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Quantum Superspin Systems from Conformal Field Theory

Jochen Christian Peschutter

Published Apr 1, 2017• 0 citations
OpenAlex

Abstract

Phase transitions of second order play an important role in solid state physics. It is in their nature that they occur only in systems that are composed from an infinite number of components since, only that way, the necessary conditions for this (for example, scale invariance) are granted. Such a system at a second order phase transition is called critical. The infinite number of particles poses a particular challenge, even for the description of classical systems. Quantum mechanical systems, however, are distinctly more difficult to treat in this context, since the dimension of their state space grows exponentially with the number of their particles, not to mention the fact that only a few – and usually only one-dimensional – quantum systems are exactly solvable. One of these exactly solvable one-dimensional quantum systems is the SU(N) Haldane-Shastry spin chain that may be regarded as the archetype of long-range spin chains. Moreover, it is critical in the continuum limit and its effective low-energy theory is the so-called SU(N)_1 Wess-Zumino-Witten model. It is a quantum field theory which is not only conformally invariant, so, in particular, scale invariant, but, furthermore, exhibits additional symmetry in the shape of an infinite extension of the Lie algebra su(N) associated with SU(N). Recent studies show that, from these very structures of the SU(N)_1 WZW model, one can, in turn, derive spin systems, whose arrangement is not necessarily the one of a spin chain, but even two-dimensional distributions of the spins in the plane are possible. These systems are again characterized by long-range interactions, comparable to those of the Haldane-Shastry spin chain, which is also obtained as a result of an appropriate choice of parameters. \n \nIn this thesis, we extend the construction already known for the SU(N) case to the supersymmetric case GL(m|n). Here, we construct explicitly both, a special quantum state as well as a Hamiltonian that projects this quantum state to zero. We also discuss the Hamiltonian in the special case of the GL(1|1) spin chain and compare it to the respective GL(1|1) Haldane-Shastry spin chain on a bipartite state space. Both are critical and we identify the corresponding conformal field theories. Subsequently, we describe a generalization of this system in terms of two parameters and explain how its spectrum was found. It is then analyzed and its continuum limit is determined. In doing so, it shows that the system displays criticality only for generic values of one of the two parameters.

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Phase transitions of second order play an important role in solid state physics. It is in their nature that they occur only in systems that are composed from an infinite number of components since, only that way, the necessary conditions for this (for example, scale invariance) are granted. Such a system at a second order phase transition is called critical. The infinite number of particles poses a particular challenge, even for the description of classical systems. Quantum mechanical systems, however, are distinctly more difficult to treat in this context, since the dimension of their state space grows exponentially with the number of their particles, not to mention the fact that only a few – and usually only one-dimensional – quantum systems are exactly solvable. One of these exactly solvable one-dimensional quantum systems is the SU(N) Haldane-Shastry spin chain that may be regarded as the archetype of long-range spin chains. Moreover, it is critical in the continuum limit and its effective low-energy theory is the so-called SU(N)_1 Wess-Zumino-Witten model. It is a quantum field theory which is not only conformally invariant, so, in particular, scale invariant, but, furthermore, exhibits additional symmetry in the shape of an infinite extension of the Lie algebra su(N) associated with SU(N). Recent studies show that, from these very structures of the SU(N)_1 WZW model, one can, in turn, derive spin systems, whose arrangement is not necessarily the one of a spin chain, but even two-dimensional distributions of the spins in the plane are possible. These systems are again characterized by long-range interactions, comparable to those of the Haldane-Shastry spin chain, which is also obtained as a result of an appropriate choice of parameters. \n \nIn this thesis, we extend the construction already known for the SU(N) case to the supersymmetric case GL(m|n). Here, we construct explicitly both, a special quantum state as well as a Hamiltonian that projects this quantum state to zero. We also discuss the Hamiltonian in the special case of the GL(1|1) spin chain and compare it to the respective GL(1|1) Haldane-Shastry spin chain on a bipartite state space. Both are critical and we identify the corresponding conformal field theories. Subsequently, we describe a generalization of this system in terms of two parameters and explain how its spectrum was found. It is then analyzed and its continuum limit is determined. In doing so, it shows that the system displays criticality only for generic values of one of the two parameters.
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians associated with infinite matrix product states. The latter are constructed from correlators of primary fields in the SU(N)1 WZW model. Since the resulting groundstates are of Gutzwiller–Jastrow type, our models can be regarded as lattice discretizations of fractional quantum Hall systems. We then focus on two specific types of 1D spin chains with spins located on the unit circle, a uniform and an alternating arrangement. For an equidistant distribution of identical spins we establish an explicit connection to the SU(N) Haldane–Shastry model, thereby proving that the model is critical and described by a SU(N)1 WZW model. In contrast, while turning out to be critical as well, the alternating model can only be treated numerically. Our numerical results rely on a reformulation of the original problem in terms of loop models.
We show that infinite matrix product states (MPSs) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an SU(n)\mathrm{SU}(n)SU(n) generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.
We analyze the thermodynamics and criticality properties of four families of su(m|n) supersymmetric spin chains of Haldane-Shastry (HS) type, related to both the A_{N-1} and the BC_{N} classical root systems. Using a known formula expressing the thermodynamic free energy per spin of these models in terms of the Perron (largest in modulus) eigenvalue of a suitable inhomogeneous transfer matrix, we prove a general result relating the su(kp|kq) free energy with arbitrary k=1,2,⋯, to the su(p|q) free energy. In this way we are able to evaluate the thermodynamic free energy per spin of several infinite families of supersymmetric HS-type chains, and study their thermodynamics. In particular, we show that in all cases the specific heat at constant volume features a single marked Schottky peak, which in some cases can be heuristically explained by approximating the model with a suitable multilevel system with equally spaced energies. We also study the critical behavior of the models under consideration, showing that the low-temperature behavior of their thermodynamic free energy per spin is the same as that of a (1+1)-dimensional conformal field theory with central charge c=m+n/2-1. However, using a motif-based description of the spectrum we prove that only the three families of su(1|n) chains of type A_{N-1} and the su(m|n) HS chain of BC_{N} type with m=1,2,3 (when the sign ɛ_{B} in the Hamiltonian takes the value -1 in the latter case) are truly critical.
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.