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arXiv:1905.08384 · Journal of Statistical Mechanics Theory and Experiment

Some ground-state expectation values for the free parafermion Z ( N ) spin chain*

Zi-Zhong Liu, Robert A Henry, Murray T. Batchelor, Huan-Qiang Zhou

Published Dec 1, 2019• 12 citations
arXivHTMLPDF

Abstract

Abstract We consider the calculation of ground-state expectation values for the non-Hermitian spin chain described by free parafermions. For N = 2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann–Feynman theorem to obtain exact results for particular single site and nearest-neighbor ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N = 3, including how they change as a function of distance from the boundary.

DOI
10.1088/1742-5468/ab4fe1

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Abstract We consider the calculation of ground-state expectation values for the non-Hermitian spin chain described by free parafermions. For N = 2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann–Feynman theorem to obtain exact results for particular single site and nearest-neighbor ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N = 3, including how they change as a function of distance from the boundary.
We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple Z(N) model for N≥3 for which the model Hamiltonian is non-Hermitian. For N=2 the model reduces to the well-known quantum Ising model in a transverse field. For open boundary conditions, the Z(N) model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing N. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent, and the specific-heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behavior is a topological effect.
Abstract In this article we outline the historical development and key results obtained to date for free parafermionic spin chains. The concept of free parafermions provides a natural N-state generalization of free fermions, which have long underpinned the exact solution and application of widely studied quantum spin chains and their classical counterparts. In particular, we discuss the Baxter-Fendley free parafermionic Z(N) spin chain, which is a relatively simple non-Hermitian generalization of the Ising model.
We consider the non-Hermitian Z ( N ) free parafermion model from the perspective of machine learning. By neglecting the boundary contributions, key correlation functions obtained via the Hellmann–Feynman theorem are validated numerically. We implement unsupervised learning based on principal component analysis and k-means clustering to successfully detect and classify the phase transitions from the quasienergies. We also demonstrate the capability of fully connected neural networks in learning physical observables from the free parafermion formalism. Our results suggest an optimism towards finding an explicit exact formulation of the eigenvectors for N ≥ 3 .
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
Abstract Results are given for the ground state energy and excitation spectrum of a simple N -state Z N spin chain described by free parafermions. The model is non-Hermitian for N ⩾ 3 with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g. the specific heat exponent α = 1 − 2 / N and the anisotropic correlation length exponents ν ∥ = 1 and ν ⊥ = 2 / N .
Abstract We derive an exact formula for the boundary free energy of the open Heisenberg XXZ spin chain. We allow for arbitrary boundary magnetic fields, but assume zero bulk magnetization. The result is completely analogous to earlier formulas for the so-called g -function: it is expressed as a combination of single integrals and two simple Fredholm determinants. Our expressions can be evaluated easily using numerical algorithms with arbitrary precision. We demonstrate that the boundary free energy can show a wide variety of behaviour as a function of the temperature, depending on the anisotropy and the boundary fields. We also compute the low temperature limit of the boundary free energy, and reproduce the known results for the ground state boundary energy, including the case of non-diagonal fields.