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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Why are Eigenvectors of a 1D quantum ising hamiltonian real

I was modelling the 1D transverse quantum Ising model and made a Kronecker product loop to find the Hamiltonian of the system, for a given magnetic field configuration. Now, my question is that when I ...
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Is Bogoliubov transformation simply a basis transformation?

I encounter Bogoliubov transformation when I'm learning 1D transverse field Ising Model.But I think it's just a simple basis transformation, I don't understand why people give it a special name?
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Ising Model without periodic boundary conditions (PBC)

I try to calculate the correlation function $<\sigma_i \sigma_j>$ with the method of transfer matrices. I do understand how to use this method with PBC. But how can I do it without PBC? My ...
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Bragg-Williams microcanonical esemble

In this question Bragg-Williams theory of phase transition of the forum someone was asking for Bragg-Williams aprox. and how to calculate entropy. The answer is clear and correct, the Bragg-Williams ...
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Solving the quantum XX model using drone fermions

Suppose we wish to solve the XX model in 1D, which describes spin-$\frac{1}{2}$ particles interacting with their nearest neighbor. Assuming open boundary conditions for simplicity, the Hamiltonian ...
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Critical point in 2D Potts model and duality equation

I have to prove, considering the limits of high and low temperatures, duality equation: $$ Z=q e^{2 N K} F\left(e^{-K}\right)=q^{-N}\left(e^{K}+q-1\right)^{2 N} F\left(\frac{e^{K}-1}{e^{K}+q-1}\right) ...
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1D Ising model correlations

I am calculating different things in Ising 1D ring model using Trasfer matrix method: $$Z_{N}=\sum_{\left\{\sigma_{i}=\pm 1\right\}} \exp \left(K \sum_{i=1}^{N} \sigma_{i} \sigma_{i+1}+h \sum_{i=1}^{N}...
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Quantum partition of two integer spins

I am trying to write the partition function for a simple system made of two interacting bosons of spin $S=1$ having the interaction Hamiltonian $$ \hat{H} = \vec{S}_1 \cdot \vec{S}_2 $$ I think I ...
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Critical exponents and scaling dimension

It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
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How to solve this integral?

I am following some notes from a physics class on the Ising model. At some point we get to this integral \begin{equation} \frac{1}{2} \int_{\Omega_B} \frac{\text{d}^Dk}{(2\pi)^D} \ln \left[ \tau + \...
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Is the common formulation for (Ising Model) Monte Carlo simulations a bit off?

A common [Ising Model] Monte Carlo simulation repeats the following algorithm: randomly pick a [flip] event compute change in energy $\Delta E$ if new energy is lower than old energy, accept the ...
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Eigenvalue of transfer matrix using Monte Carlo in 2D Ising model

Montecarlo is an algorithm capable of numerical estimation of any quantity which can be written as the average of a state function like, for example, the magnetization or the internal energy in the 2D ...
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Mean-field theory for a heterogeneous Ising model with two types of particles

Let's say I have an Ising model, but this time with two kinds of particles, A and B. Let the spin of particle A be denoted by $\sigma ^a$, and the spin of particle B by $\sigma ^b$. Let the total ...
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Overlap of Matrix Product States (Python)

I'd like to implement the TEBD (finite, real time evolution) by hand (in python) and want to compute the overlap of a reference MPS with the time evolved MPS. I want to regard a simple Ising Chain. ...
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Entropy of a spin spontuaneous simmetry breaking?

The Shannon entropy of a probability mass function is $$H(S) = -\sum_s p(s)\log_2 p(s),$$ where the sum runs over all the configurations $s$. If a system $S$ stays half of the time in the ...
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How to take into account finite temperature in transverse Ising chain?

