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A variational investigation of the SU(N) Heisenberg model in antisymmetric irreducible representations

Jérôme Dufour

Published Jan 1, 2016• 0 citations
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Abstract

Motivated by recent experimental progress in the context of ultra-cold multi-colour fermionic atoms in optical lattices, this thesis investigates the properties of the antiferromagnetic SU(N) Heisenberg models with fully antisymmetric irreducible representations labelled by a Young tableau with $m$ boxes on each site of various lattices. Thanks to an intensive use of variational Monte Carlo method based on Gutzwiller projected fermionic wave functions, we aim to determine the ground state and its related properties of systems which are beyond the capabilities of other numerical methods. This typically means that we will study large lattices with many particles per site for $N>2$. The first part of this thesis will introduce the basic theoretical tools connected to the SU(N) Heisenberg model. A very short review of group theory introducing Young tableaux, will clarify the concept of fully antisymmetric irreducible representation of the SU(N) Lie algebra. Some properties of the Heisenberg model will be discussed and a mean-field solution will be given. The numerical methods used during this thesis and their limitations wil also be introduced. We will investigate the properties of 1-dimensional systems in the second part of this work. In particular, the chains have been studied for arbitrary $N$ and $m$ with non-abelian bosonisation leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. We have been able to verify these predictions for a representative number of cases with $N\leq10$ and $m\leq N/2$, and we have shown that the opening of a gap is associated with a spontaneous dimerisation or trimerisation depending on the value of $m$ and $N$. We have also investigated the marginal cases, where non-abelian bosonisation did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents that are consistent with conformal field theory. The other 1-dimensional system that has been studied is the ladder for which field-theory gives some predictions. For instance, the ground state of the SU(4) case in the fundamental irreducible representation is believed to consist of a 4-site plaquette state for any non-zero inter-leg coupling. We have confirmed the presence of this state, even if in the weak coupling regime, the optimal variational state is gapless. The study has been extended to various $N$ and $m$ values providing interesting results. The last part will present the results for 2-dimensional lattices. The square lattice will first be investigated. A mean-field analysis predicts a plaquette ground state for $3\leq N/m < 5$ and of a chiral ground state for $N/m\geq 5$. By systematically studying the SU(3m) and SU(6m) Heisenberg model, we determined that the mean-field solution is valid even at small $m$ ($m>1$ for SU(3m) and $m>2$ for SU(6m)). Then, for the SU(6m) Heisenberg model with $m$ particles per site on a honeycomb lattice, we make the connection between the $m=1$ case, for which the ground state has recently been shown to be in a plaquette singlet state, and the $m\rightarrow \infty$ limit, where a mean-field approach has established that the ground state has chiral order. We were able to show that this system has a clear tendency toward the chiral order as soon as $m\geq 2$. This demonstrates that the chiral phase can be stabilised for not too large values of $m$, opening the way to its experimental realisations in other lattices. We will finally present results for the SU(3m) on the triangular lattice for which we find a plaquette state, when $m>1$.

