Exploring the 2-D Ising Model & Its Continuum Limit

[hilbert2]

TL;DR Summary

About the continuum limit of Ising lattice

I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including

https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf
https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf
As far as I understood it, the state of the lattice is described with the spin variables $\sigma_\mathbf{x}$, and additionally the disorder variables $\mu_{\mathbf{x}}$, where the position vectors $\mathbf{x}$ can form either a discrete lattice or a continuum.

The fermion variables $\psi_{a,\mathbf{x}}$ are defined as

$\psi_{a,\mathbf{x}} = \sigma_{\mathbf{x}}\mu_{\mathbf{x}+\mathbf{e}_a}$

where $a$ seems to be the spinor index (taking values from 1 to 4) and the $\mathbf{e}_a$ are the four lattice unit vectors.

Some questions:

1. How can the $\psi_{a,\mathbf{x}}$ be a complex field obeying the Dirac equation in the continuum limit if $\sigma_{\mathbf{x}}$ and $\mu_{\mathbf{x}}$ are binary variables taking only values $\pm 1$ ? I would expect the components of a Dirac spinor to be able to have a plane-wave dependence on the position $\mathbf{x}$, so at least the complex phase of $\psi$ should be able to be something other than $0$ or $\pi$.

2. Are there other ways to define the $\psi_{a,\mathbf{x}}$ without changing anything physically relevant in the model? In other words, is it possible to reconstruct the $\sigma_{\mathbf{x}}$ for all $\mathbf{x}$ if I know $\psi_{a,\mathbf{x}}$ for all $\mathbf{x}$, or do several sets of $\sigma_{\mathbf{x}}$ produce the same set of $\psi_{a,\mathbf{x}}$ (or the other way around)?

 

[hilbert2]

Ok, now I think I got it, the $\sigma$:s are not just plain numbers $\pm 1$, but are related to Pauli matrices which have eigenvalues $\pm 1$... But can the $\sigma_{\mathbf{x}}$ at some $\mathbf{x}$ be a superposition of several spin matrices?

Edit: The last sentence should probably read "can the spin state at some position $\mathbf{x}$ be a superposition of several eigenvectors of a spin matrix?".

 

[king vitamin]

I have a lot of familiarity with this particular critical point from many different perspectives, but I have to admit that much of the approach in that link is not an approach I know particularly well. But there are so many ways to solve the 2D Ising model that probably very few people know all of them. I am familiar with the high/low temperature expansions and ("Kramars-Wannier") duality in the first set of lecture notes, as well as the CFT approach mentioned in the second link. But the direct mapping of 2D classical Ising variables to fermionic variables isn't something I know particularly well.

Here is my favorite perspective on this, and you can find this in a lot of sources. I'll mention ahead of time this classic review article, as well as textbooks by Sachdev, Fradkin, these McGreevy lecture notes (PDF), etc. I list so many pedagogical sources partially to support my point that I think this is generally considered the easiest way to see free fermions emerge as a description of this particular critical point.

The main point is the equivalence of the following two partition functions:
$$
\mathcal{Z}_1 = \sum_{\{ s_i = \pm 1 \}} e^{-\beta_\text{C} H_\text{C}} = \mathcal{Z}_2 = \lim_{\beta \rightarrow \infty} \mathrm{Tr} e^{- \beta H}.
$$
Here, the first partition function describes a classical statistical mechanics problem - in fact, it is an anisotropic version of the 2D Ising model which your links are discussing:
$$
\beta_\text{C} H_\text{C} = -K_x \sum_{i} s_i s_{i + x} - K_y \sum_{i} s_i s_{i + y}.
$$
I am not using sigmas because I want to stress that this is not a quantum problem, and everything commutes. The variables $s_i$ are really just numbers $\pm1$.

On the other hand, the second partition function is really the trace over a quantum density matrix, with a quantum Hamiltonian usually called the "one-dimensional transverse-field Ising model":
$$
H_\text{Q} = - J \sum_{i} \sigma^z_i \sigma^z_{i+1} - \sum_i \sigma^x_i.
$$
These two Pauli matrices do not commute with each other, as usual. In both theories, I take the number of sites to infinity.

Now, the statement of duality is that the finite-temperature classical partition function of the 2D model is equal to the zero-temperature quantum partition function of the 1D model provided one takes the following limit:
$$
K_x \rightarrow 0, \quad K_y \rightarrow \infty, \quad J = K_x e^{2 K_y}
$$
So the quantum model is really an extreme anisotropic limit of the classical one, but that doesn't end up changing any of the conceptually or qualitatively interesting aspects of the model and its phase transition. In particular, there is a zero temperature phase transition of the quantum model for some critical #J_c#, above which the ground state has nonzero polarization in the z direction, and below which there is zero polarization in the z direction. There is a lot you can do with this duality, but I will go on to fermions since this is getting long.

