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

Hutchinson, J.; Keating, J. P.; Mezzadri, F.

doi:https://doi.org/10.1155/2015/652026

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Research Article | Open Access

Volume 2015 | Article ID 652026 | https://doi.org/10.1155/2015/652026

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

J. Hutchinson,¹J. P. Keating,¹and F. Mezzadri¹

Academic Editor: Pierluigi Contucci

Received31 Aug 2015

Accepted01 Nov 2015

Published15 Dec 2015

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 thickness¹ 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 symmetry² [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 groups³, 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 model⁴where , . 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 mapped⁵ 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.

Thus we obtain a partition function equivalent to that for a class of two-dimensional classical Ising type models on a lattice with classical Hamiltonian given bywhere can have the form (22) or (23), with the additional constraint (25).

In this case we see that the classical spin variables at each site of the two-dimensional lattice only interact with other spins bordering the same shaded square, represented schematically in Figure 1, with an even number of these four interacting spins being spun up and down (from condition (25)).

This mapping holds in the limit , which would result in coupling parameters and unless we also take . Therefore this again gives us a connection between the ground state properties of this class of quantum systems and the finite temperature properties of the classical systems.

Again we have the same relationship between expectation values (15) and (16).

2.2. A Class of Classical Ising Type Models with Imaginary Interaction Coefficients

Alternatively, lifting the restriction (25), we instead can obtain a class of classical systems described by a Hamiltonian containing imaginary interaction coefficients:with parameter relations given by

To achieve this, we first apply the Trotter-Suzuki mapping to the quantum partition function divided in the following way:where this time for and .

Next insert of each of the identity operators and , which are in the and basis, respectively, into (29) obtaining

It is then possible to rewrite the remaining matrix elements in (30) as complex exponentials:where , , , and , and we have used

The classical system with Hamiltonian given by (27) can be depicted as in Figure 2, where the two types of classical spin variables and can be visualised as each representing two-dimensional lattices on two separate planes, as shown in the top diagram in Figure 2. One can imagine “unfolding” the three-dimensional interaction surface shown in the top diagram in Figure 2 into the two-dimensional plane shown in the bottom diagram with new classical spin variables labelled by .

As in previous cases, this mapping holds in the limit , which would result in coupling parameters , , and unless we also take . Therefore, it gives us a connection between the ground state properties of the class of quantum systems and the finite temperature properties of the classical ones.

We can use this mapping to write the expectation value of any function or with respect to the ground state of the class of quantum systems (9) aswhere and are the finite temperature expectation values of the equivalent function of classical spin variables with respect to the class of classical systems (27).⁶

An example of this is the two-spin correlation function between spins in the ground state of the class of quantum systems (9) in the and direction which can be interpreted as the two-spin correlation function between spins in the same odd and even rows of the corresponding class of classical systems (13), respectively:

2.3. A Class of Classical Vertex Models

Another interpretation of the partition function obtained using the Trotter-Suzuki mapping, following a similar method to that of [17], is that corresponding to a vertex model.

This can be seen by applying the Trotter-Suzuki mapping to the quantum partition function ordered as in (17) and inserting identity operators as in (18), with remaining matrix elements given once more by (19). This time, instead of writing them in exponential form as in (21), we interpret each matrix element as a weight corresponding to a different vertex configuration at every point of the lattice:

As such, the partition function can be thought of as corresponding to a class of two-dimensional classical vertex models on a lattice as shown in Figure 4, with vertices each with a weight given by one of the following:thus leading to a class of 8-vertex models with the usual 8 possible respective vertex configurations as shown in Figure 3.

The values of these weights depend upon the column of the original lattice; thus each column has its own separate set of weights, as represented by the different colours of the circles at the vertices in each column in Figure 4.

Once again, this mapping holds in the limit , which would result in weights and weights unless we also take . It thus gives us a connection between the ground state properties of the class of quantum systems and the finite temperature properties of the corresponding classical systems.

