˆ(ω + ES/In )c†k,↑|S/I, n〉. (2.1)Remember that we are in a canonical (not grand-canonical) ensemble,assuming that the carrier is injected in the otherwise undoped FM which isin thermal equilibrium. As a result, the trace is over the eigenstates |S/I, n〉of HˆS/I (in the absence of carriers, Hˆ ≡ HˆS/I).At T = 0, the trace reduces to a trivial expectation value over |FM〉, andwe recover the result from Chapter 1.5G(0)↑ (k, ω) = 〈FM|ck,↑Gˆ(ω)c†k,↑|FM〉 =1ω − E↑(k) + iη . (2.2)The eigenenergy is E↑(k) = (k) + J0 S2 for both the Heisenberg and Isingmodels. As discussed, this shows that at T = 0 a spin-up carrier propagatesfreely and acquires an energy shift from Hˆzexc.At finite temperature, we expect to find:G(k, ω) =1ω − E↑(k)− Σ(k, ω) + iη= G(0)↑ (k, ω) +G(0)↑ (k, ω)Σ(k, ω)G(0)↑ (k, ω) + . . . (2.3)Strictly speaking, the energy shift J0S2 is part of the self-energy, however itis convenient to separate it as we do here so that Σ(k, ω) contains only thefinite-T terms.372.2. The low-temperature expansionSince we are interested in the lowest-T contribution to Σ(k, ω), we con-sider only the first two terms of Eq. (1.15), i.e. the completely FM orderedstate and states with one magnon. In doing so we findG↑(k, ω) =G(0)↑ (k, ω) +∑q e−βΩqG(1)↑ (k,q,q, ω) + . . .1 +∑q e−βΩq + . . ., (2.4)where we define the new propagatorsG(1)↑ (k,q,q′, ω) = 〈Φ(q′)|ck,↑Gˆ(ω + Ωq′)c†k+q′−q,↑|Φ(q)〉. (2.5)Note that the argument of the resolvent is shifted by the magnon energy,meaning that the carrier’s energy is measured with respect to that of thestate in which the carrier is injected. Furthermore only diagonal q′ = qterms contribute to the trace. Following calculations detailed in AppendixA, we find:∑qe−βΩqG(1)↑ (k,q,q, ω) =∑qe−βΩq{G(0)↑ (k, ω)− J02N[G(0)↑ (k, ω)]21 + J0SG(0)↑ (k + q, ω + Ωq) +J02 g(k,q, ω) , (2.6)whereg(k,q, ω) =1N∑QG(0)↑ (k + q−Q, ω + Ωq − ΩQ) (2.7)is a known function. When this expression is used in Eq. (2.4), we obtainG↑(k,ω)=G(0)↑ (k,ω)(1+∑qe−βΩq+...)+[G(0)↑ (k,ω)]2Σ(k,ω)(1+...)+...1+∑qe−βΩq+...(2.8)=G(0)↑ (k,ω)+[G(0)↑ (k,ω)]2Σ(k,ω)+..., (2.9)since the terms in the brackets are the expansion of Z (to the order consid-ered here; higher order contributions will come from including many-magnonprocesses) and cancel with the denominator. This has the expected form ofEq. (2.3), so we can identify the lowest-T correction to the self-energy:Σ(k,ω)=− J02N∑qe−βΩq1+J0SG(0)↑ (k+q,ω+Ωq)+J02 g(k,q,ω)+... (2.10)382.2. The low-temperature expansionIt is important to mention that although we only considered states withzero or one magnon in our derivation, we will see some higher-order effectsin our results when using G↑(k, ω) = [ω−E↑(k)−Σ(k, ω)+ iη]−1, i.e. whenthe self-energy is placed in the denominator. These are from states wheremultiple magnons are present in the system but the carrier interacts onlywith one of them while the rest are “inert” spectators.Equation (2.10) is the main result of this chapter. The only differencebetween Heisenberg and Ising backgrounds is the expression for the magnonenergy Ωq. For the Ising case, this energy is independent of momentum,resulting in a self-energy Σ(ω) independent of k.Before presenting results, let us consider what the spectral weight givenby A↑(k, ω) = − 1pi ImG↑(k, ω) should be expected to reveal. The Lehmannrepresentation of the propagator in its expanded form is (cf. Eq. (1.20))G↑(k,ω)=1Z[1ω−E↑(k)+iη+∑α,qe−βΩq|〈Φ(q)|ck,↑|Ψ(1)α (k+q)〉|2ω+Ωq−E(1)α (k+q)+iη+...].(2.11)At T = 0 only the first term contributes, giving a single QP peak at ω =E↑(k). The second term has poles at ω = E(1)α (k + q) − Ωq. The m = 1subspace also corresponds to a spin-down carrier injected in the FM at T =0, thus we can find the energies E(1)α (k) from the spectral weight A(0)↓ (k, ω)which was discussed in Chapter 1.5.As already mentioned and further detailed below, the spectrum E(1)α (k)certainly contains an up-carrier+magnon continuum spanning the energies{E↑(k−q′)+Ωq′}q′ ; in the right circumstances, a coherent spin-polaron statewith the magnon bound to the carrier may also appear, see below. Thus, forT 6= 0, A↑(k, ω) should have weight at all energies {E↑(k + q− q′) + Ωq′ −Ωq}q,q′ . In the Ising case the magnon energies cancel out so weight shouldbe expected at all energies {E↑(q)}q in the spin-up carrier spectrum, notjust at E↑(k). This automatically implies that the T = 0 infinitely lived QPof energy E↑(k) acquires a finite lifetime at T 6= 0. This remains true for theHeisenberg case, with the added complication that now, {E↑(k + q− q′) +Ωq′ − Ωq}q,q′ will generally span a wider range of energies than {E↑(q)}q.If a spin-polaron appears in the m = 1 sector, additional weight is expectedat energies in its band minus the magnon energy. Higher order terms willcontribute similarly (remember that our solution for the propagator doesinclude partial contributions from many-magnon states). To conclude, atfinite T one can no longer assume that energies for which the spectral weightA↑(k, ω) is non-zero are necessarily in the spectrum of the momentum-k392.3. ResultsHilbert subspace. This makes the interpretation of the spectral weight lessstraightforward than it is at T = 0.2.3 ResultsWe now present and analyze low-T results for the spectral weight of thespin-up carrier. Since the calculation of G↑(k, ω) becomes numerically veryexpensive in 3d, most of our analysis is in 2d. However, we will also showa selection of 3d spectra which prove that the 3d results are qualitativelysimilar to the 2d results. The low-T expansion derived above is expected tofail in 1d where, at least for Ising coupling, the energy of magnetic domainsis independent of their length. We therefore do not show any 1d results.The spectral weight A↑(k, ω) and the self-energy Σ(k, ω) are shown forthe Heisenberg and Ising models with AFM coupling J0/t = 10 in Figs. 2.1and 2.2, respectively. In both cases the top panel is for k = (0, 0) and thebottom one is for k = (pi, pi). However, for the Ising model the self-energyis independent of k and therefore in Fig. 2.2 it is only shown beneath thek = (0, 0) spectral weight. The value of J0/t was chosen so large in orderto ensure that the different features in the spectrum are well separated, tosimplify the analysis. Results for smaller values of J0 will be shown below.From the previous discussion of the m = 1 sector at T = 0 we know thatfor smaller J0 the spin-polaron states change their nature as they becomeresonances in the c+m continuum. Similar behavior is expected to occur atfinite T and its signatures are discussed below.A↑(k, ω), which at T = 0 is the peak δ(ω−E↑(k)) (indicated by the thickgreen line), broadens into a continuum at finite T . As discussed at the endof the previous section, this continuum has its origin in the c+m continuumof the m = 1 sector, thus we continue to call it the “c+m” continuum,and should span {E↑(k + q− q′) + Ωq′ − Ωq}q,q′ . The red dashed linesshow the boundaries of this energy range, in excellent agreement with thebroadening observed in A↑(k, ω). We note that most of the spectral weightis still located near E↑(k).This broadening confirms that at finite T the QP acquires a finite lifetime(the peak at E↑(k) is now a resonance inside a broad continuum, not adiscrete state). Clearly, this is due to processes where the spin-up carrierabsorbs a thermal magnon and then re-emits it with a different momentum,thus scattering out of its original state.The finite lifetime of the carrier in the c+m continuum is also evidentin the self-energy. The inset in Fig. 2.2 shows that for energies within the402.3. Results0.↑(0,0,ω) sp1c+mE↑(k)↑(pi,pi,ω)sp1c+mE↑(k)-12 -10 -8 -6 -4 -2 0 2 4 6 8 10ω/t-30.00.0Σ(0,0,ω)ImRe-12 -10 -8 -6 -4 -2 0 2 4 6 8ω/t-Σ(pi,pi,ω) ImReFigure 2.1: Spectral weight A↑(k, ω) and the real (solid line) and imaginary(dashed line) part of the self-energy Σ(k, ω) for the 2d Heisenberg modelwith AFM J0/t = 10 and βt = 1, at k = (0, 0) (top) and k = (pi, pi)(bottom). The expected sp1 continuum boundaries are marked with dash-dotted blue lines and the expected c+m continuum boundaries with dashedred lines. The E↑(k) energy of the T = 0 δ-peak is marked with a thickgreen line. Other parameters are J/t = 0.5, S = 0.5, η = 0.02 (top) andη = 0.05 (bottom).412.3. Results0.000.010.02A↑(0,0,ω) sp1 c+mE↑(k)-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10ω/t0.↑(pi,pi,ω) sp1c+mE↑(k)-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10ω/t-60.0-Σ(ω)ImReFigure 2.2: Same as Fig. 2.1 but for the Ising model. All parameters arethe same except βt = 0.5 and η = 0.01 in both panels. Note that for theIsing model Σ(ω) is independent of k. The inset shows a zoom on Σ(ω) athigh energies.422.3. Resultsc+m continuum the imaginary part of the self-energy is finite. The same istrue for the Heisenberg model (not shown).While the broadening of the T=0 δ-peak may be thought of as quitetrivial, Figs. 2.1 and 2.2 show that it is not the only effect of the finite T :spectral weight is also transferred to a new continuum located below thec+m continuum. We attribute this continuum to the sp1 state. Indeed, ifwe denote by Esp1(k) the energy of the sp1 polaron, we find that this con-tinuum spans {Esp1(k + q)−Ωq}q (the boundaries of this range are markedby the dashed-dotted blue lines). Its presence agrees with the Lehmann rep-resentation and reveals this spectral weight transfer to be due to processeswhere the spin-up carrier binds a thermal magnon and turns into an sp1polaron.The sp1 continuum is also where both the real and imaginary part ofΣ(k, ω) take their largest values. Consequently the lifetime of these statesis roughly two orders of magnitude smaller than that of the states withinthe c+m continuum. This is not surprising as the c+m continuum stemsfrom a δ-peak with an infinite lifetime at T=0, whereas the sp1 continuumvanishes at T=0.There is furthermore a qualitative difference between the real-part ofΣ(k, ω) in the sp1 continuum and in the c+m continuum. For the latter thereal part falls off relatively smoothly (cf. inset in Fig. 2.2), whereas for thesp1 continuum it is highly singular and almost discontinuous.Note that there are no major differences between the Heisenberg andIsing models, except for the fact that the boundaries of these continua aremomentum dependent for the former and momentum independent for thelatter, due to their different magnon dispersions.Figures 2.1 and 2.2 also show a very puzzling discrete state at low ener-gies. Before we turn our attention to the analysis of this peak, we quicklydiscuss the case with FM coupling J0 < 0. Ising and Heisenberg resultsare depicted in Fig. 2.3 for J0/t = −2 and J/t = 0.5. From the discus-sion of the T = 0 spectrum in the m = 1 Hilbert space, we know that forthese parameters the Ising model has an sp2 state below its c+m continuumand therefore expect to find its signature in the finite-T spectrum, as well.This is indeed the case, as seen more clearly in panel (b) which expandsthe low-energy part of the Ising spectrum shown in (a), revealing weight atenergies spanning {Esp2(k + q) − Ωq}q (its lower boundary is marked bydashed-dotted blue lines). Note that since the sp2 state merges with thec+m continuum (boundaries marked by red dashed lines), their correspond-ing continua also merge, but panel (b) reveals a clear discontinuity wherethey overlap. The high-energy sp1 continuum is also clearly observed in432.3. Results-4 -2 0 2↑(0,0,ω)-4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -↑(0,0,ω)-6 -4 -2 0 2 4 6ω/t0.000.01A↑(0,0,ω)sp2c+msp1sp2c+msp1(a)(b)(c)E↑(k)E↑(k)Figure 2.3: Spectral weight A↑(0, 0, ω) for the 2d Ising model (panels (a)and (b)) and 2d Heisenberg model (panel (c)) for FM J0/t = −2 at βt =0.5, η/t = 0.01 (Ising) and βt = 1, η/t = 0.02 (Heisenberg). The expectedlocation of various features are also indicated (see text for more details).Other parameters are J/t = 0.5, S = 0.5.panel (a), again merged with the c+m continuum since the sp1 state is notfully separated at such a small |J0|, either.The Heisenberg model (panel (c)) only shows the c+m and sp1 continua,since there is no sp2 polaron here. Again, agreement with the expectedboundaries is excellent (the weight seen below the c+m lower edge is dueto the finite η and the fact that we zoomed in close to the axis to make iteasier to see the sp1 continuum).It is worth noting that since for small |J0| the various features merge, itwould be easy to misinterpret the thermal broadening as being all of c+morigin, i.e. to entirely miss the role played by the spin-polaron solutions inthe m = 1 subspace. This is also illustrated in Fig. 2.4, where we returnto an AFM J0 coupling and show how the k = 0 spectra change as J0 isdecreased. All features discussed previously can be easily identified for largeJ0 but merge into one another as J0 decreases, so that by the time J0/t = 3there is only one very broad feature, albeit with a non-trivial structure, left442.3. Results0.↑(0,0,ω) IsingHeisenberg0.↑(0,0,ω)-15 -10 -5 0 5 10ω/t0.↑(0,0,ω)J0/t=10J0/t=5J0/t=3Figure 2.4: Spectral weight A↑(0, 0, ω) for the 2d Ising (dashed lines) andHeisenberg (full lines) models for J0/t = 10, 5, 3 in the top, middle andbottom panels, respectively. Other parameters are J/t = 0.5, S = 0.5 andβt = 0.5, η/t = 0.01 (Ising), and βt = 1, η/t = 0.02 (Heisenberg). Theoscillations visible especially in the sp1 continuum are due to finite-sizeeffects (we used N = 1002 and N = 5002 for Heisenberg and Ising models,respectively).in the spectrum (apart from the low-energy discrete peak, which we willdiscuss later). If one assumed that this is all of c+m origin, i.e. scatteringof the carrier on individual thermal magnons, one would infer very wrongvalues of the parameters from the boundaries’ locations.The results shown so far are for large temperatures kBT ∼ t = 2J (forour parameters), where higher order corrections should certainly becomequantitatively important. On the other hand, from the Lehmann decompo-sition we expect that the location of the various features does not dependon temperature; only how much spectral weight they carry can change withT . For a more thorough analysis we return to the case of AFM J0, using arather large value so that the various features are well separated, and plot inFig. 2.5 the spectral weight in the sp1 continuum for several different tem-peratures, for both the Ising and Heisenberg models. This confirms that,452.3. Results0.↑(0,0,ω) βt=↑(0,0,ω)βt=↑(0,0,ω)βt=14-8 -7 -6 -5ω/t0.000.050.10A↑(0,0,ω)βt=20ω/t0.000.010.02βt=βt=βt=βt=2Heisenberg IsingFigure 2.5: Spectral weight A↑(k = 0, ω) for the 2d Heisenberg (left) andIsing (right) models with AFM J0/t = 7, at different temperatures. Onlythe sp1 continuum is shown. Its edges are indicated with dot-dashed bluelines. Other parameters are J/t = 0.5, S = 0.5 and η/t = 0.01 and 0.02 forIsing and Heisenberg, respectively.462.3. Results0 2 4 6 8 10βt0.