Exact Solution for Three-Dimensional Ising Model

The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.

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Free partition functions and an averaged holographic duality

Journal of High Energy Physics ◽

10.1007/jhep01(2021)130 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nima Afkhami-Jeddi ◽

Henry Cohn ◽

Thomas Hartman ◽

Amirhossein Tajdini

Keyword(s):

Partition Function ◽

Spectral Gap ◽

Central Charge ◽

Three Dimensional ◽

Two Dimensions ◽

Three Dimensions ◽

Partition Functions ◽

General Constraints ◽

Dimensional Gravity ◽

Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

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THE EXACT SOLUTION OF SOME LATTICE STATISTICS MODELS WITH FOUR STATES PER SITE

Canadian Journal of Physics ◽

10.1139/p64-142 ◽

1964 ◽

Vol 42 (8) ◽

pp. 1564-1572 ◽

Cited By ~ 32

Author(s):

D. D. Betts

Keyword(s):

Partition Function ◽

Nearest Neighbor ◽

Quaternary Alloy ◽

Interesting Property ◽

Two Dimensions ◽

Lattice Statistics ◽

Critical Temperatures ◽

Different Types ◽

Interacting Systems ◽

Statistical Mechanical

Statistical mechanical ensembles of interacting systems localized at the sites of a regular lattice and each having four possible states are considered. A set of lattice functions is introduced which permits a considerable simplification of the partition function for general nearest-neighbor interactions. The particular case of the Potts four-state ferromagnet model is solved exactly in two dimensions. The order–disorder problem for a certain quaternary alloy model is also solved exactly on a square net. The quaternary alloy model has the interesting property that it has two critical temperatures and exhibits two different types of long-range order. The partition function for the spin-3/2 Ising model on a square net is expressed in terms of graphs without odd vertices, but has not been solved exactly.

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Order-disorder statistics. I

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1949.0012 ◽

1949 ◽

Vol 196 (1044) ◽

pp. 36-50 ◽

Cited By ~ 26

Keyword(s):

Ising Model ◽

Partition Function ◽

Statistical Mechanics ◽

Three Dimensional ◽

Infinite Matrix ◽

Characteristic Structure ◽

Single Function ◽

Simple Characteristic ◽

Distant Neighbour ◽

Function Of Two Variables

In 1941 Kramers & Wannier discussed the statistical mechanics of a two-dimensional Ising model of a ferromagnetic. By making use of a ‘screw transformation’ they showed that the partition function was the largest eigenvalue of an infinite matrix of simple characteristic structure. In the present paper an alternative method is used for deriving the partition function, and this enables the ‘screw transformation’ to be generalized to apply to a number of problems of classical statistical mechanics, including the three-dimensional Ising model. Distant neighbour interactions can also be taken into account. The relation between the ferromagnetic and order-disorder problems is discussed, and it is shown that the partition function in both cases can be derived from a single function of two variables. Since distant neighbour interactions can be taken into account the theory can be formally applied to the statistical mechanics of a system of identical particles.

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Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions

Physical Review Letters ◽

10.1103/physrevlett.36.1326 ◽

1976 ◽

Vol 36 (22) ◽

pp. 1326-1328 ◽

Cited By ~ 48

Author(s):

Zvi Friedman

Keyword(s):

Renormalization Group ◽

Ising Model ◽

Critical Exponents ◽

Three Dimensional ◽

Real Space ◽

Two Dimensions ◽

The Real ◽

Real Space Renormalization Group

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Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

Materials Science Forum ◽

10.4028/www.scientific.net/msf.845.150 ◽

2016 ◽

Vol 845 ◽

pp. 150-153

Author(s):

Andrey N. Vakilov

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Ising Model ◽

Partition Function ◽

Critical Behavior ◽

Critical Exponents ◽

Three Dimensional ◽

Finite Size ◽

Finite Size Scaling ◽

Zeros Of Partition Function

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

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Accurate estimate ofνfor the three-dimensional Ising model from a numerical measurement of its partition function

Physical Review Letters ◽

10.1103/physrevlett.59.803 ◽

1987 ◽

Vol 59 (7) ◽

pp. 803-806 ◽

Cited By ~ 67

Author(s):

Gyan Bhanot ◽

Román Salvador ◽

Steve Black ◽

Paul Carter ◽

Raúl Toral

Keyword(s):

Ising Model ◽

Partition Function ◽

Three Dimensional ◽

Accurate Estimate

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MONTE CARLO STUDIES OF THE THREE-DIMENSIONAL ISING MODEL IN EQUILIBRIUM

International Journal of Modern Physics C ◽

10.1142/s0129183101002383 ◽

2001 ◽

Vol 12 (07) ◽

pp. 911-1009 ◽

Cited By ~ 54

Author(s):

MARTIN HASENBUSCH

Keyword(s):

Monte Carlo ◽

Ising Model ◽

Monte Carlo Simulations ◽

Three Dimensional ◽

Finite Size ◽

Two Dimensions ◽

Three Dimensions ◽

Amplitude Ratios ◽

Interface Tension ◽

Monte Carlo Studies

We review Monte Carlo simulations of the Ising model and similar models in three dimensions that were performed in the last decade. Only recently, Monte Carlo simulations provide more accurate results for critical exponents than field theoretic methods, such as the ∊-expansion. These results were obtained with finite size scaling and "improved actions". In addition, we summarize Monte Carlo results for universal amplitude ratios, the interface tension, and the dimensional crossover from three to two dimensions.

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