Band theory and topology: From adiabatic particle transport

to quantum Hall effect to topological insulators

Plan• Topological insulators: introduction

• Topology in pictures

• Berry phase

• Band theory

• Topology of band insulators in 1D• Adiabatic charge transport (Thouless)• Polarization in a 1D insulator (Resta, King-Smith and Vanderbilt)• Su-Schrieffer-Heeger and Rice-Mele models, solitons

• Quantum Hall effect• Anomalous velocity contribution• TKNN model• Laughlin argument• Haldane model

• Time-reversal invariance

• topological insulators in two dimensions

• Spin-orbit coupling, quantum spin Hall effect, and realistic materials

• Topological insulators in three dimensions

• Topological superconductors

• Classification

Some useful references

• Books1. B. A. Bernevig and T. L. Hughes, Topological Insulators And Topological

Superconductors, (Princeton University Press, 2013).2. M. Franz and L. Molenkamp (Eds.), Topological Insulators, (Elsevier, 2013);

especially contributions by C. L. Kane and J. Moore. 3. Shun-Qing Shen, Topological Insulators, (Springer, 2012).4. E. Fradkin, Field Theories Of Condensed Matter Physics, Second edition, (Cambridge

University Press, 2013).5. A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, The Geometric

Phase in Quantum Systems, (Springer, 2003).6. G. Morandi. Quantum Hall effect (Bibliopolis, Naples, 1988).

• Reviewes• Di Xiao, Ming-Che Chang, and Qian Niu, Berry phase effects on electronic properties,

Rev. Mod. Phys. 82, 1959 (2010).• M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82,

3045 (2010).• Xiao-Liang Qi and Shou-Cheng Zhang, Topological insulators and superconductors,

Rev. Mod. Phys. 83, 1057 (2011).

Band theory and topology

Part 1Introduction and topology

Topological insulators: introduction

What is a topological insulator?

• Electronic system that is an insulator: gap in the energy spectrum.

• Band structure is characterized by a topological invariant, a quantized number.

• Robust against local perturbations of finite size (disorder and interactions).

• Can be characterized at the level of free-fermion systems: a generalization ofthe integer quantum Hall states.

• (Simplest) topological insulators can be characterized by their band structure.

• Consequence of topology: edge states, gapless excitations at the edges of thesystem.

• Edge states also robust against the effects of disorder and interactions. Inparticular, cannot be localized by disorder.

• Topological states of quasiparticles in superfluids and superconductors, as well as in special condensates in the particle–hole channel.

• A complete “10-fold way” classification of non-interacting topological phases

Topology in pictures

• Topology is the study of continuity

• Some shapes can be continuously deformed

into each other: they are topologically equivalent

• Can classify inequivalent shapes

• E.g. 2D closed orientable surfaces are classified by the number of handles,

the genus

Topology in pictures

• More complicated and interesting object to classify

• Mappings between manifolds (homotopy groups)

• Fiber bundles

• Connections (gauge fields) on fiber bundles and characteristic classes

Topology and geometry: Gauss-Bonnet formula

• Topological (quantized) numbers can be written as integrals of local quantities

• Gauss-Bonnet theorem. The Gaussian curvature of a 2D surface of genusintegrated over the surface gives the Euler characteristic

• Can locally change the curvature, but the integral is quantized

• Generalization for fiber bundles (Chern classes) is central to topological phases

Gauss-Bonnet formula: examples

• A sphere of radius

• A torus

Topology and band theory

• Two insulators are topologically equivalent if one can be continuously changed into the other by slowly changing the Hamiltonian such that the system always remains in the ground state. Need a gap that stays finite throughout the change

• Connecting topologically distinct insulators necessarily involves a phase transition where the energy gap vanishes

• Finite gap ensures stability of a topological insulator to interactions and disorder

• Consider only non-interacting electrons, add assume translation invariance. Add interactions and disorder later

Topology and band theory

• Bloch theorem: states are labeled by a crystal momentum :

• are lattice-periodic and are eigenstates of the Bloch Hamiltonian

• labels bands. Fully filled bands are separated by a gap from empty bands

• Lattice symmetry implies periodicity in the reciprocal (momentum) space

• Crystal momenta lie in a periodic Brillouin zone

Topology and band theory

• Need to classify mappings from a torus to the space of gapped Bloch Hamiltonians

• A consequence of existence of distinct band topological insulators: if two inequivalent insulators are in contact with each other, the gap must vanish at the boundary. One of the insulators may be “trivial” – empty space

• Gapless states at the boundary between inequivalent insulators

• The gapless states can also be classified topologically using the bulk-boundary correspondence

Band theory and topology

Part 2Adiabatic quantum evolution

and Berry phase

Adiabatic evolution

• Consider a Hamiltonian that depends on parameters (points of a manifold )

• In what follows will be a point in a Brillouin zone

• Adiabatic (“slow” compared with energy gaps) evolution of parameters

• Instantaneous orthonormal eigenstates

• Adiabatic theorem: as long as the motion is “slowan eingenstate remains an eigenstate but acquires a phase:

• Dynamic phase:

• Geometric phase

• Two implicit assumptions: states are non-degenerate, and for all values of are in the same Hilbert space

Adiabatic evolution: derivation

• Plug into Schrodinger equation

• Scalar product with , use normalization

• Integrate with initial condition , get

• Notice that is real

Adiabatic perturbation theory

• We need the following result for the wave functions under adiabatic motion to first order in slowness (suppress the dependence on parameters for brevity)

