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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Mathematical Rigorousness of Taking the Thermodynamical limit of a finite size quantum model
Suppose I have nearest Neighbour Quantum Ising model with a transverse field.
$$\hat{H} = \sum_{i}S^{x}_iS^{x}_{i+1} + h\sum_i S^{z}_i$$
Through Jordan-Wigner and Bogoliubov transformation, one finds ...
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Quantum Monte Carlo Loop Algorithm for quantum spin: why is the freezing graph present in ferromagnetic Ising model?
I study the loop algorithm (Evertz et al).
I cannot understand, why the freezing graph type where we have to flip all 4 spins together is not present for the quantum-XY model and the anti-/...
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Correlation length anisotropy in the 2D Ising model
In the Ising model, the two-spin correlation function is
$$
C(\vec{r}) = \langle \sigma_{\vec{r}_0+\vec{r}}\sigma_{\vec{r}_0}\rangle - \langle \sigma_{\vec{r}_0+\vec{r}}\rangle \langle \sigma_{\vec{r}...
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RG of Ising model whose Hamiltonian is represented with Kronecker delta
Let $H$ be hamiltonian, $i$ the index of a spin, and $S_i = \pm 1 $ the $i$-th spin's value. When 1D Ising model's hamiltonian is represented as
$$ H = - J \sum _i S_i S_{i + 1}\ \ \ (J > 0), $$
...
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Why are the autocorrelations larger for the energy at the critical temperature?
Considering a simulation with the Swendsen-Wang algorithm for the 3-D cubic lattice I wanted to have a look at the auto-correlations, and expecting it to be quite small considering Swendsen-Wang is a ...
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References and papers to distinguish between the Heisenberg and Ising Model
Does anybody have any good papers or references to explain the differences between the Heisenberg model and Ising model?
To the best of my knowledge, I am aware that the Hamiltonians are similar, ...
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Why do the Binder Cumulants of different system sizes intersect at the critical point?
When Monte Carlo simulations are performed for spin models (Ising model etc.) the critical temperature can be found by simulating for different lattice sizes and plotting the Binder Cumulant for them. ...
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Different concepts of phase transitions in spin models
I am currently revising the lecture notes in which different spin systems
are analyzed, focussing on the occurrence (or absence) of phase transitions.
Different techniques are applied to analyze the ...
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Questions on mean field theory and the enforcement of local spin conserving constraints
Hello Physics StackExchange community,
I've recently been working on a problem that seems like it should be straightforward, but I can't seem to overcome what seems to be a basic obstacle. I'll try ...
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How to get mean field critical exponents for this Hamiltonian?
$$ \mathcal{H} = -J \sum_{\langle ij\rangle} \sum_{\alpha=1}^N s_i{}^\alpha s_j{}^\alpha -g \sum_{\langle ij\rangle} \sum_{\alpha\beta} (s_i{}^\alpha s_j{}^\alpha) (s_i{}^\beta s_j{}^\beta) $$
Above ...
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Order parameter for spin-1 ising model
I'm curious what the order parameter would be for a spin-1 Ising model for magnetism and anti-ferromagnetism. So if spin's can take the value's -1,0 and 1.
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Derivation of the Ising free energy close to a critical point
In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with,
$$
f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right),
$$
wherein $t=\vert T-T_c\vert/T_c$ ...
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What are the excitations in the near critical 2D-Ising model in a magnetic field?
Apparently it is well known that the 2D Ising model with $T=T_C$ in a small magnetic field has a mass gap and correlation length $\xi \sim h^{- \frac{8}{15}} $. Further, in a paper in 1989 ...
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Asymmetric hysteresis loop in Ising Model
I am doing simulations using Monte Carlo of the 2 dimensional square lattice with periodic boundary conditions Ising model, and i obtain hysteresis loops, which are asymmetric. Meaning i obtain a ...
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How to calculate the autocorrelation function of magnetic susceptibility for the Ising model?
In the paper Wolff U. 1989. Physics Letters B. 228(3):379–82, the autocorrelation time of susceptibility, $\tau_\chi$ was calculated, but the way to do so was not clearly explained in the paper.
To ...
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Water boiling and 3D Ising model
I've been told for a long time that water boiling near critical temperature and the 3D Ising model near critical temperature are described by the same laws, and give a CFT. This is usually mentioned ...
