Lecture 2: Berry phase

Berry phase

What is it?

Anholonomy effect in a cyclic adiabatic evolution of a closed quantum system

Time-dependent Schrödinger equation

slow cyclic changeof parameters

How to capture and exploit slowness in description of system’s dynamics?

Expand in instantaneous eigenbasis

Completeness at every t

For convenience introduce

dynamical phaseand insert into TD-SE equations for

Exercise 4: Show that

For convenience introduce

dynamical phaseand insert into TD-SE equations for

Exercise 4: Show that

Solution:

Use and project on

Exercise 5: Show that for it holds

Exercise 5: Show that for it holds

Solution: Differentiate instantaneous eigenvalue problem

and project on

Exercise 5: Show that for it holds

Solution: Differentiate instantaneous eigenvalue problem

and project on

Conclusion:

Adiabatic approximation

If at every t

(slow driving does not excite

transitions between levels)

approximate

Adiabatic approximation

If at every t

(slow driving does not excite

transitions between levels)

approximate Initial conditions:

Solution of dynamical problem

geometric phase

Geometric properties

After one cycle

Accumulated geometric phase can not be gauged away!

Moreover, it can be observed (interferometry!) as it is a relative phase.

Geometric properties

After one cycle

Accumulated geometric phase can not be gauged away!

Moreover, it can be observed (interferometry!) as it is a relative phase.

How to distinguish between dynamical and geometric phases?

They have different properties. In particular,

does not depend on velocity of parameter changeloop in parameter space

Geometric properties

Berry connection

circulation of Berry connection

Geometric meaning of : anholonomy in parallel transport

Vector is transported along the loop on parameter manifold

and does not return to itself

Geometric properties

Gauge freedom: re-define

This implies

– arbitrary differentiable function

transformation law

of a vector potential

Parallel transport gauge

or

Can we choose continuous globally ? NO! (otherwise )

Geometric properties

What is an obstacle? Berry curvature and its flux (both gauge-invariant)

Stokes theorem (d = 3)

In d > 3

Explicitly:

Important conclusions:

● Geometric description of a (quantum closed) system in adiabatic regime:

System follows instantaneous eigenstate (well-separated from other states).

Effect of other states is condensed to a single number – geometric phase

● Geometric description is associated with some gauge freedom: It is

necessary to identify it to pursue such a description

● Despite gauge freedom, anholonomy effect (geometric phase) is independent

of a chosen gauge. Moreover, geometric phase is observable

Examples

1) Spin ½ in slowly rotating magnetic field

Instantaneous EV problem:

Solution:

In both gauges wavefunctions are single-valued

Northern gauge

Southern gauge

Berry connections:

Northern gauge

Southern gauge

Dirac monopole potential with

Gauge transformation:

Berry curvature:

Berry phase:

Analogously for excited state Note that

(for even N)

N sites, PBCM orbitals on each site

Examples

2) Berry-Zak phase of 1d crystal band structure

Tight-binding model of 1d crystal

Basis of Bloch waves

In TD limit : first Brillouin zone (1BZ)

Inverse transformation

Examples

2) Berry-Zak phase of 1d crystal band structure

Bloch Hamiltonian

(for NN hoppings)

(in second quantization)

Diagonalization gives band structure

Periodicity:

For each band:

Berry connection Zak phase

and

Label discrete k from left to right of BZ by

Define trace of projector’s product:

Discrete Berry phase

Discrete Berry phase

Equivalence to continuous Zak phase in TD limit

Understanding ahholonomy in parallel transport gauge

continuous discrete

Discrete Berry phase

Explicit construction

definition

property

Parallel transport gauge: whole phase accumulated in last jump

Summary of this lecture

● Geometric description of adiabatic unitary evolution:

– gauge freedom as a prerequisite

– Berry phase (anholonomy in parallel transport)

– observable (cannot be gauged away)

● Berry-Zak phase in 1d crystal band structure:

– gauge freedom, k-vector periodicity

● Discrete Berry phase: explicit check of gauge invariance