My reason for preferring the quantum model is that the Kramars-Wannier duality and the emergence of fermions is a lot easier to see here in my opinion. This was first noted by Lieb, Schultz, and Mattis ([1], [2]).
The disorder operators are obtained by the following transformation of the quantum model:
$$
\mu^z_{i+1/2} = \prod_{j\leq i} \sigma^x_j, \qquad \mu^{x}_{i+1/2} = \sigma^z_i \sigma^z_{i+1}.
$$
Here, I'm defining new variables on the links of the 1D lattice. You can check that these satisfy the same commutation relations as $\sigma^z$ and $\sigma^x$, and that the transformation maps the Ising model to itself:
$$
H_\text{Q} = - \sum_{i} \mu^z_{i-1/2} \mu^z_{i+1/2} - J \sum_i \mu^x_{i+1/2},
$$
but now the coupling $J$ appears on the transverse field term. So now the ordered phase is at small $J$ and the disordered phase is at large $J$. You can probably also see that the operator $\mu^z_{i+1/2}$ creates a "kink" in the spins between the sites $i$ and $i+1$ (it flips the z direction of all spins $\sigma^z$ to the left of the operator). So $\mu^z$ is the disorder operator.

We now show the emergence of fermions. As your link says, this is obtained by taking a product of the spin operator and the disorder operator:
$$
a_i = \mu^z_{i - 1/2} \sigma^z_i \\
b_i = i \mu^z_{i - 1/2} \sigma^z_i \sigma^x_i
$$
These are actually two independent Majorana operators, $\{ a_i, a_j \} = \{ b_i , b_j \} = 2 \delta_{ij}$, $\{a_i,b_j\} = 0$. After writing this in terms of a single complex fermion $c_i = a_i + i b_i$, the Hamiltonian is completely quadratic and may be diagonalized rather trivially. This transformation from strings of spins to fermions is known as the Jordan-Wigner transformation.

Notice that I have not taken the continuum limit of the quantum model. I have instead derived a theory of fermions hopping on a discrete 1D lattice. Now, if I want, I may take the continuum limit if I am near the phase transition, and then I can write this as a continuum quantum field theory of fermions. This is one way to obtain the fermionic field theory given in your notes, and only then do I get a continuum Dirac equation rather than a tight-binding band dispersion.

One can use this theory to obtain essentially all of the universal properties of the original 2D classical Ising model (the anisotropic limit really doesn't change anything). There are some nontrivial (and rather beautiful) mathematical details which go into computing the correlation functions of the $\sigma^z$ variables (which map to correlation functions of the classical variables $s_i$), but it can be done.

Ok, now I think I got it, the $\sigma$:s are not just plain numbers $\pm 1$, but are related to Pauli matrices which have eigenvalues $\pm 1$... But can the $\sigma_{\mathbf{x}}$ at some $\mathbf{x}$ be a superposition of several spin matrices?

In my opinion, in application to the classical Ising model (which is what your link is about), you should really think of those $\sigma_{\mathbf{X}}$'s as just numbers (which is why I avoided using sigma above).

 

[hilbert2]

Thanks a lot for the comprehensive answer. Another thing that I was wondering was what happens if the continuum Ising lattice has a totally discontinuous spin as a function of position, i.e. every area/interval contains an infinite number of points with both spin up and spin down. Would the total energy of the system be infinite in that case?

 

[king vitamin]

I do not know of a good way to write down the Ising model "continuously," which is to say, I do not know of a good way to take the continuum limit of the model while the variables take the discontinuous discrete values $s_i = \pm1$. I can show you how to rewrite the Ising partition function in terms of an auxiliary continuous real scalar field and then take the continuum limit, or I can do the transformations in my last post which then allow a continuum limit, but I would say that the Ising model in its usual definition cannot be taken to the continuum limit right away - you need to transform to new continuous variables first.

In fact, even in the 2D XY model (which does have a continuously variable spin direction), you famously get in trouble for taking the continuum limit too quickly. You may have heard of the Kosterlitz-Thouless transition, which occurs in the 2D XY model - this transition can only be obtained if you allow particular "vortex" configurations, but you will miss these configs if you take the continuum limit in the wrong way, as these configurations become divergent. In some sense, this is the problem with a continuum limit of the Ising model - a domain wall in the continuum will result in a divergent contribution to the energy.

 

[hilbert2]

My project leader at work approaches this by thinking of the 2d lattice positions as forming a network in complex plane and the fermion field obeying a finite difference version of Cauchy-Riemann equations. Then you can assume the continuum limit of the field to be a holomorphic function and prove statements with this assumption.