THERMODYNAMICS OF A PLANAR DIPOLE LATTICE

Yu. B. Rumer

INTRODUCTION

The application of the methods of statistical mechanics to systems capable of undergoing second-order phase transitions encounters serious mathematical difficulties. In order to discern the cause of the difficulties that arise, let us briefly recall what constitutes the basic difference between first- and second-order phase transitions.

In the case of first-order transitions, which proceed with the release of heat, on both sides of the transition point there exist metastable, thermodynamically unstable states. Therefore, over the entire range of temperature variation the system can be described by two thermodynamic potentials $\Phi_A(p,T)$, $\Phi_B(p,T)$; below the transition point $\Phi_A < \Phi_B$, above the transition point $\Phi_A > \Phi_B$.

It follows from this that the transition point $T = T_0$, at which $\Phi_A(p,T_0) = \Phi_B(p,T_0)$, is, in the mathematical sense, an ordinary point for the thermodynamic potentials, and the functions $\Phi_A(p,T)$ and $\Phi_B(p,T)$ need not have a singularity there.

The situation is different in the case of a second-order phase transition, which proceeds without the release of heat. Here there are no metastable states in the thermodynamic sense, although supercooled states of partial equilibrium, associated with a large difference in relaxation times, may exist.

Well-known second-order transition points are the Curie points in ferromagnets, the transition point of liquid helium into the superfluid state, and the transition point of a metallic conductor into the superconducting state.

An equilibrium system is described over the entire range of temperature variation by a single thermodynamic potential $\Phi(p,T)$, which has a continuous derivative at the transition point. The transition is revealed in the presence of discontinuities in higher derivatives (for exam-

for example, in a jump of the heat capacity or in a jump of the derivative of the heat capacity with respect to temperature). It follows from this that the point of a phase transition of the second kind is, in the mathematical sense, a singular point of the thermodynamic potential.

Herein lies the cause of the mathematical difficulties that arise. As is known, any approximate representations of an analytic function become inapplicable in a neighborhood of a singular point. Therefore, in order to describe a system near a transition point of the second kind, we are forced to use only exact expressions for the thermodynamic functions and cannot employ approximate methods. This circumstance greatly narrows the class of systems for which a theoretical treatment can be carried through to the end.

The general theory of phase transitions of the second kind was given by L. D. Landau^1. In his theory the system is described by the thermodynamic potential \(\Phi(p,T,\xi)\), which depends on three variables. The additional variable \(\xi\) has in the first phase the value zero, and in the second phase values different from zero, positive or negative.

The variable \(\xi\) is in a certain sense not on an equal footing with the variables \(p,T\); whereas the pressure and temperature may be specified arbitrarily, the actually realized value of \(\xi\) itself must be determined from the condition of thermal equilibrium, i.e., from the condition that \(\Phi\) be minimal for given \(p\) and \(T\).

The basic assumption of Landau’s theory is that near a point of a phase transition of the second kind, where \(\xi\) takes an arbitrarily small value, the function \(\Phi(p,T,\xi)\) can be expanded in a series in powers of \(\xi\).

However, as L. D. Landau himself emphasizes, the possibility of such an expansion is by no means obvious. Moreover, since, as has already been indicated, the point of a transition of the second kind must be a certain singular point of the thermodynamic potential, there is every reason to expect that such an expansion cannot be carried out up to terms of arbitrary order, and that the coefficients of the expansion may have singularities as functions of \(p\) and \(T\).

A complete elucidation of the nature of the singularity of the thermodynamic potential at the transition point is associated with great difficulties. Therefore, consideration of special models that admit phase transitions of the second kind and whose investigation can be carried through to the end is of considerable interest. The only such model known at the present time is the special model of a plane dipole lattice, considered in detail in Onsager’s work^2 and in a number of related works.

Applying a new original method, based on the broad use of matrix calculus, Onsager succeeded in calculating the exact distribution function for the model under consideration.

over the entire range of temperature variation and to investigate the character of the singularity of the thermodynamic functions at the transition point.

Onsager’s original method is based on the use of very complicated and specialized algebraic methods, which makes it not very accessible to physicists. Considerable progress is found in Kaufman’s work³. In that work the goal is achieved by bringing in the theory of spinors in multidimensional spaces, which is likewise not very accessible to physicists.

The purpose of the present review is to set forth all the material as elementarily as possible and to make it accessible to a broader circle of theoretical physicists.

Unfortunately, the method presented here, which in the case of a plane lattice solves the problem to the end, does not generalize directly to the case of a three-dimensional lattice. One should think, however, that further progress is not excluded and that investigations in this area may prove promising.

We have used a recently published review⁴, from whose contents our article differs substantially in the selection and exposition of the material.

We shall have to deal with the notions of the direct sum $\hat A \dot{+} \hat B$ and the direct product $\hat A \times \hat B$ of two matrices. The reader will find brief information on these operations in the appendix; more complete information, for example, is in⁵.

We have also placed in the appendix the problem of decomposing an arbitrary rotation in a $2n$-dimensional complex space into successive plane commuting rotations in $n$ mutually perpendicular planes.

§ 1. THE DISTRIBUTION MATRIX $\hat P$

Consider a plane two-dimensional square lattice consisting of $m$ columns, in each of which $n$ sites are placed. We shall speak of a lattice composed of $m$ columns and $n$ rows. Impose on the lattice the condition of cyclicity by closing it in both directions, i.e. identify the $(m+1)$-st column with the first column and the $(n+1)$-st row with the first row.

At each site of the lattice place one dipole with axes directed perpendicular to the plane of the lattice. Each of the dipoles may have one of two possible mutually opposite orientations. It is obvious that the total number of possible configurations of the dipoles in the lattice is equal to $2^{mn}$.

To describe the various configurations we proceed as follows. Assign to each dipole a discrete variable $\sigma$, capable of taking only two values: $\sigma=+1$ if the dipole is oriented to the right, and $\sigma=-1$ if the dipole is oriented to the left.

Next introduce \(m\) functions \(\nu_k\) of \(n\) discrete variables \(\sigma_1, \sigma_2, \ldots, \sigma_n\),

\[ \nu_k=\nu_k(\sigma_1,\sigma_2,\ldots,\sigma_n). \tag{1,1} \]

Each of the functions \(\nu_k\) is capable of assuming \(2^n\) discrete values, and each of these may be used to enumerate the configurations of dipoles in the \(k\)-th column. Finally, introduce a function \(C\) of \(m\) variables \(\nu_1,\nu_2,\ldots,\nu_m\),

\[ C=C(\nu_1,\nu_2,\ldots,\nu_m). \tag{1,2} \]

It is evident that the function \(C\) is capable of assuming \(2^{nm}\) discrete values, each of which may be used to enumerate the configurations of dipoles in the lattice.

For a thermodynamic description of the lattice we must calculate the distribution function:

\[ Z(T)=\sum_{(C)} \exp\{-E(C)/kT\}, \tag{1,3} \]

where \(E(C)\) is the energy of their interaction, depending on the configuration of the dipoles, \(T\) is the absolute temperature, and the sum is extended over all \(2^{nm}\) possible configurations of dipoles in the lattice. It is evident that the expression for \(Z(T)\) may be rewritten in the form

\[ Z(T)=\sum_{\nu_1}\sum_{\nu_2}\ldots\sum_{\nu_m} \exp[-E(\nu_1,\nu_2,\ldots,\nu_m)/kT], \tag{1,4} \]

where each of the summations is carried out over all \(2^n\) possible configurations of dipoles in the corresponding columns.

If we confine ourselves only to taking into account the interaction energy between nearest neighbors and neglect the interaction between more distant dipoles, then the configurational energy \(E(\nu_1,\nu_2,\ldots,\nu_m)\) may be represented in the form

\[ E(\nu_1,\nu_2,\ldots,\nu_m) = \sum_{k=1}^{k=m} \{V_1(\nu_k)+V_2(\nu_k,\nu_{k+1})\}, \tag{1,5} \]

where \(V_1(\nu_k)\) is the interaction energy of dipoles situated in the \(k\)-th column, and \(V_2(\nu_k,\nu_{k+1})\) is the interaction energy between the dipoles of two neighboring columns. Then \(Z(T)\) is represented in the form

\[ Z(T)= \sum_{(\nu_1)}\sum_{(\nu_2)}\ldots\sum_{(\nu_m)} \prod_{k=1}^{k=n} \exp\left\{-\frac{V_1(\nu_k)}{kT}\right\} \exp\left\{-\frac{V_2(\nu_k,\nu_{k+1})}{kT}\right\}. \tag{1,6} \]

If we now introduce two matrices \(\hat P_1\) and \(\hat P_2\) of order \(2^n\) (of which \(\hat P_1\) is diagonal), with elements

\[ \left. \begin{aligned} (\nu|\hat P_1|\nu) &= \exp\{-V_1(\nu)/kT\},\\ (\nu|\hat P_2|\nu') &= \exp\{-V_2(\nu,\nu')/kT\}, \end{aligned} \right\} \tag{1,7} \]

and take into account the formula of matrix calculus

\[ \operatorname{Sp}\hat A^m=\sum_{(n_1)}\sum_{(n_2)}\cdots\sum_{(n_m)} A_{n_1 n_2}A_{n_2 n_3}\cdots A_{n_m n_1}, \tag{1,8} \]

then the distribution function may be written in the form

\[ Z(T)=\operatorname{Sp}(\hat P_1\hat P_2)^m. \tag{1,9} \]

If all \(2^n\) eigenvalues of the matrix

\[ \hat P=\hat P_1\hat P_2, \]

which we shall denote by \(E(\nu)\), are computed, then the distribution function is written in the form

\[ Z(T)=\sum_{(\nu)}\{E(\nu)\}^m. \tag{1,10} \]

For further calculations it is necessary to write the interaction energies \(V_1(\nu)\) and \(V_2(\nu,\nu')\) in explicit form. For this purpose we denote by \(I'_1\) and \(I'_2\) the interaction energies of two neighboring dipoles with identical and different orientations in one and the same column. Consider the function

\[ V'(\sigma_k,\sigma_{k+1})=\frac{I'_1+I'_2}{2}+\frac{I'_1-I'_2}{2}\sigma_k\sigma_{k+1}, \tag{1,11} \]

which can take only two values: \(I'_1\), if both dipoles are oriented identically, and \(I'_2\), if they are oriented differently.

For the function \(V_1(\nu)\), expressing the interaction energy of dipoles in one and the same column, we have:

\[ V_1(\nu)=\sum_{k=1}^{k=n} V'(\sigma_k,\sigma_{k+1}) =\frac{n(I'_1+I'_2)}{2} +\frac{I'_1-I'_2}{2}\sum_{k=1}^{k=n}\sigma_k\sigma_{k+1}. \tag{1,12} \]

Next, denote by \(I_1\) and \(I_2\) the interaction energies of two neighboring dipoles with identical and different orientations in one and the same row. The function

\[ V(\sigma_k,\sigma'_k)=\frac{I_1+I_2}{2}+\frac{I_1-I_2}{2}\sigma_k\sigma'_k \tag{1,13} \]

is able to take only two values: \(I_1\), if both neighboring dipoles are oriented in the same way, and \(I_2\), if they are oriented differently.

For the function \(V_2(\nu,\nu')\), expressing the energy of interaction of the dipoles of two neighboring columns, we have:

\[ V_2(\nu,\nu')=\sum_{k=1}^{k=n} V(\sigma_k,\sigma'_k) =\frac{n(I_1+I_2)}{2} +\frac{I_1-I_2}{2}\sum_{k=1}^{k=n}\sigma_k\sigma'_k . \tag{1,14} \]

We obtain for the matrix elements of the matrices \(\hat P_1\) and \(\hat P_2\), according to formulas (1,7), the expressions:

\[ \left. \begin{aligned} (\nu|\hat P_1|\nu)&=\exp n\theta'_0 \prod_{k=1}^{k=n}\exp(\theta'\sigma_k\sigma_{k+1}),\\ (\nu|\hat P_2|\nu')&=\exp n\theta_0 \prod_{k=1}^{k=n}\exp(\theta\sigma_k\sigma'_k), \end{aligned} \right\} \tag{1,15} \]

where, for brevity, we have introduced the following notation:

\[ \left. \begin{aligned} -\theta'_0&=(I_1+I_2)/2kT; & \theta'&=(I_2-I_1)/2kT,\\ -\theta_0&=(I_1+I_2)/2kT; & \theta&=(I_2-I_1)/2kT. \end{aligned} \right\} \tag{1,16} \]

Our problem now consists in expressing both \(2^n\)-row matrices \(\hat P_1\) and \(\hat P_2\), with elements (1,15), in terms of a system of simpler \(2^n\)-row matrices. We shall proceed as follows. We shall regard a function \(\varphi(\sigma)\) of one discrete variable \(\sigma\) as a two-component quantity

\[ \varphi=\{\varphi(+1),\ \varphi(-1)\}. \]

