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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?

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  • $\begingroup$ there are recent studies on 3+1D topological order in this Ref: arxiv.org/abs/1404.7854. Using Modular SL(3,Z) generalized from Modular SL(2,Z) of 2D; and using Dijkgraaf-Witten lattice TQFT, and string-string braidings to characterize 3+1D topological order. $\endgroup$ – wonderich Jun 24 '14 at 2:08
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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.

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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.

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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.

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