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arXiv:1504.07224 · Physical Review B

Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

Hong-Hao Tu, Germán Sierra

Published Jul 29, 2015• 18 citations
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Abstract

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 $\mathrm{SU}(n)$ generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.

DOI
10.1103/physrevb.92.041119

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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 generalize the matrix product states method using the chiral vertex operators of conformal field theory and apply it to study the ground states of the XXZ spin chain, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model and random Heisenberg models. We compute the overlap with the exact wave functions, spin-spin correlators, and the Renyi entropy, showing that critical systems can be described by this method. For rotational invariant ansatzs we construct an inhomogenous extension of the Haldane-Shastry model with long-range exchange interactions.
In this paper, we study an su(m)\mathrm{su}(m)su(m)-invariant open version of the Haldane-Shastry spin chain whose ground state can be obtained from the chiral correlator of the $c=m\ensuremath{-}1$ free boson boundary conformal field theory. We show that this model is integrable for a suitable choice of the chain sites depending on the roots of the Jacobi polynomial ${P}_{N}^{\ensuremath{\beta}\ensuremath{-}1,{\ensuremath{\beta}}^{\ensuremath{'}}\ensuremath{-}1}$, where NNN is the number of sites and $\ensuremath{\beta},{\ensuremath{\beta}}^{\ensuremath{'}}$ are two positive parameters. We also compute in closed form the first few nontrivial conserved charges arising from the twisted Yangian invariance of the model. We evaluate the chain's partition function, determine the ground-state energy, and deduce a complete description of the spectrum in terms of Haldane's motifs and a related classical vertex model. In particular, this description entails that the chain's level density is normally distributed in the thermodynamic limit. We also analyze the spectrum's degeneracy, proving that it is much higher than for a typical Yangian-invariant model.
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 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.
Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.