∫ c+mdω A↑(k,ω)Ising k=(0,0)Ising k=(pi,pi)Heisenberg k=(0,0)Heisenberg k=(pi,pi)Figure 2.6: Integrated spectral weight in the c+m continuum as a functionof β. Lines are fits described in the text. Parameters are: J0/t = 10, J/t =0.5, S = 1/2, η/t = 0.01 (Ising), η/t = 0.05 (Heisenberg).indeed, the weight in this continuum decreases fast as T → 0 while its loca-tion is not affected (the location of the low-energy peak shifts with T , butas we argue below, we do not believe that this is a physical feature).To quantify the spectral weight transferred, we calculate∫c+m dωA↑(k, ω),i.e. how much is in the c+m continuum. Since at T = 0 all the weight isin the δ-peak at E↑(k) located inside the c+m continuum, this value startsat 1 and decreases with increasing T , as weight is transferred into the sp1continuum; one can easily check that the spectral weight obeys the sum rule∫∞−∞ dωA↑(k, ω) = 1.The results are shown in Fig. 2.6 for both models, both at the centerand at the corner of the Brillouin zone. Note that because of the finite valueof η, some spectral weight “leaks” outside the continuum’s boundaries. Thisproblem is more severe at lower T because E↑(k) is located very close toan edge of the continuum; this explains why the value saturates below 1 asβ → ∞. This explanation is also consistent with the observation that theamount of “missing weight” as T → 0 is of order η.Two features are immediately apparent. First, there is a substantialdifference in the amount of spectral weight transferred out of the c+m con-tinuum at k = (0, 0) vs. k = (pi, pi). This is expected for the Heisenberg472.3. Resultsmodel where the location of all features changes with k, but may come as asurprise for the Ising model where their location is independent of k. How-ever, for both models E↑(k), where most of the weight is found, moves fromthe lower edge of the c+m continuum when k = 0, to the upper edge fork = (pi, pi). As a result, it is reasonable that weight is transferred into thelow-energy sp1 continuum more efficiently at k = (0, 0) than at k = (pi, pi),since in the former case the “effective” energy difference between the twofeatures is smaller.The second observation is that spectral weight is transferred into the sp1continuum more efficiently in the Heisenberg model than in the Ising model.This difference is also clearly visible in Fig. 2.5, where the weight in the sp1continuum of the Heisenberg model is still respectable at βt = 20, while forthe Ising model this weight is already negligible at βt = 8.An explanation for this difference comes from assuming that the weightin the sp1 continuum is proportional to the average number of thermalmagnons, since no sp1 polaron can appear in their absence. Because theIsing magnon spectrum is gapped, at low T this number is proportional tothe Boltzmann factor e−βΩ. This suggests an integrated weight in the c+mspectrum of a− be−β4JS , where a = 1−O(η) is the limiting value as T → 0.We fitted the data points for βt > 5 with this form and found a very goodfit (solid lines), which moreover works well for a larger range of β valuesthan used in the fit.Magnons of the Heisenberg model are gapless so their number increasesmuch faster with T . A simple estimate for a 2d unbounded parabolic dis-persion suggests 〈n〉 ∼ kBT .1 The lines shown for the Heisenberg model inFig. 2.6 are fits to a− b/β for the data points with βt > 5. The fit is againreasonable over a wider range, and much superior to other simple functionalforms we tried, such as a− b/βn, n > 1 or a− be−βcJ (the former assumingthat we misidentified the power law, the second to see if Ising-like fits mightbe more appropriate). Of course, one can find excellent fits for all datausing more complicated functions with additional parameters, but they aremuch harder to justify physically than our simple hypothesis resulting in aneffectively one parameter fit.Let us now discuss the discrete peak appearing below the sp1 continuumfor both models, for AFM J0. After carefully investigating many of itsproperties, such as how its energy and the region in the Brillouin zone where1The prefactor contains the Riemann number ζ(1) = ∞. However, as mentionedin Section 1.3 we assume that long range magnetic order is stabilized by an externalmechanism such as weak interlayer coupling. The same mechanism should remove thesingularity.482.3. Results-10 -5 0 5↑(0,0,0,ω)c+msp1-15 -10 -5 0 5 10ω/t0.↑(0,0,0,ω)c+msp1E↑(k)E↑(k)Figure 2.7: Spectral weight A↑(k = 0, ω) for the 3d Heisenberg model atβt = 1 for FM J0/t = −3 (top) and AFM J0/t = 10 (bottom) couplings.The edges of the c+m continuum (dashed red lines) and sp1/sp2 continuum(dot-dashed blue lines) are indicated, as is E↑(0) (thick green line). Otherparameters are J/t = 0.5, S = 0.5, η = 0.1.it exists depend on various parameters including T , we believe that this isan unphysical artefact of our approximation. Arguments for this are: (i)the temperature dependence of its location, clearly visible in Fig. 2.5 (notethat for the Ising model, the peak only separates below the sp1 continuumat higher T . At βt = 2 one just starts to see weight piling up near the loweredge, in preparation for this). According to the Lehmann decomposition,the ranges where finite spectral weight is seen cannot vary with T ; (ii) thefact that the problem is worse at higher T , where we know that higher ordercorrections ought to be included in the self-energy; these could easily removean unphysical pole; (iii) the fact that this is a discrete peak, not a resonanceinside a continuum (this can be easily verified by checking that its lifetime isset by η). According to the Lehmann decomposition, discrete peaks cannotappear in the T 6= 0 spectral weight. Even if the carrier binds all thermalmagnons in a coherent QP, the finite-T spectral weight would reveal only acontinuum associated with it, as is the case for the sp1 and sp2 polarons.To summarize, we believe that this discrete peak is an artefact and that in492.3. Results-10 -5 0↑(0,0,0,ω)-15 -10 -5 0 5 10ω/t0.00.10.2A↑(0,0,0,ω)c+msp1sp2c+msp1E↑(k)E↑(k)Figure 2.8: Spectral weight A↑(k = 0, ω) for the 3d Ising model at βt = 0.5for FM J0/t = −3 (top) and AFM J0/t = 10 (bottom) couplings. Theedges of the c+m continuum (dashed red lines) and sp1/sp2 continuum(dot-dashed blue lines) are indicated, as is E↑(0) (thick green line). Otherparameters are J/t = 0.5, S = 0.5, η = 0.01.reality, its weight is part of the sp1 continuum from which it came.Ideally, these arguments would be strengthened by a calculation of thenext correction to the self-energy, to check its effects. We found the exactcalculation of the two-magnon term to be daunting even for the Ising model.The difficulty is not so much in evaluating different terms, but in tracingover all possible contributions – so far we did not find a sufficiently efficientway to do this. One can use approximations to speed things up, but thatdefeats the purpose since it would not be clear if the end results are intrinsicor artefacts, as well. Given this, we cannot entirely rule out that the discretepeak is a (precursor pointing to a) real feature, but we believe that to bevery unlikely.So far we have done the whole analysis in 2d, simply because the calcu-lation of Σ(k, ω), especially for the Heisenberg model, is numerically muchfaster 2. However, we did investigate the 3d models and found essentially2All calculations were performed on a personal computer. For 2d spectra the calculation502.4. Conclusionsthe same physics. As examples, in Figs. 2.7, 2.8 we show spectra for bothFM J0/t = −3 and AFM J0/t = 10, for both models. These spectra displayexactly the same features as the corresponding 2d spectra. For the Heisen-berg model we chose a larger η = 0.1 and decreased the linear system sizedrastically to keep the computational time reasonable. Consequently, thecontinuum edges are more difficult to discern, while finite size effects aremore pronounced. In any event, the knowledge accumulated from analyzingthe 2d data is fully consistent with all features we observed in all 3d datawe generated.2.4 ConclusionsTo summarize, we calculated analytically the lowest-T correction to theself-energy of a spin-up carrier injected in a FM background. We used bothHeisenberg and Ising couplings to describe the background, to understandthe relevance of gapped vs. gapless magnons. These results show how thespectral weight evolves from a discrete peak at T = 0 to a collection ofcontinua for T 6= 0 (these can merge, in the appropriate circumstances),and explain their origin and how their locations can be inferred.We were aided in this task by the fact that this model conserves thez-component of the total spin, allowing us to consider the contribution tothe spectral weight coming from Hilbert subspaces with different numbersm of magnons when the carrier has spin up. Although we focused on them = 1, lowest-T contribution, based on the knowledge we acquired we canextrapolate with some confidence to higher T , as we discuss now.One definite conclusion of this work is that knowledge of the T = 0carrier spectrum (in the m = 0 sector) E↑(k), and of the magnon disper-sion, Ωq, is generally not sufficient to predict a priori all features of thefinite-T spectral weight, although a fair amount can be inferred from them.To see why, let us assume that magnons do not interact with one another.(This is not true for either model, for example due to their hard-core repul-sion; we will return to possible consequences of their interactions below.) Ifmagnons were non-interacting, then Lehmann decomposition of the higher-order contributions in Eq. (1.15) would predict finite-T spectral weight forall intervals {E(m)α (k +∑mi=1 qi)−∑mi=1 Ωqi}q1+···+qm , m = 0, 1, . . . .Since we move from the m to the m+ 1 subspace by adding a magnon,and given that total momentum is conserved, we know that the spectrum insubspace m + 1 necessarily includes the convolution between the spectrumtime is on the order of hours, whereas for 3d spectra it is on the order of days.512.4. Conclusionsof the subspace m and the magnon dispersion, i.e. {E(m)α (k− q) + Ωq}q ispart of the spectrum E(m+1)α (k) (these are the scattering states between theextra magnon and any eigenstate in the m spectrum).This observation allows us to infer the location of some of the finite-Tspectral weight, by recurrence. E(1)α (k) must include all scattering states{E↑(k − q) + Ωq}q, so the m = 1 contribution to the spectrum must span{E(1)α (k + q′)−Ωq′}q′ = {E↑(k−q + q′)−Ωq′ + Ωq}q,q′ . We called this thec+m continuum and verified that it is indeed seen in the finite-T spectralweight. Knowledge of this part of the m = 1 spectrum allows us to inferscattering states that are part of the m = 2 spectrum and therefore theirLehmann contribution, etc. The conclusion is that all intervals {E↑(k +∑mi=1 q′i−∑mi=1 qi)−∑mi=1 Ωq′i+∑mi=1 Ωqi}q1,...,q′m will contain some spectralweight at finite T . For the dispersionless Ising magnons this interval is thesame for all m. For dispersive Heisenberg magnons this interval broadenswith m. For very small J , the additional broadening as m increases is verysmall and moreover one would expect little spectral weight in the high-msectors if T is not too large. Thus, we expect weight to be visible in thec+m continuum up to high(er) temperatures; its boundaries may also slowlyexpand with T , for a Heisenberg background, as higher m subspaces becomethermally activated.Apart from these scattering states, E(m+1)α (k) might also contain boundstates where the extra magnon is coherently bound to all the other particles.The existence and location of such coherent states cannot be predicted apriori, as they depend on the details of the model (however, they certainlycannot appear unless coherent states exist in the m space). An example isthe E(1)α (k) spectrum which indeed contains the scattering states discussedabove, but also contains the sp1 and/or sp2 discrete polarons states. Thesegive rise to their own continua of scattering states in higher m subspaces,whose locations can be inferred by recurrence.The question, then, is if it is likely to find such new, bound coherentstates for all values of m, i.e. if the number of additional continua becomesarbitrarily large with increasing T . Generally, the answer must be “no”,since this requires bound states between arbitrarily large numbers of objects.For the problem at hand, we believe that it is quite unlikely that they appeareven in the m = 2 subspace, since that would involve one carrier bindingtwo magnons. This is a difficult task given the weak nearest-neighbourattraction of order J between magnons (due to the breaking of fewer FMbonds), and the fact that the carrier can interact with only one magnonat a time. The exception is likely to be in 1d systems where magnons can522.4. Conclusionscoalesce into magnetic domains.Let us now consider the role of magnon interactions. Because of them,many-magnon states are not eigenstates of the Heisenberg Hamiltonian sohigher-order terms are not obtained by tracing over states with many inde-pendent magnons (in the Ising model this complication can be avoided byworking in real space). If the attraction between magnons is too weak tobind them, this is not an issue since their spectra will still consist of scatter-ing states spanning the same energies like for non-interacting magnons. Asa result, the location of various features is not affected, but the distributionof the spectral weight inside them will be since the eigenfunctions are dif-ferent. Magnon pairing is unlikely for d > 1 unless the exchange is stronglyanisotropic. However, if it happens and if the spectrum of the magnon pairsis known, one could infer its effects on the carrier spectral weight just likeabove.Based on these arguments, we expect the higher-T spectral weight toshow the same features we uncovered at low T (the distribution of the weightbetween them might be quite different, though). These expectations couldbe verified with numerical simulations (conversely, our low-T results canbe used to test codes). Such simulations would also solve the issue of thediscrete peak that we observed for AFM J0, and which we argued to be anartefact of our low-T approximation.To conclude, although quantitatively our results are only valid at ex-tremely low T , we believe that this study clarifies qualitatively how thespectral weight of a spin-up carrier evolves with T . Our arguments can bestraightforwardly extended to predict what features appear in the spectralweight of a spin-down carrier, as well.A general feature demonstrated by our work is that finite T does notresult in just a simple thermal broadening of the QP peak, as it becomesa resonance inside a continuum. Spectral weight can also be transferred toquite different energies if the QP can bind additional magnons into coherentpolarons. When this happens, interpretation of experimentally measuredand/or of computationally generated spectra could become difficult, unlessone is aware of this possibility.53Chapter 3Local environment effects ona charge carrier injected intoan Ising chain3.1 IntroductionIn this Chapter we try to answer the question of what role, if any, is playedby the local environment. Is there a class of configurations that contributesmore than others? How