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### 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

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)

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

• 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

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