The three Pauli matrices:

\[ \hat a= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}; \qquad \hat b= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}; \qquad \hat c= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \tag{1,17} \]

acting on \(\varphi(\sigma)\), give:

\[ \left. \begin{aligned} \hat a \begin{pmatrix} \varphi^{(+1)}\\ \varphi^{(-1)} \end{pmatrix} &=\{\varphi^{(+1)},\ -\varphi^{(-1)}\},\\[4pt] \hat b \begin{pmatrix} \varphi^{(+1)}\\ \varphi^{(-1)} \end{pmatrix} &=\{\varphi^{(-1)},\ \varphi^{(+1)}\},\\[4pt] \hat c \begin{pmatrix} \varphi^{(+1)}\\ \varphi^{(-1)} \end{pmatrix} &=\{-i\varphi^{(-1)},\ i\varphi^{(+1)}\}. \end{aligned} \right\} \tag{1,18} \]

The formulas (1,18) can be written briefly in the form:

\[ \hat a \varphi(\sigma)=\sigma\varphi(\sigma);\qquad \hat b \varphi(\sigma)=\varphi(-\sigma);\qquad \hat c \varphi(\sigma)=i\sigma\varphi(-\sigma). \tag{1,18′} \]

Let us now introduce a system of \((2^n\)-row) matrices

\[ \begin{array}{cccc} \hat A_1,\ \hat A_2,\ldots,\ \hat A_n,\\ \hat B_1,\ \hat B_2,\ldots,\ \hat B_n,\\ \hat C_1,\ \hat C_2,\ldots,\ \hat C_n, \end{array} \left. \right\} \tag{1,19} \]

acting on the function \(\varphi\) of \(n\) discrete variables \(\sigma_1,\sigma_2,\ldots,\sigma_{n-1},\sigma_n\). We shall regard the function \(\varphi\) as a \(2^n\)-component quantity with components of the form

\[ \varphi(\pm 1,\pm 1,\ldots,\pm 1). \]

We define the matrices \(\hat A_k,\ \hat B_k,\ \hat C_k\) as operators performing the following operations on functions \(\varphi(\sigma_1,\sigma_2,\ldots,\sigma_n)\):

\[ \begin{aligned} \hat A_k \varphi(\sigma_1,\sigma_2,\ldots,\sigma_k,\ldots,\sigma_n) &=\sigma_k\varphi(\sigma_1\sigma_2\ldots\sigma_k\ldots\sigma_n),\\ \hat B_k \varphi(\sigma_1,\ldots,\sigma_k,\ldots,\sigma_n) &=\varphi(\sigma_1\ldots(-\sigma_k)\ldots\sigma_n),\\ \hat C_k \varphi(\sigma_1,\ldots,\sigma_k,\ldots,\sigma_n) &=i\sigma_k\varphi(\sigma_1\ldots(-\sigma_k)\ldots\sigma_n). \end{aligned} \left. \right\} \tag{1,20} \]

Taking (1,18′) into account, it is easy to see that the matrices \(\hat A_k,\ \hat B_k,\ \hat C_k\) can be written as direct products of the following matrices:

\[ \begin{aligned} \hat A_k&=\hat \varepsilon\times\hat \varepsilon\times\ldots\times\hat a\times\hat \varepsilon\times\ldots\times\hat \varepsilon,\\ \hat B_k&=\hat \varepsilon\times\hat \varepsilon\times\ldots\times\hat b\times\hat \varepsilon\times\ldots\times\hat \varepsilon,\\ \hat C_k&=\hat \varepsilon\times\hat \varepsilon\times\ldots\times\hat c\times\hat \varepsilon\times\ldots\times\hat \varepsilon, \end{aligned} \left. \right\} \tag{1,21} \]

where by \(\hat \varepsilon\) is denoted the unit two-row matrix, and the matrices \(\hat a,\hat b,\hat c\) stand in all products (1,21) in the place of the \(k\)-th factor.

From the definitions (1,21) it follows that all matrices \(\hat A_k,\hat B_k,\hat C_k\) with different indices commute with one another, while with identical indices they are related to one another, just as the Pauli matrices are, by the formulas

\[ \hat A_k\hat B_k=i\hat C_k;\qquad \hat B_k\hat C_k=i\hat A_k;\qquad \hat C_k\hat A_k=i\hat B_k. \tag{1,22} \]

We shall now express the matrices \(\hat P_1\) and \(\hat P_2\) in terms of the matrices \(\hat A_k,\hat B_k,\hat C_k\).

The matrix \(\hat P_1\) is diagonal. Therefore, since from (1,20) it follows for any function \(f(\hat A_k \hat A_{k+1})\)

\[ f(\hat A_k \hat A_{k+1})\,\varphi(\sigma_1 \ldots \sigma_n) = f(\sigma_k \sigma_{k+1})\,\varphi(\sigma_1 \ldots \sigma_n), \tag{1,23} \]

we can immediately represent it in the form

\[ \hat P_1=\exp n\theta'_0 \prod_{k=1}^{k=n}\exp(\theta'\hat A_k\hat A_{k+1}). \tag{1,24} \]

The matrix \(\hat P_2\) is not diagonal and may be represented as the product of the number \(\exp(n\theta_0)\) by the direct product of \(n\) two-row matrices:

\[ \hat P_2=\exp n\theta_0 \begin{pmatrix} e^\theta & e^{-\theta}\\ e^{-\theta} & e^\theta \end{pmatrix} \times \begin{pmatrix} e^\theta & e^{-\theta}\\ e^{-\theta} & e^\theta \end{pmatrix} \times \cdots \times \begin{pmatrix} e^\theta & e^{-\theta}\\ e^{-\theta} & e^\theta \end{pmatrix} = \]

\[ =\exp n\theta_0\,(\hat e e^\theta+\hat b e^{-\theta})\times (\hat e e^\theta+\hat b e^{-\theta})\times \cdots \times (\hat e e^\theta+\hat b e^{-\theta}). \tag{1,25} \]

Using formula (1,21), we obtain for \(\hat P_2\) the expression

\[ \hat P_2=\exp n\theta_0 \prod_{k=1}^{k=n}(e^\theta+\hat B_k e^{-\theta}). \tag{1,26} \]

It is obvious that, by virtue of (1,22), the matrices \(\hat P_1\) and \(\hat P_2\) do not commute.

However, since the trace of the product of two matrices \(\hat P_1\) and \(\hat P_2\) does not depend on the order of the factors, when calculating the distribution function \(Z(T)=\operatorname{Sp}(\hat P_1\hat P_2)^m\), the order of the factors is immaterial.

Thus, for the distribution matrix we have obtained the expression

\[ \hat P=\exp(n\theta'_0+n\theta_0) \prod_{k=1}^{k=n}\exp(\theta'\hat A_k\hat A_{k+1}) \prod_{k=1}^{k=n}(e^\theta+\hat B_k e^{-\theta}). \tag{1,27} \]

Let us note that the presence in the distribution function of the factor of the form \(\exp(C/kT)\) is inessential, and it may be discarded.

Indeed, by virtue of \(E=kT^2\dfrac{d\ln Z}{dT}\), the presence of such a factor denotes the addition to the energy \(E\) of the system of an inessential additive constant \(C\). Therefore, in what follows, in the distribution matrix we everywhere omit the factor \(\exp(n\theta'_0+n\theta_0)\).

and write it in the form

\[ \hat P=\prod_{k=1}^{k=n}\exp\left(\theta' \hat A_k \hat A_{k+1}\right) \prod_{k=1}^{k=n}\left(e^\theta+\hat B_k e^{-\theta}\right). \tag{1,27'} \]

Let us carry out one more formal transformation of expression (1,26) for the matrix \(\hat P_2\).

Put

\[ e^\theta=\rho \operatorname{ch}\theta^*;\qquad e^{-\theta}=\rho \operatorname{sh}\theta^*, \tag{1,28} \]

whence we find:

\[ \rho=\sqrt{2\operatorname{sh}2\theta};\qquad \operatorname{th}\theta^*=e^{-2\theta};\qquad \theta^*=\frac12\ln\operatorname{cth}2\theta. \tag{1,29} \]

Since \(\hat B_k^2=1\), we have:

\[ e^\theta+\hat B_k e^{-\theta} =\rho\left(\operatorname{ch}\theta^*+\hat B_k\operatorname{sh}\theta^*\right) =\sqrt{2\operatorname{sh}2\theta}\,\exp\left(\theta^*\hat B_k\right) \tag{1,30} \]

and, consequently, finally

\[ \hat P_2=(2\operatorname{sh}2\theta)^{n/2} \prod_{k=1}^{k=n}\exp\left(\theta^*\hat B_k\right). \tag{1,31} \]

The final expression for the matrix \(\hat P=\hat P_1\hat P_2\) has the form:

\[ \hat P=\hat P_1\hat P_2 =(2\operatorname{sh}2\theta)^{n/2} \prod_{k=1}^{k=n}\exp\left(\theta'\hat A_k\hat A_{k+1}\right) \prod_{k=1}^{k=n}\exp\left(\theta^*\hat B_k\right). \tag{1,32} \]

In conclusion we derive several formulas connecting \(\theta\) and \(\theta^*\), which will be useful later.

From formulas (1,28) we find:

\[ \left. \begin{aligned} \frac12\left(\operatorname{cth}\theta^*-\operatorname{th}\theta^*\right) &=\frac{1}{\operatorname{sh}2\theta^*} =\operatorname{sh}2\theta,\\ \frac12\left(\operatorname{cth}\theta^*+\operatorname{th}\theta^*\right) &=\frac{\operatorname{ch}2\theta^*}{\operatorname{sh}2\theta^*} =\operatorname{ch}2\theta \end{aligned} \right\} \tag{1,33} \]

and obtain the relations important for what follows:

\[ \left. \begin{aligned} \operatorname{sh}2\theta^*\,\operatorname{sh}2\theta&=1,\\ \operatorname{ch}2\theta^*\,\operatorname{sh}2\theta&=\operatorname{ch}2\theta. \end{aligned} \right\} \tag{1,34} \]

When the temperature changes in the interval \(0<T<\infty\), the quantities \(\theta'\) and \(\theta^*\) vary monotonically, with
\(\theta'=\dfrac{I'_2-I'_1}{2kT}\) decreasing monotonically from \(\infty\) to \(0\), while the quantity
\(\theta^*=\dfrac12\ln\operatorname{cth}\dfrac{I_2-I_1}{kT}\) monotonically in-

increases from \(0\) to \(\infty\). There exists, therefore, a certain temperature \(T=T_0\) at which \(\theta^*=\theta'\). At this temperature, according to (1,34),

\[ \operatorname{sh} 2\theta'\,\operatorname{sh} 2\theta = 1 . \tag{1,35} \]

We shall see below that the temperature \(T_0\) is precisely the temperature at which a phase transition of the second kind occurs in the plane dipole-lattice model under consideration.

In what follows we shall distinguish between low temperatures \(T<T_0\) \((\theta'>\theta^*)\) and high temperatures \(T>T_0\) \((\theta^*>\theta')\).

We proceed to the determination of the eigenvalues of the matrix \(\hat P\) and to the calculation of the distribution function \(Z(T)\).

§ 2. CALCULATION OF THE DISTRIBUTION FUNCTION \(Z(T)\)

Here we shall carry out the calculation of the eigenvalues of the matrix

\[ \hat M= \prod_{k=1}^{k=n}\exp\left(\theta'\hat A_k\hat A_{k+1}\right) \prod_{k=1}^{k=n}\exp\left(\theta^*\hat B_k\right), \tag{2,1} \]

which differs from the distribution matrix \(\hat P\) by the numerical factor \((2\operatorname{sh}2\theta)^{n/2}\).

