Connection formulas for the lambda generalized Ising correlation functions

TL;DR

This paper derives and proves connection formulas for the lambda generalized diagonal correlation functions in the Ising model, enhancing understanding of their mathematical structure.

Contribution

It introduces new connection formulas for lambda generalized Ising correlation functions, providing a rigorous mathematical foundation.

Findings

Derived explicit connection formulas

Proved the formulas rigorously

Enhanced understanding of correlation functions

Abstract

We derive and prove the connection formulas for the lambda generalized diagonal Ising model correlation functions.

Equations

$$C^{-}(N,t;\lambda )=(1-k^2{)}^{1/4}\exp \sum _{n=1}^{\infty }{\lambda }^{2n}{F}_N^{(2n)}$$

$$F_N^{(2n)}=\frac{(-1{)}^{n+1}}{n(2\pi {)}^{2n}}\oint \prod _{j=1}^{2n}\frac{d{z}_j{z}_j^N}{1-z_j{z}_{j+1}}\prod _{j=1}^{n}Q(z_{2j-1})Q(z_{2j-1}^{-1})P(z_{2j})P(z_{2j}^{-1})$$

$$P(z)=1/Q(z)=(1-kz{)}^{1/2}$$

$$t=k^2$$

$$\displaystyle C(N,t;1)=\begin{array}[]{|llll|}{a}_{0}&{a}_{-1}&\cdots&{a}_{-N+1}\\ {a}_{1}&{a}_{0}&\cdots&{a}_{-N+2}\\ \vdots&\vdots&&\vdots\\ {a}_{N-1}&{a}_{N-2}&\cdots&{a}_{0}\end{array}$$

$$a_n=\frac{1}{2\pi }{\int }_0^{2\pi }d\theta e^{-in\theta }{\left[\frac{1-k{e}^{-i\theta }}{1-k{e}^{i\theta }}\right]}^{1/2}$$

$$k=(\sinh 2E_v/kT\sinh 2E_h/kT{)}^{-1}$$

$${\left(t(t-1)h^{\prime \prime }\right)}^2+4h^{\prime }\left((t-1)h^{\prime }-h-1/4\right)\left(t{h}^{\prime }-h\right)=N^2{\left((t-1)h^{\prime }-h\right)}^2$$

$$h(t)=t(t-1)\frac{d}{dt}\ln C^{-}-\frac{t}{4}\mathrm{with}t=(\sinh 2E_v/kT\sinh 2E_h/kT{)}^{-2}$$

$$C^{-}(N,t;\lambda )=(1-t{)}^{1/4}\{1+{\lambda }^2\frac{(1/2{)}_N(3/2{)}_N}{4[(N+1)!{]}^2}{t}^{N+1}(1+O(t))\}$$

$$t(t-1)\frac{d}{dt}\ln (1-t{)}^{1/4}-\frac{t}{4}=0$$

$$h(t)\to -{\lambda }^2{t}^{N+1}\frac{(1/2{)}_N(3/2{)}_N}{4N!(N+1)!}$$

$$h(t)=-\frac{1-{\sigma }^2}{4}+\frac{1}{8}(1-{\sigma }^2)(1-t)$$

$$+\frac{s}{16\sigma }(1+\sigma )(2N+\sigma )(1-t{)}^{1-\sigma }$$

$$-\frac{s^{-1}}{16\sigma }(1-\sigma )(2N-\sigma )(1-t{)}^{1+\sigma }$$

$$C^{-}(N,t;\sigma ,\hat{s})=K(N;\sigma )(1-t{)}^{{\sigma }^2/4}[1-\frac{1-{\sigma }^2}{8}(1-t)$$

$$+\frac{\hat{s}(N;\sigma )}{16\sigma }(2N+\sigma )(1-t{)}^{1-\sigma }$$

$$-\frac{\hat{s}(N;\sigma {)}^{-1}}{16\sigma }(2N-\sigma )(1-t{)}^{1+\sigma }+O((1-t{)}^{2-\sigma })]$$

