An Exact Closed Formula of a Spin-spin Correlation Function of the Two-Dimensional Ising Model with Finite Size

Abstract

Some expressions of spin-spin correlation functions of the two-dimensional Ising model in the thermodynamic limit have been obtained, however, so far, no one correlation function that can show long range order of the model with finite size has been obtained. The purpose of this paper is to fill this gap. We present an exact closed formula for the correlation functions of the farthest pairs of spins in a column of the two-dimensional rectangular Ising model with periodic-free boundary condition and finite size by the spinor analysis method. The concrete expression correct to $\mathrm {e}^{-LC_{0} }$ order of magnitude of such correlation functions is presented, from which it is shown clearly how long range order emerges as the decrease of the temperature. Some properties are discussed.

Appendices

Appendix A: The proof of detξ = 1

At first, according to (2.16) and (2.19), the matrix ξ is expressed in terms of 2 × 2 blocks ξ _{r s} , 1 ≤ r, s ≤ N, given by

$$\begin{array}{@{}rcl@{}} \begin{array}{llll} {\boldsymbol{\xi }}_{1r} =\varOmega_{r-1} \left[ {{\begin{array}{lllll} 1 & {-1} \\ {\frac{\sinh \gamma_{r-1} }{\sinh 2{K}^{\prime}\cosh K}} & {\frac{\sinh \gamma_{r-1} }{\sinh 2{K}^{\prime}\cosh K}} \\ \end{array} }} \right] , \\ {\boldsymbol{\xi }}_{nr} =\varOmega_{r-1} \left[ {{\begin{array}{lllll} {G_{n-1} (x_{r-1} )\cosh K} & {-G_{n-1} (x_{r-1} )\cosh K} \\ {\frac{\sinh \gamma_{r-1} g_{n-1} (x_{r-1} )}{\sinh 2{K}^{\prime}\cosh K}} & {\frac{\sinh \gamma_{r-1} g_{n-1} (x_{r-1} )}{\sinh 2{K}^{\prime}\cosh K}} \\ \end{array} }} \right] , \\ {\boldsymbol{\xi }}_{Nr} =\varOmega_{r-1} \left[ {{\begin{array}{lllll} {G_{N-1} (x_{r-1} )\cosh K} & {-G_{N-1} (x_{r-1} )\cosh K} \\ {\frac{\sinh \gamma_{r-1} g_{N-1} (x_{r-1} )}{\sinh 2{K}^{\prime}}} & {\frac{\sinh \gamma_{r-1} g_{N-1} (x_{r-1} )}{\sinh 2{K}^{\prime}}} \\ \end{array} }} \right] . \\ {\kern50pt}(n=2,3,{\cdots} ,N-1 ; r=1,2,\cdots ,N) \\ \end{array} \end{array} $$

Using some basic properties of determinant, for example, “If the elements of a line are added to the elements of another parallel line previously multiplied by a real number, the value of the determinant is unchanged” (When we use this property for ξ, we must notice the fact that all x _{r − 1} and γ _{r − 1} in one column in all elements are the same but vary with the element in one row of ξ), etc, for ξ we obtain

$$\det {\boldsymbol{\xi }}=(\cosh K)^{N}\left( {\prod\limits_{r=1}^{N} {\frac{2\sinh \gamma_{r-1} }{\sinh 2{K}^{\prime}\cosh K}} } \right)\left( {\prod\limits_{r=1}^{N} {\varOmega_{r-1}^{2} } } \right)\det {\mathbf{{\xi }^{\prime}}}, $$

where the matrix ξ′ is expressed in terms of 2 × 2 blocks ξ′ _{r s} , 1 ≤ r, s ≤ N, given by

$${\mathbf{{\xi }^{\prime}}}_{1r} =\left[ {{\begin{array}{lllll} 1 & 0 \\ 0 & 1 \\ \end{array} }} \right] , {\mathbf{{\xi }^{\prime}}}_{nr} =\left[ {{\begin{array}{lllll} {G_{n-1} (x_{r-1} )} & 0 \\ 0 & {g_{n-1} (x_{r-1} )} \\ \end{array} }} \right] (n=2,3,{\cdots} ,N ; r=1,2,\cdots ,N). $$

