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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 ...


8

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 ...


5

The reason that the systems energy is lowered when spins are aligned comes from the coulomb (electrostatic) potential, not magnetism. The details are non-trivial, but basically if you combine the Pauli Exclusion Principle with the Coulomb Potential you find that the ground state occurs when spins are aligned. The repulsion with two macroscopic magnets is a ...


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 ...


4

After thinking about it I must say it is not as simple as I thought it would be. The JW transformation on the transverse Ising model contains quite a few subtleties. So to proceed, 1) Take your ground state for ANY $h$ expressed in the spinless fermion language. I stress ANY because this condition is true always - it's not just for $h<1$. Now this is ...


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

Onsager found the partition function of the 2D periodic square lattice (toroidal boundaries) Ising model. It is arguably one of the most elegant proof of modern statistical mechanics. The original paper can be found here (you will need institutional access) L. Onsager, "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", ...


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 ...


3

This doesn't seem quite like an appropriate question for a math site, so I guess it will be transferred shortly, but anyway, the answer to your question is that the dipolar interaction is by no means neglected in theories of ferromagnetism, it is the long-range interaction that governs the appearance of domain walls. The short-range exchange interaction, on ...


3

It is only exactly at the critical temperature that this CFT result works. You haven't mentioned if you have used the critical temperature when you did the monte-carlo. At/near critical point, autocorrelation time becomes huge. (If I am not mistaken, autocorrelation time must blow up exactly at critical temperature, however it is cut-off due to finiteness ...


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

Let me repeat/reproduce some of the most important definitions. $d~=~$ dimension of lattice. $n~=~$ number of blocks between block $I$ and block $J$. $r~=~$ length of a block measured in units of lattice spacings. $r^d~=~$ number of lattice points in a block. $nr~=~$ distance between between block $I$ and block $J$ measured in units of lattice spacings. ...



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