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UBC Theses and Dissertations

Field induced phase transition in one dimensional Heisenberg antiferromagnet model studied using density… Gustainis, Peter 2017

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FIELD INDUCED PHASE TRANSITION IN ONE DIMENSIONAL HEISENBERGANTIFERROMAGNET MODEL STUDIED USING DENSITY MATRIX RENORMALIZATIONGROUPbyPETER GUSTAINISB.Sc., The University of Waterloo, 2015A THESIS SUBMITED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2017© Peter Gustainis, 2017AbstractThis thesis examines the Heisenberg antiferromagnetic spin chain in one dimension (1D) witha crystal field splitting term and applied magnetic field term. We use theoretical techniques fromquantum field theory and conformal field theory (CFT) to make predictions about the excitationspectrum for our model. We then use Density Matrix Renormalization Group (DMRG) numericaltechniques to simulate our spin chain and extract the energy spectrum as we vary our crystal fieldsplitting and magnetic field terms. These results are compared and we examine where theoreticalcalculations accurately describe our system. This work is motivated by recent experimental work doneon SrNi2Vi2O8by Bera et al. [1] which is a quasi-1D material with weakly coupled spin chains in thebulk. These 1D chains are expected to be described by the Hamiltonian we study in this thesis, and weneglect interchain coupling. We first consider our system where the crystal field splitting term is setto zero, which can be described theoretically using a mapping to the non linear sigma model (NLSM).Near the critical field, it undergoes a Bose condensation transition whose excitation spectrum can bemapped to non-interacting fermions in 1D. We then consider large negative crystal field splitting, andfind that near small applied magnetic field we can describe some excited states using Landau-Ginsburgtheory. Near critical field, we show that the transition is in the Ising universality, and use results fromCFT to predict the spectrum for finite size systems. This allows us to make predictions about wherethe transition field would be for very large or infinite system size. Finally, we examine our crystal fieldsplitting tuned to the value obtained in Ref. 1, which is a small, negative value. We observe qualitativeelements in this spectrum from the spectra obtained at zero and large negative crystal field splitting.iiPrefaceThis dissertation is original, unpublished, independent work by the author, P. Gustainis.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Theory of Heisenberg antiferromagnet with no crystal field splitting (D=0) . . . . . . . 32.2 Theory of Heisenberg antiferromagnet with crystal field splitting . . . . . . . . . . . . . 82.3 Description of numerical simulation techniques . . . . . . . . . . . . . . . . . . . . . . . 112.4 Theory of electron spin resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Previous experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 D < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 D > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31ivList of Tables1 D = 0 energy spectrum near critical point . . . . . . . . . . . . . . . . . . . . . . . . . 62 Ising conformal tower structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Minimal character for each highest weight in Ising class . . . . . . . . . . . . . . . . . . 104 Energy spectrum from conformal towers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10vList of Figures1 Schematic of Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Sketch of D = 0 ground state after transition . . . . . . . . . . . . . . . . . . . . . . . . 73 Schematic of excitation spectrum with momentum dependence . . . . . . . . . . . . . . 74 Sketch of D < 0 ground state after transition . . . . . . . . . . . . . . . . . . . . . . . . 115 AKLT Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Energy Spectrum for D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Rescaled spectrum for D = 0 near critical point . . . . . . . . . . . . . . . . . . . . . . . 168 Energy spectrum for D = −0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Energy spectrum for D = −0.2 for small system size . . . . . . . . . . . . . . . . . . . . 1910 Rescaled spectrum for D = −0.2 near critical point . . . . . . . . . . . . . . . . . . . . . 2111 Energy spectrum for D = −0.037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212 Energy spectrum for D = −0.037 near zero field . . . . . . . . . . . . . . . . . . . . . . . 2413 Energy spectrum for D = −0.037 near zero field with theoretical fits . . . . . . . . . . . 2514 Energy spectrum for D = −0.037 with theoretical fits . . . . . . . . . . . . . . . . . . . 2615 Energy spectrum for D = −0.2 with theoretical fits . . . . . . . . . . . . . . . . . . . . . 2716 Ising-rescaled spectrum for D = −0.037 near critical field . . . . . . . . . . . . . . . . . 2817 Condensation-Rescaled spectrum for D = −0.037 near critical field . . . . . . . . . . . . 29viI would like to thank my supervisor Dr. Ian Aeck at the University of British Columbia. He hashelped me throughout my research and assisted in the editing of this thesis. I would also like to thankMiles Stoudenmire for allowing me to use his DMRG software ITensor, and for helping me modifythe simulation code to complete my research. I would like to thank Armin Rahmani for helping meduring the start of my project and helping me get set up on the WestGrid cluster. I would also like tothank WestGrid for providing cluster time and troubleshooting resources so I could run my simulationssuccessfully.viiFor my family, who have always believed in me.viii1 IntroductionQuantum spin chains in one dimension (1D) are of great interest both theoretically and experimentallydue to the enhancement of quantum effects that occur in 1D. Experimentally, they often provide verygood models of quasi-1D solids with weak exchange interactions between chains [1]. Analytic predic-tions can be made about them using quantum field theory, including conformal field theory (CFT).Furthermore, they can be modeled using a variety of methods including density matrix renormaliza-tion group (DMRG) techniques. This numerical procedure, developed by Steve White in 1992 [3], isresponsible for numerous advancements because it accurately and efficiently solves one dimensionalquantum systems, and even some two dimensional systems.The S = 1 Heisenberg antiferromagnetic chain,H =∑jJ ~Sj · ~Sj+1, J > 0has been studied intensely both experimentally and theoretically since Haldane's discovery that thereis an excitation gap (denoted ∆) from the ground state to a triplet of excited states, which is quitedifferent from the S = 1/2 case which is gapless [4].Further, the ground state of the system cannot be the Néel state, since continuous symmetriescannot be spontaneously broken in 1D, as was proven rigorously by the Mermin-Wagner-ColemanTheorem [5]. The ground state of the system, called the Haldane phase, has correlations and sym-metries that are quite different from simple Néel order. It has full SO(3) symmetry that is robustto