For the calculation it is convenient, instead of the system of \(2^n\)-row Pauli matrices \(\hat A_k,\ \hat B_k,\ \hat C_k\), satisfying the relations

\[ \begin{aligned} \hat A_k\hat B_k &= i\hat C_k,\\ \hat B_k\hat C_k &= i\hat A_k,\\ \hat C_k\hat A_k &= i\hat B_k, \end{aligned} \tag{2,2} \]

to introduce a system of \((2n+1)\) anticommuting \(2^n\)-row Dirac matrices by the formulas:

\[ \begin{aligned} \hat\Gamma_1 &= \hat C_1, & \hat\Gamma_2 &= \hat A_1,\\ \hat\Gamma_3 &= \hat C_2\hat B_1, & \hat\Gamma_4 &= \hat A_2\hat B_1,\\ \hat\Gamma_5 &= \hat C_3\hat B_1\hat B_2,& \hat\Gamma_6 &= \hat A_3\hat B_1\hat B_2,\\ \cdots\quad & & \cdots\quad & \\ \hat\Gamma_{2n-1} &= \hat C_n\hat B_1\hat B_2\cdots\hat B_{n-1}, & \hat\Gamma_{2n} &= \hat A_n\hat B_1\hat B_2\cdots\hat B_{n-1}, \end{aligned} \tag{2,3} \]

\[ \hat U=\hat B_1\hat B_2\cdots\hat B_n =\prod_{k=1}^{k=n}\left(i\hat\Gamma_{2k}\hat\Gamma_{2k-1}\right). \tag{2,3'} \]

It is easy to verify, using (2.2), that all the \((2n+1)\) matrices of the system (2.3) anticommute:

\[ \left. \begin{aligned} \hat{\Gamma}_{\alpha}\hat{\Gamma}_{\beta}+\hat{\Gamma}_{\beta}\hat{\Gamma}_{\alpha}&=2\delta_{\alpha\beta},\\ \hat{\Gamma}_{\alpha}\hat U+\hat U\hat{\Gamma}_{\alpha}&=0,\\ U^2&=1. \end{aligned} \qquad (\alpha,\ \beta=1,2,\ldots,2n). \right\} \tag{2.4} \]

Taking (2.2) into account, we compute:

\[ \left. \begin{aligned} \hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k}&=i\hat B_k,\\ \hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k+2}&=-\,i\hat A_k\hat A_{k+1}, \end{aligned} \qquad k=1,2,\ldots,(n-1) \right\} \]

\[ \hat{\Gamma}_{2n-1}\hat{\Gamma}_{2} =\hat C_n\hat B_1\hat B_2\ldots\hat B_{n-1}\hat A_1 =-\,i\hat A_n\hat B_n\hat B_1\ldots\hat B_{n-1}\hat A_1 =i\hat A_n\hat A_1\hat U \tag{2.5} \]

and, consequently:

\[ \left. \begin{aligned} \hat B_k&=-\,i\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k},\\ \hat A_k\hat A_{k+1}&=i\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k+2},\\ \hat A_n\hat A_1&=-\,i\hat{\Gamma}_{2n-1}\hat{\Gamma}_{2}\hat U, \end{aligned} \qquad k=1,2,\ldots,(n-1), \right\} \tag{2.5'} \]

Substituting (2.5′) into (2.1), we obtain for the matrix \(\hat M\) the expression

\[ \hat M(\hat U)= \exp\!\left(-i\theta\,\hat{\Gamma}_{2n-1}\hat{\Gamma}_{2}\hat U\right) \prod_{k=1}^{k=n-1} \exp\!\left(i\theta\,\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k+2}\right) \times \]

\[ \times \prod_{k=1}^{k=n} \exp\!\left(-i\theta^{*}\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k}\right). \tag{2.6} \]

Let us introduce for consideration the additional matrix

\[ \hat M(-\hat U)= \exp\!\left(i\theta\,\hat{\Gamma}_{2n-1}\hat{\Gamma}_{2}\hat U\right) \prod_{k=1}^{k=n-1} \exp\!\left(i\theta\,\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k+2}\right) \times \]

\[ \times \prod_{k=1}^{k=n} \exp\!\left(-i\theta^{*}\hat{\Gamma}_{2k-1}\hat{\Gamma}_{2k}\right). \tag{2.6'} \]

We note that the matrix \(\hat U\) commutes with both matrices \(\hat M(\hat U)\) and \(\hat M(-\hat U)\).

Therefore all eigenvalues \(\eta(\nu)\) and eigenvectors of the matrix \(\hat M(\hat U)\) split into two classes:

1) positive, for which \(\hat U=+1\):

\[ \hat M(1)\psi^{+}(\nu)=\eta^{+}(\nu)\psi^{+}(\nu), \tag{2,7} \]

2) negative, for which \(\hat U=-1\):

\[ \hat M(-1)\psi^{-}(\nu)=\eta^{-}(\nu)\psi^{-}(\nu). \tag{2,7'} \]

According to \((2,7')\), the negative eigenvalues of the matrix \(\hat M(\hat U)\) are the positive eigenvalues of the matrix \(\hat M(-\hat U)\). Thus, we obtain all \(2^n\) eigenvalues of the matrix \(\hat M\equiv \hat M(\hat U)\), if we find all \(2^{n-1}\) positive eigenvalues of the matrix:

\[ \hat M^{+}\equiv \hat M(1)= \exp(-i\theta^{*}\hat\Gamma_{2n-1}\hat\Gamma_{2}) \prod_{k=1}^{k=n-1} \exp(i\theta\hat\Gamma_{2k-1}\hat\Gamma_{2k+2}) \times \]

\[ \times \prod_{k=1}^{k=n} \exp(-i\theta^{*}\hat\Gamma_{2k-1}\hat\Gamma_{2k}). \tag{2,8} \]

and all \(2^{n-1}\) positive eigenvalues of the matrix

\[ \hat M^{-}\equiv \hat M(-1)= \exp(i\theta'\hat\Gamma_{2n-1}\hat\Gamma_{2}) \prod_{k=1}^{k=n-1} \exp(i\theta\hat\Gamma_{2k-1}\hat\Gamma_{2k+2}) \times \]

\[ \times \prod_{k=1}^{k=n} \exp(-i\theta^{*}\hat\Gamma_{2k-1}\hat\Gamma_{2k}). \tag{2,8'} \]

For the computation we shall use the following method.

The Dirac matrices \(\hat\Gamma_\alpha\), satisfying the commutation conditions \((2,4)\), may be subjected to an arbitrary complex orthogonal transformation:

\[ \hat\Gamma'_\alpha=\sum_{\beta} L_{\alpha\beta}\hat\Gamma_\beta, \]

where \(L_{\alpha\beta}\) is an arbitrary orthogonal matrix of order \(2n\), moreover

new \(\hat{\Gamma}_\alpha\) satisfy the same permutation conditions. Indeed, we have:

\[ \hat{\Gamma}'_\alpha\hat{\Gamma}'_\beta+\hat{\Gamma}'_\beta\hat{\Gamma}'_\alpha = \sum_{\sigma,\tau} \{(L_{\alpha\sigma}\hat{\Gamma}_\sigma)(L_{\beta\tau}\hat{\Gamma}_\tau) +(L_{\beta\tau}\hat{\Gamma}_\tau)(L_{\alpha\sigma}\hat{\Gamma}_\sigma)\} = \]

\[ = \sum_{\sigma,\tau} L_{\alpha\sigma}L_{\beta\tau} (\hat{\Gamma}_\sigma\hat{\Gamma}_\tau+\hat{\Gamma}_\tau\hat{\Gamma}_\sigma) = 2\sum_{\sigma,\tau}\delta_{\sigma\tau}L_{\alpha\sigma}L_{\beta\tau} = 2\sum_\sigma L_{\alpha\sigma}L_{\beta\sigma}. \]

But by the orthogonality condition

\[ \sum_\sigma L_{\alpha\sigma}L_{\beta\sigma}=\delta_{\alpha\beta}, \]

and therefore:

\[ \hat{\Gamma}'_\alpha\hat{\Gamma}'_\beta+\hat{\Gamma}'_\beta\hat{\Gamma}'_\alpha = 2\delta_{\alpha\beta}, \]

i.e. the assertion is proved.

On the other hand, let us show that if the Dirac matrices \(\hat{\Gamma}_\alpha\) are subjected to a similarity transformation by means of the matrices \(\hat{M}^{+}\) or \(\hat{M}^{-}\), then the new matrices \(\hat{\Gamma}^{+}_\alpha\) and \(\hat{\Gamma}^{-}_\alpha\):

\[ \hat{\Gamma}^{+}_\alpha = \hat{M}^{+}\hat{\Gamma}_\alpha(\hat{M}^{+})^{-1} \quad\text{and}\quad \hat{\Gamma}^{-}_\alpha = \hat{M}^{-}\hat{\Gamma}_\alpha(\hat{M}^{-})^{-1} \tag{2,9} \]

are related to the old matrices \(\hat{\Gamma}_\alpha\) by orthogonal transformations:

\[ \hat{\Gamma}^{+}_\alpha=\sum_\beta L^{+}_{\alpha\beta}\hat{\Gamma}_\beta \quad\text{and}\quad \hat{\Gamma}^{-}_\alpha=\sum_\beta L^{-}_{\alpha\beta}\hat{\Gamma}_\beta . \tag{2,9'} \]

For this it is enough to note that the matrices \(\hat{M}^{+}\) and \(\hat{M}^{-}\) are products of factors of the form

\[ \exp(\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta) = \cos\lambda+\hat{\Gamma}_\alpha\hat{\Gamma}_\beta\sin\lambda, \]

where \(\lambda\) is an arbitrary complex parameter, and to verify the validity of the following formulae:

\[ \left. \begin{aligned} \exp(\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta)\, \hat{\Gamma}_\alpha\, \exp(-\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta) &= \hat{\Gamma}_\alpha\cos 2\lambda-\hat{\Gamma}_\beta\sin 2\lambda, \\ \exp(\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta)\, \hat{\Gamma}_\beta\, \exp(-\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta) &= \hat{\Gamma}_\alpha\sin 2\lambda+\hat{\Gamma}_\beta\cos 2\lambda, \\ \exp(\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta)\, \hat{\Gamma}_\gamma\, \exp(-\lambda\hat{\Gamma}_\alpha\hat{\Gamma}_\beta) &= \hat{\Gamma}_\gamma \quad(\gamma\ne \alpha,\beta). \end{aligned} \right\} \tag{2,10} \]

We arrive, therefore, at the basic formulas:

\[ \hat{\Gamma}_{\alpha}^{+} = \hat{M}^{+}\Gamma_{\alpha}(\hat{M}^{+})^{-1} = \sum_{(\beta)} L_{\alpha\beta}^{+}\Gamma_{\beta}, \tag{2,11} \]

\[ \hat{\Gamma}_{\alpha}^{-} = \hat{M}^{-}\Gamma_{\alpha}(\hat{M}^{-})^{-1} = \sum_{\beta} L_{\alpha\beta}^{-}\Gamma_{\beta}, \tag{2,11'} \]

which underlie the method, set forth below, for computing the eigenvalues of the matrices \(\hat{M}^{+}\) and \(\hat{M}^{-}\).

Let us note that the matrices \(\hat{M}^{+}\) and \(\hat{M}^{-}\) are special cases of the more general \(2^{n}\)-row matrix of the form

\[ \hat{S} = \prod_{\alpha,\beta} \exp\left(\Lambda_{\alpha\beta}\hat{\Gamma}_{\alpha}\hat{\Gamma}_{\beta}\right), \tag{2,12} \]

where \(\Lambda_{\alpha\beta}=-\Lambda_{\beta\alpha}\) are arbitrary numbers, in general complex. We shall show how to compute the eigenvalues of such a matrix \(\hat{S}\), starting from the general formula

\[ \hat{\Gamma}_{\alpha}^{\prime} = \hat{S}\hat{\Gamma}_{\alpha}\hat{S}^{-1} = \sum_{\beta} L_{\alpha\beta}\hat{\Gamma}_{\beta}. \tag{2,13} \]

If one introduces the \(2^{n}\)-dimensional matrix vector \(\hat{\Gamma}\) with components \(\hat{\Gamma}_{\alpha}\), then formula (2,12) can be written in condensed form as

\[ \hat{\Gamma}^{\prime} = \hat{S}\hat{\Gamma}\hat{S}^{-1} = \hat{L}\hat{\Gamma}. \tag{2,12'} \]

The simple geometric meaning of expression (2,12′) is that to one and the same rotation of a \(2^{n}\)-dimensional space \((\hat{\Gamma}_{\alpha}\to \hat{\Gamma}_{\alpha}^{\prime})\) there corresponds, on the one hand, a \(2n\)-row orthogonal matrix \(\hat{L}\) (by the formula \(\hat{\Gamma}^{\prime}=\hat{L}\hat{\Gamma}\)), and, on the other hand, a \(2^{n}\)-row matrix \(\hat{S}\) (by the formula \(\hat{\Gamma}^{\prime}=\hat{S}\hat{\Gamma}\hat{S}^{-1}\)). Since to the product of two matrices \(\hat{L}_{1}\) and \(\hat{L}_{2}\) there corresponds, by the formula

\[ \hat{\Gamma}^{\prime} = \hat{S}_{2}\hat{S}_{1}\hat{\Gamma}\hat{S}_{1}^{-1}\hat{S}_{2}^{-1} = (\hat{S}_{2}\hat{S}_{1})\hat{\Gamma}(\hat{S}_{2}\hat{S}_{1})^{-1} = \hat{L}_{2}\hat{L}_{1}\hat{\Gamma}, \tag{2,13'} \]

the product of the corresponding matrices \(\hat{S}_{1}\) and \(\hat{S}_{2}\), we are dealing with a \(2^{n}\)-row representation of the rotation group in \(2n\)-dimensional space. The study of this representation constitutes the content of the theory of spinors in multidimensional spaces.

In connection with the theory of spinors we shall note only the following. In Dirac’s theory the wave function is four-component

complex quantity \(\psi\)—the so-called spinor. Upon passing to a new reference system

\[ x'_1,\ x'_2,\ x'_3,\ x'_4=ict', \]

effected by the complex orthogonal Lorentz matrix

\[ x'_i=\sum_{(k)} L_{ik}x_k \qquad (i,k=1,2,3,4), \tag{2,14} \]

the four components of the spinor \(\Psi_\sigma\) are transformed according to the formulas:

\[ \psi'_\sigma=\sum_{(\tau)} S_{\sigma\tau}\psi_\tau \qquad (\sigma,\tau=1,2,3,4), \tag{2,14'} \]

where

\[ \hat S\,\hat\gamma_k\,\hat S^{-1}=\sum_{(i)} L_{ki}\hat\gamma_i, \tag{2,15} \]

where \(\hat\gamma_k\) are the four Dirac matrices. Here we have a special case of formula (2,12) for \(n=2\). In this special case both matrices, the \(2^n\)-row matrix \(\hat S\) and the \(2n\)-row matrix \(\hat L\), turn out to be four-row matrices. For more details see, for example, \({}^{6}\).

