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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2 dimensional Ising model: How do we visualize the Hamiltonian for interacting spins, but with no external magnetic field?

Suppose we don't have any external magnetic field, so that the Hamiltonian is given by $H=-J\sum_{i,j}s_is_j$. If we have an $n\times n$ 2D lattice of spins. Then does the $H$ correspond to one whole ...
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Onsager's solution vs Mean Field Theory

This is a question on the reliability of the Mean Field approach. I have been studying the Ising model recently and have come across 2 approaches to solve the Ising model. For simplicity, I set $k_{B}=...
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Systems with a $\mathbb{Z}_2$ spontaneous symmetry breaking transition?

I am studying spontaneous $\mathbb{Z}_2$ symmetry breaking transition for some time now in quantum (transverse-field) and classical Ising systems. I would like to look beyond my little box in terms of ...
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Bethe's mean field approximation and general cluster treatment of Ising model

In Bethe's mean field approximation the Hamiltonian describes only the energy of a central spin $\sigma_0$ and its $q$ nearest neighbors: $$ H_{BMF}=−h\sigma_0−J\sigma_0\sum_{i=1}^{q}\sigma_i−(h+h')\...
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Why the Helmholtz free energy is minimized when states obey the Boltzmann distribution?

The Helmholtz free energy is defined by the difference between the internal energy and the entropy of the system, $$F_{T} = U_{T} + kTH_{T},$$ where $U_{T} = \sum_s P_{T}(s)E(s)$ is the internal ...
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Is there a 3D spin glass that's paramagnetic at all finite temperatures?

Here, I'm defining spin glass where the couplings $\mathbf{J}$ are sampled from some product measure (say Gaussian or $\pm 1$ random bond). And by paramagnetic, I mean that the overlap distribution ...
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Magnetization ($z$-basis) of a 1D Transverse Ising Model

I'm trying to find the magnetization $\langle\sigma_{z} \rangle$ of a 1D transverse Ising chain and plot it as a function of the transverse field $\lambda$. More specifically, I want to plot this for ...
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1D ising system: reducing configuration with respecet to symmetry and total magnetization

this question is mathematical in its sense and considers the following 1D ising spin model $$s_1s_2s_3....s_{n-1}$$ where $s_i=\pm 1$. I would like to find the total number of different configurations ...
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Kramers-Wannier duality high and low temperature expansions confusion

I am reading the section on the 2D Ising model Krammer-Wannier duality in the book Exactly Solved Models in Statistical Mechanics (pg. ~76) by R.J. Baxter. I have two questions: What was the ...
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How to obtain a nonzero order parameter for a symmetry-breaking quantum phase transition?

If $\hat{m_z}=\frac{1}{N}\sum_i \hat{\sigma^z_i}$ is an order parameter for finite quantum system (transverse Ising model, say), then it will never break the $\mathbb{Z}_2$ symmetry since $\langle\...
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Ising model 2D and mean field theory

Consider the 2D Ising model. Now, let's divide it into 4-spins blocks and treat the interaction inside each block exactly, while applying the mean-field approximation to the interaction between blocks....
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Variational principles for approximating thermodynamic potentials in the inverse Ising problem: How to go from double to single extremum?

I'm trying to wrap my head around section 2.2.6 (on variational principles) in the following paper (on the inverse Ising problem): https://arxiv.org/abs/1702.01522 Here the authors explain how to use ...
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2D ising model susceptibility - different values

As I've been simulating Ising model in 2D, I searched the Internet for some plots to check whether my simulation is working correctly. I stumbled upon two versions of magnetic susceptibility vs ...
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Proving this modified classical 2D Ising Hamiltonian has a phase transition

Consider the following Hamiltonian with a local Hilbert space $\mathcal{H}=\mathcal{H}_\Delta\otimes \mathcal{H}_{Ising}\cong\mathbb{C}^2\otimes\mathbb{C}^2 $. Denote an $L\times L $ square lattice as ...
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Is this a common shape of graph in Thermodynamics?

