Complex-Temperature Singularities of Ising Models

TL;DR

This paper explores the complex-temperature phase diagrams and singularities of various two-dimensional Ising models, including higher-spin and external magnetic field cases, using partition function zeros and series analysis.

Contribution

It provides new insights into the complex-temperature properties and phase diagrams of 2D Ising models, including exact results and zero analyses for higher-spin and magnetic field cases.

Findings

Elucidation of complex-T phase diagrams for higher-spin Ising models.

Exact results for phase diagrams at specific magnetic field values.

Determination of susceptibility exponents at singularities.

Abstract

We report new results on complex-temperature properties of Ising models. These include studies of the $s=1/2$ model on triangular, honeycomb, kagom\'e, $3 \cdot 12^2$, and $4 \cdot 8^2$ lattices. We elucidate the complex--$T$ phase diagrams of the higher-spin 2D Ising models, using calculations of partition function zeros. Finally, we investigate the 2D Ising model in an external magnetic field, mapping the complex--$T$ phase diagram and exploring various singularities therein. For the case $\beta H=i\pi/2$, we give exact results on the phase diagram and obtain susceptibility exponents $\gamma'$ at various singularities from low-temperature series analyses.

Equations

$$\mathcal{H}=-J\sum _{<ij>}{\sigma }_i{\sigma }_j-H\sum _i{\sigma }_i$$

$$K=\beta J,h=\beta H$$

$$z=e^{-2K},u=e^{-4K},\mu =e^{-2h}$$

$$1/u_e\le u\le u_e$$

$$u_e=-(3-2^{-3/2})$$

$${\gamma }_e^{\prime }=\frac{5}{4}$$

$${\gamma }_1^{\prime }=\frac{5}{2}$$

$${\gamma }_s^{\prime }=1$$

$$u=\frac{1}{2}(-1+e^{i\theta }),{\theta }_{ce}\le |\theta |\le \pi$$

$$-\infty \le u\le -1/2$$

Full text

1 Introduction

During the past year, in collaboration with Victor Matveev, we have carried out a number of studies of complex-temperature (CT) properties of Ising models [1]-[7]. We report on some results from these here (see also [8, 9]). There are several reasons for studying the properties of statistical mechanical models with the temperature variable generalized to take on complex values. First, one can understand more deeply the behavior of various thermodynamic quantities by seeing how they behave as analytic functions of complex temperature. Indeed, complex–$T$ singularities have important influences on the behavior of such functions on the real $K$ axis (see definition in eq. (2)). Second, one can see how the physical phases of a given model generalize to regions in appropriate complex-temperature variables. Third, a knowledge of the complex-temperature singularities of quantities which have not been calculated exactly helps in the search for exact, closed-form expressions for these quantities. This applies, in particular, to the susceptibility of the 2D zero-field Ising model, and to the Ising model for higher spin or external magnetic field, neither of which has been solved above $d=1$. Some of the earliest works on complex–$T$ properties of the Ising model include Refs. [10]-[13]. For comparison, a different direction is to study the case in which the temperature is kept physical but the external field $H$ is complex [14]. We also report new results which both $K$ and $H$ are complex.

2 Model

The Ising model, and our notation, are specified as follows: the partition function is $Z=\sum_{\{\sigma_{i}\}}e^{-\beta{\cal H}}$ with

[TABLE]

where $\sigma_{i}=\pm 1$, $\beta=(k_{B}T)^{-1}$, and $H$ denotes a possible external field (zero unless otherwise indicated). We use the standard notation

[TABLE]

[TABLE]

$v=\tanh K$, and the elliptic modulus variable $k_{<}=4u/(1-u)^{2}$. We define the reduced free energy as $f=-\beta F=\lim_{N\to\infty}N^{-1}\ln Z$ and the reduced susceptibility as $\bar{\chi}=\beta^{-1}\chi$.

3 Phase Diagrams (Homopolygonal Lattices)

Owing to the symmetries under the shifts $K\to K+ni\pi$ and $K\to K+(2n+1)(i\pi/2)$ [8], there is an infinite replication of complex phases as functions of $K$; this is reduced to a single set by using the variables $v$ or $z$. For a bipartite lattice, the phase boundaries are invariant under $z\to-z$, so that the most compact way to show them is in terms of the variable $u=z^{2}$. In Ref. [1] we showed that for the square lattice the phase boundaries in the $u$ plane are given by a Pascal limaçon. The inner and outer branches of this limaçon cross at the point $u=-1$.

