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Ising model
About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.
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TL;DR: The Quantum Ising model is an exactly solvable model of quantum phase transition and this Letter gives an exact solution when the system is driven through the critical point at a finite rate.
Abstract: The Quantum Ising model is an exactly solvable model of quantum phase transition. This Letter gives an exact solution when the system is driven through the critical point at a finite rate. The evolution goes through a series of Landau-Zener level anticrossings when pairs of quasiparticles with opposite pseudomomenta get excited with a probability depending on the transition rate. The average density of defects excited in this way scales like a square root of the transition rate. This scaling is the same as the scaling obtained when the standard Kibble-Zurek mechanism of thermodynamic second order phase transitions is applied to the quantum phase transition in the Ising model.
530 citations
TL;DR: The solution of the zero-field "eight-vertex" model is presented in this article, which includes the square lattice Ising, dimer, ice, $F, and KDP models as special cases.
Abstract: The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, $F$, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.
530 citations
TL;DR: In this article, a new type of effective field theory of the Ising model is presented and the differential operator method is introduced into the exact spin correlation function identity obtained by Callen.
Abstract: A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.
529 citations
TL;DR: The zero-temperature random-field Ising model is used to study hysteretic behavior at first-order phase transitions using mean-field theory and results of numerical simulations in three dimensions are presented.
Abstract: We use the zero-temperature random-field Ising model to study hysteretic behavior at first-order phase transitions. Sweeping the external field through zero, the model exhibits hysteresis, the return-point memory effect, and avalanche fluctuations. There is a critical value of disorder at which a jump in the magnetization (corresponding to an infinite avalanche) first occurs. We study the universal behavior at this critical point using mean-field theory, and also present results of numerical simulations in three dimensions.
518 citations
TL;DR: In this article, the authors present a pedagogical review of duality and its application to a wide range of field theories and statistical systems, including the Ising model, the vector Potts model and the Wilson lattice gauge theory with a discrete or continuous Abelian symmetry.
Abstract: This paper presents a pedagogical review of duality (in the sense of Kramers and Wannier) and its application to a wide range of field theories and statistical systems. Most of the article discusses systems in arbitrary dimensions with discrete or continuous Abelian symmetry. Globally and locally symmetric interactions are treated on an equal footing. For convenience, most of the theories are formulated on a $d$-dimensional (Euclidean) lattice, although duality transformations in the continuum are briefly described. Among the familiar theories considered are the Ising model, the $x\ensuremath{-}y$ model, the vector Potts model, and the Wilson lattice gauge theory with a ${Z}_{N}$ or $U(1)$ symmetry, all in various dimensions. These theories are all members of a more general heirarchy of theories with interactions which are distinguished by their geometrical character. For all these Abelian theories it is shown that the duality transformation maps the high-temperature (or, for a field theory, large coupling constant) region of the theory into the low-temperature (small coupling constant) region of the dual theory, and vice versa. The interpretation of the dual variables as disorder parameters is discussed. The formulation of the theories in terms of their topological excitations is presented, and the role of these excitations in determining the phase structure of the theories is explained. Among the other topics discussed are duality for the Abelian Higgs model and related models, duality transformations applied to random systems (such as theories of a spin glass), duality transformations in the "lattice Hamiltonian" formalism, and a description of attempts to construct duality transformations for theories with a non-Abelian symmetry, both on the lattice and in the continuum.
500 citations