does an AFM ordered environment compare to aFM ordered one, and are there any similarities between the two? Answer-ing such questions is crucial for the development of a better understanding(and hopefully better approximations) for the finite-T spectral weight of thecarrier. In order to do this one must move away from mean-field like ap-proximations such as the CPA [53] and the method of Nolting et. al. [81](see Section 2.1 for a brief discussion of their approaches).Our approach is to study a simplified model which allows us to obtain thefirst (to the best of our knowledge) finite-T , exact numerical results in thethermodynamic limit. These teach us valuable lessons about the physics ofthis simpler problem; some of these are relevant for more complex models,too. The simplification consists in neglecting all spin-flip processes, i.e.assuming that all interactions between the lattice spins, and between themand the carrier, are Ising-like. While, compared to the model discussedin the previous chapter, the T = 0 behavior of this model is trivial, wefind that it produces rich physics at finite T , which needs to be understoodbefore considering the additional complications introduced by allowing spin-flip processes.Apart from making an exact solution possible, this simple model also al-lows us to consider both FM and AFM backgrounds on equal footing, whichis not possible in general. Interestingly, we find considerable similarities be-tween the results for the two cases, which we are able to explain as being dueto similarities between the local background configurations that control the543.1. Introduction(at least low and medium-T ) carrier dynamics. We then use these insightsto construct an analytical approximation which accurately reproduces thefeatures of the exact solution in the range of low to medium temperatures.By ignoring spin-flip processes, we create a situation that is in some wayssimilar to that of disorder binary alloys of the type AxB1−x. A carrier inthe conduction band of such an alloy experiences a different on-site energyif it sits on an A or a B site. In our model the on-site carrier energy is±J0, depending on whether the local spin at that site points up or down.Thus, any configurations of the alloy can be exactly mapped into a spinconfiguration of the magnetic background.These binary alloys have been extensively studied. An exact numericalsolution can be obtained for the disorder-averaged GFs, as pointed out bySchmidt[93] and Dyson [25]. It was used for numerical studies of the phononDOS [21] and the DOS of carriers in the conduction band [1, 28]. Manyapproximations have been proposed, the most well-known and used being theCPA[100]. Diagrammatic expansions of the self-energy have been developedby several authors, see for instance Refs. [12, 75, 108], but were shownto result in non-analytic behavior and thus unphysical GFs [74]. Laterthis problem was resolved by a careful consideration of which diagrams tosum[66].This knowledge cannot be directly used in our problem, despite the simi-larities between the models, because for binary alloys one performs a disorderaverage whereas we use a thermal average. The disorder average keeps theconcentration x fixed; all configurations consistent with it are equally likely(one generally ignores correlations in the disorder), all the others are forbid-den. In contrast, a thermal average includes all possible configurations butwith a Boltzmann factor that controls the extent of spin-spin correlations.These correlations are key, as no magnetic order can exist in their absence.The problems also have different symmetry properties. Our problem is trulytranslationally invariant, whereas for binary alloys translational invarianceis only restored by averaging over all possible disorder configurations. Forexample, while all eigenstates are localized in a 1d disordered alloy model,the eigenstates of our model remain extended at any T . Despite these dif-ferences, there are similarities between the resulting spectral weights whichhelp us understand our results, as discussed below.553.2. Simplified model of a 1d magnetic semiconductor3.2 Simplified model of a 1d magneticsemiconductorWe consider a single spin-12 charge carrier which interacts with a chain ofIsing spins, also of magnitude 12 . Note that the exact method that we use canbe generalized straightforwardly to higher dimensions. The advantage of 1dchains (apart from speed of computations) is that a host of analytic resultsare available for the undoped case [42, 55, 84]. We therefore limit ourselveshere to Ising chains with periodic boundary conditions after N → ∞ sites.The nth site is located at Rn = na, and we set a = 1.The exchange between lattice spins is Ising-like:HˆI = −J∑iσˆiσˆi+1 − h∑iσˆi, (3.1)and is FM (J > 0) or AFM (J < 0). The second term describes the effectof an external magnetic field h. The spin operator at site i is Szi =12 σˆi,with the prefactors absorbed into the coupling constants. The eigenstatesHˆI|{σ}〉 = EI{σ}|{σ}〉 are described by the set {σ} ≡ {σ1, σ2, . . . , σN} ofeigenvalues σi = ±1 of each spin, and EI{σ} = −J∑i σiσi+1 − h∑i σi is theeigenenergy.The kinetic energy of the carrier is described by a tight-binding modelTˆ = −t∑i,σ(c†i,σci+1,σ + h.c.)=∑k,σε(k)c†k,σck,σ, (3.2)where c†i,σ is the creation operator of a spin-σ carrier at site i. The cre-ation operator in momentum-space is c†k,σ =1√N∑n eikRnc†n,σ, and ε(k) =−2t cos(k).The exchange between the carrier and the lattice spins is also of Isingtype:Hˆexc = J0∑i,σσc†i,σci,σσˆi, (3.3)where again a factor 1/4 is absorbed into J0. Since no spin-flips are allowedin this model, from now on we assume without loss of generality that thecarrier has spin-up, σ =↑, and do not write it explicitly. We also set J0 > 0.Results for a spin-down carrier are obtained from these by switching J0 →−J0.563.3. MethodFor the undoped Ising chain, an exact solution at finite T is possibleand reviewed in Appendix B. Note that a 1d chain has long-range magneticorder only at T = 0. We can, however, mimic a finite-T , ordered state byturning on the magnetic field h. This leads to a finite, long-range spin-spincorrelation.3.3 MethodWe want to calculate the finite-T , retarded single-particle GF, which in thefrequency-domain and canonical ensemble is defined as (c.f. Eq. (1.29)):G(k, ω) =∑{σ}e−βEI{σ}Z〈{σ}|ckGˆ(ω + EI{σ})c†k|{σ}〉, (3.4)where Gˆ(ω) = (ω−Hˆ+ iη)−1 is the resolvent of Hˆ. The small, real quantityη > 0 ensures retardation and sets a finite carrier lifetime 1/η. Note thatthe argument of the resolvent in Eq. (3.4) is shifted by HˆI . This means thatthe energy is measured from that of the Ising chain at the time of injection.This becomes clear when using a Lehmann representation by projecting onthe one-carrier eigenstates Hˆ|n〉 = En|n〉:G(k, ω) =∑n∑{σ}e−βEI{σ}Z|〈n|c†k|{σ}〉|2ω + EI{σ} − En + iη(3.5)G(k, ω) has poles at energies ω = En − EI{σ} that measure the change intotal energy due to the carrier’s injection. The weights correspond to theoverlap between the true eigenstates |n〉 and the free-carrier states c†k|{σ}〉.For any configuration {σ}, the contribution to the thermal average canbe evaluated using continued fractions.[6] First, we need to shift to the real-space representation. Making use of the translational invariance we thenobtainG(k, ω) =∑neikRn∑{σ}e−βEI{σ}Zg0,n(ω, {σ}), (3.6)where we define:gm,n(ω, {σ}) = 〈{σ}|cmGˆ(ω + EI{σ})c†n|{σ}〉. (3.7)573.3. MethodPhysically gm,n(ω, {σ}) is related to the probability that the carrier is in-jected at site n and propagates to site m. For any state {σ} (except thefully ordered ones) the translational invariance is broken, gm,n(ω, {σ}) 6=g0,n−m(ω, {σ}). This symmetry is only restored by the ensemble average,which then leads to Eq. (3.6).To obtain the GF EOM we use Dyson’s identity Gˆ(ω) = Gˆ0(ω) +Gˆ(ω)Vˆ Gˆ0(ω) where Hˆ = Hˆ0+Vˆ and Gˆ0(ω) is the resolvent for Hˆ0. ChoosingHˆ0 = HˆI + Hˆexc, we find:g0,n(ω, {σ}) =G0(ω − J0σn) (δ0,n − tg0,n−1(ω, {σ})−tg0,n+1(ω, {σ})) , (3.8)where G0(ω) = (ω+iη)−1. Since the EOM do not change the values of ω and{σ}, to shorten notation we do not write them explicitly in the following.The EOM are solved with the ansatz [6] g0,n = Ang0,n−1 for n > 0and g0,n = B−ng0,n+1 for n < 0 (N → ∞ is assumed and implemented asexplained below). ThenAn =−tG0(ω − J0σn)1 + tG0(ω − J0σn)An+1 , (3.9)and similarly for Bn. We now introduce a cutoff Mc  1 at which wetruncate these relations by setting AMc+1 = BMc+1 = 0. The justificationis provided by the finite lifetime 1/η of the carrier, which prevents it frompropagating arbitrarily far from its injection site. As a result g0,n, whichmeasures the amplitude of probability that the carrier propagates betweensites n and 0, must vanish for sufficiently large |n| [6].It is then straightforward to calculate all A1, . . . , AMc and B1, . . . , BMcfor the configuration {σ} and a given ω, to find:g0,0 =1ω − J0σ0 + t(A1 +B1) (3.10)g0,n = AMc . . . A1g0,0, if n > 0 (3.11)g0,−n = BMc . . . B1g0,0, if n < 0. (3.12)Let us now discuss the cutoff Mc in more detail. In practice Mc isincreased until convergence is reached. Since g0,0 for the fully ordered con-figuration is known analytically, it can be used to verify the convergence.We need to have Mc  N/2, otherwise the carrier may travel be-tween sites n and 0 on both sides of the closed loop, which is at oddswith the ansatz chosen above. This condition is automatically satisfied583.3. Methodif N → ∞. To take this limit, we note that g0,n only depend on thespins σ−Mc , σ−Mc+1, . . . , σMc . We make use of this by splitting the fullset {σ} into the set {Mc} containing just the aforementioned spins, andthe complementary set {Mc}C . The energy of the Ising chain is also split:EI{σ} = EI{Mc} + EI{Mc}C ,σ−Mc ,σMc , whereEI{Mc} = −JMc−1∑n=−Mcσnσn+1 − hMc∑n=−Mcσn (3.13)and EI{Mc}C ,σ−Mc,σMc contains the energy of all other bonds and spins, in-cluding the “boundary” bonds σMcσMc+1 and σ−Mc−1σ−Mc . This is why italso depends on σ±Mc , not just on the {Mc}C spins.Eq. (3.6) can then be rewritten as:G(k, ω) =∑neikRn∑{Mc}e−βEI{Mc}Zg0,n(ω, {Mc})× Zbath(β, σ−Mc , σMc), (3.14)whereZbath(β, σ−Mc , σMc) =∑{Mc}Ce−βEI{Mc}C,σ−Mc,σMc . (3.15)Zbath is the partition function of the complementary set of spins {Mc}C , forset values of its “boundary” spins σ−Mc and σMc . Using transfer matrices(see Appendix B and Ref. [84]), we find that:Zbath(β, σMc , σ−Mc) =(T N−2Mc)σMc ,σ−Mc. (3.16)Thus, limN→∞ Zbath(β, σMc , σ−Mc)/Z is known analytically. The average inEq. (3.14) now involves only the spins {Mc} in the chain sector that can beexplored by the carrier within its finite lifetime 1/η. Effectively, the rest ofthe infinite chain is treated as a bath that is integrated out analytically. Weuse the Metropolis algorithm 3 to estimate the sum over the {Mc} set. Theresults are discussed next.3The autocorrelation time ranges from 4 Monte Carlo (MC) steps (βt = 1) to 6630 MCsteps (βt = 5). During one MC steps an attempt to flip each of the 2Mc + 1 Ising spinsis made exactly once. For most spectra 204800 MC measurements were used. The timewaited between measurements is identical to the autocorrelation time. More details canbe found in Appendix C.593.4. Results3.4 ResultsWe begin by briefly reviewing the T = 0 solution which can be calculatedexactly and serves as a useful reference.For FM coupling J > 0, at T = 0 all Ising spins point either up ordown, m = ±1. Then Hˆexc simply shifts the energy of the carrier by J0m:EFMm (k) = εk+J0m. As T → 0 and for h = 0, an infinite chain will arbitrar-ily choose as its ground state one of these two possible FM configurations.One can control which configuration is chosen by cooling the system in asmall magnetic field, which is then switched off. However, as long as thetemperature is not exactly zero and if h = 0, then the presence of largedomains with opposite order is possible, especially in 1d. If the carrier isinjected into one of these large domains it is unable to leave it within itsfinite lifetime 1/η. Consequently as T → 0 and for h = 0 we expect to seecontributions from both subspaces m < 0 and m > 0, and the GF becomes:GFM(k, ω) =12(1ω − EFM+ + iη+1ω − EFM− + iη). (3.17)For AFM coupling there are also two possible ground states: either allspins of the even sublattice point up and all spins of the odd sublatticepoint down, or vice versa. The doubling of the unit cell results in theappearance of two bands in the reduced Brillouin zone (−pi/2, pi/2], withenergies EAFM± (k) = ±√J20 + ε2k. Averaging over both contributions for thereasons discussed above, the T → 0, h = 0 GF is found to be:GAFM(k, ω) =ω + εk + iη(ω − EAFM+ + iη)(ω − EAFM− + iη)(3.18)for any k ∈ (−pi, pi].Contour plots of the spectral function A(k, ω) = − 1pi ImG(k, ω) for theseGFs are shown in Figs 3.1 and 3.2. As expected from applying the Lehmannrepresentation, Eq. (3.5), to these GFs, the spectrum consists of two bandsfor both FM and AFM coupling. For FM coupling, the bands have band-widths of 4t, are centered at ±J0 and have equal weights for all k. For AFMcoupling, the bands span [−√J20 + 4t2,−J0] and [J0,√J20 + 4t2], respec-tively. The eigenenergies show the pi periodicity expected for the two-siteunit cell. Spectral weight is transferred from the lower to the upper band as|k| increases, because the GF combines contributions from both sublatticeswith a k-dependent phase factor.Note that for J0/t ≤ 2 the two FM bands overlap. In order to simplifyfuture analysis, we set J0/t = 2.5 from now on. We also set |J |/t = 0.5,603.4. Results−1.0 −0.5 0.0 0.5 1.0k/pi−4−2024ω/tFigure 3.1: Contour plot of the T = 0 spectral functions for FM coupling,|J |/t = 0.5. Other parameters are J0/t = 2.5, h = 0, η/t = 0.04.