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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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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28 views
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
19 views
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
52 views
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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1answer
150 views
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
80 views
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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0answers
21 views
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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40 views
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
77 views
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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31 views
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
32 views
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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45 views
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
95 views
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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1answer
26 views
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
85 views
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
85 views
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$$
is assumed ($\sum _{<ij>}$ is the summation over all the bonds or adjacent pairs of sites, $\...
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1answer
65 views
Fixed boundaries in 1D Ising model
What are the differences for solving the one dimensional Ising model for fixed boundaries using the transfer matrix, compared with periodic boundaries?
this picture show the solution for periodic ...
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1answer
40 views
What is this secondary transition in the simulation of the Ising model?
Here, the horizontal axis is the strength of the ambient magnetic field. The Hamiltonian I used is $$H = -h\sum_i \sigma_i - J\sum_{\langle i \, j \rangle}\sigma_i\sigma_j.$$ The horizontal axis is $h$...
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0answers
20 views
Sherrington-Kirkpatrick model with negative mean $J_0$
In the Sherrington-Kirkpatrick (SK) model, one considers an Ising Hamiltonian
$$H = -\sum_{i<j}J_{ij}s_is_j$$
where $J_{ij}$ are drawn independently from a Gaussian distribution with mean $J_0$ ...
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1answer
56 views
Fitting an Ising Model with Probabilities
Question
How to use the observations to fit an Ising model?
After self-studying for several days, my current guess is:
$\theta_{ii} = \log[P(X_{i} = 1)]$
$\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$
...
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2answers
200 views
How can I explicitly express the Ising Hamiltonian in matrix form?
I am reading this book about numerical methods in physics. It has the following question:
Consider the Ising Hamiltonian defined as following $$H=-\sum_ {i=1}^{N-1}
\sigma_i^x \sigma_ {i+1} ^x + h ...
2
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1answer
113 views
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:
$$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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1answer
75 views
Mean field theory formulation of Ising model
I am having a bit of trouble with basic combinatorics pertaining to the Ising model and mean field theory. Specifically, I get that the Hamiltonian can be written
$H=-\frac{J}{2}\Sigma_{i,j} s_is_{i+...
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0answers
85 views
Ising model with quantum magnetic field
Hamiltonian of the Ising model with an external magnetic field is written as
$$H=-J\sum_{\langle i,j \rangle} s_i s_j + h\sum_j s_j$$
where $J$ is nearest neighbor coupling constant and $h$ is the ...
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1answer
67 views
Ising model and duality
I'm studying the Ising model in 2 dimensions in an approximative way. Now my professor has written this formula that links the dual space and the "normal" space:
$$\sinh(2 K) \sinh(2 K^*) = 1 $$
Do ...
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2answers
72 views
Product Rule for Partition Sums $Z_N=(Z_1)^N$
For the 1D Ising model with the Hamiltonian
$$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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1answer
44 views
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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1answer
55 views
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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1answer
94 views
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 $$\chi (t) = \frac{1}{t_{max}-t} \sum_{t' = 0}^{t_{max}-t-1} ...
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1answer
78 views
One-dimensional Ising Model in a three spin chain
I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is:
$$H=J[S_z(1)S_z(2) ...
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3answers
190 views
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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1answer
113 views
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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1answer
61 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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1answer
90 views
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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1answer
107 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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2answers
151 views
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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1answer
74 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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1answer
85 views
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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1answer
102 views
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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0answers
84 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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1answer
65 views
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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1answer
62 views
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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1answer
176 views
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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0answers
145 views
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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0answers
70 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-...
