Correspondence between spanning trees and the Ising model on a square lattice

TL;DR

This paper reveals a deep mathematical connection between the number of spanning trees, the Ising model partition function, and Mahler measures on a square lattice, extending known relations across all temperatures.

Contribution

It generalizes the relationship between spanning trees and the Ising model partition function to all temperatures on the square lattice, linking them through Mahler measures and the random walk structure function.

Findings

Established a formula relating Z(K) and T(k) for all temperatures.

Connected Mahler measure to lattice Green functions and spanning tree enumeration.

Showed the relation does not extend straightforwardly to non-planar lattices.

Abstract

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function $T(z)$ gives the spanning tree constant when evaluated at $z=1$, while giving he lattice green function when differentiated. It is known that for the infinite square lattice the partition function $Z(K)$ of the Ising model evaluated at the critical temperature $K=K_c$ is related to $T(1)$. Here we show that this idea in fact generalizes to all real temperatures. We prove that $ ( Z(K) {\rm sech~} 2K ~\!)^2 = k \exp\big[ T(k) \big] $, where $k= 2 \tanh(2K) {\rm sech}(2K)$. The identical Mahler measure connects the two seemingly disparate quantities $T(z)$ and $Z(K)$. In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.

Equations

$${\lambda }_{\mathcal{L}}=\underset{N\to \infty }{\lim }\frac{\log n_{\mathcal{L}}(N)}{N}\text{leavevmode}.$$

$$\lambda =\log {\mathrm{z}}_{\mathcal{L}}+\frac{1}{(2\pi {)}^d}{\int }_{-\pi }^{\pi }\dots {\int }_{-\pi }^{\pi }\log [1-{\Lambda }_{\mathcal{L}}(\vec{k})]\text{leavevmode}d{k}_1\dots d{k}_d\text{leavevmode}.$$

$$\Lambda (\vec{k})=\frac{1}{2}(\cos k_1+\cos k_2)\text{leavevmode}.$$

$$\lambda$$

$$=\frac{4G}{\pi }\approx 1.166$$

$$\log Z(K)$$

$$+\frac{1}{8{\pi }^2}\underset{-\pi \text{leavevmode}}{\stackrel{\text{leavevmode}\pi }{\iint }}\log [1-\frac{k}{2}(\cos k_1+\cos k_2)]\text{leavevmode}d{k}_{1}d{k}_2$$

$$k=2\tanh 2K\mathrm{sech}2K\text{leavevmode}.$$

$$\log Z(K_c)=\frac{1}{2}\log 2+\frac{2G}{\pi }\text{leavevmode}.$$

$$[Z(K_c){]}^2=2e^{\lambda }\text{leavevmode}.$$

$$T_{\mathcal{L}}(z)=\log {\mathrm{z}}_{\mathcal{L}}+\frac{1}{(2\pi {)}^d}{\int }_{-\pi }^{\pi }d{k}_1\dots {\int }_{-\pi }^{\pi }d{k}_d\text{leavevmode}\log [1/z-{\Lambda }_{\mathcal{L}}(\vec{k})]\text{leavevmode}.$$

$$-z\frac{d{T}_{\mathcal{L}}(z)}{dz}=\frac{1}{(2\pi {)}^d}{\int }_{-\pi }^{\pi }d{k}_1\dots {\int }_{-\pi }^{\pi }d{k}_d\text{leavevmode}\frac{1}{1-z{\Lambda }_{\mathcal{L}}(\vec{k})}\text{leavevmode}.$$

$$T(z)=\log 4+\frac{1}{4{\pi }^2}\underset{-\pi \text{leavevmode}}{\stackrel{\text{leavevmode}\pi }{\iint }}\log [\frac{1}{z}-\frac{1}{2}(\cos k_1+\cos k_2)]d{k}_{1}d{k}_2\text{leavevmode}.$$

$$\big{(}Z(K)\operatorname{sech}2K\big{)}^{2}=k\exp\big{[}T(k)\big{]}\leavevmode\nobreak\ ,$$

$$\rm m[P]=\int_{0}^{1}\ldots\int_{0}^{1}\log\big{|}P(e^{2\pi i\theta_{1}},\ldots,e^{2\pi i\theta_{n}})\big{|}\leavevmode\nobreak\ d\theta_{1}\ldots d\theta_{n}\leavevmode\nobreak\ .$$

$$p(\zeta )=\zeta +z_1+z_1^{-1}+z_2+z_2^{-1}$$

$$\mathrm{m}[p(\zeta )]=\frac{1}{4{\pi }^2}\underset{-\pi \text{leavevmode}}{\stackrel{\text{leavevmode}\pi }{\iint }}\log [\zeta +2(\cos k_1+\cos k_2)]d{k}_{1}d{k}_2\text{leavevmode}\text{leavevmode}.$$