However, for our purpose it proves superfluous to use the ready-made theory of spinors in multidimensional spaces, as Kaufman \({}^{3}\) did in her work. We shall obtain everything needed more simply, using only the elementary methods of matrix algebra.

Suppose we have succeeded in finding, in the \(2n\)-dimensional complex space, a system of coordinates in which the complex orthogonal matrix \(\hat L\) is represented as a direct sum of \(n\) two-row matrices. In the Appendix it is shown that this problem is equivalent to the problem of determining the complex eigenvalues of the matrix \(\hat L\).

Denoting by \(\Gamma^*_\alpha\) the components of the matrix vector \(\Gamma\) in such a coordinate system, and by \(\exp(\pm 2\lambda_k)\) the eigenvalues of the matrix \(\hat L\), we have:

\[ \hat L^*= \begin{pmatrix} \cos 2i\lambda_1, & \sin 2i\lambda_1\\ -\sin 2i\lambda_1, & \cos 2i\lambda_1 \end{pmatrix} + \begin{pmatrix} \cos 2i\lambda_2, & \sin 2i\lambda_2\\ -\sin 2i\lambda_2, & \cos 2i\lambda_2 \end{pmatrix} +\cdots \]

\[ \cdots+ \begin{pmatrix} \cos 2i\lambda_n, & \sin 2i\lambda_n\\ -\sin 2i\lambda_n, & \cos 2i\lambda_n \end{pmatrix}. \tag{2,16} \]

In such a coordinate system the matrix \(\hat S\) will, according to the formula

(2,13) is represented as a product of \(n\) commuting matrices

\[ \hat S^*=\hat S(\lambda_1)\hat S(\lambda_2)\ldots \hat S(\lambda_n), \tag{2,17} \]

where, according to formulas (2,10),

\[ \hat S(\lambda_k)=\exp(-i\lambda_k\hat\Gamma^*_{2k-1}\hat\Gamma^*_{2k}). \tag{2,18} \]

In the special coordinate system under consideration we therefore have:

\[ \hat S^*=\prod_{k=1}^{k=n}\exp(-i\lambda_k\Gamma^*_{2k-1}\Gamma^*_{2k}). \tag{2,17'} \]

Taking into account that the eigenvalues of all the commuting matrices
\(i\hat\Gamma^*_{2k-1}\hat\Gamma^*_{2k}\) will be \(\pm1\), we can at once write the eigenvalues of the matrix \(\hat S\) in the form

\[ (\nu|\hat S|\nu)=\exp(\pm\lambda_1\pm\lambda_2\pm\ldots\pm\lambda_n). \tag{2,19} \]

We shall obtain all \(2^n\) eigenvalues of the matrix by choosing all possible \(2^n\) combinations of signs in (2,19).

Thus, we have arrived at the following simple result. In order to find the eigenvalues of the \(2^n\)-row matrix \(\hat S\), satisfying conditions (2,12), it is sufficient to find \(2n\) eigenvalues of the \(2n\)-row matrix \(\hat L\).

We now pass to the particular problem of computing the eigenvalues of the matrices \(M^+\) and \(M^-\), which is equivalent, as we have shown, to computing the eigenvalues of the corresponding matrices \(\hat L^+\) and \(\hat L^-\).

Instead of the matrices \(\hat L^+\) and \(\hat L^-\) themselves, it is more convenient for us to deal with the corresponding orthogonal transformations of the \(2n\)-dimensional complex space

\[ R_{2n}=\{x_1,y_1,x_2,y_2,\ldots,x_n,y_n\}. \]

Taking into account definitions (2,8) and (2,8′) and formulas (2,10), we can write the orthogonal transformations of interest to us, of the spaces \(R_{2n}\to R'_{2n}\) and \(R_{2n}\to R''_{2n}\), in the form

\[ \left. \begin{aligned} x'_k&=x_k\cos(-2i\theta^*)-y_k\sin(-2i\theta^*),\\ y'_k&=x_k\sin(-2i\theta^*)+y_k\cos(-2i\theta^*),\\ x''_k&=x'_k\cos(2i\theta')-y'_{k+1}\sin(2i\theta'),\\ y''_{k+1}&=x'_k\sin(2i\theta')+y'_{k+1}\cos(2i\theta'). \end{aligned} \right\} \tag{2,20} \]

where, according to (2.8) and (2.8′), we must set:

\[ \begin{array}{r} \text{In the case of the matrix } L^{+}: \quad y_{n+1}=-y_1,\\[3pt] \text{“ \hspace{1.5em} ” \hspace{1.5em} ” } L^{-}: \quad y_{n+1}=+y_1. \end{array} \tag{2.21} \]

We shall compute the eigenvalues of the matrices \(L^{+}\) and \(L^{-}\) in two successive stages. First we represent these matrices in the form of direct sums of two-row unimodular matrices, and then reduce each of the two-row matrices to diagonal form.

In order to satisfy the “boundary conditions” (2.21), we perform the transformation:

\[ x_k=X(\beta)\exp(ik\beta);\qquad y_k=Y(\beta)\exp(ik\beta). \tag{2.22} \]

Substituting (2.22) into (2.21), we obtain for \(\beta\) the following sequence of \(2n\) discrete values:

\[ \begin{array}{ll} \text{in the case of the matrix } L^{+}:& \displaystyle \beta_{2k-1}=\frac{\pi}{n}(2k-1);\\[8pt] \text{in the case of the matrix } L^{-}:& \displaystyle \beta_{2k}=\frac{\pi}{n}\,2k. \end{array} \tag{2.23} \]

Substituting (2.22) into (2.20) and denoting \(X\!\left(\dfrac{\pi\alpha}{n}\right)=X_{\alpha}\), \(Y\!\left(\dfrac{\pi\alpha}{n}\right)=Y_{\alpha}\), we find:

\[ \begin{aligned} X'_{\alpha}&=X_{\alpha}\cos 2i\theta^{*}+Y_{\alpha}\sin 2i\theta^{*},\\ Y'_{\alpha}&=-X_{\alpha}\sin 2i\theta^{*}+Y_{\alpha}\cos 2i\theta^{*}, \end{aligned} \qquad \begin{aligned} X''_{\alpha}&=X'_{\alpha}\cos 2i\theta' -Y'_{\alpha}\exp\!\left(\frac{i\pi\alpha}{n}\right)\sin 2i\theta',\\ Y''_{\alpha}&=X'_{\alpha}\exp\!\left(-\frac{i\pi\alpha}{n}\right)\sin 2i\theta' +Y'_{\alpha}\cos 2i\theta'. \end{aligned} \tag{2.24} \]

The transformation taking \(\{X_{\alpha},\,Y_{\alpha}\}\to\{X''_{\alpha},\,Y''_{\alpha}\}\) is written in the form

\[ \begin{aligned} X''_{\alpha} &=X_{\alpha}\left\{\cos 2i\theta'\cos 2i\theta^{*} +\exp\!\left(\frac{i\pi\alpha}{n}\right)\sin 2i\theta'\sin 2i\theta^{*}\right\}+\\ &\quad +Y_{\alpha}\left\{\cos 2i\theta'\sin 2i\theta^{*} -\exp\!\left(\frac{i\pi\alpha}{n}\right)\sin 2i\theta'\cos 2i\theta^{*}\right\},\\[6pt] Y''_{\alpha} &=X_{\alpha}\left\{-\cos 2i\theta'\sin 2i\theta^{*} +\exp\!\left(-\frac{i\pi\alpha}{n}\right)\sin 2i\theta'\cos 2i\theta^{*}\right\}+\\ &\quad +Y_{\alpha}\left\{\cos 2i\theta'\cos 2i\theta^{*} +\exp\!\left(-\frac{i\pi\alpha}{n}\right)\sin 2i\theta'\sin 2i\theta^{*}\right\}. \end{aligned} \tag{2.25} \]

Denoting the unimodular matrix of this transformation by

\[ \begin{pmatrix} A_\alpha & B_\alpha\\ C_\alpha & D_\alpha \end{pmatrix}, \tag{2,25'} \]

we can write \(L^+\) and \(L^-\) in the form of a direct sum of two-row unimodular matrices; moreover, according to (2,23),

\[ \left. \begin{aligned} L^+&= \begin{pmatrix} A_1 & B_1\\ C_1 & D_1 \end{pmatrix} + \begin{pmatrix} A_3 & B_3\\ C_3 & D_3 \end{pmatrix} +\ldots+ \begin{pmatrix} A_{2n-1} & B_{2n-1}\\ C_{2n-1} & D_{2n-1} \end{pmatrix},\\ L^-&= \begin{pmatrix} A_2 & B_2\\ C_2 & D_2 \end{pmatrix} + \begin{pmatrix} A_4 & B_4\\ C_4 & D_4 \end{pmatrix} +\ldots+ \begin{pmatrix} A_{2n} & B_{2n}\\ C_{2n} & D_{2n} \end{pmatrix}. \end{aligned} \right\} \tag{2,26} \]

The first stage of our calculations is finished. It remains now to reduce each of the matrix summands to diagonal form.

The characteristic equation

\[ \xi^2-(A_\alpha+D_\alpha)\xi+1= \]

\[ =\xi^2-2\left\{\operatorname{ch}2\theta'\operatorname{ch}2\theta^* -\cos\frac{\pi z}{n}\operatorname{sh}2\theta'\operatorname{sh}2\theta^*\right\}+1=0 \tag{2,27} \]

has real roots, which we shall denote by \(e^{2\gamma_\alpha}\) and \(e^{-2\gamma_\alpha}\), where

\[ \operatorname{ch}2\gamma_\alpha = \operatorname{ch}2\theta'\operatorname{ch}2\theta^* -\cos\frac{\pi}{n}\alpha\,\operatorname{sh}2\theta'\operatorname{sh}2\theta^* = \cos 2i\gamma_\alpha . \]

Knowing now the eigenvalues of the matrices \(\hat L^+\) and \(\hat L^-\), we can rewrite them in such a coordinate system \(x_k^*\) and \(y_k^*\) in which they will be represented as direct sums:

\[ \left. \begin{aligned} L^+&= \begin{pmatrix} \cos 2i\gamma_1 & \sin 2i\gamma_1\\ -\sin 2i\gamma_1 & \cos 2i\gamma_1 \end{pmatrix} + \begin{pmatrix} \cos 2i\gamma_3 & \sin 2i\gamma_3\\ -\sin 2i\gamma_3 & \cos 2i\gamma_3 \end{pmatrix} +\ldots\\ &\quad\ldots+ \begin{pmatrix} \cos 2i\gamma_{2n-1} & \sin 2i\gamma_{2n-1}\\ -\sin 2i\gamma_{2n-1} & \cos 2i\gamma_{2n-1} \end{pmatrix},\\ L^-&= \begin{pmatrix} \cos 2i\gamma_2 & \sin 2i\gamma_2\\ -\sin 2i\gamma_2 & \cos 2i\gamma_2 \end{pmatrix} +\ldots+ \begin{pmatrix} \cos 2i\gamma_{2n} & \sin 2i\gamma_{2n}\\ -\sin 2i\gamma_{2n} & \cos 2i\gamma_{2n} \end{pmatrix}. \end{aligned} \right\} \tag{2,26'} \]

In this coordinate system the matrices \(M^+\) and \(M^-\) are represented

in the form of a product of commuting matrices

\[ \begin{aligned} M^{*+} &= \prod_{k=1}^{k=n} \exp\left(-i\gamma_{2k-1}\Gamma^{*}_{2k-1}\Gamma^{*}_{2k}\right),\\ M^{*-} &= \prod_{k=1}^{k=n} \exp\left(-i\gamma_{2k}\Gamma^{*}_{2k-1}\Gamma^{*}_{2k}\right), \end{aligned} \tag{2,28} \]

and their eigenvalues by

\[ \begin{aligned} \eta^{+}(\nu) &= \exp\left(\pm\gamma_1 \pm\gamma_3 \pm \ldots \pm\gamma_{2n-1}\right),\\ \eta^{-}(\nu) &= \exp\left(\pm\gamma_2 \pm\gamma_4 \pm \ldots \pm\gamma_{2n}\right). \end{aligned} \tag{2,29} \]

It remains for us now to select the positive eigenvalues of the matrices \(\hat M^{+}\) and \(\hat M^{-}\). For this, we note that the operator \(\hat U^{*}\), defined by formula \((2,3')\),

\[ \hat U^{*}=\prod_{k=1}^{k=n}\left(i\hat\Gamma^{*}_{2k}\hat\Gamma^{*}_{2k-1}\right), \tag{2,30} \]

is a product of \(n\) commuting matrices, each of which has eigenvalues \(\pm 1\).

Taking into account (2,28) and (2,29), we see that \(\hat U^{*}\) will have the value \(+1\) if the choice of signs in (2,29) is restricted by the requirement that the number of minus signs be even. Imposing this requirement, we select \(2^{n-1}\) positive eigenvalues of the matrix \(\hat M^{+}\) and \(2^{n-1}\) positive eigenvalues of the matrix \(\hat M^{-}\), i.e. we obtain all \(2^n\) eigenvalues of the matrix \(\hat M\), which differs from the distribution matrix \(\hat P\) only by the factor \((2\operatorname{sh}2\theta)^{n/2}\).