2.4. Algebraic Form for the Classical Partition Function

Finally one last form for the partition function can be obtained using the same method as in Section 2.1.2 such that the quantum partition function is mapped to one involving entries from matrices given by (20). This time however, instead of applying the extra constraint (25), we can write the partition function as

2.5. Longer Range Interactions

The Trotter-Suzuki mapping can similarly be applied to the class of quantum systems (1) with longer range interactions, to obtain partition functions equivalent to classical systems with rather cumbersome descriptions, examples of which can be found in Appendix B.

3. Method of Coherent States

An alternative method to map the partition function for the class of quantum spin chains (1) as studied in [6] onto that corresponding to a class of classical systems with equivalent critical properties is to use the method of coherent states [18].

To use such a method for spin operators , we first apply the Jordan-Wigner transformations (10) once more to map the Hamiltonian (1) onto one involving Pauli operators , :

We then construct a path integral expression for the quantum partition function for (38). First we divide the quantum partition function into pieceswhere and .

Next we insert resolutions of the identity in the infinite set of spin coherent states between each of the factors in (39), such that we obtain Taking the limit such that we can rewrite (40) aswhere now has the form of a Hamiltonian corresponding to a two-dimensional classical system andappears through the overlap between the coherent states at two infinitesimally separated steps and is purely imaginary. This is the appearance of the Berry phase in the action [18, 19]. Despite being imaginary, this term gives the correct equation of motion for spin systems [19].

The coherent states for spin operators, labeled by the continuous vector in three dimensions, can be visualised as a classical spin (unit vector) pointing in direction such that they have the propertyThey are constructed by applying a rotation operator to an initial state to obtain all the other states as described in [18] such that we end up withwith s given by

Thus when our quantum Hamiltonian is given by (38), in (42) now has the form of a Hamiltonian corresponding to a two-dimensional classical system given by

4. Simultaneous Diagonalisation of the Quantum Hamiltonian and the Transfer Matrix

This section presents a particular type of equivalence between one-dimensional quantum and two-dimensional classical models, established by commuting the quantum Hamiltonian with the transfer matrix of the classical system under certain parameter relations between the corresponding systems. Suzuki [2] used this method to prove an equivalence between the one-dimensional generalised quantum model and the two-dimensional Ising and dimer models under specific parameter restrictions between the two systems. In particular he proved that this equivalence holds when the quantum system is restricted to nearest neighbour or nearest and next nearest neighbour interactions.

Here we extend the work of Suzuki [2], establishing this type of equivalence between the class of quantum spin chains (1) for all interaction lengths when the system is restricted to possessing symmetries corresponding to that of the unitary group only⁷ and the two-dimensional Ising and dimer models under certain restrictions amongst coupling parameters. For the Ising model we use both transfer matrices forming two separate sets of parameter relations under which the systems are equivalent. Where possible, we connect critical properties of the corresponding systems, providing a pathway with which to show that the critical properties of these classical systems are also influenced by symmetry.

All discussions regarding the general class of quantum systems (1) in this section refer to the family corresponding to symmetry only, in which case, we find thatunder appropriate relationships amongst parameters of the quantum and classical systems, when is the transfer matrix for either the two-dimensional Ising model with Hamiltonian given byor the dimer model.

A dimer is a rigid rod covering exactly two neighbouring vertices either vertically or horizontally. The model we refer to is one consisting of a square planar lattice with rows and columns, with an allowed configuration being when each of the vertices is covered exactly once such thatwhere and are the number of horizontal and vertical dimers, respectively. The partition function is given bywhere and are the appropriate “activities” and .

The transform used to diagonalise both of these classical systems as well as the class of quantum spin chains (1) can be written aswhere the s are the Fermi operators in which the systems are left in diagonal form. This diagonal form is given by (3) for the quantum system and for the transfer matrix for the Ising model by⁸ [20]where and is the positive root of⁹

The dimer model on a two-dimensional lattice was first solved by Kasteleyn [21] via a combinatorial method reducing the problem to the evaluation of a Pfaffian. Lieb [22] later formulated the dimer-monomer problem in terms of transfer matrices such that is left in the diagonal form given bywith

For the class of quantum spin chains (1) as well as each of these classical models we have that the ratio of terms in transform (52) is given bywhich as we show in the following sections will provide us with relationships between parameters under which these classical systems are equivalent to the quantum systems.

