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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1answer
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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 ...
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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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Solution of 2D Ising model using the transfer matrix
I'm following Huang's book solution of the 2D Ising model using the transfer matrix. The partition function is given by:
$$Z_{n} = \sum_{\mu_{1}}\cdots\sum_{\mu_{n}}\langle \mu_{1}| P| \mu_{1}\rangle\...
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Does the 3D Ising model violate the hyperscaling relation?
The hyperscaling relation relates two critical exponents $\{\alpha,\nu\}$ (the power the divergence of the specific heat $c$ and correlation length $\zeta$ near a critical temperature $T_c$) as ...
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1answer
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Fermionic solution of 2D Ising
I'm trying to understand the discussion in this book on the fermionization of the 2D Ising model. The transfer matrix for this model becomes $T = \theta\tilde{\theta}$ where:
$$\theta = e^{\beta \sum_{...
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1answer
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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 - ...
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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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1answer
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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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Decorrelation times for a 2D Ising Model over a range of temperatures
So, I'm trying to simulate the Ising Model on a 2D square lattice of spins. When exploring the auto correlation of the magnetisation:
Where the auto covariance: $$A(T) = \langle(M(t)\ - \langle M\...
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Mean field theory of three-state Potts model
I was trying to solve 3-state Potts model and got eventually stuck when the problem approached mean-field theory. I have managed to show that
$H=-\frac{3J}{2}\sum_{(ij)}{\sigma_i \sigma_j}$
is ...
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0answers
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One dimensional Ising model MCMC simulation strong fluctuations in autocorrelation functions
I'm performing some MCMC simulation of the 1D Ising model, but when it comes to computing the integrated autoroccelation time I observe strong fluctuations in the autocorrelations functions for long ...
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Determining critical exponent in spontaneous magnetization of 2D Ising model
I have been solving a problem related to Onsager's relation in Ising model. The initial relation is:
$<s>^8=1-sinh(\frac{2J}{k_BT})^{-4}$
I got that critical temperature is $\frac{2J}{k_BT}=sinh^...
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Why do degenerate ground states mix in perturbation theory?
I'm interested in the Ising model and I'm reading ahead a little about it. I picked up this article that states that the Ising model has degenerate ground states (all up and all down), which is good ...
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1answer
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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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1answer
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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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0answers
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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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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 ...
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
68 views
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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1answer
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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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1answer
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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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1answer
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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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1answer
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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} $$
The spins $\sigma_{i}$ are arranged linearly and have values $1$ ...
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1answer
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Using renormalization group theory on the Ising model using decimation transformations
Consider a 1d Ising model with no external magnetic field $(h=0)$ and adopt a decimation transformation in which every other spin is traced out.
So the Hamiltonian $H$ is given by
$$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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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}...
