# Tag Info

## Hot answers tagged ising-model

12

The Ising model is a model, originally developed to describe ferromagnetism, but subsequently extended to more problems. Basically, it is an interaction model for spins. Imagine you have a system which is a collection of $N$ spins. Each spin $S_i$ has two possible states $+1$ or $-1$. Here you can imagine already a possible extension to more states. You can ...

10

There is a result I only heard about recently: it has been proven that computing partition functions for the Ising-model in dimensions > 2 is NP-complete. (The paper can be found at http://www.cs.brown.edu/people/sorin/pdfs/Ising-paper.pdf; a more readable one is here http://www.siam.org/pdf/news/654.pdf - both can be found on the Wikipedia on the Ising ...

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There are two problems at low temperatures. One is intrinsic to the Ising model, the other to the Metropolis algorithm. The Ising one is a bit more serious. Problem with the Ising Model at low temperatures The Ising model of ferromagnetism posits that the electrons in a model can be modeled as little magnetic dipoles, confined to only two quantized spin ...

6

I think that the most prominent example of "prediction before observation" in statistical physics is the Bose-Einstein condensate. It was predicted in ~1925 by, well, Bose and Einstein, obviously. Then after more than ten years it was proposed as an explanation for superfluidity and superconductivity. And the actual BEC of atoms (as a new state of matter) ...

5

Yes mean-field theory is wrong for the one-dimensional case (and wrong for the two and three dimensional cases as well, where the transition exists but the mean-field approximation gets the wrong critical temperature and exponents). In fact it's a typical first year exercise to solve the 1D Ising model exactly using transfer matrices, and I suggest you look ...

5

It seems pretty clear that if you take a very diluted subset of, say, the horizontal line through $0$, then you'll be able to make a Peierls argument. For example, put $h=+\infty$ (worst possible case, amounting to fixing the corresponding spins to $+1$) at all vertices with coordinates (10^k,0), with $k\geq 1$. Then, when removing a contour surrounding a ...

5

1) The Metropolis algorithm is more general than just the sampling scheme used: the sampling scheme that you describe is a "local update," however there exist more general sampling schemes ("cluster updates" like Wolff algorithm, Swendsen-Wang, etc.). The point is that local update sampling isn't inefficient at low temperatures, per se, it's inefficient near ...

5

Maybe I didn't get the question, but the whole point of the discussion of Ising model is not getting zero magnetization. This is a topic of the spontaneous symmetry breaking (further "SSB"). The symmetry you are talking about is broken spontaneously leading to nonzero magnetization. I must admit that I never got into details of Onsager solution, but I know ...

5

Off the top of my head, the example I can think of is the whole work that Boltzmann did. He based his entire theory of statistical mechanics on the concept of indivisible particles (i.e. that all matter is made up out of atoms). Doing this, his theory (using theoretical mathematical methods as you said) was able to predict how the atoms determine the visible ...

4

Let me ask a seemingly naive question first: what is actually a statistical ensemble? Mathematically, it's a prescription of an assignment of probability to every microstate. So this is a probability measure. But we have to be careful with measures defined on infinite systems. You certainly can't blindly work with $\exp(-\beta E)$ factors as energy will ...

4

I am about halfway the most important part of Onsager's paper, so I'll try to summarize what I've understood so far, I'll edit later when I have more to say. Onsager starts by using the 1D model to illustrate his methodology and fix some notations, so I'm gonna follow him but I'll use some more "modern" notations. In the 1D Ising model, only neighbouring ...

4

Seeing as no one is trying to give an answer, I'll take a stab at it myself. Shortly after writing this question I learned (in this cute answer of Raskolnikov's) about Baxter's wonderful book on exact solutions in statistical mechanics. Slowly but surely I realized that Ising model has been solved so many times by some many different methods by virtually ...

4

This is mostly a question of definitions: Spontaneous symmetry breaking occurs when the underlying laws of a physical system have a symmetry, but the ground state does not. For an Ising system with $B=0$, $$H = \sum_{i,j} J_{ij} s_i s_j$$ we can see explicitly that the energy of a state $\{s_i\}$ is precisely the same as the energy of the state with every ...

4

For nearest-neighbor interactions in 1D and 2D, the free energy of the system can be computed analytically. We can then check that this free energy is at its global minimum for a certain state. In 3D, we do not know the free energy analytically, so we have to resort to some kind of simulation (Monte-Carlo probably). If you reach a final state of your ...