DOI
10.5075/epfl-thesis-7309

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Motivated by recent experimental progress in the context of ultra-cold multi-colour fermionic atoms in optical lattices, this thesis investigates the properties of the antiferromagnetic SU(N) Heisenberg models with fully antisymmetric irreducible representations labelled by a Young tableau with mmm boxes on each site of various lattices. Thanks to an intensive use of variational Monte Carlo method based on Gutzwiller projected fermionic wave functions, we aim to determine the ground state and its related properties of systems which are beyond the capabilities of other numerical methods. This typically means that we will study large lattices with many particles per site for N>2N>2N>2. The first part of this thesis will introduce the basic theoretical tools connected to the SU(N) Heisenberg model. A very short review of group theory introducing Young tableaux, will clarify the concept of fully antisymmetric irreducible representation of the SU(N) Lie algebra. Some properties of the Heisenberg model will be discussed and a mean-field solution will be given. The numerical methods used during this thesis and their limitations wil also be introduced. We will investigate the properties of 1-dimensional systems in the second part of this work. In particular, the chains have been studied for arbitrary NNN and mmm with non-abelian bosonisation leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. We have been able to verify these predictions for a representative number of cases with N≤10N\leq10N≤10 and m≤N/2m\leq N/2m≤N/2, and we have shown that the opening of a gap is associated with a spontaneous dimerisation or trimerisation depending on the value of mmm and NNN. We have also investigated the marginal cases, where non-abelian bosonisation did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents that are consistent with conformal field theory. The other 1-dimensional system that has been studied is the ladder for which field-theory gives some predictions. For instance, the ground state of the SU(4) case in the fundamental irreducible representation is believed to consist of a 4-site plaquette state for any non-zero inter-leg coupling. We have confirmed the presence of this state, even if in the weak coupling regime, the optimal variational state is gapless. The study has been extended to various NNN and mmm values providing interesting results. The last part will present the results for 2-dimensional lattices. The square lattice will first be investigated. A mean-field analysis predicts a plaquette ground state for 3≤N/m<53\leq N/m < 53≤N/m<5 and of a chiral ground state for N/m≥5N/m\geq 5N/m≥5. By systematically studying the SU(3m) and SU(6m) Heisenberg model, we determined that the mean-field solution is valid even at small mmm (m>1m>1m>1 for SU(3m) and m>2m>2m>2 for SU(6m)). Then, for the SU(6m) Heisenberg model with mmm particles per site on a honeycomb lattice, we make the connection between the m=1m=1m=1 case, for which the ground state has recently been shown to be in a plaquette singlet state, and the m→∞m\rightarrow \inftym→∞ limit, where a mean-field approach has established that the ground state has chiral order. We were able to show that this system has a clear tendency toward the chiral order as soon as m≥2m\geq 2m≥2. This demonstrates that the chiral phase can be stabilised for not too large values of mmm, opening the way to its experimental realisations in other lattices. We will finally present results for the SU(3m) on the triangular lattice for which we find a plaquette state, when m>1m>1m>1.
Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have investigated the properties of the SU(N)\mathrm{SU}\phantom{\rule{0.16em}{0ex}}{(N)}_{}SU(N)​ Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with m<Nm<Nm<N particles per site. These models have been studied for arbitrary NNN and mmm with non-Abelian bosonization [I. Affleck, Nucl. Phys. B 265, 409 (1986); I. Affleck, Nucl. Phys. B 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with $N\ensuremath{\le}10$ and $m\ensuremath{\le}N/2$, and we have shown that the opening of a gap is associated to a spontaneous dimerization or trimerization depending on the value of mmm and NNN. We have also investigated the marginal cases where Abelian bosonization did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents consistent with conformal field theory.
This thesis explores various approaches of studying the long-range colour order of antiferromagnetic SU(N) Heisenberg models with the linear flavour-wave theory (LFWT). The LFWT is an extension of the well-known SU(2) spin-wave theory to SU(N), and this semi-classical method has been used to study SU(N) models with colour-ordered ground states in the fundamental irreducible representation (irrep). The aim of this thesis is to study various SU(N) Heisenberg models in different irreps and in various colour configurations, in part by extending the LFWT method to different irreps of SU(N). This will be achieved by using three different methods all using different bosonic representations. First, the LFWT willbe performed using the multiboson method that introduces a boson for each state of a given irrep. Then, a different SU(3) bosonic representation first introduced by Mathur & Sen will be presented and used to perform the LFWT calculations. Finally, another way of applying the LFWT will be shown using the bosons first used by Read & Sachdev for SU(N) irreps with rectangular Young tableaux. The specific models that will be treated by these methods are the fully antisymmetric SU(N) models on the square, honeycomb, and triangular lattices with m>1 particles per site, as well as the bipartite SU(3) Heisenberg chain in the adjoint irrep to show how the LFWT can be used for mixed symmetries. The last chapter of the thesis will be dedicated to another problem related to the accidental line of zero modes in the harmonic spectrum of the antiferromagnetic SU(3) Heisenberg model on the square lattice with one particle per site in a three-sublattice order. The objective will be to try to lift the accidental zero modes in the spectrum.
Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg SU(N)\mathrm{SU}(N)SU(N) Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of SU(N)\mathrm{SU}(N)SU(N). The method, based on the use of standard Young tableaux, takes advantage of the full SU(N)\mathrm{SU}(N)SU(N) symmetry, allowing one to work directly in each irreducible representation of the global SU(N)\mathrm{SU}(N)SU(N) group. Since the SU(N)\mathrm{SU}(N)SU(N) singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to N=10N=10N=10 and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [SU(2)\mathrm{SU}(2)SU(2) with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the SU(N)\mathrm{SU}(N)SU(N) level kkk Wess-Zumino-Witten universality class, where kkk is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the SU(N)\mathrm{SU}(N)SU(N) level 1 WZW universality class.
The extension of the linear flavor-wave theory to fully antisymmetric irreducible representations (irreps) of SU(N)\mathrm{SU}\left(N\right)SU(N) is presented in order to investigate the color order of SU(N)\mathrm{SU}\left(N\right)SU(N) antiferromagnetic Heisenberg models in several two-dimensional geometries. The square, triangular, and honeycomb lattices are considered with mmm fermionic particles per site. We present two different methods: the first method is the generalization of the multiboson spin-wave approach to SU(N)\mathrm{SU}\left(N\right)SU(N) which consists of associating a Schwinger boson to each state on a site. The second method adopts the Read and Sachdev bosons which are an extension of the Schwinger bosons that introduces one boson for each color and each line of the Young tableau. The two methods yield the same dispersing modes, a good indication that they properly capture the semiclassical fluctuations, but the first one leads to spurious flat modes of finite frequency not present in the second one. Both methods lead to the same physical conclusions otherwise: long-range N\'eel-type order is likely for the square lattice for SU(4) with two particles per site, but quantum fluctuations probably destroy order for more than two particles per site, with N=2mN=2mN=2m. By contrast, quantum fluctuations always lead to corrections larger than the classical order parameter for the tripartite triangular lattice (with N=3mN=3mN=3m) or the bipartite honeycomb lattice (with N=2mN=2mN=2m) for more than one particle per site, m>1m>1m>1, making the presence of color very unlikely except maybe for m=2m=2m=2 on the honeycomb lattice, for which the correction is only marginally larger than the classical order parameter.
SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.