I am calculating the acceptance ratios of the local and non-local Kawasaki algorithm for the Ising model and generated the following graph: This shape of graph comes over as kind of familiar, is this ...
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Generalized Ising model

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
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Partition function for 2D classical XY model

Studying the classical XY model (https://en.wikipedia.org/wiki/Classical_XY_model), I wish to compute the partition function: \begin{equation} Z=\int \mathrm{d}\mathbf{s}\; e^{-\beta H(\mathbf{s})} \...
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Is there any point in doing Monte Carlo on classical 2D Ising spin systems? [closed]

The partition function of a classical Ising spin system with arbitrary bonds on any planar graph can be evaluated in polynomial time, through the FKT algorithm. And if I understand correctly, this ...
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What is the critical coupling constant in an Ising model and how to spot it?

Consider a zero-field Ising model with $N$ spins and periodic boundary conditions, with the Hamiltonian given by $$H = -K \sum _{(ij)} s_i s_j$$ in 1D and 2D, where $K = \frac{J}{k_BT}$, where $J$ is ...
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Coupling between external field and internal parts in Ising model

If we consider the Ising model: $$H(\sigma)= - \sum_{ij}J_{ij}\sigma_{i}\sigma_{j} - \mu \sum_{j}h_{j}\sigma_{j} $$ where $h_{j}$ is the external magnetic field. The fields that we have are the $\...
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How models become gapless in the thermodynamic limit?

Given an Hamiltonian on some finite lattice, it lives in a finite-dimensional Hilbert space with a finite number of eigenvalues, so obviously there is a gap between the lowest value of an eigenvalue ...
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Continuous phase transitions in statistical models with binary degrees of freedom

Except from the Ising, is there any other statistical mechanics model that exhibits second order phase transition in two dimensions that has binary degrees of freedom?
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Evolving policy of Ising models?

Setup Let the Hamiltonian of the Ising model be $$ H_{J,h}(\sigma) = \Sigma_{i, j} J\sigma_i \sigma_j + \Sigma_{i} h \sigma_i.$$ Then the Gibbs partition function for the pair $(J,h)$ is given by $$ ...
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Flipping probabilities using the Glauber update algorithm for $1$ Dimensional Ising Model

My question relates to one dimensional Glauber Dynamics: If you are using the Glauber update algorithm and the current value is $\sigma_{\alpha} = −1$, what are the probabilities for flipping and for ...
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Probabilities for Ising Model using a Heat Bath Algorithm

I am considering the one-dimensional ferromagnetic Ising model where: $$ H = - \sum_{i}\sigma_{i}\sigma_{i+1} -B\sum_{i} \sigma_{i} $$ The spins $\sigma_{i}$ are arranged linearly and have values $1$ ...
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Using renormalization group theory on the Ising model using decimation transformations

Consider a 1d Ising model with no external magnetic field $(h=0)$ and adopt a decimation transformation in which every other spin is traced out. So the Hamiltonian $H$ is given by $$H = -J\sum_{(i,j)} ...
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Magneto-caloric effect in the transverse field Ising model

I'm trying to solve the ODE of the magneto-caloric effect (MCE) in the disordered phase of the transverse field Ising model (TFIM), i.e. for $H>H_c$: $$ \frac{T_\mathrm{bath}-T}{\alpha\tau} = \frac{...
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Positivity of correlation functions in the ferromagnetic Ising model

Is it true that all correlation functions of any even number of spins in the ferromagnetic Ising model with nearest neighbors interaction are nonnegative in any spatial dimension? In the one-...
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How to perform discrete functional derivative

If I have some action that depends on a set of variables like so: $S(\{\phi_{i} \}) = \phi_{i}A^{-1}_{ij}\phi_{j} + g(\phi_{i})$ (where einstein summation notation is being used for the term $\phi_{i}...
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How to solve the 1D Ising model with periodic boundary conditions?

I am trying to solve the 1D Ising model with periodic boundary conditions and no external magnetic field. Solving the Open Boundary Problem seems pretty straightforward. $$Q = \sum_{s_1,s_2,...,s_N} \...
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Specific Heat Ising and Heisenberg Model

I have found when temperature limits to zero specific heat of Ising model results as 0. However, for Heisenberg model for limiting to 0 gives a constant. Why do these two models read different result ...
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Ising model with open boundary condition

Can you give me any reference about solving 1 D Ising model with open boundary condition, magnetic field and nearest interaction?
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Argument on why spin correlation functions in Ising model decay exponentially with a correlation length?