4.1. The Ising Model with Transfer Matrix

We see from (57) that the Hamiltonian (1) commutes with the transfer matrix if we require that

This provides us with the following relations between parameters under which this equivalence holds¹⁰:or inversely aswhere

From (60) we see that this equivalence holds when

For , we also have the added restrictions on the parameters thatwhich implies that all coefficients of for in (63) and of for in (64) are zero¹¹.

When only nearest neighbour interactions are present in (1) (), with for , we recover Suzuki's result [2].

The critical properties of the class of quantum systems can be analysed from the dispersion relation (4), which under the above parameter restrictions is given bywhich is gapless for for all parameter values.

The critical temperature for the Ising model [20] is given bywhich using (59) and (60) gives

This means that (65) becomeswhich is now gapless for all , and, for , (67) is the well known critical value for the external field for the quantum model.

The correlation function between two spins in the same row in the classical Ising model at finite temperature can also be written in terms of those in the ground state of the quantum modelusing the fact that , for , andfrom (3), (48), and (53), where is the eigenvector corresponding to the maximum eigenvalue of and is the ground state eigenvector for the general class of quantum systems (1) (restricted to symmetry).

This implies that the correspondence between critical properties (i.e., correlation functions) is not limited to quantum systems with short range interactions (as Suzuki [2] found) but also holds for a more general class of quantum systems for a fixed relationship between the magnetic field and coupling parameters as dictated by (64) and (63), which we see from (65) results in a gapless system.

4.2. The Ising Model with Transfer Matrix

From (57) the Hamiltonian for the quantum spin chains (1) commutes with transfer matrix if we set

This provides us with the following relations between parameters under which this equivalence holds when the class of quantum spin chains (1) has an interaction length : or inversely as where

From (73) we see that this equivalence holds when

When , we have further restrictions upon the parameters of the class of quantum systems (1); namely,This implies that coefficients of for in (76) and of for in (77) are zero.

Under these parameter restrictions, the dispersion relation is given bywhich is gapless for for all parameter values.

The critical temperature for the Ising model (66) becomesusing (72) and (73).

Substituting (79) into (78) we obtainwhich we see is now gapless for all (for this clearly corresponds to a critical value of causing the energy gap to close).

In this case we can once again write the correlation function for spins in the same row of the classical Ising model at finite temperature in terms of those in the ground state of the quantum model as where is the eigenvector corresponding to the maximum eigenvalue of and

Once more this implies that the correspondence between critical properties such as correlation functions is not limited to quantum systems with short range interactions; it also holds for longer range interactions, for a fixed relationship between the magnetic field and coupling parameters which causes the systems to be gapless.

4.3. The Dimer Model with Transfer Matrix

In this case, when the class of quantum spin chains (1) has a maximum interaction length , it is possible to find relationships between parameters for which an equivalence is obtained between it and the two-dimensional dimer model. For details and examples see Appendix C. When , for , we recover Suzuki's result [2].

Appendices

A. Symmetry Classes

See Table 1.

B. Longer Range Interactions

B.1. Nearest and Next Nearest Neighbour Interactions

The class of quantum systems (1) with nearest and next nearest neighbour interactions can be mapped¹² ontowhere and using the Jordan Wigner transformations (10).

We apply the Trotter-Suzuki mapping to the partition function for (B.1) with operators in the Hamiltonian ordered aswhere again and are the set of odd and even integers, respectively, and and , for , and once more denotes either or .

For this model we need to insert identity operators into (B.2). We use in each of the and bases and in the basis in the following way:

For this system it is then possible to rewrite the remaining matrix elements in (B.3) in complex scalar exponential form by first writingwhere , , and . We can then evaluate the remaining matrix elements as

Thus we obtain a partition function with the same form as that corresponding to a class of two-dimensional classical Ising type systems on lattice with classical Hamiltonian given by

A schematic representation of this model on a two-dimensional lattice is given in Figure 5, with a yellow border representing a unit cell which can be repeated in either direction. The horizontal and diagonal blue and red lines represent interaction coefficients , and , , respectively, and the imaginary interaction coefficients are represented by the dotted green lines. There is also a complex magnetic field term applied to each site in every second row as represented by the black circles.

