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Impurity effects in antiferromagnetic quantum spin-1/2 chains Eggert, Sebastian 1994

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IMPURITY EFFECTS IN ANTIFERROMAGNETIC QUANTUM SPIN—1/2 CHAINSBySebastian EggertM. Sc. (Physics) University of Wyoming, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conforming1994© Sebastian Eggert, 1994THE UNIVERSITY OF BRITISH COLUMBIAIn presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)The University of British ColumbiaVancouver, CanadaDate ZSDepartment of y/AbstractWe calculate the effects of a single impurity in antiferromagnetic quantum spin-1/2 chainswith the help of one-dimensional quantum field theory and renormalization group techniques in the low temperature limit. We are able to present numerical evidence from exact diagonalization, numerical Bethe ansatz, and quantum Monte Carlo methods, whichsupport our findings. Special emphasis has been put on impurity effects on the localsusceptibility in the chain, because of the experimental relevance of this quantity. Wepropose a muon spin resonance experiment on quasi one-dimensional spin compounds,which may show some of the impurity effects.11Table of ContentsAbstract iiTable of Contents jjjList of Tables viList of Figures viiAcknowledgements xvI Introduction 11.1 The Hamiltonian 11.2 Impurities 21.3 Experimental Relevance . 51.4 Outline 62 Theoretical Background 82.1 From the Lattice Model to the Quantum Field Theory 82.2 Symmetries 122.3 Correlation Functions 133 Scaling and Finite Size Effects 153.1 Boundary Conditions 153.1.1 Periodic boundary conditions 153.1.2 Open boundary conditions 161113.2 Scaling and Irrelevant Operators3.3 Finite-size Spectrum3.3.1 Periodic boundary conditions3.3.2 Open boundary conditions182222254 Impurities 314.1 One Perturbed Link4.2 Two Perturbed Links4.3 Relation to Other Problems5.4.15.4.2Impurity Susceptibility Effects6.1.1 One weak link6.1.2 Two weak links4344455152565861636565693136385 Susceptibilities5.1 Periodic Chain Susceptibility5.1.1 Contributions from the leading irrelevant operator5.2 Open Chain Susceptibility5.2.1 Contributions from the boundary condition5.2.2 Contributions from the leading irrelevant boundary5.3 Susceptibility Contributions from Perturbations5.3.1 Two perturbed links5.3.2 One perturbed link5.4 A Muon Spin Resonance ExperimentExperimental SetupField Theory Analysis6 Monte Carlo Results6.1operator71717275iv6.1.3 Alternating Parts 766.2 Muon Knight Shift 806.2.1 One perturbed link 846.2.2 Two perturbed links 926.3 Conclusions 96A Field Theory Formulas 106B Exact Diagonalization Algorithm 109C Monte Carlo Algorithm 111Bibliography 115VList of Tables3.1 Low energy spectrum for periodic boundary conditions. Relative parityand total spin are given 243.2 Low energy spectrum for open boundary conditions. Relative parity andtotal spin are given 28viList of Figures1.1 An impurity breaks conformal invariance and renormalizes to a boundarycondition 31.2 An analytic continuation of left movers in terms of right movers to thenegative half axis effectively removes the boundary and restores conformalinvariance 43.3 Numerical low energy spectrum for periodic, even length 1 = 20 spin chain.The integer values El/irv of the numerically accessible states agree withthe theoretical predictions. The velocity vr = 3.69 was used (see figure3.5 ) 253.4 Numerical low energy spectrum for periodic, odd length 1 = 19 spin chain(vir = 3.69) 263.5 Renormalization group flow towards the asymptotic spectrum of the periodic chain. The lowest excitation gap 0, 1 is fitted to 1 E = a + b/i2 foreven lengths (a = 3.69, b = 3.94) 273.6 Numerical low energy spectrum for the open, even length 1 = 20 spin chain(v7r = 3.42) 293.7 Numerical low energy spectrum for the open, odd length 1 = 19 spin chain(vir = 3.42) 293.8 Renormalization group flow towards the asymptotic spectrum for the openchain. The lowest excitation gap E(i/Trv) is fitted to 1 E = a+b/l for botheven and odd length chains (a = 3.65, b = —4.6) 30vii4.9 A quantum spin chain with one altered link 324.10 Flow away from the periodic chain fixed point due to one altered link foran odd length chain with 7 i 23. The lowest excitation gap isfitted to IE = a 11/2, which is the predicted scaling 354.11 Renormalization group flow towards the open chain fixed point due to oneweak link for an odd length chain with 7 1 23. The corrections to thelowest excitation gap , is fitted to lE = a/I + b/i2, exhibiting thepredicted 1/i scaling up to higher order 364.12 A quantum spin chain with two altered links 374.13 Flow towards the periodic chain fixed point for two altered antiferromagnetic links. The , gap is fitted to lE = a/i’!2,which is the predictedscaling 384.14 Flow away from the open chain fixed point for two weak antiferromagneticlinks. Corrections to the , gap are fitted to E/E = (a+b/i+clnl),demonstrating relevant logarithmic scaling (ac > 0). The dotted line isthebestfitforc=0 394.15 Flow towards the open chain fixed point for two weak ferromagnetic links.Corrections to the t[ gap are fitted to /E/E = (a + b/i + clnl),demonstrating irrelevant logarithmic scaling (ac < 0). The dotted line isthe best fit for c 0 394.16 The equivalent spin chain model to the two impurity Kondo problem. . 415.17 (T) from the Bethe ansatz. (0) = 1/Jir2 is taken from equation (5.67) 485.18 Field theory [equation (5.76 ), T0 7.7J] versus Bethe ansatz results for(T) at low temperature 49viii5.19 Estimates for the effective coupling g from lowest order perturbation theory correction to the finite-size energy of ground-state, first excited tripletstate, first excited singlet state[19] and to the susceptibility, using 1 —* v/T.The renormalization group prediction of equation (5.75 ) is also shown. 505.20 The open ends of the broken chain are expected to be more susceptible. 525.21 The local susceptibility near open ends from Monte Carlo simulations for= 15/J 545.22 The uniform and alternating parts of the local susceptibility near open ends 555.23 Renormalization group analysis of the cross-over from an unstable to astable fixed point as the temperature is lowered 595.24 Schematic Muon Spin Resonance setup: One Muon at a time enters throughthe thin timer and stops inside the sample where its spin precesses. Decaypositrons are detected in the two counters 665.25 The location of the muon relative to the chain for the link parity symmetriccase 676.26 The open chain impurity susceptibility as a function of temperature. Thesolid line is only drawn for visual guidance and does not necessarily reflectan accurate estimate 736.27 The impurity susceptibility for a small coupling J’ across the open endsas a function of temperature 746.28 The impurity susceptibility for a small perturbation 6J of one link in thechain as a function of temperature. The solid lines are only drawn forvisual guidance and do not necessarily reflect an accurate estimate. . . 756.29 The impurity susceptibility for a small perturbation SJ of two links in thechain as a function of temperature 77ix6.30 The local susceptibility correction of the central spin closest to the impurity for a small perturbation 6J on two links in the chain as a function oftemperature 776.31 The impurity susceptibility for a small coupling J’ of the open ends to animpurity spin as a function of temperature 786.32 The local susceptibility of an impurity spin coupled with a small perturbation J’ to the open ends of the chain as a function of temperature. . 786.33 The local susceptibility as a function of distance from a weakened linkJ’=0.75JatT=J/15 796.34 The alternating part of the local susceptibility as a function of distancefrom the weakly coupled link J’ across the open ends at T = J/15.. ... 806.35 The local susceptibility as a function of distance with the open ends coupled with J’ = 0.1J to an impurity spin at the first site at T = J/15. . 816.36 The local susceptibility as a function of distance with the open ends coupled with J’ = 0.25J to an impurity spin at the first site at T = J/15. . 816.37 The local susceptibility as a function of distance with the open ends coupled with J’ = 0.5J to an impurity spin at the first site T = J/15 826.38 The local susceptibility as a function of distance from two slightly weakened links J’ = 0.75J at T = J/15 826.39 The effective normalized susceptibility in a powdered sample for smallperturbations on one link and d± = 0.5 as a function of temperature. . 846.40 The effective normalized susceptibility in a powdered sample for smallperturbations on one link and d± = 1 as a function of temperature. . . 856.41 The effective normalized susceptibility in a powdered sample for one strengthened link and d± = 0.5 as a function of temperature 86x6.42 The effective normalized susceptibility in a powdered sample for one strengthened link and d± = 1 as a function of temperature 876.43 The effective normalized susceptibility in a powdered sample for strongperturbations on one link and d± = 0.5 as a function of temperature. . 876.44 The effective normalized susceptibility in a powdered sample for strongperturbations on one link and d± = 1 as a function of temperature. . . 886.45 The effective normalized susceptibility for an applied field perpendicularto the chain, small perturbations on one link and d± = 0.5 as a functionof temperature 896.46 The effective normalized susceptibility for an applied field perpendicularto the chain, small perturbations on one link and d± = 1 as a function oftemperature 896.47 The effective normalized susceptibility for an applied field perpendicular tothe chain, one strengthened link and d = 0.5 as a function of temperature. 906.48 The effective normalized susceptibility for an applied field perpendicularto the chain, one strengthened link and d± = 1 as a function of temperature. 906.49 The effective normalized susceptibility for an applied field perpendicularto the chain, strong perturbations on one link and d± = 0.5 as a functionof temperature 916.50 The effective normalized susceptibility for an applied field perpendicularto the chain, strong perturbations on one link and d± = 1 as a function oftemperature 916.51 The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on one link and d± = 0.5 as a function oftemperature 92xi6.52 The effective normalized susceptibility for an applied field parallel to thechain, one strengthened link and d± = 0.5 as a function of temperature. 936.53 The effective normalized susceptibility for an applied field parallel to thechain, strong perturbations on one link and d± = 0.5 as a function oftemperature 936.54 The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on one link and d± = 1 as a function of temperature 946.55 The effective normalized susceptibility for an applied field parallel to thechain, one strengthened link and d1 = 1 as a function of temperature. . 946.56 The effective normalized susceptibility for an applied field parallel to thechain, strong perturbations on one link and d± = 1 as a function of temperature 956.57 The effective normalized susceptibility in a powdered sample for smallperturbations on two links and d± = 0.5 as a function of temperature. . . 966.58 The effective normalized susceptibility in a powdered sample for smallperturbations on two links and d± = 1 as a function of temperature. . . 976.59 The effective normalized susceptibility in a powdered sample for strongperturbations on two links and d1 = 0.5 as a function of temperature. .. 976.60 The effective normalized susceptibility in a powdered sample for strongperturbations on two links and d± = 1 as a function of temperature. . . . 986.61 The effective normalized susceptibility in a powdered sample for two strengthened links and d± = 0.5 as a function of temperature 986.62 The effective normalized susceptibility in a powdered sample for two strengthened links and d1 = 1 as a function of temperature 99xii6.63 The effective normalized susceptibility for an applied field perpendicularto the chain, small perturbations on two links and d1 = 0.5 as a functionof temperature 996.64 The effective normalized susceptibility for an applied field perpendicularto the chain, small perturbations on two links and d± = 1 as a function oftemperature 1006.65 The effective normalized susceptibility for an applied field perpendicular tothe chain, two strengthened links and d1 = 0.5 as a function of temperature. 1006.66 The effective normalized susceptibility for an applied field perpendicularto the chain, two strengthened links and d± = 1 as a function of temperature. 1016.67 The effective normalized susceptibility for an applied field perpendicularto the chain, strong perturbations on two links and d± = 0.5 as a functionof temperature 1016.68 The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d = 0.5 as a function oftemperature 1026.69 The effective normalized susceptibility for an applied field parallel to thechain, two strengthened links and d± = 0.5 as a function of temperature 1026.70 The effective normalized susceptibility for an applied field parallel to thechain, strong perturbations on two links and d1 = 0.5 as a function oftemperature 1036.71 The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d± = 1 as a function of temperature 1036.72 The effective normalized susceptibility for an applied field parallel to thechain, two strengthened links and d± = 1 as a function of temperature. . 104xiii6.73 The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d± = 1 as a function of temperature 104XWAcknowledgementsI would like to give my special thanks to my advisor, Ian Affleck for his patience inmany long and helpful discussions. Without him and his extraordinary ability to passon his vast knowledge this thesis would not have been possible. I am also very gratefulfor interesting discussions with Eugene Wong, Rob Kiefi, Bill Buyers, and Philip Stampwhich were helpful in preparing this thesis. Special thanks also go to a number of peoplein the physics department with whom I had the pleasure to interact with in the pastyears: Junwu Gan, Arnold Sikkema, Jacob Sagi, Gordon Semenoff, Erik Sørensen, BirgerBirgerson, and Michel Gingras.At this point I would also like to acknowledge some of my former advisors, mentors,and teachers which made a special contribution in my course of studies: Herr Unger, EbsHilf, Alexander Rauh, Glen Rebka, Lee Schick, and Ramarao Inguva.xvChapter 1IntroductionConsiderable attention has been focused on spin-1/2 chains since Bethe’s original workmore than 60 years ago[1}. The large interest in these relatively simple many-body quantum mechanical systems is no surprise, since they exhibit many fascinating cooperativephenomena which may be shared by more complex models. The Bethe ansatz has beenrefined over the years[2], and thermodynamic quantities can be calculated exactly for awide parameter range [3]. With the advent of conformal field theory and more computingpower, we are now able to understand the model even on a more detailed level as presented in this thesis. Another goal of this thesis is to link this theoretical knowledge toreal experimental systems with special emphasis on impurity effects.1.1 The HamiltonianWe can model the magnetic properties of insulators very well by describing the exchangecoupling between orbital spins in terms of an anisotropic Heisenberg coupling. It ispossible to have quasi one-dimensional spin systems, in which the spins form “chains”along one crystal axis in the sense that the exchange coupling is much stronger betweenneighboring spins within the chain compared to the coupling J1 between spins of different chains. If we neglect this interchain coupling we can describe the model by theHamiltonian1—1 j+ JSS+1], (1.1)1=11Chapter 1. Introduction 2where S, S are the usual spin-1/2 raising and lowering operators at site i, 1 is the totalnumber of sites, and J is taken to be positive. We may choose open boundary conditionswhere the ends at the and 1th site are free, or we may impose periodic boundaryconditions where the ends are coupled with the same coupling constants, J and J.Some materials are known to exist for spin-1/2 which exhibit this one-dimensionalbehavior to various degrees (e.g. KCuF3[4] and CPC[5j). The ratio J/J is a measureof the one-dimensional properties of the material since three dimensional Néel orderingwill occur for low temperatures T < TN, TN cx J [6]. In some materials a spin-Peierlstransition to a dimer phase may occur instead if the phonon-spin coupling is strong.Typically the exchange coupling J is of the order of 20 — 1000K, while the orderingtemperature Tr is at least one order of magnitude smaller. Experimental results inKCuF3[4] and CPC[5] are reported to agree well with the prediction of the Hamiltonianin equation (1.1) at the isotropic point J J. Both J and J arise from an exchangeintegral since the dipole-dipole interaction is only in the mK range. We also neglectedthe spin-orbit coupling which is generally also much smaller than the exchange coupling.The effect of a spin-orbit coupling can be described by a single-ion anisotropy of the form(Sz)2 in the Hamiltonian, which reduces to a trivial c-number for spin-1/2.1.2 ImpuritiesThe main goal of this thesis is to provide a good understanding of impurity effects inspin-1/2 chains. The study of impurities has always been a large part of solid statephysics, because there are many cases where impurities produce very interesting effectsand may even dominate the behavior of the system. The best known examples may besemiconductor doping, the Kondo effect, and high temperature superconductors.Chapter 1. Introduction 3bulkv/TK.jboundaryFigure 1.1: An impurity breaks conformal invariance and renormalizes to a boundarycondition.Recently, there has been an increased theoretical interest in quantum impurity problems which can be described by (1+ 1) dimensional conformal field theories. The resultingtheory of boundary critical phenomena proved to be very successful in treating a variety of problems[8], with the Kondo problem being probably the most famous. It turnsout that our model system can be regarded as one example of this technique, so it isinstructive to present the central ideas of this approach at this point (see also reference[8]).Let us start with some gapless, scale invariant system that can be described by aconformally invariant field theory. We may introduce a local, time-independent perturbation as shown in figure 1.1, which represents the impurity in the system and breaks theconformal invariance. It also creates a new energy scale in the system, which depends onthe initial strength of the perturbation and on the scaling dimensions of the perturbingoperators in the field theory. This energy scale is defined as the temperature where weexpect a breakdown of perturbation theory, which is often called TK in analogy with theKondo effect. It is reasonable to assume that the system will still be described by theconformal field theory far away from the impurity, i.e. outside a “boundary layer” whichis defined by the new energy scale in the system i.e. with width v/TK, where v is theIIChapter 1. Introduction 4><R L LLboundary “no boundary”Figure 1.2: An analytic continuation of left movers in terms of right movers to thenegative half axis effectively removes the boundary and restores conformal invariance.effective speed of light of the field theory.The system outside the boundary layer may still be affected by the impurity in auniversal way, however, since it may effectively introduce a boundary condition on