# Lecture 3: Berry connection and topological invariants Lecture 3: Berry connection and topological...

date post

30-May-2020Category

## Documents

view

2download

0

Embed Size (px)

### Transcript of Lecture 3: Berry connection and topological invariants Lecture 3: Berry connection and topological...

Lecture 3: Berry connection and topological invariants

[Northern gauge]

Two-band models

Two-band Bloch Hamiltonian and EVP in arbitrary dimension

conduction band

Berry connection and curvature (for valence band)

valence band

Eigenstates

Sublattice (chiral) symmetry

trivial phase

topological phase

D = 1 and chiral symmetry

Su-Schrieffer-Heeger (SSH) model: staggered hopping

Intracell hopping Intercell hopping

Sublattice symmetry: leads to

PBC:

D = 1 and chiral symmetry

satisfies

trivial topological

D = 1 and chiral symmetry

Su-Schrieffer-Heeger (SSH) model: bands

Zak phase

What is physically different?

Example: SSH model with OBC (half-infinite) and edge states

Cut one w-link and try Bloch-wave ansatz (j > 0)

boundary condition

condition like in PBC case

a) Extended states: combine and to satisfy BC,

same dispersion as in PBC case (only with )

Two types of solutions:

Example: SSH model with OBC (half-infinite) and edge states

Cut one w-link and try Bloch-wave ansatz (j > 0)

boundary condition

condition like in PBC case

b) Satisfy BC by . Hence and

Two types of solutions:

Bulk-boundary correspondence!

Localized solution with and exists for

condition for topological phase

Note: Here . Previously

Mapping:

D = 2, no symmetries: Chern insulator

Berry connection (valence band, northern gauge):

Mapping:

D = 2, no symmetries: Chern insulator

Berry curvature (valence band):

Jacobian

Alternative expression:

Stokes theorem

must be regular everywhere on

If not, switch to another gauge [“northern” → “southern”]

Suppose: regular on

regular on

winding number (integer)

Chern number topological

invariant

Chern number (index)

– Alternative way to see that it is integer

“pullback”sphere’sarea degree of mapping: how many times torus wraps around sphere

“many-to-one”

– Robustness: Chern number changes only at gap closing (topological phase transition)

– Physical significance: predicts number of edge states in a system with OBC

Bulk-boundary correspondence!

Example: Qi-Wu-Zhang model

no symmetries (sublattice, time-reversal, particle-hole)

PBC:

Energy bands:

Gap closes at:

Example: Qi-Wu-Zhang model Gap closes at: Phases:

trivial

topological

topological

Example: Qi-Wu-Zhang model

Exercise:

Gap closes at: Phases: trivial

topological

topological

Evaluate Chern numbers in all phases [Hint: use winding numbers]

Reminder:

Exercise: Evaluate Chern numbers in all phases [Hint: use winding numbers]

Reminder:

Solution:

at singular at singular

Example: Qi-Wu-Zhang model – edge states

OBC (half-infinite system): cut along x-axis and consider

Bloch-wave ansatz

Example: Qi-Wu-Zhang model – edge states

Two types of solutions:

a) Extended states: combine and to satisfy BC,

same dispersion as in PBC case (only with )

b) Edge states: BC satisfied by

substitute in

Dispersion of edge state: Does it exist for all ?

decisive condition

Example: Qi-Wu-Zhang model – edge states

dispersion crosses through dispersion crosses through

Example: Qi-Wu-Zhang model – edge states (finite system)

dispersion crosses through dispersion crosses through

Topological pumping

1d crystal potential with PBC (in space)

Additionally modulate periodically in time

In k-space

Thouless (1983): Current flows through any cross section of a lattice; a number of particles transferred in one period is integer!

Nontrivial statement for quantum- and time-averaged quantity

NOTE: for this to happen, pumping must be adiabatic!

Topological pumping

Significance of this model: correspondence

2d stationary 1d time-modulated

dimensional reduction/extension

Method to construct higher-d TI on basis of lower-d counterparts

Time-periodic Rice-Mele model

SSH model

staggered potential (breaks sublattice symmetry)

In k-space

Choose pumping protocol

equivalent to stationary Qi-Wu-Zhang model up to

Current operator through w-link between B-site of m-cell and A-site of (m+1)-cell:

Average over m translationally invariant current operator

Mean number of electrons pumped through cross section in one period

Many-body state: completely filled

valence band

Adiabaticity condition: gap is always larger than pumping frequency

Time-evolution of single-particle states – Schrödinger equation:

Make use of instantaneous eigenbasis :

expanding

Equations for are known from Berry phase consideration

Equations for are known from Berry phase consideration

Geometric phases irrelevant (go to parallel transport gauge)

Equations for are known from Berry phase consideration

Leading order analysis:

Leading order analysis:

Identities:

(obtained by differentiation of instantaneous EV problem wrt k)

Use periodicity of and completeness

Berry curvature

Must be quantized!

Summary of this lecture

● Berry connection and curvature winding number, Chern number

● Topological invariants characterize symmetry protected topological phases

(sometimes without symmetry, e.g. in 2d)

● Bulk-boundary correspondence: bulk top. invariant = number of edge states

● Robustness: no effect of small perturbations, only at gap closing

● Topological pumping dimensional reduction/extension

Slide 1 Slide 2 Slide 3 Slide 4 Slide 5 Slide 6 Slide 7 Slide 8 Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Slide 25 Slide 26 Slide 27 Slide 28 Slide 29 Slide 30 Slide 31 Slide 32 Slide 33 Slide 34 Slide 35 Slide 36

Recommended

*View more*