4

This is an assumption, but in the case of the of the Ising model it is an obviously correct one. The statement that $m$ is the same at all sites is the assumption that the state of the system respects translational invariance. In general a system may spontaneously break translational symmetry, just as it breaks the $s\rightarrow-s$ symmetry. So when you do ...

3

If I understand you correctly, you seem to be asking the fundamental question of how on earth broken symmetries are possible. After all, there is an apparent contradiction: on one hand, the thermal average $\langle M \rangle= \sum_{\{s_i\}} M \exp(-\beta E)$ must vanish due to symmetry, while on the other hand, physicists claim that the 2D Ising model gives ...

3

It is a direct consequence of the usual (weak) spatial Markov property enjoyed by the Ising model. Namely, if $\Lambda$ is a (deterministic) finite subset of $\mathbb{Z}^d$, and $f$ a local function with support inside $\Lambda$, then the expected value of $f$ in the box $\Lambda$, with a given frozen configuration $\omega$ outside $\Lambda$ only depends on ...

3

I wish I could do your question justice, but I will content myself with a remark on the connection between two of the solution methods mentioned in Barry McCoy's article, namely Baxter's commuting transfer matrix method, and Onsager's original algebraic approach. In a certain sense, these methods have to be considered distinct since Baxter's method is ...

3

The simplest model is the spin-1/2 chain with Majumdar–Ghosh interaction: $$H=\sum_i P_{3/2}(i-1,i,i+1),$$ where $P_{3/2}(i,j,k)$ is the projection operator that projects a state onto the subspace with total spin-3/2 on sites $i,j,k$. The ground states are two dimer states (see the figure on wikipedia Majumdar–Ghosh model): ...

3

Yes there is. Such Ising models are called 'Potts models' (due to the arbitrary number of possible spins). This can be mapped exactly to a graph or network, for which partitioning algorithm or 'community detection' algorithms are useful. For appropriate choices of objective function, the optimal partition will correspond to the ground state of the Potts ...

2

Good question! The reponse really depends on what you 'need': if you're sampling a small system and you're lucky enough to actually be able to count (or estimate...) the number of states $\mathcal{N}(E)$ for a given energy $E$, then you can explicitly work out $Z$. Such algorithms exist for the Ising model, but in the generic case you're not this lucky. ...

2

Well, if it is not too immodest to answer my own question by referring to my own papers, I'd like to suggest reading of the following papers, where the answer is essentially spelled out for a specific model (i.e., the sequence of interactions $\{J_i\}$ following a quasiperiodic substitution). This, I suspect, can be extended to the general case. I'd like ...

2

Should have read the cross-list first - you are already aware of the reference below :) This might be of some use: B P Dolan, D A Johnston and R Kenna The information geometry of the one-dimensional Potts model J. Phys. A: Math. Gen. 35 (2002) 9025–9035 [arXiv:cond-mat/0207180] An information geometry metric is calculated for real h for 1D Potts/Ising ...

2

There is, because there is not a unique way to define a renormalised coupling constant. In general, if you have a system with one bare coupling $g_0$, you define a renormalised coupling in a sensible way, i.e. such that $g = g_0 + \text{loop/quantum corrections} \sim \mathcal{O}(g_0^2).$ However, there is not a unique way to renormalise a theory. In a ...

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As far as I can tell, the only cases in which analytic computations are possible are when: The system is very (very!) small; Dimension = 1, and $h_i$ and $J_{ij}=J_{|j-i|}$ are periodic and of finite-range (and both must probably be small, if you want to be able to get explicit expressions...); Dimension = 2, $h_i\equiv 0$ and $J_{ij}=J$ if $i$ and $j$ are ...

2

That is not a definition of correlation length. (It is a definition of the critical exponent.) The correlation length is defined in terms of the 2-point correlation function of spin observables. Pick points $x$ and $y$ on the lattice, and consider the expectation value $\langle s(x) s(y) \rangle$ of the product of the spin observable at $x$ and the spin ...

2

The energy and the particle number should be both extensive (i.e. $E/N\to\rm{const}$ in the thermodynamic limit, $N\to\infty$). If you calculate the energy of the above system for $H=0$ and, for example, all spins aligned, you get something like $E \propto J_0 N$. Thus, $J_0$ must be $\propto 1/N$.

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