thesystem. The effective boundary conditions are created quite naturally, because the usualrenormalization group ideas of perturbations in scale invariant systems apply. We expecta relevant impurity perturbation to renormalize from a weak coupling limit to a strongor intermediate coupling limit as the temperature is lowered. The weak coupling limitrecovers the original boundary condition of the unperturbed system, while the strong(infinite) coupling limit can most likely be described by some other (e.g. fixed) boundarycondition as indicated in figure 1.1. In this case we expect to find universal correlationfunctions for points close to the impurity compared to their relative distance (but outsidethe boundary layer v/TK as shown in figure 1.1). These boundary correlation functionsare in general different from the correlation functions in the bulk. The cross-over temperature between the two boundary conditions is simply given by the original energy scaleTK that has been created by the perturbation.One important point of the theory is the fact that a fixed boundary condition is stillconsistent with conformal invariance, although it seems to break translational invariance.As an example consider a fixed boundary condition where some quantum field has beenset to zero (O) = L(O) + (°) 0. An analytic continuation to the negative half axisChapter 1. Introduction 5of the left moving field in terms of the right moving field allows us to effectively get rid ofthis boundary condition for the newly defined left moving field qL(—x) (x), x > 0as shown in figure 1.2.Although we will not use the theory of boundary critical phenomena to its full extent,we will recover the same results in our analysis of impurities. The reader may understandsome of the presented ideas better once they are explained with the example of the spin-1/2 chain later in this thesis.As will be shown in chapter 4, we can understand the effect of impurities in thesesystems very well with the help of the field theoretical analysis. In all cases, we find thatany impurity renormalizes to an effective boundary condition on the bulk system at zerotemperature. An effectively decoupled spin may be left over and there might be impuritycorrections to thermodynamic quantities. While extensive quantities generally scale withthe size of the system, the impurity contributions are independent of the length 1 of thechain. These findings are analogous to those of the Kondo effect to some extent.1.3 Experimental RelevanceTo detect these impurity effects in experimental systems, we have to overcome somedifficulties. Since the predicted corrections to thermodynamic quantities scale with theimpurity density, we will generally need a macroscopic number of defects, and even then itwill be difficult to extract the part of the signal which is due to the impurities. Moreover,we expect that impurities will affect each other in a strongly correlated system like thespin-1/2 chain[7J. Instead of making a global measurement on thermodynamic quantities,we would therefore ideally like to make a measurement only locally, close to an isolatedimpurity. In this case, we expect a strong effect since the impurity will effectively play therole of a boundary condition on an otherwise unperturbed system at low temperatures.Chapter 1. Introduction 6Correlation functions will be changed drastically in this case as we will see later.Out of the motivation to make a local measurement and perturbation, we developedthe idea of a Muon Spin Resonance (tSR) experiment on quasi one-dimensional spincompounds. In this case the electric charge of the muon creates a defect in the material,while the muon also makes a measurement of the local susceptibility in its vicinity. Theidea of the experimental setup will be discussed in more detail in section 5.4.Even assuming that we are able to create an idealized impurity system, we stillhave the serious problem that our field theory predictions are strictly valid only at lowtemperatures where experimental materials might already behave three dimensionally.This problem can be overcome only to some extent by selecting materials that have verypronounced one-dimensional behavior (i.e. a large ratio J/Jj.To give a more complete prediction of the outcome of the 1iSR experiments andto link theoretical calculations to the experimentally accessible temperature range, weperformed extensive quantum Monte Carlo simulations. We can recover the predictedscaling at low temperatures, which we can link to the predicted experimental signal athigher temperatures. This gives some very encouraging results for the possible iSRexperiment. The presented setup for the ,uSR experiment is of course only one possibleway of detecting the predicted effects of impurities, which will be present in any quasione-dimensional compound. This thesis will provide some interesting Monte Carlo datafor the local susceptibility near an impurity. The reader is encouraged to use this datato develop other experimental methods to probe the predicted effects.1.4 OutlineThis thesis is organized as follows: A review of the derivation of the quantum field theorytreatment for spin-1/2 chains is given in chapter 2 which is largely based on previousChapter 1. Introduction 7references. We are able to extend this analysis to derive some results for finite sizesystems in chapter 3. Some renormalization group ideas will also be presented in chapter3 in connection with finite size scaling. We use the field theory treatment to study theeffects of impurities in the chain as discussed in chapter 4 in some detail, which is basedon some of my previous work with Ian Affleck in reference [9]. Some finite size scalingresults from numerical exact diagonalization studies are also presented to confirm ourresults.The most recent results of our field theory analysis are predictions for the local andthe bulk susceptibility of the spin chain in chapter 5. The impurity contributions tothe susceptibility will be discussed in the language of boundary critical phenomena asdescribed above. In the last chapter 6, we present the promising data from our MonteCarlo simulations, which also establishes our predictions for the 1tSR experiments. Theresults are discussed in the context of the expectations from the field theory.Chapter 2Theoretical BackgroundTo establish our notation, we will review the field theoretical treatment of the spin-1/2chain in this chapter. While we attempt to give a complete outline, it might be necessaryto refer to reference [10] or appendix A in some cases, since it is not the primary goal ofthis thesis to present research on this aspect. We consider the antiferromagnetic spin-1/2xxz chain with 1 sites, which is described by the Hamiltonian in equation (1.1).2.1 From the Lattice Model to the Quantum Field TheoryWe first apply the Jordan-Wigner transformation by expressing the spin operators interms of spinless fermion annihilation and creation operators at each site[11]:S =SI = (-1)exp(i) (2.2)The exponential string operator cancels for nearest neighbor interactions on a chain, andwe are left with a local Hamiltonian for interacting Dirac fermions by direct substitutioninto equation (1.1):H= (+ + h.c.) + Jz - - )] (2.3)For J = 0, this is just a Hamiltonian for free fermions on a lattice. For this case,we obtain a cosine dispersion relation, and the ground state is a half-filled band withthe Fermi points at kf = ±7r/2. Expanding around this ground state, we can restrict8Chapter 2. Theoretical Background 9ourselves to low energy excitations by only considering those fermions which have wave-vectors close to kf =(x) e12bL(x) +e2R(x) (2.4)The coordinate x is measured in units of the lattice spacing, and L and bR contain onlylong wavelength Fourier modes.We now take the continuum limit, and, up to terms with higher order derivatives,we are left with a (1+1) dimensional relativistic field theory of left- and right-movingfermions. The resulting Hamiltonian for the case J,,, = 0 is:H = vfdx (2.5)The Jr-interaction can be reintroduced in terms of the fermion currents Jj = :4t’1,I = L, R by use of equation (2.4):J : : : J fdx[J! + 4+ 4JLJR — {(: 4R :)2 + h.c.}} (2.6)Because of Fermi statistics, we can drop the last term for now. The first two terms canbe rewritten to first order with the help of ‘Wick’s formula:JL(x)JL(x+6) :JL(x)JL(x): +const.+ —[4(x + 6)bL(x)—4(x)’ibL(x + 6)]i d+ const.K dxiJR(x)JR(x+6) —b—bR + const. (2.7)rith the use of those relations, we can rewrite the complete Hamiltonian in terms of theFermion currents with a renormalized “speed of light” v:H = vKfdx[4+J+-JLJR] (2.8)Chapter 2. Theoretical Background 10This model can now be transformed using the usual abelian bosonization rules[i0]:1JL =1JR = -=(fl +1/JR = const.exp(i1/R)1/iL = const.exp(—iv’cbL), (2.9)where the constant of proportionality can be taken to be real, but cut-off dependent.The fields cL and /iR are the left and right-moving parts of q which can be defined in aninfinite system as1L(X) = x)+fH(y)dy1 is= (x)-fH(y)dy, (2.10)where f1, is the momentum variable conjugate to . Left moving operators are functionsof only x + vt, while right moving operators are functions of only x — vt. A dual fieldcan also be defined in terms of those components:L/iR (2.11)The resulting Hamiltonian is a non-interacting boson theory= v [(1 — 2JZ)112 + (1 + 2JZ)()2]. (2.12)However, the boson operators now have to be transformed by a canonical transformationto obtain a conventionally normalized theory:fl, —* fl (2.13)2 I7rv+2J 1R = —/ —+ . (2.14)4?r V 7rv — 2J 4w 2virChapter 2. Theoretical Background 11This gives us the usual free boson Hamiltonian:= + ()] = v [TL + TR]. (2.15)Here TL,R are the left- and right-moving parts of the free HamiltonianTR,L(ORL)2 =. ( ± (2.16)By combining the spin to fermion and fermion to boson transformations, we obtainthe continuum limit representation for the spin operators:1 OSi— + (—1) const. cosSJ cx e_i27 [cos () + const.(_1Y] (2.17)Altogether, this is a very nice result, because we are now in the position to calculate anyexpectation value of spin operators in terms of free boson Green’s functions.Note, that all physical operators are invariant under a shift of the boson+2irR+ 1/R. (2.18)Therefore, the boson must be thought of as a periodic variable measuring arc-lengthon a circle of radius R.So far, we have treated the J interaction perturbatively so that the rescaling equations (2.14) are only accurate to lowest order in J/J. Fortunately, the “boson radius”R and the “spin-wave velocity” v have been analytically determined with the help of theBethe ansatz[12, 13]. After defining a new variable 8,JzcosO—-(2.19)Chapter 2. Theoretical Background 12the two quantities are conveniently expressed asJirsinO= 20R= -(2.20)which agrees to first order in J/J r/2 — 6 with the perturbative field theory calculations in equation (2.14).2.2 SymmetriesThere are two independent discrete symmetries of the spin chain which we can identify inthe continuum limit. The first one is translation by one site, T. This appears as a discretesymmetry independent of translation in the continuum limit, simply interchanging evenand odd sublattices. By comparing with equation (2.17) we see that it corresponds to:T: —* + rR, T: —* + 1/2R. (2.21)The second one is site parity, Ps, i.e. reflection of the whole chain about a site. Note,that this does not interchange even and odd sub-lattices. Thus it must map the spinoperators into themselves. Since parity interchanges left and right, and g transformoppositely. We see that the correct transformation isPs : ‘—,Ps : ‘ . (2.22)There is a third discrete symmetry, link parity, PL, i.e. reflection about a link.However, this is not independent, but is a product of Ps and T. It corresponds toPL :—i —+irR, PL :q—*q+1/2R. (2.23)Chapter 2. Theoretical Background 132.3 Correlation FunctionsOne of the first[13] and most important results of the field theory treatment is the calculation of spin correlation functions, which is not possible with Bethe ansatz methods.Using equation (2.17) and some results from appendix A it is straight forward to expressthe SZ Green’s function asG2(x,t) <S(O)S(t) >1 Oc(O,O)Oq(x,t) (O,O) (x,t)= 4ir2R > + const.(_1)x <cosRcosR>= 163R2 ((X +vt)2+ ( t)2) + const. (2 22)1/4R• (2.24)The separation into uniform and alternating parts is taken from equation (2.17), whichalso implies that the spin operators can be separated into uniform and alternating parts.This seems to be a valid assumption in the long wave-length limit, but the separationis not unique on small length scales. Note, that the cross terms of the alternating anduniform parts of S in equation (2.17) have a vanishing expectation value as they should.In a scale invariant system we can define a scaling dimension d dL + dR of an operator0 = OLOR by the auto-correlation function<O(x, t)O(O, 0) >=< OL(X+Vt)OL(0) >< OR(XVt)OR(0)> x + Vt2dL x — vt2dR(2.25)According to equation (2.24) the scaling dimension of the uniform part of the SZ operatoris always one, while the exponent of the alternating part decreases with anisotropy. Atthe isotropic point the alternating scaling dimension is d = 1/2, while we recover d = 1at the xx point (free fermions).Likewise, we can calculate the S± Green’s function:G(x,t) < S(0)S(t) > (2.26)Chapter 2. Theoretical Background 14oc (x — v2t2 )_(1/R_2R)/4(( +vt)2 + ( ) + const. (x2 v2t)RNow the uniform scaling dimension decreases from d = 5/4 at the xx-model, to d = 1 atthe Fleisenberg point, while the alternating dimension increases from 1/4 to 1/2. At theHeisenberg point, the expressions for the two Green’s functions GZ and G± are identical,as expected.The scaling dimensions at the xx-point (free fermions) agree with previous results fromrigorous methods[14]. Extensive numerical studies at the Heisenberg point show that thepredicted exponents are correct there as well[15, 16] up to logarithmic corrections. Theconstant of proportionality of the alternating part in equation (2.24) has been estimatednumerically to be const. O.5[16].Chapter 3Scaling and Finite Size EffectsSo far we have treated the spin chain with a theory which used the implicit assumptionthat we are in the limit of infinite length and very low temperatures. It is now useful toextend this theory to make useful predictions on finite size systems.3.1 Boundary ConditionsTo identify possible fixed points, we need to uncover the corresponding boundary conditions on the boson in the continuum limit.3.1.1 Periodic boundary conditionsTo get periodic boundary conditions, we can define g0 and let the sum in equation (1.1) run from 0 to 1. For the fermions, this condition translates into periodic orantiperiodic boundary conditions, depending on the total number of fermions[11]. It isclear from equation (2.17) that the boundary conditions on the boson are given byq5(l) = (0) + 2IrRSZ(l)=(O) + m/R, (3.27)where m and SZ have to be integer for even length 1 and half-odd-integer for odd length 1.We can identify $Z to be the z-component of the total spin by integrating equation(2.17):S = -((l)—q(0)) (3.28)15Chapter 3. Scaling and Finite Size Effects 16As expected, S is integer or half-odd-integer for an even or odd length chain, respectively.There is no immediate physical interpretation for m other than that it represents aconserved quantity with integer or half-odd-integer value [see also equation (3.42) later].3.1.2 Open boundary conditionsThe case of free ends is slightly more subtle. One way of dealing with it is to introducetwo additional “phantom sites” at 0 and 1+1 and let the sum of the first term in equation(2.3) run from 0 to 1, and then impose vanishing boundary conditions on b0 andThis imposes conditions on the continuum limit left and right moving Fermion fields:= 0bL(l + 1) + (—1)’bR(l + 1) = 0 (3.29)Using equations (2.4) and (2.9) and taking into account the correct commutation relationsin equation (A.106), we conclude that the correct boundary conditions on the bosons are(O) = irR/2(l + 1) = irR/2 + 271R8z, (3.30)where SZ is integer for 1 even or half-odd-integer for 1 odd. As expected, this conditionis not compatible with site (link) parity for an even (odd) number of sites.At first sight, these conditions do not seem to correspond to conformally invariantboundary conditions because they break translational invariance, but we can rewritethem in terms of left- and right-movers75L(0,t) = 7rR/2—q5R(0,t). (3.31)Since c5L is a function only of x + vt and only of x — vt, we can define qj for negativevalues of x by regarding q as an analytic continuation:L(X,t) —j?(x,t) + 7rR/2, x > 0. (3.32)Chapter 3. Scaling and Finite Size Effects 17The condition at 1 + 1 then becomesL(l + 1, t) = —cR(l + 1, t) + irR/2 + 2KRSZ = — 1, t) + 27rRSZ. (3.33)We therefore recover the usual periodic or antiperiodic boundary conditions, dependingon whether 1 is even or odd. This is in complete agreement with the discussion in section1.2 and figure 1.2. The right moving channel c’R has been replaced by an analyticalcontinuation of the left moving field ç to the negative half axis. Since L has now twicethe range 21 we have the same degrees of freedom as before, but the left moving field hasthe usual periodic (conformally invariant) boundary conditions. It appears as if we havegotten rid of the fixed boundary condition altogether.One may ask at this point how the boundary correlation functions can be differentafter we have effectively recovered periodic boundary conditions and translational invariance. The reason is that all physical operators that were previously expressed in termsof left and right movers are now written in terms of left-movers only. The spin operatorstherefore become non-local expressions because they will be a function of both L(x) and5L(—x). To understand the effect on the boundary scaling dimensions, it is instructive toconsider the staggered part of the spin-spin correlation function at the Heisenberg pointas an example. This is most easily calculated for S— by usingS(x) cc (—1 (3.34)The two-point Green’s function for < SS+ > now becomes a four-point function for theleft-moving boson, giving, according to equation (A.105):x12<S(t1,x). S(t2,x)>cc (_1)x2[(— x2) — t?2][(xi + x2)—t2j’ (3.35)where we have set the spin-wave velocity to one and t12 t1 — t2. Note, that farfrom the boundary, when x12 >> I(xi — x2) — t, we recover the bulk correlationChapter 3. Scaling and Finite Size Effects 18function 1//(xi— x2) — t2, corresponding to a scaling dimension of d = dL + dR =1/2 for the staggered spin operator. This also fixes the constant of proportionality inequation (3.35) to be const. 