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 does the Ising model at the critical point have scale invariance?
If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and ...
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Critical exponent $\nu$ of 3d Ising model
this question comes from an exercise of Sethna's book "Statistical Mechanics: Entropy, Order Parameters and Complexity". it is in page 282 question 12.2
In 3d Ising model, the spin ...
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Mean Field Phase Transition of Spin Systems on a irregular graph
In the Ising model w/o external field, if we use mean-field approximation, and have the regular graph lattice...
then we can use the symmetry argument to recover the fact that
$$\bar{x} = \tanh{(\beta ...
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Legendre transforms for the Ising model
I'm reading a review on the Ising model and came across a section where they discuss Legendre transforms of thermodynamic potentials. Now I'm familiar with the classical thermodynamic relations such ...
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2D Ising model and FK-percolation
Consider the 2D Ising model on the finite lattice $\Lambda$ with $+$ boundary conditions, i.e., all spins outside of $\Lambda$ are $=+1$. Let $\mathscr{E}_\Lambda^b$ denote the edges in $\Lambda$ and ...
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Physical explanation of the non-analyticities in the ising model in the vicinity of zero external field
In the Ising model when $T<T_C$ with $T_C$ being the Curie temperature, there is a finite jump of the mean magnetization per spin, $m$, as the external field crosses $0$ (goes from negative values ...
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Generating Ising model steady state configurations
What is the most efficient way to simulate steady state configurations of the Ising model? I am just interested in having a large set of random steady state configurations of the 1D Ising model (with ...
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Plotting the magnetic susceptibility of the mean field Ising model
I am struggling to understand how they have ploted some functions regarding the Ising model in the mean field approximation (Curie-Weiss model) in my lecture notes
For more details,you can see them ...
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Confusion about units of physical magnitudes in the Hamiltonian of the Ising model
I am having trouble with the units used in the Hamiltonian of the Ising model. I have search several notes, I have three examples in the picture below
No one states explicitly what the units of the ...
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From where does the Ising Hamiltonian come?
So in my Stat Mech course, we were introduced to the classical Ising Model:
$$H = -J\Sigma _{<ij>}S_iS_j - K\Sigma_i S_i$$
But from where does this come from? Is there any rationale behind this? ...
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What happens to the energy when a system following Ising Model goes to its ground state?
I'm a computer scientist and new to Ising Model.
I've learned that if such a system is left to itself it will converge to its minimum energy state. Here are the questions I have:
As the system is ...
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Wolff algorithm for anisotropic Ising model
Question
Is there way to simulate a 2D Ising model with anisotropic coupling parameter $J_{ij}$ and zero external field using Wolff algorithm?
Particularly, I am looking for when coupling parameter $...
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Simulation of Quantum Ising Model
I curious to know if there is a way to do simulation of quantum ising model with transverse field.
The method I know is - do classical ising model simulation in d+1 dimension which essentially maps to ...
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Gaussianity of lattice models in statistical mechanics
Recently there was a result on triviality (or Gaussianity) of the Ising model and $\phi^4$-theory in dimension $d=4$. This therefore holds in any dimension $d \geq 4$.
We also know that the 2D Ising ...
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2D Ising model exact expression for two-point function
The Ising model on $\mathbb{Z}^2$ is given by the Hamiltonian $$ H(\sigma)=-\sum_{\{x,y\}}\sigma_x\sigma_y $$ and the Gibbs measure as $$ \frac{\exp(-\beta H(\sigma))}{Z_\beta}\,. $$
There exists an ...
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Why is this the Helmholtz Free Energy for the Onsager Ising Model?
I'm reading through Kerson Huang's presentation of the Onsager solution. We end up determining that the natural log of the partition function is
$$\ln Z = \frac{1}{2}\ln (\frac{2 \cosh^2(2 \beta \...
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1answer
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Phase transition in parameter (and not temperature) for a classical system
Consider the $q$-state Potts model on $\mathbb{Z}^d$ for some integer $q$ - this also has an FK-representation for any real number $q$.
For $d = 2$ the model is exactly solvable and has a critical ...
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Estimating the heat capacity of ising model
I am have written a Metropolis-Hastings algorithm and am currently trying to compare it to the analytical results for the 2D Ising model. The free energy seems reasonable but the heat capacity I'm ...
