Questions tagged [ising-model]

The Ising model of ferromagnetism in statistical mechanics consists of discrete bimodal (+1 or −1) "spin" (moment) variables in a simple Hamiltonian interacting with their next neighbors on a lattice. The one-dimensional variant does not evince a phase transition, but the two-dimensional square-lattice one does. Use for analog and generalized discrete models on several lattices and dimensions.

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3answers
1k views

Critical 2d Ising Model

The 2d Ising model is extremely well studied, nevertheless I have encountered two facts which seem to contradict one another, and I have not been able to find the resolution in the literature. The ...
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1answer
984 views

Describe Ising model dynamics in stochastic differential equation or stochastic process

The Ising model is described by the Hamiltonian $$ H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j, $$ and is treated extensively by equilibrium statistical ...
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Zero magnetization of spin model without external magnetic field

For a given Hamiltonian with spin interaction, say Ising model $$H=-J\sum_{i,j} s_i s_j$$ in which there are no external magnetic field. The Hamiltonian is invariant under transformation $s_i \...
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Ising model for dummies

I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge. Can you direct me to some documentation on it ...
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1answer
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What is the difference between classical and quantum Ising model?

The Ising model is defined with the Hamiltonian: $$ H = -\sum_{<i,j>}S_i^z\cdot S_j^z $$ What is the difference between quantum version and classical version? My intuition is that the ...
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What is the definition of correlation length for the Ising model?

The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$ where $\nu$ is the critical exponent. Is this the formal definition of correlation length? ...
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1answer
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How to measure the spin-spin correlation in a Monte Carlo simulation of the Ising model?

I'm simulating the Ising Model in 2D up to 5D and I want to calculate the spin-spin correlation, correlation length, and critical exponent of the system. What is a good way to go about doing this? ...
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1answer
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Critical temperature and lattice size with the Wolff algorithm for 2d Ising model

When I run my implementation of the Wolff algorithm on the square Ising model at the theoretical critical temperature I get subcritical behaviour. The lattice primarily just oscillates between mostly ...
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Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?

Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk (...
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1answer
125 views

Mean field theory formulation of Ising model

I am having a bit of trouble with basic combinatorics pertaining to the Ising model and mean field theory. Specifically, I get that the Hamiltonian can be written $H=-\frac{J}{2}\Sigma_{i,j} s_is_{i+...
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What are alternative ways to think about transfer matrix as used in Ising model?

I recently learned about how to find the partition function of Ising model using Transfer Matrix method. At my level of understanding things, it is a trick that happens to work! I would like to ...
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1answer
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Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$. If $L=M$ (2-dimensional square lattice), it is known (e.g. by Peierls argument or Onsager explicit ...
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Confusion about duality transformation in 1+1D Ising model in a transverse field

In 1+1D Ising model with a transverse field defined by the Hamiltonian \begin{equation} H(J,h)=-J\sum_i\sigma^z_i\sigma_{i+1}^z-h\sum_i\sigma_i^x \end{equation} There is a duality transformation which ...
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Examples of important known universality classes besides Ising

I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical ...
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Why is it hard to solve the Ising-model in 3D?

The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model ...
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Subtleties in the exact solution to the 1D quantum XY model, in particular the Bogoliubov transformation

I am writing programs to construct the spectra of models with known exact solutions, and soon noticed some subtleties that are not often mentioned in most references. These subtleties are not ...
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3answers
825 views

Are the first order phase transitions always associated with a latent heat?

Is the first order ferromagnetic transition below the critical temperature associated with latent heat? For example, the transition of ferromagnetic configuration with all its spins aligned up to a ...
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5answers
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Mean field theory Vs Gaussian Approximation?

I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the ...
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1answer
3k views

Why do spin correlation functions in Ising Models decay exponentially below the critical temperature?

I'm trying to form a better understanding of the 2D Ising Model, in particular the behaviour of the correlation functions between spins of distance $r$. I've found a number of explanatory texts that ...
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1answer
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Continuum Field Theory for the Ising Model

My problem is to take the $d$-dimensional Ising Hamiltonian, $$H = -\sum_{i,j}\sigma_i J_{i,j} \sigma_j - \sum_{i} \tilde{h}_i \sigma_i$$ where $J_{ij}$ is a matrix describing the couplings between ...
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A simple model that exhibits emergent symmetry?

