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-...

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Why is it hard to solve the Ising-model in 3D?

The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model ...
21
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1answer
816 views

List of known universality classes

I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical ...
18
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2answers
2k views

Ising model for dummies

I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge. Can you direct me to some documentation on it ...
14
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1answer
208 views

Phase Transition in the Ising Model with Non-Uniform Magnetic Field

Consider the Ferromagnetic Ising Model ($J>0$) on the lattice $\mathbb{Z}^2$ with the Hamiltonian with boundary condition $\omega\in\{-1,1\}$ formally given by $$ H^{\omega}_{\Lambda}(\sigma)=-J\sum_{...
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4answers
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How many Onsager's solutions are there?

Update: I provided an answer of my own (reflecting the things I discovered since I asked the question). But there is still lot to be added. I'd love to hear about other people's opinions on the ...
10
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1answer
340 views

2d Ising model in CFT and statistical mechanics

When I recently started to read about conformal field theory, one of the basic examples there is the so called Ising model. It is characterized by certain specific collection of fields on the plane ...
10
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0answers
205 views

Measure of Lee-Yang zeros

Consider a statistical mechanical system (say the 1D Ising model) on a finite lattice of size $N$, and call the corresponding partition function (as a function of, say, real temperature and real ...
9
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3answers
2k views

Zero magnetization of spin model without external magnetic field

For a given Hamiltonian with spin interaction, say Ising model $$H=-J\sum_{i,j} s_i s_j$$ in which there are no external magnetic field. The Hamiltonian is invariant under transformation $s_i \...
9
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2answers
887 views

Confusion about duality transformation in 1+1D Ising model in a transverse field

In 1+1D Ising model with a transverse field defined by the Hamiltonian \begin{equation} H(J,h)=-J\sum_i\sigma^z_i\sigma_{i+1}^z-h\sum_i\sigma_i^x \end{equation} There is a duality transformation which ...
9
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3answers
603 views

Estimating Partition functions

I have a finite state ensemble with an energy functional (you can think of it as an ferromagnetic Ising model if you like), and I need very careful estimates of the partition function. What methods ...
8
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1answer
136 views

What is the momentum canonically conjugate to spin in QM?

In Kopec and Usadel's Phys. Rev. Lett. 78.1988, a spin glass Hamiltonian is introduced in the form: $$ H = \frac{\Delta}{2}\sum_i \Pi^2_i - \sum_{i<j}J_{ij}\sigma_i \sigma_j, $$ where the ...
7
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2answers
1k views

Numerical Ising Model - Wolff algorithm and correlations

I'm doing some numerical Monte Carlo analysis on the 2 dimensional Ising model at the critical point. I was using the Metropolis 'single flip' evolution at first with success, though it suffers from ...
7
votes
1answer
768 views

Mean-field theory in 1D Ising model

A mean-field theory approach to the Ising-model gives a critical temperature $k_B T_C = q J$, where $q$ is the number of nearest neighbours and $J$ is the interaction in the Ising Hamiltonian. Setting ...
7
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2answers
244 views

Is there a spin glass version of Prince Rupert's Drop?

Spin Glasses are known to converge to their ground state under Simulated Annealing. The word choice is especially interesting since annealing is also the name of a process performed on actual glass. ...
7
votes
1answer
396 views

What is the information geometry of 1D Ising model for a complex magnetic field?

Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by $$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
7
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0answers
357 views

Information geometry of 1D Ising model in complex magnetic field regime

Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by $$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
6
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2answers
1k views

Efficiency of Metropolis algorithm

Context is 1D Ising model. Metropolis algorithm is used for simulate that model. Among all possible spins configurations (states) that algorithm generates only states with the desired Boltzmann ...
6
votes
1answer
198 views

Ising model for ferromagnetism is not intuitive

In the Ising model for ferromagnetism a lower energy is assumed when two spin magnetic dipoles are aligned parallel to each other and the energy is higher when they are antiparallel. If I take two ...
6
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2answers
375 views

Ising model observables

Is there a formula or equation relating $\langle E\rangle$ and $\langle M\rangle$ (average spin per site) and $\langle E^2\rangle$ to temperature $T$ for the square lattice Ising model at zero ...
6
votes
1answer
597 views

Interpretation of the 1D transverve field Ising model vacuum state in a spin-language

The 1D transverse field Ising model, \begin{equation} H=-J\sum_{i}\sigma_i^z\sigma_{i+1}^z-h\sum_{i}\sigma^x_i, \end{equation} can be solved via the Jordan-Wigner (JW) transformation (for further ...
6
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1answer
139 views

Wick's theorem and transverse field Ising model

I am trying to understand calculation of correlation function in the ground state of the Transverse Field Ising model, from the following book, which is freely available: http://link.springer.com/book/...
6
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2answers
8k views

What is the definition of correlation length for the Ising model?

The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$ where $\nu$ is the critical exponent. Is this the formal definition of correlation length?...
6
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1answer
748 views

Percolation in a 2D Ising model

For a 2D Ising model with zero applied field, it seems logical to me that the phases above and below T_c will have different percolation behaviour. I would expect that percolation occurs (and hence ...
6
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1answer
208 views

Why are large scale structures isotropic in the Ising model?

I have at least a qualitative understanding of why the critical state of the Ising model is scale invariant, by arguments to do with renormalisation, which I understand only very roughly. However, in ...
5
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2answers
149 views

Time reversal symmetry of transverse field Ising model

Is the transverse field Ising model time-reversal invariant? Specifically consider a non-integrable variant: \begin{equation} H = -J \sum_i^{L-1} \sigma_i^z \sigma_{i+1}^z + g \sum_i^L \sigma_i^x + h ...
5
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2answers
1k views

Subtleties in the exact solution to the 1D quantum XY model, in particular the Bogoliubov transformation

I am writing programs to construct the spectra of models with known exact solutions, and soon noticed some subtleties that are not often mentioned in most references. These subtleties are not ...
5
votes
1answer
233 views

What are alternative ways to think about transfer matrix as used in Ising model?

I recently learned about how to find the partition function of Ising model using Transfer Matrix method. At my level of understanding things, it is a trick that happens to work! I would like to ...
5
votes
1answer
747 views

A simple model that exhibits emergent symmetry?

In a previous question Emergent symmetries I asked, Prof.Luboš Motl said that emergent symmetries are never exact. But I wonder whether the following example is an counterexample that has exact ...
5
votes
1answer
339 views

The strong Markov property of Gibbs measures in 2D Ising Model

My background is that of a mathematician. I have a question about the two Dimensional Ising Model. I think the terminology I use is similar to the physical. I'm trying to understand the following ...
5
votes
2answers
302 views

Toric Code and Random Bond Ising Model

It was established by Dennis, Kitaev et al. that the 2D Toric Code can be mapped to a 2D Random Bond Ising Model. The original derivation was given in the paper "Topological quantum memory" which ...
4
votes
1answer
274 views

Percolation and number of phases in the 2D Ising model

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive. After a long time I came back to try to understand an article on the Ising model. ...
4
votes
2answers
112 views

Determining the probability of a particular site having a particular spin in an Ising model

Given an Ising model, we have the energy formula: $E= - \sum_i h_i S_i - \sum_{i \neq j} J_{ij} S_i S_j$ and we have the probability of a given state, given the energy of that state and the ...
4
votes
2answers
547 views

Mean field theory = large-N approximation?