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Eigenvector problem for a block toeplitz matrix [closed]
Consider a 2*2 block toeplitz real matrix A . The (i,j) block is given by
$$ \begin{matrix}
-f_{i-j} & g_{i-j} \\
-g_{j-i} & f_{i-j}
\end{matrix}
$$
where i,j go from 1 to L.
f satisfies $f_{-...
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Energy in the Ising model for liquid-gas transition
I am teaching myself statistical mechanics using the lecture notes of Professor Leonard Susskind. I am confused about a statement 'reducing $h$ reduces the density' at page $555$. I know the statement ...
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2D Ising model on curved surface
What will be the sensible extension of the 2D Ising to some curved surface - for instance, for a sphere or even something non-orientable?
For the flat space energy is given by well-known expression:
$$...
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Ising model critical temperature from divergence of the high temperature series
Consider the Ising model on a d-dimensional hypercubic lattice with nearest-neighbor coupling $J=1$: $$H=-\sum_{\langle ij \rangle}\sigma_i\sigma_j$$
For small $\beta$ (corresponding to high ...
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2D Ising model entropy as a function of external field
I am trying to figure out what is the entropy expression as a function of the external field in a 2D Ising model with nearest neighbour interaction.
My Hamiltonian is the following:
$\mathcal{H}=-\...
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1answer
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Sum of all two point correlation functions in the Ising model
The two point correlation function for the Ising model is defined as $\left[\langle\sigma_i\sigma_j\rangle -\langle\sigma_i\rangle\langle\sigma_j\rangle\right]$. Then the sum over $i$ $j$ of that ...
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Obtain Mean Field Equations for Spin Models using a uniform Ansatz
I would like to see how my model I am working on behaves in the limit of infinite dimensions so I get a little bit of intuition for the low dimensional case. In the paper I am reading they have a ...
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What is value of critical temperature?
What is the value of critical temperature in the 2D classical ising model?
My Understanding
Suppose one can write the partition function for the 2D classical ising model in high-temperature expansion ...
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Transverse field Ising in 2 dimensional lattice - kronecker product
Assume we have a transverse field Ising chain (1D):
$\hat H =-J\sum_{i=1}^{N}\sigma^z_i\sigma^z_{i+1}-h\sum_{i=1}^{N}\sigma^x_i$,
where $\sigma^{\alpha}_i$ are the local spin operators at site i ...
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2D ising model explains phase transition (Para-ferro) in 3D space How?
2D ising model explains phase transition (Para-ferro). How come a 2-D model explain a system where spins are distributed in all three directions?
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Is the fully connected Potts model exactly solvable?
Suppose that we have "spins" $\sigma_1,\dots,\sigma_N$, with $\sigma_i\in\{1,\dots,q\}$, for $i=1,\dots,N$, and that our Hamiltonian is
$$
H = -\frac{J}{N} \sum_\stackrel{i,j=1}{i\ne j}^N \...
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What methods can I use to find the minimum of a tranverse field Ising model?
I am trying to solve for the minimum of the hamiltonian of the form:
$$
H = \sum_{i,j} J_{ij}q(i)q(j) + g_i\sum_i x(i)
$$
where q(i) is the operator (I + z(i))/2 and z(i) and x(i) are pauli operators ...
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What is the relationship between the Ising model and a magnet's corresponding magnetic field at varying temperatures?
I understand that ferromagnetism occurs when all the spins in a magnet are aligned, leading to the magnetic vector field that we are familiar with. What happens qualitatively to the magnetic field as ...
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Emergent supersymmetry in tricritical Ising model
In TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems there is a statement that 2d supersymmetry can can emerge from the dilute Ising model:
$$
\beta H = ...
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Confusion about notation for block transformation in Ising model
I'm going through Cardy's "Scaling and Renormalization in Statistical Physics", and I've run across a notational confusion. Consider a 2D Ising system with the following Hamiltonian
$$\...
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1D Ising Model: spin interaction in $z$-direction, magnetic field in $x$-direction
I've stumbled across an interesting extra question in an old exam. In my own words:
Consider the Magnetization of the 1D Ising model $$H=-J\sum_iS_{z,i}S_{z,i+1}-B\sum_iS_{x,i}$$ at $T=0$. We know ...