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How to deduce the formula of the Correlation Length on a periodic lattice?
Sometimes in Monte Carlo simulations we need to compute the correlation length, but this is a hard task without a formula. However, for instance, in an periodic cubic lattice of $L^3$ spins, some ...
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Magnetic susceptibility vs Monte Carlo step
I have some difficulties in understanding how to compute the magnetic susceptibility from a Monte Carlo simulation of the Ising model. I know that it is related to the magnetisation of the system by $\...
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1answer
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Ginzburg Criterion (Ising model)
In my statistical field theory class, we were told that we want the magnetization fluctuations in the Ising model to be smaller than their background. Specifically this was written as $$\langle\phi^2\...
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1answer
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Average magnetisation in the Ising Model
The Ising Model has energy given by
$$
E=-B \sum_{i} s_{i}-J \sum_{\langle i, j\rangle} s_{i} s_{j}
$$
where $\langle i, j\rangle$ indicates that the second sum is over each pair of nearest ...
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Renormalization Group - Scaling fields and physical critical exponents (1D Ising model)
This is related to this question: Critical exponents and scaling dimensions from RG theory.
TLDR: How to compute physical critical exponents $\alpha, \beta, \gamma, etc$ from the RG exponents when ...
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Length of domain wall in Ising model
A subquestion of a homework for my statistical mechanics class this week asked of the 2d Domain wall Ising model approximation:
"Now argue that for the formation of a domain wall separating the ...
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Evaluating the quality of Monte Carlo simulations for 3D Ising model
Suppose I have developed a new Monte Carlo method, and I plan to test this method on studying the magnetization of a 3D Ising model at some non-zero temperature $T$. The coupling is nearest neighbor, ...
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1answer
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Correlation function at zero distance
I'm confused about the definition of the correlation function (at equal time). I know it is defined from the thermal average of the scalar product of two random variables (for example the spins of a ...
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Scaling limit of the Ising model with nonzero order parameter
I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$
In one dimension I can fix average magnetization $m=\langle s\rangle$ and ...
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1answer
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Magnetic susceptibility error by binning Monte Carlo
I am studying the 2D Ising model using Monte Carlo simulations and I have learned the binning (or batching) method for the error statistical analysis. Following this discussion https://books.google.it/...
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1answer
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Microscopic origin of Ising model
A typical Hamiltonian for Ising model is
$$ H=-\sum_{i,j} J_{ij}S_iS_j - K \sum_i S_i.$$
In many references we can find exact solutions for special cases, mean-field approach, phase transition, and ...
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Symmetry transformations that are self-inverse and global symmetries of the Hamiltonian
I have the simplified Ising model. The Hamiltonian is given by
$$
\mathcal{H} = -\mathrm{J}\sum_{<ij,i' j'>} \sigma_{ij} \sigma_{i'j'}.
$$
Where the sum over $<ij,i'j'>$ means just the ...
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1answer
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Ferromagnetism - computational physics
The autocorrelation of the magnetisation is plotted for the Ising model of a ferromagnet.
The critical temperature is 2.3 J/k
Is this the expected behaviour? as in, it decays super fast for ...
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Critical parameter for 1D quantum system corresponding to $T_c$ of 2D Classical model
Utilizing the fact that there is a correspondence between a $d$ dimensional quantum system and a $d+1$ dimensional classical system (c.f. Trotter Decomposition), my question regards what the critical ...
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1answer
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Dynamics of the Ising model and its Monte Carlo sampling
The Ising model is a statistical mechanical model of ferromagnetism that defines the energy of a collection of magnetic dipoles arranged in a lattice, hence, through the Boltzmann distribution, also ...
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1answer
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What is the Kitaev Model and why it became so popular? [closed]
I am seeing Kitaev Model everywhere. It feels like the spin-glass model of our time. How the Kitaev model differ from spin-glass and why it can be used everywhere? Looking at equation 1 here suggests ...
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Total momentum of multiparticle eigenstates of discrete translation operator
I will try to frame my question using the transverse field Ising model in the low spin-coupling limit as motivation. I'll begin by discussing a case I believe I understand, that of eigenstates of ...
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1answer
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Does the critical dynamical exponent z of a 2D Ising model (simulated with Metropolis) vary with the temperature?