$$p(\zeta )=\zeta +4\Lambda (\vec{z})\text{leavevmode},$$

$$\log (\frac{1}{2}{Z}_{\text{mbox}Tri})$$

$$-{\sinh }^{2}2K(\cos k_1+\cos k_2)$$

$$\displaystyle\quad\quad\quad-\sinh^{2}2K\cos(k_{1}+k_{2})\bigg{\}}\leavevmode\nobreak\ dk_{1}dk_{2}\leavevmode\nobreak\ .$$

$$T_{\text{mbox}Tri}$$

$$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\cos(k_{1}+k_{2})\big{)}\bigg{\}}\leavevmode\nobreak\ dk_{1}dk_{2}\leavevmode\nobreak\ .$$

$${\Lambda }_{\text{mbox}Tri}(\vec{k})=\frac{1}{3}(\cos k_1+\cos k_2+\cos (k_1+k_2))\text{leavevmode}.$$

$$p_{\text{mbox}Tri}(\zeta )=\zeta +z_1+z_1^{-1}+z_2+z_2^{-1}-z_1{z}_2-(z_1{z}_2{)}^{-1}$$

$$\Lambda (k_1,k_2,k_3)=\frac{1}{3}(\cos k_1+\cos k_2+\cos k_3)\text{leavevmode}.$$

$$s(\zeta )=\zeta +z_1+z_1^{-1}+z_2+z_2^{-1}+z_3+z_3^{-1}\text{leavevmode}.$$

$$T(z)=\log\frac{4}{z}-{z^{2}\over 8}{{}_{4}}F_{3}\left[\begin{array}[]{c}{1,1,{3\over 2},{3\over 2}}\\ {2,2,2}\end{array};z^{2}\right]\leavevmode\nobreak\ .$$

$$\ln Z(K)$$

$$2\kappa =\tanh (2K)\mathrm{sech}(2K)\text{leavevmode}.$$

$$-\kappa^{2}{{}_{4}}F_{3}\left[\begin{array}[]{c}{1,1,{3\over 2},{3\over 2}}\\ {2,2,2}\end{array};16\kappa^{2}\right]=\frac{1}{2}\big{(}T(4\kappa)+\log\kappa\big{)}\leavevmode\nobreak\ .$$

$$Z(K)$$

Full text

I Introduction

There is continued interest in exploring the fascinating connections between partition functions of lattice models in statistical physics on the one hand and graph theoretical enumeration problems on the other. In the 1960s, the seminal chapter by Kasteleyn in Harary’s Graph Theory and Statistical Physics gttp led to widespread recognition of the effectiveness of graph enumeration methods for solving combinatorial problems in statistical mechanics. In the 1970s, Wu showed that the number of spanning trees is related to the partition function of ice-type models wu77 . On the square lattice, for example, it is known that the spanning tree constant is related to the partition function of the critical Ising model, since both can be expressed in terms of Catalan’s constant. By the late 1980s, however, interest in exactly solvable models had to some extent become overshadowed by the appeal of concepts such as universality, scaling, and the renormalization group stanley-rmp1999 ; fisher-rmp1998 , which led to advances not only within the field of statistical physics but more generally in the interdisciplinary study of complex systems csbook1 ; csbook2 and complex networks cnbook1 . Nevertheless, there remain questions concering exactly solvable models in statistical mechanics (and their graph theoretical analogs) that cannot be tackled using scaling laws and related methods.

Here we revisit the spanning tree problem and ask whether or not the relationship between spanning tree enumeration and the Ising model can be generalized to non-critical temperatures. This question is (almost certainly) impossible to answer using renormalization group methods or scaling analysis, due to breakdown of scale invariance symmetry far from critical points. We instead take a different approach. In a sense that we make precise below, we directly compare the generating functions related to certain types of graphs, viz., spanning trees and multipolygons.

The main results we report here represent an advance for the following reasons: (i) Theorem 1 below sheds light on how the partition function of the Ising model on the square lattice is related to a corresponding generating function for spanning trees, via the random walk structure function and associated Mahler measures; (ii) we report preliminary results that strongly suggest that the theorem generalizes to other planar lattices; and finally (iii) our results have a bearing on the question of why the Ising problem is notoriously difficult for non-planar lattices.

In Section II we report the key advances. In Section III we interpret the results in terms of the underlying physics. Finally in Section IV we discuss possible generalizations.

