13

This is a very good question. The same operator algebra does not imply that $H(J,h)$ and $H(h,J)$ have the same spectrum. As has been mentioned in Dominic's answer, even the ground state degeneracy is different under the interchange of $J$ and $h$ ($J\gg h$: symmetry-broken two-fold degeneracy, and $J\ll h$ unique ground state), therefore it is impossible to ...


10

After days of thinking, searching, discussions, and testing, I can finally answer my own question now. The answer is much more involved than I expected from such a "simple" XY model (even just for the Ising model)! All "correct solutions/spectrum" stated below are checked against results from exact diagonalization. Simply put, there's nothing wrong with ...


10

I'm not a 100% this will address the question, so this might be more of a long comment than an answer. My main point is, perhaps, that given a Hamiltonian, you'll still want to specify the dynamics to do simulations. Even the Ising model can be simulated in several different ways, all satisfying detailed balance, where the relaxation towards equilibrium is ...


9

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 ...


9

Even though the explicit commutator you wrote is wrong--you should not have conjugated $\Pi$ in the second term-- your conclusion is sound that you cannot possibly satisfy the Born-Heisenberg commutation relation with 2x2 matrices. In fact, there is a general Theorem: The Heisenberg algebra does not admit faithful finite-dimensional (matrix) representations....


8

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 ...


8

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 ...


8

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 ...


8

I don't think that there is a direct, natural physical interpretation (one can of course always cook up something ex post facto). There are however close relations with other topics. Here, I'll try to explain some close links with Markov chains. I'll stick to the case of the one-dimensional Ising model, to keep things concrete, but it should be clear from ...


7

The first thing to realize is that there are no "true" phase transitions (in the sense of non-analytic behaviour of thermodynamic potentials) in finite systems. This is the main difficulty one faces when analysing phase transitions using (most) computer simulation schemes. In particular, such simulations are only reliable as long as the observed correlation ...


7

As this is a list-like question, let me list a few things (without much discussion -- feel free to ask specific questions about individual points). Each item mentions what the Heisenberg model (HM) has as opposed to the Ising model (IM). continuous symmetry vs. discrete symmetry as a consequence: gapless excitations whenever the symmetry is broken (i.e. in ...


7

The correlation length of the 2d Ising model has been computed explicitly. You can find the expression in the famous book by McCoy and Wu. Here's a plot of the inverse correlation length (i.e., $1/\xi$) at various temperatures, taken from this recent review paper: This is only to show the directional dependence, as the radial scale is not the same for all ...


6

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 ...


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) ...


6

Onsager computed 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 is available on the APS website below: (you will need institutional access) L. Onsager, "Crystal Statistics. I. A Two-Dimensional Model with an Order-...


6

I would argue that this maybe due to the way you calculate your autocorrelation. An autocorrelation like that straight line is the result of a large square signal. The Ising model has a phase transition at the critical temperature. Above it, it's disordered; below it, it becomes ordered, which means that the magnetization stops flipping back and forth. This ...


6

Here is how Chandler does his counting: Take the (square) lattice to be of infinite extent or a finite lattice with periodic boundary conditions in both directions. The total number of edges is equal to $4N/2=2N$ if there are $N$ sites. The ground state corresponds to all spins being in the same state (all up or all down). The ground state energy is $-J$ ...


6

I will explain how I measured the spin-spin correlation function for the 2d Ising model. Generalization to more than 2 dimensions should be straightforward as long as you have hypercubic lattices. Just to get the notation straight: Let's use the name $\sigma_{(i,j)}$ for the spin at position $\vec r_{(i,j)} = \begin{pmatrix} i \\ j \end{pmatrix}$. Let's ...


6

Basically, the answer is yes: $H$ is TRI because it is real. Reality condition really means that the Hamiltonian obeys a certain anti-unitary symmetry. In this case, the time-reversal operation is simply $T=K$ where $K$ is the complex conjugation. It is not the usual one($T=K\prod_i i\sigma^y_i$), and in particular $T^2=1$, so there is no Kramers' theorem ...


6

The Ising model is indeed very interesting! In 2-dimensions, there is an analytical solution, in the case of no applied field. It is very complicated, and when it first came out, it consisted of 30 pages of very challenging maths. Only to be 'simplified' down to about 15 pages in the 60s. With an applied field, or in higher dimensions, it is typical to ...


6

The Lattice gas hamiltonian is: $$H = A\sum_{i=1}^N n_i + \frac{1}{2}\sum_{i,j} B_{ij}n_in_j $$ If we want to map this into Ising we just need to take a lattice such that the occupation number can only be $n_i=0,1$ (which is always possible) and then substitute: $$n_i = \frac{1}{2} (1+S_i)$$ $S_i$ can assume values $-1$ or $+1$. The first term of the ...


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

As the size of the lattice gets large, the superposition $$\vert \psi \rangle = \frac{1}{\sqrt{2}}\left(\vert \uparrow \uparrow \uparrow \cdots \rangle + \vert \downarrow \downarrow \downarrow \cdots \rangle\right)$$ in the Ising model becomes unstable to environmental perturbation. Just like Schrodinger's cat (a superposition $\vert \mathrm{live} \...


5

I think that you are really interested in the $q$-state clock model, which is similar to the Potts model, and is defined as follows. Fix an integer $q\geq2$. For each $i\in\mathbb{Z}^d$, let $$ \theta_i \in \bigl\{\frac{2\pi}{q} k\,:\, k\in\{0,1,\ldots,q-1\}\bigr\}, $$ and define the spin at site $i$ by $$ \mathbf{S}_i = (\cos\theta_i,\sin\theta_i) . $$ The ...


5

wsc's answer (i.e., Onsager's computation of the free energy) provides one alternative road to a proof of a phase transition in the Ising model. It implies the existence of a phase transition in dimension 2 (for the nearest-neighbor model). Combined with correlation inequalities, this implies existence of a phase transition in any dimension d≥2, and ...


5

Eq. 2.A.30 is a somewhat non-trivial identity for the ground-state $|\psi_0\rangle$ which only uses the ground-state property that $\eta_q|\psi_0\rangle =0$. (Of course, as the OP has noted, $c_i|\psi_0\rangle \neq 0$.) What we need to show is that $$I=\langle \psi_0 | (c_j+c_j^\dagger)(c_i+c_i^\dagger)|\psi_0\rangle =\delta_{ij}.$$ Using Eq. 2.A.37a, we ...


5

You are correct that for $h=0$ the quantum Ising model reduces to the classical model. Assuming a 2D square lattice this model has been solved exactly by Onsager. It undergoes a phase transition at a certain critical temperature which is signaled by the order parameter $M_2 = \left( \frac{1}{N} \sum_i S_i^z \right)^2$. Below this temperatur the system breaks ...


5

This is an extremely deep question that still isn't fully understood. Such a "macroscopic superposition" $|\uparrow \uparrow \uparrow \uparrow \dots \rangle + |\downarrow \downarrow \downarrow \downarrow \dots \rangle$ is a perfectly valid state in the Hilbert space, and yet we never see it experimentally (at least not without a lot of careful work to ...


5

Here is a book chapter to solve your problem: Kopietz et al. “Mean-Field Theory and the Gaussian Approximation”. Lect. Notes Phys. 798, 23–52 (2010) [PDF].


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