A similar question has already been asked here What I'm wondering is how to take into account finite temperature in the transverse Ising chain and see how that affects the magnetization. The reason ...
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Meaning of 'thermalization' in Markov Chain Monte Carlo simulations

In performing MCMC simulations, it is standard practice to 'equilibriate' or 'thermalize' the system and then discard the initial data before useful sampling is done. My question is about the concept ...
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Reference request: Kawasaki dynamics for Ising model

I want to learn more about Glauber and Kawasaki dynamics which, by my understanding, are used to model lattice spin systems for the pre-equilibrium Ising model. There seems to be quite a few ...
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Transverse field Ising model quantum phase transition

I am looking at the quantum phase transition of the transverse field Ising model. Let: \begin{equation} H = -J \sum_{x=1}^{N-1} \sigma_x^3\sigma_{x+1}^3 - B \sum_{x=1}^N \sigma_x^1 \end{equation} Once ...
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Ground state energy from Hamiltonian [closed]

It has been a long time since I did QM, and I am getting stuck at the most basic stuff. Assume I have a Hamiltonian: \begin{equation} H = \int_{-\pi}^\pi f(q) \left[\alpha^\dagger_q, \alpha_q \right]\...
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Scaling of "non-reduced" parameters in RG theory

I'm studying quantum phase transitions using the Renormalization Group (RG) method. In Continentino's book "Quantum Scaling in Many-Body Systems: An Approach to Quantum Phase Transitions" ...
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Anticommutation and Bogoliubove transformation

I am given the following transformation: \begin{equation} \begin{bmatrix} ...
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Applications of real-space renormalizaton group (RG)

I'm looking for lattice models on which real-space RG can be applied fairly simply to get decent results. In particular, I'm looking for something like the classical 2D Ising model on a triangular ...
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What is the order of the transition for a 2D Ising model?

I have been running around the block trying to find answers for this question, and I keep running into caveats. So, I just want to write down the list of things I want to know: Given that the order ...
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Quantum vs. Classical Ising model and the Bohr van Leeuwen Theorem [duplicate]

The classical Ising model is often described as a simplistic model for ferromagnetism, and the Bohr-Van Leeuwen theorem is understood to preclude classical physics origins of magnetization in matter. ...
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Frustrated classical field theory

The frustrated Ising model (see e.g. this answer) is an example of a system that shows no unique ground state and many metastable states (its "energy landscape" is extremely complex). ...
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First-order phase transition in the Ising model?

I am doing a simulation of the 2D Ising model with a Monte Carlo algorithm. I think that the model should exhibit a second order phase transition at $\beta=\beta_c$, but when I try to plot the ...
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Concavity of magnetization for Potts model

For the Ising model the magnetization $ \langle \sigma_x \rangle_{\beta,h} $ is concave in the variable $h$. This means that \begin{align*} \frac{ \partial^2 \langle \sigma_x \rangle_{\beta,h} }{\...
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Is the free energy of the 3 and 4 state Potts model in a positive magnetic field analytic?

For the Ising model in a magnetic field $h>0$ the Lee Yang theorem ensures that the free energy is analytic. The $1 \leq q \leq 4$ Potts model the phase transition is continous. Is it known ...
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Interaction term in the Hamiltonian of transverse-field Ising model

I got a question about quantum transverse-field Ising model. The Hamiltonian has two terms which are external field term and interaction term. Why are there only $Z$-$Z$ interactions between ...
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Why does the chain be at equilibrium in the MH algorithm?

I'm implementing the Metropolis algorithm to solve the 2D Ising model. I've understood how to implement it and now I'm trying to understand a bit of how the algorithm works. In the site I'm reading it ...
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Mean field approximation in the Ising model

I am studying statistical physics for an exam scheduled next week and there's something I really do not get about mean field approximation in the Ising model. The situation In the lesson, we defined ...
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Role of chemical potential in Ising model

I know that when modelling a lattice gas through the Ising model, it is possible to draw parallels from the chemical potential $\mu$ of the gas to the role coverede by an external magnetic field $H$: ...
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Simulation time for Ising model of large systems

I have tried to run simulation for Ising model of large-size square lattices at the critical point. Mostly I use Python optimized with numba decorator for $L=256$ it takes approx 2.5 min with ...
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Tensor product operator of 2D ( for Tensor Network)