−1.0 −0.5 0.0 0.5 1.0k/pi−4−3−2−101234ω/tFigure 3.2: The same as in Fig. 3.1, but for AFM coupling.613.4. Results-4 -2 0 2 4ω/t0246810A(0,ω)βt=1βt=2βt=3βt=4βt=5(a) J > 0Figure 3.3: Spectral function A(0, ω) for different temperatures and FMcoupling. Parameters are |J |/t = 0.5, J0/t = 2.5, h = 0, η/t = 0.04.although we note that if h = 0, J only appears in conjunction with β so achoice for J simply sets the temperature scale. We use a cutoff of Mc = 400,which for η=0.04 is sufficient for convergence for the fully ordered FM chain(this is the most slowly converging case).We now discuss the results of the Monte Carlo simulations. In Figs 3.3and 3.4 we plot k = 0 spectral functions for different values of β for FM andAFM couplings, respectively. In both cases the βt = 5.0 results are in verygood agreement with those at T = 0, defining “low-temperatures” to meanβt ≥ 5. As β decreases (T increases), the sharp peaks broaden considerablyand new peaks appear. For FM coupling, the lowest energy state still lies atthe bottom of the low-energy T = 0 band. For AFM coupling, however, newstates appear below the T = 0 spectrum. This is expected since at finite-T ,FM domains can form in the AFM background and the carrier lowers itsenergy when located in such domains. Of course, these energies are boundedfrom below by the lowest FM eigenenergy.At first sight the appearance of these new peaks (in fact resonances, asdiscussed below) may seem to signal lack of convergence of the MC simu-lations, or finite-size issues related to a Mc cutoff that is not big enough.However, the results are converged and do not change upon further Mcincrease; these features are real.623.4. Results-4 -2 0 2 4ω/t0246810A(0,ω)βt=1βt=2βt=3βt=4βt=5(b) J < 0Figure 3.4: The same as in Fig. 3.3, but for AFM coupling.The appearance of similar features is a well documented phenomenonfor the disordered binary alloys with which our problem has similarities, asdiscussed above. Studies of binary alloys have revealed that these peaks(which are truly discrete states, in that context) mark the appearance ofbound states where the carrier is trapped by small clusters of atoms of thesame type. [21, 28]. As we show now, the resonances we observe have similarorigin. For instance, in the FM case they are due to the charge carrierbeing trapped into spin-down domains formed into an otherwise spin-upbackground, or vice versa.The eigenenergies for trapping the carrier inside several such short do-mains embedded in an otherwise ordered FM or AFM background can beobtained by calculating the real-space GFs g0,n(ω, {σ}) where {σ} corre-sponds to the state of the Ising-chain with said domain centered at site0. The simplest case is that of a single flipped spin inside an otherwiseFM ordered background. We denote the corresponding real-space GFs bygFM,10,n (ω). Assuming that all the spins, except for the flipped spin at theorigin, are up and suppressing the ω-dependence the EOM are (cf. Eq.(3.8))gFM,10,0 [ω + iη + J0] = 1− tgFM,10,1 − tgFM,10,−1 (3.19)gFM,10,n [ω + iη − J0] = −tgFM,10,n+1 − tgFM,10,n−1 n 6= 0 (3.20)633.4. Results-5 -4 -3 -2 -1 0ω/t0123A(0,ω)1 223 3 444 45 5 5k=0k=pi/2k=piFigure 3.5: Spectral functions A(k, ω) for βt = 1 and k = 0, pi/2, pi, for FMcoupling. Solid vertical lines mark the trapping energies of the carrier insmall domains. The corresponding numbers show the length of the domain(see Table 3.1). The dashed vertical lines mark the band-edges of the T =0 low-energy band. Other parameters are |J |/t = 0.5, J0/t = 2.5, h =0, η/t = 0.04.Since the carrier has a finite lifetime ∼ 1/η we must have gFM,10,n → 0 for|n| → ∞. Consequently, for n > 0, the EOM can be solved with the ansatzgFM,10,n = zFMgFM,10,n−1, where |zFM| < 1. A similar ansatz can be made forn < 0. Plugging this ansatz back into the EOM above, we obtaingFM,10,0 =1ω + iη + J0 − 2tzFM , (3.21)zFM =ω + iη − J02t±√[ω + iη − J02t]2− 1. (3.22)From this all the other GFs can be obtained with the relation gFM,10,n =zFMgFM,10,n−1. The procedure can easily be generalized to longer domains.Once the expression for the real-space GFs gFM,l0,n of a domain of length lis known the eigenenergies of the trapped carrier states for this domain canbe obtained by finding the poles of these GFs. This can be done by explic-643.4. Results-5 -4 -3 -2 -1 0ω/t0123A(0,ω)1 123 3’3’ 2’ 2’ 2’2’ 5453534 52k=0k=pi/2k=piFigure 3.6: The same as in Fig. 3.5, but for AFM coupling.itly setting the denominators of the GFs to zero, or perhaps more easily byplotting the LDOS −Im[gFM,l0,0 ]/pi and reading off the ω-values of the poles 4.The eigenenergies for AFM domains can be obtained in a similar manner.The eigenvalues obtained in this fashion (various lines) are compared tothe spectral weights obtained for βt = 1 and k = 0, pi/2, pi in Figs 3.5 and3.6. The integers labelling the lines show the length of the correspondingdomains, also see Table 3.1. The agreement between these trapping energiesand the location of the resonances in A(k, ω) is very good. The weights ofthese resonances vary strongly with k but their energies are nearly disper-sionless. For AFM coupling, the trapping energies in different domains aresometimes very similar, suggesting that here trapping occurs at the bound-aries of the domain, not inside its bulk; this explains the broader featuresat ω/t ≈ −3.7,−2.9 and −1.7.To better understand the momentum dependence, complete sets of spec-tral weights are shown in Figs. 3.7 and 3.8 for FM and AFM coupling,respectively. For simplicity we do not mark the trapping energies of thecarrier in the various domains, but we have checked that the agreement isas good as in Figs 3.5 and 3.6 at all k.4Note that with the latter procedure one only finds the eigenenergies of trapped stateswhich have a finite probability of the carrier being at the center of the domain.653.4. ResultsFor FM coupling, we see the two bands that have evolved from theT = 0 peaks moving with increasing k in a way that roughly mirrors theT = 0 dispersions shown in Fig. 3.1. As k increases, spectral weight issystematically shifted from the lower to the upper edge of each band. Inaddition we notice a small spectral weight transfer from the lower band tothe upper band. This is in contrast to the FM T = 0 solution, where bothpeaks have equal spectral weight. For AFM coupling, the pi-periodicity ofthe T = 0 dispersion is partially masked by the many additional resonancesthat appear on both sides of the T = 0 peaks, and the significant transferof spectral weight from the lower to the upper band. The latter is similarto the behavior observed for the T = 0 solution. In both cases, the locationof the various resonances does not change appreciably with k.To understand the physical origin of these resonances, consider the anal-ogy with the binary alloy model, which also shows such “peaky” structuresin its total density of states (DOS), marking the bound states of the carrierinside small clusters of like-atoms [21, 28]. As is well known, in the pres-ence of any amount of on-site disorder all eigenstates of a 1d chain becomelocalized. To find the DOS one can formally calculate the disorder averagedGreen’s function (which regains invariance to translations) but this quantityhas no physical meaning. This is because in any real system there is a givendisorder distribution, and if all eigenstates are localized then the carrier oc-cupies forever the same small region of space and self-averaging does notoccur. In other words, if the carrier is trapped in a cluster of atoms it willstay trapped indefinitely.Each configuration of the binary alloy can be mapped into a spin con-figuration of the Ising chain, by replacing atoms A/B by spins up/down.Color Length FM domain AFM domainblack 1 . . . ↑↑ ↓ ↑↑ . . . . . . ↑↓ ↓ ↓↑ . . .blue 2 . . . ↑↑ ↓↓ ↑↑ . . . . . . ↑↓ ↓↑ ↑↓ . . .red 3 . . . ↑↑ ↓↓↓ ↑↑ . . . . . . ↓↑ ↑↓↑ ↑↓ . . .green 4 . . . ↑↑ ↓↓↓↓ ↑↑ . . . . . . ↓↑ ↑↓↑↓ ↓↑ . . .magenta 5 . . . ↑↑ ↓↓↓↓↓ ↑↑ . . . . . . ↑↓ ↓↑↓↑↓ ↓↑ . . .cyan 2’ - . . . ↑↓ ↓↓ ↓↑ . . .yellow 3’ - . . . ↑↓ ↓↓↓ ↓↑ . . .Table 3.1: List of the shortest domains (underlined spins) that form inotherwise ordered backgrounds. The energies for trapping the carrier inthese domains are shown in Figs 3.5 and 3.6.663.4. Results-4 -2 0 2 4ω/t01234567891011A(k,ω) 3.7: A(k, ω) for different values of k and FM coupling, at |J |/t =0.5, J0/t = 2.5, βt = 1.0, h = 0, η/t = 0.04.Small clusters of like-atoms then map into magnetic domains, and thereare trapped states of the carrier inside them, as already shown. However,unlike the fixed disorder configuration, the spin configuration changes con-tinuously through thermal fluctuations. A trapped carrier therefore has afinite lifetime linked to the persistence of that domain: eventually the lo-cal spin configuration changes and the carrier is released to move alongthe chain. This is why here the “peaky” structures indicate actual finitelife-time resonances for extended eigenstates of well-defined momentum, notinfinitely-lived localized states like in the alloy model. This is a significantqualitative difference.We conclude this section by considering the role of the external magneticfield h. Its effects on the k = 0 spectral weight for FM coupling are shownin Fig. 3.9. We find that as h increases, the initially large feature at thebottom of the lower band starts to disappear and most of its weight ismoved to the bottom of the upper band. This is expected since the externalfield favors/disfavors the spin-up/down background responsible for this main673.4. Results-4 -2 0 2 4ω/t01234567891011A(k,ω) 3.8: The same as in Fig. 3.7, but for AFM coupling.feature (evolved from the T = 0 peak) of the upper/lower band. Interesting,however, most of the resonances in the lower band remain almost unchanged.The reason is that while βh 1, the energy cost for a small domain is verylow so their appearance is very likely. If βh  1 the Ising chain is forcedinto the m > 0 ground state and only the higher T = 0 peak survives (notshown).For AFM coupling the effect of h is more dramatic (see Fig. 3.10), asit forces the system into the m > 0 FM ground-state if βh becomes largeenough. Indeed, for large h most of the weight is moved into the upper FMpeak and most of the resonances disappear, except for the one at ω/t ≈ −2.9that is still quite large. Its energy is very close to that of the first FM clusterlisted in Table 3.1, and indeed it seems plausible that this is due to a singleflipped spin which entraps the carrier. This domain is disfavored by h, butactually lowers the exchange energy with its neighbors.Note that we used different field strengths for the FM and AFM cases.In the latter case much higher fields are needed to produce long range cor-relations since there is a competition between the exchange energy of neigh-683.4. Results-4 -2 0 2 4ω/t02468A(0,ω)h=0h=0.05h=0.1h=0.15h=0.2h=0.25h=0.3Figure 3.9: Spectral functions A(0, ω) for different values of h, for FM cou-pling. Parameters are |J |/t = 0.5, J0/t = 2.5, βt = 1.0, η/t = 0.04.boring spins and the external field. A measure for this is the spin-spincorrelation function 〈σ−McσMc〉, which for the parameters used in Figs 3.9and 3.10 equals 0.41 (FM, if h/t = 0.3) and 0.64 (AFM, if h/t = 2.0). Avalue of 〈σ−McσMc〉 = 1 means that the chain is completely ordered.3.4.1 Analytic approximationWe now use the insights gained from the Monte Carlo results to propose ananalytic approximation for the GF at low and medium temperatures. Wepresent the derivation only for the case of FM coupling when h = 0; theother cases can be treated similarly.The main idea is to only consider a limited number of spin configurationswhen performing the thermal average, to allow for its (quasi)analytic eval-uation. Since our numerical results show the importance of small domains,the configurations we select are the two ordered configurations |FM, σ〉 withall spins pointing up or down, σ =↑, ↓ together with the one-domain config-urations:|n, n+ l, ↑〉 = 4−(l+1)σˆ−n · σˆ−n+1 · . . . σˆ−n+l|FM, ↑〉 (3.23)|n, n+ l, ↓〉 = 4−(l+1)σˆ+n · σˆ+n+1 · . . . σˆ+n+l|FM, ↓〉, (3.24)693.4. Results-4 -2 0 2 4ω/t02468A(0,ω)h=0h=0.5h=1.0h=1.5h=2.0Figure 3.10: The same as in Fig. 3.9, but for AFM coupling.where the domain starts at site n and ends at n + l. The operator σˆ±i isthe raising/lowering operator for the ith Ising spin. To preserve translationalinvariance we need to consider all possible locations of the domain within theIsing chain. All these one-domain states are weighed by the same Boltzmannfactor e−4βJ (we take the energy of the fully ordered FM states as reference).As discussed, a physically meaningful result has equal contributions fromthe spin-down and spin-up sectors. We now discuss the spin-up contribution,which we denote by G↑(k, ω). The spin-down contribution G↓(k, ω) is thenobtained by simply letting J0 → −J0, and the GF is given by G(k, ω) =[G↑(k, ω) +G↓(k, ω)]/2. By itself, the decomposition into an up-part and adown-part is not an approximation. The approximation stems from the factthat we are only considering the one-domain configurations when calculatingG↑(k, ω), G↓(k, ω).By only considering domains up to a maximal length L (for reasonsdiscussed below), we thus approximate:G↑(k, ω) =1Z[GFM↑ (k, ω) + e−4βJL−1∑l=0G(l+1)↑ (k, ω) + . . .](3.25)where Z = 1 + e−4β|J |L · N + . . . . The thermodynamic limit N → ∞ will703.4. Resultsbe taken at a later stage. The first contribution, from the ordered state, isGFM↑ (k, ω) =〈FM, ↑ |ckGˆ(ω + HˆI)c†k|FM, ↑〉=1ω − ε(k)− J0 + iη . (3.26)To find the contributionsG(l+1)↑ (k, ω) =∑n〈n, n+ l, ↑ |ckGˆ(ω + HˆI)c†k|n, n+ l, ↑〉 (3.27)from the states with a domain of length l, we have to work harder. UsingDyson’s identity once we obtain:G(l+1)↑ (k, ω) = GFM↑ (k, ω)[N − 2J0l∑m=0f(l+1)↑,k (m,ω)], (3.28)where we defined the auxiliary GFs:f(l+1)↑,k (m,ω) =∑neikRn+m√N〈n, n+ l, ↑ |ckGˆ(ω + HˆI)c†n+m|n, n+ l, ↑〉.(3.29)Using Dyson’s equation again we find:f(l+1)↑,k (m,ω) =− 2J0l∑m′=0gFM↑ (m′ −m,ω)eik(Rm−R′m)f (l+1)↑,k (m′, ω)+GFM↑ (k, ω), (3.30)where gFM↑ (m′ −m,ω) = 1N∑q eiq(Rm′−Rm)GFM↑ (q, ω) are easy to find ana-lytically. This is a linear system of l + 1 equations that is solved to find allf(l+1)↑,k (m,ω), which are then used in Eq. (3.28). Note that all f(l+1)↑,k (m,ω)are proportional to GFM↑ (k, ω), since the latter quantity provides the inho-mogeneous terms in this linear system.When Eq. (3.28) is inserted in Eq. (3.25), if we group all terms pro-portional to GFM↑ (k, ω) we see that its factor 1/Z is canceled. Higher orderterms corresponding to states with two or more domains (not included inthis calculation) should similarly cancel the factor 1/Z for the remainingterms in Eq. (3.25), or at least make the thermodynamic limit of the ratiomeaningful. To O(e−4βJ) order and for N →∞, we therefore find:G↑(k, ω) ≈[1− 2J0e−4βJL−1∑l=0l∑m=0f(l+1)↑,k (m,ω)]GFM↑ (k, ω). (3.31)713.4. Resultsω/tA(0,ω)00.20.4 MCL=5L=10L=30-5 -4 -3 -200.40.8 MCL=2L=3L=51 2 3βt=3βt=2Figure 3.11: Comparison between the Monte Carlo (MC) results (shadedarea) and the analytic approximation of Eq. (3.31) for domains with amaximal length of L, for FM coupling |J |/t = 0.5, at J0/t = 2.5, h =0, η/t = 0.04, k = 0 and βt = 3 (top) and βt = 2 (bottom).Eq. (3.31) obeys the sum rule∫ +∞−∞ dωA(k, ω) = 1 if the second term has nopoles in the upper half of the complex plane. This is because the second termis proportional to [GFM↑ (k, ω)]2 and therefore falls of like 1/ω2 as |ω| → ∞.One may use Eq. (3.31) to extract a low-T approximation for the selfenergy:Σ↑(k, ω) ≈ −2J0e−4βJL−1∑l=0l∑m=0f(l+1)↑,k (m,ω)GFM↑ (k, ω), (3.32)and define G↑(k, ω) ≈[[GFM↑ (k, ω)]−1 − Σ↑(k, ω)]−1instead of G↑(k, ω) ≈GFM↑ (k, ω)[1 + Σ↑(k, ω)GFM↑ (k, ω)]of Eq. (3.31). At low enough temper-atures both give the same results, but at higher temperatures the formerapproximation leads to spurious poles in the spectral weight, as discussedin Chapter 3, so we use Eq. (3.31) in the following.Results for FM and AFM coupling are shown in Figs. 3.11 and 3.12, re-spectively, for various lengths L of the largest domain included, at two tem-peratures. For comparison, the Monte Carlo results are also shown (shaded723.4. Resultsregions). The quality of the approximation varies substantially with L. Thetop panel of Fig. 3.11 shows that for βt = 3 and L = 5, the weight ofthe resonances is underestimated and not all of them are reproduced by theapproximation. For L = 10 the agreement between the approximation andthe exact results is very good, but it worsens again for L = 30. Not onlydoes the latter overestimate the weight of the resonances, but it also predictsnegative spectral weight just below the bands. While this negative weightis needed to satisfy the sum rule, its presence is unphysical and signals afailure of the approximation. The same trends are observed for βt = 2 in thebottom panel of Fig. 3.11. Here the best agreement is obtained at L = 3,although resonances associated with longer domains are missing. They ap-pear for L = 5, however so does the unphysical behavior. For even lowervalues of β the approximation fails completely to capture the correct weightof the resonances, although, as shown in Figs 3.5 and 3.6, their locations aredue to carrier trapping in domains.The AFM approximation yields very similar results. The top panel ofFig. 