This mapping holds in the limit , which would result in coupling parameters , , , , and unless we also take . Therefore this gives us a connection between the ground state properties of the class of quantum systems and the finite temperature properties of the classical systems.

Similarly to the nearest neighbour case, the partition function for this extended class of quantum systems can also be mapped to a class of classical vertex models (as we saw for the nearest neighbour case in Section 2.1) or a class of classical models with up to 6 spin interactions around a plaquette with some extra constraints applied to the model (as we saw for the nearest neighbour case in Section 2.1). We will not give the derivation of these as they are quite cumbersome and follow the same steps as outlined previously for the nearest neighbour cases, and instead we include only the schematic representations of possible equivalent classical lattices. The interested reader can find the explicit computations in [23].

Firstly in Figure 6 we present a schematic representation of the latter of these two interpretations, a two-dimensional lattice of spins which interact with up to 6 other spins around the plaquettes shaded in grey.

To imagine what the corresponding vertex models would look like, picture a line protruding from the lattice points bordering the shaded region and meeting in the middle of it. A schematic representation of two possible options for this is shown in Figure 7.

B.2. Long-Range Interactions

For completeness we include the description of a classical system obtained by applying the Trotter-Suzuki mapping to the partition function for the general class of quantum systems (1) without any restrictions.

We can now apply the Trotter expansion (7) to the quantum partition function with operators in the Hamiltonian (38) ordered as where for and .

For this model we need to insert identity operators, in each of the , , and bases, into (B.7) in the following way:

For this system it is then possible to rewrite the remaining matrix elements in (B.8) in complex scalar exponential form by first writingwhere , , and . Finally evaluate the remaining terms as

The partition function now has the same form as that of a class of two-dimensional classical Ising models on a lattice with classical Hamiltonian given by

A schematic representation of this class of classical systems on a two-dimensional lattice is given in Figure 8 where the blue and red lines represent interaction coefficients and , respectively, the black lines are where they are both present, and the imaginary interaction coefficients are given by the dotted green lines. The black circles also represent a complex field acting on each individual particle in every second row.

This mapping holds in the limit , which would result in coupling parameters , , and unless we also take . Therefore this gives us a connection between the ground state properties of the quantum system and the finite temperature properties of the classical system.

C. Systems Equivalent to the Dimer Model

We give here some explicit examples of relationships between parameters under which our general class of quantum spin chains (1) is equivalent to the two-dimensional classical dimer model using transfer matrix (55).(i)When , from (57), we have therefore it is not possible to establish an equivalence in this case.(ii) When from (57) we have Thus the systems are equivalent under the parameter relations(iii) When from (57) we have Thus the systems are equivalent under the parameter relations

Therefore we find that in general when , we can use (57) to prove that we have an equivalence ifWe can write the sum in the denominator of (C.6) as

This gives us the following conditions:

We can then rewrite the remaining terms in the denominator (C.7) as

Finally we equate coefficients of matching powers of in the numerator in (C.6) and denominator (C.9). For example, this demands that .

Disclosure

No empirical or experimental data were created during this study.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Shmuel Fishman for helpful discussions and to Professor Ingo Peschel for bringing some references to their attention. J. Hutchinson is pleased to thank Nick Jones for several insightful remarks, the EPSRC for support during her Ph.D., and the Leverhulme Trust for further support. F. Mezzadri was partially supported by EPSRC research Grant EP/L010305/1.

Endnotes

The thickness of a band matrix is defined by the condition if , where is a positive integer.
For the other symmetry classes, see [8].
This is observed through the structure of matrices and summarised in Table 1 inherited by the classical systems.
We can ignore boundary term effects since we are interested in the thermodynamic limit only.
Up to an overall constant.
Recall, from the picture on the right in Figure 2, that the and represent alternate rows of the lattice.
Thus matrices and have Toeplitz structure as given by Table 1.
The superscripts represent anticyclic and cyclic boundary conditions, respectively.
This is for the symmetrisation of the transfer matrix; the other possibility is with , where , , and .
Here we have used De Moivre's Theorem and the binomial formula to rewrite the summations in and (5) as
For example, setting the coefficient of to zero implies that .
Once again we ignore boundary term effects due to our interest in phenomena in the thermodynamic limit only.

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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.

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