$$\sigma =(2/\pi )\arccos \lambda$$

$$\hat{s}(N,\sigma )=16^{\sigma }\prod _{n=1}^N\frac{1-\sigma /2n}{1+\sigma /2n}$$

$$K(N;\sigma )=2^{-{\sigma }^2}{\left(\frac{\sigma }{\sin \pi \sigma /2}\right)}^N\prod _{m=1}^{N-1}{\left(1-\frac{1}{4m^2}\right)}^{m-N}\prod _{m=1}^{N-1}{\left(1-\frac{{\sigma }^2}{4m^2}\right)}^{N-m}$$

$$t\to 1,N\to \infty ,\mathrm{with}N(1-t)=r\mathrm{fixed}$$

$$G_{-}(r;\lambda )=\underset{\mathrm{scaling}}{\lim }(1-t{)}^{-1/4}{C}^{-}(N,t;\lambda )$$

$${\theta }_2(u;q)=2\sum _{n=0}^{\infty }{q}^{(n+1/2{)}^2}\cos [(2n+1)u]$$

$${\theta }_3(u;q)=1+2\sum _{n=1}^{\infty }{q}^{n^2}\cos 2nu$$

$${\theta }_4(u;q)=1+2\sum _{n=1}^{\infty }(-1{)}^n{q}^{n^2}\cos 2nu$$

$$q=e^{-\pi K^{\prime }(k)/K(k)}$$

$$C^{-}(0,t;\lambda )=\frac{{\theta }_3(u;q)}{{\theta }_3(0,q)}$$

$$C^{-}(1,t;\lambda )=\frac{-{\theta }_2^{\prime }(u;q)}{\sin (u){\theta }_2(0,q){\theta }_3(0;q{)}^2}$$

$$\lambda =\cos u$$

$${\theta }_3(\frac{uK(k)}{i{K}^{\prime }(k)};e^{-\pi K(k)/K^{\prime }(k)})e^{-u^{2}K(k)/\pi K^{\prime }(k)}={\left(\frac{K^{\prime }(k)}{K(k)}\right)}^{1/2}{\theta }_3(u;e^{-\pi K^{\prime }(k)/K(k)})$$

$${\theta }_4(\frac{uK(k)}{i{K}^{\prime }(k)};e^{-\pi K(k)/K^{\prime }(k)})e^{-u^{2}K(k)/\pi K^{\prime }(k)}={\left(\frac{K^{\prime }(k)}{K(k)}\right)}^{1/2}{\theta }_2(u;e^{-\pi K^{\prime }(k)/K(k)})$$

$$\lambda =\cos m\pi /n$$

$$C^{-}(0,t;\cos (\pi /4))=2^{-1/4}(1-t{)}^{1/16}[1+(1-t{)}^{1/2}{]}^{1/4}$$

$$C^{-}(1,t;\cos (\pi /4))=2^{-3/4}(1-t{)}^{1/16}[1+(1-t{)}^{1/2}{]}^{3/4}$$

$$C^{-}(2,t;\cos (\pi /4))=2^{-5/4}(1-t{)}^{1/16}[1+(1-t{)}^{1/2}{]}^{5/4}[5-(1-t{)}^{1/2}]/4$$

Full text

Connection formulas for the $\lambda$ generalized Ising correlation functions

1 Introduction

The $\lambda$ generalized Ising model correlations may be defined for $T<T_{c}$ by the Fredholm determinant expression [1]

[TABLE]

[TABLE]

where the contours of integration are $|z_{j}|=1-\epsilon,~{}z_{2n+1}\equiv z_{1}$ and

[TABLE]

The parameter $k$ satisfies $0\leq k\leq 1$ and we will use

[TABLE]

It should be noted that an equivalent form of (1)-(2) was given in the 1976 paper of [2]. The integrals of the two expressions are presumably seen to be equal by adding appropriate total derivatives to the integrands but this has never been explicitly demonstrated.