We therefore have detξ′ = (detp)(detq), the elements of the two N × N matrices p and q are

$${\mathbf{p}}_{1r} =1, \quad {\mathbf{p}}_{nr} =G_{n-1} (x_{r-1} ); \quad {\mathbf{q}}_{1r} =1, \quad {\mathbf{q}}_{nr} =g_{n-1} (x_{r-1} ) (n=2,3,\cdots ,N ; r=1,2,{\cdots} ,N). $$

We now cite a mathematical theorem:

For n polynomials $P_{i} (x) = \sum \limits _{j=0}^{i} {a_{ij} x^{i-j}}$, a _i0 ≠ 0, i = 0, 1, ⋯ , n − 1, the determinant

$$\det \left[ {{\begin{array}{lllll} {P_{0} (x_{1} )} & {P_{0} (x_{2} )} & {\cdots} & {P_{0} (x_{n-1} )} & {P_{0} (x_{n} )} \\ {P_{1} (x_{1} )} & {P_{1} (x_{2} )} & {\cdots} & {P_{1} (x_{n-1} )} & {P_{1} (x_{n} )} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & \vdots \\ {P_{n-2} (x_{1} )} & {P_{n-2} (x_{2} )} & {\cdots} & {P_{n-2} (x_{n-1} )} & {P_{n-2} (x_{n} )} \\ {P_{n-1} (x_{1} )} & {P_{n-1} (x_{2} )} & {\cdots} & {P_{n-1} (x_{n-1} )} & {P_{n-1} (x_{n} )} \\ \end{array} }} \right]=\left( {\prod\limits_{i=0}^{n-1} {a_{i0} } } \right)\left( {\prod\limits_{1\le j<i\le n}^{n} {\left( {x_{i} -x_{j} } \right)}} \right). $$

The above theorem is an exercise in a textbook and can be proved easily based on some basic properties of determinant and the Vandermonde determinant.

According to (2.12) and (2.17), both g _{n − 1}(x _{r − 1}) and G _{n − 1}(x _{r − 1}) are (n − 1)−th order monic polynomials in x _{r − 1}, hence, using the above mathematical theorem, we obtain $\det {\mathbf {p}}=\det {\mathbf {q}}=\prod \limits _{1\le r<{r}^{\prime }\le N}^{N} {\left ({x_{{r}^{\prime }-1} -x_{r-1} } \right )} .$

Combining the above results, we obtain

$$\det {\boldsymbol{\xi }}=(2\text{csch}2{K}^{\prime})^{N}\left( {\prod\limits_{r=1}^{N} {\sinh \gamma_{r-1} } } \right)\left( {\prod\limits_{r=1}^{N} {\varOmega_{r-1}^{2} } } \right)\left( {\prod\limits_{1\le r<{r}^{\prime}\le N}^{N} {\left( {x_{{r}^{\prime}-1} -x_{r-1} } \right)} } \right)^{2}>0. $$

On the other hand, since ξ is orthogonal, detξ = ±1, we therefore conclude that detξ = 1.

Appendix B: Some Properties of the Equation (2.15)

Substituting (2.10) to (2.15), (2.15) becomes

$$\begin{array}{@{}rcl@{}} d^{-N}f(d)\!&=&\! 0; \end{array} $$

(B.1)

$$\begin{array}{@{}rcl@{}} f(d)\!&=\!&\frac{d^{2N}\left({d^{2}\! -\! 2d\coth 2{K}^{\prime}\tanh K\! +\!\tanh^{2}K} \right)\! -\!\left( {d^{2}\tanh^{2}K\! -\! 2d\coth 2{K}^{\prime}\tanh K\! +\! 1} \right)}{d^{2}-1} \end{array} $$

(B.1)

$$\begin{array}{@{}rcl@{}} &=\!&\frac{d^{2N}-1}{d^{2}-1}\left( {d-\coth {K}^{\prime}\tanh K} \right)\left( {d-\tanh {K}^{\prime}\tanh K} \right)+\text{sech}^{2}K \end{array} $$

(B.1)

$$\begin{array}{@{}rcl@{}} &=&\!\left( {\sum\limits_{k=0}^{N-1} {d^{2k}} } \right)\left( {d-\coth {K}^{\prime}\tanh K} \right)\left( {d-\tanh {K}^{\prime}\tanh K} \right)+\text{sech}^{2}K . \end{array} $$

(B.1)

At first, according to (B.1), it is easy to prove the following property:

• P1. If d is a root of the equation (B.1), then d ⁻¹ is also a root of the equation (B.1).