Knowing the eigenvalues of the matrix \(\hat P\), we can write the eigenvalues of the matrix \(\hat P^m\) in the form

\[ \begin{aligned} \{E^{+}(\nu)\}^{m} &= (2\operatorname{sh}2\theta)^{\frac{nm}{2}} \exp m\left(\pm\gamma_1 \pm\gamma_3 \pm \ldots \pm\gamma_{2n-1}\right),\\ \{E^{-}(\nu)\}^{m} &= (2\operatorname{sh}2\theta)^{\frac{nm}{2}} \exp m\left(\pm\gamma_2 \pm\gamma_4 \pm \ldots \pm\gamma_{2n}\right). \end{aligned} \tag{2,31} \]

and compute the partition function by formula (1.10)

\[ Z(T)=(2\operatorname{sh}2\theta)^{\frac{mn}{2}} \left\{ \sum_{(v)} \exp m(\pm\gamma_1\pm\ldots\pm\gamma_{2n-1})+ \right. \]

\[ \left. +\sum_{(v)} \exp m(\pm\gamma_2\pm\ldots\pm\gamma_{2n}) \right\}, \tag{2.32} \]

where the summation is over all allowed sets of signs. It is easy to verify that, as a result of the calculation, we obtain:

\[ Z(T)=\frac{1}{2}(2\operatorname{sh}2\theta)^{\frac{mn}{2}} \left\{ \prod_{k=1}^{k=n}(2\operatorname{ch}m\gamma_{2k-1})+ \prod_{k=1}^{k=n}(2\operatorname{sh}m\gamma_{2k-1})+ \right. \]

\[ \left. +\prod_{k=1}^{k=n}(2\operatorname{ch}m\gamma_{2k})+ \prod_{k=1}^{k=n}(2\operatorname{sh}m\gamma_{2k}) \right\}. \tag{2.32'} \]

The partition function \(Z(T)\) is an analytic function of the argument \(T\) throughout the entire range of temperature variation, and the point \(\theta'=\theta^*\) is an ordinary point for it. We arrive at the important conclusion that a lattice consisting of a finite number of dipoles does not exhibit a second-order phase transition.

Let us now consider the case of a lattice infinite in only one direction, i.e. let \(m\to\infty\), keeping \(n\) finite. For \(m\to\infty\), from formula (1.10) we have

\[ Z(T)=E_1^m\left\{1+\left(\frac{E_2}{E_1}\right)^m+\ldots+\frac{E(2n)^m}{E_1}\right\}\to E_1^m, \tag{2.33} \]

where \(E_1\) denotes the maximal eigenvalue of the matrix \(\hat P\). Introduce the “partition function \(\Lambda(T)\) per one column” as the limit:

\[ \Lambda(T)=\lim_{m\to\infty}\sqrt[m]{Z(T)}=E_{\max} \tag{2.34} \]

and, using expressions (2.31), we find:

\[ \Lambda(T)=(2\operatorname{sh}2\theta)^{\frac{n}{2}}\exp(\gamma_1+\gamma_3+\ldots+\gamma_{2n-1}). \tag{2.34'} \]

The partition function \(\Lambda(T)\) is also an analytic function of the argument \(T\) throughout the entire range of temperature variation. We see that an infinite lattice consisting of a finite number of rows likewise does not exhibit a second-order phase transition. We shall see in § 4 that a second-order phase transition appears only in the case of a lattice infinitely extended in both directions.

§ 3. Spectrum of the distribution matrix \(\hat P\)

To understand the difference in the physical properties of the model of a plane dipole lattice at low and high temperatures, it is useful to form an idea of the character of the spectrum of the matrix \(\hat P\) and of the dependence of its terms on temperature. Here we shall be interested in the case of large values of \(n\), in the limit asymptotically tending to infinity.

We have obtained expressions for the terms of the matrix \(\hat P\):

positive:
\[ E^{+}(\nu)=(2\operatorname{sh}2\theta)^{n/2} \exp\{\pm\gamma_1\pm\gamma_3\pm\cdots\pm\gamma_{2n-1}\}, \tag{3,1} \]

negative:
\[ E^{-}(\nu)=(2\operatorname{sh}2\theta)^{n/2} \exp\{\pm\gamma_0\pm\gamma_2\pm\cdots\pm\gamma_{2n-2}\}, \tag{3,1'} \]

where
\[ \gamma_\alpha=\gamma_{2n-\alpha}, \tag{3,2} \]
\[ \operatorname{ch}2\gamma_\alpha = \operatorname{ch}2\theta'\operatorname{ch}2\theta^* - \cos\frac{\pi\alpha}{n}\operatorname{sh}2\theta'\operatorname{sh}2\theta^*; \tag{3,3} \]
moreover, the choice of signs is restricted by the requirement that in both expressions the number of minus signs be even.

According to formulas (3,2) and (3,3), all quantities
\[ \gamma_0<\gamma_1<\cdots<\gamma_n \]
lie in the range
\[ \gamma_0=\theta'-\theta^*<\gamma_\alpha<\gamma_n=\theta'+\theta^*, \tag{3,4} \]
and all of them, with the exception of \(\gamma_0=\theta'-\theta^*\), retain their sign throughout the whole range of temperature variation. Only \(\gamma_0\) changes sign when \(\theta'=\theta^*\), becoming negative at high temperatures. The dependence of the quantities \(\gamma_\alpha\) on temperature for the case of an isotropic lattice \(\theta'=\theta\) is shown in Fig. 1.

The difference in the sign of \(\gamma_0\) determines, as we shall see, the difference in the character of the spectrum at low and high temperatures. Let us consider the two ranges separately.

Low temperatures \(T<T_0;\ (\theta^*<\theta')\).

Taking into account that all \(\gamma_\alpha\) are positive and that, for
\[ n\to\infty \]
\[ |\gamma_\alpha-\gamma_{\alpha-1}|\to O\left(\frac{1}{n}\right), \tag{3,4'} \]
we conclude that the spectrum of the matrix \(\hat P\) consists of an alternating sequence of positive and negative terms forming narrow doublets, the distance between the components of whi-

which asymptotically, as \(n \to \infty\), tend to zero. We speak of a twofold quasi-degeneracy of each term.

For the maximal doublet we have, choosing in (3,1) all signs positive:

\[ \begin{aligned} E_1^+&=E_{\max}^{+}=(2\operatorname{sh}2\theta)^{n/2} \exp\{\gamma_1+\gamma_3+\ldots+\gamma_{2k-1}\},\\ E_1^-&=E_{\max}^{-}=(2\operatorname{sh}2\theta)^{n/2} \exp\{\gamma_0+\gamma_2+\ldots+\gamma_{2n-2}\}. \end{aligned} \tag{3,5} \]

For the next, second doublet we obtain, observing the rule for the choice of signs:

\[ \begin{aligned} E_2^+&=(2\operatorname{sh}2\theta)^{n/2} \exp\{-\gamma_1+\gamma_3+\ldots-\gamma_{2n-1}\},\\ E_2^-&=(2\operatorname{sh}2\theta)^{n/2} \exp\{\gamma_0-\gamma_2+\ldots-\gamma_{2n-2}\}. \end{aligned} \tag{3,6} \]

Taking into account that \(\gamma_1=\gamma_{2n-1}\), \(\gamma_2=\gamma_{2n-2}\), we conclude,

Fig. 1.

that the relative distance between the first two doublets tends, as \(n\to\infty\), to a finite limit:

\[ (E_1^+ - E_2^+)/E_{\max}^{+} \to 1-\exp(-4\gamma_0)=1-\exp 4(\theta^*-\theta'). \tag{3,7} \]

Applying the rule for the selection of signs, we note that to each term \(E_2^+\) and \(E_2^-\) there adjoins a sequence of terms \(E_k^+\) and \(E_k^-\), whose logarithms will be:

\[ \ln E_k^+ = \ln E_{\max}^{+}-4\gamma_{2k-1} \quad \left(k=2,3,\ldots,\left[\frac n2\right]\right), \]

\[ \ln E_k^- = \ln E_{\max}^{-}-4\gamma_{2k}. \tag{3,8} \]

Relative distances between them are given by the formulas:

\[ \begin{aligned} \left(E_k^+ - E_{k-1}^+\right)/E_{\max} &= \exp(-4\gamma_{2k-3})-\exp(-4\gamma_{2k-1}),\\ \left(E_k^- - E_{k-1}^-\right)/E_{\max} &= \exp(-4\gamma_{2k-2})-\exp(-4\gamma_{2k}) \end{aligned} \tag{3,9} \]

and as \(n \to \infty\) tend to zero. We conclude that adjoining \(E_2^+\) and \(E_2^-\)

Fig. 2.

there is a sequence of closely spaced terms, and that the spectrum becomes quasi-continuous. The character of the spectrum is shown in Fig. 2,a.

High temperatures: \(T > T_0;\;(\theta' < \theta^*)\).

Taking into account that at \(\theta'=\theta^*\) the quantity \(\gamma_0=\gamma_{2n}\) changes its sign and becomes negative, we have, as \(n \to \infty\):

\[ \begin{aligned} \gamma_\alpha &\to -\gamma_1 \quad (\alpha=2,3,\ldots,2n),\\ |\gamma_\alpha-\gamma_{\alpha-1}| &\to 0\left(\frac{1}{n}\right). \end{aligned} \tag{3,10} \]

For the maximal terms we now have:

\[ \begin{aligned} E_1^+ = E_{\max}^+ &= (2\,\operatorname{sh} 2\theta)^{n/2} \exp\{\gamma_1+\gamma_3+\cdots+\gamma_{2n-1}\},\\ E_1^- = E_{\max}^- &= (2\,\operatorname{sh} 2\theta)^{n/2} \exp\{-|\gamma_0|+\gamma_2+\cdots+\gamma_{2n-2}\}. \end{aligned} \tag{3,11} \]

We see that the twofold quasi-degeneracy of the maximal term is lifted, and the relative distance between its components as \(n \to \infty\) tends to the finite limit

\[ \begin{aligned} \left(E_{\max}^+ - E_{\max}^-\right)/E_{\max}^+ &= 1-\exp(-2|\gamma_0|)\\ &= 1-\exp 2(\theta^*-\theta'). \end{aligned} \tag{3,12} \]

Applying the sign-selection rule, we note that the sequence of nearby terms \(E_k^{-}\) adjoins the term \(E_{\max}^{-}\), whose logarithms will be

\[ \ln E_k^{-}=\ln E_{\max}^{-}+2|\gamma_0|-2\gamma_{2k}, \tag{3.13} \]

the relative distances between these terms being given by the formulas:

\[ \left(E_k^{-}-E_{k-1}^{-}\right)/E_{\max}^{-} =\exp(-2\gamma_{2k})-\exp(-2\gamma_{2k-2}) \tag{3.14} \]

and, as \(n\to\infty\), tend to zero. We note that the spectrum becomes quasi-continuous and acquires the character shown in Fig. 2, б.

Finally, at the transition point \(\theta'=\theta^*\), as formulas (3.7) and (3.12) show, the whole spectrum becomes quasi-continuous (Fig. 2, в).