II Method and Results

We first review the definition of the spanning tree constant. Trees are connected graphs that contain no loops (i.e., polygons). A spanning tree on a given graph is a tree that connects every node on that graph. The number $n_{\mathcal{L}}$ of spanning trees on a regular lattice $\mathcal{L}$ grows exponentially in the number $N$ of lattice sites. Hence, the following limit is well defined:

[TABLE]

The number $\lambda_{\mathcal{L}}$ is known as the spanning tree constant for the lattice ${\mathcal{L}}$.

The spanning tree constant also bears a connection with lattice models. Aiming to study phase transitions, Fortuin and Kasteleyn in the 1970s introduced a model that covers many others as special cases, including the Ising model, the Ashkin-Teller-Potts model and percolation fortuin-kasteleyn . Using this approach, Wu subsequently showed that $\lambda_{\mathcal{L}}$ is expressible in terms of a partition function of a lattice model, and thereby was able to calculate the exact spanning tree constant on several planar lattices wu77 . In what follows, we adopt some of the notation of ref. arXiv:1207.2815 . Let $\Lambda_{\mathcal{L}}$ denote the random walk structure function for the lattice $\mathcal{L}$, i.e., the Fourier transform of the discrete step probability distribution, and let ${\rm z}_{\mathcal{L}}$ denote the coordination number of the lattice. Then the spanning tree constant of a $d$-dimensional regular lattice can be expressed as

[TABLE]

Specializing to the infinite square lattice, the random walk structure function is given by

[TABLE]

Hence, on the square lattice, the spanning tree constant is given by

[TABLE]

where $G\approx 0.915$ is Catalan’s constant.

In 1944, Onsager had already shown that Catalan’s constant appears in the expression for the partition function of the critical Ising model ons . Let $Z(K)$ denote the partition function of the Ising model on the infinite square lattice:

[TABLE]

where

[TABLE]

The critical temperature $K_{c}=\operatorname{arctanh}(\sqrt{2}-1)$ was first found by Kramers and Wannier kw , and corresponds to the condition $k=1$. Subtituting $K=K_{c}$ into (6) gives us

[TABLE]

Eliminating Catalan’s constant from (5) and (8), we find that $\lambda$ and $Z(K_{c})$ are related by

[TABLE]

This result is of course not new. Indeed, it is well known to specialists that the critical Ising model is related to the problem of enumerating of spanning trees (see, e.g., ref. deTiliere01 and Eq. (57) in ref. arXiv:1207.2815 ).

The question we address here is whether the above relation holds only at the critical point or whether it generalizes in some way. Specifically, we ask here whether or not (9) can be extended to non-critical temperatures. It is well known that the partition function of the Ising model is closely related to the generating function of “loops” or multipolygons on the square lattice, i.e. those graphs all whose nodes have even degree and whose edges connect only nearest neighbors. The critical temperature $K_{c}$ correponds to weighting the edges of multipolygons in such a way that multipolygons of small and large size make comparable overall contributions. The form of Eq. (9) strongly suggests that spanning tree constant is related to some “spanning tree generating function” evaluated at a special critical value of some tunable parameter. This generating function must generalize the spanning tree constant, such that when evaluated at other values of the parameter, it should be related to (6) with $K\neq K_{c}$. What is the correct or “natural” way to define this spanning tree generating function that generalizes the spanning tree constant? Fortunately, the answer is known rosengren ; arXiv:1207.2815 .

The spanning tree generating function $T_{\mathcal{L}}(z)$ was defined by Guttmann and Rogers arXiv:1207.2815 as

[TABLE]

Clearly, $T_{\mathcal{L}}(1)=\lambda_{\mathcal{L}}$. Moreover, differentiating one obtains

[TABLE]

This integral is the Lattice Green function lgf , whose value at $z=1$ is related to the probability of a random walker to return to the origin.

On the square lattice, the spanning tree generating function is given by

[TABLE]

The integral that appears in the above expression is, up to constants, identical to the one in (6). It thus follows that $\log(Z(K)\operatorname{sech}2K)=\tfrac{1}{2}(T(k)+\log k)\leavevmode\nobreak\ ,$ from which we immediately obtain the following new result:

Theorem 1**.**

Let $T(z)$ be the the spanning tree generating function given by (12) for the infinite square lattice and let $Z(K)$ be the partition function of the Ising model $Z(K)$ given by (6). Then, for all $K\in\mathbb{R}$, the following identity holds,

[TABLE]

with $k$ given by (7).

The identity (13) is the correct generalization of (9) to non-critical temperatures $K\neq K_{c}$. We originally arrived at Theorem 1 by noting that $T(z)$ and $Z(K)$ can both be written in terms of the same ${}_{4}F_{3}$ generalized hypergeometric function. See the appendix for details.