small symmetry breaking terms. In particular, this work studies a modified version of the famousHeisenberg antiferromagnetic spin chain for S = 1 spins in 1D with the following Hamiltonian,H =∑jJ ~Sj · ~Sj+1 +D (Sxj )2 + hSzjThis Hamiltonian is of interest since it used to describe many quasi-1D materials, which containeffective 1D chains with weak interchain coupling [1,15,16,17,18]. We neglect the interchain couplingin this study. We have modified the Heisenberg Hamiltonian with a crystal field splitting term (Sxj )2which is a property of the material studied, and can in general have positive or negative sign depending1on the material. For this study, we focus onD ≤ 0. In addition, we apply a magnetic field perpendicularto the crystal field, which can vary in magnitude, corresponding to coefficient h which will generallybe considered to be positive (h ≥ 0).We study this quantum spin chain using DMRG methods to numerically simulate the spin chain.Since we consider a spin 1 chain with 3 states per site (thus the matrix that must be diagonalizedis of size 3L where L is the number of sites), exact diagonalization techniques are exhausted aroundL = 20− 30 [6]. The correlation length ξ for the Heisenberg antiferromagnet has been estimated to beξ ∼ 6, thus we must use a more powerful numerical technique to obtain system sizes much larger thanthe correlation length [7]. DMRG can model systems with more than 100 sites with excellent accuracyand efficiency.We are interested in a phase transition that occurs as we increase the magnetic field, where thesymmetry of the Haldane phase is spontaneously broken. This phase transition is predicted to be inthe Ising universality class for crystal field splitting D < 0 because the SO(3) symmetry is brokendown to a Z(2) symmetry by the introduction of the crystal field term and magnetic field, and thissymmetry is spontaneously broken at the critical point. We may use conformal field theory (CFT)to predict the spectra at the critical point for finite sized systems. As will be shown, the predictedfinite size spectra corresponds to the numerical data quite closely, and can be used to accurately findthe critical point. We also have quantum field theory predictions from Landau-Ginsburg theory thatdescribe the system for small magnetic field [8]. One goal of this work will be the study the systemas it goes from small fields, understood by Landau-Ginsburg, up to the critical field where the phasetransition occurs, which is understood by CFT.Electron-spin resonance (ESR) is an experimental technique used to probe the spectra of quantumsystems by bombarding them with photons of an appropriate energy to excite electron spin flips inthe material [9]. The absorption peaks indicate resonances where these transitions occur in the energyspectra. ESR has been used to study systems similar to the one studied in Ref. 5, and could be used tocompare theoretical spectra and the spectra in materials to test correspondence with theoretical models.In quantum spin chains, spin flip excitations can be understood as transitions between different magnonstates with the same momentum but energy that differs by the energy of the photon. By studying themomentum of excited states of our system, we can predict the excitations that would be detected byESR on systems described by this model.22 Background2.1 Theory of Heisenberg antiferromagnet with no crystal field splitting(D=0)First, we consider the Heisenberg antiferromagnet with only an external magnetic field applied,H =∑jJ ~Sj · ~Sj+1 + hSzjThis Hamiltonian conserves total spin in the z direction, 〈Sztot〉, which allows us to characterize statesby their 〈Sztot〉 quantum number. This model Hamiltonian has been extensively studied, both usingnumerics as well as analytic techniques [7,10]. This system can be understood quite simply, sinceeach state of the unperturbed Heisenberg antiferromagnet has a fixed 〈Sztot〉. When a magnetic fieldis applied, the states will Zeeman split according to their 〈Sztot〉. The ground state, in the Haldanephase, has 〈Sztot〉 = 0, and the first triplet of single magnon excited states has 〈Sztot〉 = −1, 0, 1 atenergy ∆ ≈ 0.41J above the ground state. Thus, for positive h, the 〈Sztot〉 = −1 state will meet theground state energy at h = ∆, and a phase transition occurs. In fact, it can be shown that the lowestenergy two magnon state has energy 2∆ above the ground state, and minimum 〈Sztot〉 = −2, and so onfor higher number of magnon states [11]. Thus, at h = ∆ we have a state of n magnons matching theground state energy for every n > 0, nI. This results in an effective Bose condensation of magnonsin the 1D chain, as shown in Figure 1.30.0 0.2 0.4 0.6 0.8 1.0 1.2Applied Magnetic Field/hc0.00.51.01.52.02.53.03.5Energy/∆Figure 1: Schematic of Zeeman splittingTheoretical expectation for the Zeeman splitting for each lowest energy multiplet up to S=3.However, these magnons experience effective short range repulsive interactions [7]. At large dis-tances, the short range interactions can be neglected and the magnons are effectively non-interacting.At short distances, these repulsive interactions act like the Pauli exclusion principle since the magnonsdo not want to be near to one another. For a small number of magnons in a large system (dilute)the magnons are seldom close together and the low energy spectrum is made up of states where thesemagnons are very far from one another, but cannot occupy the same state since there would be astrong repulsive interaction between them. This allows us to understand this condensate of magnonsas a system of dilute non-relativistic non-interacting fermions [10]. This is valid for an arbitrarily weakrepulsive interaction, as long as the intermagnon spacing is sufficiently large. Thus, our multimagnonstate can be written as a sum of products of single, non-interacting fermion wavefunctions ψi(xj),multiplied by a sign function (x1, x2, ..., xn) that changes the symmetry of the wavefunction fromfermionic (antisymmetric) to bosonic (symmetric). The magnon wavefunction ΨM for n magnons can4be written as,ΨM (x1, x2, ..., xn) =1√n!(x1, x2, ..., xn)[ ∑P (i1...in)n∏i=1ψ1(xi1)ψ2(xi2)...