§ 4. LIMITING CASE OF AN INFINITE PLANE LATTICE

Let us perform the limiting transition to the case of an infinite lattice, making the number of columns \(m\) and the number of rows \(n\) tend to infinity. Introduce the “distribution function per dipole” \(\lambda(T)\) by the formula

\[ \lambda(T)=\lim_{\substack{m\to\infty\\ n\to\infty}}\sqrt[mn]{Z(T)}. \tag{4.1} \]

The free energy \(f\), internal energy \(\varepsilon\), and heat capacity \(c\), calculated per dipole, are given by the expressions:

\[ f=-kT\ln\lambda;\qquad \varepsilon=kT^2\frac{d\ln\lambda(T)}{dT};\qquad c=kT\frac{d^2(T\ln\lambda)}{dT^2}. \tag{4.2} \]

Taking formulas (2.33) and (2.34) into account, we obtain:

\[ \left. \begin{aligned} \ln\lambda(T) &=\frac12\ln(2\,\operatorname{sh}2\theta) +\lim_{n\to\infty}(\gamma_1+\gamma_3+\cdots+\gamma_{2n-1})/n \\ &=\frac12\ln(2\,\operatorname{sh}2\theta) +\frac1\pi\int_0^\pi \gamma(\omega)\,d\omega, \end{aligned} \right\} \tag{4.3} \]

where, according to (3.3),

\[ \operatorname{sh}2\gamma(\omega)=\operatorname{ch}2\theta'\operatorname{ch}2\theta^* -\cos\omega\,\operatorname{sh}2\theta'\operatorname{sh}2\theta^*. \tag{4.4} \]

Using the formula (see Appendix)

\[ 2\gamma=\frac1\pi\int_0^\pi \ln(\operatorname{ch}2\gamma-\cos\omega')\,d\omega'+\ln2, \]

THERMODYNAMICS OF A PLANE DIPOLE LATTICE

we can rewrite the integral in (4,3) in the form of a double integral and obtain:

\[ \ln \lambda = -\frac{1}{2}\ln(2\sinh 2\theta) +\frac{1}{2}\ln 2 + \frac{1}{2\pi^2} \int_0^\pi\!\!\int_0^\pi \ln\bigl(\cosh 2\gamma(\omega)-\cos\omega'\bigr)\,d\omega\,d\omega' . \tag{4,5} \]

Finally, writing the first term in (4,5) in the form of the integral

\[ \frac{1}{2\pi^2} \int_0^\pi\!\!\int_0^\pi \ln(2\sinh 2\theta)\,d\omega\,d\omega', \]

we can rewrite (4,5) in the form

\[ \ln\lambda = \frac{1}{2}\ln 2 + \frac{1}{2\pi^2} \int_0^\pi\!\!\int_0^\pi \ln\bigl(\sinh 2\theta\,\cosh 2\gamma(\omega)-\cos\omega'\,\sinh 2\theta\bigr) \,d\omega\,d\omega' . \tag{4,5'} \]

Taking into account (4,3) and (1,34), we have:

\[ \sinh 2\theta\,\cosh 2\gamma(\omega) = \cosh 2\theta'\,\cosh 2\theta - \sinh 2\theta'\,\cos\omega \]

and we obtain for \(\ln\lambda(T)\) an expression symmetric in \(\theta\) and \(\theta'\):

\[ \ln\lambda(T)=\frac{1}{2}\ln 2+ \]

\[ + \frac{1}{2\pi^2} \int_0^\pi\!\!\int_0^\pi \ln\bigl(\cosh 2\theta'\,\cosh 2\theta - \sinh 2\theta'\,\cos\omega - \sinh 2\theta\,\cos\omega'\bigr) \,d\omega\,d\omega' . \tag{4,6} \]

Let us show that at the temperature \(\theta^*=\theta'\) the function \(\ln\lambda(T)\) has a singularity, and let us investigate the nature of this singularity. From formula (4,4) we see that for \(\theta'=\theta^*\) and \(\omega=0\) the function \(\cosh 2\gamma(\omega)=1\), and therefore, by formula (4,5), the double integral at \(\theta'=\theta^*\) has a singularity at the lower limit \(\omega=0;\ \omega'=0\).

In order to investigate this singularity in the general form, let us write expression (4,5) in the form

\[ \ln\lambda(T) = A(T) + \frac{1}{2\pi^2} \int_0^\alpha\!\!\int_0^\beta \ln\bigl(\cosh 2\gamma(\omega)-\cos\omega'\bigr)\,d\omega\,d\omega', \tag{4,7} \]

where \(A(T)\) is a function regular at \(T=T_c\); \(\alpha,\beta\) are small quantities.

Expanding the integrand in a Taylor series in powers of the small quantities \(\tau=\theta'-\theta^*;\ \omega,\ \omega'\), we obtain:

\[ \ln\lambda(T) = A(T) + \frac{1}{2\pi^2} \int_0^\alpha\!\!\int_0^\beta \ln\left( 2\tau^2 + \frac{1}{2}\sinh^2 2\theta^*\,\omega^2 + \frac{1}{2}\omega'^2 \right) \,d\omega\,d\omega' . \tag{4,8} \]

Performing a change of variables of integration according to the formulas

\[ \frac{\omega}{\sqrt{2}}\sinh 2\theta^* = \rho\cos\varphi, \qquad \frac{\omega'}{\sqrt{2}} = \rho\sin\varphi, \tag{4,9} \]

we obtain:

\[ \ln \lambda (T)=A(T)+\frac{2}{\operatorname{sh}2\theta^{*}} \int_{0}^{\varphi'} d\varphi \int_{0}^{\rho'} \rho \ln(2\tau^{2}+\rho^{2})\,d\rho, \tag{4,10} \]

where \(\varphi'\) and \(\rho'\) are new upper limits of integration, already depending on the temperature. Carrying out the integration, we finally obtain for \(\ln \lambda(T)\) an expression of the form

\[ \ln \lambda(T)=P(T)+Q(T)\tau^{2}\ln \tau, \tag{4,11} \]

where \(P(T)\) and \(Q(T)\) are two functions regular at \(T=T_{0}\).

From expression (4,11) we conclude that the expansion of the energy \(\varepsilon\) near the singular point \(\tau=0\) has a term proportional to \(\tau\ln\tau\), while the expansion of the specific heat \(c\) has a term proportional to \(\ln\tau\). Consequently, the energy \(\varepsilon\) is continuous at \(T=T_{0}\), while the specific heat has a logarithmic singularity.

We have verified that the temperature \(T_{0}\) is the temperature of a second-order phase transition for an infinitely extended planar dipole lattice.

The integral entering formula (4,6) in the expression for \(\ln\lambda\) is not expressible in terms of known functions. However, as Onsager\(^2\) showed, the integrals entering formulas (4,2) for the energy \(\varepsilon\) and the specific heat \(c\) are expressible in terms of elliptic integrals. We shall give this transformation for the case of an isotropic lattice \(\theta=\theta'\). From formulas (1,16), (4,2), and (4,3) we have:

\[ \left. \begin{aligned} \varepsilon&=-\frac{I_{2}-I_{1}}{2}\frac{d\ln\lambda}{d\theta},\\[6pt] \frac{d\ln\lambda(\theta)}{d\theta} &=\operatorname{cth}2\theta+\frac{1}{\pi}\int_{0}^{\pi} \frac{d\gamma(\omega)}{d\theta}\,d\omega . \end{aligned} \right\} \tag{4,12} \]

Taking into account formulas (1,34) and (4,4), at \(\theta=\theta'\) we have

\[ \operatorname{ch}2\gamma(\omega)=\operatorname{ch}2\theta\,\operatorname{cth}2\theta-\cos\omega. \tag{4,13} \]

Denoting

\[ \operatorname{ch}2\theta\,\operatorname{cth}2\theta=\frac{2}{k}, \tag{4,14} \]

we compute:

\[ \frac{d\gamma(\omega)}{d\theta} = \frac{ 2\left(\operatorname{ch}2\theta-\frac{1}{k}\operatorname{cth}2\theta\right) }{ \sqrt{\left(\frac{2}{k}-\cos\omega\right)^{2}-1} }. \tag{4,15} \]

Substituting (4.15) into (4.12), after the usual transformation we obtain:

\[ \varepsilon=-\frac{(I_2-I_1)}{2}\left\{\operatorname{cth}2\theta+ \frac{(4\operatorname{th}2\theta-2\operatorname{cth}2\theta)}{\pi} \int\limits_0^{\pi/2}\frac{d\varphi}{\sqrt{1-k^2\sin^2\varphi}}\right\}. \tag{4.16} \]

In the general case of an anisotropic lattice, the elliptic integrals become considerably more complicated (see (2) in the Appendix).

§ 5. HIGHER ORDER

We have shown that the distribution function \(Z(T)\) for a planar lattice is expressed in terms of the eigenvalues of the matrix \(\hat P\).

The question arises as to what physical meaning the off-diagonal elements of this matrix have:

\[ \left\langle \nu' \middle| \hat P \middle| \nu \right\rangle = \exp\left(-\frac{V_1(\nu')+V_2(\nu',\nu)}{kT}\right). \tag{5.1} \]

In order to answer this question, let us consider a lattice consisting of only two columns. Let the zero-th column have configuration \(\nu^0\). Then, by Boltzmann’s principle, the probability \(w^{(1)}(\nu',\nu^0)\), depending on \(\nu^0\), that the first column will have configuration \(\nu'\) will be

\[ w^{(1)}(\nu',\nu^0) = C^{(1)}\exp\left(-\frac{V_1(\nu')+V_2(\nu',\nu^0)}{kT}\right) = \]

\[ = C^{(1)}\left\langle \nu' \middle| \hat P \middle| \nu^0 \right\rangle, \tag{5.2} \]

where \(C^{(1)}\) is a normalizing factor determined from the condition

\[ \sum_{(\nu')} w^{(1)}(\nu',\nu^0)=1 \]

and equal to

\[ C^{(1)} = \frac{1}{\sum\limits_{(\nu')}\left\langle \nu' \middle| \hat P \middle| \nu^0 \right\rangle}. \tag{5.3} \]

Thus, the matrix element \(\left\langle \nu' \middle| \hat P \middle| \nu^0 \right\rangle\) is proportional to the probability \(w^{(1)}(\nu',\nu^0)\).

Next let us consider a lattice consisting of three columns. By the theorems of multiplication and addition of probabilities, we find for

of the probability \(w^{(2)}(\nu'', \nu^0)\) that the second column will have the configuration \(\nu''\), under the condition that the zero-th has the configuration \(\nu^0\), the expression

\[ w^{(2)}(\nu'', \nu^0)=\sum_{(\nu')} w^{(1)}(\nu'', \nu')\,w^{(1)}(\nu', \nu^0) = C^{(2)}\left\langle \nu'' \left| \hat P^{2}\right| \nu^0 \right\rangle , \tag{5,4} \]

where the normalizing factor \(C^{(2)}\) is determined from the requirement

\[ \sum_{(\nu'')} w^{(2)}(\nu'', \nu^0)=1 \]

and is equal to

\[ C^{(2)}=\frac{1}{\displaystyle \sum_{(\nu'')}\left\langle \nu'' \left| \hat P^{2}\right| \nu^0 \right\rangle }. \tag{5,5} \]

Continuing in the same way, we arrive at the general formula expressing the probability \(w^{(k)}(\nu^{(k)}, \nu^0)\) that the \(k\)-th column will have the configuration \(\nu^{(k)}\), under the condition that the zero-th column has the configuration \(\nu^0\):

\[ w^{(k)}(\nu^{(k)}, \nu^0) = \frac{ \left\langle \nu^{(k)} \left| \hat P^{k}\right| \nu^0 \right\rangle }{ \displaystyle \sum_{(\nu^{(k)})} \left\langle \nu^{(k)} \left| \hat P^{k}\right| \nu^0 \right\rangle }. \tag{5,6} \]

The connection of the preceding with the theory of discrete Markov chains is obvious, since \(w(\nu'',\nu')\) are elements of a second-order stochastic matrix.

By definition, we say that the lattice has long-range order if the probability of the configuration \(\nu^{(k)}\) in the \(k\)-th column continues to depend on the configuration \(\nu^0\) in the zero-th column as \(k\to\infty\). If this dependence disappears in the limit, then we say that there is only short-range order.

To clarify the criterion for the existence of long-range order, we must transform expression (5,6), in which the matrix \(\hat P\) is given to us in the representation corresponding to the diagonal nature of \(\hat A_k\), into an expression in which the matrix \(\hat P\) is diagonal. Let the formulas for passing from the eigenvectors \(\nu^k\) of the matrices \(\hat A_k\) to the eigenvectors \(\psi(E)\) of the matrix \(\hat P\) have the form:

\[ \nu^{(k)}=\sum_{(E)} C(\nu^{(k)}, E)\,\psi(E). \tag{5,7} \]

Let us note that, since the matrix \(\hat P=\hat P_1\hat P_2\) is nonsymmetric, the system of vectors \(\psi(E)\) will not be orthogonal.

Then for (5.6) we have the expression

\[ w^{(k)}(\nu^{(k)},\nu^0)= \frac{\sum\limits_{(E)} E^k C(\nu^{(k)},E)C(\nu^0,E)} {\sum\limits_{(\nu^k)}\sum\limits_{(E)} E^k C(\nu^{(k)},E)C(\nu^0,E)}, \tag{5.8} \]

where the summation \(\sum\limits_{(E)}\) is carried out over all eigenvalues of the matrix \(\hat P\). For large values of \(k\) this expression asymptotically goes over into

\[ \lim_{k\to\infty} w^{(k)}(\nu^{(k)},\nu^0)= \frac{E_{\max}^k C(\nu^{(k)},E_{\max})C(\nu^0,E_{\max})} {E_{\max}^k \sum\limits_{(\nu^{(k)})} C(\nu^{(k)},E_{\max})C(\nu^0,E_{\max})}, \tag{5.9} \]

and we see that the dependence on \(\nu^0\) drops out if the largest eigenvalue of the matrix \(\hat P\) is nondegenerate.

The situation is different in the case of a twofold degeneracy of the largest eigenvalue. Denoting the two degenerate eigenvalues by \(E_1=E_2=E_{\max}\), we obtain in this case from (5.8), asymptotically:

\[ \lim_{k\to\infty} w^{(k)}(\nu^{(k)},\nu^0)= \]

\[ = \frac{C(\nu^{(k)},E_1)C(\nu^0,E_1)+C(\nu^{(k)},E_2)C(\nu^0,E_2)} {\sum\limits_{(\nu^k)}\bigl(C(\nu^k,E_1)C(\nu^0,E_1)+C(\nu^{(k)},E_2)C(\nu^0,E_2)\bigr)}. \tag{5.10} \]

That is, in the limit \(k\to\infty\) the probability continues to depend on the configuration \(\nu^0\).

We have shown, following work\({}^7\), that at low temperatures \(T<T_0\), when the largest eigenvalue is asymptotically twofold degenerate, long-range order exists. For \(T>T_0\) this degeneracy is lifted and the long-range order disappears. Thus, the second-order phase transition is associated with the disappearance of long-range order.