2[16] as mentioned at the end of section 2.3. However,the correlation function near the boundary (i.e. when 1t2 >> x1,x2) takes the form/T/It12I2,corresponding to a scaling dimension of d = 1 for the staggered boundaryspin operator. In this case the scaling dimensions of the original left and right movers nolonger add, since they are no longer independent as x1,x2 —* 0. In this case the differentscaling dimension can formally be derived by the operator product expansion[17, 18].3.2 Scaling and Irrelevant OperatorsAlthough we were able to arrive at a free Hamiltonian in equation (2.15), it is importantto realize that we neglected all terms which involved higher order derivatives or powersof fermions. These terms are irrelevant at low temperatures and long wavelengths, butthey will give some corrections with characteristic scaling relations.We can study these corrections systematically by classifying operators in the Hamiltonian density by their scaling dimension. We see that the free Hamiltonian density has ascaling dimension of d = 2 as it should since its integral has to have units of energy. Thisis assuming that in a scale invariant theory the scaling dimension d in equation (2.25) isthe only quantity that determines the units of the corresponding operator. If we wantto consider perturbing operators with scaling dimension other than d = 2, we need toconsider that this operator must contain the appropriate powers of the ultraviolet cutoffA in so that its overall units work out to that of the Hamiltonian density. We may chooseto define a dimensionless coupling constant .XAd_2 by absorbing the appropriate powersof the cutoff. The renormalized coupling constant of an operator with scaling dimensiond 2 is therefore proportional to )A’2, where A is the original coupling parameter inChapter 3. Scaling and Finite Size Effects 19the Hamiltonian density. We therefore conclude that operators with dimension d > 2are irrelevant when the cutoff is lowered, while operators with d < 2 will be relevant. If relevant operators are present we expect a breakdown of perturbation theory, andthe system either develops a mass gap or renormalizes to a different fixed point. Theultraviolet cutoff A may be reduced to the larger of the temperature T or the inversesystem size v/i. We expect that this results in an effective Hamiltonian that describesthe macroscopic physics correctly and oniy depends on the energy scale T or v/i and therenormalized coupling constants.Since the coupling constants always appear in the combination )Ad_2 it is sufficientin most cases to only consider the perturbing operator with the lowest scaling dimensiond (the “leading” operator). This determines the leading correction to the spectrum andother quantities which will be proportional to T”2 or 12—d to first order in perturbationtheory. This can be generalized to higher orders in )A2 if higher order perturbationtheory should be necessary to calculate the corrections.We still need to consider the special case of perturbing operators with scaling dimension d = 2, which are marginal and can be categorized into three different cases.Sometimes we may absorb the operator exactly into the free Hamiltonian as we didwith the J interaction in equations (2.6-2.8). We then refer to the operator as exactlymarginal. If we cannot absorb the operator in the free Hamiltonian, we can calculatethe rescaling equations from perturbation theory. If we have only one marginal couplingconstant .\, the so called a-function has the generic formdlogA = (3.36)where b is determined by the perturbation expansion. We see that whether the perturbation is relevant or irrelevant depends on the sign of b and ). In particular, assumingChapter 3. Scaling and Finite Size Effects 20b is positive, we see that A decreases when the cutoff is lowered, making a positive coupling A marginally irrelevant and a negative coupling A marginally relevant. It thereforedepends on the initial sign of the bare coupling constant if the perturbation is relevantor irrelevant. Integrating equation (3.36) givesA—1—AoblogA’ (3.37)where A A(A = 1). For the irrelevant case A0b > 0, the renormalized coupling Abecomes smaller when the cutoff is lowered and “universal” logarithmic corrections oforder —1/binA arise which are independent of A0 as A —+ 0. If A0b < 0, however,the perturbation is relevant and we expect a breakdown of perturbation theory whenA0blnA —* 1 (i.e. TK cx e11° in terms of the cross-over energy scale).Since the above arguments rely on the dimensional analysis of the operators, we canimmediately deduce that a 6-function increases the scaling dimension by one. Thereforelocal operators are regarded to be marginal for d = 1 and irrelevant for d> 1. Likewisea derivative will always increase the effective scaling dimension of operators by one.Let us study these renormalization group concepts with the example of the spin-1/2chain. One perturbing operator comes from the last term in the J interaction of equation(2.6), which represents an Umklapp process for the fermions. We expect this to be theleading irrelevant operator, because it is the only four-Fermion operator, which we havenot taken into account which does not include higher derivatives. After direct substitutionof the bosonization formulas in equation (2.9) and the rescaling in equation (2.14), thisoperator is given by A cos(2/R) with some non-universal coupling constant A. Accordingto equation (A.104) its scaling dimension is given by d = 1/7rR2, which decreases withJ and becomes d = 2 at the isotropic (“Heisenberg”) point J = J, R =corresponding to a marginal irrelevant operator. For J > J, the operator will be relevantand drive the system into the Néel ordered phase.Chapter 3. Scaling and Finite Size Effects 21Since the operator is marginally irrelevant at the Heisenberg point, the effective coupling constant )‘eff(l) scales to zero only logarithmically slowly with length 1, and logarithmic corrections arise[19]. This seems to make an accurate determination of the criticalbehavior from numerical finite-size scaling essentially hopeless, unless exponentially largechains can be studied. However, it is known from numerical studies that the marginalcoupling constant ). can be decreased by adding a positive next-nearest-neighbor couplingH—+ + JSi . S+2). (3.38)The coupling ) passes through 0 at a critical point, which has been estimated numericallyto be at J2/J 0.24[20]. For larger J2/J the operator is marginally relevant and thesystem renormalizes to a spontaneously dimerized phase. In particular, at J2/J = 1/2,the exact ground-states are the nearest neighbor dimer states. Right at the critical point,the marginal operator is absent, and hence finite-size scaling becomes very accurate evenwith chains of modest lengths 1 < 30 since corrections drop off at least as fast as 1/1. Themodel with the critical value of J2/J represents the critical point to which the nearestneighbor model and all models with J2 less than the critical value flow logarithmicallyslowly under renormalization. Therefore, we expect the behavior to be the same for thenearest neighbor model up to logarithmic corrections.Now that we have managed to get rid of the leading irrelevant operator, we can consider higher order corrections. Since translational symmetry is broken for open boundaryconditions, the local energy boundary operators TL(0) and TL(l) are allowed as a perturbation with some unknown coupling constant cc TL(0) = C (aL)2(0) = ((o) - H(O)), (3.39)with an analogous expression for TL(l). Here, we have also used the analytic continuationin equation (3.32), which allows a description in terms of left-movers only. Local operatorsChapter 3. Scaling and Finite Size Effects 22are multiplied by a 5-function in the Hamiltonian density and will therefore be marginalfor scaling dimension d = 1 and relevant for d < 1. The operator in equation (3.39) hasdimension d = 2 for all values of J and should give corrections of order 1/i to the finitesize spectrum, which will be discussed in the next section.There is no such local operator for periodic boundary conditions, and the lowestdimension bulk operator is TLTR of dimension d = 4, where TL,R have been defined inequation (2.16). Corrections to the finite size spectrum of the periodic chain shouldtherefore be at least of order 1/12. Note that the operator cos 4q/R is also allowed by theoriginal symmetries of the Hamiltonian, but its scaling dimension of d = 4/?rR2 makes itmuch more irrelevant for all values of J.3.3 Finite-size SpectrumAs discussed above, there are four different fixed points to consider, corresponding to thefour different possible boundary conditions: periodic or open with even or odd length(i.e. SZ integer or half-odd-integer).3.3.1 Periodic boundary conditionsWe first consider the case of periodic boundary conditions on a spin-chain. This implies periodic boundary conditions on as in equation (3.27) and determines the modeexpansion:(x, t) = o + + + )__[e_ta + e_t_a + h.c.]. (3.40)This implies that has the mode expansion(x,t)= + + f1 + {e_ta — e_ t_X)a + h.c.]. (3.41)Chapter 3. Scaling and Finite Size Effects 23The a’s are bosonic annihilation operators. fi and are canonically conjugate to theperiodic variables and,respectively. Hence their eigenvalues are quantizedfT = m/R, = 27rRSz, (3.42)with SZ and m given in equation (3.27). Note, that is also periodic with radius 1/27rR,as already mentioned in equation (2.18). The Hamiltonian can now be written as[21]H= j ‘Hdx = j1 [n + () 2]=+ç + , (3.43)with the resulting excitation spectrumE = [ (2 (SZ)2 + 2R2) + n(m + me)]. (3.44)The corresponding wave-function isei(8z20+m0) fl(at)m (at)m 0> (3.45)We see from equation (2.22) and (3.40) that site-parity takes m —* —m and m —* m.It also multiplies the wave-function in equation (3.45) by e(+m).Here and in what follows, we always measure parity relative to that of the ground-state. The ground-state parity itself for an even length chain is (_1)h/2. At the pointJZ= 0, R = 1/’, this spectrum is that of free fermions with anti-periodic (periodic)boundary conditions for even (odd) particle number. At the Heisenberg point J =J, R = 1/’, the spin of left and right-movers is separately conserved and the zcomponents are given by(3.46)are either both integer or both half-odd-integer for even length 1. For odd lengthone quantum number is half-odd-integer while the other one is integer valued (i.e. SSis half-odd-integer valued).Chapter 3. Scaling and Finite Size Effects 24[ - Even periodic Odd PeriodicL7íi+ i+i_2 ‘21 O,1—2 1, 1 2 x (c), 2 x () r3 O, 0, 1, 14 2 x (Of), 0-, 1, 2 x (lj, 2 4 x (c), 4 x (p), 3 x (r) 3 x (fl5 2 x (0), 0, 2 x (lj, 3 x (lj, 2, 2Table 3.1: Low energy spectrum for periodic boundary conditions. Relative parity andtotal spin are given.The energy can then be written asE = [(Sfl2 + ($)2+n(m + rn)j. (3.47)This spectrum has SU(2)L x SU(2)R symmetry for this value of R. Note, for instance,that for even 1 the lowest four excited states have total spin quantum numbers (SL, SR) =(1/2, 1/2), corresponding to a degenerate triplet and singlet under diagonal SU(2). Wecan take higher values of SL and S and can always find degenerate states to group theexcited states into SU(2) multiplets. It is useful to divide the spectrum into four sectorscorresponding to SL integer or half-odd-integer and 8R integer or half-odd-integer. Wethe write (SL,SR) = (Z,Z) + (Z + 1/2,Z + 1/2) for 1 even and (Z,Z + 1/2) + (Z +1/2, Z) for 1 odd where Z represents the integers. Parity interchanges all left and rightquantum numbers and multiplies wave-functions by (—1) in the (Z + 1/2, Z + 1/2) and(Z + 1/2, Z) sectors. Although periodic boundary conditions for even or odd lengthchains give identical equations (3.40 - 3.47), we can clearly distinguish two different fixedpoints with different excitation energies for the two cases, S’ integer or half-odd-integer.The states of the first six energy levels have been worked out in table 3.1 for the periodicchain with even and odd length 1 at the Heisenberg point. We can test the predictedspectrum numerically by exact diagonalization on a finite system with the algorithm inChapter 3. Scaling and Finite Size Effects 2554LU 210-0.5Figure 3.3: Numerical low energy spectrum for periodic, even length 1 = 20 spin chain.The integer values El/7rv of the numerically accessible states agree with the theoreticalpredictions. The velocity v = 3.69 was used (see figure 3.5).appendix B. To get rid of the logarithmic correction, we chose a next nearest neighborcoupling of J2 = 0.24J as discussed above. Figures 3.3 and 3.4 show the excellentagreement for all the states that were accessible with our algorithm (see appendix B).Moreover, we can see in figure 3.5 that for this choice of the Hamiltonian, the correctionsto the spectrum E(l/K)v drop off exactly as 1/12 as predicted in section 3.2 for periodicboundary conditions.3.3.2 Open boundary conditionsWe now turn to the case of free boundary conditions on the spins corresponding to fixedboundary conditions on g, as in equation (3.30). The mode expansion is now:(x, t) = 2irR ( + s) + ,L sin() [e nt/lan + h.c.j0 0.5 1 1.5 2 2.5Total Spin(3.48)Chapter 3. Scaling and Finite Size Effects 2676543LU2103Figure 3.4: Numerical low energy spectrum for periodic, odd length 1 = 19 spin chain(v7r = 3.69).with SZ integer (half-odd-integer) for 1 even (odd). The spectrum now takes the form[22]E=[27rR2 ($z)2 + nmn]. (3.49)These results can also be derived when we consider a single left-moving boson on twicethe range —l to 1 and periodic or antiperiodic boundary conditions as in equation (3.33).Note, that parity [i.e. x —* 1 — x for fixed boundary conditions or x—+ —x for thesingle boson] takes am —+ (1)fham. It also multiplies wave-functions by (_1)SZ for 1even (for odd 1 we only have site-parity, which does not change the phase of the wavefunction). ThusP = (_l)=om21+82 = (_l)ElPmP+(Sz) (3.50)for 1 even. For 1 odd, (Sz)2 — 1/4 is even, so we may write a similar formula:0 0.5 1 1.5 2 2.5Total SpinP =(_)lPmP+(S)2_1/4 (3.51)Chapter 3. Scaling and Finite Size Effects 273.763.753.743.73Ui3.723.713.703.690.016Figure 3.5: Renormalization group flow towards the asymptotic spectrum of the periodicchain. The lowest excitation gap 0, 1 is fitted to 1 E = a + b/i2 for even lengths(a = 3.69, b = 3.94).At the Heisenberg point, this can be expressed in terms of the excitation energyp = (_i)lEez/vlr, (3.52)where the ground-state energy of irv/41, for 1 odd, is subtracted from Eex; i.e. theenergy levels are equally spaced, and the parity alternates. Again we measure parityrelative to the ground-state, which is (_1)h/2 or +1 for an even or odd-length openchain, respectively. There is now a single SU(2) symmetry at the Heisenberg pointcorresponding to two possible sectors with total spin s integer for i even or s half-oddinteger for 1 odd.The states of the first six energy levels have again been worked out in table 3.2 for openboundary conditions at the Heisenberg point. We can test this spectrum numerically withthe algorithm in appendix B and find excellent agreement at the critical point J2 = O.24J0 0.004 0.008 0.0121/Length2Chapter 3. Scaling and Finite Size Effects 28Even Open Odd Open0 0 121 1— 20+, + 1+ 3+3 0—, 2 x (1-) 2 x E4 2 x (0), 2 x (lj, 2 3 x (j, 2 x ()5 2 x (0—), 4 x (1—), 2— 4 x (p), 3 x (E)Table 3.2: Low energy spectrum for open boundary conditions. Relative parity and totalspin are given.(see figures 3.6 and 3.7). Figure 3.8 shows that corrections to the energy gaps E(l/?rv)now drop off’ as 1/i, as expected for open boundary conditions. Note, however, that thisis a length dependent renormalization of the velocity v, since the corrections come fromthe boundary energy operator in equation (3.39). [see also the discussion before equation(5.83)]. Therefore we estimate the velocity as vir = 3.65 — 4.6/i in figures 3.6 and 3.7,which gives good results.>LiiTotal SpinFigure 3.6: Numerical low energy spectrum for the open, even length 1(vr = 3.42).876‘—‘ 5> 4LU 3210= 20 spin chainFigure 3.7: Numerical low energy(vr = 3.42).spectrum for the open, odd length 1 = 19 spin chainChapter 3. Scaling and Finite Size Effects 29I I I I I I I I I I I I I I I I10--8-6 --__-4 -- a-2 —--0 I I I I I I I I I I I-1 0 1 2 3 40 0.5 1 1.5 2 2.5Total Spin3Chapter 3. Scaling and Finite Size Effects 3090705030105 15 25LengthFigure 3.8: Renormalization group flow towards the asymptotic spectrum for the openchain. The lowest excitation gap E(l/irv) is fitted to 1 E = a + b/i for both even and oddlength chains (a = 3.65, b = —4.6).111111 I I I I I I I I I I I I10 20Chapter 4ImpuritiesWe are now in the position to calculate the effect of any perturbation on the chain interms of our renormalization group and scaling analysis. Since we are able to expressperturbing spin operators in terms of the boson fields, we can determine the relevance ofthe perturbation by looking at the corresponding scaling dimension.Although a variety of possible perturbations can be considered[9], we will only considera local change of coupling constants between spins within the chain, which can modelmany kinds of defects or impurities. We can obtain a more or less complete pictureby focusing on two simple cases, which can later be generalized with respect to theirsymmetry properties: The perturbation of one coupling constant (“link”) within thechain and the equal perturbation of two neighboring links. We will mainly considerthe isotropic model J = J unless otherwise indicated, since it seems to be the mostinteresting case experimentally[4J.4.1 One Perturbed LinkA generic local lattice distortion in a quasi one-dimensional spin compound could bedescribed by the small perturbation of one link in a periodic chain J’m m+1, as shownin figure 4.9. Although such a perturbation has special link parity symmetry, we will arguelater that the following renormalization group analysis is valid for all local perturbationthat are not site parity symmetric. The SS1 + SS1 part of the interaction is31Chapter 4. Impurities 32Figure 4.9: A quantum spin chain with one altered link.given bysS1+ SS1 j(_1)m(4bR — h.c.), (4.53)ignoring derivative terms of higher dimension as well as the uniform part which has thesame form as the free Hamiltonian density with scaling dimension d = 2. Using thebosonization formula of equation (2.9) and rescaling the boson field following equation(2.13), we obtainSS1+ SS1 (—1)mconst. sin . (4.54)which has scaling dimension d = 1/4irR2 and is therefore more relevant than the uniformcontribution for all positive values of J.The uniform part of S,S1 is known to have a scaling dimension of d = 2 from thefree Hamiltonian. The staggered part, i.e. the cross-term between uniform and staggeredparts of SZ can be written in terms of the fermion currents:[JL + JR] (m) [R(m +1) + h.c.] + {4R(m) + h.c.j [JL + JR] (m +1) (4.55)The individual terms can be written as a completely normal ordered four-Fermion operator together with an additional term from Wick-ordering of the formJL(m)4bR(m +1) -+ : 4bL4bR: - (4.56)We can ignore all normal ordered terms since they reduce to irrelevant derivative operators. After combining all remaining terms together, we obtain the same operator as in<> <>Chapter 4. Impurities 33equations (4.53) and (4.54), for all values of R(—1)mconst. sink. (4.57)While this follows from symmetry at the Heisenberg point, it is not a priori obvious inthe general case. These results can also be obtained by using the bosonic representationof the spin operators of equation (2.17) and the operator product expansion. In the bulk,the operator in equations (4.54) and (4.57) corresponds to a staggered interaction. Thescaling dimension is given by d = 1 /47rR2 and a staggered interaction is therefore relevantfor all positive values of J Such a staggered interaction may be induced by phonons,which leads to the so called spin-Peierls transition to a spontaneously dimerized, orderedphase.Since this operator has a scaling dimension of d = 1/2 at the Heisenberg point, weconclude that it is relevant even as a local perturbation. Under the presence of such aperturbation, the energy corrections to the periodic chain spectrum should increase aslzE cx as the cutoff is lowered according to the discussion in section 3.2. While thisestablishes that the periodic chain fixed point is unstable under this perturbation, wealso know that a local perturbation should not affect the bulk behavior of the system.In particular, we expect that correlation functions of points which are