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We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many-body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter.

My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this?

Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"?

Edit:

While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena.

However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this.

On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?

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Topological order is called topological because the ground state degeneracy depends only on the topology of the space, see en.wikipedia.org/wiki/Topological_degeneracy –  Luboš Motl Jul 11 '13 at 13:59
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Strictly speaking, describe "topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this" is not quite correct, since such a description only apply to free fermions. The real topological insulators in experiments have Coulomb interaction. –  Xiao-Gang Wen Jul 26 '13 at 17:12

3 Answers 3

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As you have mentioned, topological insulators (TI) are "topological" because they can not be smoothly connected to trivial band insulators without closing the band gap (and without breaking certain symmetry). Simply generalize this to the many-body case, we may say that the topologically ordered states are called "topological" because they can not be smoothly connected to the trivial product state without closing the many-body gap.

To gain a better understanding, one should realize that "topology" is a complement to "geometry". By geometry, we mean that there is a sense of measurement of the distance and angle etc., and the shape and the size of the object matters. While by topology, we mean that one can continuously deform the object, and the shape or the size does not matter. So the topological properties are those properties that can endure continuous deformation of the state (by continuity we mean without encountering a quantum phase transition). To protect the topological properties against deformation, a gap between the ground states and the excited states is always required. So topological property is only defined for gapped quantum matters (both TI and topological order are within this scope). On the other hand, gapless quantum matters do not have topological properties, and their properties are geometrical.

The topological/geometrical distinction is also reflected from the mathematical tools we used to study the physics. For gapped quantum matters, we use topological tools like homotopy, cohomology, K-theory, category theory etc. For gapless quantum matters, we use geometrical tools like gravity theory (AdS/CFT).

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It seems that topology is not relevant in the gapless case, but, however, could we imagine that topology could change in the gapless case - that is, degenerate ground states corresponding to different topologies ? –  Trimok Jul 11 '13 at 18:54
    
@Trimok The ground state degeneracy is ill-defined for gapless system. The gapless "topological order" may be called the quantum order. –  Everett You Jul 12 '13 at 1:30

The 'topological' in topological order means 'robust against ANY local perturbations'.

According to such a definition, topological insulator is not 'topological' since its properties are not robust against ANY local perturbations, such as the perturbation that break the U(1) and time reversal symmetry. So a more proper name for topological insulator is 'U(1) and time-reversal symmetry protected insulator', which is one example of SPT order.

Some example of topologically ordered states (in the sense of 'robust against ANY local perturbations'):

1) $\nu=\frac{1}{3}$ FQH state

2) $Z_2$ spin liquid state

3) $\nu=1$ IQH state

4) $E_8$ bosonic QH state

The example 3) and 4) have no non-trivial topological quasi-particles (ie no non-trivial statistics, no non-trivial topological degeneracy), but have gapless edge state that is 'robust against ANY local perturbations'.

-- Edit -- (I lifted some discussions below to here):

There are two kinds of topology in math. The "topology" in "topological order" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category. The "topology" in "topological order" is different from the "topology" in "topological insulator". The "topology" in "topological insulator" is related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical".

The correct way to describe any gapped phases (such as topological orders and topological insulators) is to use the first kind of topology -- "quantum" topology, because the gapped phases are usually interacting. The second kind of topology -- the "classical" topology -- can be used to describe the one-body physics (include free fermion systems). The "classical" topology cannot be used to describe interacting many-body systems, which need "quantum topology".

One needs to go beyond "filling energy level" picture to understand topological order (the first kind of topology). Our education in traditional condensed matter physics (or traditional many-body physics) is almost all about "filling energy levels" (such as Landau Fermi liquid theory, band theory, etc), which is a trap that limit our imagination. The second kind of topology (the "topology" in "topological insulator") can be understood within the framework of "filling energy level" picture.

To answer the question What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this? I like to say that topological order is algebraic, not geometric. So the topological invariants of topological order are very different from Chern numbers. The robust ground state degenercy and the robust non-Abelian geometric phases of the degenerate ground states are the topological invariants of topological order (which are the analogues of the Chern number).

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'ANY local perturbations' are hermitian operators of the form $\delta H=\sum_i H_i$ where $H_i$ acts on a finite region. So ANY really means "any" in the usual way. –  Xiao-Gang Wen Jul 25 '13 at 23:45
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There are two kinds of topology in math. This physics kind of "topology" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category. This physics kind of "topology" is not related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical". –  Xiao-Gang Wen Jul 26 '13 at 23:11
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The correct way to describe any gapped phases (such as topological orders and topological insulators) is to use the first kind of topology -- "quantum" topology. The second kind of topology -- the "classical" topology -- can be used to describe the one-body physics (include free fermion systems). The "classical" topology cannot be used to describe interacting many-body systems, which need "quantum topology". On the other hand, "quantum" topology, in some sense, can be viewed as "classical" topology for infinite dimensional manifolds. –  Xiao-Gang Wen Jul 26 '13 at 23:19
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One needs to go beyond "filling energy level" picture to understand topological order. Our education in traditional condensed matter physics (or traditional many-body physics) is almost all about "filling energy levels" (such as Landau Fermi liquid theory, band theory, etc), which is a trap that limit our imagination. Also Everett You's answer explains the two kinds of topology very well. –  Xiao-Gang Wen Jul 26 '13 at 23:33
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The theory of 'topological insulators' has two versions, one for free fermions and one for interacting fermions. Only the version for interacting fermions apply for real experiments since the electrons in real samples do interact. The version for free fermions is described by classical topology while the version for interacting fermions is described by quantum topology (with symmetry). –  Xiao-Gang Wen Jul 27 '13 at 5:04

The 'topological' in topological order can refer to:

  • The fact that the ground state degeneracy is sensitive to the the topology of the manifold (as mentioned by Motl).
  • The low energy, effective theory is a Topological Field Theory.
  • The low energy excitations are anyons which obey a generalized form of exchange statistics. This steps into the realm of knot theory and related topics, which is quite "topological".
  • In fact, these anyons are better understood as examples of topological defects, instead of simple energetic excitation. In the case of the quantum Hall effect they are vortices in the quantum Hall liquid.
  • Many properties of the system do not depend on the microscopic details (e.g. the conductivity in the case of the quantum Hall effect). The system is insensitive to local perturbations.
  • These systems are characterized by topological entanglement entropy. The entanglement entropy contains a term which does not scale with the system size. It's constant and can be used to partially identify the topological order.
  • The system exhibits bulk-edge duality. Again, the exact shape of the edge is irrelevant.
  • In some sense these topological ordered system are so special because the ground state configuration has an "enhanced" symmetry compared to the excited (topological-order breaking) states.
  • The name 'topological order' also shows the difference between the more conventional order used in Landau-Ginzburg theory. Topologically ordered systems cannot be described through use of local order parameters, since the low energy effective theory does not have local degrees of freedom.
  • The Hilbert space associated with anyons has some non-locality built into it. For instance, a configuration of multiple anyons is not described by a direct sum of single particle Hilbert spaces (one assigned to each anyon). The dimensionality of many-anyon state follows very non-trivial rules (e.g. in the case of Fibonacci anyons the dimension of the n-particle Hilbert space grows according to the Fibonacci sequence when we introduce more anyons to the system).

Feel free to comment.

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One comment: the listed criterion are not all necessary to have a topological order. Especially the ones concerning anyons, am I correct ? Thanks for your answer. –  FraSchelle Jul 11 '13 at 16:33

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