3.12 shows that again for βt = 3 excellent agreement with the exactsolution is reached for L = 10, while for larger values of L the weight ofthe resonances is overestimated and unphysical behavior occurs if L = 30.For βt = 2 the agreement with the exact solution is best for L = 4 andunphysical behavior already occurs at L = 6. Again the approximation failsbadly to capture the proper weight of various features, for smaller values ofβ (higher T ).Naively, one may expect the approximation to improve when L is in-creased since this means that a larger fraction of the possible configurationsis considered. However, there are two factors which determine how a domaincontributes to the thermal average. One is the additional energy cost of adomain, which is accounted for by the Boltzmann factor and in 1d does notdepend on the domain’s length. The other is the increase in entropy withincreasing number of domains. As the temperature increases, minimizationof the free energy F = U − TS is increasingly driven by entropy maximiza-tion, resulting in more domains and thus shorter correlations. The order ofmagnitude for the maximal domain size should be given by the spin-spincorrelation length ξ, which for h = 0 is given by ξ−1 = − log(tanh(β|J |))(see Appendix B for a derivation). Indeed, we obtain ξ ≈ 4 and ξ ≈ 10 forβt = 2 and βt = 3, respectively (cf. Fig. 3.13). This compares well withthe values of L where the approximation performs well, see Figs. 3.11 and3.12.Another way to see why the approximation with L → ∞ is bound tobecome wrong is to realize that all domains whose length is longer than the733.4. Resultsω/tA(0,ω) MCL=10L=20L=30-5 -4 -300.40.81.2 MCL=2L=4L=62 3 4βt=3βt=2Figure 3.12: The same as in Fig. 3.11 but for AFM coupling.distance explored by the carrier within its lifetime are actually indistinguish-able from the “other” ordered FM configuration, from the point of view ofthe carrier. In other words, all these configurations essentially contributea GFM↓ (k, ω), and their inclusion gives the wrong weighting to the |FM, ↓〉contribution. Similarly, configurations with two long domains placed rela-tively close together will have states where the carrier is trapped in the shortregion between the domains, indistinguishable from having a short domainformed in the “other” ordered FM configuration. Adding many such contri-butions will affect the weights of these one-domain contributions, etc. Thesearguments suggest that a better approximation is:G↑(k, ω) =1− N/2∑l=0wl(β)l∑m=0f(l+1)↑,k (m,ω)GFM↑ (k, ω) (3.33)and G↓(k, ω) = G↑(k, ω)|J0→−J0 , where wl(β) are adjusted to capture accu-rately the weight of resonances due to trapping into short domains. For lowand medium temperatures we showed that wl(β) = 2J0e−4βJ if l ≤ ξ, andzero otherwise, gives very decent predictions. Clearly this cannot work athigh temperatures of order βt = 1 where ξ → 0. So far we have been unableto think of a reasonable form of wl(β) in this regime, but the comparisonsdisplayed in Figs 3.5 and 3.6 suggest that it should exist.743.5. Conclusions0 1 2 3β02.557.51012.51517.5ξ(β)Figure 3.13: The correlation length ξ for |J | = 0.5. Note that the correla-tion length is the same for FM and AFM J , but for AFM J the spin-spincorrelation function 〈σnσm〉T contains an additional factor (−1)|m−n|.3.5 ConclusionsTo summarize, we obtained numerically exact spectral functions for a sim-plified model of a carrier injected into a 1d Ising chain at finite T . Theresults highlight the importance of small domains that can trap the carrier,which were shown to be responsible for the resonances that appear as Tincreases. A simple analytic approximation based on these ideas was foundto perform well at low and medium temperatures. With further insights, itmay be possible to generalize it to high temperatures, as well. Interestingly,chains with both FM and AFM coupling can be understood in similar terms,although generically one expects quite different phenomenology for a carrierinjected into a FM vs AFM background.As highlighted throughout, there are parallels between this problem andthat of a carrier moving in a random binary alloy, where the importance ofsmall clusters of like-atoms, equivalent to the small domains of our model,is well documented [1, 21, 28]. There are, however, also major differences:finite J maps into correlations between the atoms of the alloy (usually theseare ignored). The thermal average is also very different, both qualitativelyand quantitatively, from a disorder average. It is therefore not a priori clearhow much of the considerable amount of work devoted to finding analyticapproximations for binary alloys can be used for the magnetic problem.In terms of generalizations, one direction is to see how far these insightscarry over to higher dimensions, where long-range magnetic order survives at753.5. Conclusionsfinite T . For binary alloys it is known that the fine-structure of the spectralfunction is most dominant in 1d [28]. For our model, the energy cost scaleswith the domain size in 2d and 3d, unlike in 1d and unlike for a binary alloy.This may well lead to a behavior that is different from that of the binaryalloys and could cause the fine-structure associated with very small domainsto remain present in higher dimensions, at least at low temperatures.Another interesting direction is to allow for Heisenberg coupling betweenthe carrier and the local moments (still with Ising coupling between the lat-ter). In this case spin-flip processes become possible. The carrier can movesmall domains around (in fact, the spin-polaron[7, 95] can be thought of asa mobile bound state between the carrier and a one-site domain), or splitlonger domains into several smaller ones, etc. Understanding the conse-quences of such processes and their effect on the finite-T spectral functionwould be very useful. Even more complicated is the case with Heisenbergcoupling between the lattice spins. Clearly there is a lot of work, bothnumerical and analytical, to be done before these problems are solved.Our results underline the importance of the local environment for thebehavior of a charge carrier in a magnetic background, at least for this modeland in low dimension. Incorporating these effects is difficult since mean-fieldapproximations cannot capture them. The only real alternative is to obtainan understanding of which states of the environment contribute most to thetemperature average, and to propose approximations based on taking theaverage over this limited set of states. Our work presents the first step inthis process.76Chapter 4On the equivalence of modelswith similar low-energyquasiparticles4.1 IntroductionAll physics knowledge is built on the study of models. Formulating a modelfor the system of interest is thus a key step in any project. Of course,“all models are wrong, but some of them are useful” [11]. This is becauseideally, a model incorporates all relevant physics of the studied system sothat its solution is useful to gain intuition and knowledge regarding someproperties of interest. At the same time, models discard details assumed tobe irrelevant for these properties. Even though this makes them “wrong”, itis a necessary and even desirable step if the solution is to not be impossiblycomplicated.How to decide where lies the separation line between relevant and ir-relevant aspects for a given system and set of properties of interest, is stillan art. A general guiding principle, based on perturbation theory, is thathigh-energy states can be discarded (integrated out) if one is interested inlow-energy properties. Consequently, it is assumed that models with iden-tical low-energy spectra provide equivalent descriptions of a system, andtherefore the simplest of these models can be safely used.A prominent example is the modeling of cuprates. It is widely believedthat the Emery model [29] can be replaced by the simpler t-J model tostudy their low-energy physics [18, 59]. The justification was provided byZhang and Rice [114] who argued that the low-energy states of the Emerymodel are singlets formed between the spin of a doping hole hosted on thefour oxygens surrounding a copper and the spin of that copper, and thatthe resulting QP is described accurately by the t-J model [4]. Whether thisis true is still being debated [26, 27, 56].In this Chapter we show that by itself, the condition that two models774.2. Modelshave the same low-energy spectrum is not sufficient to guarantee that theydescribe similar low-energy properties, despite widespread belief to the con-trary. Indeed, we identify three models that have identical T = 0 QPs yethave very different low-energy behavior at any temperature T 6= 0. Thequalitative differences are due to rare events controlled by thermal fluctua-tions, which lead to a pseudogap-type of phenomenology.While our argument takes the form of a “proof by counterexample”,we also provide arguments that our findings are not merely an “accident”caused by our specific choice of models, but are more general in nature.Specifically, we comment on its validity in arbitrary dimensions and also forother types of magnetic coupling which differ from the examples that areour main focus.4.2 ModelsThe models of interest are sketched in Fig. 4.1. Note that Model II wasstudied in detail in Chapter 3. All three models describe the interactionof the carrier with a background of local moments, and as such bear somesimilarity to those used in the Zhang-Rice mapping mentioned above. ModelI is the parent two-band model, from which Models II and III are derived asincreasingly simpler effective models. In Model I, one band hosts the spin−12magnetic moments and a second band, located on a different sublattice,hosts the carrier. Model II is also a two-band model, but the carrier andlocal moments are located on the same sites. One can think of the statesoccupied by the carrier in this model as being local linear combinations ofthe carrier states in Model I, each centered at a spin site. In Model III,the carrier is locked into a singlet with its lattice spin, forming a “spinlesscarrier” analogous to the Zhang-Rice singlet.Model I Model II Model IIIFigure 4.1: Sketch of the three models. Large, red arrows represent thelocal magnetic moments, empty (filled) blue circles represent empty (filled)carrier sites. For Models I and II the carrier spin is represented by a bluearrow, for Model III the carrier is a spinless “hole” in the Ising chain.784.2. ModelsThere are also significant differences between our models and the Zhang-Rice mapping: (i) we restrict ourselves to one dimension as this suffices toprove our claim. However, some comments on the extension of our resultsto higher dimensions can be found below; (ii) We concentrate on the case ofa FM background because for models with an AFM background the T = 0spectra are not identical. However, we also present some AFM results lateron, to demonstrate that some of the features we discuss here are generic,not FM-specific; and (iii) all spin exchanges are Ising-like, i.e. no spinflipping is allowed. The latter constraint allows us to find numerically theexact solutions using a Metropolis algorithm, to uncover a surprising finite-Tbehavior for Model III.In all three cases, the interactions between the local moments are de-scribed by the Ising Hamiltonian:HˆI = −J∑iσˆi+δσˆi+1+δ − h∑iσˆi+δ, (4.1)where δ = 1/2 for Model I and δ = 0 for Models II and III, and σˆi+δ is theIsing operator for the local magnetic moment located at Ri+δ = i + δ (weset a = 1). Its eigenvalues are σi = ±1. For J > 0 the ground state of HˆIis FM, and it is AFM for J < 0. In the case of FM coupling, the externalmagnetic field h can be used to favor energetically one of the two possibleFM ground states of the h = 0 case.For Models I and II, the kinetic energy of the carrier is described by anearest-neighbor hopping Hamiltonian:Tˆ = −t∑i,σc†i,σci+1,σ + h.c. =∑k,σ(k)c†k,σck,σ (4.2)where c†i,σ is the creation operator for a spin-σ carrier at site Ri and c†k,σ =1/√N∑i eikRic†i,σ are states with momentum k ∈ (−pi, pi) and eigenenergy(k) = −2t cos k. The interaction between the carrier and the local momentsis an AFM Ising exchange:Hˆ(I,II)exc =J02∑i,σσc†i,σci,σ(σˆi−δ + σˆi+δ). (4.3)Note that flipping the sign of the carrier spin corresponds to letting J0 →−J0, so we can assume without loss of generality that the carrier has spin-upand suppress the spin index. The total Hamiltonian for Models I and II isthus given by Hˆ(I,II) = HˆI + Tˆ + Hˆ(I,II)exc .794.3. MethodModel III is the FM (J > 0) or AFM (J < 0) Ising version of the one-band t-J model discussed extensively in the cuprate literature [18, 59]. Thecase of interest now has N + 1 electrons in the N site system (N → ∞),and double occupancy is forbidden apart from the site where the addi-tional carrier is located and which can be viewed as hosting a “spinlesscarrier” whose motion shuffles the otherwise frozen spins. The Hamiltonianis Hˆ(III) = PTˆP + HˆI , where the operator P projects out additional dou-ble occupancy. It is important to note that in contrast to Models I and II,here the spin-operators σˆi are related to the electron creation/annihilationoperators via σˆi =∑σ σc†i,σci,σ.4.3 MethodWe calculate the finite-T spectral weight A(k, ω) = − 1pi ImG(k, ω), whereG(k, ω) is the one-carrier propagator. If the carrier is injected in the mag-netic background equilibrated at temperature T , the GF is given by cf. Eq.