According to (B.1) we have

$$ f(\tanh {K}^{\prime}\tanh K) = f(\coth {K}^{\prime}\tanh K) = \text{sech}^{2}K>0, $$

(B.5)

combining (B.1) and (B.5) we conclude that when d ≤ tanhK′ tanhK, or d ≥ cothK′ tanhK, f(d) > 0. Hence, only in the range tanhK′ tanhK < d < cothK′ tanhK, maybe f(d) ≤ 0.

Therefore, if d ₀ is a real root of the equation (B.1), i.e., f(d ₀) = 0, then tanhK′ tanhK < d ₀ < cothK′ tanhK, from which we obtain $\tanh {K}^{\prime }\coth K<d_{0}^{-1} <\coth {K}^{\prime }\coth K$. On the other hand, since $d_{0}^{-1}$ is also a root of (B.1), for which we still have $\tanh {K}^{\prime }\tanh K<d_{0}^{-1} <\coth {K}^{\prime }\tanh K$. Combining these inequalities, we conclude that if the equation (B.1) has two positive real roots d ₀ and $d_{0}^{-1}$, then

$$ \tanh {K}^{\prime}\coth K<d_{0}^{-1} <1<d_{0} <\coth {K}^{\prime}\tanh K. $$

(B.6)

When K′ ≥ K, we have tanh2K′ ≥ tanh2K, i.e., tanhK′ cothK ≥ cothK′ tanhK, this inequality is just contrary to (B.6). We conclude that

• P2. If K′ ≥ K, then (B.1) has not real root.

Therefore, if the (B.1) has at least two positive real roots d ₀ and $d_{0}^{-1}$, then K > K′. For this case, by assuming d ₀ = e^χ, χ > 0, according to (2.10), $g_{n} =\frac {\sinh (n+1)\chi }{\sinh \chi }$, and, further,

$$ \begin{array}{l} g_{N} -2g_{N-1} \coth 2{K}^{\prime}\tanh K+g_{N-2} \tanh^{2}K \\ =\frac{\sinh N\chi \left( {\cosh \chi \sinh 2{K}^{\prime}\cosh 2K-\cosh 2{K}^{\prime}\sinh 2K} \right)+\cosh N\chi \sinh \chi \sinh 2{K}^{\prime}}{\sinh \chi \sinh 2{K}^{\prime}\cosh^{2}K} . \\ \end{array} $$

(B.7)

Hence, through introducing two functions

$$f_{1} (\chi ) = \frac{\tanh N\chi }{\tanh \chi }, \quad f_{2} (\chi ) = \frac{\sinh 2{K}^{\prime}}{\sinh 2K\cosh 2{K}^{\prime}\text{sech}\chi -\cosh 2K\sinh 2{K}^{\prime}}, $$

according to (B.7), the eigenvalue equation (2.15) is written to the form

$$ f_{1} (\chi ) = f_{2} (\chi ). $$

(B.8)

Notice that f ₁(χ) > 0, from (B.8) we have f ₂(χ)( = f ₁(χ)) > 0, according to the above definition of f ₂(χ), we obtain

$$ \cosh \chi <\coth 2{K}^{\prime}\tanh 2K. $$

(B.9)

This is a limitation condition of the real roots of the eigenvalue equation (2.15).

Considering (B.9), some special values of f ₁(χ) and f ₂(χ) are

$$\begin{array}{@{}rcl@{}} f_{1} (0) = N, \quad &&\lim\limits_{\chi \to \infty } f_{1} (\chi ) = 1; \quad f_{2} (0) = \sinh 2{K}^{\prime}\text{csch}2(K-{K}^{\prime}), \quad \\&&\lim\limits_{\chi \to \mathrm{arccosh}(\coth 2{K}^{\prime}\tanh 2K)^{\pm }}f_{2} (\chi ) = \mp \infty ; \end{array} $$

On the other hand, it is easy to prove that f ₁(χ) is strictly monotone decreasing in [0, ∞) and f ₂(χ) is strictly monotone increasing in [0,arccosh(coth2K′ tanh2K)). Combining these results we obtain

• (i) If the two curves f ₁(χ) and f ₂(χ) have intersection points, then there is only one intersection point, which is in [0, arccosh(coth 2K′ tanh 2K));

• (ii) f ₁(χ) and f ₂(χ) have (one) intersection point if and only if

$$ N\ge \sinh 2{K}^{\prime}\text{csch}2(K-{K}^{\prime}). $$

(B.10)

The above properties can be observed visually in Fig. 1.