far away fromthe impurity compared to their relative distance are not affected by the presence of theimpurity and can still be calculated by the field theory as presented in chapters 2 and3. A reasonable conclusion to draw from this scenario is that the system renormalizes toanother, more stable conformal fixed point, which is characterized by a different boundarycondition, but uses the same field theory description. This is in analogy with the ideaswhich were discussed in section 1.2 (see also figure 1.1).In the case of one perturbed link, we can easily analyze where the periodic chain fixedpoint will renormalize to. Since we expect a slightly weakened link to become weakerChapter 4. Impurities 34under renormalization, the obvious guess for the stable fixed point is the chain with openboundary conditions. A slightly strengthened link will grow as the cutoff is lowered, andthe two strongly coupled spins will eventually lock into a singlet which decouples fromthe rest of the chain, so that we expect the stable fixed point again to be the open chain,but now with two sites removed.To test this assumption, we analyze the scaling dimension of a weak link across theopen ends. In the field theory, this is described by the product of two independentboundary operators at the weakly coupled endsss( ObL(1)OcbL(SjS o e4’’e’ (4.58)where we have used the boundary condition in equation (3.32). Sincethe scaling dimensions of independent operators simply add, we find that the Sf S[ operator has a scalingdimension of d = 2 independent of J, while the SjS1 part has a scaling dimension ofd = 4irR2 which is also irrelevant as a local perturbation for all positive values of J.Thus, a weak link across the open ends renormalizes to zero. The open chain fixed pointis therefore indeed stable, and our assumptions above are consistent. At the Heisenbergpoint all operators from a perturbation of one weak link across the open ends of a chainhave d = 2, and we expect that corrections to the open chain spectrum should flow tozero as 1/i according to the discussion in section 3.2.We can generalize these findings since the operator in equation (4.54) is always themost relevant operator that can be produced by a local perturbation. A general perturbation in the chain will therefore produce this operator unless special symmetries (e.g.site parity) are present. Since relevant coupling constants will in general renormalize tozero or infinity, we conclude that the periodic chain fixed point flows to the more stableopen chain fixed point as the temperature is lowered, if a local perturbation is present.Chapter 4. Impurities 350.180.160.14>w 0.120.10.08Figure 4.10: Flow away from the periodic chain fixed point due to one altered link foran odd length chain with 7 1 23. The lowest excitation gap , is fitted tolE = a 11/2, which is the predicted scaling.At the open chain fixed point, decoupled singlets may also be left over, and the effectivelength of the chain may be reduced by an even number of sites.We can test the scaling analysis from above numerically by looking at the energycorrections lE of the energy gaps if a small perturbation is present. Since we wantto determine the scaling exponents accurately, all numerical simulations were done witha next nearest neighbor coupling J2 = 0.24J present, to avoid logarithmic corrections.Perturbations from the periodic point should correspond to a scaling with lE cx v”1 asmentioned above, which is indeed the case as shown in figure 4.10. A weak coupling acrossopen ends of the chain should lead to corrections to the spectrum lE cx 1/i + 0(1/12),which is demonstrated nicely in figure 4.11.7 9 11 13 15 17 19 21 23LengthChapter 4. Impurities 360.20.180.160.140.120.10.080.06Figure 4.11: Renormalization group flow towards the open chain fixed point due to oneweak link for an odd length chain with 7 1 23. The corrections to the lowestexcitation gap , is fitted to iE = a/i + b/i2, exhibiting the predicted 1/i scalingup to higher order.4.2 Two Perturbed LinksSince the operator in equation (4.54) does not respect site parity, we expect fundamentallydifferent behavior for site parity symmetric perturbations. As a generic case of a siteparity symmetric perturbation, we will consider the equal perturbation of two neighboringlinks in the chain as shown in figure 4.12. Since the operator in equation (4.54) isalternating, we immediately find that the most relevant site parity invariant operator isthe derivatived.q—sin—. (4.59)dx RBecause the derivative increases the scaling dimension by one, this operator has scalingdimension d = 1 + 1/4rR2, corresponding to d = 3/2 at the Heisenberg point. It is7 9 11 13 15 17 19 21 23LengthChapter 4. Impurities 37Figure 4.12: A quantum spin chain with two altered links.therefore irrelevant as a local perturbation. The uniform parts of the interaction havedimension d = 2 as discussed before and are even more irrelevant.The situation at the open chain fixed point is completely different, however, becausethe open ends are now coupled to an external spin-1/2. The corresponding boundaryoperators at 1srs— + —4lriP.4L(l)Q+I imp e impnow have scaling dimensions of d = 1 and d = 27rR, respectively, since the impurity spinis dimensionless. The boundary operators at x = 0 have corresponding expressions. Thecoupling of open ends to an external spin is therefore found to be marginally relevantat the Heisenberg point for anti-ferromagnetic coupling and marginally irrelevant forferromagnetic sign. We therefore expect the open chain to renormalize to the morestable periodic chain if a site symmetric antiferromagnetic perturbation is present. Thechain effectively “heals” in this scenario.These predictions can be tested numerically as before. Figure 4.13 shows the predicted scaling with i//i for site symmetric perturbations from the periodic chain whichcorrespond to the scaling dimension of d = 3/2 as discussed in section 3.2. Figure 4.14establishes the marginally relevant scaling for two weak anti-ferromagnetic links of theopen ends to an additional spin, while the marginally irrelevant case of two ferromagnetic<>Chapter 4. Impurities 380.20.180.16W 0.140.120.10.08Figure 4.13: Flow towards the periodic chain fixed point for two altered antiferromagneticlinks. The , gap is fitted to 1E = a/i’!2, which is the predicted scaling.links is shown in figure 4.15. We can see that the logarithmic corrections increase anddecrease relative to the original coupling in the two cases. Since logarithmic scaling isslow we have to also consider the irrelevant 1/i contribution in the two figures to obtaina good fit. Although this three parameter fit is not entirely convincing, we can show thatthe logarithmic contribution is essential to achieve a good fit as presented in figures 4.14and 4.15. Even other three parameter fits which we tested cannot reproduce an equallygood fit without taking the logarithmic contribution into account.4.3 Relation to Other ProblemsBoth link and site parity symmetric impurities that we have discussed above correspondto special cases of models studied in the context of defects in one-dimensional quantumwires[23, 24]. In these papers spinless fermions were considered, which are equivalent7 9 11 13 15 17 19 21 23LengthChapter 4. ImpuritiesLULU-0.13-0.135-0.14-0.145-0.15-0.1557 9 11 13 15 17 19 21 23Length39Figure 4.14: Flow away from the open chain fixed point for two weak antiferromagneticlinks. Corrections to the , gap are fitted to E/E = (a+b/i+clni), demonstratingrelevant logarithmic scaling (ac> 0). The dotted line is the best fit for c = 0.7 9 11 13 15 17 19 21 23LengthFigure 4.15: Flow towards the open chain fixed point for two weak ferromagnetic links.Corrections to the [ gap are fitted to i.E/E = (a + b/i + clni), demonstratingirrelevant logarithmic scaling (ac < 0). The dotted line is the best fit for c = 0.Fr I I I I r1I I I I I J I I I I I I I III IIIJ’=OlJ\,twoutIorLULU-0.068-0.07-0.072-0.074-0.076-0.078Chapter 4. Impurities 40to the xxz spin chain by the Jordan-Wigner transformation. The Heisenberg modelcorresponds to a particular value of the repulsive interaction. The flow of a single modifiedlink to the open chain fixed point corresponds to the perfectly reflecting fixed point [23].The “healing” discussed here corresponds to resonant tunneling[24]. In that work, it wasnecessary to adjust one parameter to achieve the resonance condition (even with exactsite-Parity maintained). This parameter, a local chemical potential at the impurity site,corresponds to an external magnetic field term h S at the impurity site. In the spinproblem this is naturally set to zero by spin-rotation symmetry or time-reversal. Thusresonance (healing) occurs without fine-tuning in the spin chain.The case of two equally perturbed links J’ also has an interesting equivalent in thetwo-channel Kondo problem [9], which has been realized independently in reference [25].The description in the field theory language is completely analogous in the spin sectorof the Kondo problem if we identify the marginal coupling constants J’ of the spins atthe ends to the impurity spin in figure 4.12 with the Kondo coupling. The operators inequation (4.60) can be identified with the spin currents that couple to the impurity in theKondo effect. The two ends of the spin chain are effectively independent of each otherand can be regarded as being described by two independent fields which play the roleof the two spin channels in the Kondo effect. The central spin in figure 4.12 representsthe impurity spin in the Kondo case. The overscreened Kondo problem is known torenormalize to a non-trivial intermediate coupling fixed point [26], which corresponds tothe healed chain. For higher impurity spin, we recover the normal underscreening ofthe Kondo problem in the spin chain model as well. It is, however, important to realizethat the spins in the chain are not equivalent to the spins of the electrons in the Kondoproblem, although the boson theory describing them is identical. The Kondo problemadditionally has charge excitations which do not couple to the impurity and can thereforebe neglected.Chapter 4. Impurities 41We can carry this analogy over to the single channel Kondo problem as well. In thiscase, we imagine a semi-infinite spin chain with the open end coupled to the impurityspin. As before, anti-ferromagnetic coupling J’ is marginally relevant, and the stablefixed point is the open end with one extra site added (i.e. coupling J’ = J). Thiscorresponds to the strongly coupled fixed point in the Kondo effect. We see that werecover the famous ir/2 phase shift of the fermions according to equation (2.4). As in theKondo effect, the effect of the renormalization can be described by the so called fusionwith a s = 1/2 primary field[8]. For higher impurity spin we obtain the correct size ofthe effective left-over spin s — 1/2 at the strong coupling fixed point J’ — oc as well, andthe usual underscreening of the Kondo problem is recovered[26].Recently, it has been discovered that the two-impurity Kondo problem has an analogyin a spin chain problem as well[8, 27]. In the equivalent spin chain model, two spin-1/2impurities are coupled together with an RKKY coupling JRKKY, and each of the spinsis coupled separately to one of the open ends of the chain with a Kondo coupling JKas shown in figure 4.16. In this case, both coupling constants are relevant in the weakcoupling limit, but a healed chain unstable fixed point JRKKY = JK = J separates thetwo strong coupling stable phases. The two open ends of the spin chain now play therole of the electron spin sectors at the two impurity sites in the Kondo problem. Thisscenario is just another case of a link symmetric perturbation as discussed in section 4.1.The two stable fixed points correspond to open chains with a possible decoupled singletRKKYFigure 4.16: The equivalent spin chain model to the two impurity Kondo problem.chapter 4. Impurities 42and two sites removed if JRKKY scales to infinity. From the analysis above, we concludethat the non-trivial fixed point should be unstable with scaling dimension d = 1/2.These analogies are very useful since the spin chain Hamiltonians can be treated witha variety of numerical techniques, which might give some better insight in correlationfunctions and the size of the screening cloud in the Kondo effect. Moreover, it is easyto visualize non-trivial fixed points for spin chain problems, and additional fixed pointsmay be found by simply adjusting the corresponding coupling constants.Chapter 5SusceptibilitiesNow that a good theoretical understanding of the effect of impurities is established,it is necessary to make clear predictions on how this can be seen in an experiment. Asindicated above, a iSR experiment on quasi one-dimensional magnetic compounds seemsto be a good choice to observe the predicted effects. In this setup, we expect that thetemperature dependence of the effective susceptibility at the muon site will be differentfrom the results of a conventional susceptibility measurement due to the renormalizationof the induced perturbation. In section 5.4 we will present the idea of the experimentalsetup in more detail. But before we can make any definite predictions for an experiment,it is necessary to take a closer look at the susceptibility of the spin chain and the effectof perturbations on this susceptibility in general. In this chapter we primarily discussthe field theory predictions for the susceptibility, while most of the Monte Carlo resultsare presented in chapter 6.The local susceptibility Xi 111 the z-direction at site i is given byx(T) <Si> h0 = <SJ(r)S(O)> = <S1S>, (5.61)where h is the uniform magnetic field on the chain. We can drop the time dependence ofthe expectation value, because the total spin-z S commutes with the Hamiltonian. Wealso have set the Bohr magneton times the gyromagnetic ratio gu and the Boltzmannconstant kB to one. The susceptibility is then measured in units of 1/J. This localsusceptibility will be different in the bulk and near a boundary since it is directly related43Chapter 5. Susceptibilities 44to the correlation function. In some cases we will refer to the total susceptibility Xtotal ofthe chain, which is simply the sum of all local susceptibilitiesXtotal Xi (5.62)The following calculations assume that we are in the scaling limit T << J, 1 >> 1, sothat the field theory analysis is valid. A “thermal length” is naturally defined by v/T,and the width of the boundary layer is given by v/TK, where TK has been mentioned insection 1.2. The meaning of the thermal length will become clearer shortly in equation(5.64). For our analysis, we require that the system size 1 is very large compared to allother scales in the system, in particular compared to v/T, i.e. Ti/v >> 1. The ultravioletcutoff can then always be reduced down to the temperature T.5.1 Periodic Chain SusceptibilityBecause the periodic chain is translationally invariant, the local susceptibility will be independent of the site index i. All spins in the chain therefore have the same susceptibilityx which defines the bulk susceptibility per site. Using equation (2.24) we find that thesum over the SZ correlation function in equation (5.61) is given by<sjsj>= l6 3R21 dx ( t)2 + ‘t)2). (5.63)This expression is useful for calculating the susceptibility of the free boson model, i.e.ignoring all irrelevant operators of the theory. Note, that the alternating part in equation(2.24) does not contribute when summed over and can therefore be neglected. To get afinite temperature result, we first make a Wick rotation to imaginary time T, which weassociate with the inverse temperature /. A conformal transformation of the space timecontinuum onto a cylinder with circumference /3 and length i —* cc then yields the finiteChapter 5. Susceptibilities 45temperature correlation functions according to equation (A.107)—iv/3 . (ixvT)Kx±vt—*x±zvr--÷ sin . (5.64)v3The bulk susceptibility for the free boson model is thus given by= 167r3R2L dx { [ sin ((VT _ix)T)] -2 + sin ((VT Iix)7r)] _2}(5.65)The integral can be done by the change of variables: u = tan and w = —itangiving—2 2I = / dx sin (ix + vr) / dw (1 + = , (5.66)i_co \\ ‘ir v/3 ,) i-i V/3(u + iw) Viiwhich is independent of T or u as it should be since the total spin-z SZ is conserved. Thefinal result for the bulk susceptibility per site of the free boson model is therefore[29]X= v(2R)2’ (5.67)which is independent of temperature. This expression agrees with the zero temperatureexpressions from the analytical Bethe ansatz[12}. We can improve this formula for thespin chain in the field theory treatment by using perturbation theory in the irrelevantoperators, which will give us temperature dependent corrections to this expression. Weknow from conformal invariance that finite size scaling should be analogous to finitetemperature scaling upon identifying 1 = V/T, so that we can use the same analysis asin section 3.2.5.1.1 Contributions from the leading irrelevant operatorThe first order contribution to the bulk susceptibility from the leading irrelevant operator cos(2ç/R) vanishes. The second order contribution is determined by our scalingChapter 5. Susceptibilities 46arguments in section 3.2 and the scaling dimension d = 1/7rR2,giving:(T) 2 + const.T2(1R2). (5.68)v(2irR)For 1/rR2 > 3, J < J/2 the exponent will get replaced by 2, because in this case TLTRfrom equation (2.16) is the leading irrelevant operator. For J > Jcos’(ir/5) .809Jwe notice that the correction leads to an infinite slope at zero temperature.At the Heisenberg point J = J, R = i//, we expect a logarithmic correctionaccording to section 3.2, because the leading irrelevant operator is marginal. These corrections have been calculated by using the non-abelian bosonization of the spin chain[10],in which the low energy effective field theory description is given by the k = 1 WessZumino-Witten (WZW) non-linear g-model. The uniform part of the spin density isgiven by the conserved current operators, i, iz for left and right-movers:YL(x) + ?1?