When $\lambda=1$ the Fredholm determinant (1) reduces to the diagonal correlation function of the Ising model given by the $N\times N$ Toeplitz determinant [3]

[TABLE]

with

[TABLE]

and

[TABLE]

The expression (1)-(2) is obtained in [1] for $\lambda=1$ by extending to all orders the proceedure used by Wu [4] to compute the leading order expansion of (9) for $k$ fixed and $N\rightarrow\infty$.

The generalized correlation (1) for $0<t<1$ satisfies the sigma form of the Painlevé VI equation first derived by Miwa and Jimbo [5] for the diagonal Ising correlation

[TABLE]

where

[TABLE]

It is readily seen from the definition (1) that these generalized correlations are analytic at $t=0$ and that for $t\rightarrow 0$

[TABLE]

where by noting that

[TABLE]

we have for $t\rightarrow 0$ the one parameter boundary condition for (11)

[TABLE]

The question of interest is to determine the behaviour of this one parameter family at $t=1$. A local analysis of the nonlinear equation (11) gives the result that as $t\rightarrow 1$ that

[TABLE]

and thus to order $(1-t)^{1+\sigma}$

[TABLE]

where $\sigma(\lambda)$ and ${\hat{s}}(N,\sigma)$ are two integration constants for the second order equation (11) and $K(N,\sigma)$ is a normalizing constant which can be determined from the original definition (1). The computation of $\sigma(\lambda),{\hat{s}}(N,\sigma),K(N,\sigma)$ is the purpose of this note.

The results are as follows

[TABLE]

In section 2 we briefly discuss the history of this connection problem and in section 3 we present special cases which confirm the results (18)-(20). A proof of (19) and (20) is given in section 4 by use of the Toda-like relation of Mangazeev and Guttmann [6].

2 History

In the scaling limit

[TABLE]

the scaling function

[TABLE]

was shown in 1976 in [2] for $\lambda=1$ to be expressed in terms of a Painlevé III function and the connection formulas for $\sigma(\lambda)$ and ${\hat{s}}(N,\sigma)$ for the $\lambda$ generalized scaling function were computed the next year in [7] by a direct expansion of the integrals in the scaled version of (1)-(2). The normalization constant was computed in 1991 by Tracy in [8].

For the generic case of the Painlevé VI function the connection formulas for $\sigma$ and ${\hat{s}}$ were computed in 1982 by Jimbo in [9] by means of deformation theory and the normalizing constant $K$ was computed in 2018 by Its, Lisovyy and Prokhorov [10].

In the generic case there is nonanalytic behavior at all three points $t=0,1,\infty$. However, the Painlevé VI for the Ising model is not generic and therefore the results of [9], while still relevant at $t=1$ where the correlation function is singular, do not hold at $t=0,\infty$ where the correlation function is analytic. Nongeneric cases have been studied by Guzzetti [11]) but the case relevant for the generalized Ising correlations seems not to have been investigated.

3 Special Cases

We here present several special cases of computations which confirm the results (18)-(20).

3.1 $C^{-}(N,t;\lambda)$ for $N=0,1$

In [12] a prescription is given to express $C^{-}(N,t;\lambda)$ in terms of the theta functions

[TABLE]

where $q$ is related to the variable $t$ by

[TABLE]

with $k^{2}=t$ and $K(k)$ and $K^{\prime}(k)$ are the complete elliptic integrals of the first kind. In [13] and [14] this prescription was used to obtain explicit expressions for $N=0,1$

[TABLE]

where prime indicates the derivative with respect to $u$ and

[TABLE]

The expressions (27) and (28) are expanded at $t=0$ by the direct use of (23) and (24) to obtain the form (13). The expansion at $t=1$ is obtained from (27) and (28) by use of the identities (on page 370 of [15] with $v\rightarrow u/\pi)$

[TABLE]

The results (18)-(20) are obtained by comparing these explicit expansions at $t=1$ with the general form (17).

