How to see that the Ising CFT has $c = 1/2$ while the quantum XY CFT has $c = 1$ via Jordan-Wigner?

Question

It is well known that the CFT at the critical point of the 1+1d transverse field Ising model has central charge 1/2. This can be attributed to the fact that, after a Jordan-Wigner transformation, the critical point consists of a Majorana mode which goes gapless.

On the other hand, the quantum XY model can also be solved by the Jordan-Wigner transformation and becomes a theory of free complex fermions hopping on the lattice. The CFT for this model is equivalent to a free boson (for example, this can be seen via bosonization. Another version consists of taking the classical 2d XY model and taking its continuum limit, which becomes a Gaussian scalar field). It then makes sense that the CFT describing this model has $c = 1$.

My question is the following: how do I see the difference in central charge from the lattice fermion theories? After doing Jordan-Wigner on both models, they both consist of the same degrees of freedom: complex fermions on each lattice site. However, the above implies that in the Ising case, only one of the underlying Majorana species participates in the long-distance physics, whereas in the XY case, they both participate and add their central charges. This isn't obvious to me from the forms of the fermion Hamiltonians. For concreteness, these are the Jordan-Wigner duals of the models which I use. For Ising, I have $$ H_{Ising} = - J \sum_{j} i \beta_{j+1} \alpha_j - h \sum_{j} i \beta_j \alpha_j$$ and for XY I have $$ H_{XY} = - J \sum_{j} i \beta_{j+1} \alpha_j + i \beta_{j} \alpha_{j+1}$$ where $\alpha_j, \beta_j$ are a pair of Majorana fermions associated with site $j$.

Answer 1

Try drawing a picture.

In a lattice of fermionic sites, there are two Majoranas $\alpha_j$ and $\beta_j$ on each site. Draw a dot for each of these Majoranas.

Connect two Majorana "dots" $\gamma_1,\gamma_2$ via a "bond" if there is a term in the Hamiltonian of the form $i\gamma_1 \gamma_2$. Using this prescription, "draw" the two fermion Hamiltonians that you wrote down.

The picture is not that in the Ising case, only one of the Majoranas participate in the long distance physics. Instead, in the Ising case, you have a single Majorana chain whereas in the XY case you have two decoupled Majorana chains of half the original size. The presence of two Majorana chains immediately tells you that $H_\text{XY}$ is two copies of $H_\text{Ising}$. This means that the central charge must be doubled.

Answer 2

I think the following quote from your post is the incorrect way of thinking about things:

However, the above implies that in the Ising case, only one of the underlying Majorana species participates in the long-distance physics, ...

This is incorrect. In the scaling limit $h \to J$, the Jordan-Wigner'ed TFIM (sometimes called the "Kitaev chain") is described by a two-component Majorana spinor $\psi = (\psi_1, \psi_2)^T$. If you carefully take the continuum limit of your lattice Hamiltonian, you will get a continuum Hamiltonian of the form $$ H = \frac{v}{2} \int dx \, \psi^T \left[ -i \sigma^x \partial_x + m \sigma^y \right] \psi $$ where $v = 2Ja$ is the velocity of low-energy excitations at the critical point, and $m \propto |h - J|$ is zero at the critical point. If you follow this step-by-step, you will see that $\psi_1$ and $\psi_2$ are just the continuum analogs of your $\alpha$ and $\beta$ respectively. (CFT books commonly perform the change of basis $\psi \to \mathsf{H} \psi$, where $\mathsf{H}$ is the Hadamard matrix, so that the resulting fermions are decoupled. The two components of $\tilde{\psi} = \mathsf{H} \psi$ are just the left and right-moving fermions.) The point is that both species of microscopic Majorana operators are important at the critical point, and contribute to the $c = \frac{1}{2}$ Majorana fermion CFT. Of course, it would be very weird if this were not the case.

On the other hand, you know that the XY model Jordan-Wigners to a simple tight-binding model, and the low-energy excitations are just the left and right-moving complex fermions near the two Fermi points. So it's clear that the XY model is described at long wavelengths by the complex fermion CFT with $c=1$.

The upshot of this is that you should not infer the CFT just from which microscopic operators enter the Hamiltonian; indeed, for any 1d spin-chain with at least a global $\mathbb{Z}_2$ symmetry, you can Jordan-Wigner into some sort of Majorana chain, and this faulty logic would imply that they all would be described by the same CFT. Instead, since CFTs in condensed matter describe low-energy physics, it is crucial to think about the structure of low-energy excitations to guess the correct CFT for a given lattice model.