(x). (5.69)In the WZW model J and are uncorrelated and their self-correlations are given by6ab<J(r,x)J(0,0)>= 8ir2(vT—ix)<J(T,x)4(0,0)>= 8K2(vr+ix) (5.70)The leading irrelevant operator is marginal and is now given by—8K2v -. -.gJLJft. (5.71)Using first order perturbation theory gives a correction to x that involves four currentoperators1 82 1/2 1/2x = + v2 L112dy< [J + J](0)[J + J](x)JL . JR(y)>. (5.72)Chapter 5. Susceptibilities 47Due to the fact that the left and right currents are uncorrelated, this expression factorizesinto a product of two two-point Green’s functions, one for left-movers and one for right-movers according to equation (5.70). Using translational invariance, the spatial integralsfactorize into two independent integrals of the form of equation (5.66), givingx = +v28f2dxfy(<J(0)J(y) >< J(x)J(y)>+ < J(0)J(y) >< Jfr)J(y)>)2— 1 ( 1 21r\ ( 1 2K— 2Kv+ 8K2v) +82=-—+-. (5.73)Again, the correction is naively temperature-independent, since g is dimensionless. However, this formula can be improved by replacing g with g(T), the effective renormalizedcoupling at temperature T. By integrating the lowest order -function, g(T) is givenby[19]g(T) g1 (5.74)1 + 4Kg1 ln(Tj/T)/’JHere g is the value of the effective coupling at some temperature T1. Both g and g(T)must be small for this formula to be valid. We may write this more compactly, asg(T)4Kln(To/T)’ (5.75)for some temperature T0. Thus we obtain the leading T-dependence of x1 1(T) = + . (5.76)2Kv 4Kv ln(To/T)As mentioned before, we can adjust gi and T0 by introducing a next nearest neighborcoupling J2, and at J2 0.24J we expect T0 — oc. However, for the model in equation(1.1), J = J, J2 = 0, we expect a finite T0 and a logarithmic divergence of the slope atT=0.Chapter 5. Susceptibilities 480.14— 0.120.1Figure 5.17: (T) from the Bethe ansatz. x(O) = 1/J’’r2 is taken from equation (5.67).As shown in figure 5.17, we have calculated the complete bulk susceptibility curvefor this case by use of the numerical Bethe ansatz of Takahashi[3, 30]. The bulk susceptibility obeys Curie’s law for large temperatures and goes through a maximum atT 0.640824J, x 0.147/J before the slope starts to increase again below an inflectionpoint at T 0.087J. We will use this curve as the unperturbed reference point for ourMonte Carlo simulations, which of course give identical results for this case, but withlarger error-bars. The existence of the maximum has been known for a long time[31] andis used in experiments to establish the one-dimensional characteristics and determine thecoupling strength J[4, 5]. Our calculations give an additional prediction of an inflectionpoint at T 0.087J, which might be observable in some very highly one-dimensionalmaterials.0.5 1 1.5 2T/JChapter 5. Susceptibilities 49Figure 5.18: Field theory [equation (5.76), T0 7.7J] versus Bethe ansatz results for(T) at low temperature.According to figure 5.18 our field theory predictions agree reasonably well with thelow temperature susceptibility behavior for a value of T0 7.7J. The curve starts todeviate from our first order calculation as corrections of order (1 / in T)3 and T2 becomeimportant. The magnitude of these corrections are not known, but inT0/T>> 1 shouldbe a sufficient condition for the validity of equation (5.76) which is consistent with figure5.18. The divergent slope is not accessibie by experiments, because of a finite orderingtemperature in quasi one-dimensional spin compounds, but the onset of this behaviormay have been observed in some cases[5].The logarithmic scaiing behavior has been observed before in numerical Bethe ansatzcalculations for finite size systems[19]. ‘We can compare the result in the finite length0.1150.110.1050.1T/J0 0.02 0.04 0.06 0.08 0.1C8Chapter 5. Susceptibilities 500.0550.050.0450.040.0350.030.0250.020.01510 1000Figure 5.19: Estimates for the effective coupling g from lowest order perturbation theorycorrection to the finite-size energy of ground-state, first excited triplet state, first excitedsinglet state[19] and to the susceptibility, using 1 —* v/T. The renormalization groupprediction of equation (5.75) is also shown.100LengthChapter 5. Susceptibilities 51limit iT/v — 0 from reference [19] directly with our calculation in the finite temperaturelimit iT/v —+ cc upon identifying 1 = v/T, which appears to be the appropriate relation(see figure 5.19).It is possible to continue the expansion in the irrelevant operators to higher orders inperturbation theory which will give higher order corrections in T. In the limit iT/v >> 1the system size is always much larger than the finite temperature correlation length, andwe expect that finite length corrections to the periodic chain susceptibility are exponentially small in iT/v. Because we have translational invariance we can therefore write thetotal susceptibility of the periodic chain as Xtotal = 1X(T), where x is independent of thesystem size 1 up to exponentially small corrections.5.2 Open Chain SusceptibilityThe situat.ion is somewhat different for the open chain since translational invariance isbroken. In this case it is possible to have an additional impurity contribution to the totalsusceptibility which is independent of lengthXimp lirn ( Xi — iX) (5.77)where is now the local susceptibilities at site i of the open chain and x is the bulksusceptibility per site from the previous section. This impurity susceptibility comes fromlocal irrelevant operators as in equation (3.39) in the field theory, which will be discussedin section 5.2.2. The boundary condition condition itself does not contribute to theuniform part of the susceptibility and therefore does not affect the impurity susceptibilityas will be explained at the end of section 5.2.1. An impurity susceptibility is also presentfor any local perturbation on the periodic chain, and if no local operators were present,neither fixed point in the field theory would have an impurity susceptibility contribution.(Note, that the open chain in the lattice model corresponds to the open field theoryChapter 5. Susceptibilities 52TBFigure 5.20: The open ends of the broken chain are expected to be more susceptible.fixed point from section 3.3.2 only up to irrelevant operators and therefore always has animpurity susceptibility).It seems intuitively clear that the open chain will be more susceptible at the endsthan bulk spins in the periodic chain as indicated in figure 5.20. To calculate this effect,we have to take into account that the correlation functions will be different near theboundary as described in section 3.1.2 and that there will be a correction due to theleading irrelevant operator in equation (3.39).5.2.1 Contributions from the boundary conditionLet us first consider the effect of the boundary condition itself. Although the boundarycondition is not responsible for the impurity susceptibility as mentioned above, we doexpect that the correlation functions and therefore the local susceptibility will be affected.According to equation (5.61) one index j in the expectation value < SS7 > is alwayssummed over, so that the alternating part of the second operator S7 does not contributeto the local susceptibility. Because the boundary condition relates left and right moversaccording to equation (3.32), there is now however a possibility for a non-zero cross termof the uniform part of S7 and the alternating part of S, which gives an alternatingcontribution to the local susceptibility as a function of the site index i relative to theChapter 5. Susceptibilities 53boundary. We therefore choose to separate the local susceptibility into a sum of analternating and a uniform part, corresponding to the alternating and uniform parts ofSz(x) in equation (2.17):fdy<Sz(x)Sz(y)>= <{S1(x) + (_1)xSit(x)} fdyS(y)>= X + (1)xxt (5.78)This separation should be valid in the scaling limit, but at short distances it is of coursesomewhat ambiguous (e.g. within a few lattice spacings of the boundary).The corresponding expression for Xt can be non-vanishing, because it is now expressed in terms of a three point Green’s function (without the boundary condition, it isa vanishing two point function). After expressing the cosine in equation (2.17) in termsof exponentials and using equation (3.32) we findx1t 3( dy x’t’)_L(Y, t))x /3i dy , (5.79)J—oo (y+vt—x—vt’)(y+vt+x—vt’)according to equation (A.108) with scaling dimensions d1 = d2 = 1/4 and d3 = 1 at theHeisenberg point. Here, the index x measures the distance from the boundary in units ofthe lattice spacing. We can evaluate this expression for finite T by using equation (5.64)4/v3sinh1 dy v v/3X J_ 43sinh (y + x + ivr)sinh y — x + ivr)iJv,6 sinhL (cosh2 (y + ivT) + sinh2 (y + ivr) — cosh (5.80)This integral can be done if we set u = coth y + ivT), du = sinh2 y+ivr) Thefinal result isalt d/v13 sinhcx J Uu2+1_(u2_1)coshrIChapter 5. Susceptibilities 540.8 I I I I0.60.40.2 1 z.:-0.4 I-0.6 I I I0 5 10 15 20 25 30 35 40 45 50SitesFigure 5.21: The local susceptibility near open ends from Monte Carlo simulations for= l5/j.xcx . (5.81)/inhNote, that the alternating part may increase as we increase the distance x from theboundary if we are at low temperatures. In particular at T = 0 we expect the alternating part to increase exactly with which seems very counter-intuitive. (Note,however, that experimental systems always contain the exponential drop-off from finitetemperatures.) This surprising effect has been checked by Monte Carlo simulations usingthe algorithm in appendix C. The local susceptibility as a function of distance from theopen ends is shown in figure 5.21 for i = 15/J.Using this Monte Carlo data, we can extract the uniform and alternating parts asshown in figure 5.22 (as mentioned before this is somewhat ambiguous very close to theboundary). The alternating part fits the predicted form 0.52(x+2)//8sinh2K(x + 2)/v/up to a constant a = 0.52 and a shift of two sites. This functional dependence holdsChapter 5. Susceptibilities 55C.EC0 5 50Figure 5.22: The uniform and alternating parts of the local susceptibility near open endsfrom Monte Carlo simulations for = 15/J. We compare this data to the theoreticalprediction for the alternating part O.52(x + 2)///3 sinh 2ir(x + 2)/v/3.rather well for all temperatures that were sampled.It is important to notice that the uniform part of the susceptibility does not acquireany change from the boundary condition in equation (3.32) because the uniform part ofS is a sum of left and right movers and not a product. In particular we can rewrite theintegral over the correlation function of the uniform part with the use of equation (3.32)1 f°° /Oq aqXtotal= 42Ri dxj0 dy1 [°° f’° / (0L ‘\ (0L= 42Rj dx dy a(x) + -(_x)j --(,i) += 2irR L dxf dy K)a’) (5.82)10 15 20 25 30 35 40 45Seswhich gives the same result as the periodic case in equations (5.63-5.67).Chapter 5. Susceptibilities 565.2.2 Contributions from the leading irrelevant boundary operatorAlthough the local corrections due to the boundary condition are large, there will beno impurity contribution to the total susceptibility since the alternating part does notcontribute under the integral and the uniform part does not change. This is in completeagreement with the statement that the impurity susceptibility comes entirely from theirrelevant boundary operators. In this section we will consider the contribution from theleading irrelevant boundary operator in equation (3.39) with some unknown couplingconstant c. This boundary operator is present at each end, TL(O) and TL(l), but the twooperators are independent at the open chain fixed point and we may consider only theoperator at the origin and then generalize our findings for both ends.The impurity correction in equation (5.77) to the susceptibility from TL(O) can becalculated to first order with a simple trick, which has been used before in the contextof the Kondo prohlem[32j. The operator in equation (3.39) is proportional to the localenergy density, so that its effect to first order in perturbation theory is simply a lengthdependent renormalization of the velocity v for any bulk quantity. In particular for thebulk susceptibility to first order we can writecvJdxfdyfdz K6(z)TL(z)(x)7(y))- v f dx f dyf dz (TL(z)-)-Y)) (5.83)The integral over dz can now be reabsorbed to first order in the free Hamiltonian byrescaling v —* v(1 + c/i). In the thermodynamic limit, the effect for all translationalinvariant quantities is then simply a rescaling of the temperature T —* T/(1 + c/i) inthe partition function. In particular, the susceptibility per site can be calculated as afunction of the coupling constant c(c,T) =Chapter 5. Susceptibilities 57= f dx= X(c=Oi/1) 1—c/ix+E. (5.84)At T = 0 the correction corresponds to a shift cx/i in the susceptibility per site. Thesum over all sites gives us the impurity correction to the susceptibility Ximp cx, whichis independent of length as it should be in the limit 1 —* in equation (5.77).Although this argument can give us the impurity correction to the total susceptibility,it is not sufficient to predict the local corrections 6x due to the irrelevant operator inequation (3.39). These can be calculated by doing first order perturbation theory in thecoupling constant c of the operator in equation (3.39)xx c/32j dy (:[(0)]: &L()0L()) (5.85)where we again only consider the boundary operator at 0 for simplicity. After replacing ,8with an integral over dr to take the proper time-correlations into account we can calculatethis expression by using the usual boson Green’s functions and Wick ordering:/ OcL ,, \ / . ,+ zv’r )ã(zvT) -—(y + ZVT )——(zvT)(x — iVT + iVT”) 2(y + ivY — ivTr2. (5.86)The finite temperature result for the local susceptibility correction can now be derivedfrom equation (5.64)çi3 /vj3 iK(x — ZVT + ivT”) i°° (v3 . ir(iy — vr’ + vr)’26xx cx c13 j dT i — sin I I dy i — sinJO \1 v/3 j J— v3/ ,, . \—2i3,‘v/3 . r(vr — VT + ix) \x c I dr I —sin I , (5.87)Jo ir v/3 jwhere we also used equation (5.66). The integral over dT gives —rcot[ir(vr — vr” +ix)//3v]/v23,which cancels for the given limits of integration and any finite x. For x = 0,Chapter 5. Susceptibilities 58however, we can get a non-zero value, which can be determined by doing the integralover all x using equation (5.66). Therefore, the local correction to the uniform part ofthe susceptibility is given by6xx = c6(x)x. (5.88)This result simply says that the impurity susceptibility Ximp = cx from equation (5.84) isonly added directly at the open ends, but we have to keep in mind that our field theoryanalysis is valid only in the scaling limit so that we have to allow some finite width ofa few lattice spacings for the delta-function. In addition we still have the alternatingpart from equation (5.81), which does not contribute when summed over x, but gives alarge local contribution. This picture agrees with the numerical findings in figure 5.22,although the separation into the alternating and uniform part is somewhat ambiguousright at the origin.In chapter 4 we discussed some cases in which the boundary conditions did not onlycorrespond to the usual field theory fixed points, but also had sites removed or additionaldecoupled impurity spins. In particular, the periodic chain with one strengthened linkrenormalizes to the open chain with two sites removed. The impurity correction to thesusceptibility is now 2(c— 1)x, which may be positive or negative. Likewise, the unstablefixed point in section 4.2 represents an open chain with a decoupled impurity spin. Inthis case the impurity susceptibility has an additional Curie contribution proportional to1/T from the decoupled spin.5.3 Susceptibility Contributions from PerturbationsTo study the effect of perturbations on the lattice Hamiltonian, it is useful to reviewthe renormalization group arguments which have been successfully applied in the Kondoproblems[32, 33]. These renormalization group arguments are directly related to theChapter 5. Susceptibilities 59unstable stableFigure 5.23: Renormalization group analysis of the cross-over from an unstable to astable fixed point as the temperature is lowered.discussion in section 1.2, which might be helpful in understanding the general ideas. Atthe fixed points the system is completely scale invariant, but as soon as we introduce aperturbation we expect that an energy scale TK is introduced, which simply correspondsto the temperature where we expect the breakdown of perturbation theory. For stable(unstable) fixed points the description of the free Hamiltonian is then approximatelyvalid for temperatures below (above) TK. If there is a renormalization from an unstableto a stable fixed point, we expect the energy scales at the two fixed points to be related,so that there is really only one energy scale TK that governs the cross-over.In the case where a small perturbation creates all local operators near the fixedpoint, we usually only have to consider the leading operator with the smallest scalingdimension d, since all corrections due to the more irrelevant local operators vanish underrenormalization. We can then determine the relation between TK and the bare couplingconstant A by a simple renormalization argument. According to the discussion in sectionT/TK= 1Chapter 5. Susceptibilities 603.2 we expect the breakdown of perturbation theory when )Td becomes of order onefor a local perturbation, so thatT_d (5.89)For marginally relevant perturbations (i.e. b\ < 0) with d = 1 this formula is replacedby TK Dc e11 due to equation (3.37). The cross-over from an unstable to a stable fixedpoint is universal since it is only governed by one energy scale, and impurity correctionsshould be functions only of the dimensionless ratio T/TK. This idea is illustrated infigure 5.23, assuming that there is one relevant operator at the unstable fixed point andone leading irrelevant operator at the stable fixed point. If we ignore all other higherirrelevant terms the cross-over is described by one universal trajectory, which representsthe x-axis in figure 5.23. Each point on the x-axis is labeled by only one parameter T/TKwhich is decreasing along the x-axis and all impurity corrections are functions only ofthis parameter. In the limit T << J we can write the impurity susceptibility asXirnp (5.90)where the factor of 1/TR has to be inserted for dimensional reasons. The other higher order irrelevant operators are represented by the y-axis in figure 5.23 and can be neglectedfor most purposes. Consider for example if we start close to the unstable fixed point, i.e.with only a small relevant coupling constant as well as other arbitrary irrelevant couplingconstants as indicated by the trajectory (1) in figure 5.23. The irrelevant coupling constants become small very quickly and the actual renormalization trajectory (1) followsthe x-axis very closely for several orders of magnitude in the parameter T/TK. In thissense the cross-over function in equation (5.90) is universal. This equation is also usefulfor an arbitrary perturbation which may start closer to the stable fixed point as indicatedby trajectory (2), corresponding to a larger value of TK. The trajectory still approachesthe universal cross-over function represented by the x-axis, but equation (5.90) is nowChapter 5. Susceptibilities 61valid only in an asymptotic region T/TK —* 0.This analysis fails if there is more than one leading irrelevant (or relevant) couplingconstant, since each coupling constant may set an independent cross-over energy scale.In the case of two coupling constants c and \ with a leading scaling dimension d equation(5.90) must be replaced byXimp = g (cTd_1, Td_1), (5.91)This expression becomes equivalent to equation (5.90) if one of the coupling constants isset to zero, which can be seen by using equation (5.89).For the two-channel spiri-1/2 Kondo problem the asymptotic behavior has been determined to be[34, 35]Ximp ln(TK/T)/TK, (5.92)for T << TK. Curie’s law has to be recovered for T >> TK, so that Ximp 1/T flthat regime. If the initial Kondo coupling J’ is weak, equation (5.92) reduces to Xirnpln(J/T)e/’/J at low temperatures, which is less divergent than the unperturbed Curielaw behavior for any finite J’. The scaling of TK with J’ follows from the discussionthat lead to equation (5.89) and the fact that the corresponding operator is marginallyirrelevant (b = 1/v7r).5.3.1 Two perturbed linksThe case of two weak links of the spin chain is very closely related to the two-channelspin-1/2 Kondo effect as discussed in section 4.3, and we therefore expect the samelogarithmic divergence as in equation (5.92) of the impurity susceptibility at low temperatures. Starting from the weak coupling limit J’ — 0 equation (5.92) is valid for T << TK.In this limit we can keep only the most divergent part of the impurity contribution (i.e.Chapter 5. Susceptibilities 62drop any constant terms) which results in the relationJXimp in () (5.93)since the marginal coupling constant of the “Kondo interaction” in equation (4.60) isgiven by J’ to first order. The complete analytic form of the cross-over function isdifficult to determine, but we can easily see that