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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Evaluating the quality of Monte Carlo simulations for 3D Ising model

Suppose I have developed a new Monte Carlo method, and I plan to test this method on studying the magnetization of a 3D Ising model at some non-zero temperature $T$. The coupling is nearest neighbor, ...
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Correlation function at zero distance

I'm confused about the definition of the correlation function (at equal time). I know it is defined from the thermal average of the scalar product of two random variables (for example the spins of a ...
48 views

Scaling limit of the Ising model with nonzero order parameter

I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$ In one dimension I can fix average magnetization $m=\langle s\rangle$ and ...
49 views

Magnetic susceptibility error by binning Monte Carlo

I am studying the 2D Ising model using Monte Carlo simulations and I have learned the binning (or batching) method for the error statistical analysis. Following this discussion https://books.google.it/...
62 views

Microscopic origin of Ising model

A typical Hamiltonian for Ising model is $$H=-\sum_{i,j} J_{ij}S_iS_j - K \sum_i S_i.$$ In many references we can find exact solutions for special cases, mean-field approach, phase transition, and ...
157 views

Symmetry transformations that are self-inverse and global symmetries of the Hamiltonian

I have the simplified Ising model. The Hamiltonian is given by $$\mathcal{H} = -\mathrm{J}\sum_{<ij,i' j'>} \sigma_{ij} \sigma_{i'j'}.$$ Where the sum over $<ij,i'j'>$ means just the ...
97 views

Ferromagnetism - computational physics

The autocorrelation of the magnetisation is plotted for the Ising model of a ferromagnet. The critical temperature is 2.3 J/k Is this the expected behaviour? as in, it decays super fast for ...
29 views

Critical parameter for 1D quantum system corresponding to $T_c$ of 2D Classical model

Utilizing the fact that there is a correspondence between a $d$ dimensional quantum system and a $d+1$ dimensional classical system (c.f. Trotter Decomposition), my question regards what the critical ...
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Dynamics of the Ising model and its Monte Carlo sampling

The Ising model is a statistical mechanical model of ferromagnetism that defines the energy of a collection of magnetic dipoles arranged in a lattice, hence, through the Boltzmann distribution, also ...
904 views

What is the Kitaev Model and why it became so popular? [closed]

I am seeing Kitaev Model everywhere. It feels like the spin-glass model of our time. How the Kitaev model differ from spin-glass and why it can be used everywhere? Looking at equation 1 here suggests ...
59 views

Total momentum of multiparticle eigenstates of discrete translation operator

I will try to frame my question using the transverse field Ising model in the low spin-coupling limit as motivation. I'll begin by discussing a case I believe I understand, that of eigenstates of ...
148 views

Does the critical dynamical exponent z of a 2D Ising model (simulated with Metropolis) vary with the temperature?

I have found in the literature that the critical dynamical exponent $z$ of an Ising model simulated with a local algorithm (such as Metropolis) is something around 2 near the critical temperature, ...
54 views

RG of 2D Ising with nonzero magnetic field on triangular lattice

I am given the Ising Hamiltonian \begin{align} H = K \sum_{<ij>}S_i S_j + h \sum_i S_i, \quad K>0 \end{align} to set up a real-space block-spin RG, where the renormalized spins are ...
41 views

How to quantify frustration for spin models with long range interactions?

Consider the following Hamiltonian: $$H=-\sum_{i\neq j}J_{ij}S_iS_j-\sum_i H_iS_i$$ where $S_i\in\{-1,1\}$, and the summed pair $i,j$ can be any two distinct indices (not necessary adjacent spins)....
42 views

Using FFT for spins in a non-cubic crystal lattice

Classical Ising/XY/Heisenberg models on a crystal lattice are commonly used to model magnetic materials. These can be studied using Monte Carlo simulations on a computer. Magnetic systems are often ...
72 views

How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?

When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
47 views

Correlation length amplitudes in Ising 2D model

I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino. I have a question about page 3 of the paper. For a magnetic ...
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Why are simulations like Monte Carlo or Metropolis studied for Ising Models when 1d and 2d case have analytical solutions?

I know that absolute analytical solutions exist for the 1d and 2d case but need some intuition as to why these simulation algorithms are used and how do we benefit from them ?
136 views

Ising Model with site-dependent magnetic field

Consider an Ising system in an external field, which is different at different sites. The Hamiltonian of the system is given by $H = -J\sum_{<i,j>}^{}s_i s_j - \sum_{i}^{} h_i s_i$ Here each ...
142 views

Long Range order in 2D Ising model

We know from the exact solutions for 2D Ising model on square lattice the long range order appears bellow critical temperature, but how does this agree with the Mermin-Wagner theorem, from which we ...