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31
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5answers
4k views

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 ...
14
votes
1answer
131 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 $$ ...
13
votes
2answers
1k 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 ...
10
votes
4answers
967 views

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 ...
8
votes
0answers
143 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 ...
7
votes
3answers
986 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 ...
7
votes
3answers
358 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 ...
7
votes
0answers
275 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
votes
2answers
918 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
2answers
178 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 ...
6
votes
2answers
358 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 ...
6
votes
1answer
466 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 ...
6
votes
1answer
318 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 = ...
6
votes
0answers
87 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 ...
5
votes
2answers
195 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 ...
5
votes
1answer
252 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 ...
5
votes
2answers
4k 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 ...
5
votes
1answer
417 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 ...
5
votes
2answers
210 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
115 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 ...
4
votes
1answer
596 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 ...
4
votes
2answers
94 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
1answer
211 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 ...
4
votes
2answers
213 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
194 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
votes
0answers
103 views

Spin Glass Prince Rupert's Drop

Spin Glass is known to converge to its ground state under Simulated Annealing. The word choice is especially interesting since annealing is also the name of a process performed on actual glass. ...
4
votes
2answers
341 views

Renormalization Group and Ising with d=1 and D=1

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
338 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
3answers
491 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
100 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} ...
3
votes
1answer
110 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
229 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 ...
3
votes
1answer
74 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 ...
3
votes
1answer
440 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 ...
3
votes
1answer
738 views

Some limiting cases of the Heisenberg XXZ model (2/2)

NOTE: Because this was a long question I have split it up in two different questions! For a course on quantum integrability I am reading these notes. (Franchini: Notes on Bethe Ansatz Techniques. ...
3
votes
0answers
53 views

Critical temperature difference between Ising and XY model

The following formula gives the critical coupling (more precisely the ratio of the spin-spin coupling over the temperature) for $O(n)$ models on a triangular lattice: ...
3
votes
0answers
60 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 ...
3
votes
2answers
360 views

1D Ising Model with different boundary conditions

The Hamiltonian for one-dimensional Ising model is given by, \begin{equation} \mathcal{H} = -J\sum_{<ij>} S_iS_j; \quad i,j=1,2,...,N+1 \end{equation} where $<ij>$ denotes that there is ...
3
votes
0answers
89 views

Monte Carlo for Random Bond Ising ferromagnet

The set-up: Consider the Ising model on an $L \times L$ square lattice, where the coupling of each bond is chosen to be $+J$ (ferromagnetic) with probability $(1-p)$ and $-J$ (antiferromagnetic) with ...
3
votes
0answers
67 views

Question about the derivation of an equation in full replica symmetry breaking solution

Using replica method and saddle point method, the free energy of a magnetic system can be expressed as $$-\beta[f]=\lim_{n\to0}\{\frac{-\beta^2J^2}{4n}\sum_{a\ne b}q_{\alpha\beta}^2-\frac{\beta ...
2
votes
1answer
179 views

Ising model Mean field theory and translational invariance

In the Ising model the mean value of any particular spin is: $$ m = \left<s_i\right> = \frac{ \sum_{s_i}e^{-\frac{H}{T} }s_i} { \sum_{s_i} e^{-\frac{H}{T} } } .$$ I read in several ...
2
votes
1answer
178 views

Local minima in Ising model in a Monte Carlo simulation

Is there any way to check whether in a Monte Carlo simulation using Ising model is stuck in any (false) local minima of energy or not, particularly in 3D system ?
2
votes
4answers
180 views

What is the simplest system that has both, discontinous and continous phase transitions?

I am looking the simplest system that has both discontinous phase transition and a continous phase transition between the same phases (you can change one parameter). discontinous transition: first ...
2
votes
1answer
126 views

Research on ground state configuration of Ising model

I want to do mathematical research (algorithm construction and mathematical analysis) on Ising model ground state configuration. From what I know, the state of art research is using graph theory ...
2
votes
1answer
51 views

Numerical Ising Model: Swendsen–Wang algorithm, Percolation theory?

When you look at the original paper of Swendsen and Wang in 1987: "Nonuniversal critical dynamics in Monte Carlo simulations" it is somewhat mentioned that the proposed algorithm uses percolation ...
2
votes
1answer
84 views

What happens to the free energy of the two-dimensional ising model with vortices?

The classical 2d Ising model has a Hamiltonian of the form: \begin{equation} H = -\sum_{m,n = 0}^{M,N} J_1 x_{m,n}x_{m+1,n} + J_2 x_{m,n}x_{m,n+1} \end{equation} The partition function for the model ...
2
votes
1answer
153 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. ...
2
votes
1answer
194 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 ...
2
votes
1answer
43 views

Does an Ising lattice that returns to equilibrium create a current by induction?

Consider you have an Ising lattice with a dominant up component out of thermal equilibrium, that's your initial state. The down spins want to flip up and align with the ups, and they'll do so until a ...
2
votes
1answer
115 views

What's the meaning of the coupling change after a renormalization (in the 1-dim Ising Model)?

What does it mean that after the theory (1-dim Ising model here, but the question is general) is renormalized one time and $g_i\rightarrow g_i'$, that the couplings are weaker, even if the theory ...