we have to recover Curie law behaviorXirnp x 1/T in the opposite limit T >> TK, corresponding to a nearly free impurity spin.The divergent part of the impurity susceptibility is expected to come mostly from thecentral “impurity spin” in figure 4.12, which is clear from the analogy to the two-channelKondo effect and the analysis when J’ is small (i.e. when the “impurity spin” producesa divergent Curie law susceptibility).A small perturbation of two links from the periodic chain fixed point of 6J J — J’is the more interesting scenario for a SR experiment. In this case there is no localoperator in the Hamiltonian at the unperturbed fixed point, and all local field theoryoperators will have coupling constants of order 5J or higher. In this case we only haveto consider the leading irrelevant operator which is given in equation (4.59) with scalingdimension of d 3/2 at the Heisenberg point. We can use equation (5.89) to predictthe low temperature behavior of the impurity susceptibility for small perturbations 6Jon the lattice model. Since 6J produces the leading irrelevant operator to first ordercx 6J/J3/2+ O(6J2) according to the analysis in section 4.2 we can write6J2 1T\Ximp cx -__j--ln-jJ, (5.94)where the powers of J were inserted by dimensional analysis. This expression followsdirectly from equations (5.89) and (5.92). Note, that we have inserted powers of J onlyto relate coupling constants in the lattice models to coupling constants in the field theorydescription with the correct units. However, J does not appear as an energy scale in theChapter 5. Susceptibilities 63field theory description, because we lowered the ultraviolet cutoff all the way to thetemperature and J has been absorbed in the definition of the spin-wave velocity v.In summary, the impurity susceptibility Ximp increases monotonically with decreasingtemperature or increasing perturbation 6J.5.3.2 One perturbed linkA similar analysis can be applied for the perturbation of one link from the periodic chain6J with the leading operator in equation (4.54) of dimension d = 1/2. By using perturbation theory in the leading relevant operator with coupling constant of order ). xwe see that the leading order correction to the susceptibility is proportional to 6J2/J.This is because the first order contribution to the susceptibility < ODsin /R> hasa vanishing expectation value, which can be seen by equation (A.105). Since TK c5J2/Jaccording to equation (5.89), we conclude that the cross-over function in equation (5.90)has an asymptotic behavior of f(r) —* const/r2 as r —+ cc. The simplest assumption forsmall temperatures is f(0) = const.There is one complication, however, which causes a problem near the open chainfixed point, because there are now two dimension d = 2 leading irrelevant boundaryoperators. The effect of the operator from equation (3.39) with coupling constant chas been discussed above, but the operator from equation (4.58) must also be takeninto account, which has a coupling constant ) cx J’/J2 for a weak link. The relativemagnitude of the two leading operators changes with J’. For bare coupling constantsJ’ —* 0, which correspond to the open chain fixed point, the energy scale is thereforedetermined by two independent parameters c and J’. Therefore, we have to consider themore general case of equation (5.91). We can expand g to first order by using the notionthat both first order corrections are independent of temperature according to equationChapter 5. Susceptibilities 64(5.84) and the fact that f(0) = const. The resulting relation isXimp = g(cT, AT) c + const. J’/J2 (5.95)where x is the bulk susceptibility per site and c is the coupling constant of the localoperator TL in equation (3.39). We have also used A x J’/J2 and equation (5.84).Equation (5.95) reduces to equation (5.90) as A —+ 0 or c —* 0.These findings can be directly carried over to the case of a very large link J’ —f oc,which also corresponds to the open chain fixed point but with two sites removed. Inthis case there is an effective virtual coupling of order J2/J’ across the open ends, whichnow determines the leading order of A and the same analysis as above can be applied.However, it is not clear if the impurity susceptibility is positive or negative for large J’ asdiscussed at the end of section 5.2.2, and therefore we cannot make any reliable scalingarguments in this limit.For a bare coupling constant 6J — 0, which corresponds to the periodic chain fixedpoint, there is only one leading relevant operator with d = 1/2. Now, the energy scale isdetermined by only one parameter A o 6J/v’7 According to equation (5.89) this energyscale becomes smaller as we approach the periodic chain TK cc 6J2/J, which just represents an expected small cross-over temperature near the unstable fixed point. This hasa very interesting consequence for the zero temperature impurity susceptibility, becauseequation (5.90) predicts Xirnp cc l/TK cc J/6J2, which means a large impurity susceptibility close to the periodic chain. The sign of the overall constant of proportionalitycannot be determined, because we are in the open chain fixed point regime T <TK andthe two leading irrelevant operators may partially cancel. This large T = 0 impuritysusceptibility is of little experimental relevance, however, because as soon as the temperature is increased beyond 6J2/ we find ourselves in the periodic fixed point regime, andXimp cc TK/T2 cc 6J2/T which gives a small impurity susceptibility as expected.Chapter 5. Susceptibilities 655.4 A Muon Spin Resonance ExperimentNow that we have a good understanding of the local and impurity susceptibilities, it wouldbe nice to find experimental evidence for the predicted effects. As we mentioned in theintroduction one possible experiment to consider is a Muon Spin Resonance experiment onquasi one-dimensional spin-1/2 compounds, in which the muon would perturb the systemand also probe the system locally at the impurity site[36]. The following discussion willfocus on this particular experiment, but it would of course be possible to use the MonteCarlo results in chapter 6 for other kinds of experiments.5.4.1 Experimental SetupPositive muons are relatively long living particles (r 2.2 x 10—6 sec) which carry apositive charge and a spin-1/2 and have about 200 times the mass of the electron. In a[tSR experiment, a low energy beam of muons is sent into the magnetic material to besampled. The beam is polarized, i.e. all muon spins are pointing in the same directionperpendicular to a small applied magnetic field (see figure 5.24). The muons enter thesetup one at a time, and a timer is started for each muon. Once in the sample, the muonsquickly find chemically preferred sites, and the magnetic moment of the spin precesses inthe local magnetic field. Because weak interactions violate parity, the muon then decaysinto a positron and two neutrinos in a way that the positron is most likely ejected in thedirection of the muon spin. The location where the positron hits one of the detectorsgives some information about the direction of the muon spin before the decay. Giventhis direction and the time spent inside the sample, we can infer the strength of the localmagnetic field in which the muon precessed. Typically about 10,000-100,000 such eventsper second can yield sufficient data within one or two hours. If a small external magneticfield H is applied, this setup effectively measures the local susceptibility of the sample,Chapter 5. Susceptibilities 66Muon Beam with spin upFigure 5.24: Schematic Muon Spin Resonance setup: One Muon at a time enters throughthe thin timer and stops inside the sample where its spin precesses. Decay positrons aredetected in the two counters.weighted with a dipole interaction at the chemically preferred site of the muon.Unfortunately, the impurity susceptibility is not necessarily directly related to thepSR signal since the muon measures the local magnetic field ë, which is the sum of thedipole fields from all spin sites and the applied field H. The dipole moment at each sitej is proportional to the local susceptibility Xi, so that an applied magnetic field in thez-direction HZ results in a magnetic field at the muon site= HZ + >3(. xj) — ZXj (5.96)i riI Iwhere fj is the location of site j relative to the muon. The second term in equation (5.96)is proportional to the so called Knight shift. The measured signal depends crucially on theperpendicular distance d± of the muon from the chain as shown in figure 5.25 as well ason the direction of the applied field HZ relative to the chain. Although we determined thez-component of the local susceptibility, this coordinate is not related to the orientationof the chain. The field H’ can therefore be applied in any direction relative to the chain,in particular parallel, perpendicular, or on a powdered sample. In general, it is useful toCounterTimerMagnetic Field HChapter 5. Susceptibilities 67Figure 5.25: The location of the muon relative to the chain for the link parity symmetriccase.define an “effective” susceptibility at the muon site which is given by the second term inequation (5.96) divided by the applied magnetic field3(• )2 — 1XeffXj..3 (5.97)j 3The different cases for the various field directions will be discussed in chapter 6, butfor simplicity in this section we consider only the effective susceptibility XefF at the muonsite for a powdered sample as an rough indication of the muon signalXeff (5.98)Iwhere we performed an integration over the solid angle in equation (5.97).So far we have discussed how the muon is used as a magnetic probe in the system,but a very important effect of the muon is of course the fact that it acts as an impurityas well. Although we use the dipole interaction to probe the system, we do not expectthat this will produce a significant perturbation on the chain, since it is small comparedto the exchange interaction, the coulomb interaction, and even the interchain coupling.The main effect of the muon therefore is a lattice distortion from inserting a positivewIChapter 5. Susceptibilities 68charge in the system as shown in figure 5.25 for a link symmetric site. It is reasonableto assume that the lattice distortion will drop off at least as fast as the derivative of thecoulomb force i.e. as l/r3 and belongs to the same universality class as a short rangedistortion. In cases where the muon binds directly to ions in the lattice or an electronthe distortion may be even more localized. The effect on the exchange interaction J canonly be roughly estimated by a previous experiment that observed a 20% shift in thehyperfine levels under the influence of the muon[37], which indicates a distortion of theorbitals and therefore a change of J of the same order of magnitude. Lattice distortionsdue to the muon may be in excess of 30%[38]. But even small distortions are bound tocreate large changes in J, since it is known from studies on MnO[39] that 6J/J can bemore than one order of magnitude larger than the lattice distortion 5a/a, where a is thelattice spacing.The magnitude and range of the perturbation will vary from material to material. Itis therefore most appropriate to use a generic model for link and site parity symmetricperturbations, corresponding to one or two equally perturbed links of various magnitude.Although it seems unlikely that the effect of the muon will exactly correspond to one ofthose two perturbations, it is reasonable to expect that it creates a symmetric perturbation. All preceding calculations do not require the knowledge of the actual form ofthe perturbation, but only refer to the leading operator, which is directly related to thesymmetry properties of the perturbation. The renormalization group arguments shouldtherefore be valid for any generic site or link parity symmetric perturbation of the muon.Unfortunately, the muon will primarily probe the “non-universal” boundary region,which is not accessible by the field theory analysis and does depend on the specificform of the perturbation. As a first attempt to calculate the effective susceptibilityquantitatively, we consider the cases of either two equally or one perturbed isotropiccouplings. Depending on the results of the proposed experiment [36], further simulationsChapter 5. Susceptibilities 69may be appropriate. The Monte Carlo data for the iSR experiment in section 6.2 shouldtherefore be taken as a rough estimate of the predicted effects. The detailed signal ofan experiment may look different, but we expect that the size of the corrections to thelocal susceptibility will be determined by the renormalization effects and the alternatingoperator, both of which are large.5.4.2 Field Theory AnalysisLet us first consider the site parity symmetric case, which is modeled by two adjacent,equally perturbed links. We concluded that in this case the logarithmically divergentpart of the impurity susceptibility comes primarily from the local susceptibility of thecentral “impurity” spin in figure 4.12. This followed from the analogy to the two channelKondo effect and should be true for small J’ or for T > TK. Since this central spin isalso the closest to the muon we expect that the 1tSR signal is directly related to theimpurity susceptibility. The induced alternating susceptibility in the chain for small J’will be secondary because the affected sites are further away and the magnitude of ôyjwill be smaller. Nevertheless, this alternating part is interesting from a theoretical pointof view and we expect interesting behavior in the local susceptibility from the MonteCarlo simulations. For small coupling J’ and temperatures T > TK we expect openchain behavior with an induced alternating part from the boundary condition, which ispositive at the ends. The interesting effect occurs when we lower the temperature orincrease the coupling J’ so that we approach the periodic chain fixed point. The centralimpurity spin is then considered part of the chain, but still has a large local susceptibility(i.e. a large < SZ > expectation value when a small magnetic field is applied). But sincethe spin is now considered part of a periodic chain equation (2.24) is valid and this large< SZ > expectation value propagates into the chain. In particular it induces a largealternating part which is of opposite sign of that from the boundary condition, so thatChapter 5. Susceptibilities 70we should be able to observe a cross-over from a regime where the boundary alternatingpart dominates to a regime where the induced alternating part from the impurity spindominates. This behavior is discussed again with the Monte Carlo data in section 6.1.3and figures 6.35 - 6.38.A similar analysis can be applied for the case corresponding to one weak link inthe chain. We now expect that the two sites connected by the weak link dominate theKnight shift. Those two sites have a large contribution to the uniform part of the localsusceptibility which is directly related to the impurity susceptibility according to equation(5.88). However, we now also have to consider the effect of the large alternating part fromequation (5.81), which is of similar magnitude at the two sites. We are somewhat savedby the fact that there is no cancellation of the two parts in this case and the alternatingpart adds to the uniform contribution, so that the muon signal is roughly related to theimpurity susceptibility. If the link is strengthened the effective susceptibility in equation(5.98) is always decreased because the two closest sites in figure 4.9 lock into a singlet atlow temperatures. Their local susceptibility will therefore be lower and this behavior willdominate the effective susceptibility at the muon site. In this case it is not even clearif the impurity susceptibility (which involves a sum over the complete chain) should bepositive or negative as indicated at the end of section 5.2.2. The effect of the perturbationon the effective susceptibility should be large in all scenarios.Note, that only the uniform part of the local susceptibility is dependent on the barecoupling constant J’ while the alternating part is produced by the boundary conditionitself. The uniform part of the local susceptibility of the first few sites near the impuritytherefore should depend on the coupling constant J’ while the alternating part far awayshould be independent at a fixed temperature (provided that J’ is small enough that theopen fixed point description is valid).Chapter 6Monte Carlo ResultsThe field theory treatment of the spin chain is very useful in predicting impurity effectsin quasi one-dimensional spin systems, but unfortunately this analysis cannot give usany quantitative results for the experiments other than the scaling behavior in certainlimits. To tie the theoretical predictions to experiments it is therefore useful to have anindependent method of determining the quantitative effects of impurities. With the helpof the standard quantum Monte Carlo algorithm as described in appendix C we are ableto present these quantitative results, which can provide experimentalists with an initialestimate of the magnitude of the effects and should also be helpful in separating out theimpurity contribution from other effects in the analysis of the experimental data.6.1 Impurity Susceptibility EffectsMonte Carlo simulations always have an inherent statistical error associated with them.In our case there is also a critical slowing down at low temperatures, so that the simulations do not produce useful data for T < J/15 (see also appendix C). Although the localsusceptibilities and the ,uSR signal have been determined with reasonable accuracy, theerror bars of the impurity susceptibility are rather large. Since we have to sum all localsusceptibilities to calculate the overall impurity susceptibility, its error bars are larger bya factor of the square root of the system size (i.e. one order of magnitude for 1 48— 128).As an example, the temperature dependence of the open chain impurity susceptibilityand the local susceptibility correction (x— x) of the site closest to the impurity have71Chapter 6. Monte Carlo Results 72been plotted in figure 6.26. The error bars of the local susceptibility are much smaller,but unfortunately we can normally not extract the impurity susceptibility directly fromthis quantity, because it also contains an unspecified alternating contribution, which isapparently large. As predicted, the impurity susceptibility seems to approach a constantpositive value as T —k 0. The local susceptibility at the first site seems to be roughly related to this impurity susceptibility, but with smaller error bars. The alternatingcontribution adds so that the signal at the first site is larger than the uniform impuritysusceptibility. However, the accurate determination of the zero temperature impuritysusceptibility is difficult, and the computer simulations for this quantity can only givea trend and a consistency check of our analysis in sections 5.3.1 and 5.3.2. Figure 6.26gives a good estimate for the approximate error bars in general for the local and impuritysusceptibilities, respectively. To make the presentation of the Monte Carlo data less confusing, the error bars of figures 6.27 - 6.38 in this section have been omitted and shouldsimply be taken from figure 6.26.6.1.1 One weak linkFigure 6.27 shows the impurity susceptibility of an open chain which has been slightlyperturbed with a coupling J’ across the open ends. The large error bars as given infigure 6.26 make this Monte Carlo data not unambiguous, but the findings seem to beconsistent with equation (5.95) which predicts a change in the impurity susceptibilityproportional to J’. We believe that the apparent crossing for some parameters J’ is onlyproduced by the large uncertainties.The more interesting case is a small weakening by 6J of one link from the periodicchain. At intermediate temperatures TK <T < J the impurity susceptibility should beproportional to 6J2 according to section 5.3.2. Figure 6.28 confirms an increase of theimpurity susceptibility in this temperature regime, but higher order irrelevant operatorsChapter 6. Monte Carlo Results 730.50.450.40350.30.2502E0.150.10.050Figure 6.26: The open chain impurity susceptibility as a function of temperature. Thesolid line is only drawn for visual guidance and does not necessarily reflect an accurateestimate.TMChapter 6. Monte Carlo Results 740.3J.