III Discussion

The physics that underpins Theorem 1 can be understood in terms of random walks. We briefly discuss this relationship. The integrals that appear in the expressions for $T(z)]$ and $Z(K)$ can be written in terms of a Mahler measure (for $T$ and $Z$ real). The (logarithmic) Mahler measure of a Laurent polynomial $P(z_{1},\ldots,z_{n})$ is conventionally defined according to

[TABLE]

If we choose $P$ to be the symmetric 2-variable Laurent polynomial

[TABLE]

where $\zeta$ is not integrated over, we obtain the required integral:

[TABLE]

For example, we find $T(z)={\rm m}[p(4/z)]$ for $z\in\mathbb{R}\backslash\{0\}$. The reason the identical Mahler measure is found in both $T(z)$ and $Z(K)$ is that they can both be calculated in terms of a random walk on the lattice, as is well known. The polynomial $p(\zeta)$ is in fact determined by the random walk structure function $\Lambda(\vec{k})$ for the square lattice:

[TABLE]

with $\Lambda$ given by (3) and $\vec{z}$ denoting the vector with components $z_{1}$ and $z_{2}$. Hence, the connection between the spannng tree generating function and the Ising model is ultimately due to the fact that both share a mathematical relationship with the random walk structure function for the square lattice.

IV Conclusion

We have generalized the relation between the spanning tree generating function and the partition function of the Ising model to all real temperatures. We recover the known relation between the spanning tree constant and the critical Ising model as a special case of our more general result. Does Theorem 1 generalize to other lattices? For planar lattices, one expects similar results to hold (and we hope to systematically investigate such lattices when time permits).

As evidence in support of this idea, we cite the the triangular lattice. The partition function $Z_{\mbox{\tiny Tri}}$ of the Ising model on the triangular lattice is known to be given by montroll ,

[TABLE]

The spanning tree generating function for this lattice is given by arXiv:1207.2815 ,

[TABLE]

Note that if we restrict $z$ and $K$ to real values, then we can attempt to express both $Z_{\mbox{\tiny Tri}}$ and $T_{\mbox{\tiny Tri}}$ in terms of the same Mahler measure. This is possible by noting that both both integrals can be written in terms of the identical structure function

[TABLE]

A suitable Mahler measure is obtained by choosing $P$ to be the Laurent polynomial

[TABLE]

where $\zeta$ is not integrated over in (14). Hence, the correspondence between the Ising model and spanning trees generalizes to the triangular lattice (and very likely also to other planar lattices).

For nonplanar lattices, however, the issue is not at all clear. Consider for example the simple cubic lattice. The random walk structure function and the spanning tree generating function are known rosengren ; arXiv:1207.2815 . Specifically, the structure function for the simple cubic lattice is given by

[TABLE]

Hence, the spanning tree generating function for the cubic lattice can be expressed in terms of the corresponding Mahler measure. The relevant Mahler measure is the one given by choosing $P$ to be the symmetric 3-variable Laurent polynomial

[TABLE]

It is possible to show that this Mahler measure cannot be reconciled with the known series expansion of the partition function of the cubic Ising model, in a manner analogous to the two dimensional case. Specifically, it is easy to check that if $\zeta$ is chosen to be an arbitrary rational function in the high temperature variable $t=\tanh\beta J$ then one cannot recover the known high temperature series expansion for the partition function, unlike the case for the two dimensional Ising model. Hence, there can be no simple relation between the partition function of the Ising model and the spanning tree generating function on the cubic lattice. (Indeed, this should be obvious, since otherwise the cubic Ising model would have been solved long ago.)

Nevertheless, there are a number of related unanswered questions, for instance, might it be possible to rule out all Mahler measures for the Ising model on non-planar lattices? Currently, it is widely believed that the partition functions for non-planar lattice models are not D-finite, whereas Mahler measures by definition are D-finite. Without proofs, of course, such statements remain conjectural. These and related questions questions remain open and the posibilities are intriguing. We believe they merit further study and hope to investigate them in the future.

Acknowledgments

We thank CNPq for funding. We thank MGE da Luz, RTG de Oliveira and EP Raposo for very helpful comments.

Appendix A Proof of Theorem 1 via ${}_{p}F_{q}$ functions

On the one hand, it is known from Eq. (23) in ref. arXiv:1207.2815 that

[TABLE]

On the other hand, it is known from Eq. (13) in ref. viswan2015-jstat that

[TABLE]

with $\kappa$ defined as

[TABLE]

Putting $z=4\kappa$ in (24) gives us

[TABLE]

Substituting this last expression into (27) to eliminate the ${}_{4}F_{3}$ function, and then exponentiating, we obtain

[TABLE]

The claim follows from noting that $4\kappa=k$. $\square$