I'm reading Quantum Field Theory in Strongly Correlated Electronic Systems, Nagaosa. Consider 1D Ising model, $$H=J_z\sum_i S^z_iS^z_{i+1}.$$ on page 3, it says The groud stae is 2-fold degenerate ...
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For discrete optimization problems, why does continuous approximation - solving - discretization work?

I was wondering about the following question. In maths/computer science, many optimization problems ask for integer optimizations. A prominent example would be spin-glass (Ising) systems for simulated ...
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Why does the Ising model at the critical point have scale invariance?

If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and ...
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Critical exponent $\nu$ of 3d Ising model

this question comes from an exercise of Sethna's book "Statistical Mechanics: Entropy, Order Parameters and Complexity". it is in page 282 question 12.2 In 3d Ising model, the spin ...
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Legendre transforms for the Ising model

I'm reading a review on the Ising model and came across a section where they discuss Legendre transforms of thermodynamic potentials. Now I'm familiar with the classical thermodynamic relations such ...
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2D Ising model and FK-percolation

Consider the 2D Ising model on the finite lattice $\Lambda$ with $+$ boundary conditions, i.e., all spins outside of $\Lambda$ are $=+1$. Let $\mathscr{E}_\Lambda^b$ denote the edges in $\Lambda$ and ...
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Physical explanation of the non-analyticities in the ising model in the vicinity of zero external field

In the Ising model when $T<T_C$ with $T_C$ being the Curie temperature, there is a finite jump of the mean magnetization per spin, $m$, as the external field crosses $0$ (goes from negative values ...
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Generating Ising model steady state configurations

What is the most efficient way to simulate steady state configurations of the Ising model? I am just interested in having a large set of random steady state configurations of the 1D Ising model (with ...
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Plotting the magnetic susceptibility of the mean field Ising model

I am struggling to understand how they have ploted some functions regarding the Ising model in the mean field approximation (Curie-Weiss model) in my lecture notes For more details,you can see them ...
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Confusion about units of physical magnitudes in the Hamiltonian of the Ising model

I am having trouble with the units used in the Hamiltonian of the Ising model. I have search several notes, I have three examples in the picture below No one states explicitly what the units of the ...
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From where does the Ising Hamiltonian come?

So in my Stat Mech course, we were introduced to the classical Ising Model: $$H = -J\Sigma _{<ij>}S_iS_j - K\Sigma_i S_i$$ But from where does this come from? Is there any rationale behind this? ...
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What happens to the energy when a system following Ising Model goes to its ground state?

I'm a computer scientist and new to Ising Model. I've learned that if such a system is left to itself it will converge to its minimum energy state. Here are the questions I have: As the system is ...
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Wolff algorithm for anisotropic Ising model

Question Is there way to simulate a 2D Ising model with anisotropic coupling parameter $J_{ij}$ and zero external field using Wolff algorithm? Particularly, I am looking for when coupling parameter $...
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Simulation of Quantum Ising Model

I curious to know if there is a way to do simulation of quantum ising model with transverse field. The method I know is - do classical ising model simulation in d+1 dimension which essentially maps to ...
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Gaussianity of lattice models in statistical mechanics

Recently there was a result on triviality (or Gaussianity) of the Ising model and $\phi^4$-theory in dimension $d=4$. This therefore holds in any dimension $d \geq 4$. We also know that the 2D Ising ...
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2D Ising model exact expression for two-point function

The Ising model on $\mathbb{Z}^2$ is given by the Hamiltonian $$ H(\sigma)=-\sum_{\{x,y\}}\sigma_x\sigma_y $$ and the Gibbs measure as $$ \frac{\exp(-\beta H(\sigma))}{Z_\beta}\,. $$ There exists an ...
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Why is this the Helmholtz Free Energy for the Onsager Ising Model?

I'm reading through Kerson Huang's presentation of the Onsager solution. We end up determining that the natural log of the partition function is $$\ln Z = \frac{1}{2}\ln (\frac{2 \cosh^2(2 \beta \...
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Phase transition in parameter (and not temperature) for a classical system

Consider the $q$-state Potts model on $\mathbb{Z}^d$ for some integer $q$ - this also has an FK-representation for any real number $q$. For $d = 2$ the model is exactly solvable and has a critical ...

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