We have extended this analysis to the triangular and honeycomb lattices [2]. As in the case of the square lattice, the continuous locus of points where $f$ is non-analytic can be determined exactly. ($f$ is also trivially non-analytic at the points $|K|=\infty$, but since these do not separate any phases, they will not be important here.) In contrast to the square lattice, this (continuous) locus of points includes arcs or line segments which protrude into, and terminate within, certain phases and hence do not completely separate any phases. For example, for the triangular lattice, in the $u$ plane, the above locus of points consists of the union of the circle $u=(-1+2e^{i\theta})/3$ and the semi-infinite line segment $-\infty\leq u\leq-1/3$. The interior (exterior) of the above circle forms the FM (PM) phase. $M$ diverges at $u=-1/3$ with exponent $\beta=-1/8$ and is discontinuous across the line segment $-1<u<-1/3$. We have proved that such a divergence in $M$ automatically implies a divergence in $\bar{\chi}$ [2].

4 Heteropolygonal Lattices

While the square, triangular, and honeycomb lattices each involve only one type of regular polygon (and thus may be denoted “homopolygonal”), there are also regular tilings of the plane (heteropolygonal lattices) which involve more than one type of regular polygons, but still have all vertices equivalent, and all bonds of equal length. We have determined the exact complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices: $3\cdot 6\cdot 3\cdot 6$ ($=$ kagomé), $3\cdot 12^{2}$, and $4\cdot 8^{2}$ (bathroom tile), where the notation denotes the regular $n$-sided polygons adjacent to each vertex [4]. We have also worked out the exact complex–$T$ singularities of $M$. In particular, we found the first case (the $4\cdot 8^{2}$ lattice) where, even for isotropic spin-spin exchange couplings, the nontrivial non-analyticities of the free energy of the Ising model lie in a two-dimensional, rather than one-dimensional, algebraic variety in the $z=e^{-2K}$ plane.

5 Complex–$T$ Singularities

Complex–$T$ singularities exhibit a number of new features not observed for physical critical points. First, as we showed by exact results [1], universality, in the sense of lattice-independence of critical exponents, is violated at $u=-1$, since the magnetization exponent $\beta$ is 1/4 for the square lattice but 3/8 for the triangular and honeycomb lattices. Second, on the square lattice, we proved [8] that as one approaches the boundary of the complex–$T$ extension of the PM phase, $\bar{\chi}$ has at most finite non-analyticities except for its divergence at the physical critical point $u_{c}$, and hence, in particular, at $u=-1\equiv u_{s}$, it follows that $\gamma_{s}<0$. Analyses of low-temperature series indicate that $\bar{\chi}$ diverges at $u=-1$ as this point is approached from within the FM phase [15, 1] (and AFM phase [1]), with the exponent $\gamma_{s}^{\prime}$ numerically consistent with 3/2. This value also follows from the relation $\gamma_{s}^{\prime}=2(\gamma^{\prime}-1)$ [1]. Hence, $\gamma_{s}\neq\gamma_{s}^{\prime}$, which is unprecedented for physical critical points. Third, the correlation length exponent $\nu_{s}^{\prime}$ has different values depending on whether one extracts it from non-diagonal correlation functions or diagonal correlation functions. Several scaling relations are violated; for example, $\alpha_{s}+2\beta_{s}+\gamma_{s}\neq 2$ for the singularity at $u=-1$ as approached from within the PM phase. For honeycomb lattice, we have shown [2] that $\alpha^{\prime}+2\beta+\gamma^{\prime}\neq 2$ (the RHS is consistent with being equal to 4) at the CT singularity at $z=-1$, as approached from within the FM phase, and have discussed the origin of this.

6 Higher-Spin Ising Model

We have carried out a study of the complex–$T$ properties of higher-spin 2D Ising models [5]. The method which we use for this purpose is the calculation of complex–$T$ zeros of the partition function on finite lattices. It is known from exactly solved cases that as the lattice size increases, the CT zeros of $Z$ occur on, or progressively closer to, the continuous locus of points where $f$ is non-analytic in the thermodynamic limit. Hence, by studying these zeros, one can make reasonable inferences about this locus of points and the corresponding complex-$T$ phase diagram. Specific results are presented for $s=1$, 3/2, and 2 in Ref. [5]. We infer several general features of the phase diagrams and motivate a conjecture that the number of arcs protruding into the complex-temperature extension of the FM phase is $4[s^{2}]-2$ for $s\geq 1$, where $[x]$ denotes the integral part of $x$. From comparison with an analysis of low-temperature series expansions [16], further support is obtained for our conjecture that the magnetization diverges at endpoints of such arcs terminating in the FM phase. We also present an exact determination of the complex-temperature phase diagram for the 1D spin $s$ Ising model. We find that in the complex $u_{s}=e^{-K/s^{2}}$ plane (i) it consists of $N_{c,1D}=4s^{2}$ infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles $\theta_{n}=(1+2n)\pi/(4s^{2})$, for $n=0,...,4s^{2}-1$; (iii) there is a boundary curve (line) along the negative real $u_{s}$ axis if and only if $s$ is half-integral. An interesting connection is noted between some features of the 1D and 2D complex-temperature phase diagrams for higher-spin Ising models.