= 0= o1J AJ’=0.25JJ’= 0.5J0.2502ee0.15AI : ODAAAA01 Ax a Ax0.05 xX) X00 0.2 0.4 0.5 0.8 1TIJFigure 6.27: The impurity susceptibility for a small coupling J’ across the open ends asa function of temperature.seem to alter the scaling dependence, which appears to be linear with bJ. Perturbationtheory in the lattice model actually does predict a linear dependence on 6J. The scalingprediction with 6J2 comes from perturbation theory in the field theory Hamiltonian withthe leading relevant operator only, which does not seem to be valid at the intermediatetemperatures O.3J <T < J. Other irrelevant operators [e.g. TL(O) from equation (3.39)]become important in this regime, which do contribute to first order in perturbation theoryand also have coupling constants of order 6J. Hence, the linear change with 6J in theimpurity susceptibility is consistent with the field theory analysis.For temperatures below the small cross-over scale TK cc 6J2/ we expect that theimpurity susceptibility should go as J/6J2. If a negative proportionality constant isassumed, the numerical findings are consistent with this prediction. However, our MonteCarlo data is not good enough to give a reliable confirmation of the sharp cross-over atChapter 6. Monte Carlo Results 750.10.080.060.04. 0.020-0.02-0.04-0.06-0.08-0.1-0.12Figure 6.28: The impurity susceptibility for a small perturbation 6J of one link in thechain as a function of temperature. The solid lines are only drawn for visual guidanceand do not necessarily reflect an accurate estimate.very low temperatures as we approach the periodic chain.6.1.2 Two weak linksAs discussed in section 5.3.1 we expect a divergent impurity susceptibility as T —* 0for any finite weakening of two adjacent links. For small perturbations 6J, equation(5.94) predicts a scaling of the impurity susceptibility with 6J2, which is consistent withfigure 6.29. However, the large error bars in figure 6.26 also make this Monte Carlo datasomewhat ambiguous. It is therefore instructive to look at the correction to the localsusceptibility of the central spin closest to the impurity in figure 4.12, because of thesmaller error bars associated with local susceptibilities. This central spin is equivalentto the “impurity spin” in the Kondo problem and presumably carries the divergent partof the impurity susceptibility. Figure 6.30 seems to indicate a saturation of the impurity0 0.2 0.4 0.6 0.8T!JChapter 6. Monte Carlo Results 76susceptibility at TK 0.3J, 6J = 0.1J, which might be interpreted as cross-over froma Curie-law behavior to a weaker logarithmic scaling. This would imply that the crossover temperatures for all other coupling constants considered are too small to observe,because TK x J/6J2. This would indicate that the “healing” process as described insection 4.2 is very slow, as might be expected from the dimensionalities of the leadingoperators, which are only marginal relevant and only weakly irrelevant at the two fixedpoints. Figure 6.30 also seems consistent with equation (5.94).Figure 6.31 shows the impurity susceptibility for small bare coupling constants J’to the impurity spin. It is again instructive to look at the local susceptibility of the“impurity spin”, which can be compared with the Curie-law behavior of a completelydecoupled spin as shown in figure 6.32. Since the cross-over temperature is much lowerthan the lowest accessible temperature, we cannot observe the predicted behavior with_eth/J’ ln T, but the Monte Carlo data certainly is consistent with this scaling function.6.1.3 Alternating PartsPredictions for the alternating part of the impurity susceptibility can be tested much moreeasily, because it only involves local susceptibilities with much smaller uncertainties. Thefunctional dependence of the alternating part has already been confirmed in figure 5.22,but it is interesting to note that the strong staggered behavior persists even for smallperturbations of one link in the periodic chain as shown in figure 6.33. In fact, we expectthe staggered part to be asymptotically independent of the bare coupling constant inthe limit T << TK and x > v/TK, which is confirmed in figure 6.34. At T = J/15, thestaggered part is only very weakly dependent on J’ as long as J’ is small (which alsomeans v/TK is small). For perturbations of the order of J’ 0.5J we find a differentamplitude at short distances from the boundary, but far away from the boundary allChapter 6. Monte Carlo Results 770.12J’=0.75JJ 0.9J0.10.080.06•0D 0.040x •OxE 0.02 xx xx0 .. -....- .. -....- -....-x-0.02-0.040 0.2 0.4 0.6 0.8TIJFigure 6.29: The impurity susceptibility for a small perturbation 6J of two links in thechain as a function of temperature.0.2 I I++ J=0.75J +J’= 0.9J x0.18 +++0.16 +++0.14 +0.12 ++0.10.08 +0.06x xXx X X X +0.04 Xx0.020 I0 02 0.4 0.6 0.8T/JFigure 6.30: The local susceptibility correction of the central spin closest to the impurityfor a small perturbation 6J on two links in the chain as a function of temperature.•1C0>%80Figure 6.32: The local susceptibility of an impurity spin coupled with a small perturbationJ’ to the open ends of the chain as a function of temperature.Chapter 6. Monte Carlo ResultsJ,= o.1JJ’O.25JJ= O.5J1.61.41.20.80.60.40.20xxxGxxxaxaa780.6 0.8T/Ja small coupling J’ of the open ends to anxaDODxa0xa0 0.2 0.4Figure 6.31: The impurity susceptibility forimpurity spin as a function of temperature.2.521.50.50TIJChapter 6. Monte Carlo Results 790.30.25020.15:1 :0-0.05-0.150Figure 6.33: The local susceptibility as a function of distance from a weakened linkJ’ = O.75J at T = J/15.curves seem to follow a universal shape within the error bars (see figure 6.26 for anestimate of the error bars of the local susceptibility).For site parity symmetric perturbations, on the other hand, we expect the oppositebehavior in the limit T << TK and x > v/TK, corresponding to no alternating part dueto the boundary condition. Unfortunately, we cannot test this behavior because of thelow cross-over temperatures, but we can observe a competition between the alternatingpart due to the boundary condition (the “boundary contribution”) according to equation(5.81) and the alternating part which has been induced by the impurity spin accordingto equation (2.24) as discussed in section 5.4. The boundary contribution is positive atthe first site away from the impurity, while the induced alternating part from the centralspin is negative. The induced alternating part becomes stronger as the temperature islowered, but drops off fast with 1/x where x is the distance from the impurity. The0 5 10 15 20 25 30 35 40 45SitesChapter 6. Monte Carlo Results 800.51::T::<0—0 5 10 15 20 25SitesFigure 6.34: The alternating part of the local susceptibility as a function of distance fromthe weakly coupled link J’ across the open ends at T = J/15.boundary contribution decreases as J’ —f J, but always dominates for larger distancesfrom the impurity (unless T <<TK). This behavior is demonstrated in figures 6.35 - 6.38.For bare coupling constants close to the open chain J’ —* 0 the boundary alternatingpart completely dominates, but as we get closer to the periodic chain J’ —+ J we onlysee a small boundary alternating part far away from the impurity while the inducedalternating part dominates near the impurity site.6.2 Muon Knight ShiftSince the measured Knight shift depends strongly on the direction of the applied magneticfield we want to distinguish three cases of interest: field direction perpendicular or parallelto the chain or a powdered sample. The signal also depends on the perpendicular offset(distance) d1 of the muon from the chain as shown in figure 5.25, but generally it isJ,= 0 e3.. 0.1J +J’..0.25J DJ 0.5J x• D+o th,th,0+ xxxxx x0xx30 35 40 45 50Figure 6.35: The local susceptibility as a function of distance with the open ends coupledwith I’ = O.1J to an impurity spin at the first site at T = J/15.Figure 6.36: The local susceptibility as a function of distance with the open ends coupledwith J’ = O.25J to an impurity spin at the first site at T = J/15.Chapter 6. Monte Carlo Results 813.532.5211.50.50-0.50 5 10 15 20 25 30 35Sites4002.521.50.50-0.50 5 10 15 20 25 30 35 40SitesI03-JFigure 6.37: The local susceptibility as a function of distance with thewith J’ = O.5J to an impurity spin at the first site T = J/15.open ends coupledC0-JI 5 10 15 20 25 30 35 40SitesChapter 6. Monte Carlo Results 820.80.70.60.50.40.20.10-0.10 5 10 15 20 25 30 35 40Sites0.350.30.250.20.150.10.050Figure 6.38: The local susceptibility as a function of distance from two slightly weakenedlinks J’ = O.75J at T = J/15.Chapter 6. Monte Carlo Results 83sufficient to choose only two different values for d± corresponding to one-half or one unitsof lattice spacing in order to get a good impression on the impurity effects. The distance ofthe individual sites from the muon along the chain d is always taken to be either half-odd-integer or integer values for link or site parity symmetric locations, respectively. Figure5.25 shows the geometrical arrangement with the example of a link parity symmetriclocation. We can simply use the effective susceptibility in equation (5.97) to derive theexpected muon signal from the local susceptibilities in the Monte Carlo simulations. Sincethis effective susceptibility is to be compared with the unperturbed bulk susceptibility infigure 5.17, we always consider Xeff normalized by a geometrical factor ‘y, which is simplyobtained by setting all xj = 1 in equation (5.97)7— 1 (6.99)The case of the powdered sample has been discussed in section 5.4 where angular averaging yielded equation (5.98). Using equation (5.97) we can immediately compute theeffective susceptibility at the muon site for the other two cases. If the field is appliedparallel to the chain, we find:Xeff = > (3d/j1jI2 — 1) i*fr (6.100)where j is the location of the spin at site j relative to the muon and d is the horizontaloffset along the chain of site j from the impurity (i.e. I = + dy). The analogousexpression for an applied field perpendicular to the chain isXeff = (3d/I2—1) (6.101)Note, that the arithmetic mean of the expressions for the two field directions reducesto the effective susceptibility of a powdered sample. We should mention that there is apossible fourth case when the applied field is perpendicular to both the chain orientationFigure 6.39: The effective normalized susceptibility in a powdered sample for small perturbations on one link and d± = 0.5 as a function of temperature.and the vertical offset d1. In this case the normalized effective susceptibility reduces tothe effective susceptibility of the powdered sample in equation (5.98), so that we do notneed to discuss it separately.6.2.1 One perturbed linkFigures 6.39 and 6.40 show the effective susceptibility at the muon site with a verticaloffset of d = 1/2 and d = 1 in units of lattice spacing, respectively, in a powderedsample for a small perturbation on one link of the periodic chain. The error bars in figure6.39 will serve us as an estimate for the relative error of the predicted muon signal infigures 6.39- 6.73.The maximum seems to be shifted to lower temperatures with increasing perturbation, and the overall amplitude of the signal is increased. Apparently, the predictedChapter 6. Monte Carlo Results 840.190.180.170.160.150.14LU0.130.120.110.10 02 0.4 0.6 0.8TIJchapter 6. Monte Carlo Results 85Cd,LCdFigure 6.40: The effective normalized susceptibility in a powdered sample for small perturbations on one link and d± = 1 as a function of temperature.renormalization of the weakened link is responsible for the observed effects. In particular,we expect the contributions of both the impurity susceptibility and the alternating partto increase as we lower the temperature as shown in figure 6.26. In addition the renormalization of the weak link across the open ends enhances this effect. These increasedsusceptibility contributions at low temperatures are responsible for the shifted maximumand the larger overall signal. In a 1iSR experiment, the location of the maximum as wellas the ratio of the maximum susceptibility to the low temperature susceptibility will givean indication of the strength of the perturbation.The opposite effect is observed when one link is strengthened as shown in figures 6.41and 6.42. The maximum seems to be shifted to lower values and the overall amplitudeis reduced. This effect has to be attributed to the formation of a singlet by the twostrongly coupled spins. The observed upturn of the signal at low temperatures on the0 0.2 0.4 0.6 0.8TIJChapter 6. Monte Carlo Results 86wFigure 6.41: The effective normalized susceptibility in a powdered sample for onestrengthened link and d± = 0.5 as a function of temperature.other hand might be a renormalization effect of the small virtual coupling across theopen ends. When the perturbation is strong enough to produce bare coupling constantsthat correspond to the open chain (i.e. J’ small) a complete vanishing of the maximumis observed as shown in figures 6.43 and 6.44. In this case the renormalization effectsdominate even at intermediate temperatures. All observed effects become weaker as thedistance d± of the muon is increased, because bulk spins will influence the signal morefor larger distances d±.If a magnetic field is applied perpendicular to the orientation of the chain the samequalitative picture can be observed, as shown in figures 6.45 - 6.50. Equation (6.101)indicates that the spins closest to the impurity are sampled much more, because the firstterm drops off with Therefore, the effects which we observed for a powderedsample are much more pronounced. However, the upturn of the signal for a strengthened0 0.2 0.4 0.6 0.8T/JIIFigure 6.42: The effective normalized susceptibility instrengthened link and d± = 1 as a function of temperature.a powdered sample for one.0a,0w0.55 I I I P I IJ= 0J.I, O.1J +J=0.25J0.5 j= o.sJG0.45::L015 I I0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9T/J87Chapter 6. Monte Carlo Results0.150.140.130.120.110.10.090.080.070 0.2 0.4 0.6 0.8T/J0Figure 6.43: The effective normalized susceptibility in a powdered sample for strongperturbations on one link and d± = 0.5 as a function of temperature.Chapter 6. Monte Carlo Results 88C’,wFigure 6.44: The effective normalized susceptibility in a powdered sample for strongperturbations on one link and d± = 1 as a function of temperature.link at low temperatures is not observed in figures 6.47 and 6.48 since this effect comesfrom the second sites in the chain.Figures 6.51 - 6.53 show the analogous effects for an applied field parallel to thechain. The same features can be observed, but again with different magnitude (thisanisotropy effect might help to determine the actual location of the muon in the sample).If, however, d± is chosen so that the geometrical normalization factor y for equation(6.100) or (6.101) becomes very small, we can see very pronounced impurity effects.Since the absolute signal of the unperturbed susceptibility is very small in this case,small perturbation can produce big relative changes of arbitrary sign. As an example,we simulated a vertical offset of d± = 1 which reduces the geometrical factor -y for a fieldparallel to the chain by a factor of 200 compared to d = 0.5. The resulting signal isshown in figures 6.54 - 6.56 which shows completely different behavior than the previousT/JC0CuJChapter 6. Monte Carlo Results 890.24022020.180.160.140.120.10 0.2 0.4 0.6 0.8T/JFigure 6.45: The effective normalized susceptibility for an applied field perpendicular tothe chain, small perturbations on one link and d± = 0.5 as a function of temperature.0.22020.180.160.140.120.1TI.)Figure 6.46: The effective normalized susceptibility for an applied field perpendicular tothe chain, small perturbations on one link and d± = 1 as a function of temperature.Chapter 6. Monte Carlo ResultsI90Figure 6.47: The effective normalized susceptibility for an applied field perpendicular tothe chain, one strengthened link and d± = 0.5 as a function of temperature.IFigure 6.48: The effective normalized susceptibility for an applied field perpendicular tothe chain, one strengthened link and d± = 1 as a function of temperature.0 0.2 0.4 0.6 0.8TIJ0.160.140.12T/JChapter 6. Monte Carlo Results0.80.70.60.50.402020.10 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9T/J91Figure 6.49: Thethe chain, strongLiieffective normalized susceptibility for an applied field perpendicular toperturbations on one link and d± = 0.5 as a function of temperature.0.650.60.550.50.450.40.350.3025020.15Figure 6.50: The effective normalized susceptibility for an applied field perpendicular tothe chain, strong perturbations on one link and d± = 1 as a function of temperature.0 eJ.