I have found in the literature that the critical dynamical exponent $z$ of an Ising model simulated with a local algorithm (such as Metropolis) is something around 2 near the critical temperature, ...
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RG of 2D Ising with nonzero magnetic field on triangular lattice
I am given the Ising Hamiltonian
\begin{align}
H = K \sum_{<ij>}S_i S_j + h \sum_i S_i, \quad K>0
\end{align}
to set up a real-space block-spin RG, where the renormalized spins are ...
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How to quantify frustration for spin models with long range interactions?
Consider the following Hamiltonian:
$$
H=-\sum_{i\neq j}J_{ij}S_iS_j-\sum_i H_iS_i
$$
where $S_i\in\{-1,1\}$, and the summed pair $i,j$ can be any two distinct indices (not necessary adjacent spins)....
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1answer
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Using FFT for spins in a non-cubic crystal lattice
Classical Ising/XY/Heisenberg models on a crystal lattice are commonly used to model magnetic materials. These can be studied using Monte Carlo simulations on a computer. Magnetic systems are often ...
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How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?
When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
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Correlation length amplitudes in Ising 2D model
I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino.
I have a question about page 3 of the paper. For a magnetic ...
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1answer
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1d Ising model: Transfer matrices
we came across a peculiarity when calculating the partition function of $N$ spins $s_i=\pm1$ with Hamiltonian $$H=-J\sum_{i=1}^Ns_is_{i+1}$$
where we impose periodic boundary conditions such that $s_{...
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votes
1answer
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Decomposition of $\mu^{free}$ for the Ising-Dyson Model
For the nearest neighbours Ising-Model in any dimension, it is known that
$$
\mu^{free}_\beta= \frac{1}{2} \mu^{+}_\beta+\frac{1}{2} \mu^{-}_\beta
$$
for any inverse of temperature $\beta>0$.
...
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2answers
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Understanding Ising Model in Statistical Mechanics
A section on the Ising model in the text "Introduction to modern statistical mechanics" by Chandler states the following:
"We consider a system of $N$ spins arranged on a lattice. In the
...
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1answer
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Obtaining an expression for spontaneous magnetization in 1D Ising model with $H=0$ from the beginning
The usual trick to find the spontaneous magnetization for the 1D Ising model is to calculate the partition function $Z$ with the Hamiltonian
$$\mathscr{H}=-J\sum\limits_{i}S_iS_{i+1}-H\sum\limits_{i}...
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1answer
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Problem in counting bonding pairs (elementary mean-field theory on the Ising Model) [duplicate]
When the Ising model Hamiltonian
$$H=-J\sum _{<ij>} \sigma _i\sigma _j-H\sum _i \sigma _i$$
is assumed ($\sum _{<ij>}$ is the summation over all the bonds or adjacent pairs of sites, $\...
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1answer
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Fixed boundaries in 1D Ising model
What are the differences for solving the one dimensional Ising model for fixed boundaries using the transfer matrix, compared with periodic boundaries?
this picture show the solution for periodic ...
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1answer
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What is this secondary transition in the simulation of the Ising model?
Here, the horizontal axis is the strength of the ambient magnetic field. The Hamiltonian I used is $$H = -h\sum_i \sigma_i - J\sum_{\langle i \, j \rangle}\sigma_i\sigma_j.$$ The horizontal axis is $h$...
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Sherrington-Kirkpatrick model with negative mean $J_0$
In the Sherrington-Kirkpatrick (SK) model, one considers an Ising Hamiltonian
$$H = -\sum_{i<j}J_{ij}s_is_j$$
where $J_{ij}$ are drawn independently from a Gaussian distribution with mean $J_0$ ...
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1answer
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Fitting an Ising Model with Probabilities
Question
How to use the observations to fit an Ising model?
After self-studying for several days, my current guess is:
$\theta_{ii} = \log[P(X_{i} = 1)]$
$\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$
...
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2answers
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How can I explicitly express the Ising Hamiltonian in matrix form?
I am reading this book about numerical methods in physics. It has the following question:
Consider the Ising Hamiltonian defined as following $$H=-\sum_ {i=1}^{N-1}
\sigma_i^x \sigma_ {i+1} ^x + h ...
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votes
2answers
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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_{\{...