7 Finite Magnetic Field

Although the Ising model has never been solved in an arbitrary external magnetic field, Lee and Yang [14] did succeed in solving exactly for the free energy and magnetization for the model on the square lattice for $\beta H\equiv h=i\pi/2$. This solution yields further insight into the structure of the model. It is interesting to investigate the properties of this model when one also generalizes the temperature to complex values, and we have done this [3]. We have first determined the continuous locus of points where $f$ is non-analytic for this model: in the $u$ plane this is the union of (i) the unit circle and (ii) the finite line segment along the negative real axis

[TABLE]

where

[TABLE]

The complex–$T$ phase diagram of this model consists of a FM phase inside of, and an AFM phase outside of, the unit circle in the $u$ plane. We find that the specific heat $C$ has the following singularities: divergences, with exponent $\alpha_{e}^{\prime}=1$ at $u_{e}$ and $1/u_{e}$, and finite logarithmic non-analyticities at $u=-1\equiv u_{s}$ and $u=1$ ($\alpha_{s}^{\prime}=0$, $\alpha_{1}^{\prime}=0$). The uniform and staggered magnetizations $M$ and $M_{st}$ vanish with exponent $\beta_{s}=1/2$ at $u=-1$ and diverge at $u=1$ with exponent $\beta_{1}=-1/4$. $M$ and $M_{st}$ diverge, respectively, at $u_{e}$ and $1/u_{e}$, with exponent $\beta_{e}=-1/8$.

From the Baxter-Enting low-temperature, high-field series expansion of the partition function, we have extracted the low-temperature series for the susceptibility $\bar{\chi}$ to $O(u^{23})$. Analyzing this series, we find that $\bar{\chi}$ has divergent singularities (i) at $u=u_{e}$ with exponent $\gamma_{e}^{\prime}=1.25\pm 0.01$; (ii) at $u=1$, with exponent $\gamma_{1}^{\prime}=2.50\pm 0.01$; and (iii) at $u=u_{s}=-1$, with exponent $\gamma_{s}^{\prime}=1.00\pm 0.08$. We infer the exact values of these exponents to be

[TABLE]

[TABLE]

[TABLE]

We have subsequently determined the exact complex–$T$ phase diagram for the model at $h=i\pi/2$ on the honeycomb ($hc$) and triangular ($t$) lattices [7]. For the $t$ lattice, we find that this consists of a single (FM) phase, and the continuous locus of points where $f$ is non-analytic consists of the union of (i) the circular arc

[TABLE]

where $\theta_{ce}=arctan(4\sqrt{2}/7)\simeq 38.9^{\circ}$, corresponding to the endpoints $u_{ce}=e^{i\theta_{ce}}$ and $u_{ce}^{*}$, where $u_{ce}=(-1+2^{3/2}i)/9$, and (ii) the semi-infinite line segment

[TABLE]

For the $hc$ lattice, we have proved that all of our results on the complex–$T$ phase diagram and the singularities of $\bar{\chi}$ for $h=0$ on the $hc$ lattice also apply for $h=i\pi/2$ with the replacement of $z$ by $-z$.

In Ref. [7] we have recently investigated the complex–$T$ phase diagram of the square-lattice Ising model for nonzero external magnetic field $H$ in the range $-\infty\leq H\leq\infty$, i.e. $0\leq\mu\leq\infty$ where $\mu$ is given by (3). We have also carried out a similar analysis for $-\infty\leq\mu\leq 0$. Without loss of generality, one may restrict to the range $-1\leq\mu\leq 1$. The results for this interval provide a new way of continuously connecting the two known exact solutions of this model, viz., that at $\mu=1$ ($h=0$) and $\mu=-1$ ($h=i\pi/2$). Our methods include calculations of complex-temperature zeros of the partition function and analysis of low-temperature series expansions. For real nonzero $H$, we find that the inner branch of a limaçon bounding the FM phase breaks and forms two complex-conjugate arcs which move away from the real $u$ axis as $|H|$ increases. We have studied the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For $\mu<0$, there are two line segments of singularities on the negative and positive $u$ axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case.

8 Conclusions

Thus, the study of complex-temperature singularities of the Ising model provides many interesting insights. A number of topics deserve further investigation, such as related spin models and behavior in $d>2$.