ψn(xin)sgnP]where P denotes the permutation and sgnP is the sign of the permutation. This fermionic wavefunctioncan be formed using the Slater determinant to guarantee antisymmetry in the wavefunction. Withinthe square brackets is our fermionic wavefunction, and the  function outside corrects the fermionicsymmetry to bosonic symmetry and1√n!normalizes the wavefunction.Now it is quite elementary to calculate the energy of excited states at the point of condensation,for various numbers of magnons present, for a finite sized system. This can be used to directlycompare with numerical results, and will serve as an excellent point of comparison for when we moveto the Hamiltonian with crystal field splitting present. The energy of the magnon states is given byE =n∑i=1p2i2m , the energy for non-relativistic fermions at momentums pi. The allowed values for pi willbe fixed by the allowed form of the wavefunction. We begin by imposing periodic boundary conditionson the system (ΨM (0, x2, ..., xn) = ΨM (L, x2, ..., xn)). However, for fermionic wavefunctions we mustorder the fermions by their relative position. Thus, when we move a particle from position 0 to L, wemust permute n−1 times in our fermionic wavefunction, picking up a factor of (−1)n−1 = sgnP . So foran odd number of magnons, this does not give an overall sign change, and our individual wavefunctionshave the form ψi(xj) = e2piiL kixj , ki = 0,±1,±2, .... The energy is simply E = 12m(2piL)2 n∑i=1(ki)2sincepi =2piL k, kI. For an even number of magnons, we get an overall negative factor from the permutations(sgnP = −1). To resolve this and keep periodic boundary conditions for our Fermionic wavefunction,our individual wavefunctions will have the form ψi(xj) = epiiL kixj , ki = 0,±1,±2, .... Now, when we setx1 = L we get ψi(L) = epiiL kiL = −1, which cancels with sgnP = −1, satisfying our periodic boundaryconditions. The energy is given by E = 12m∑ki(pikiL)2= 12m(2piL)2 n∑i=1(ki +12)2since pi =piLk, kI.Now, we tabulate the lowest energy states for each n to develop our expected low energy spectrum,5Number of magnons Energy ÷ 12m(2piL)2Momentum ÷ 2piLn = 0 E = 0 p = 0n = 1 E = 0, 1, 1, 4, 4, ... p− pi = 0, 1,−1, 2,−2, ...n = 2 E = 12 , 212 , 212 , .. p = 0, 1,−1, ...n = 3 E = 2, 5, 5, ... p− pi = 0, 1,−1, 2,−2, ...n = 4 E = 5, ... p = 0, ...Table 1: D = 0 energy spectrum near critical pointFinite size spectrum for D = 0 near the critical point calculated from non-interacting fermion model.Energies for the various states with fixed magnetization (number of magnons) and their correspondingmomentum are included.This gives the low energy spectrum at finite length: E = 12m(2piL)2 (0, 0, 12 , 1, 1, 2, 212 , 212 , ...). It isimportant to note that the splitting of these levels goes as1L2 and even at finite length, the gap closesbetween the ground and first excited state at the critical point, then opens back up after thequantum critical point. A similar calculation is done in Ref. 7 for open boundary conditions.For finite size systems, our theoretical picture of dilute interacting magnons begins to break downas we go to higher magnon states. This causes a slight deviation from the linear energy increase ofmultiplets at zero field with the number of magnons (i.e.n∆→ n∆ + δn). Therefore, at h = ∆, we donot have all lowest energy magnon states condensing, only the n = 1 magnon state for a finite system(since a single magnon cannot repulsively interact with itself). All other magnon states will cross theHaldane ground state at field h = ∆ + δnn , where δn will increase monotonically with n. This resultsin a cascading condensation where the ground state will continually increase in 〈Sztot〉as we increasefield past the critical field. Before the phase transition, our ground state has 〈Sztot〉 = 0, with unbrokenSO(3) symmetry, despite the Hamiltonian being reduced to SO(2) symmetry because of the magneticfield. After the critical point where the ground state possesses only SO(2) symmetry, we will have auniform, integer 〈Sztot〉 that will increase with increasing field. Figure 2 shows a qualitative sketch ofthe ground state for the spin chain after the phase transition.6Z Figure 2: Sketch of D = 0 ground state after transitionQualitative sketch of spins after critical point where SO(3) symmetry of the ground state goes to SO(2)symmetry and a magnetic moment appears. The spins have uniform moment in the z direction buthave no preferred direction in the xy plane, characteristic of SO(2) symmetry.In addition, predictions have been made about the momentum and energy dependence of excitedstates of the Heisenberg antiferromagnetic chain using the non-linear sigma model (NLSM) [11]. Thismodel is exact only in the large S limit of our Hamiltonian and accurate near momentum p = 0, pi, butprovides at least a qualitative picture for the momentum of excited states that is verified by DMRGsimulation.0 0.5 1k/pi0123456ω/∆kc2 magnon 3 magnonFigure 3: Schematic of excitation spectrum with momentum dependenceThe above is a schematic of the excitation spectrum from Ref. 11 for states with 1, 2 and 3 magnons.The vertical axis indicates energy in units of the first excitation gap ∆ and the horizontal axis is thecrystal momentum in units of pi. This is a reasonable qualitative sketch for our system if D = 0.Figure 3 shows a schematic diagram of the spin chain excitation spectrum for D = 0. The solid7line indicates the one magnon spectrum, with lowest energy ∆ at wave vector pi. This line mergeswith the two magnon continuum, shown in grey, at some kc. We also show the start of the 3 magnoncontinuum at wave vector pi and energy 3∆. Higher magnon continua are not shown on this plot butfollow a similar pattern.2.2 Theory of Heisenberg antiferromagnet with crystal field splittingNow we consider the full Hamiltonian of interest,H =∑jJ ~Sj · ~Sj+1 +D (Sxj )2 + hSzjFirst, we consider the quantum number for this Hamiltonian, which is different from before (conserved〈Sztot〉). Taking the representation of (Sx)2 in the canonical Sz basis for S = 1, we find,(Sx)2= 121 0 10 2 01 0 1This now allows for transitions between local Sz = 1 and Sz = −1 states, meaning that we no longerconserve 〈Sztot〉, but can change between basis states of our Hamiltonian which differ in 〈Sztot〉 by 2.Now, we have total Sz parity (even/odd) to characterize our states. This makes our Hamiltonian blockdiagonal in these parity states. It is worth noting that 〈Sztot〉 can have any value for eigenstates of ourHamiltonian. Now the situation changes, because we can no longer think in terms of states with fixedmagnetization. Further, the states in our spectrum do not Zeeman split as simply as before, since〈Sztot〉 will change as we change magnetic field.It is predicted that this transition will be in the Ising universality class because we are breaking theSO(3) symmetry of the Haldane phase into Z(2) (Sxj → −Sxj ) with our D term. The Z(2) symmetry isspontaneously broken at the critical point, as is characteristic for an Ising type transition (see Figure4). We can now use conformal field theory, and conformal towers, to predict the finite size spectrum atthe critical point using results from Cardy [13], and the Ising conformal structure from Di Francesco[14]. Note, we have dropped the constant in the energy formulae shown below since we only consider8energy differences between the ground state and excited states, and these constants cancel out whenwe take the difference.Boundary Condition Dimension Parity EnergyPeriodic(0, 0)( 12 ,12 )( 116 ,116 )Even (+)Odd (−)2pivL (xL + xR)Free012Even (+)Odd (−)pivL xTable 2: Ising conformal tower structureAbove is shown the dimension of each conformal tower for periodic and free boundary conditions,including the parity of each conformal tower. In the final column is the formula for the energy of eachstate in the conformal tower. Information taken from Ref. 13.We show calculations for free and periodic boundary conditions. From Cardy [13], we know thedimension of each conformal tower and what parity it has (shown in Table 2). For instance, our systemwith periodic boundary conditions (PBC) has three towers, two have even parity (0, 0), ( 12 ,12 ) and onehas odd parity ( 116 ,116 ). When a conformal tower has a pair of dimensions, as is the case for PBC, theenergy is specified by a pair of numbers xR and xL, which come from a list of integers correspondingto their minimal character plus the dimension value. For example, for the ( 12 ,12 ) conformal tower welook at the minimal character in the third row of Table 3 corresponding to hr,s =12 . The exponent ofeach q term tells us which integers are allowed, and the coefficients in front of each term tell us thedegeneracy allowed. For example, the term q2 appears so we are allowed the integer a = 2 but onlyone copy. Alternatively, the term 2q4 appears, so we are allowed two copies of the integer a = 4. Theallowed pairs of (xR, xL) for this tower are given by (12 + a,12 + a) where a is taken from the list ofallowed integers for hr,s =12 , and12 comes from the dimension of the tower. So, the lowest energypair allowed would be ( 12 + 0,12 + 0) with energy E =2pivL (12 +12 ) =pivL (2). The next two states have(xR, xL) = (12 + 1,12 + 0), (12 + 0,12 + 1) with the same energy E =2pivL (12 +32 ) =pivL (4). For freeboundaries, we have towers with a single dimension, not a pair of dimensions. The procedure is thesame as shown above, only now we have a single x value instead of a pair, and a different formula forthe energy.9Highest Weights forIsing Class (hr,s)Minimal Character Allowed Integers ah1,1 = 0 1 + q2 + q3 + 2q4 + 2q5 + 3q6 + ... (0, 2, 3, 4, 4, 5, 5, 6, 6, 6, ...)h1,2 =12 1 + q + q2 + q3 + 2q4 + 2q5 + 3q6 + ... (0, 1, 2, 3, 4, 4, 5, 5, 6, 6, 6, ...)h2,1 =116 1 + q + q2 + 2q3 + 2q4 + 3q5 + 4q6 + ... (0, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, ...)Table 3: Minimal character for each highest weight in Ising classFor the highest weights for each Ising class, we list the terms in the minimal character up to order 6,as well as the corresponding allowed integers used in the calculation of the energies of states in eachconformal tower.Now we may simply use the minimal characters for each of the highest weights with the formulaefor the energy and tabulate the results in Table 4. We choose unique values/pairs of possible x valuesfor each value/pair of highest weight(s). In the third column below we show the energies for each towerseparately. In the fourth column, we order the energies from each tower, indicating their parity + or−,below each energy. This is what will be compared to numerical data obtained from DMRG.BoundaryConditionDimension Energy(in units ofpivL )Energy Spectrum(ordered, in units ofpivL )Periodic(0, 0) +( 12 ,12 ) +( 116 ,116 )−(0, 4, 4, 6, 6, ...)(2, 4, 4, 6, 6, 6, ...)( 14 , 214 , 214 , 414 , 414 , 414 , ...)(0 14 2 214 214 4 4)+ − + − − + +Free0 +12 −(0, 2, 3, 4, 4, ...)( 12 , 112 , 212 , 312 , 412 , ...)(0 12 112 2 212 3 312 )+ − − + − + −Table 4: Energy spectrum from conformal towersFor both periodic and free boundary conditions, we calculate the lowest energy states in each conformaltower. In the final column, we tabulate the lowest energy states from all conformal towers for eachboundary condition, showing the corresponding parity below each state energy.It is important to highlight that unlike the D = 0 case described before, we do not expect the gap toclose for a finite system at the critical point. Instead, the gap will scale like1L , and we will requireconformal field theory predictions to find the critical point. Further, the ground state goes fromunique to degenerate as the gap closes past the phase transition where the Z(2) symmetry isspontaneously broken. This is characteristic of an Ising transition. This is different from the D = 0behavior, where the ground state goes from SO(3) symmetric to SO(2) symmetric as we go past thecritical field. Figure 4 shows a sketch of the ground states of the system for D < 0 after the phasetransition.10Z X Z X Figure 4: Sketch of D < 0 ground state after transitionNow spins have acquired overall magnetic moment, but also have a preferred alternating directionalong the x axis. We have an degenerate ground states (both shown above) which can be mapped toone another by reversing each spin in the x direction. This shows that we have spontaneously brokenthe Z(2) symmetry after the phase transition, which is characteristic of an Ising phase transition.2.3 Description of numerical simulation techniquesThe primary numerical technique used in this study is Density Matrix Renormalization Group (DMRG)and we use the ITensor package designed by Miles Stoudenmire and Steven White [2]. DMRG is by farthe most efficient and effective numerical technique for studying 1 dimensional quantum systems. It ishowever important to note that other numerical techniques were considered. For the S = 1 Heisenbergantiferromagnet, it has been shown that the correlation length of the system ξ ∼ 6 [7]. We first triedusing exact diagonalization techniques to study this system. However, working with S = 1 means ourHilbert space grows like 3Land makes it quite difficult to go past L = 20−30, which is still quite smallcompared to the correlation length of the system.It is canonical to use open boundary conditions (OBC) when using DMRG, since the wavefunctionAnsatz is generally in an OBC form. However, for our particular system we have a number of degeneratespin -12 edge states when we consider OBC. To see this, consider a similar spin-1 system in the Aeck-Kennedy-Lieb-Tasaki (AKLT) ground state [12].11Figure 5: AKLT Ground stateThe black dots are effective spin12 's, the black circles are the spin 1 sites, the blue lines are the singletstates, the red circles indicate the two unpaired effective spin-12 's.When we join neighboring effective spin-12 's in singlets, we see that we have two unpaired spinsat the ends of the chain for OBC. This creates four degenerate ground states at zero field, and willcreate up to four nearly degenerate states for each excited state as well. We observe similar edge statespresent in our system. Thus, to understand our system and its excited states using OBC DMRG, wewould need up to four times as many excited states in our DMRG calculation as we would like tosee in our study. This puts tremendous strain on the algorithm that struggles with larger numbers ofexcited states. Moreover, it is difficult to parse through all the excited states and determine which areedge state permutations of another state or constitute new bulk eigenstates.Instead, we will consider periodic boundary conditions (PBC) for our DMRG simulation. Thistechnique carries with it other difficulties because we are using an OBC Ansatz with a PBC Hamil-tonian. This causes a large range interaction length for one hopping term that goes from site 1 to Nthat carries a large amount of entanglement entropy across all bonds in the DMRG algorithm. Thisincreases computation time considerably, compared to OBC DMRG with the same number of sites,states and excited states desired. However, since we would need four times the excited states in OBCto study the same physics, this method is still much more efficient and allows us to study the bulkproperties of our system far more easily.Further, it will also allow us