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:
\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