# Bulk-boundary correspondence in topological insulators

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.

The bulk-boundary correspondence in hermitian systems establishes a relation between a bulk property of a (translational-invariant) lattice encoded in a topological invariant (obtained from the Bloch-type eigenstates) and what happens at its boundary (surface, edge, etc).

In a simpler way. If we take the SSH model and assume translation symmetry along the chain, we can express the Hamiltonian as a $2\times2$ matrix in momentum space. With this Hamiltonian, we calculate a quantity, known as invariant. So far so good. Now, to see what this invariant means, we assume that the chain has a finite size, that is, it is constituted by N unit cells, which implies that we are going to diagonalize a $2N\times2N$ matrix. Now, in the topological phase, states will appear located at the edges, i.e., one on the right and one on the left. This relation that links the invariant calculated when the system has translational symmetry (in this case a $2\times2$ matrix) and the states that appear when we diagonalize the finite system (in this case a $2N\times2N$ matrix) is what we known as bulk-boundary correspondence in hermitian systems.

It is worth mentioning that the bulk-boundary correspondence in non-hermitian Hamiltonian systems is a subject of intense debate and controversy.

References:

Books

B.A. Bernevig, T.L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013).

J.K. Asboth, L. Oroszlany, A. Palyi, A Short Course on Topological Insulators (Springer, 2016).

Papers

T.E. Lee, Phys. Rev. Lett. 116(13), 133903 (2016).

Y. Xiong, Journal of Physics Communications 2(3), 035043 (2018).

V.M. Martinez Alvarez, J.E. Barrios Vargas, L.E.F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018).

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.