to measure the momentum of each state, which will be of greatimportance for making experimental predictions for electron spin resonance experiments. Considerthe translation operator T which translates our wavefunction by one site. This operator will commutewith the Hamiltonian for PBC [T,H] = 0. This means that for the eigenstates |ψi〉 of the HamiltonianH |ψi〉 = Ei |ψi〉 will also be eigenstates of T , T |ψi〉 = eikia |ψi〉 . Setting the lattice spacing a = 1, and12noting that due to PBC TN = I, ki =2pinL , nZ. The ki can be associated with the crystal momentumof the state. We may now calculate momentum easily by taking a copy of our wavefunction, thenshifting the tensor index label one site over for each local wavefunction 〈ψi|T |ψi〉 = 〈ψi| eikia |ψi〉 =eikia. Contracting the shifted state with the original wavefunction will give us the momentum of thestate. This technique is only possible with PBC.2.4 Theory of electron spin resonanceElectron Spin Resonance (ESR) is an experimental technique which uses photons to excite spin flips ina quantum system. The strength of the absorption signal from ESR is proportional to the imaginarypart of the susceptibility I(ω) ∝ ωχ′′(ω); where2~χ′′(ω) =(1− e−~ω/T ) ´∞−∞ dt eiωt 〈Sxtot(t)Sxtot(0)〉T , Sxtot = ∑iSxiwhere we have chosen the microwave field used to excite the spins polarized in the x direction [8]. Inprinciple, the matrix element depends on the direction of the alternating field and could be in the yor z direction. In ESR experiments, we assume zero-momentum transitions since our Brillouin zonewidth should be much larger than the photon wavevector (p = ~ωv =~λ , where v = c for the photonand vcrystal  c). Assume we have two states |1, k1〉 , |2, k2〉 with energies, E1(k1, h), E2(k2, h) thatcan depend on momentum and applied magnetic field. Further, assume we have an allowed transitionbetween these states at momentum k0. Thus, |E1(k0, h)− E2(k0, h)| = ω~, and it can be shown [8],2~χ′′(ω) = L~(1− e−~ω/T ) e−Emin(k0,h)/T |∂E1(k0, h)/∂k − ∂E2(k0, h)/∂k|−1 |〈2, k0|Sxtot |1, k0〉|2,Emin ≡ min{E1, E2}Thus, to make predictions about the ESR peak intensity, we must be able to predict the matrixelement between states as well as the momentum and field dependent dispersion relations of the twostates E1,2(k, h). To do this theoretically we must use an approximate model. We model the firstexcited triplet of magnons in our system as a triplet of bosonic fields φ in the Landau-Ginsburg model,described by the Lagrangian,13L = 12v∣∣∣∂φ∂t + h× φ∣∣∣2 − v2 (∂φdx)2 − 12v∑i∆2iφ2iWe solve the Euler-Lagrange equations for this field theory, fixing h = (0, 0, h), and obtain the fieldand momentum dependence energy dispersions for the three bosonic fields with k shifted by pi,ω23 = ∆23 + v2k2ω22,1 = (∆22 + ∆21 + 2v2k2)/2 + h2 ± [2h2(∆22 + ∆21 + 2v2k2) + (∆22 −∆21)/4]1/2The values of ∆i are generally taken to be phenomenological parameters when comparing to experi-mental or numerical results. We will compare numerical results to these predictions.2.5 Previous experimental resultsThere has been much experimental interest in quasi-1D materials with weak interchain coupling,since they are ideal for studying 1D quantum systems experimentally. One of the earlier exam-ples of a quasi-1D S=1 antiferromagnetic material is CsNiCl3[15,16]. Recently, another materialNi(C5H14N2)2N3(PF6) (NDMAP) was found to also be in this class of materials [17], along withanother material Ni(C2H8N2)2NO2ClO4(NENP) [18]. For our study, we focus on the model com-pound SrNi2V2O8[1]. Using inelastic neutron scattering, the researchers in Ref. 1 estimated theeffective crystal field splittingDJ ≈ −0.037. In our investigation we will ignore interchain couplingand next-nearest neighbor interactions. This is an acceptable simplification since these couplings aremeasured to be 2 orders of magnitude smaller than the primary coupling J between nearest neighborsfor SrNi2V2O8.143 Results3.1 D = 0We begin by showing results for our Hamiltonian in the case where D = 0. Though this system hasbeen studied extensively both using numerical [7] and analytic techniques [10], we show new results forthe system and reproduce a few older results. Further, it is a simple starting point for understandingthe full behavior of the system with D 6= 0. We start with a quick overview of the low energy spectrum,then look at the rescaled spectrum near the critical point.0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6Applied Magnetic Field (h)0.10.00.10.20.30.40.50.6Energy (Relative to ground state)D = 0.0  L = 70  PBCFigure 6: Energy Spectrum for D = 0Blue lines indicate even parity states, green lines indicate odd parity states. We use the quantumnumbers (parity) from D 6= 0 in our DMRG simulations for D = 0 because this will allow us to moreeasily compare graphs throughout the paper, though we could use Sztot quantum numbers for D = 0.Figure 6 shows us the energy of excited states relative to the ground state. We see the excitedtriplet at energy ∆ = 0.41 that Zeeman splits with the applied magnetic field. We also observe thereis another triplet above the first triplet with energy difference at zero field ∆E ≈ v22∆(2piL)2. Thisexpression comes from recognizing that in this PBC system, we are restricted to momentum p = 2piL n.15We then look at the spectral function from the NLSM and expand around momentum pi in a relativisticapproximation where E =√m2v4 + p2v2 ≈ mv2+ p22m , where now p is measured relative to momentumpi. We identify the mass term mv2 = ∆ → m = ∆v2 and we know v ≈ 2.49 from a paper by I. Aeckand E. Sørensen [7]. This allows us to predict the gap between the triplets as ∆E ≈ v22∆(2piL)2 ≈ 298.5L2 .This shows us that there will be two copies of this second triplet at the same energy above the firsttriplet since we can have momentum p = ± 2piL (relative to pi) that give the same energy.Near the critical point at h = ∆ (indicated by a vertical red line), we now see evidence of thecascading condensation in finite size systems. Notice the Sz = −2 state is at a gap δ from the groundstates at the critical field. After the transition, the graph becomes very messy since we are measuringenergy relative to the ground state, and now with the cascading transitions occurring, the ground stateis changing to higher magnon states as we increase field.We now study the system near the critical field and the behavior of the spectrum near this point.0.400 0.402 0.404 0.406 0.408 0.410 0.412 0.414 0.416 0.418Applied Magnetic Field (h)0.00.51.01.52.02.53.0Energy (Relative to ground state) x 2∆(L/2piv)2D = 0.0  L = 40, 50, 60 70 80 PBCFigure 7: Rescaled spectrum for D = 0 near critical pointSystem size L = 40, 50, 60, 70, 80 are shown in red, green, blue, violet, light blue, respectively. Thevertical purple line is the critical field at h = ∆ = 0.41. The black horizontal lines are the theoreticalpredictions from dilute repulsive bosons in 1D.16In Figure 7 we have taken DMRG data from system sizes L = 40, 50, 60, 70, 80 and plotted themon the same graph. We multiply each energy by the length of the system squared to remove theexpected scaling behavior of the energies near the critical field, thus we