For the generalized Ising models with all possible interactions within a face of the square lattice the formulas for finding partition function and free energy per lattice site in the thermodynamic limit were derived on a certain, in the general case, 8-dimensional subset of exact disordered solutions of 10-dimensional set of the Hamiltonian's parameters. When a part of parameters are set to zero, as a consequence, the disorder solutions were got for the models with nearest, next-nearest-neighbor and the interaction of four spins in an external field and without an external field, triangular and "checkerboard-triangular" Ising models with triple interactions in an external magnetic field.
Graph theory and Pfaffian representations of Ising partition function
Kasteleyn的一个著名定理指出，任意平面图上的伊辛模型的配分函数可以表示为与该图相关联的斜对称矩阵的Pfweian。这一结果既体现了任何平面伊辛模型的自由费米性质，又给出了计算其配分函数的一种有效的封闭形式。这个结果的扩展到非平面模型表示的配分函数的Pfcans的数目是相关的有向表面上的图形可以嵌入的属的总和。在图论中，McLane定理（1937）给出了平面性作为图的循环空间的一个性质的特征，最近，Diestel等人（2009）将这种方法扩展到任意曲面中的嵌入。在这里，我们表明，麦克莱恩的方法自然会导致Kasteleyn的结果：麦克莱恩表征的平面图正是需要把一个伊辛配分函数到Pfidian。使用这种方法，我们证明了伊辛配分函数的任意非平面图可以被写为\emph{the real part}的Pfuman的一个单一的矩阵的系数在一个多复代数$\C_{\tilde g}$，其中$\tilde g$是不可定向的亏格，或crosscap数，嵌入表面。从这个结果可以得到已知的表示法，即Pfweians和。特别地，Kasteleyn关于实系数$4^g$矩阵，$g$可定向亏格的结果也通过一些代数约化得到了恢复.
The zero-field partition function of two-dimensional nearest neighbor Ising models of square lattices is derived in terms of the generalized hypergeometric series by evaluating the integral in the exact solution of Onsager. An approximate equation for the partition function in terms of Chebyshev polynomials is also provided.
Employing the exact solution of Onsager for two-dimensional Ising models, simple expressions are proposed for computing the partition function, magnetization, specific heat and susceptibility for non-zero magnetic fields of square lattices. The partition function in zero fields is also estimated and excellent agreement with the values arising from the exact solution of Onsager is demonstrated.
Employing heuristic susceptibility equations in conjunction with the well-known critical exponents, the magnetization and partition function for two-dimensional nearest neighbour Ising models are formulated in terms of the Gauss hypergeometric functions. The isomorphism existing between the Bragg-Williams approximation and the exact solution of Onsager is pointed out. The precise manner in which the critical exponents influence the partition functions is pointed out.
We consider the $\pm J$ Ising model on a cubic lattice with a gauge-invariant disorder distribution. Disorder depends on a parameter $β_G$ that plays the role of a chemical potential for the amount of frustration. We study the model at a specific value of the disorder parameter $β_G$, where frustration shows long-range correlations. We characterize the universality class, obtaining accurate estimates of the critical exponents: $ν= 0.655(15)$ and $η_q = 1.05(5)$, where $η_q$ is the overlap susceptibility exponent.
Motivated by the compound ${\rm LiHo}_x{\rm Y}_{1-x}{\rm F}_4$, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attr…
We consider the three-dimensional site-diluted Ising model with power-law correlated defects and study the critical behavior of the second-moment correlation length and the magnetic susceptibility in the high-temperature phase. By comparing, for various defect correlation strengths, the extracted critical exponents $ν$ and $γ$ with the results of our previous finite-size scaling study, we consolidate the exponent estimates.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic fu…
While the Ising model remains essential to understand physical phenomena, its natural connection to combinatorial reasoning makes it also one of the best models to probe complex systems in science and engineering. We bring a computational lens to the study of Ising models, where our computer-science perspective is two-fold: On the one hand, we consider the computational complexity of the Ising partition-function problem, or #Ising, and relate it to the logic-based counting of constraint-satisfaction problems, or #CSP. We show that known dichotomy results for #CSP give an easy proof of the hard…
Dynamical structure factor (S(k,$ω$)) is calculated for the one-dimensional (1D) transverse field Ising model, and its recent extension to include a three spin term, with quenched binary disorder. We study the low energy modes for lattices as large as 256 sites. We show that the intense zero energy modes appear whenever the binary disorder straddles two different winding numbers. We argue that these are Majorana modes, which reside on the boundaries of the rare regions. The size distribution of Majorana pairs has a fractal behavior at the critical points. With the longer ranged interactions a …
We obtain exact closed-form expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble, for three commonly-used boundary conditions: periodic, antiperiodic and Dirichlet. These expressions allow for the determination of fluctuation-induced forces in the canonical ensemble, which we term Helmholtz forces. The thermodynamic expressions and the calculations flowing from them should provide insights into the nature and behavior of fluctuation induced forces in interesting and as-yet unexplored regimes.
We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law-correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as benchmark systems both for generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-valu…
The partition function of a factor graph and the partition function of the dual factor graph are related to each other by the normal factor graph duality theorem. We apply this result to the classical problem of computing the partition function of the Ising model. In the one-dimensional case, we thus obtain an alternative derivation of the (well-known) analytical solution. In the two-dimensional case, we find that Monte Carlo methods are much more efficient on the dual graph than on the original graph, especially at low temperature.
We present an analytical and numerical study of the Ising model on a bilayer honeycomb lattice including interlayer frustration and coupling with an external magnetic field. First, we discuss the exact $T=0$ phase diagram, where we find finite entropy phases for different magnetisations. Then, we study the magnetic properties of the system at finite temperature using complementary analytical techniques (Bethe lattice), and two types of Monte-Carlo algorithms (Metropolis and Wang-Landau). We characterize the phase transitions and discuss the phase diagrams. The system presents a rich phenomenol…
