Advances in Mathematical Physics

Volume 2015, Article ID 652026, 18 pages

http://dx.doi.org/10.1155/2015/652026

## On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK

Received 31 August 2015; Accepted 1 November 2015

Academic Editor: Pierluigi Contucci

Copyright © 2015 J. Hutchinson et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.

#### 1. Introduction

Mappings between statistical mechanical models have provided new pathways to compute thermodynamic properties of systems which were previously intractable [1–3]. In particular, critical phenomena in -dimensional quantum systems have been investigated by mapping them to -dimensional classical systems for which there are better developed techniques, such as Monte Carlo simulations [4, 5]. For example, one well known connection is that between the one-dimensional model and the two-dimensional zero-field eight-vertex model; namely, the Hamiltonian of the quantum model and the transfer matrix of the classical model have the same eigenvectors. Baxter [1] found the ground state energy for the model by first finding the partition function of the eight-vertex model and then showing that the quantum Hamiltonian is effectively the logarithmic derivative of the transfer matrix for the classical system.

In this paper we exploit these quantum to classical (QC) mappings for the opposite reason: to take advantage of known ground state critical behaviour in a general class of quantum spin chains with long-range interactions to determine the finite temperature critical properties of an equivalent class of classical spin systems.

In a companion paper to appear in [6] we computed the critical exponents , , and , corresponding to the energy gap, correlation length, and dynamic exponent, respectively, for a class of quantum spin chains, establishing universality for this class of systems. We also computed the ground state correlators , , and for this class of systems when translation invariance is imposed. These correlators were found to exhibit quasi-long-range order behaviour when the systems are gapless, with a critical exponent dependent upon the system parameters.

The class of quantum spin chains studied in [6] consists of spin-1/2 particles in an external field , with a Hamiltonian quadratic in Fermi operators given bywhere the s are the Fermi operators satisfying the usual Fermi commutation relationsThe measure of anisotropy is real, with ; the matrix must be Hermitian and antisymmetric, both containing only real entries without loss of generality; and periodic boundary conditions are assumed. We can think of and as band matrices, whose thickness^{1} determines the length of the spin-spin interaction.

This model can be diagonalised [7] so thatwith the dispersion relation determined by matrices and ; the s are Fermi operators, and is a constant.

In [8, 9] Keating and Mezzadri restricted the Hamiltonian (1) to possess symmetries corresponding to the Haar measure of each of the classical compact groups , , , , , and , enabling the calculation of using techniques from random matrix theory. This corresponds to a symmetry classification of spin chains similar to that introduced for disordered systems by Altland and Zirnbauer [10–12]. These symmetry properties were encoded into the structure of the matrices and , as summarised in Appendix A. For example, when restricted to symmetry^{2} [8, 9]where , with denoting the integer part of its argument. The real and imaginary parts of (4) areFurthermore,In general, the symmetry constraints were achieved using real functions and , even and odd functions of , respectively, to dictate the entries of matrices and , as reported in Appendix A.

Exploiting the formalism developed in [8, 9] enabled us to compute the critical properties of this class of spin chains [6], demonstrating a dependence of the critical exponents on system symmetries and establishing universality for this class of quantum systems.

Having established universality for the above class of quantum spin chains in [6], we now make use of QC mappings to obtain a class of classical systems with equivalent critical properties, establishing universality for this class of classical systems as well by extension. This is our main goal.

There is no systematic technique to construct a classical -dimensional lattice system from a quantum -dimensional one; there is no alternative but to develop an ad hoc approach for each case. This is usually a major challenge. Furthermore, such a mapping is not unique. However, over the years, several models have been proved equivalent. Suzuki [3] introduced a powerful method based on Trotter's formula. Another technique exploits the fact that if the quantum Hamiltonian commutes with the transfer matrix of a classical system, then they are equivalent. This idea was used by Suzuki [2] to prove the equivalence between the generalised quantum model and the two-dimensional Ising and dimer models. Krinsky [13] showed that the eight-vertex free fermion model with an electric field is equivalent to the ground state of the model in the presence of a magnetic field. Peschel [14] demonstrated that the quantum model can be mapped to Ising type models with three different frustrated lattice structures. Lifting the translational invariance, Minami [15] proved the equivalence of the model to a class of two-dimensional Ising models with nonuniform interaction coefficients. Iglói and Lajkó [16] showed that the quantum Ising model with site-dependent coupling parameters in a transverse magnetic field is equivalent to an Ising model on a square lattice with a diagonally layered structure. Some of these systems are not translation invariant, but they all have nearest neighbour spin-spin interactions; to our knowledge there is no system with long-range interactions for which classical-quantum equivalence has previously been proved.