I have difficulty representing Tensor Product Operators (TPO) of 2D in a concrete form. For example in 1D case, according to the tutorial in ITensor , the TPO of the Hamiltonian for the simple Ising ...
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What is after the Ising model? [closed]

If this model is solved in three dimensions, Will there be additional research on it? Like what? Does this open the way to solve other models? Like what?
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Phase space of Ising minimal model + deformations

Consider the Ising field theory, a conformal field theory in 2 dimensions which corresponds to the minimal model $\mathcal{M}_{4,3}$ and it's perturbations by the relevant operators $\epsilon, \sigma$ ...
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Why do the critical exponents at the Gaussian fixed point coincide with mean field theory?

In the Ising model, we know that in dimensions higher than the upper critical dimension, $d_u=4$, the critical exponents can be found from mean field theory. We also know that for the same dimensions, ...
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Exact Diagonalization, Jordan-Winger Transformation and the Second Quantization

I am currently study the quantum Ising model with the guide of my supervisor in an undergraduate project. Since my undergraduate courses didn't cover this, I use this paper to as the main material ...
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Are phase transitions in one-dimensional random-field Ising model possible?

Translationally invariant one-dimensional models, with interactions of finite range and a finite number of states at the site, don't allow phase transitions at positive temperatures. This fact is a ...
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How to determine the distance between any two sites of a finite lattice subjected to periodic boundary conditions?

I want to study the Ising model on a finite kagome lattice assuming periodic boundary conditions (PBC) and long range interactions. More specifically, all spin pairs contribute to the total energy, so ...
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Latent heat in the Ising model

I am confused about the (non)-existence of latent heat in the first order phase transition of the Ising model. Most textbooks talk about latent heat as a signature sign of a first order transition, ...
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Exact heat capacity of the 2-dimensional Ising model

The following is a section from the book Newman, M., and G. Barkema. "Monte carlo methods in statistical physics" New York, USA (1999). and then: From those two quotes, it seems that there ...
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Rewriting of a occupation based Hamiltonian to an spin based Ising Hamiltonian

I run in to the following problem of rewriting a hamiltonian derived in an earlier question to an Ising hamiltonian. (b) Identify your result in (a) with the Hamiltonian and the partition function of ...
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Ferromagnet : $N$ Spin-$1/2$ Particle on Circle

Consider one-dimensional ferromagnet namely $N$ spin-$1/2$ objects placed around a circle with the Hamiltonian $$\mathscr{H}=-\mathcal{J}\sum_{n=1}^N\vec{\mathcal{S}}_n\cdot \vec{\mathcal{S}}_{n+1}$$ ...
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Legendre Transformation of Landau Free Energy

I am trying to get an intuition for the Legendre Transformation of a generic Landau free energy, e.g. for the Ising model with magnetization $m$ given by $$F(m) = \frac{a}{2} m^2 + \frac{b}{4} m^4 + \...
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Why Can Electrons be Modelled as Classical Spins?

Although electrons are spin $1/2$ particles described by the Pauli matrices, the Ising model treats electrons as classical spins. As a result, the Ising model does not describe anything physical, but ...
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Absence of Symmetry Breaking in 1D Ising Model--Continuum Version

I have seen arguments for why there is no symmetry breaking in the 1D Ising model--for example, using the transfer matrix method to explicitly solve the model, and another of energy-entropy arguments ...
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Interpretation of "pressure" as the logarithm of the partition function

Consider the Ising model on a subset $\Lambda\subset\mathbb{Z}^d$, with partition function $Z_{\Lambda; \beta , h}$ where $\beta$ is the inverse temperature and $h$ the external magnetic field. The ...
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High temperature expansion of the Ising model - proof of extensivity of free energy

Consider the $2d$ Ising model, which has partition function $$ Z = \sum_{\{S_i\}}\exp\left[J\sum_{\langle ij\rangle}S_iS_j \right], $$ where $\langle ij\rangle$ denotes nearest neighbours, and $\sum_{...
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