Fig. 1

The two curves f ₁(χ) and f ₂(χ)

The (B.8) has solution is equivalent to that the two curves f ₁(χ) and f ₂(χ) have intersection point, we therefore have

• P3. When K > K′, the equation (B.1) has no negative real root and has at most two positive real roots d ₀ and $d_{0}^{-1}$, which satisfy (B.6).

On the other hand, notice N ≥ 1, hence, when K ≥ 2K′, (B.10) holds absolutely, we therefore have

• P4. If 2K′ > K > K′, two positive real roots of (B.1) arise if and only if (B.10) holds.

• P5. If K ≥ 2K′, then (B.1) has always two positive real roots, no matter what value of N takes.

According to P3 we see that most of the roots of (B.1) are complex. On the other hand, since eigenvalue of the Hermitian matrix ω must be real, according to (2.14), any root $x=\frac {d+d^{-1}}{2}$ of the equation (2.15) must also be real, hence, if d is a complex root of (B.1), then d must has the form e^iϕ, that is to say,

• P6. All those complex roots of (B.1) are on the unit circle.

If d = e^iϕ is a complex root of (B.1), then $x=\frac {d+d^{-1}}{2}=\cos \phi$, this is just (2.22).

If d ₀ is a positive real root of (B.1), then we can use the following method to obtain the approximate value of d ₀.

For this case we have concluded K > K′, the quantity

$$\kappa =\coth K\tanh {K}^{\prime} $$

(B.11)

thus satisfies 0 < κ < 1. According to (B.6), $1<d_{0} <\coth {K}^{\prime }\tanh K=\frac {1}{\kappa }$, we can try to find the value of d ₀ shaped as

$$d_{0} =\coth {K}^{\prime}\tanh K\left( {1-\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} } \right) = \frac{1}{\kappa }\left( {1-\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} } \right). $$

(B.12)

According to (B.1) and (B.1), d ₀ must satisfy

$$\frac{{d_{0}^{2}} -2d_{0} \coth 2{K}^{\prime}\tanh K+\tanh^{2}K}{{d_{0}^{2}} \tanh^{2}K-2d_{0} \coth 2{K}^{\prime}\tanh K+1}=\frac{1}{d_{0}^{2N} }. $$

(B.13)

Substituting (B.12) to (B.13), we obtain

$$\begin{array}{l} \frac{\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} -\cosh^{2}{K}^{\prime}\left( {\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} } \right)^{2}}{2\frac{\cosh 2K-\cosh 2{K}^{\prime}}{\sinh^{2}2K}\left( {1-2\frac{\sinh^{2}K\left( {\cosh 2{K}^{\prime}-\sinh^{2}K} \right)}{\cosh 2K-\cosh 2{K}^{\prime}}\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} -\frac{2\cosh^{2}{K}^{\prime}\sinh^{4}K}{\cosh 2K-\cosh 2{K}^{\prime}}\left( {\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} } \right)^{2}} \right)} \\ =\frac{\kappa^{2N}}{\left( {1-\sum\limits_{n=1}^{\infty} {a_{n} \kappa^{n}} } \right)^{2N}}.\\ \end{array} $$

(B.14)

We see that the first a _n that does not vanish is $a_{2N} =2\frac {\cosh 2K-\cosh 2{K}^{\prime }}{\sinh ^{2}2K}$, accordingly,

$$d_{0} \approx \coth {K}^{\prime}\tanh K\left( {1-2\frac{\cosh 2K-\cosh 2{K}^{\prime}}{\sinh^{2}2K}\kappa^{2N}} \right). $$

(B.15)

κ ^2N is a very small quantity when N is very large, d ₀ thus very nears to cothK′ tanhK when N is very large. On the other hand, it is not difficult to obtain higher approximation value of d ₀ in terms of (B.14) for finite value of N.

For such d ₀, according to (B.9), $x_{0} =\frac {d_{0} +d_{0}^{-1}}{2}=\cosh \chi$ satisfies 1 < x ₀ < coth2K′ tanh2K, of which the approximation value is given by (2.24).