A quantum and a classical system are equivalent if their partition functions are the same; such a correspondence, however, is not unique as different classical systems can be equivalent to the same quantum system. We will here adopt the following different approaches to map the partition functions of the quantum spin chains (1) onto those of a general class of two-dimensional classical systems:(i)The Trotter-Suzuki formula (Section 2).(ii)The method of coherent states (Section 3).(iii)The simultaneous diagonalisation of the quantum Hamiltonian and the transfer matrix for the classical system (Section 4).

#### 2. Trotter-Suzuki Mapping

This approach was developed by Suzuki [3], who applied the Trotter product formulato map the partition function for a -dimensional quantum system to that for a -dimensional classical one. In particular he applied it to the partition function of a -dimensional quantum Ising model in a transverse magnetic field, mapping it to that of a -dimensional classical Ising model [3]. He then proved the equivalence of the critical properties of the ground state of the quantum system and the finite temperature properties of the classical system.

Here we harness this technique to supply us with a class of two-dimensional classical systems with critical properties equivalent to those of the ground state of the quantum spin chains (1). Like the original quantum system, the classical counterparts are also able to possess symmetries reflected by those of the Haar measure of each of the different classical compact groups^{3}, enabling the dependence of critical properties on system symmetries to be observed.

There are many ways to apply the Trotter-Suzuki mapping to the partition function for the class of quantum spin chains (1), resulting in different classical partition functions. Those that we obtain are of the formwhere is the effective classical Hamiltonian. In (8a) and (8b), is a real function of the classical spin variables and in (8d) it is a complex function of the classical spin variables , which represent the eigenvalues of the Pauli matrices , , and , respectively. The second index in the classical variables is due to the extra dimension appearing when applying the Trotter formula (7). The function is also a real function of the classical spin variables , and we find that if , then (8a) has the familiar form of a classical partition function, with representing the Hamiltonian describing the effective classical system. The same is true for (8b) and (8d), but (8b) has additional constraints on the spin states and (8d) involves imaginary interaction coefficients. The form in (8c) is that of a vertex model with vertex weights given by . Examples of equivalent partition functions with each of these forms will be given in the following sections.

We begin to present our results by first restricting to quantum systems with nearest neighbour interactions only. The extensions to longer range interactions are detailed in Appendix B.

##### 2.1. Nearest Neighbour Interactions

Restricting (1) to nearest neighbour interactions gives the well known one-dimensional quantum model^{4}where , . This mapping is achieved by using Jordan-Wigner transformations:whereor inversely as The s are thus Hermitian and obey the anticommutation relations .

###### 2.1.1. A Class of Classical Ising Type Models with Nearest Neighbour Interactions

When we restrict to and , (9) becomes a class of quantum Ising type models in a transverse magnetic field with site-dependent coupling parameters. Suzuki showed [3] that the partition function for such a system can be mapped^{5} to that for a class of two-dimensional classical Ising models with Hamiltonian given bywith parameter relationswhere is the inverse temperature of the quantum (classical) system.

Thus we have an equivalence between our class of quantum spin chains under these restrictions and a class of two-dimensional classical Ising models also with site-dependent coupling parameters in one direction and a constant coupling parameter in the other. From (14) we see that the magnetic field driving the phase transition in the ground state of the quantum system plays the role of temperature driving the finite temperature phase transition of the classical system.

This mapping holds in the limit , which would result in anisotropic couplings for the class of classical Ising models, unless we also take . This therefore provides us with a connection between the ground state properties of the class of quantum systems and the finite temperature properties of the classical systems.

In this case we can also use this mapping to write the expectation value of any function with respect to the ground state of the class of quantum systems aswhere is the finite temperature expectation of the corresponding function of classical spin variables with respect to the class of classical systems (13).

Some examples of this are the spin correlation functions between two or more spins in the ground state of the class of quantum systems in the direction, which can be interpreted as the equivalent correlator between classical spins in the same row of the corresponding class of classical systems (13):

###### 2.1.2. A Class of Classical Ising Type Models with Additional Constraints on the Spin States

Similarly, the Trotter-Suzuki mapping can be applied to the partition function for the model (9) in full generality. In this case we first order the terms in the partition function in the following way:where for , , and denotes either or , which are the sets of odd and even integers, respectively.

We then insert copies of the identity operator in the basis; where between each of the terms in (17):

The remaining matrix elements in (18) are given bywhere

It is then possible to write the terms (19) in exponential form aswhere can be written asor more symmetrically aswhereas long as we have the additional restriction that the four spins bordering each shaded square in Figure 1 obeyThis guarantees that each factor in the partition function is different from zero.