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Relating scaling and critical exponents in the Ising model
I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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Boltzmann distribution in Ising model
I've written in Matlab a code for a Ising model in 1 dimension with 40 spins at $k_{B}T=1$.
I record the energy of every step in a Metropolis Monte Carlo algorithm, and then I made an histogram like ...
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Relevant operators in Ising model
Why in 3d Ising near critical point there are only two relevant deformations?
I am interested in experimental arguments and also in theoretical explanation.
For example, in 3D Ising Model and ...
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A closer look on the derivation of the susceptibility in the Ising-Model
The susceptibility $\chi$ can be defined as
$$ \chi = \frac{\partial \langle M \rangle}{\partial H}, \tag{1}$$
where $\langle M \rangle$ is defined as the average magnetization and thus can be written ...
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Ising model autocorrelation of magnetisation per state
I am trying to simulate Ising model, and have learnt that we get more accurate results when we take correlation time $\tau$ into account. I.e, decrease the correlations between observing samples only ...
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Ising spin are bosonic?
I was going through the following paper The 1D Ising model and the topological phase of the Kitaev chain. They call ising spins to be bosonic. If I the quote the paper:
Hard-core bosons realized ...
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Correlation length for ising-Kitaev chain: Coincidence or are they same?
The correlation length for the two-dimensional classical ising model goes as
$$\xi_{ising}(T)\sim |T-T_c|^{-\nu};\qquad \nu=1$$
We can map the classical ising model to its quantum cousin, one-...
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1answer
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Value of critical exponent $\alpha$ for 2D ising model
The Onsager solution for specific heat is
$$C\approx -Nk\frac{2}{\pi}\bigg(\frac{2J}{kT_c}\bigg)^2\ln\Big|1-\frac{T}{T_c} \Big|\qquad (T \textrm{ near } T_c)$$
Critical exponent $\alpha\neq 0$. ...
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Autocorrelation function problem in Monte Carlo simulation of 2D Ising model
Currently, I did a Monte Carlo simulation with the local update and Wolff cluster updated in 2D classical Ising model. I use the autocorrelation function to compare 2 different algorithm in critical ...
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Nonzero spontaneous magnetization in two-dimensional Ising model
The two-dimensional Ising model with the nearest-neighbour interactions enjoys a $\mathbb{Z}_2$ symmetry under $S_i\to -S_i$; it displays sponatebous symmetry breaking at a finite temperature $T_C=2J[...
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Applications of Sampling from SK Ising Model
I have written a program for Monte Carlo sampling from Sherrington-Kirkpatrick (SK) Ising model. I have two questions about it:
1- What are some applications of it? I already know training Boltzmann ...
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Simple Quantum Monte Carlo question
Currently I am doing some simple simulation of 1D Transverse field Ising model. I map the quantum mechanical problem into classical 2D classical Ising model with different horizontal interaction and ...
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Numerical renormalization of 2D Ising lattice
I'm trying to make some toy computations on the $2D$ Ising model on a square lattice. I want to apply a renormalization transformation, and try to estimate observables on the renormalized lattice ...
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What is the ground state wavefunction of $\hat{H}=-J\sum\limits_{\langle i,j\rangle}\hat{S}_i^z\hat{S}_j^z,~~ (J>0)$?
The hamiltonian of a collection of noninteracting quantum spin-$1/2$ operators $\hat{S}_i$ fixed at each lattice site $i(=1,2,..., N)$ in presence of magnetic field ${\bf B}=B\hat{{\bf z}}$ $$\hat{H}=-...
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Averages of absolute values in Monte Carlo simulation of Ising Model
Consider the 2D Ising model in $0$ field, with Hamiltonian
$$ H=J\sum_{\langle i,j\rangle}\sigma_i\sigma_j$$
The magnetization per spin is defined as
$$M=\frac{1}{N}\sum_i \sigma_i $$
Where $N$ is ...
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Needed Clarification: In Calculation of specific heat of Ising model (Simulation)
I am want to calculate the specific heat for 2D 100x100 square lattice ising model. I have calculated the correlation time, viz., $\tau$. Now I want to calculate the specific heat and error in ...