expect that at the criticalfield all the field lines will converge on top of one another. We then use the predictions made beforefor the spectrum in units of12m(2piL)2= v22∆(2piL)2. The data follows the predicted spectrum well,particularly for lower energies. At higher energies where the magnon number is larger, there are likelyinteraction effects that are not accounted for in the theoretical model of dilute repulsive bosons, sincewe are working in systems with finite size. We would expect these deviations to decrease for largersystem sizes. In fact, we see this clearly as our energy spectral lines approach the expected values aswe increase the system size. Further evidence to support this interpretation is the increasing splittingas we go to higher magnon states. So long as we approach the theoretical prediction for increasingsystem size, it is consistent with our expectations.Overall, this numerical evidence supports our theoretical predictions of finite size spectra near thecritical point and we can easily match energy and momentum of each state to what was predicted. Thedifferences between data and model can be easily explained by finite size effects and seem to convergeto theoretical expectation as we go to larger system sizes.3.2 D < 0Experimentally, neutron scattering experiments can measure the coefficient of the crystal field splitting.The ratio of this coefficient D relative to the primary coupling J can vary dramatically from materialto material, and can be positive or negative. We focus on negative D since this is what is observed forSrNi2V2O8. We will first investigate a moderately large D relative to what was measured by Bera etal. The intention is to investigate the system in a regime that is dominated by the crystal field, whereit is not a small perturbation on the system. However, we know from work done by Z.C. Gu and X.G.Wen [19] that there is a transition out of the Haldane phase at zero applied magnetic field h as we goto larger negative D. From the graph in their paper we can estimate that this transition occurs aroundDJ ≈ −0.3 so we will choose D = −0.2 and investigate the phase transition as we increase the appliedmagnetic field. We start with an overview of the low energy spectrum, then consider the spectrumwith additional excited states near zero applied field. Finally, we examine the rescaled spectrum nearthe critical field to compare with theoretical predictions.170.0 0.1 0.2 0.3 0.4 0.5Applied Magnetic Field (h)0.10.00.10.20.30.4Energy (Relative to ground state)D = -0.2  L = 80  PBCFigure 8: Energy spectrum for D = −0.2Parity even states are shown in blue, parity odd in green. As can be seen, we have curves instead ofstraight lines as we increase our magnetic field. The vertical purple line is where the gap appears toclose and we would expect the critical field is reached. However, the vertical red line is the transitionfield as predicted using CFT and finite size spectra (shown in Figure 10) which we assert is moreaccurate for the true infinite size transition field.Comparing Figure 8 to the D = 0 data, we see a number of clear differences. Now, since magnonnumber is not a good quantum number, 〈Sztot〉 will change as we change the applied field. This resultsin curves which do not simply Zeeman split as we saw previously. We also see that the ground stateand first excited state cross at a much smaller hc, and become degenerate after the transition insteadof crossing and splitting away from each other as compared to D = 0.While many features of our system have changed drastically both quantitatively and qualitatively,we find parallels between the two systems that deepen our understanding and allow us to think interms of our original picture with differences due to the crystal field splitting.180.0 0.1 0.2 0.3 0.4 0.5Applied Magnetic Field (h)0.10.00.10.20.30.40.50.60.70.8Energy (Relative to ground state)D = -0.2  L = 40  PBCFigure 9: Energy spectrum for D = −0.2 for small system sizeNumerous excited states are shown for parity even (in blue) and parity odd (in green)As was shown in the background section, we predict the triplet to split into a singlet below ∆ anddoublet above the singlet. We study a small system to allow us to get many excited states from DMRG,shown in Figure 9. By measuring momentum using the method described before and measuring 〈Sztot〉for each state, we can try to map these states to those seen in the D = 0 case with some perturbationcaused by the crystal field splitting. Looking at the curves at zero field, the first state above the groundstate is simply the bottom singlet from our triplet that has been split, with momentum p = pi. Thesecond green line is actually two data sets exactly on top of one another. These are the two singletsfrom two triplets at momentum p = pi± 2piL , again split down similarly to the first excited state. Noticetheir curvature is quite similar to the curvature of the first excited state as well. The first blue lineabove the ground state has momentum p = 0, and near the middle of the curve, 〈Sztot〉 ≈ −2. Thissuggests that this is the bottom state from the S = 2 multiplet originally at energy Eh=0 = 2∆ thathas been split down by the large crystal field term. The next green and blue curve is the doublet (theother two states of the triplet) at momentum p = pi, and are degenerate at zero field as expected.19These two states are intersected by higher energy states near field h = 0.2. The green curve at thispoint has momentum p = pi, and 〈Sztot〉 ≈ −3, which suggests it is the bottom state from the S = 3multiplet in the D = 0 picture, perturbed by crystal fields. The blue curve at this point is actuallytwo curves with momentum p = 0 ± 2piL and 〈Sztot〉 ≈ −2. These are the higher momentum states forthe S = 2 multiplet, analogous to what was seen for the S = 1 triplet.Interpreting these states as perturbations of the original picture constructed for D = 0 is helpfulso long as we keep clear that these states are related but not the same. Their magnetization varieswith applied field, they have different quantum numbers, and are split from their original positions atzero applied magnetic field in the D = 0 diagram because of the crystal field. However, when we takethese into account and measure the momentum of the states, we can clearly see how the crystal fieldchanges our qualitative picture. This understanding will be useful when we consider much smallerDJas seen in SrNi2V2O8.Near the critical field, we will use CFT predictions developed beforehand and compare them to thespectra we obtain. Similar to the section before for D = 0, we will rescale our energies and plot thespectrum for multiple lengths to find the critical field where all these curves converge. However, wenow rescale our energy by L instead of L2.200.24 0.26 0.28 0.30 0.32 0.34 0.36Applied Magnetic Field (h)012345Energy (Relative to ground state) x L/(piv)D = -0.2  L = 40, 50, 60  PBCFigure 10: Rescaled spectrum for D = −0.2 near critical pointRed lines indicate data for L = 40, green lines for L = 50, and blue lines for L = 60. Vertical magentaline is the transition and the horizontal black lines show the theoretical