I'm not an expert in the area; just recently I checked this paper because of my research. What puzzles me a lot is the so called bulk-boundary correspondence. Can anyone explain in succint terms what's about? References to books (courses, monographs, etc.) will be much appreciated.
[Physics] Bulk-boundary correspondence in topological insulators
topological-insulators
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There is no proof of bulk-boundary correspondence for topological phases in general. In fact, topological phases like toric code model does not have gapless excitations on the boundary.
For non-interacting fermion systems protected by internal symmetries (as in the "periodic table" classification), bulk-boundary correspondence holds. For non-interacting fermion systems protected by a spatial symmetry, gapless surface states also exist on those crystal surfaces preserving the symmetry.
In some sense, the existence of some kind of boundary states is all there is about topology in non-interacting fermions.
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).
Best Answer
A bulk-boundary correspondence connects a physically measurable quantity to a topological bulk invariant. The most well-known bulk-boundary correspondence explains the Integer Quantum Hall Effect: \begin{align*} \sigma^{xy}_{\mathrm{edge}} \overset{(1)}{=} T_{\mathrm{edge}} \overset{(2)}{=} \tfrac{e^2}{h} \, \mathrm{Ch}(P_{\mathrm{F}}) \overset{(3)}{=} \sigma^{xy}_{\mathrm{bulk}} \end{align*} It consists of three equalities, and (1) and (2) are due to Hatsugai while (3) is what Thouless has received his Nobel Prize for in 2016. (1) and (3) connect a physical observable, the transverse conductivity at the edge and in the bulk, to a topological invariant. And (2) tells us that the edge and bulk invariant necessarily agree. Note that in many cases there is no physical bulk observable.
The topological edge invariant can be defined in many ways. Hatsugai defined it as a winding number associated to a two-dimensional Riemann surface where the remaining periodic direction is augmented by loops in energy space. Another way is to define it as a Chern number (I can provide references if necessary).
A bulk-boundary correspondence now states the following: if you have an insulating bulk (i. e. the Fermi energy lies in a spectral gap) and the Chern number in non-zero, then if you cut the insulator its surface will become conductive. What is more, the transverse is quantized and you can compute it just knowing the bulk material. This is due to the appearance of states in the bulk band gap. Due to the bulk spectral gap you know that these must be surface states as there are no states in the bulk to populate. So any of these extra states cannot propagate into the bulk and therefore must remain localized near the boundary.
Topological phenomena are robust for the following reason: if you deform physical systems in a continuous fashion, many physical observables should also change in a continuous fashion. On the other hand, you know that topological invariants are proportional to integers, so they only way to change to another integer is to make a discontinuous jump. If you look at the definition of topological invariants it becomes clear that in the periodic case the spectral bulk band gap must close. Or, in case the system is random, an Anderson localization-delocalization transition must take place.
Derivations of (1) and (3) for the Quantum Hall Effect
Equations (1) and (3) are often derived using linear response theory: where you expand the conductivity as a power series with respect to the applied electric field and the coefficients from the Taylor expansion is nothing but the conductivity tensor. You can then check that the transverse conductivity in the bulk is given in terms of the Green-Kubo formula, which is just the formula for the Chern number expressed in terms of projections. There are other approaches, e. g. semiclassical derivations.
Derivation of (2) for the Quantum Hall Effect
A derivation of this equality is a highly non-trivial affair and involves very advanced mathematics. To my knowledge there are no really simple derivations that show that (2) holds as a matter of course. One of the more accessible sources here is the book of Prodan and Schulz-Baldes. They present the necessary mathematics in a rather pedagogical way, although the mathematical concepts they rely on are still advanced. There is currently no way around that.
Other systems
For “standard” models physicists (usually correctly) assume that the bulk-boundary correspondence applies, so most of the time everything works just fine. Problems arise when researchers use heuristic characterizations of topologically non-trivial systems (I have seen many papers that have appeared in well-respected journals, which claim that a system is topological when a quick glance at the band spectrum tells you it can't be). Or when researchers want to derive new bulk-boundary correspondences, where they perhaps only know the bulk invariant.