3.2 Algebraic cases for $\lambda=\cos(m\pi/n)$ where

$\sigma=2m/n$

When

[TABLE]

the function $C^{-}(N,t;\lambda)$ is an algebraic function and in [13] the explicit results are given for $N=0,1,2$ that for $\lambda=\cos(\pi/4)$ with $\sigma=1/2$

[TABLE]

When $t=0$ these expressions are expanded as

[TABLE]

which agree with the expansion at $t=0$ of (13) with $\lambda=\cos(\pi/4)=2^{-1/2}$.

When $t\rightarrow 1$ these expressions are expanded as

[TABLE]

The only other case where an explicit result is given in [13] is for

[TABLE]

where for $N=0$ the function $C^{-}(0,t;\cos(\pi/3)$ satisfies

[TABLE]

The expansion near $t=0$ of the solution of (43) which does not vanish at $t=0$ is

[TABLE]

and for $t\rightarrow 1$

[TABLE]

These results for $\lambda=\cos(\pi/4)~{}{\rm and}~{}\cos(\pi/3)$ all agree with (18)-(20).

3.3 The Ising case $\lambda=1$ ($\sigma=0$)

The Ising case $\lambda=1$ has been extensivly studied in [12] where it is shown that as $t\rightarrow 1$

[TABLE]

with

[TABLE]

which agrees with (18)-(20) in the limit $\sigma\rightarrow 0$.

3.4 The case $\lambda=0$ ($\sigma=1$)

When $\sigma=1$ we find from (19) and (20) that

[TABLE]

and thus (17) reduces to

[TABLE]

as required by (1).

3.5 $N\rightarrow\infty$ for $K(N,\sigma)$

To obtain the behavior of $K(N;\sigma)$ for $N\rightarrow\infty$ we use the identity

[TABLE]

to write

[TABLE]

which we use in (20) to obtain

[TABLE]

To now expand $K(N;\sigma)$ for $N\rightarrow\infty$ we use for the products running $N$ to infinity

[TABLE]

For the products from $1$ to $N-1$ we write

[TABLE]

where for $N\rightarrow\infty$ the second product converges and the first product is expanded using the definition of Eulers constant $\gamma$

[TABLE]

to find for $N\rightarrow\infty$

[TABLE]

Thus we have for $N\rightarrow\infty$

[TABLE]

When this is rewritten in terms of Barnes G functions and the derivative of the zeta function at $-1$ this agrees with the result obtained by Tracy [8] for the scaling limit of $C^{-}(N,t;\lambda)$

4 The Toda-like equation

In 2010 Mangazeev and Guttmann [6] proved that $C^{-}(N,t;\lambda)$ satisfies the following Toda-like equation

[TABLE]

The verification of this identity in the limit $t\rightarrow 1$ using the expansion (17) with (18)-(20) combined with the previous results for $N=0,~{}1$ constitutes an inductive proof of the connection formulas (19) for ${\hat{s}}(N,\sigma)$ and (20) for $K(N;\sigma)$.

It is straigntforward to see from (20) that

[TABLE]

and thus for $x=1-t\rightarrow 0$ the right hand side of (58) to order $x^{1-\sigma}$ is

[TABLE]

Using (17) we find to order $O(x^{1-\sigma})$ that

[TABLE]

and thus the leading order terms in (58) cancel because of the connection formulas (20) and from the terms of order $x^{1-\sigma}$ we obtain the recursion relation which must be satisfied by ${\hat{s}}(N;\sigma)$

[TABLE]

which by direct substitution is easily seen to be satisfied by the expression (19) for ${\hat{s}}(N;\sigma)$. Thus we have proven by induction that (19) and (20) are correct.