Johan Félisaz

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The Kitaev-Preskill cut

Context

The results of this article holds for a specific class of quantum systems: topologically ordered two-dimensional media with a mass gap.

Mass gap

The setup is analogous to the harmonic crystal operator ladder. Instead of seeing the quantum system as an Hamiltonian acting on a lattice with a simpler system at each site (i.e. spin \(\frac12\)), one can see the ground state of the system like the vacuum state. This vacuum can be populated by (quasi)particles. Each particle can either put the system in an excited state (higher eigenenergy), in a less excited state (if it was acting on an excited state ; on the vacuum it just yields zero), or leave it in a different state of same energy. If there is a gap in the energy spectrum of the system between the ground state manifold and the first excited state, then this is a mass gap. This means that the quasiparticles have a mass (the jump in energy cannot be arbitrarily small yet nonzero).

Topological order

The order of the system is topologically protected. This means that any weak perturbation cannot destroy the mass gap, or change the order of the system to another order, even if it could be possible to have it in the first place.

Main result of the paper

\begin{equation*} S_{\text{topo}} = - \log(D) \end{equation*} with \(D\) the total quantum dimension of the medium: \begin{equation*} D = \sqrt{\sum_a d^2_a} \end{equation*}

Prerequisites

The von Neumann entropy

It is defined for a specific region, that is a (usually connected) subset of the sites. Let's assume \(\rho_{\text{total}}\) is the density matrix of the whole system. It is trivial as this is a pure state. However, by tracing out all of the sites outside of the region of interest, \(\rho\) is no longer a pure state density matrix. The von Neumann entropy is defined as followed: \begin{equation*} S(\rho) = \Tr\rho \log \rho \end{equation*}

A few remarks:

  • The logarithm is defined by diagonalizing the operator and taking

the logarihm of the eigenvalues.

  • It is likely that the density operator has zero eigenvalues. We

choose the convention \(0\log 0 = 0\) to avoid singularities.

Main text

Definition of the topological entanglement entropy

Let's assume we are interested in the von Neumann entropy of a given region of boundary length \(L\), large compared to the correlation length. It follows the following infinite size expansion: \begin{equation*} S = \alpha L - \gamma + \ldots \end{equation*}

\(\alpha\) is not universal, and depends on the hamiltonian and geometry of the system. However, as we'll show soon, \(-\gamma\) is universal and topological. It is called the *topological entanglement entropy*. In practise, this expansion is complicated to use to compute the entropy. It is easier to use another expression for it. Consider the following system, composed of four regions A, B, C and D:

\begin{figure}[htp] \centering \begin{tikzpicture} \draw (0, 0) circle(1); \draw (0, 0) -- (0, 1); \draw[rotate=-120] (0, 0) -- (0, 1); \draw[rotate=120] (0, 0) -- (0, 1); \draw (-1.5, -1.1) rectangle (1.5, 1.1); \node at (-0.45, 0.4) {A}; \node at (0.45, 0.4) {B}; \node at (0, -0.5) {C}; \node at (-1.2, 0.8) {D}; \end{tikzpicture} \end{figure}

The topological entanglement entropy is defined as such: \begin{equation*} S_{topo} = S_A + S_B + S_C - S_{AB} - S_{BC} - S_{AC} + S_{ABC}, \end{equation*} where \(AB = A \cup B\) and so on.

Proof that \(S_{topo}\) is topological invariant

Let's assume that the boundary between C and D is slightly deformed. The change in entropy can be written as follow: \begin{equation*} \Delta S_{topo} = (\Delta S_{ABC} - \Delta S_{BC}) - (\Delta S_{AC} - \Delta S_C) \end{equation*} As the regions are large compared to the correlation length, appending the region A to BC have little effect on the change of entropy. Thus the first term on the right hand side vanishes. Similarly, the second one does too, and the entropy remains unchanged.

Now, let's see what happens when a triple point, like the one between B, C and D is deformed. Because the ground state is a bipartite pure state, entropies of both subsystems are equal. That is, \(S_{ABC} = S_{D}\) and \(S_{BC} = S_{AD}\). Thus the change in entropy can be expressed as \begin{equation*} \Delta S_{topo} = (\Delta S_B - \Delta S_{AB}) + (\Delta S_C - \Delta S_{AC}) + (\Delta S_D - \Delta S_{AD}) \end{equation*} Using the same argument as before, the three terms vanishes.

Proof that \(S_{topo}\) is universal

A universal quantity does not change when the hamiltonian is smoothly deformed. Let's assume that the hamiltonian is a sum of local terms, and that the correlation length stays small compared to the regions defined along the deformation. Two cases here:

  • If the changes are far away from a given region, its entropy is not

affected.

  • If the changes are close to a region, using the topological

invariance, we are free to move back the boundaries until it's far enough to neglect the effect of the deformation. After the hamitonian changes, the boundaries can be put back were they were.

A more topological formulation of \(S_{topo}\)

It helps to glue the medium with its time-reversal conjugate at spatial infinity. It can now be seen as a sphere with 4 holes. We can now exploit the composition property of the entropy. On one side, both the medium and its conjugate carry an entropy \(S_{topo}\). On the other hand, the usual formula for the entropy, and the fact that the entropy depends only on the topology (number of punctures) of each region: \begin{equation*} 2S_{topo} = S_A + S_B + S_C - S_{AB} - S_{BC} - S_{AC} = 3S_4 - 4S_3, \end{equation*} where \(S_3\) and \(S_4\) are the entropies of a sphere with 3 or 4 punctures.

Computing \(S_{topo}\) using TQFT methods

Standard proof

Simpler proof using CFT

Questions

  • why does equation 1 hold ?
  • what is a superselection sector ?
  • what is an abelian anyon ?
  • what is a \(SU(2)_k\) Chern-Simons theory ?
  • why ambiguity between terms that scales with \(L\) and constant ones

?

  • what is a mass gap ?
  • chirality in this context ?
  • topological S-matrix ?
  • what's a Wilson loop ?
  • what's a conformal field theory ?
  • modular transformation ?

© 2021, Johan Félisaz. Generated using Gatsby