predictions from CFT. Noticethat for higher energy states we have a slight discrepancy between the theory and results. However,as we go to larger system sizes (red to green to blue) the data is moving towards the theoreticalexpectation at the critical field. We have used v ≈ 1.57 to best fit the data.In Figure 10, the curves seem to converge quite nicely near the critical field when rescaled appro-priately, especially those with lower energies. An important difference between this plot and the oneshown for D = 0, is that now we expect there to be a finite gap between the ground state and firstexcited state for a finite system. In the D = 0 system the gap closed at the same time as the energiesconverged, which meant either method of estimating the critical field gave similar results. Now the gapscales like1L , and we can no longer estimate the critical field using the closing of the gap. Instead, CFTpredictions give us a clear point at which the fields converge indicating the critical field. In theory, thisconvergence should happen for all excited states. However, this field theory is a low energy effectivetheory and begins to breakdown at higher energies. This breakdown should be resolved as we go tolarger system sizes, and we can see in all the plots that there is a trend towards the expected energyscaling as we go to larger system sizes.213.3 D > 0Now we consider a small negative crystal field splitting, which is what is expected for SrNi2V2O8.We use the results from Bera et al. for the ratioDJ ≈ −0.037. We shall see that in this regime thetransition is driven primarily by the magnetic field, but many of the features from the crystal fielddominated regime will persist. We start with an overview of the low energy spectrum, then considerour spectrum near zero field, comparing to Landau-Ginsburg theory. Finally, we examine the rescaledspectrum near the critical point.0.0 0.1 0.2 0.3 0.4 0.5Applied Magnetic Field (h)0.10.00.10.20.30.40.50.6Energy (Relative to ground state)D = -0.037  L = 80  PBCFigure 11: Energy spectrum for D = −0.037The first two states for each parity for small negative crystal field splitting. Parity even states areshown in blue, parity odd in green.It is insightful to compare and contrast Figure 11 with Figure 6 for D = 0. Past small fields butbefore the critical field, the graphs look almost indistinguishable qualitatively. We see the same 3curves: the first being the bottom of the S = 1 triplet, the second being the same state at momentump = pi ± 2piL , and the third being the Sz = 0 part of the triplet, that is intersected by the bottom stateof the S = 2 multiplet. Notice that unlike the large D case, these spectral lines in the intermediate22region are almost perfectly straight with slopes corresponding to what would be expected for D = 0.The main differences are evident at fields near zero and above the critical field. Near zero field, thegraphs are not straight lines as they are in D = 0. Instead, they have some curvature, and in fact startfrom Sztot = 0 at h = 0. This is to be expected since at zero field, the z direction is not preferred, butrather the x direction is preferred due to the crystal field. As we turn on the field in the z direction,the spins align to produce an overall moment in the z direction for certain states. This curvature ismost evident for the first excited state as it curves from zero slope to a slope of -1. Further, aroundthe critical point we see another change from the behavior of D = 0. The first excited state and theground state merge and stay degenerate for all fields above the critical field. What's more, we do notsee the same cascading condensation effect here as we saw in D = 0. In fact, since states are notconstrained by an Sztot quantum number, they can have a magnetic moment that continually increaseswith increasing field. This results in a degenerate ground state that is not intersected by other excitedstates as field is increased, similar to the plots for large D.Next we will look at our system near zero field and near the critical field since this is where oursimple picture from D=0 does not work without modification.230.00 0.05 0.10 0.15 0.20Applied Magnetic Field (h)0.10.00.10.20.30.40.50.60.7Energy (Relative to ground state)D = -0.037  L = 40  PBCFigure 12: Energy spectrum for D = −0.037 near zero fieldWith a smaller system size, we may examine many more excited states. Using our understanding ofhow the spectrum changes with system size we can easily map this to larger system sizes which areharder to simulate with the same number of excited states. Parity even states are shown in blue, parityodd in green.For small crystal field it is much easier to understand our spectrum as a modification of the D=0system. In Figure 11 near zero field, the first 3 states are the first excited triplet from the D=0system, but now the bottom state of the triplet has been split down as expected from our field theorypredictions. The next two green lines are the bottom states from the triplets at momentum p = pi± 2piL ,that are degenerate as expected. The final blue lines are also degenerate and part of the triplet. Athigher field, this final blue line is intersected by the bottom part of the S = 2 multiplet. The onlylarge differences between this graph and the D = 0 graph seen before are the curvature near zero fieldand the splitting of the triplets at zero field.We may now use the theory developed by I. Aeck for ESR spectra that should match our spectrumat small fields. We will fit using the energies at zero field. Notice the equations can be written entirelyusing variables αi ≡ ∆2i + v2k2, and h. Setting h = 0 we find ω2i = αi, allowing us to easily fit our24spectrum to theoretical predictions.0.00 0.05 0.10 0.15 0.20Applied Magnetic Field (h)0.10.00.10.20.30.40.50.60.7Energy (Relative to ground state)D = -0.037  L = 40  PBCFigure 13: Energy spectrum for D = −0.037 near zero field with theoretical fitsThe red lines are fits for both the lowest and next lowest triplet. Each fit lines corresponds to the dataquite convincingly. The final red line trending upwards with no data matching it is the upper part ofthe higher momentum triplet, which was not captured by DMRG due to limited number of excitedstates. Parity even states are shown in blue, parity odd in green. ∆1 = 0.36155, ∆2 = ∆3 = 0.43627In Figure 13, the red lines now indicate the fits obtained using the theory described beforehand.As we can see, the agreement is very good for fields in this range, and works well for the highermomentum triplet as well. We can use this theory to calculate the velocity for this system as well.Taking α2piLi − α0i = v2( 2piL )2 (the superscript now indicates the momentum relative to pi), we cancalculate the velocity which has the value v ≈ 2.45 for all branches, which is similar to the valuev ≈ 2.49 obtained by I. Aeck and E. Sørensen for D = 0 [7]. To see where this fit begins tobreakdown we need to look closer to the critical field.250.0 0.1 0.2 0.3 0.4 0.5Applied Magnetic Field (h)0.10.00.10.20.30.40.50.6Energy (Relative to ground state)D = -0.037  L = 80  PBCFigure 14: Energy spectrum for D = −0.037 with theoretical