The Puzzling Boundaries of Topological Quantum Matter Michael Levin Collaborators: Chien-Hung Lin (University of Chicago) Chenjie Wang (University of Chicago) Zheng-Cheng Gu (Perimeter Institute)

Protected boundary modes Gapped excitations Gapless excitations

Outline I. Example: 2D topological insulators II. General non-interacting fermion systems III. The puzzle of interactions

Model Non-interacting electrons in a periodic potential a

Solving the model Discrete translational symmetry  k x, k y are good quantum numbers (mod 2  /a) k -  /a aa aa

Energy spectrum aa-  /a E kxkx

Energy spectrum aa-  /a E kxkx  Conduction band Valence band

Energy spectrum aa-  /a E kxkx  Band insulator Conduction band Valence band

1928: Theory of band insulators (Bloch + others)

2005: Two fundamentally distinct classes of time reversal invariant band insulators: 1. Conventional insulators 2. “Topological insulators” (Kane, Mele + others)

Vacuum Insulators with an edge

Energy spectrum with an edge aa-  /a E kxkx Conduction Valence 

Energy spectrum with an edge aa-  /a E kxkx Edge modes Conduction Valence 

Two types of edge spectra aa kxkx 0 E

aa kxkx 0 E

aa kxkx 0 E aa kxkx 0 E

aa kxkx 0 E aa kxkx 0 E

aa kxkx 0 E aa kxkx 0 E

aa kxkx 0 E aa kxkx 0 E EFEF EFEF

aa kxkx 0 E aa kxkx 0 E Conventional insulator“Topological insulator” EFEF EFEF

Experimental Realization (HgCdTe) (Koenig et al, Science, 2007) I. d < d c II. d < d c III. d > d c IV. d > d c

Experimental Realization (HgCdTe) (Koenig et al, Science, 2007) I. d < d c II. d < d c III. d > d c IV. d > d c

Other examples 3D topological insulators “Topological superconductors” (1D/2D/3D) Quantum Hall states (2D) Many others…

The main question What is the general theory of protected boundary modes? In general, which systems have protected boundary modes and which do not?

Formalism for non-interacting case Step 1: Specify symmetry and dimensionality of system e.g. “2D with charge conservation symmetry” Step 2: Look up corresponding “topological band invariants” e.g. “Chern number”

Formalism for non-interacting case Bulk band structure Topological band invariant Boundary is not protected Boundary is protected

Formalism for non-interacting case Bulk band structure Topological band invariant Boundary is not protected Boundary is protected

Topological insulator boundaries are also protected against interactions! Arbitrary local interactions xxxxxxxxx

Theory of interacting boundaries

Boundary is not protected Boundary is protected

Bulk Hamiltonian Theory of interacting boundaries Boundary is not protected Boundary is protected

Bulk Hamiltonian ??? Theory of interacting boundaries Boundary is not protected Boundary is protected

Phase space of interacting systems with energy gap No symmetry Anti-unitary symmetry No fractional statistics Fractional statistics Statistics of excitations Symmetry Unitary symmetry

Phase space of interacting systems with energy gap No symmetry Anti-unitary symmetry No fractional statistics Fractional statistics Statistics of excitations Symmetry Unitary symmetry Case 1

Phase space of interacting systems with energy gap No symmetry Anti-unitary symmetry No fractional statistics Fractional statistics Statistics of excitations Symmetry Case 2 Unitary symmetry

Case 1: No symmetry

Example: Integer quantum Hall states B

Energy spectrum kxkx E cc

kxkx E cc

Integer quantum Hall edge

A more general result n R = 2 n L = 1

A more general result n R = 2 n L = 1 (Kane, Fisher, 1997)

Yes! (ML, PRX 2013)

Examples Superconductor

Examples Superconductor

Fractional statistics in 2D

General criterion for protected edge l m (ML, PRX 2013)

Connection between braiding statistics and protected edges

Phase space of interacting systems with energy gap No symmetry Anti-unitary symmetry No fractional statistics Fractional statistics Statistics of excitations Symmetry Case 2 Unitary symmetry

Case 2: No fractional statistics Focus on two toy models with Z 2 (Ising) symmetry: One model has protected edge; the other does not Analogues of topological insulator and ordinary insulator

The models Symmetry: Hamiltonians:

The models Symmetry: Hamiltonians:

The models Symmetry: Hamiltonians: p

The models Symmetry: Hamiltonians: p qq’

The basic question How can we see that H 1 has a protected edge mode while H 0 does not?

Step 1: Couple to a Z 2 gauge field Z 2 gauge field: Replace:

Step 2: Compute braiding statistics of  -flux excitations  flux:

Result for statistics H 0 : Find   -fluxes are bosons or fermions H 1 : Find   -fluxes are “semions” or “anti-semions” (ML, Z. Gu, PRB, 2012)

Generalizing from 2D to 3D

2D3D (C. Wang, ML, PRL 2014) (C.-H. Lin, ML, in preparation)

Bulk Hamiltonian ??? Summary Boundary is not protected Boundary is protected

Bulk Hamiltonian Braiding statistics data + … Summary Boundary is not protected Boundary is protected (ML, PRX 2013) (ML, Z. Gu, PRB 2012) (C. Wang, ML, PRL 2014) (C.-H. Lin, ML, in preparation)

Open questions Anti-unitary symmetries? General bulk-boundary correspondence? Connection to anomalies in QFT?