§ 6. PLANE LATTICE IN AN EXTERNAL FIELD

Let us now generalize our consideration by taking into account an external field directed perpendicular to the plane of the lattice. Each dipole, relative to the external field \(F\), may assume two opposite orientations. The configuration energy \(E(\nu_1,\nu_2,\ldots,\nu_n)\) in formula (1,6) acquires an additional term \(\Delta E(\nu_1,\nu_2,\ldots,\nu_n)\), depending on the field strength \(F\):

\[ \Delta E(\nu_1,\nu_2,\ldots,\nu_n)=\prod_{k=1}^{k=m} V_0(\nu_k); \]

\[ V_0(\nu)=-p_0 F \sum_{k=1}^{k=n} \sigma_k, \tag{6,1} \]

where \(p_0\) denotes the dipole moment. The distribution function according to formula (1,6) is written in the form

\[ Z(T)= \sum_{(\nu_1)}\sum_{(\nu_2)}\cdots\sum_{(\nu_m)} \prod_{k=1}^{k=m} \exp\left(\frac{-V_0(\nu_k)}{kT}\right) \exp\left(\frac{-V_1(\nu_k)}{kT}\right) \times \]

\[ \times \exp\left(\frac{-V_2(\nu_k,\nu_{k+1})}{kT}\right). \tag{6,2} \]

If, in addition to the former matrices \(\hat P_1\) and \(\hat P_2\), we introduce the matrix \(\hat P_0\):

\[ \hat P_0= \prod_{k=1}^{k=n} \exp\left(\beta \hat A_k\right), \tag{6,3} \]

where \(\beta=\dfrac{p_0F}{kT}\), then the distribution function in the presence of an external field can be written as

\[ Z(T,\beta)=\operatorname{Sp}\left\{\hat P_0\hat P_1\hat P_2\right\}^{m} \tag{6,4} \]

and the problem again reduces to the computation of the eigenvalues of a certain \(2^n\)-order matrix \(\hat P_0\hat P_1\hat P_2\). Knowing the distribution function \(Z(T,\beta)\), we can compute the mean dipole moment \(\langle p\rangle\) from the formula

\[ \langle p\rangle= \frac{p_0}{mn}\, \frac{\partial}{\partial\beta}\ln Z(T,\beta). \tag{6,5} \]

If we regard the number of columns \(m\) in the lattice as large, then (6.5) may approximately be written in the form

\[ \langle p\rangle=\frac{p_{0}}{n}\frac{\partial}{\partial \beta}\ln E_{\max}(\beta), \tag{6.5′} \]

where \(E_{\max}(\beta)\) is the maximum eigenvalue of the matrix \(\hat P_{0}\hat P_{1}\hat P_{2}\).

In the general case, the problem of determining the eigenvalues of the matrix \(\hat P_{0}\hat P_{1}\hat P_{2}\), or at least its maximum eigenvalue \(E_{\max}(\beta)\), has not yet been solved. The methods we used earlier are not applicable, since the matrix

\[ \hat P_{0}=\prod_{k=1}^{k=n}\exp(\beta \hat A_{k}) \]

cannot be represented as a rotation operator in a \(2n\)-dimensional space.

However, using perturbation-theory methods, it turns out to be possible to find an expression for the mean moment \(\langle p\rangle\) when the external-field strength tends to zero.

For this purpose, let us expand the matrix \(\hat P_{0}\hat P_{1}\hat P_{2}\) in powers of \(\beta\), and obtain:

\[ \hat P_{0}\hat P_{1}\hat P_{2} = \hat P_{1}\hat P_{2} +\beta\left(\sum \hat A_{k}\right)\hat P_{1}\hat P_{2} +\cdots . \tag{6.6} \]

In perturbation theory it is more convenient to deal with a symmetric unperturbed matrix, since in this case the system of eigenvectors proves to be orthogonal. Therefore we perform in (6.6) a similarity transformation:

\[ \hat P_{2}^{1/2}\left(\hat P_{0}\hat P_{1}\hat P_{2}\right)\hat P_{2}^{-1/2} = \hat P_{2}^{1/2}\hat P_{1}\hat P_{2}^{1/2} + \beta \hat P_{2}^{1/2}\left(\sum \hat A_{k}\right)\hat P_{1}\hat P_{2}^{1/2}. \tag{6.7} \]

Let us find the perturbed value of the maximum eigenvalue \(E_{\max}(0)\) of the unperturbed matrix \(\hat P_{2}^{1/2}\hat P_{1}\hat P_{2}^{1/2}\).

Denoting by \(\Psi\) the eigenvector of the unperturbed matrix corresponding to this term, we have, by the general rules of perturbation theory,

\[ E_{\max}(\beta) = E_{\max}(0) + \beta\sum_{k=1}^{k=n} \left( \Psi \left| \hat P_{2}^{1/2}\hat A_{k}\hat P_{1}\hat P_{2}^{1/2} \right| \Psi \right). \tag{6.8} \]

Since \(\Psi\) is at the same time an eigenvector of the matrix \(\hat U\), we have:

\[ \hat U\Psi=\Psi. \tag{6.9} \]

Substituting (6,9) into (6,8), we have, since \(\hat U\) commutes with \(\hat P_1\) and \(\hat P_2\) and anticommutes with all \(\hat A_k\):

\[ \left(\Psi\left|\hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2}\right|\Psi\right) = \left(\Psi\left|\hat U\hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2}\hat U\right|\Psi\right) = \]

\[ = -\left(\Psi\left|\hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2}\right|\Psi\right)=0, \tag{6,10} \]

and, consequently, in the first approximation of perturbation theory \(E_{\max}(\beta)\) does not depend on \(\beta\). By formula (6,5) it follows from this that \(\langle p\rangle=0\).

The situation is different if the maximal eigenvalue \(E_{\max}^{+}(0)=E_{\max}^{-}(0)\) is doubly degenerate as \(n\to\infty\). In this case the perturbation removes the degeneracy, and the term is split. According to the general rules of perturbation theory, for the split terms we have:

\[ E_{\max}^{+}(\beta)=E_{\max}(0)+ \beta\sum_k \left( \frac{\Psi^{+}+\Psi^{-}}{\sqrt{2}} \left| \hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2} \right| \frac{\Psi^{+}+\Psi^{-}}{\sqrt{2}} \right) = \]

\[ =E_{\max}(0)+ \beta\sum_k \left( \Psi^{-} \left| \hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2} \right| \Psi^{+} \right), \]

\[ E_{\max}^{-}(\beta)=E_{\max}(0)+ \beta\sum_k \left( \frac{\Psi^{+}-\Psi^{-}}{\sqrt{2}} \left| \hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2} \right| \frac{\Psi^{+}-\Psi^{-}}{\sqrt{2}} \right) = \]

\[ =E_{\max}(0)- \beta\sum_k \left( \Psi^{-} \left| \hat P_{2}^{1/2}\hat A_k\hat P_1\hat P_{2}^{1/2} \right| \Psi^{+} \right). \tag{6,11} \]

Taking into account

\[ \left(\hat P_{2}^{1/2}\hat P_1\hat P_{2}^{1/2}\right)\Psi^{+} = E_{\max}(0)\Psi^{+}, \]

we obtain:

\[ E_{\max}^{+}(\beta)= E_{\max}(0) \left\{ 1+\beta\sum_k \left( \Psi^{-} \left| \hat P_{2}^{1/2}\hat A_k\hat P_{2}^{-1/2} \right| \Psi^{+} \right) \right\}, \]

\[ \ln E_{\max}^{+}(\beta)= \ln E_{\max}(0)+ \beta\sum_k \left( \Psi^{-} \left| \hat P_{2}^{1/2}\hat A_k\hat P_{2}^{-1/2} \right| \Psi^{+} \right). \tag{6,12} \]

Noting that the matrix \(P_{2}^{1/2}\) is cyclic in the indices and that, consequently, all \(n\) terms in the last sum are equal, we can, by formula (6,5), write:

\[ \langle p\rangle = p_0 \left( \Psi^{-} \left| \hat P_{2}^{1/2}\hat A_1\hat P_{2}^{-1/2} \right| \Psi^{+} \right). \tag{6,13} \]

We have:

\[ \hat P_2^{1/2}\hat A_1=\hat A_1 e^{-\vartheta^*\hat B_1}\hat P_2^{1/2} \]

and, consequently,

\[ \hat P_2^{1/2}\hat A_1 \hat P_2^{-1/2} =\hat A_1 e^{-\vartheta^*\hat B_1} =\hat A_1\left(\operatorname{ch}\vartheta^*-\hat B\,\operatorname{sh}\vartheta^*\right) = \]

\[ =\hat A_1\operatorname{ch}\vartheta^* - i\hat C_1\operatorname{sh}\vartheta^*. \tag{6,14} \]

Substituting into (6,13), we obtain:

\[ \langle p\rangle =p_0\left\{\operatorname{ch}\vartheta^*\,(\Psi^-|\hat A_1|\Psi^+) -i\operatorname{sh}\vartheta^*\,(\Psi^-|\hat C_1|\Psi^+)\right\}. \tag{6,13'} \]

In principle the matter is as follows: expressions are given for the matrices \(\hat A_1\) and \(\hat C_1\) in the representation in which \(\hat A_1\) is diagonal; it is required to write them in the representation in which the matrix \(\hat P_2^{1/2}\hat P_1\hat P_2^{1/2}\) is diagonal. However, the calculations turn out to be very laborious, since the explicit expressions for the eigenvectors \(\Psi^+\) and \(\Psi^-\) are very complicated. Yang\(^8\) managed to overcome this difficulty by means of very complicated calculations. Since the result proves to be simple, one should think that there exists a simpler method of calculation, not yet found.

According to Yang we have (in the case of an isotropic lattice \(\theta=\theta'\)):

\[ (\Psi^-|\hat A_1|\Psi^+) =\frac{1}{\operatorname{ch}\vartheta^*}\sqrt[8]{1-\operatorname{sh}^4 2\theta^*}, \]

\[ (\Psi^-|\hat C_1|\Psi^+)=0, \]

and, consequently, finally:

\[ \begin{aligned} \langle p\rangle&=p_0\sqrt[8]{1-\operatorname{sh}^4 2\theta^*}, &&\theta^*<\theta \quad (T<T_0),\\ \langle p\rangle&=0, &&\theta^*>\theta \quad (T>T_0). \end{aligned} \]

We see that the disappearance of the spontaneous moment at the temperature \(T>T_0\) is connected with the disappearance of long-range order, which in turn is connected with the lifting of the twofold degeneracy of the maximal term of the distribution matrix.

§ 7. BINARY PLANE LATTICE

Consider a plane quadratic lattice consisting of atoms of two kinds, located at its sites. We shall write the distribution function in the form

\[ Z(T)=\sum_{N=n_1+n_2}\sum_{(v)} \exp(n_1\xi_1+n_2\xi_2-E(v))/kT, \tag{7,1} \]

where \(n_1\) and \(n_2\) denote the numbers of atoms of the different kinds, the variable \(\nu\) enumerates all possible configurations of atoms in the lattice, \(\zeta_1\) and \(\zeta_2\) are the chemical potentials, and \(E\) is their interaction energy, depending on the configuration of the atoms in the lattice.

Let us assign to each site \((i)\) a discrete variable \(\sigma_i\), capable of taking two values: \(\sigma_i=+1\), if the site is occupied by an atom of the first kind, and \(\sigma_i=-1\), if it is occupied by an atom of the second kind. We have:

\[ \left. \begin{aligned} n_1 &= \frac{1}{2}\sum_{i=1}^{i=N}(1+\sigma_i),\\ n_2 &= \frac{1}{2}\sum_{i=1}^{i=N}(1-\sigma_i). \end{aligned} \right\} \tag{7,2} \]

If \((i)\) and \((k)\) denote two neighboring sites in the lattice, then the interaction energy \(V(\sigma_i,\sigma_k)\) between two atoms may be written in the form

\[ V(\sigma_i,\sigma_k) = \frac{1}{4}I_{11}(1+\sigma_i)(1+\sigma_k) + \frac{1}{4}I_{22}(1-\sigma_i)(1-\sigma_k) + \frac{1}{2}I_{12}(1-\sigma_i\sigma_k), \tag{7,3} \]

where \(I_{11}\) is the interaction energy between two neighboring atoms of the first kind, \(I_{22}\) is the interaction energy between two neighboring atoms of the second kind, and \(I_{12}\) is the interaction energy between two neighboring atoms of different kinds.

Introduce the symbol \(a_{ik}\), defined as follows:

\[ \left. \begin{aligned} a_{ik}&=1,\quad \text{if } i \text{ and } k \text{ are neighboring sites,}\\ a_{ik}&=0\quad \text{in all other cases.} \end{aligned} \right\} \tag{7,4} \]

Obviously, \(\sum_{k=1}^{k=N} a_{ik}\) is the number of nearest neighbors of the given site, equal to four for a square lattice.

