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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70 views

What is the probability distribution for a subsystem in canonical ensemble?

Suppose we have a 3d Ising Model(NN interaction) in simple cubic lattice, if we define a subsystem of it to be a 2d plane of spins(for example all sites with z = L/2, L being the linear system size) ...
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How does the ground state of the quantum Ising model relate to Schrodinger equation?

The Hamiltonian $$H = -\sum_{i\in V} h_i \sigma_i^z -\sum_{(i,j)\in E} J_{ij} \sigma_i^z\sigma_j^z - \Gamma\sum_{i\in V} \sigma_i^x$$ is kind of the cost function of the quantum annealing optimization ...
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123 views

Weird results of Monte Carlo simulation

I'm simulating the 3D Ising Model using the Wolff update algorithm. I am using the Mersenne Twister RNG. When the lattice size is $L = 50$, the specific heat curve looks very weird!! I want to ...
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APS $\eta$-invariant and spin-Ising TQFT

I am interested in the relation between the Atiyah-Patodi-Singer-$\eta$ invariant and spin topological quantum field theory. In the paper Gapped Boundary Phases of Topological Insulators via Weak ...
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118 views

Frustrated Ising model

Consider a 2D Ising model with nearest neighbour, and second nearest neighbour interactions $\mathcal{H}= -J_1\sum_{\langle ij\rangle}\sigma_i \sigma_j-J_2\sum_{\langle\langle ik\rangle\rangle}\...
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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 ?
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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 ...
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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 ...
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146 views

Feynman tricks to reproduce Onsager's solution of the 2D Ising model

I found the following quote in this paper: Wilson, Kenneth G. "The renormalization group and critical phenomena." Reviews of Modern Physics 55.3 (1983): 583. Later, Jon Mathews explained some of ...
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Transverse field Ising model - why is the sum restricted to half of the Brillouin zone?

I am reading Coleman's "Introduction to many-body physics" and am working on problem 4.2, which involves calculating the spectrum of the transverse-field Ising model. We start with the Hamiltonian $...
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Critical Ising T transformation

How can the following be consistent? The $T$ transformation of a Virasoro character $\chi(q)$ of central charge $c$ and weight $h$ is given by $$\chi(q+1) =e^{2 \pi i (h - c/24)}\chi(q)$$ for ...
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Classical Heisenberg Model Using Mean Field Approximation

Suppose we have the Classical Heisenberg Model of N spins $\vec S_i$ (unit vectors), external field $\vec H // \hat z $ $$ {\cal{H}} = -2J \sum_{<i,j>} \vec S_i \cdot \vec S_j - gμ_B \sum_i \...
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Difference between classical, semi-classical and quantum Ising model

I have a question I have been struggling about for two weeks now and would be very happy for any advice or direct help here in this forum. The question is: What is the difference between classical, ...
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108 views

Mean-field approximation in quantum all-to-all connected Ising model

I was struggling on a topic, namely the application of the mean-field approximation to the Ising model where all spins are connected to each other. In literature and internet I just find the mean-...
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Why is there only one critical point in Ising model?

While reading about Kramers-Wannier duality in "Statistical Field Theory" by Giuseppe Mussardo, I read that the hypothesis that there is only one critical point is fully justified from the physical ...
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How can I say whether a Hamiltonian is integrable or not?

The transverse field Ising Hamiltonian $$ H = J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x} $$ is integrable because it can be exactly solved using Jordan Wigner ...
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Elitzur theorem and the Ising model

I was recently studying the Elitzur theorem and its application to the Ising model on Kogut: An introduction to lattice gauge theories and spin systems, chapter $5$C. I was wondering how he obtain $\...
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Practical/experimental difference between (quantum) Heisenberg and (classical) Ising model

I have read a few discussions about the difference between the Heisenberg model (using quantum spin operators) and Ising model (with spins $\pm 1$), notably this one or this Quora post. All the ...
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Is it possible to efficiently update the ground state of an Ising lattice after a local change in the fields?

The Hamiltonian of an Ising model can be written as: $$H(\mathbf s) = \sum_{i<j}J_{ij}s_i s_j + \sum_i h_i s_i$$ where $s_i \in \{0,1\}$ are the spins on each site. The ground state is the spin ...
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Transverse Ising model in continuum limit

Recently I have read "Analyzing the two dimensional Ising model with conformal fi eld theory" by Paolo Molignini, but I don't understand clearly manipulations in the section about continuum limit of ...
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Work performed by ramping of magnetic field (non-interacting Ising model) [closed]

Consider the following hamiltonian $$H=-h\sum_{i=1}^N\sigma_i$$ where $\sigma_i=\pm1$ and $h$ is the magnetization. Let us assume that the system is equilibrated with a bath at temperature $T$ with ...
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What's The 2pt Correlation Function For The Spin Fields For The 3D Critical Ising Model?

The title it self explanatory. What's The 2pt Correlation Function For The Spin Fields For The 3D Ising Model? I know the form of the four point function and have worked out how to express it in terms ...
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Free energy in statistical mechanics

While studying topics related to the Ising model I stumbled upon two different definitions of the free energy, first I was presented this one: $\Phi=E-TS$ and from this, not deriving it: $\Phi=-\frac{...
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Ising 2d Using Montecarlo Metropolis (Markov chain method)

I'm doing as a personal training the 2d Square lattice Ising model. I decided to go with metropolis Monte Carlo method using Markov chain. I'm not into this methods, but I'm just using them as a tool (...
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Question about Landau theory of Phase Transitions

The landau theory makes a mean-field approximation on the order parameter, which assumes that there are no fluctuations in the value of the order parameter at different sites (neglects the effects of ...
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251 views

Flipping more than one spin in Metropolis Monte Carlo algorithms

In lattice systems such as Ising model or spin glasses, the standard Monte Carlo simulation with Metropolis algorithm works by proposig a single spin flip and then accepting or rejecting the proposal ...
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What happens to the dynamical critical exponent in the quantum-classical mapping?