= 0.1J +J-0.25J 111’— 0.5.1 xS.V.1Chapter 6. Monte Carlo Results 92JFigure 6.51: The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on one link and d± = 0.5 as a function of temperature.figures.6.2.2 Two perturbed linksLet IIS now consider the analogous cases for a site symmetric perturbation on two adjacentlinks in the chain. The muon signal will be dominated by the strong local susceptibilityof the impurity spin for any weakened coupling 0 < J’ < J. Monte Carlo simulationsconfirm this picture for a powdered sample in figures 6.57 - 6.60 (see also figures 6.30 and6.32). The measured effect again decreases with distance d±. The maximum can onlybe observed for small perturbations and is shifted by large amounts even for J’ = 0.9Jsince the strong logarithmic impurity susceptibility dominates the behavior for any largerperturbation. This behavior is much stronger than the corresponding cases for one weaklink, which may be somewhat surprising, since we expect healing of the chain in this0.2 0.4 0.6 0.8TMChapter 6. Monte Carlo Results.0I0.150.140.130.120.110.10.090.0893Figure 6.52: The effective normalizedchain, one strengthened link and d1 =.0Liisusceptibility for an applied field parallel to the0.5 as a function of temperature.Figure 6.53: The effective normalized susceptibility for an applied field parallel to thechain, strong perturbations on one link and d1 = 0.5 as a function of temperature.T/J0.2 0.4 0.6 0.80 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9TMChapter 6. Monte Carlo Results 94IFigure 6.54: The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on one link and d± 1 as a function of temperature..0IFigure 6.55: The effective normalizedchain, one strengthened link and d± =susceptibility for an applied field parallel to the1 as a function of temperature.TIJTIJ02 0.4 0.6 0.8Chapter 6. Monte Carlo Results 950-0.5-1-1.5-2-2.5Figure 6.56: The effective normalized susceptibility for an applied field parallel to thechain, strong perturbations on one link and d± = 1 as a function of temperature.scenario. However, as we saw in section 6.1.2 the healing process is very slow so that theimpurity susceptibility is initially Curie like and apparently becomes very large beforethe renormalization effects contribute. Even after complete renormalization to a healedchain the remaining impurity susceptibility is still logarithmic divergent.A strengthening of two links also produces a significant change in the signal, corresponding to a shifted maximum to higher temperatures, an overall lowered signal, and acurious downturn at low temperatures as shown in figures 6.61 and 6.62. Although wedo not have any reliable renormalization arguments for this case, the Monte Carlo dataprovides an interesting estimate for the pSR signal.The Monte Carlo data for the two field directions perpendicular and parallel to thechain in figures 6.63 - 6.73 give virtually identical results because it is always the impurityspin that dominates the behavior. For a special choice of d = 1 and a field parallel to0 0.2 0.4 0.6 0.8T/JChapter 6. Monte Carlo Results 9602 I I I1= J —J’=0.75.i 00 0.4 0.6 0.8 1T/JFigure 6.57: The effective normalized susceptibility in a powdered sample for small perturbations on two links and d± = 0.5 as a function of temperature.the chain, the geometrical factor ‘.y is again very small. The effect of perturbations istherefore artificially inflated as shown in figures 6.71 - 6.73. The effect is only largerelative to the unperturbed signal, but small in absolute terms.6.3 ConclusionsIn summary we have managed to analyze the interesting renormalization behavior of impurities in quantum spin-1/2 chains and their effects on the susceptibility (which turnedout to be quite exotic in some cases). Numerical simulations are always consistent withour analysis and support the validity of the methods of boundary critical phenomena.We were able to propose a tSR experiment on quasi one-dimensional spin-1/2 compounds, which might be able to show some of the predicted effects. Since the impurityeffect of the muon will be strongly dependent on the particular material, we do not expectDCoLiiChapter 6. Monte Carlo Results 97020.190.180.170.160.150.140.130.120.110.10Figure 6.58: The effective normalized susceptibility in a powdered sample for small perturbations on two links and d± = 1 as a function of temperature.02 0.4 0.6 0.8T/J3.532.521.50.500.5T/JFigure 6.59: The effective normalized susceptibility in a powdered sample for strongperturbations on two links and d± 0.5 as a function of temperature.0.2 0.4 0.6 0.8TIJChapter 6. Monte Carlo Results 982.521.5110.500 1Figure 6.60: The effective normalized susceptibility in a powdered sample for strongperturbations on two links and d± = 1 as a function of temperature.j0:-0.05-0.1 I I0.8 1T/JFigure 6.61: The effective normalized susceptibility in a powdered sample for twostrengthened links and d± = 0.5 as a function of temperature.//0 0.2 0.4 0.6Chapter 6. Monte Carlo Results 99I0.160.140.120.10.080.060.040.0200.40.350.30.250.20.150.10Figure 6.63: The effective normalized susceptibility for an applied field perpendicular tothe chain, small perturbations on two links and d± = 0.5 as a function of temperature.:1Figure 6.62: The effectivestrengthened links and d =0.2 0.4 0.6 0.8T/.Jnormalized susceptibility in a1 as a function of temperature.powdered sample for two0.2 0.4 0.6 0.8T/J=.00=wnjChapter 6. Monte Carlo Results 100.0IFigure 6.64: The effective normalized susceptibility for an applied field perpendicular tothe chain, small perturbations on two links and d± = 1 as a function of temperature.0.15 IT/JFigure 6.65: The effective normalized susceptibility for an applied field perpendicular tothe chain, two strengthened links and d1 = 0.5 as a function of temperature.0 02 0.4 0.6 0.9Figure 6.67: The effective normalized susceptibility for anthe chain, strong perturbations on two links and d± = 0.5applied field perpendicular toas a function of temperature.Chapter 6. Monte Carlo Results 101I0.150.10.050-0.05-0.10 0.2 0.4 0.6 0.8 1T/JFigure 6.66: The effective normalized susceptibility for an applied field perpendicular tothe chain, two strengthened links and d± = 1 as a function of temperature.LU0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9TIJChapter 6. Monte Carlo ResultsFigure 6.68: The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d± = 0.5 as a function of temperature.Figure 6.69: The effective normalized susceptibility for an applied field parallel to thechain, two strengthened links and d± = 0.5 as a function of temperature.102I0.50450.40.350.30.250.20.150.10 0.2 0.4 0.6 0.8T/J0.8LU0 02 0.4 0.6 0.8T/J.0wChapter 6. Monte Carlo Results 103Figure 6.70: The effective normalized susceptibility for an applied field parallel to thechain strong perturbations on two links and d± = 0.5 as a function of temperature.1.81.61.41.20.80.60.40200 0.2 0.4 0.6 0.8T/JFigure 6.71: The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d± = 1 as a function of temperature.Chapter 6. Monte Carlo Results 104.0I0 0.2 0.4 0.8 0.8T/JIFigure 6.72: The effective normalized susceptibility for an applied field parallel to thechain, two strengthened links and d1 = 1 as a function of temperature.302520Is10500 02 0.4 0.6 0.8T/JFigure 6.73: The effective normalized susceptibility for an applied field parallel to thechain, small perturbations on two links and d1 = 1 as a function of temperature.Chapter 6. Monte Carlo Results 105that the numerical simulations of our generic models will be able to describe the expectedsignal in full detail. However, we are confident that our Monte Carlo data provides arough quantitative estimate of the effects and describes the generic renormalization behavior, which seems to depend only on the symmetry properties of the perturbation (e.g.the shift of the maximum and the increase of the signal can most likely be observed formost any link symmetric weakening of links in the chain.) The Monte Carlo simulationscan be extended for any material once the location of the muon and its effect on thechain have been estimated.The main conclusion which can be drawn from the field theory analysis as well asfrom the numerical simulations is the fact that impurities have a surprisingly large effecton the susceptibility in these one-dimensional systems. The impurity effects on themeasured Knight shift for a muon in the chain are of the same order of magnitude asthe unperturbed susceptibility and may even dominate the signal. On the other hand, intSR experiments on higher dimensional magnetic compounds the impurity effect of themuon is neglected in most cases.Appendix AField Theory FormulasIn this appendix we would like to give a short summary of the central (1+1) dimensionalfield theory formulas, which have been used in this thesis. These formulas are explainedand derived in references [18] and [40] in a pedagogical way. Reference [10] may also beof some use. vVe will not attempt to reproduce the derivation in a compressed form, sinceit is not possible without assuming a common background.Green’s functions of left- and right-movers always separate into a product, unlessthey are related by a boundary condition. We can therefore look at the expressions forleft- and right-movers separately and simply add the scaling dimensions d = dL + dR.The two-point boson Green’s function can be calculated directly from the infinite lengthmode expansion:<L(x,t)L(0,0)> = —lni(x+vt)+const.<R(X,t)R(0,0)> = —lni(x—vt) +const., (A.102)where the constant is cutoff dependent. The Green’s function for the boson derivative isgiven by the second derivative of equation (A.102)/8L \ 11 = —__________\ Ox Ox / 4ir(x+vt)2= 47r(x— vt)2 (A.103)so that the usual fermion current scaling dimension d = 1 is recovered. Equation (A.102)106Appendix A. Field Theory Formulas 107also implies a scaling dimension of dL 72/8 for eL, since(ei71 (2,t) e_i7:00)) x e[,t)_(0,0)] :)e72<,t)(0,0)>/. 72/47re_1 +vt)-y2/4ir=—z (A.1o4)\x + vtlwhich is in complete agreement with bosonization in equation (2.9) and fermionic Green’sfunctions. This equation can be generalized to/ . \ 7i7k/8r—z 1 , (A.105)jk \Z ZkJwhere z = x + vt2. This relation is useful for equation (3.35). We also see that singlepowers of e’ have a vanishing expectation value.From equation (2.10) it is clear that the commutator of the left- and right-movingbosons is i/4 which is essential for the bosonization formulas (2.9) to work. For fixedboundary conditions the commutation relation is modified at the origin because left-and right-movers are related there. It takes some careful analysis of the finite lengthmode expansion in equation (3.48) to obtain the correct value of the commutator atthe boundary. This calculation has been done by Eugene Wong (unpublished), and theresults are[L,R]= 0,,x=y=lelse (A.106)Conformal transformations are represented by analytic functions in the complex plane(z), and chiral primary operators °L are defined as operators which transform as/ \dLi dw \(DL(z) —*—) OL(w), (A.107)Appendix A. Field Theory Formulas 108where dL is the scaling dimension of (DL. In this sense conformal transformations generalize scale transformations, Lorentz transformations and translations all of which can berepresented by an analytic function w(z) in the complex plane for (1+1) dimensions. Oneapplication of equation (A.107) is the the transformation w(z) =e2’’ which gives thewell known result in equation (5.64) for finite temperature correlation functions. Note,that is periodic in imaginary time with radius 6.Conformal invariance dictates a unique form for three-point Green’s functions of primary operators(O(zi)O(z3) (z cc (zi —z2)_1 2_d3)(zi —3)_13_d2)(z2—z3)_23_d1)(A.108)where d, i 1,2,3 are the scaling dimensions of the three primary operators Of andz=z + vt.Appendix BExact Diagonalization AlgorithmThe algorithm we used starts from a normalized initial trial vector ‘I’ and successively minimizes the energy expectation value in each iteration step by forming the linearcombination I’2 = bill1 + Hil’1. The explicit formulae areb=— + 424), (B.109)so that the normalized improved ground state is given by= b1+H1• (B.11O)(2+2b<H>+<H>)rwhere= <H2>—<H>= <H><H>—<H>= <H2—<H><H3 (B.111)Here, we used the notation < H > = < ‘T! H’j li >. The new energy expectationvalue is given by+ 4z24= <H> — (B.112)b2+2b<H>+<HThe algorithm terminates when we are close enough to the ground state so that theenergy cannot be lowered much further. Clearly, all symmetries of il’ are preserved ineach step, so the algorithm can be used to find ground states in different sectors of H,which are characterized by convenient quantum numbers.109Appendix B. Exact Diagonalization Algorithm 110We decided to work in the orthonormal Si-basis because the next nearest neighborcoupling requires excessive computations in the valence bond basis[41], which keeps trackof the total spin. The basis states can be represented by integer bit strings, and theHamiltonian was implemented as a procedure that manipulates and then stores the bitstrings and their coefficients as they are created. For numerical convenience we usedthe Exchange Hamiltonian, which differs by a factor of two and a constant from theHeisenberg Hamiltonian in equation (1.1) with J = J. The various tricks to optimizethe algorithm include a hashing technique[15], extrapolation to the exact ground state,and reusing previously created information on how to update basis states. The resultingground state can be used as an initial starting state for a similar Hamiltonian with onlyslightly modified parameters. The extrapolation is based on the fact that the actualground state energy is approached exponentially and simply uses the last three iterationvalues to find an improved result. This can give reliably at least two more digits accuracy.In all problems the translational invariance is broken by the impurity. Taking intoaccount the limited remaining available symmetries of our problem, we can handle onlyabout 22 sites on a SUN workstation (about 8 sites less than what can be done for aperiodic chain in the valence bond basis). Some calculations were done on a NEC SX3/44supercomputer because supercomputers generally allow for about four more sites. Working in the valence bond basis with s = 0 and using translational and parity invariance wecan find the exact ground state to 8 digits accuracy of a periodic chain of 24 sites in only15 seconds CPU time on a NEC SX3/44 supercomputer. This needs to be comparedto 20 mm CPU time on a SPARCstation2 when working in the Se-basis on the sameproblem.Appendix CMonte Carlo AlgorithmThe goal is to make predictions of measurements of physical quantities expressed in termsof quantum mechanical operators at finite temperatures. The expectation value of anyoperator A is given in terms of the partition function Z = Ek <kIelmIk>:<A> = k < kAeIk> (C.113)where the sum is over all basis states k > of the system and = 1/T. For example, thelocal susceptibility at site i would be given by:Xi== <SS> (C.114)To calculate these sums exactly, we would first need to diagonalize H in the Hubertspace to express exp(—,3H) and then sum over all eigenstates. Since the dimension ofthe Hilbert space grows exponentially with the system size 1, the exact diagonalization ofH cannot be done for more than N 30 spins as indicated in appendix B. In additionwe would have to sum over all eigenvalues to get a finite temperature expression.To get around this problem of large Hubert spaces we use the quantum Monte Carlomethod. We first transform the expression in equation (C.113) so that the exponential ofthe Hamiltonian can be evaluated approximately by use of the Trotter formula. We cansplit the Hamiltonian (1.1) into two parts H He + H0, where the two parts involve onlythe links.with i even or odd, respectively. The two parts are easily diagonalizedseparately, but they do not commute. Although exp(—H) exp(—3He) exp(—3H0,111Appendix C. Monte Carlo Algorithm 112we can write an approximate (Trotter) formula for the expression in (C.113):<klAe”Ik> (C.115)< kAe2 e/mlli >< lile_20/mIl >< 121 u1rn_i >< lm_ile_2o/mIk>This approximation becomes exact in the limit m — oo. We have inserted rn—i completeset of states l > between each of the exponentials and we have to sum over all possiblecombinations. While it is now relatively easy to compute the exponentials, we now havethe formidable task of summing over(2N)m possible configurations. Here we make use ofthe principle of Monte Carlo integration: we statistically evaluate the sum by randomlysumming expression (C.115) over a lot of different configurations l > and assume thatthe average value will give us a good estimate of the complete sum.This method is assisted by the fact that most configurations give a zero contributionbecause the expectation values < lle1I +i > are only non-zero for states that obeycertain selection rules. In fact we can evaluate the sums by going between allowedconfigurations by using “moves” in configuration space which explicitly obey the selectionrules. By just using two different “moves” we can reach any allowed configuration, whichis sufficient to assure a complete sampling.Furthermore, by assigning each “move” from an old configuration to a new one aprobability which is proportional to the change in the exponential<newIe”lnew+i ><oldIemlold+i >we automatically take care of evaluating the expression (C.115) without having to weigheach contribution separately. This method is called importance sampling. The physicalidea is that the system spends most time in configurations for which the partition function Z is large. After a certain number of moves we can make a “measurement” of aquantity by applying the corresponding operator A on the given configuration. SuccessiveAppendix C. Monte Carlo Algorithm 113measurements may not be completely independent of each other unless a large numberof moves have been made. Just like in a physical system, the measurement of a quantityA has a statistical distribution, and the mean would correspond to the expectation value<A> in equation (C.113).The only approximation we have made corresponds to equation (C.115), where alarger m. will yield better results. Fortunately, it is known[42] that the measurementsof < A > will have a leading correction of order 1/rn2, so that we can extrapolateanalytically to the infinite m limit. This result follows directly from the accuracy of theTrotter formula itself.On the computer this method is implemented in a very straightforward way. Asindicated in equation (C.115) a complete configuration consists of m states, each ofwhich has N spins which can point up or down. A coñplete configuration is thereforesimply represented by an N x in array of boolean variables (“spins”). We select twopossible moves: local and global. The local move may flip 4 adjacent spins dependingon their configuration, and the global move may flip all spins at a given site, but forall in states simultaneously. The probabilities of these moves are known for a giventemperature T = 1//, and a good random number generator determines if a possiblemove will be executed. To derive the detailed nature of the moves and probabilities ittakes some straightforward, but lengthy analysis of the physical system. This is omittedhere because it is rather technical and not important for the understanding of the method.For more details see for example reference [42].The update of the array is done in “sweeps” which consist of offering moves at allpossible locations in the array. The actual measurements of Xi from equation (C.114)take less time and are done after each sweep. Good results at lower temperatures requirelarger arrays (generally 2m N > 1OJ/T is a good estimate). Typically the array hasdimensions of N = 48 — 128 by m = 24 — 64. Both m and AT are even numbers, becauseAppendix C. Monte Carlo Algorithm 114periodic boundary conditions in the m direction are required. The number of offeredmoves per sweep is roughly given by m x N.Starting from any allowed configuration, a few thousand sweeps are done to bring thesystem into equilibrium. To get one good average value we then did 2,000,000 sweepsand measurements for a given array. For a choice of coupling and temperature, 3-5 ofthose average values have to be taken at different values for m, so that a reasonableextrapolation m —* can be done. For one measurement point a total of roughly3 x 1010 moves have to be checked and will be offered if possible. How many of thesepossible moves will actually be executed strongly depends on the temperature. At highertemperatures moves are more likely to be performed, which results in a faster updateand lower statistical errors.It is obvious that the largest possible number of sweeps is desired, so that statisticalerrors can be kept to a minimum. For this reason, I ported my program to the vectorsupercomputer Fujitsu VPX24O of the HPC Centre, Calgary. The sweeps to update thearray take most of the computer time, and it is highly desirable to offer and performthe moves in a vectorized way. This is not straight forward, because each move altersthe array and this may affect the outcome of a successive move. It took some extensiveanalysis of the detailed nature of the moves to identify the dependencies, but finally itwas possible to separate the array into sublattices, so that all moves can be offered andperformed in vectorized loops.Bibliography[1] H. Bethe, Z. Physik, 71, 205 (1931)[2] C.N. Yang, C.P. Yang, Phys. Rev. , 327 (1966).[3] M. Takahashi, Prog. Theor. Phys. 4, 401 (1971); M. 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