Then the interaction energy \(E(\nu)\) may be written in the form

\[ E(\nu)=\frac{1}{2}\sum_{i,k} a_{ik}V(\sigma_i\sigma_k). \tag{7,5} \]

Substituting expressions (7,2)—(7,5) into (7,1), we obtain:

\[ Z(T)=\sum_{(\sigma_1)}\sum_{(\sigma_2)}\ldots\sum_{(\sigma_N)} \left[n_1\zeta_1+n_2\zeta_2-E(\nu)\right]/kT, \tag{7,6} \]

where the summation over all \(n_1+n_2=N\) and \(\nu\) has been replaced by the equivalent summation over all values of the variables

\[ \sigma_1,\ \sigma_2,\ldots,\sigma_N . \]

We have:

\[ \begin{aligned} n_1\zeta_1+n_2\zeta_2-E(\nu) &=\frac{N}{2}\left[\zeta_1+\zeta_2-I_{11}-I_{22}-2I_{12}\right]+ \\ &\quad+\frac{1}{2}\left[\zeta_1+\zeta_2-2I_{11}+2I_{22}\right]\sum_{k=1}^{k=N}\sigma_k - \\ &\quad-\frac{1}{4}\left(I_{11}+I_{22}-2I_{12}\right)\frac{1}{2}\sum_{i,k}a_{ik}\sigma_i\sigma_k . \end{aligned} \tag{7,7} \]

Let us denote:

\[ \left. \begin{aligned} \frac{1}{2kT}\left(\zeta_1+\zeta_2-2I_{11}+2I_{22}\right)&=\beta,\\ -\frac{1}{4kT}\left(I_{11}+I_{22}-2I_{12}\right)&=\theta,\\ \frac{1}{2kT}\left(\zeta_1+\zeta_2-I_{11}-I_{22}-2I_{12}\right)&=\theta_0, \end{aligned} \right\} \tag{7,8} \]

and obtain:

\[ Z(T)=\exp N\theta_0 \sum_{(\sigma_1)}\sum_{(\sigma_2)}\cdots\sum_{(\sigma_N)} \prod_{k=1}^{k=N}\exp(\beta\sigma_k) \prod_{i,k}\exp\left(\frac{\theta}{2}a_{ik}\sigma_i\sigma_k\right). \tag{7,9} \]

The inessential factor \(\exp N\theta_0\) may be discarded, and we obtain the distribution function in the form

\[ Z(T)= \sum_{(\sigma_1)}\sum_{(\sigma_2)}\cdots\sum_{(\sigma_N)} \prod_{k=1}^{N}\exp(\beta\sigma_k) \prod_{i,k}\exp\left(\frac{\theta}{2}a_{ik}\sigma_i\sigma_k\right). \tag{7,10} \]

It may be written as follows

\[ Z(T)=\operatorname{Sp}_{r}\left(\hat P_0\hat P_1\hat P_2\right)^m, \tag{7,10'} \]

where the distribution matrix is given by the formula

\[ \hat P_0\hat P_1\hat P_2= \]

\[ =(2\operatorname{sh}2\theta)^{n/2} \prod_{k=1}^{n}\exp(\beta\hat A_k)\times \prod_{k=1}^{k=n}\exp(\theta\hat A_k\hat A_{k+1}) \prod_{k=1}^{k=n}\exp(\theta^{*}\hat B_k). \tag{7,11} \]

We see that the problem of a plane binary lattice is equivalent to the problem considered earlier of a plane lattice in an external field \((\beta \ne 0)\).

Computing the averages by formulas (7,2), we find:

\[ \left. \begin{aligned} N_1 &= \langle n_1\rangle=\frac{N}{2}(1+\langle \sigma\rangle),\\ N_2 &= \langle n_2\rangle=\frac{N}{2}(1-\langle \sigma\rangle). \end{aligned} \right\} \tag{7,12} \]

But the distribution matrix \(\hat P\) for \(\beta=0\) is invariant under the replacement of \(\sigma_i\) by \(-\sigma_i\), and, consequently, in this case

\[ \langle \sigma\rangle=0, \]

whence

\[ N_1=N_2=N/2. \]

Thus, the problem of a binary lattice with equal numbers of atoms of both types is equivalent to the problem of a plane dipole lattice in the absence of an external field.

APPENDIX

1. DIRECT SUM AND DIRECT PRODUCT OF MATRICES

Let an \(n\)-th order matrix \(\hat A\) and an \(m\)-th order matrix \(\hat B\) be given.

The direct sum of the matrices \(\hat A\) and \(\hat B\) is the \((n+m)\)-th order matrix

\[ \hat A \dotplus \hat B = \begin{pmatrix} \hat A & 0\\ 0 & \hat B \end{pmatrix}. \tag{1} \]

Since

\[ \hat B \dotplus \hat A = \begin{pmatrix} \hat B & 0\\ 0 & \hat A \end{pmatrix}, \tag{1'} \]

the direct sum is noncommutative. From the definition the following properties of the direct sum are easily obtained:

\[ \left. \begin{aligned} (\hat A \dotplus \hat B)\dotplus \hat C &= \hat A \dotplus (\hat B \dotplus \hat C),\\ (\hat A_1 \dotplus \hat B_1)(\hat A_2 \dotplus \hat B_2) &= \hat A_1\hat A_2 \dotplus \hat B_1\hat B_2,\\ \operatorname{Sp}(\hat A \dotplus \hat B) &= \operatorname{Sp}\hat A+\operatorname{Sp}\hat B,\\ \operatorname{Det}(\hat A \dotplus \hat B) &= \operatorname{Det}\hat A\cdot \operatorname{Det}\hat B. \end{aligned} \right\} \tag{2} \]

The direct product of the matrices \(\hat A\) and \(\hat B\) is the \(mn\)-row matrix

\[ \hat A \times \hat B = \begin{pmatrix} AB_{11} & \ldots & AB_{1m}\\ AB_{m1} & & AB_{mm} \end{pmatrix}. \tag{3} \]

Since

\[ \hat B \times \hat A = \begin{pmatrix} BA_{11} & \ldots & BA_{1n}\\ BA_{n1} & & BA_{nn} \end{pmatrix}, \tag{3'} \]

the direct product is noncommutative. From the definition the following properties of the direct product readily follow:

\[ (\hat A \times \hat B)\times \hat C=\hat A\times(\hat B\times \hat C), \]

\[ (\hat A_1\times \hat B_1)(\hat A_2\times \hat B_2) = \hat A_1\hat A_2\times \hat B_1\hat B_2, \]

\[ \operatorname{Sp}(\hat A\times \hat B)=\operatorname{Sp}\hat A\cdot \operatorname{Sp}\hat B, \]

\[ \operatorname{Det}(\hat A\times \hat B)=(\operatorname{Det}\hat A)^n(\operatorname{Det}\hat B)^m. \tag{4} \]

Under similarity transformations

\[ \left. \begin{aligned} \hat A'&=\hat S_1^{-1}\hat A\hat S_1,\\ \hat B'&=\hat S_2^{-1}\hat B\hat S_2 \end{aligned} \right\} \tag{5} \]

direct sums and the direct product are transformed according to the formulas:

\[ \hat A' + \hat B' = (\hat S_1^{-1}\hat A\hat S_1) + (\hat S_2^{-1}\hat B\hat S_2) = \]

\[ = (\hat S_1^{-1}+\hat S_2^{-1})(\hat A+\hat B)(\hat S_1+\hat S_2), \]

\[ \hat A'\times \hat B' = (\hat S_1^{-1}\hat A\hat S_1)\times(\hat S_2^{-1}\hat B\hat S_2) = \]

\[ = (\hat S_1^{-1}\times \hat S_2^{-1})(\hat A\times \hat B)(\hat S_1\times \hat S_2). \tag{6} \]

Consequently, if the transformations \(\hat S_1\) and \(\hat S_2\) bring the matrices \(\hat A\) and \(\hat B\) to diagonal form, then the transformations \(\hat S_1+\hat S_2\) and \(\hat S_1\times \hat S_2\) bring the matrices \(\hat A+\hat B\) and \(\hat A\times \hat B\) to diagonal form.

2. REDUCTION OF A COMPLEX \(2n\)-ROW ORTHOGONAL MATRIX \(\hat L\) TO A DIRECT SUM OF TWO-ROW ORTHOGONAL MATRICES

The elements of the orthogonal complex matrix \(\hat L\) satisfy the conditions:

\[ L_{\alpha\beta}=L_{\beta\alpha}^{-1}. \tag{7} \]

Let \(\psi\) and \(\varphi\) be two proper complex vectors of the matrix \(\hat L\), belonging to the complex proper values \(\lambda\) and \(\mu\):

\[ \sum_{(\sigma)} L_{\alpha\sigma}\psi_\sigma=\lambda\psi_\alpha;\qquad \sum_{(\sigma)} L_{\alpha\sigma}\varphi_\sigma=\mu\varphi_\alpha. \tag{8} \]

Taking (7) into account, it follows from this that:

\[ \sum_\alpha \psi_\alpha L_{\alpha\sigma}=\frac{1}{\lambda}\psi_\sigma;\qquad \sum_\alpha \varphi_\alpha L_{\alpha\sigma}=\frac{1}{\mu}\varphi_\sigma. \tag{8'} \]

From (8) and (8′) we derive:

\[ (\varphi_\alpha L_{\alpha\sigma}\psi_\sigma)-(\psi_\alpha L_{\alpha\sigma}\varphi_\sigma) =(\lambda-\mu)(\psi_\alpha\varphi_\alpha)= \]

\[ =\left(\frac{1}{\mu}-\frac{1}{\lambda}\right)(\psi_\alpha\varphi_\alpha) =\frac{\lambda-\mu}{\lambda\mu}(\varphi_\alpha\psi_\alpha), \tag{9} \]

whence it follows that either the vectors \(\psi\) and \(\varphi\) are orthogonal, or \(\lambda\mu=1\).

Thus, the complex proper values and their corresponding complex proper vectors split into \(n\) pairs

\[ \left(\lambda_1,\frac{1}{\lambda_1}\right),\qquad \left(\lambda_2,\frac{1}{\lambda_2}\right),\ldots, \left(\lambda_n,\frac{1}{\lambda_n}\right), \tag{10} \]

\[ \psi_1,\varphi_1;\qquad \psi_2,\varphi_2,\ldots,\psi_n,\varphi_n, \tag{10'} \]

lying in \(n\) mutually orthogonal planes:

\[ \left. \begin{array}{l} (\psi_m\psi_n)=0;\\ (\varphi_m\varphi_n)=0;\\ (\psi_m\varphi_n)=0;\\ (\psi_n\varphi_n)\ne 0. \end{array} \right\} \tag{11} \]

Denoting:

\[ \left. \begin{array}{l} \lambda_k=\exp \Lambda_k,\\ \psi_k=x_k+i y_k,\\ \varphi_k=x_k-i y_k, \end{array} \right\} \tag{12} \]

where \(\Lambda_k, x_k, y_k\) are complex, we obtain:

\[ \left. \begin{aligned} \widehat L x_k &= x_k \cos(i\Lambda_k) + y_k \sin(i\Lambda_k),\\ \widehat L y_k &= -x_k \sin(i\Lambda_k) + y_k \cos(i\Lambda_k), \end{aligned} \right\} \tag{13} \]

whence it follows that any complex orthogonal matrix of order \(2^n\) can be represented as a direct sum of \(n\) matrices of order two,

\[ \widehat L = \begin{pmatrix} \cos i\Lambda_1, & \sin i\Lambda_1 \\ -\sin i\Lambda_1, & \cos i\Lambda_1 \end{pmatrix} + \cdots + \begin{pmatrix} \cos i\Lambda_n, & \sin i\Lambda_n \\ -\sin i\Lambda_n, & \cos i\Lambda_n \end{pmatrix}. \tag{14} \]

In the real domain the decomposition (14) corresponds to the decomposition of an arbitrary rotation in \(2n\)-dimensional space into a sequence of commuting rotations in \(n\) mutually perpendicular planes.

3. COMPUTATION OF THE INTEGRAL

\[ 2x = \frac{1}{\pi}\int_0^\pi \ln(\operatorname{ch} 2x - \cos\omega)\,d\omega + \ln 2. \]

Expanding \(2\operatorname{sh} 2nx\) into simple factors, we have:

\[ 2\operatorname{sh} 2nx = (e^x)^{2n} - (e^{-x})^{2n} = \prod_{k=1}^{k=n} (e^x - \eta_k e^{-x})(e^x - \bar{\eta}_k e^{-x}), \tag{15} \]

where \(\eta_k = \exp \frac{\pi i}{n}\) is a root of unity of degree \(2n\). From (15) we have:

\[ 2\operatorname{sh} 2nx = \prod_{k=1}^{k=n} 2\left(\operatorname{ch} 2x - \cos \frac{\pi k}{n}\right). \tag{15'} \]

Taking logarithms, we obtain:

\[ \ln 2 + \ln \operatorname{sh} 2nx = n\ln 2 + \sum_{k=1}^{k=n} \ln\left(\operatorname{ch} 2x - \cos \frac{\pi k}{n}\right). \tag{16} \]

Passing to the limit \(n \to \infty\), we obtain:

\[ 2x = \ln 2 + \frac{1}{\pi}\int_0^\pi \ln(\operatorname{ch} 2x - \cos\omega)\,d\omega. \tag{16'} \]

References

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