Wikipedia entry of 1/N expansion (or 't Hooft large-N expansion) mentions that It (large-N) is also extensively used in condensed matter physics where it can be used to provide a rigorous basis ...
4
votes
1answer
372 views

Scaling with the Ising Model

I am stuck with one formula in the CFT book by Di Francesco and al. Chapter 3. Equation 3.46 third step, for those who don't have the book, he integrates out degrees of freedom from the Ising Model by ...
4
votes
1answer
201 views

Convergence of periodic single fermion operators

First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would ...
4
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0answers
74 views

${\phi}^4$ description of Ising ferromagnet

Suppose the coupling between two spins is $C_{i,j}<0$, then the classical partition function is given by $$Z=\sum_{\{s_i\}}e^{\sum_{i,j}s_iK_{ij}s_j+h\sum_{i}s_i}$$ where $K_{ij}=-\beta C_{ij}$ and ...
4
votes
1answer
205 views

Temperature in the Hamiltonian limit

There is a well known connection between statistical mechanics in D spatial dimensions and quantum field theory in D-1 spatial dimensions. Changing the temperature in statistical mechanics corresponds ...
4
votes
0answers
153 views

What real experimental systems are well-described by Glauber-Ising spins?

I'm hoping for references to actual physical systems in which all or at least most of the following can be simultaneously characterized: the spin flip rate, the temperature, and a relaxation or ...
4
votes
2answers
480 views

Renormalization Group and Ising with d=1 and D=1 [closed]

I have a question about the results of RG on Ising model. I know it's possible to obtain two couple of relations $K'(K)$, $q(K')$ $K(K')$, $q(K)$ between the coupling costants. My problem arise ...
3
votes
3answers
358 views

Results of Statistical Mechanics first obtained by formal mathematical methods

I have a question that seems natural in Physics and Mathematics mainly in Statistical Mechanics of Equilibrium. Results that are proven by formal mathematical methods that were already seem intuitive ...
3
votes
1answer
183 views

Critical temperature and lattice size with the Wolff algorithm for 2d Ising model

When I run my implementation of the Wolff algorithm on the square Ising model at the theoretical critical temperature I get subcritical behaviour. The lattice primarily just oscillates between mostly ...
3
votes
1answer
171 views

Does the q-states Potts become the XY model in large q state?

I have met several times in papers, the order of the phase transition of the $q$-state Potts model depends on $q$. E.g., in two dimensions, for $q = 2$ (the Ising model), $3$, $4$ the order-disorder ...
3
votes
1answer
115 views

Why we never observe superposition of up and down ferromagnetic ground state of Ising model?

I thought it is due to spontaneous symmetry breaking. But isn't that because we never observe the superposition states, then we claim that there is spontaneous symmetry breaking. It looks like ...
3
votes
3answers
1k views

Ising Ferromagnet: Spontaneous symmetry breaking or not?

In explaining/introducing second-order phase transition using Ising system as an example, it is shown via mean-field theory that there are two magnetized phases below the critical temperature. This ...
3
votes
1answer
328 views

NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms

From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other ...
3
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1answer
130 views

Freedom in the Choice of a Beta Functions in RG

Assume we're given a certain statistical model, say the infinite range Ising model \begin{equation} H_{N}\{\vec\sigma_{N}\}~=~ - \frac{x_{N}}{2N} \sum_{i,j =1}^{N} \sigma_{i} \sigma_{j} \end{...
3
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1answer
131 views

Random bond Ising model and computational efficiency

If you want to find the ground state of the 2d random bond Ising model (no field), a computationally efficient algorithm exists to do it for you (based on minimum weight perfect matching). What about ...
3
votes
1answer
5k views

Force from solenoid

I'd like to approximate the force from a solenoid, or at the very least find a formula which is proportional to the force so that I can experimentally find the constant for my particular case. ...
3
votes
1answer
96 views

Explicit definition of the energy operator in the Ising model

I've simulated a few 2d Ising models at critical temperature on triangular lattice and I'm now trying to check that the correlation functions are right. I alraedy did it for the spin operator ($\sigma$...
3
votes
1answer
735 views

1D Ising Model (NN and NNN interactions) with 2 transfer matrices

I've tried an alternative solution for finding the partition function of this model. So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a ...