Universal transport at the edge: Disorder, interactions, and topological protection
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1 Universal transport at the edge: Disorder, interactions, and topological protection Matthew S. Foster, Rice University March 31 st, 2016
2 Universal transport coefficients at the edges of 2D topological insulators and 3D topological superconductors Hong-Yi Xie, Yang-Zhi Chou, Heqiu Li, and Matthew S. Foster
3 Quantum Topology in Materials Topological phases in materials physics: Bulk electronic structure is insulating, knotted Knot unties at the surface: metallic (gapless) surface states with anomalous properties Surface or edge properties tied to anomalous symmetry: Holographic encoding of bulk topology J. Moore Surface state Bulk Vacuum
4 Integer Quantum Hall Effect K. von Klitzing 1980, Noble Prize in Physics D electron gas, large magnetic field Anomalous chiral edge state Topologically quantized Hall resistance Dean et al. (2010)
5 Normal 1D quantum wire (e.g., carbon nanotube) Left, right moving electrons Scattering (e-e, e-impurity) can change left-mover into rightmover, vice-versa Left and right movers are not separately conserved Total charge (left + right) is conserved, but no guarantee it will flow (Anderson localization)
6 Chiral edge state: Half of a normal quantum wire Chiral edge states in the quantum Hall effect Left, right moving electrons separated by macroscopic bulk Scattering ineffective: left mover cannot be scattered into right Left and right movers separately conserved [anomalous U(1) symmetry] No scattering: Left, right edges form a perfect quantum wire
7 Part I: Ballistic transport in 1D edge states of 2D topological insulators with Rashba spin-orbit coupling
8 How to protect a time-reversal invariant quantum wire? Time-reversal invariant: need both left and right-movers! Total charge (left + right) is conserved, but no guarantee it will flow (Anderson localization)
9 Helical quantum wire Time-reversal invariant: need both left and right-movers! Solution: add spin! (Left-up, Right-down: Perfect spin-momentum locking) No impurity or e-e backscattering without time-reversal symmetry breaking ( or spin symmetry?) How to realize?
10 Helical quantum wire Time-reversal invariant: need both left and right-movers! Solution: add spin! (Left-up, Right-down: Perfect spin-momentum locking) No impurity or e-e backscattering without time-reversal symmetry breaking ( or spin symmetry?) How to realize? At the edge of a 2D topological insulator!
11 Quantum spin Hall 2D insulator Ideal case: Invariant under time-reversal and spin S z rotations No backscattering whatsoever: perfect ballistic conduction! (per edge Landauer formula) Weak (activated) contamination by bulk at non-zero, but low T Kane and Mele 2005
12 2D Topological insulator with Rashba spin-orbit coupling More realistic: Time-reversal only. Rashba spin-orbit coupling (RSOC) destroys spin symmetry Kane and Mele 2005 Slow twisting of spin quantization axis, Xie, Li, Chou, Foster (2016) Strength of RSOC: Schmidt, Rachel, von Oppen, Glazman (2012)
13 2D Topological insulator with Rashba spin-orbit coupling Slow twisting of spin quantization axis Density operator acquires backscattering part:
14 2D Topological insulator with Rashba spin-orbit coupling Slow twisting of spin quantization axis Density operator acquires backscattering part: Attributes of a generic helical edge (Luttinger) liquid: 1. Potential disorder that couples to induces random one- particle backscattering ( random RSOC ) 2. Screened Coulomb interactions include usual Luttinger interaction, plus (irrelevant) backscattering Stroem, Johannesson, Japaridze (2010) Geissler, Crepin, Trauzettel (2014)
15 Edge transport with random Rashba spin-orbit coupling (RSOC) Effects of disorder Single-particle Hamiltonian: 1D Dirac fermion Random current operator (due to disorder and RSOC) Components
16 Edge transport with random Rashba spin-orbit coupling (RSOC) Schroedinger equation Re-write in terms of rescaled wavefunction, Equivalent to time-dependent Schroedinger equation for a spin-1/2 moment in a fluctuating magnetic field!
17 Spin-1/2 moment in a fluctuating field: The simplest unsolvable quantum mechanics problem? Want to solve Randomly rotating field: Try to solve by transforming into the rotating frame
18 Spin-1/2 moment in a fluctuating field: The simplest unsolvable quantum mechanics problem? Want to solve Randomly rotating field: Try to solve by transforming into the rotating frame Now we have to solve Typically as hard to solve as the original problem!
19 Topological protection: Integrable spin-1/2 dynamics Helical edge state with random Rasba spin-orbit coupling: Fluctuating magnetic field has two components 1. First component: completely random (due to RSOC and disorder)
20 Topological protection: Integrable spin-1/2 dynamics Helical edge state with random Rasba spin-orbit coupling: Fluctuating magnetic field has two components 1. First component: completely random (due to RSOC and disorder) 2. Second component: slaved to the first
21 Topological protection: Integrable spin-1/2 dynamics Helical edge state with random Rasba spin-orbit coupling: Fluctuating magnetic field has two components 1. First component: completely random (due to RSOC and disorder) 2. Second component: slaved to the first Perfect adiabatic counterterm! Can show that a state initially aligned along exactly follows it for all subsequent times! Xie, Li, Chou, Foster (2016)
22 Topological protection: Integrable spin-1/2 dynamics Xie, Li, Chou, Foster (2016)
23 Topological protection: Integrable spin-1/2 dynamics Xie, Li, Chou, Foster (2016) Interpretation: Right-mover entering from ideal left lead exits as rightmover through right lead with 100% probability. Perfect transmission! Explicit mechanism for ballistic transport predicted by Kane/Mele 2005
24 Random RSOC and Luttinger interactions: Still protected?? Generic edge theory with disorder, interactions* Transformation using the (integrable) transfer matrix: Homogeneous kinetic term, inhomogeneous interaction:
25 Random RSOC and Luttinger interactions: Still protected?? Generic edge theory with disorder, interactions* Transformation using the (integrable) transfer matrix: Homogeneous kinetic term, inhomogeneous interaction: Equivalent to a free boson! Xie, Li, Chou, Foster (2016) Guarantees ballistic conduction through ideal leads Maslov & Stone (1995), Ponomarenko (1995), Safi & Schultz (1995) * Neglecting irrelevant umklapp interactions, which give corrections at finite temperature. Schmidt et al. 2012; Kainaris et al. 2014; Chou, Levchenko, and Foster 2015
26 Summary 1D Helical edge states of 2D topological insulators Rashba spin-orbit coupling backscattering disorder, interactions Random Rashba disorder = integrable spin-1/2 dynamics, no backscattering through ideal leads Unchanged by Luttinger liquid interactions
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