# 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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### Correlations in Ising mean-field theory

I am reading the book "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior" (sections 1.1.2 and 1.1.3) and have been somewhat confused about the ...
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### Can we use Bosonization to study models without $U(1)$ symmetry?

When studying lattice models using bosonization, we expect the charge is preserved so that the elementary excitation is particle-hole like bosonic degrees of freedom. How about models without $U(1)$ ...
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### Energy for a Lattice gas model

The energy for a Lattice gas model can be written as \begin{equation} E_{LG} = \frac{\epsilon}{4} \sum_{<i,j>} (s_{ij} + s_i +s_j +1) - \frac{\mu}{2}\sum_i (s_i +1) \\ = \frac{\epsilon}{4} \sum_{...
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### Why is there no kinetic term in the Hamiltonian of the Ising model?

I am used to the Hamiltonian formalism in the context of (quantum) field theory, where as far as I can remember it always has the form of a kinetic term + a potential term. For me the absence of ...
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### Finite range 1D Ising model vs. infinite range Ising model

Ising model is defiend as $$\mathcal{H}=-H\sum_i S_i -\frac{1}{2}\sum_{i,j}J_{ij}S_i,S_j$$ In 1D we assume that indices $i,j$ are integers, $i,j\in\mathbb{Z}$, and that the coupling depends only on ...
1answer
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### Ising universallity class on triangular lattice

Is Ising universallity class on triangular or hexagonal lattice different from universallity class on rectangular lattice? Is universallty class depends on type of microscopical graph or topology on ...
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### How do I find the kernel of the shift operator in the solution of 2D Ising model?

Okay, this is a second part of my previous question. Again, I'm following Itzykson's book. The fermionic solution for the 2D Ising model is described in terms of a matrix $T = \theta \tilde{\theta}$, ...
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### Reference request for path integral representation of 2D Ising model

I'm looking for references that discuss the path integral approach for the two-dimensional Ising model, constructed from its transfer matrix. The only reference I know on the topic is this book, but ...
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### On the transfer matrix formalism

Classical solutions/studies of the one-dimensional and two-dimensional Ising model make use of the transfer matrix. The following is based on Huang's book. Let us consider a two-dimensional Ising ...
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### 1D transverse-field Ising model - what is the difference between its classical and quantum treatment?

The 1D transverse field Ising model: $$H(\sigma)=-J\sum_{i\in Z} \sigma^x_i \sigma^x_{i+1} -h \sum_{i \in Z} \sigma^z_i$$ is usually solved in quantum way, but we can also solve it classically - ...
1answer
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### Ising model: How is $|\langle\sigma\rangle|^{2}=\lim _{r \rightarrow \infty} G^{(2)}(r)$?

In the book of Statistical Field Theory by Giuseppe Mussardo, on page 51, it is given while talking about Ising model that One arrives to the same conclusion by analysing the possibility of a non-...
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### Ising model: typical values of $J$

In a typical (simple, 1D) Ising model, we have $$\mathcal{H}=-\frac{J}{2} \sum_{\langle i, j\rangle} \sigma_{i} \sigma_{j}-B \sum_{i} \sigma_{i}, \quad \sigma_{i}=\pm 1$$ However, what is the typical ...
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### Is spin-1 Ising model exactly solvable (one dimension and two dimension)?

I am working on spin-1 Ising model and I am new in this field. it seems that spin-1 Ising model in one dimension can be exactly solved by transfer matrix similar with spin 1/2 Ising model, am I right ...
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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 ...
1answer
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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 ...
1answer
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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 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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### 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{...
1answer
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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}...