How to understand topological order at finite temperature?

Question

I have heard that in 2+1D, there are no topological order in finite temperature. Topological entanglement entropy $\gamma$ is zero except in zero temperature. However, we still observe some features of topological order in fractional quantum Hall effect, such as fractional statistics, symmetry factionalization. So what is the meaning when people say "there is no topological order at finite temperature in 2+1D"? And what is the situation in 3+1D?

Answer 1

Here is a field theory point of view:

A gapped system has topological order if in the IR it flows to a nontrivial TQFT. We can model temperature in Euclidean field theory by using a circular (imaginary) time. The circumference of this circle is the inverse temperature $\beta$. Thus, the limit when the circle is large is the zero temperature limit, and when the circle is small is the high temperature limit.

The question is therefore whether there are $d$-dimensional TQFTs which compactify along a circle to give a nontrivial $d-1$-dimensional TQFT. This is impossible for invertible TQFTs (ie. SPT or other short-range entangled phases), since they are cobordism invariant, and we may fill in the circle as long as there is no flux about it (ie. the chemical potentials are all zero). However, something noninvertible like a $\mathbb{Z}_2$ gauge theory in 3+1d (3d toric code) can persist to finite temperature. Indeed, it will compactify to a 2+1d $\mathbb{Z}_2$ gauge theory (2d toric code).

One important caveat to this argument is that in talking about these things we take all gauge symmetries as sacrosanct. In reality, thermal flucations destroy topological order in 3+1d toric code, but this sort of argument ensures that it happens at a nonzero temperature. (Also the factor of two they discuss in the topological entanglement entropy is the gauge holonomy around the thermal circle.)

Also note that topological degeneracy is indistinguishable from symmetry breaking in 1+1d, so there is no 2+1d topological order at finite temperature (and infinite system size). Otherwise, compactification would give us a nontrivial 1+1d TQFT. Fractional quantum Hall systems are able to avoid this by being finite size and being chiral.

Also, if you like, you can think about (abelian) topological order (like toric code) as a spontaneous breaking of a 1-form symmetry, for which a Mermin-Wagner theorem prevents a finite temperature ordered phase in 2+1d.

Answer 2

In 2+1D, it is a question of time scales. For instance, as a thought experiment, consider a fractional quantum Hall phase on a torus coupled to a bath. At zero temperature, there is a topological degeneracy which can be used to encode quantum information with an infinite lifetime. If the bath is at finite temperature, it will lead to thermally activated processes in which quasiparticles are created and diffuse around the cycles of the torus, changing the state. This will lead to a finite lifetime (decoherence) of the qubit. But as the temperature goes to zero the timescale for these processes will diverge, so practically the qubit can remain experimentally detectable. Similar reasoning applies to the degeneracy of a pair of non-Abelian quasiparticles probed by a braiding experiment, or the quantization of the Hall conductance.

Fortunately, the rate of these errors should decay exponentially with temperature - at least if equilibrium conditions hold. The large ratio of the quantum Hall gap (many Kelvin) compared with experimental temperatures of 10s of mK explains the precise quantization features observed thus far.

Answer 3

I would define a Hamiltonian $H$ to be topologically ordered at a finite temperature $T$ if its thermal density matrix $\rho = e^{-H/T}/Z(T)$ is topologically ordered at that temperature.

This of course simply pushes the question back to the question of how to definite topological order for a mixed rather than pure state. As discussed in section 4 of this blog post, this is a rather subtle question and I don't think there's a universally agreed-upon definition. One possible definition is that a mixed state is topologically trivial iff it can be written as a convex combination of (i.e. probability distribution over) topologically trivial pure states. Another possible definition is that a mixed state is topologically trivial iff its purification is. Physically, this roughly means that the entire "universe" of the system together with the heat bath to which it's coupled is topologically trivial. I'm not sure if these two definitions are equivalent.

Answer 4

I want to add another perspective, which has been hinted to in Ryan's and mike's responses.

There is no 2D topological order at finite temperature.

This refers to that 2D topological orders are not stable (loosely speaking, they don't last) at finite temperature. The non-stability can be attributed to the existence of string-like operators, which create anyons. The creation of anyons by these string operators only involves a constant energy cost regardless of the length of the string. Hence, at finite temperature (meaning that you have energy to randomly trigger the creation of anyons), anyons will proliferate. The proliferation destroys the topological order.

By the same logic, 3D topologically ordered systems are not stable either, because of the existence of string operators.

On the other hand, the toric code in 4D is stable at finite temperature, because its excitations are created by operators that incur energy costs proportional to the size of the operators.

One caveat in 3D is the Haah codes, which are fractonically ordered, not topologically ordered. But, they are still considered not stable at finite temperature in terms of a self-correcting code.

A thorough discussion can be found in this review paper.