(1.29)G(k, ω) =∑{σ}e−βEI{σ}Z〈{σ}|ck↑Gˆ(ω + EI{σ})c†k↑|{σ}〉The sum is over all configurations {σ} = (σ1, . . . , σN ) of the Ising chain, withcorresponding energies HˆI |{σ}〉 = EI{σ}|{σ}〉, and Z =∑{σ} exp(−βEI{σ}).The resolvent is Gˆ(ω) = [ω− Hˆ + iη]−1, where η → 0+ ensures retardation.The shift by EI{σ} in the argument of the resolvent shows that the poles ofthe propagator mark the change in the system’s energy, i.e. the differencebetween the eigenenergies of the system with the carrier present, and thoseof the undoped states into which it was injected. This reflects the well-knownfact that electron addition states have poles at energies EN+1,α−EN,β [63].After Fourier transforming to real space and using the invariance totranslations of the thermally averaged system, we arrive at:G(k, ω) =∑neikRn∑{σ}e−βEI{σ}Zg0,n(ω, {σ}), (4.4)where g0,n(ω, {σ}) = 〈{σ}|c0,↑Gˆ(ω+EI{σ})c†n,↑|{σ}〉 is the Fourier transformof the amplitude of probability that a state with configuration {σ} andthe carrier injected at site n evolves into a state with the carrier injectedat site 0. These real-space propagators are straightforward to calculate,804.3. Methodas they correspond to a single particle (consistent with our assumption ofa canonical ensemble with exactly one extra charge carrier in the system)moving in a frozen spin background. We emphasize that this is true onlybecause of the Ising nature of the exchange between the background spins.Heisenberg coupling, on the other hand, would lead to spin fluctuations thatwould significantly complicate matters. Below we present the calculation ofthese real-space propagators for Model III. For Model II, the solution isdescribed in detail in Chapter 3, and the same approach, with only minormodifications, applies to Model I.It is convenient to introduce the following notation. When an extraelectron is injected at site n of Model III it effectively removes the spinat this site. The spin σn will therefore be missing from the set {σ} whichdescribes the state of the spin-chain before injection. Consequently we labelthe new state, after injection, as |{σ} \ σn〉 = | . . . σn−1 ◦ σn+1 . . . 〉, where ◦denotes the effective “hole” created by the injection of the extra electron.The “hole” can propagate along the chain and in doing so reshuffles thespins. To capture the propagation of the “hole” we introduce a new index jcorresponding to the number of sites that the “hole” has hopped to the left(j < 0) or right (j > 0). A general state is therefore given by |{σ} \σn, j〉 =| . . . σn−1σn+1 . . . σn+j ◦ σn+j+1〉. Note that this way of labelling states isnot unique. For instance, if σ0 = σ1 = · · · = σn, then |{σ} \ σ0, 0〉 =|{σ} \ σn,−n〉.With this notation, the real-space propagators are g0,n(ω, {σ}) = 〈{σ} \σ0, 0|Gˆ(ω + EI{σ})|{σ} \ σn, 0〉. Their EOM are obtained by splitting theHamiltonian in two parts, Hˆ = Hˆ0 + Vˆ , and repeatedly using Dyson’s iden-tity Gˆ(ω) = Gˆ0(ω) + Gˆ(ω)Vˆ Gˆ0(ω). Choosing Hˆ0 = HˆI and suppressing theω and {σ}-dependence we obtaing0,0 = G0(ω + ∆0)[1− tf0,1 − tf0,−1], (4.5)f0,n = −tG0(ω + ∆n)[f0,n+1 + f0,n−1], (4.6)where G0(ω) = (ω + iη)−1, ∆n = EI{σ} − EI{σ}\σ0,n and f0,n = 〈{σ} \σ0, 0|Gˆ(ω + EI{σ})|{σ} \ σ0, n〉. Note that f0,0 = g0,0. The exact form of∆n depends on the sign of n:∆0 = −Jσ0(σ−1 + σ1) (4.7)∆n = ∆0 + Jσ−1σ1 − Jσnσn+1, for n > 0 (4.8)∆n = ∆0 + Jσ−1σ1 − Jσnσn−1, for n < 0. (4.9)The EOM (4.6) can be solved with the ansatz f0,n = Anf0,n−1, for n > 0and f0,n = Bnf0,n+1 for n < 0. Since the “hole” has a finite lifetime ∝ 1/η814.4. Resultsand f0,n measures the probability that the “hole” injected at site 0 moves tosite n, one expects f0,n → 0 for n→∞. We therefore introduce a sufficientlylarge cutoff Mc and require AMc = 0 = B−Mc . It is then straightforward toobtainAn =−tω + ∆n + iη + tAn+1, (4.10)Bn =−tω + ∆n + iη + tBn−1, (4.11)g0,0 =1ω + ∆0 + tB−1 + tA1, (4.12)f0,n = An . . . A1g0,0 for n > 0, (4.13)f0,n = Bn . . . B1g0,0 for n < 0. (4.14)To calculate the g0,n we make use of the fact that hopping reshuffles thespins. Therefore g0,n 6= 0, only if σ0 = σ1 = · · · = σn. In that case, asmentioned above, the states |{σ} \ σ0, n〉 and |{σ} \ σn, 0〉 are equal whichmeans that g0,n = f0,n.The thermal average in Eq. (4.4) is then calculated for the infinitechain with a Metropolis algorithm which generates configurations {σ} ofthe undoped chain. To summarize, our method of solution consists of thefollowing steps: (i) generate a configuration {σ} of the Ising chain usinga Metropolis algorithm; (ii) Calculate all the g0,n(ω, {σ}) propagators forthat specific configuration, and perform the sum over n in Eq. (4.4); (iii)repeat steps (i) and (ii) until convergence is reached. The details of thisprocedure were discussed in the previous chapter for Model II and additionalinformation can be found in Section 1.6 and Appendix C; the generalizationto Models I and III is straightforward. For Model III it is convenient toinject the carrier with an unpolarized total spin, to ensure that a “hole” isalways created. Since for each configuration {σ} there is a configuration {σ¯}with all the spins flipped, injecting an unpolarized carrier does not changethe results, but merely speeds up the numerics.4.4 Results4.4.1 FM resultsAt T = 0, the undoped Ising chain is in its FM ground state. The QPs ofModels I and II have energy ∓J0 + (k) if the carrier is injected with itsspin antiparallel/parallel to the background. Only the former case can be824.4. Resultsmeaningfully compared with Model III, which has a QP of energy 2J + (k)(2J is the cost of removing two FM Ising bonds). Thus, apart from trivialshifts, the three models have identical QPs, namely carriers free to move inthe otherwise FM background.Finite-T spectral weights A(k = 0, ω) for the different models are shownin Fig. 4.2. We emphasize that only the electron-addition part is discussedhere. We do not consider the electron-removal states, which lie at energieswell below those of the electron-addition states and must be identical forall three models because in all cases, one of the electrons giving rise to themagnetic moments is removed.For Models I and II, shown in panels (a) and (b), at the lowest tem-perature one can see two peaks marking the contributions from injection ofthe carrier into the two ground states of the Ising chain (all spins up andall spins down, respectively). Indeed, these peaks are located at ±J0 − 2t,the lower one of which is marked by the vertical line. Note that we chosea large J0 value to keep different features well separated and thus easier toidentify. The insets show the spectral weights for h = −0.1t, which at lowT suppresses the contribution from the up-spin FM state so that only thelower peak remains visible.With increasing T , both peaks broaden considerably on their higher-energy side, and many resonances become visible. As demonstrated in theprevious chapter for Model II, these resonances are due to temporal trappingof the carrier inside small magnetic domains that are thermally generatedat higher T . The presence of these domains also explains the decreasingdifference between the h = 0 and h = −0.1t curves at higher T . For βJ = 0.5both curves are shown in the main panels (the finite h curve is shaded in).Indeed, the resonances appear in the same places and with equal weight inboth curves, the only difference being a small spectral weight transfer fromJ0 − 2t to −J0 − 2t, i.e. from the FM ground-state disfavored by h < 0 tothe one favored by it. The weight for the former is no longer zero like forT → 0, showing that at higher T the carrier is increasingly more likely toexplore longer domains of spin-up local moments.The main difference between Models I and II is that the latter also has athird finite-T continuum, centered around ω = 0. It corresponds to injectingthe carrier in small AFM domains, where its exchange energy vanishes be-cause it sits between a spin-up and a spin-down local moment. Such energydifferences are not possible in Model II, where the carrier interacts with asingle moment so its exchange energy is ±J0.From the analytical approximation of Section 3.4.1 we know that in thelow-T limit the spectrum of Model II is well described by only considering834.4. Results-5 0 5ω/t012345A(k=0,ω)-5 0 5ω/t024A(k=0,ω) h = -0.1tβJ  =0.5βJ = 1.0βJ = 1.5(a)-5 0 5ω/t012345A(k=0,ω)-5 0 5ω/t024A(k=0,ω) h = -0.1tβJ  =0.5βJ = 1.0βJ = 1.5(b)-10 -5 0 5ω/t0246A(k=0,ω)-12 -9 -6 -3ω/t0.,ω)βJ = 0.5βJ = 1.0βJ = 1.5(c)βJ = 2.5Figure 4.2: Spectral weight at k = 0 for a FM background and three differenttemperatures for (a) Model I with J0/t = 5, J/t = 0.5; (b) Model II withJ0/t = 5, J/t = 0.5; (c) Model III with J/t = 2.5. Insets in panels (a) and(b) show the spectral weight in the presence of a magnetic field, while in (c)it shows the two continua appearing at low energies, for βJ = 0.5. In allcases, the broadening is η/t = 0.04. The vertical lines show the energy ofthe T = 0 QP peak.844.4. Results-8 -6 -4 -2 0 2 4 6 8ω/t02468A(k,ω) -6 -4 -2 0 2 4 6 8ω/t02468A(k,ω) 0 2.5 5 7.5ω/t012345A(k,ω) 4.3: A(k, ω) for the three models with FM background at βJ = 0.5.Other parameters are as in Fig. 4.2. The dispersionless low energy, lowweight part of the spectrum of Model III is not shown. Red, vertical linesindicate the location of the T = 0 QP peaks.854.4. Resultscontributions from small domains of flipped spins with length L. Since onlythe FM bonds at the ends of the domain are broken the energy cost of such adomain is 4J , independent of the domain length. Exactly the same domainsexist in Model I, where they also cost an energy of 4J and are expected togive the largest contribution to the low-T spectrum. To compare the effectof injecting the carrier in such a domain in Model I and Model II it is usefulto rewrite H(I)exc as followsH(I)exc = J0∑ic†iciσˆ(I)i , (4.15)where we defined σˆ(I)i = (σˆi−δ+σˆi+δ)/2. Note that σˆi+δ belongs to both σˆ(I)iand σˆ(I)i+1 which makes it difficult to reverse this transformation. Furthermorewe suppressed the spin index of the carrier since as mentioned above weassume that is has spin-up.Formally H(I)exc in the rewritten form looks exactly the same as H(II)exc ,the difference between them is that the new spin operator σˆ(I)i can take thevalues 1,0 and -1. With this transformation it becomes clear that when acarrier is injected into one of the domains mentioned above there is a smalldifference between the potential it experiences in Model I and in Model II.In Model II the carrier sees a potential step from +J0 to −J0 (or vice versa)at the edge of the domain. In Model I on the other hand the domain edgesare stretched out over two sites and the carrier sees a potential step from+J0 to 0 to −J0 (or vice versa). The special case of a single site domainwhich corresponds to a potential dip from +J0 to −J0 and back to +J0 inModel II becomes a potential dip from +J0 to 0 and back to +J0 in ModelI. This explains the spectral weight at ω = 0 in Model I. However, thesharpness of the domain edges in Model II compared to Model I does notchange the nature of the low energy states. Consequently, if one is interestedin the low-energy behavior, Models I and II are equivalent because their low-energy continua have similar origins and evolve similarly with T . This istrue in the whole Brillouin zone (BZ), as can be seen from comparing panels(a) and (b) of Fig. 4.3.The finite-T evolution of the spectral weight of Model III is very different.Consider first the k = 0 case, shown in Fig. 4.2(c). The T = 0 peak at 2J−2t(marked by the vertical line) evolves with T very similarly to the low-energypeaks of the other two models, broadening on its high-energy side and againdisplaying resonances due to temporal trapping inside small domains. Thek evolution of this feature, shown in Fig. 4.3(c), is also very similar to thelow-energy continua of the other two models.864.4. Results(a) (b)Figure 4.4: When doping “removes” a thermally excited spin-down, theenergy variation upon doping is Eb − Ea = −2dJ + (k) and lies (at leastpartially) below the T = 0 QP ground-state energy of 2dJ − 2dt.However, for Model III this continuum is not the low-energy feature.Instead, in 1d there are three lower-energy continua centered at 0,−2J and−4J , all of which are due to injection of the carrier into specific, thermallyexcited configurations of the background. For example, consider the −2dJcontinuum which also appears in dimensions d > 1. As sketched in Fig.4.4, it corresponds to the carrier being paired with a thermally excited spin.This lowers the exchange energy by 2dJ , as 2d AFM bonds are broken. Incontrast, T = 0 doping always leads to loss of exchange energy, because onlyFM bonds can be broken. This is why in Model III it can cost less energyto dope from a thermally excited state rather than the ground-state, andtherefore why its finite-T , low-energy properties are not controlled by theT = 0 QP.The weight of these low-energy continua is very small, see inset of Fig.4.2(c), because they are controlled by thermal activation. For example, inthe limit T → 0 the spectral weight of the continuum centered at −2dJ canbe calculated to first order, as was shown in Chapters 2 and 3, by expandingEq. (4.4) in powers of e−β4dJ . The lowest order terms correspond to the twoFM ground states which have all spins aligned, |{↑}〉 and |{↓}〉, respectively.The first order terms are given by states with a single flipped spin and aredenoted by |{↑, σm =↓}〉 and |{↓, σm =↑}〉, where m indicates the location ofthe flipped spin. Since the flipped spin can be anywhere in the system thereare N of these states for each ground state configuration. For simplicitywe assume that the spin of the extra carrier is unpolarized, then it sufficesto consider only |{↑}〉 and |{↑, σm =↓}〉, the contribution from the otherground state will be exactly the same. Considering only these states in thetrace of Eq. (4.4) we obtain:G(k, ω) =1Z ′[G(0)(k, ω) + e−β4dJG(1)(k, ω)]+O((e−β4dJ)2), (4.16)874.4. ResultswhereG(0)(k, ω) =[ω − k − 2dJ + iη]−1 (4.17)G(1)(k, ω) =∑n,σeikRn∑mg0,n(ω, {↑, σm =↓}) (4.18)Z ′ =Ze−βEFM= (1 +Ne−β4dJ + . . . ) (4.19)Note that G(0)(k, ω) is identical to the T = 0 solution.To evaluate G(1)(k, ω) we need to treat the casem = 0, separately. In thiscase the extra carrier removes the flipped spin. This results in the breakingof 2d AFM bonds and therefore an energy gain of 2dJ . Furthermore aspointed out above only g0,0(ω, {↑, σ0 =↓}) contributes to the sum since theextra carrier was injected into a domain of length 1. Since the flipped spinwas removed and all the remaining spins are aligned it is easy o calculateg0,0(ω, {↑, σ0 =↓}) which in the limit N →∞ becomesg0,0(ω, {↑, σ0 =↓}) =∫dq(2pi)d1ω − q + 2dJ + iη , (4.20)i.e. a continuum of states centered at ω = −2dJ .We are now left with calculating the remaining contributions to G(1)(k, ω)for which m 6= 0. This is not a trivial problem, but since there is only oneflipped spin in the system and we are summing over all “hole” locations, wecan approximate g0,n(ω, {↑, σm =↓}) ≈ g0,n(ω, {↑}). In doing so we neglectthat the energy is lowered when the “hole” is adjacent to the flipped spinσm. Reinserting into Eq. (4.18) we obtainG(1)(k, ω) ≈ g0,0(ω, {↑, σ0 =↓}) + (N − 1)G(0)(k, ω), (4.21)where the factor N − 1 in front of G(0) is due to the sum over m. Notethat this factor ensures that the Z ′ in the Eq. for G(k, ω) is approximatelycanceled. Similarly one expects contributions from states with two or morewell-separated flipped spins to cancel the Z ′ in front of g0,0(ω, {↑, σ0 =↓}) (see the discussion in Chapters 2 and 3). Consequently the low-T expansionof the Green’s function gives:G(k, ω) ≈ G(0)(k, ω) + e−β4dJg0,0(ω, {↑, σ0 =↓}). (4.22)i.e. the spectral weight below the T = 0 QP which is given by g0,0(ω, {↑, σ0 =↓}) vanishes like the probability e−β4dJ to find a flipped spin.884.4. Results-6 -4ω/t00.511.5Rescaled Spectral WeightβJ=2.5βJ=2.0βJ=1.5-Im[g0,0(ω,{↑,σ0=↓})]/piFigure 4.5: Rescaled spectral weight eβ4J [A(0, ω) − A(0)(0, ω)] in the re-gion of the continuum centered at −2J , for Model III with FM backgroundand different values of β. For comparison the dashed, black line shows−Im[g0,0(ω, {↑, σ0 =↓})]/pi calculated with Eq. (4.20). Other parametersare J/t = 2.5 and η/t = 0.04.To verify this behavior we show in Fig. 4.5 the rescaled spectral weighteβ4J [A(0, ω) − A(0)(0, ω)], where A(0)(k, ω) = δ(ω − k − 2J)/pi is the T =0 QP peak. From Eq. (4.22) it is clear that at sufficiently low T theresulting curves should equal −Im[g0,0(ω, {↑, σ0 =↓})]/pi, which is shown bythe dashed, black line in Fig. 4.5. Indeed we find that the three curves inFig. 