In a previous question Emergent symmetries I asked, Prof.Luboš Motl said that emergent symmetries are never exact. But I wonder whether the following example is an counterexample that has exact ...
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2answers
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Why correlation length diverges at critical point?

I want to ask about the behavior near critical point. Let me take an example of ferromagnet. At $T < T_c$, all spins are aligned to the same direction thus it is in the ordered state, scale ...
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2answers
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Numerical Ising Model - Wolff algorithm and correlations

I'm doing some numerical Monte Carlo analysis on the 2 dimensional Ising model at the critical point. I was using the Metropolis 'single flip' evolution at first with success, though it suffers from ...
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1answer
556 views

What is the information geometry of 1D Ising model for a complex magnetic field?

Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by $$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
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Why symmetry breaking?

Let me elaborate the question by using 2D Ising model without external magnetic field. When we lower the temperature and pass $T_c$ a little bit, the theory of spontaneous symmetry breaking tells us ...
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Mean field theory = large-$N$ approximation?

Wikipedia entry of $1/N$ expansion (or 't Hooft large-N expansion) mentions that It (large-$N$) is also extensively used in condensed matter physics where it can be used to provide a rigorous basis ...
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1answer
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Manipulations with Traces: Saddle point integration in Large-$N$ model

For reference I am trying to work out the derivation in this paper, in which the partition function for an Ising model is approximated by replacing the Ising variables $\sigma_i$ with $N$ component ...
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What new features does the Heisenberg Model have compared to the Ising Model?

Both the Ising and the Heisenberg Models describe spin lattices with interaction on first neighbors. The Hamiltonian in each case is quite similar, despite the fact of treating de spins as Ising ...
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1answer
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Percolation in a 2D Ising model

For a 2D Ising model with zero applied field, it seems logical to me that the phases above and below T_c will have different percolation behaviour. I would expect that percolation occurs (and hence ...
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1answer
293 views

Wick-rotated quantum computers e.g. to be realized with Ising-like systems?

Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble - and e.g. Ising model is a basic condensed matter model, which is assumed to use ...
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1answer
424 views

Universality classes

I would like to ask about the universality classes. I know that these are the statistical models which describes different phase transitions with different critical exponents. But I would like to know ...
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1answer
295 views

Research on ground state configuration of Ising model

I want to do mathematical research (algorithm construction and mathematical analysis) on Ising model ground state configuration. From what I know, the state of art research is using graph theory ...
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1answer
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Is there a spontaneous symmetry breaking in finite-size Ising model with + boundary condition?

My question is related to Chapter 3 of Prof. Yvan Velenik's book "Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction". For an Ising model defined on a finite volume $\...
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2answers
100 views

Ignoring $(\sigma_i-M)(\sigma_j-M)$ in mean field theory?

A way to do mean field theory for the Ising model is as follows. First take the Ising Hamiltonian: $$H=-J \sum_{\left<i,j\right>} \sigma_i\sigma_j$$ Let $\sigma_i=\sigma_i-M+M$ and likewise for ...
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1answer
91 views

Various questions on renormalization in lattice systems

Forgive the long, multi questioned-question. The setting of this question is inspired by this answer. Consider some theory on a lattice, for example the 2D $0$-field Ising model $$H=-K\sum_{\langle i,...
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2answers
106 views

Is the energy of the Ising model random or deterministic?

I'm reading about the Ising model and I'm confused by the following point. It says that the probability of finding the model in a given state $s$ is proportional to $e^{-E(s)/T}$. Where $E(s)$ is the ...
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0answers
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Critical slowing down in Monte-Carlo algorithm for classical 2D Ising Model

Why do local updates (i.e. local spin flips) near the phase transition in MC algorithm for classical 2D Ising model are said to be not "effective" and lead to incorrect critical indices? I understand ...
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1answer
266 views

Can Chaos Theory be used to explain the Ising model in paramagnetic phase?

Is it possible? How can I explain the randomness of spins in the paramagnetic phase with chaos theory? In this case, is the randomness apparent? If yes, I think the temperature would be a reasonable ...
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1answer
153 views

Finite temperature spontanous symmetry breaking and Goldstone bosons

I recently asked (and then attemped to answer) a question about spontaneous symmetry breaking in the Heisenberg model: Spontanous symmetry breaking in the Heisenberg model? The question and then the ...