fitsWe now go to a larger system size and look at only the first two excited states to evaluate the qualityof the theoretical model for larger applied fields. We have used theoretical predictions and fit to ourdata (fits shown in red) and see excellent agreement between the two until we near the critical point.Again we are missing the upper part of the triplet because we have limited the number of excitedstates, though we still show the expected fit in red. Parity even states are shown in blue, parity oddin green. ∆1 = 0.36102, ∆2 = ∆3 = 0.43604Figure 14 shows that the fit works quite well until very close to the transition for the first excitedstate, where it curves down too sharply before the transition. It is interesting to notice that for a smalldifference between α1and α2 we simply get Zeeman splitting from the theory. Since the differenceterm in the square root can be neglected, α1 + α2 ≈ 2α1 ≈ 2α2 and we simplify into the formω22,1 = (α2,1 ± h)2. This is why for the intermediate region, where the curves look almost like Zeemansplitting, the fit works very well.We use this theory for our previous D = −0.2 and see that theory breaks down much sooner forthe first excited state.260.0 0.1 0.2 0.3 0.4 0.5Applied Magnetic Field (h)0.10.00.10.20.30.40.50.60.70.8Energy (Relative to ground state)D = -0.2  L = 40  PBCFigure 15: Energy spectrum for D = −0.2 with theoretical fitsThe triplet is now split by a large amount, and the fits no longer match the data past small fields,particularly for the first excited state. The fits for the triplet states are shown in red. Parity evenstates are shown in blue, parity odd in green. ∆1 = 0.15548, ∆2 = ∆3 = 0.56763To obtain the fits in Figure 15, we use the same fitting procedure described above, with fits in redfor our triplet states. The fits for the upper two states in the triplet work quite well, though the fitfor the lowest excited state quickly diverges from the data. This is not surprising since the theorydeveloped only works for small applied fields. It also assumes the velocity of each state in the tripletis the same, which may not be the case any longer, and might be part of the reason for the largediscrepancy in this case.Finally, we consider the system near the critical point for multiple system sizes for D = −0.037.We would expect this system to have a finite size spectra that scales similarly to the spectrum forD = −0.2 near the critical point. So we rescale our energies by L and plot the energies for differentlength scales.270.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43Applied Magnetic Field (h)0246810Energy (Relative to ground state) x L/(piv)D = -0.037  L = 40, 50, 60  PBCFigure 16: Ising-rescaled spectrum for D = −0.037 near critical fieldWe show L = 40, 50, 60 data in red, green and blue, respectively. The horizontal black lines are a bestguess for the fit based on CFT predictions, but these should not be taken seriously. They are meantto show that our data is very far from Ising-like behavior near the critical point.The spectrum in Figure 16 seems quite different from what was expected. First, the energies seemto converge as the first excited state and ground state become degenerate, which is more consistentwith the spectrum for D = 0 near the critical point. Further, the spacing of the higher energy statesdo not follow the pattern we would expect for an Ising transition. Instead, we analyze the spectrumnear the critical point using the methods for D = 0.280.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43Applied Magnetic Field (h)0.00.51.01.52.02.53.03.5Energy (Relative to ground state) x 2∆(L/2piv)2D = -0.037  L = 40, 50, 60  PBCFigure 17: Condensation-Rescaled spectrum for D = −0.037 near critical fieldWe show L = 40, 50, 60 data in red, green and blue, respectively. Using the finite size spectrum for aBose condensation transition, we can match our data to theoretical predictions, using the same velocitycalculated before using Landau-Ginsburg theory fits (v ≈ 2.45)Surprisingly, in Figure 17 we see the spectrum at the transition is much more accurately describedby this non-interacting fermion model instead of the Ising model. We suspect that this is because offinite size effects. If we consider the renormalization flow of the system near the critical point, wewould expect that for small D and small system size we have not flowed far from the D = 0 system.For small D, we expect that we would see Ising like behavior as we increase our system size. However,the required system size is likely much larger than we can obtain using PBC DMRG forDJ = −0.037.294 ConclusionsIn this thesis, we examine the Heisenberg antiferromagnetic spin chain in 1 dimension with crystalfield splitting and applied magnetic field terms using theoretical and numerical techniques. By firststudying the system with zero crystal field splitting, we can characterize our system by thinking aboutstates with different numbers of magnons present since we have conserved 〈Sztot〉. We use results fromthe non-linear sigma model (NLSM) to predict the structure of the excitation spectrum and how theexcited states will behave when we apply a magnetic field. Near the critical point h = ∆, we canmap our system to non-interacting fermions in 1D and find a finite size spectra near the critical pointwhich we compare to our numerical data. The data follows the theoretical predictions quite closelybut deviate at higher energy, which corresponds to states with higher magnon numbers where ourapproximate mapping breaks down.We then consider our system with a large negative crystal field splitting term D < 0. This termbreaks 〈Sztot〉 conservation and we can no longer think in terms of states with a conserved numberof magnons. This term also splits our excited multiplets at zero magnetic field, and the picture weobtain from the NLSM must be modified. At small field, we may use the Landau-Ginsburg model toapproximate how the first excited triplet behaves as we increase from zero magnetic field. Near thecritical field, we must now use conformal field theory to predict the finite size spectra for an Isingtransition. We compare our theoretical predictions to numerical results obtained using DMRG andfind good agreement near small and critical magnetic fields.Finally, we studyDJ = −0.037 which corresponds to experimental data obtained by Bera et alfor SrNi2V2O8[1]. Here, the excitation spectrum exhibits qualitative elements from both the D = 0and D < 0 excitation spectra. We would expect that near the critical point we would see behaviorcharacteristic of an Ising transition, but when we examine the data, it more closely follows the non-interacting fermion model. This is likely because we are examining small system sizes as well as asmall crystal field term, meaning that our renormalization flow is not far from the D = 0 system. Forlarger L at this fixed D we would expect to see Ising behavior near the critical point.30Bibliography1. A.K. Bera, B. Lake, A.T.M.N. Islam, et al. Phys. Rev. B 91, 144414 (2015)2. Calculations were performed using the ITensor Library. http://itensor.org/3. S.R. 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