4.5 for βJ = 1.5, 2.0 and 2.5, respectively, collapse onto each other andonto the curve for −Im[g0,0(ω, {↑, σ0 =↓})]/pi. Close to the upper edge ofthe continuum the agreement starts to falter. This is because in additionto the T = 0 peak there are other peaks in the spectral weight (see Fig.4.2) whose tails contribute to the −2J continuum and are not subtracted.Multiplying with eβ4J amplifies these tails. Similarly the oscillating featuresin Fig. 4.5 are finite size effects which are amplified by the factor eβ4J .Similar calculations can be performed for the other low-energy features.All their spectral weights vanish as T → 0 because they all originate fromdoping the carrier into a thermally excited environment, which become lessand less likely to occur in this limit.The finite-T behavior of Model III is thus qualitatively different fromthat of Models I and II. For the latter, the T = 0 QP peak also marks thelowest energy for electron-addition at any finite temperature, whereas forModel III we observe the appearance of electron-addition states well belowthe T = 0 QP peak. Their spectral weight vanishes as T → 0, which is very894.4. Results-4 -2 0 2 4ω/tA(k=0,ω)J = 0.1J = 0.2J = 0.3J = 0.4J = 0.5J = 0.6J = 0.7J = 0.8x60x60x60x60x60x60x30x20 x40x30x30x30x20x20x20x20Figure 4.6: A(k = 0, ω) for Model III with FM background, for differentvalues of J at a temperature βJ = 0.5. The dashed red lines show thelocation of the T = 0 QP peak. Full blue lines mark the energies −2J ± 2t.Parts of the spectra have been rescaled for better visibility.reminiscent of pseudogap behavior and offers a simple and general scenariofor how it can be generated 5. These low-energy states vanish from thespectrum as the temperature is lowered not because a gap opens and/orthe electronic properties are somehow changed, but simply because thesestates describe doping into thermally excited local configurations, and theprobability for the doped carrier to encounter them vanishes as T → 0.As should be clear from these arguments, the appearance of these low-energy continua is not a consequence of the large J/t values used so farfor Model III. Indeed, Fig. 4.6 shows that similar behavior is observed forsmaller J values (parts of these spectral weights have been rescaled for bettervisibility). With decreasing J the different continua overlap, but shouldersmarking some of their edges are still clearly visible and marked by dashedlines. In all cases, at finite T spectral weight appears below the T = 0 QP5For the experimental observation of pseudogap behavior it is of importance whetherthe chemical potential falls into the region of the spectrum that exhibit pseudogap behavioror remains at the T = 0 peak. Since we are working in a canonical ensemble this is nota priori clear. However, we expect that at finite T the chemical potential will move awayfrom the T = 0 peak towards the low-energy states that become available.904.4. Results8 6 4 2 0 2 4 6 6 4 2 0 2 4 6 2 4 6 8 1012140. 6 4 2 0 2 4 6 8ω/t012345678A(0,ω)(d)8 6 4 2 0 2 4 6 8ω/t0123456(e)0 2 4 6 8 101214ω/t012345678(f)Figure 4.7: T = 0, AFM solutions. Top panels: contour plots of A(k, ω).Bottom panels: Cross sections at k = 0. (a) and (d) Model I with J0/t = 5,|J |/t = 0.5; (b) and (e) Model II with J0/t = 5, |J |/t = 0.5; (c) and (f)Model III with |J |/t = 2.5. To improve visibility of the continuum a hardcutoff at A(k, ω) = 0.1 was used for the Model III contour plot. In all casesη/t = 0.04.peak, marked by the full line.It should also be clear that this phenomenology is not restricted to FMbackgrounds, either: one can easily think of excited configurations in anAFM background whose exchange energy would be lowered through doping,in a t-J model similar to Model III. We have verified numerically that atfinite T , features lying below the corresponding T = 0 QP peak indeedappear in the spectral weight of AFM chains. These results are presentedbelow. This phenomenology is therefore quite general.4.4.2 AFM resultsJust like in the FM case, there are also two ground states of the undopedAFM Ising chain: either the odd or the even lattice hosts the up localmoments. Of course, both AFM ground states yield the same QP properties.However, the QPs that result when a carrier is injected in the three modelsare different even at T = 0, for the AFM backgrounds. This is shown in914.4. ResultsFig. 4.7, where contour plots of the T = 0 spectral weight A(k, ω), and crosssections at k = 0, are shown.For Model I, the energy shifts due to the Ising exchange with the spinsto the left and right of the extra electron exactly cancel out and the QPbehaves like a free electron with dispersion (k). For Model II, interactionwith the AFM background opens a gap in the QP spectrum and halves itsBZ. The upper and lower bands have dispersion ±√J20 + 2(k), respectively.For Model III, the T = 0 spectral function is independent of k and has acoherent QP peak at ω = 4|J | − 2√J2 + t2 and a continuum for 4|J | − 2t <ω < 4|J | + 2t. The QP peak corresponds to a bound state with the extraelectron confined at its injection site. Propagation of the extra electron alongthe chain reshuffles the Ising spins and gives rise to the continuum centeredat 4|J |. One can therefore think of this continuum as the electron+magnoncontinuum. Mathematically, the k-independence follows directly from thefact that for Model III, g0,n(ω, {σ}) = 0 when n > 0, if {σ} is the AFMground state.The finite-T spectral functions for k = 0 are shown in Fig. 4.8. For allthree models the peaks broaden and spectral weight appears below, as wellas above the T = 0 QP peak. This is in contrast to the FM case, where inModels I and II, at k = 0, spectral weight appears only above the T = 0 QPpeak. For Model I the energy difference between the low-energy states andthe T = 0 peak is controlled by J0, because at finite T the extra electron canbe injected into small domains where the Ising exchange interaction with thespin to the left no longer cancels that with the spin to the right. For ModelII on the other hand injecting the extra electron into a small FM domainmerely enhances its mobility and leads to an energy change of the order of t.As the temperature increases more weight is transferred to these low-energyfeatures, and the low-energy behavior of Model II starts to resemble that ofModel I even though they have different T = 0 QPs.For Model III, at finite T new features appear, centered at −2|J | and0. Just as for the FM case, they are due to the injection of the extra elec-tron into specific, excited, local configurations of the chain. If the electronis injected into a small FM domain embedded in an otherwise AFM or-dered background, the energy is lowered by 2d|J |. As long as the electronstays within the FM domain reshuffling of the spins does not result in afurther change in energy. This explains the appearance of spectral weightat ω ∼ −2d|J |. This is true for any dimension d and consequently theappearance of spectral weight at −2d|J | is a generic feature of Model III.Coming back to the specific case of d = 1, if the electron leaves the FM do-main, then reshuffling of the spins recreates an FM bond and destroys one924.4. Results-10 -5 0 5 10ω/t012345A(k=0,ω)β|J|=0.5β|J|=1.0β|J|=1.5(a)J<0β|J|=2.5-8 -6 -4 -2 0 2 4 6 8ω/t012345A(k=0,ω)β|J|=0.5β|J|=1.0β|J|=1.5(b)J<0β|J|=2.5-5 0 5 10ω/t012345A(k=0,ω)β|J|=0.5β|J|=1.0β|J|=1.5(c)J<0β|J|=2.5-6 -4 -2 0 2 4ω/t00.010.02A(k=0,ω)Figure 4.8: AFM Spectral weight at k = 0 for three different temperaturesfor (a) Model I with J0/t = 5, |J |/t = 0.5; (b) Model II with J0/t = 5, |J |/t =0.5; (c) Model III with |J |/t = 2.5, the inset shows spectral weight belowthe T = 0 QP peak for β|J | = 2.5. In all cases,the broadening is η/t = 0.04.The vertical lines show the energy of the T = 0 QP peak.934.5. Discussion and conclusionsFigure 4.9: Spiral spin configuration in a 1d XY model.of the AFM bonds. The total change in energy (injection and reshuffling)is therefore zero, explaining the continuum centered around ω ∼ 0. Besidesthe appearance of these new, low-energy continua, resonances appear closeto the T = 0 QP peak and within the high-energy continuum (not visible inFig. 4.8 due to the scale). They are likely caused by injection of the electroninto an AFM domain and subsequent scattering off domain walls which canonly exist at finite T .4.5 Discussion and conclusionsIn this work, we identified models that have identical T = 0 low-energy QPs(for couplings favoring a FM background), and yet exhibit very differentlow-energy behavior at finite T , proving that the former condition does notautomatically guarantee the latter.In particular, the finite-T behavior in Model III is controlled by rareevents, where the carrier is injected into certain magnetic configurationscreated by thermal fluctuations. Their energies are higher than that of theundoped ground-state, however the spectral weight measures the change inenergy upon carrier addition (or removal), and this may be lower at finite Tthan at T = 0. This is the case for Model III because here doping removes amagnetic moment from the background while its motion reshuffles the otherones. It is not the case for Models I and II where the carrier can do neitherof these things. This difference is irrelevant at T = 0 because of the simplenature of the undoped FM ground-state, but becomes relevant at finite T .We showed that such transfer of finite-T spectral weight well below theT = 0 QP peak is independent of the size of the magnetic coupling J and oc-curs for both FM and AFM coupling. Furthermore, we provided argumentsthat this behavior is expected to occur in any dimension.One may wonder how much of an influence the Ising nature of the spinsplays for the spectral weight transfer to energies below the T = 0 QP peakin Model III. A way to get an estimate of this is by using a FM XY modelwhere the angles between the spins can take any value. The Hamiltonian of944.5. Discussion and conclusions0 0.05 0.1 0.15 0.2 0.25θ/pi-1-0.50EXY(θ)-EXY(0)(a)J/t = 2.0J/t = 2.5J/t = 3.0J/t = 4.02 4 6 8 10J/t0.θ c(b)Figure 4.10: (a) energy difference between the spiral state and the FM stateas a function of θ for different values of J . (b) the dependence of θc on J .In both panels t = 1.the classical XY chain is given byHˆXY = −J∑icos(θi+1 − θi) (4.23)For this model the GS of the spin chain is the state in which all spins arecompletely aligned, i.e. all angles θi = 0. This state is identical to the GSof Model III. There are many types of excited states in the XY model, butif one injects the ”hole” into a spiral configuration where neighboring spinsare rotated with respect to each other by a fixed angle θ (see Fig. 4.9) onecan use the same formalism as above to calculate the real-space GF gXY0,0 (ω)for this configuration. By setting the denominator of gXY (ω) to zero theQP energy of the lowest ”hole” state is obtained. It is given byEXY (θ) = 3J cos(θ)− J cos(2θ)−√J2(cos(θ)− cos(2θ))2 + 4t2 (4.24)Note that for θ = 0 we recover the result of the FM, T = 0 QP peak whoseenergy is EXY (0) = 2J − 2t.The question we are trying to answer is whether there are values of θfor which EXY (θ) < EXY (0). Such states would have the same effect asthe single spin down domains which we identified as giving contributions954.5. Discussion and conclusionsto the spectral weight below the T = 0 peak for Model III. Of course,this is trivially satisfied by θ = pi which produces AFM order. The realquestion is therefore whether there are also smaller values of θ for whichthis occurs. That this is indeed the case is verified in Fig. 4.10(a) where weshow EXY (θ)−EXY (0) for different values of J . Clearly a critical value, θc,exists for which the energy of a ”hole” injected into a spiral is identical tothat of a ”hole” injected into the ordered XY chain. Chains with θ > θc willcontribute to spectral weight below the T = 0 QP peak. The dependence ofθc on J is shown in panel (b) of Fig. 4.10. One can show that θc → 0 in thelimit J →∞. Note that θc < pi/2 even for relatively small values of J .Note that we did not consider the energy it requires to create such aspiral which will scale with the system size, since every spin is rotated byan angle θ. The Boltzmann weight of such a spiral will consequently bevery small. However, the above discussion does show that domains whichcan lead to spectral weight below the T = 0 QP peak also exist for morecomplicated types of coupling, as long as the carrier is injected into the sameband which hosts the spins.It is also interesting to discuss in how far these results would be altered ifthe interaction between the extra electron and the local magnetic momentswas of Heisenberg instead of Ising type. For Models I and II we showed inChapter 2 that for Heisenberg J0 and FM coupling between local magneticmoments, the formation of a spin-polaron becomes possible. As we pointedout this can also lead to the transfer of spectral weight to energies far awayfrom the T = 0 QP peak. However, this mechanism is quite different fromthe mechanism that leads to the appearance of spectral weight below theT = 0 peak in Model III. First of all the spin-polaron only exists when |J0|is sufficiently large, whereas in Model III spectral weight appears below theT = 0 peak for all values of J . Another difference is that the spin-polaronstates in Models I and II are not necessarily the low-energy features, evenat T = 0. Instead it is possible that the T = 0, low-energy states belong tothe c+m continuum. In this case a mapping from Model I/II to Model IIIbecomes impossible, because the T = 0, low-energy feature of Model III isa single QP peak instead of a continuum. If the extra electron is injectedwith its spin parallel to the local magnetic background the T = 0 spectrumfor Models I and II is a single QP peak, but again the mapping to Model IIIis impossible, because Model III requires the extra electron to have oppositespin, otherwise it cannot be hosted by the half-filled band. Consequently themapping from Model I/II to Model III is only meaningful if the low-energyfeature of Model I/II is the spin-polaron. The spin-polaron then correspondsto the effective “hole” in Model III and as discussed above, at finite T , Model964.5. Discussion and conclusionsIII predicts additional features to appear below it. For Model I/II, on theother hand, we cannot completely rule out