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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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Math problem in Kaufmann-Onsager exact solution to 2D Ising model [on hold]

So, I've been following Huang book in Statistical mechanics for the 2D exact solution to the Ising model (chapter 15). During the solution he has to solve an eigenvalue problem, that is: $$(A+z_kB+...
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Sherrington-Kirkpatrick model with negative mean $J_0$

In the Sherrington-Kirkpatrick (SK) model, one considers an Ising Hamiltonian $$H = -\sum_{i<j}J_{ij}s_is_j$$ where $J_{ij}$ are drawn independently from a Gaussian distribution with mean $J_0$ ...
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Fitting an Ising Model with Probabilities

Question How to use the observations to fit an Ising model? After self-studying for several days, my current guess is: $\theta_{ii} = \log[P(X_{i} = 1)]$ $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$ ...
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How can I explicitly express the Ising Hamiltonian in matrix form?

I am reading this book about numerical methods in physics. It has the following question: Consider the Ising Hamiltonian defined as following $$H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h ...
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How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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Mean field theory formulation of Ising model

I am having a bit of trouble with basic combinatorics pertaining to the Ising model and mean field theory. Specifically, I get that the Hamiltonian can be written $H=-\frac{J}{2}\Sigma_{i,j} s_is_{i+...
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Ising model with quantum magnetic field

Hamiltonian of the Ising model with an external magnetic field is written as $$H=-J\sum_{\langle i,j \rangle} s_i s_j + h\sum_j s_j$$ where $J$ is nearest neighbor coupling constant and $h$ is the ...
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Ising model and duality

I'm studying the Ising model in 2 dimensions in an approximative way. Now my professor has written this formula that links the dual space and the "normal" space: $$\sinh(2 K) \sinh(2 K^*) = 1 $$ Do ...
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Product Rule for Partition Sums $Z_N=(Z_1)^N$

For the 1D Ising model with the Hamiltonian $$H=const.-\mu h' \sum_i S_i^z$$ we can write the canonical partition sum as $$Z_N = \sum_{ \{ S_i^z \}_N } e^{-\beta \mu h \sum_i S^z_i} = \sum_{ \{ S_i^...
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Ising Model Error Propagation

If I have the statistical uncertainties of the ensemble average magnetisation and the average energy from a monte carlo simulation of an Ising Model, how do I find the errors on the specific heat ...
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How are the coefficients determined in the high temperature expansion of the 2D Ising model?

I have been studying the 2D Ising model lately and have been looking at high and low temperatures. But I'm having problems when trying to understand the high temperature one. The final expansion looks ...
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Periodic autocorrelation function for Ising model?

I am trying to calculate the autocorrelation time for a 2-D Ising model Monte Carlo simulation. As the autocorrelation function, I am using $$\chi (t) = \frac{1}{t_{max}-t} \sum_{t' = 0}^{t_{max}-t-1} ...
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One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
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Confusion on real space renormalization group for Ising model on lattice

For the Ising model with only nearst neighbor interaction on square lattice, if we do the RG by integrating out half degree of freedom, then we would get a new Ising model with many kind of ...
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How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$. I know conceptually how to compute the ground state of the Ising model at zero ...
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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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How to show second quantized Hamiltonian is Hermitian?

Consider fermionic Hamiltonian: $$H= i A \sum_k c_{-k} c_{k} + c_{-k}^{\dagger} c_{k}^{\dagger} $$ with annilinting operator $c_{k}$, creating operator $c_{k}^{\dagger}$, wavevector k and constant A....
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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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Finding the correlation function using the expression of the free energy (Ising model, Landau theory)

I am working on a homework problem regarding the Lee Yang theorem, though my issue already exist using only the standard approach to the Ising model. Simply put, i have no idea how to explicitly ...
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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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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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Broken symmetry Ising model

Consider an Ising model for a 2D square lattice where we have interactions for, let's say nearest neighbours and second nearest neighbours $$\mathcal{H}=-J_1\sum_{\langle ij\rangle}\sigma_i\sigma_j-...
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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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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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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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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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How does the Dirac equation follow from the algebra of order and disorder operators in the Ising model?

In his book "Statistical Field Theory", Mussardo derives the Dirac equation for the Majorana fermions in the scaling limit of the Ising model. He does this by defining order operators (the Pauli ...
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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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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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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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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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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 ...