It is well-known that one can, e.g., map the classical 2D Ising model to the 1D quantum Ising chain. Moreover, their critical points are related. Hence, if one is interested in critical exponents ...
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164 views

How are correlation length and cluster size related in the 2D Ising model?

What is the relationship between correlation length and cluster size? Does the correlation length give the average cluster size, or is the cluster size something different?
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374 views

Why is the upper critical dimension of the Ising model 4?

I have read in various sources, that the critical exponents of the Ising Model are identical to the meanfield ones for dimensions $d \geq 4$. In trying to understand this better I came across the ...
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288 views

Why does a vanishing energy gap indicate a phase transition?

More concretely: When looking at the Ising model in the description of Bogoliubov fermions, we get an explicit expression for the energy gap, that vanishes for a particular value of the magnetic field....
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Block diagonalizing a Hamiltonian using a symmetry

I have the following Hamiltonian, describing the 3 state Chiral clock model in 1D: $$H = -f \sum_{j=1}^L (\tau_j^\dagger e^{-i \phi}+h.c.)-J\sum_{j=1}^{L-1}(\sigma_j^\dagger\sigma_{j+1}e^{-i\hat{\phi}}...
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Transfer matrix of 1D Ising model

I'm sorry if this is trivial, I've been stuck on a definition in Yeomans, Statistical mechanics of phase transitions. In chapter five she describes the transfer matrix of the 1D Ising model with ...
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89 views

Why does the free energy depend on the configuration, in the Peierls argument?

The origin of my doubt lays on the Peierls argument to show that there is no phase transition in the 1D Ising Model. I understand that the Helmholtz Potential is the partial Lengendre transform of $U$...
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231 views

Transverse field Ising model with open boundary conditions

what is the energy dispersion of the transverse field Ising model looks like in the case of open boundary conditions? In the case of periodic boundary, the energy takes the form of and the ground ...
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164 views

Problem comparing Ising model with Mean Field Theory

After implementing the Metropolis algorithm for the Ising model, I tried comparing my values with the MFT predictions. Shouldn't the phase change occur at T/Tc? I would guess that my mistake is in ...
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Analytic methods for studying thin films (approximately 2D spin systems)

I am working on a project involving thin films of a magnetic material represented by essentially an Ising model on a 3 dimensional lattice. The bulk case is relatively straight forward, the ...
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305 views

Understanding entanglement entropy in the transverse field Ising model

I'm working on a project studying the transverse field Ising model in 1D with periodic boundary conditions, given by $$H = - J \sum_{i=1}^{N} \sigma^z_i \sigma^z_{i+1} - h \sum_{i=1}^N \sigma^x_i$$ I ...
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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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Will the equilibrium energy remain same for different temperatures in the 2d-Ising model?

I am doing a numerical simulation of the 2D-Ising model using the Metropolis algorithm. One of the plots that we were asked to make was Energy vs Monte-Carlo Steps for different starting Temperatures. ...
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Sums of Ising Variables

I happened to come across the following term while doing an excercise on perturbation theory \begin{equation} H^2=J^2\sum_{<i,j>}\sum_{<k,l>}\sigma_i\ \sigma_j\ \sigma_k\ \sigma_l \end{...
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Lowest energy state possess a symmetry but Gibbs state does not?

I believe the Bose condensate to be a system where above the critical temperature the only lowest energy eigenstate breaks the U(1) symmetry but the Gibbs state does not. On the contrary; Can there ...
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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 ...
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211 views

Interpretation of Ising model simulations

I've been working on numerically solving the Ising model in a study of phase transitions, but I'm having difficulty finding material to help me discuss the results. I'm studying spontaneous ...
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274 views

Block diagonalizing a spin-chain Hamiltonian

$\newcommand{\ket}[1]{\left|#1\right>}$ I am learning about exact diagonalization methods, currently following this explanation. My question is in regards to the part where we utilize the fact ...
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Help with Unusual Mean Correlation

I've been working on a variation of the Ising Model that tries to account for quantum fluctuations in the lattice. After doing a some calculations I calculated the mean correlation $<\sigma_i \...
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1D Ising Model in B-Field Partition Function (2 Spins)

I have a one-dimensional system of two Spins ( $\sigma_1$, $\sigma_2$) in a magnetic field B with energy: \begin{equation} E = -J \sigma_1 \sigma_2 -B (\sigma_1 + \sigma_2) \end{equation} each $\sigma$...
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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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165 views

What physically are the interactions of the Ising model?

The Ising model is defined as $$H(\sigma)=-\sum_{<i j>}J_{ij}\sigma_i\sigma_j$$ what are the coupling/interaction terms physically? Also how can you manufacture a set of spins to have definite ...
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248 views

Information from four point correlation functions in Ising model

For a one-dimensional classical Ising model with the Hamiltonian $$H=-J \sum_{i}\sigma_{i} \, \sigma_{i+1}$$ where $\sigma=\left\{+1,-1\right\}$ one can calculate two point correlation for the spins $$...
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808 views

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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