the possibility of additional fea-tures appearing below the spin-polaron state, but we discussed in Chapter2 that such features would correspond to a bound state between the extraelectron and multiple magnons and are extremely unlikely. The main con-clusions from this Chapter should therefore remain true for Heisenberg J0,as well.While far from being a comprehensive study, these results and our dis-cussion clearly demonstrate that the appearance of finite-T spectral weightwell below the T = 0 QP peak, due to the injection of the carrier into athermally excited local environment making it behave very unlike the T = 0QP, is a rather generic feature for t-J like models. The weight of these finite-T , low-energy features must vanish when T → 0 because the probability forsuch excited environments to occur vanishes, therefore these models exhibitgeneric pseudogap behavior.97Chapter 5Summary and outlookThe theme of this thesis is unusual finite-T signatures in the one-electronspectral function of models of doped magnetic insulators. In all the mod-els studied in this thesis a single charge carrier interacts with a magneticbackground. The most sophisticated model we studied is the so-called s-f(d) model, in Chapter 2. There we used a low-T expansion to investigatehow thermal magnons change the spectrum of a spin up carrier when itis injected into a magnetic background with positive magnetization. Wefound that apart from the usual broadening of the T = 0 QP peak, spectralweight is shifted to spin-polaron states. In the case of AFM J0 this resultsin the transfer of spectral weight to a continuum at energies well below theT = 0 QP peak. Furthermore spectral weight was transferred to the c+mcontinuum spanned by the energies {E↑(k − q + q′) − Ωq′ + Ωq}q,q′ andwe discussed that in general one expects continua to appear at all energies{E↑(k +∑mi=1 q′i −∑mi=1 qi)−∑mi=1 Ωq′i +∑mi=1 Ωqi}q1,...,q′m .Unfortunately we were unable to calculate higher order corrections tothe low-T expansion from Chapter 2. It was furthermore found that the ap-proximation breaks down when T becomes too large and produces a spuriouspole in the spectral function.Consequently we chose a very different approach in Chapter 3. By sim-plifying the model to a 1d chain for which both the spin-spin interaction, aswell as the carrier-spin interaction are of Ising type we were able to obtainexact, numerical results. These results highlight the importance of smallmagnetic domains which can temporarily entrap the carrier and lead to adistinctive fine-structure in the carrier spectrum. As discussed, a somewhatsimilar situation occurs in binary alloys, where charge carriers can becometrapped in small domains of like atoms. However, spin-spin correlations andthe absence of true disorder set the magnetic case apart from binary alloys.The results from the Monte Carlo simulation allowed us to develop ananalytical approximation which takes into account the contributions of thesimplest magnetic domains of length L. For this approximation to work oneneeds to introduce a cutoff for L which should be of the order of the spin-spin correlation length. The numerical results furthermore suggest that a98Chapter 5. Summary and outlooksimilar approximation should work even at higher T if one takes into accounta T dependent prefactor for the contribution of each domain. However, theform that such a prefactor should have is not obvious. It is likely that asimilar approach could work for the low-T expansion of Chapter 2, i.e. bymultiplying the self-energy Eq. (2.10) with a T dependent prefactor oneshould be able to account for the incomplete cancellation of the factor 1/Zand avoid the appearance of the spurious pole. But of course the sameproblem of guessing the form of the prefactor exists here, as well. A possibleextension to the work in this thesis could be to try to improve on theseapproximations.The study of simplified 1d models was continued in Chapter 4 where wefound that at finite T , t− J-like models in which the carrier is injected intothe same band as the spins are generically different from multi-band modelsin which the carrier is hosted by a separate band. The reason for this is thatin t − J-like models the injected carrier removes a spin and can thereforelower the energy of the spin chain. Since the spectral function measuresenergy differences, this results in the appearance of spectral weight belowthe T = 0 QP peak. The crucial point here is that this difference may not beobvious at all from the T = 0 spectrum. Of course we showed in Chapter 2that spectral weight below the T = 0 QP peak can also appear in a two-bandmodel, but there the mechanism for this was a Heisenberg exchange whichallowed for the emission and absorption of magnons. As discussed, this isquite different from the mechanism that leads to low-energy QP weight inthe t− J-like model from Chapter 4.Another important point raised in Chapter 4 is that of pseudogap-likebehavior. Since the spectral weight below the T = 0 QP weight in t−J-likemodels depends on the presence of certain thermally excited configurations(domain walls, in our case) it will disappear as T → 0. One can think ofthis as a type of pseudogap opening up.To summarize, in this thesis new methods for the study of finite-T effectsin doped magnetic insulators were developed. All of these methods explicitlytake into account the local environment of the carrier and its interactionswith magnons or magnetic defects. This sets them apart from mean-field likeapproaches, but also limits their applicability to more complicated models.A recurring theme was the expansion in powers of the Boltzmann factorwhich we used for the analytical approximations in Chapters 3 and 4 andfor the low T expansion in Chapter 2. Furthermore two different mechanismsfor the transfer of spectral weight to energies below the T = 0 QP peak werefound. This shows that finite-T effects in doped magnetic insulators can leadto quite unusual behavior.99Chapter 5. Summary and outlookLet us now comment on possible extensions of this work. As alreadymentioned one could try to obtain higher order terms for the low-T expan-sion from Chapter 2 which hopefully fix the appearance of the spurious pole.However, including states with more than one magnon is very difficult sinceone needs to take into account magnon-magnon interactions. Even for theIsing model these interactions exist since for S = 1/2 two magnons cannotbe on the same site. One could try to avoid this problem by transformingthe spins to bosonic operators, this would be similar to the approach fromRef. [5].Instead we believe that it is more promising to further investigate theclaims we made in Chapter 4. There we provided arguments which suggestthat even though our calculations were carried out in 1d and the exchangewas of Ising-type our main conclusions remain true for higher dimensionsand more complicated magnetic backgrounds, as well. The claim that thisis true for higher dimensions can be checked with a very similar calculationto that in Chapter 4. However, numerically a calculation in 2d or even3d will be much more costly and one will need to restrict the system sizesubstantially. A quantum Monte Carlo simulation could be used to test ourclaim for Heisenberg exchange interactions, but to ensure that the resultsare comparable one needs to work in the canonical ensemble, while oftenquantum Monte Carlo simulations use the grand canonical ensemble. Adifferent approach could be to do a similar calculation for a more complicatedmodel with classical spins, e.g. a Potts model [55, 85, 107] or the XY -model.While in this thesis we focused on the interaction of the extra chargecarrier with magnons and domain walls, more sophisticated models of themagnetic background would also allow one to study its interaction with othertypes of magnetic excitations such as magnetic vortices or skyrmions whichappear in higher dimensions. 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The secondterm accounts for the energy shift that occurs when the up-carrier is on thesame site as the magnon, and the third term contains a new propagatorF (k,q′, ω) = 〈Φ(q′)|ck,↑Gˆ(ω+Ωq′)c†k+q,↓|FM〉. This term accounts for spin-flip processes where the up-carrier absorbs the magnon, turning into a down-carrier with momentum k + q. Using Dyson’s identity again, we get an EOMfor F (k,q′, ω):F (k,q′, ω) =J0√S2NG(0)↑ (k + q, ω + Ωq′ + J0S)×∑QG(1)↑ (k,Q,q′, ω). (A.2)The diagonal element vanishes since the bra and ket are orthogonal. Theenergy shift −J0S/2 of the spin-down carrier is absorbed into the argumentof G(0)↑ , leaving only the spin-flip process which links F back to G(1)↑ .These two coupled equations can now be solved as follows. We insert111Appendix A. Derivation of the lowest T 6= 0 self-energy termEq. (A.2) into Eq. (A.1) to obtain:G(1)↑ (k,q,q′, ω) = G(0)↑ (k + q′ − q, ω + Ωq′ − Ωq){δq,q′−J02f(k,q′, ω)[1− J0SG(0)↑ (k + q, ω + Ωq′ + J0S)]}, (A.3)where f(k,q′, ω) = 1N∑QG(1)↑ (k,Q,q′, ω). Using Eq. (A.3) in the defini-tion of f(k,q′, ω) yields:f(k,q′, ω) =1NG(0)↑ (k, ω)[1 +J02g(k,q′, ω)×(1− J0SG(0)↑ (k + q′, ω + J0S))]−1,with g(k,q′, ω) = 1N∑QG(0)↑ (k + q′ −Q, ω+Ωq′−ΩQ).Note that g(k,q′, ω)can be calculated numerically since G(0)↑ (k, ω) is a known function.All that is left to do is to insert the above expression into Eq. (A.3) andcalculate∑q e−βΩqG(1)↑ (k,q,q, ω), to find the expression listed in SectionIII.112Appendix BExact solution of theundoped Ising chainHere we review the exact solution of the undoped Ising chain. All quantitiesof interest to us are obtained from the partition function:Z =∑{σ}e−βEI({σ}), (B.1)where the sum is over all configurations of lattice spins. The sum can berewritten as:Z = Tr(T N ) = λN+ + λN− ,where the transfer matrix is Tσ,σ′ = eβ(Jσσ′+h2 (σ+σ′)), and its eigenvalues are:λ± = eβJ[cosh(βh)±√sinh2(βh) + e−4βJ]. (B.2)The bulk value of the magnetization m = 1N∑i〈σˆi〉T is:m = limN→∞1Nβ∂ lnZ∂h=sinh(βh)√sinh2(βh) + e−4βJ. (B.3)The correlation between spins is given by〈σnσm〉T = sinh2(βh) + e−4βJe−|m−n|/ξsinh2(βh) + e−4βJ, (B.4)where the correlation length is ξ = −1/ log(λ−/λ+). (For AFM couplingone needs to factor out (−1)|m−n| to ensure the real-valuedness of ξ).113Appendix CImplementation of theMetropolis algorithmThe general theory behind the Metropolis algorithm is discussed in Chapter1.6. In this appendix we merely discuss the specifics of our implementation.All the data that is shown in this appendix is for the model from Chapter 3which is identical to Model II from Chapter 4.Let us start by commenting on the effect of integrating out the compli-mentary set of spins {Mc}C with the partition function Zbath(β, σ−Mc , σMc).Since it only depends on σ−Mc and σMc Zbath(β, σ−Mc , σMc) only changesthe acceptance probability Pacc when a proposal to flip one of these twospins is made. It can therefore be incorporated into the general metropolisalgorithm quite easily.The thermal average in Eq. (4.4) is then calculated for the infinitechain with a Metropolis algorithm which generates configurations {σ} ofthe undoped chain.We now discuss the measures taken to reduce correlations between indi-vidual measurements and to ensure convergence. In our implementation ina single Monte Carlo step an attempt to flip each of the 2Mc + 1 spins ismade exactly once. To do this a list with the site indices in random order isused. After every Monte Carlo step the list is randomized again. The goalof this procedure is to reduce correlations between states.To reduce correlations even further we investigate the autocorrelationfunction of the magnetization Cmn . To do this we run a Monte Carlo sim-ulation for N steps and measure the magnetization at every step. Theautocorrelation function can than be calculated as followsCmn =(N − l)−1∑N−ni=1 (mi − m¯)(mi+n − m¯)N−1∑Ni=1(mi − m¯)2, (C.1)where m¯ denotes the mean of the magnetization. Note that for AFM Jthe magnetization needs to be calculated separately for the odd and evensublattice.114Appendix C. Implementation of the Metropolis algorithm0 200 400 600 800 1000 1200Number of Monte Carlo steps n00. function C nmβt=3.0βt=3.5βt=4.0Figure C.1: The autocorrelation of the magnetization for different values ofβ and FM coupling. The dashed lines indicate when it has fallen off to 0.1.Parameters are |J |/t = 0.5, J0 = 2.5 and h = 0.The autocorrelation function is then used to determine how many MonteCarlo steps need to be performed between two consecutive measurementsto ensure that they are no longer correlated. In Fig. C.1 we show theautocorrelation function for different values of β and FM J . Note that forlarge β correlations die off very slowly. When calculating the GF we onlyperform measurements after every nth Monte Carlo step, where n is chosenin such a way that correlations have fallen off to at least 0.1, in some caseseven 0.05. It needs to be pointed out that different observables may havedifferent autocorrelation times. However, in a simple model like ours it isunlikely that the autocorrelation time of the GF varies drastically from thatof the magnetization.To assess the convergence of the spectral weight we perform a blockinganalysis as is shown in Fig. C.2 for the prominent feature of the FM spec-tral weight at ω/t = −4.5 (cf. Fig. 3.4). In this case the smallest blockscontained 800 measurements, but we also used blocks which contained only100 measurements and in turn increased the number of blocks. The datafrom Fig. C.2 shows that the spectral weight at this value of ω is well con-verged (the relative error is ∼ 0.6%) and that correlations are insignificant.Note that for less prominent features the relative errors can be much larger.This is unavoidable since these features occur when the carrier is trapped in115Appendix C. Implementation of the Metropolis algorithm16 32 64 128 256Number of Blocks00. Error (%)βt = 1.0βt = 2.0βt = 3.0βt = 4.0βt = 5.0Figure C.2: Blocking analysis of the relative error of A(k = 0, ω = −4.5t)for FM |J | = 0.5 (cf. Fig. 3.3). Other parameters are k = 0, J0/t = 2.5,h = 0 and η/t = 0.04.-4 -2 0 2 4ω/t00.51A(0,ω)J0 < 0J0 > 0DifferenceFigure C.3: The FM spectral function A(0, ω) for |J0|/t = 2.5, |J |/t = 0.5,h = 0 and η/t = 0.04. The error bars correspond to the largest error fromthe blocking method.116Appendix C. Implementation of the Metropolis algorithmmagnetic clusters which are unlikely to occur and therefore longer samplingtimes are needed to increase their accuracy.To provide the reader with a better feeling of the convergence of thewhole spectrum we show in Fig. C.3 a plot of A(0, ω) for β/t = 1.0 witherrorbars corresponding to the largest error from the blocking method. Thisfigure also shows the same spectrum, but with a negative value for J0. Phys-ically the sign of J0 is irrelevant and therefore we can use a comparisonbetween the two spectra as a consistency check. Note that A(k, ω) obeysthe sum rule∫∞−∞ dω A(k, ω) = 1. Consequently the difference between thetwo spectra which is purely of statistical nature must average to zero.117


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