• Let us use our previous notation for the dynamic and the geometric phases

• Start in an eigenstate:

• Then at a later time (exercise)

• The first term is the usual adiabatic theorem

• Resembles the usual time-independent perturbation theory, where the role of the perturbation is played by

Adiabatic perturbation theory: derivation

• Schrodinger eq. and initial condition

• Ansatz:

• Substitute, multiply on the left by

• Separate from the rest and use the initial values. In other words, keep only in the RHS. For this gives

• For we get

• Only the exponential here is a rapid function of time, treat the prefactors as constants. This gives

• Collect all terms:

Geometric phase

• Geometric phase does not depend on the parametrization

• Berry vector potential

• In the language of differential forms

where the Berry one-form is

Aside on differential forms

• Exterior derivative of a function is simply the differential (small change)

• For example, let us differentiate

• Differentiate

• Multiply by :• Hellmann–Feynman theorem

• Multiply by :

• Applied twice, exterior derivative gives zero:

• Differential forms simplify and streamline equations, but are not really necessary: one can always go back to components

Gauge transformations

• States are not uniquely defined, there is a gauge freedom:

• Convenient choice: are smooth and single-valued for all

• This is often not possible even in principle when has a non-trivial topology

• Gauge transformation of the Berry vector potential and one-form

• Gauge transformation of the geometric phase

• If is an arbitrary single-valued function, we can cancel the geometric phase, and only the dynamic phase survives (Fock)

Berry phase and gauge invariance

• Crucial observation (Berry): gauge freedom is limited if the path in the parameter space is a closed loop:

• The geometric phase around a closed loop is the Berry phase

• Berry phase is gauge invariant and cannot be removed!

• Pancharatnam phase (1955) in optics, Aharonov-Anandan non-adiabatic phase

Gauge-independent formulation of Berry phase

• Analogy with E&M. Field strength: an antisymmetric tensor

• The corresponding two-form is the Berry curvature form (use )

• Berry curvature is gauge-invariant:

• Alternative formula for Berry phase using Stokes theorem

• Here is an arbitrary surface whose boundary is the loop

Gauge-independent formulation of Berry phase

• Completeness of . Insert

• Use the antisymmetry of the wedge product to remove the diagonal term

• Use

• Manifestly gauge-independent, can be used in numerical simulations

Berry phase in 3 dimensions

• If , we can use vector notation

• Berry field strength (curvature): three-component vector dual to

• Relation to the Berry vector potential

• Berry phase for a closed loop

• Berry phase for a loop = flux of the Berry curvature

Degeneracies and two-level systems

• Berry curvature is large near degeneracy points where energy levels cross

• Generically two levels cross, denote them , so that

• Expand

• Consider a generic two-level system

where are the Pauli matrices

• Eigenvalues

• Eigenvectors depend only on the unit vector so neglect

Berry curvature in a two-level system

• Shift the degeneracy point to

• Near this point can take

• Berry curvature

• Trick: choose the axis along

• Then

• See that (proportional to )

• Rotation invariance implies

• Monopole with charge at the degeneracy point

• Berry phase is related to the solid angle swept by

• For a real Hamiltonian ( ) the curve

is confined to the plane

• Then

• Same result for any plane through the degeneracy point

• Quite generally, Berry curvature integrated over a closed 2-surface is quantized

• (First) Chern class (number) is an integer = number of monopoles inside

Berry phase in a two-level system and Chern numbers

Chern number as the degree of a mapping

• Can show (exercise) that for a generic 2-level system and an arbitrary number of parameters the Berry curvature is

• This is the solid angle on the unit sphere of per unit area in the manifold of parameters

• We can view as a mapping

• Then integrating the Berry curvature over a closed 2-surface in gives the Chern number as the degree of the mapping , that is, how many times wraps around the sphere when parameters go over

• Example:

Exercises

• Work out the first order wave functions of the adiabatic perturbation theory

• Work out the eigenstates of a two-level system where the vector

• Find the components of the Berry vector potential

• Find the component of the Berry curvature

• Show that

• Show that for a generic 2-level system and an arbitrary number of parameters the Berry curvature is

Exercises

• Degenerate states and non-abelian Berry phase

• Consider (orthonormal) states that form a degenerate multiplet separated from other states by a gap for any value of the parameters Show that adiabatic transport in the parameter space rotates the state into

where the non-abelian Berry vector potential matrix is

• Obtain the non-abelian Berry curvature involving a matrix commutator

Aside on notations

• No imaginary unit:

• Very often see a less precise but more streamlined notation

• States

• Berry vector potential

• Berry curvature

• Alternative notation for the Berry curvature

• For example

Band theory

• Single electron in a periodic potential

• Bloch wave functions are not periodic:

• Different boundary conditions (Hilbert spaces) for different

• Perform a unitary transformation to the “ -picture”:

• The new wave functions are lattice-periodic

• They are “instantaneous” eigenstates of :

HgTeSi

Exercises

• Diagonalize the simplest tight-binding Hamiltonian

• Diagonalize the Hamiltonian with alternating bond strengths

• Now add staggered sublattice potential and diagonalize:

• For the last Hamiltonian find for which values of the parameters this is a metal, and for which it is an insulator (at half filling)?

• What is the relation between second-quantized Hamiltonians (like the ones above) and in the first-quantized picture?

Berry phase in band theory• If traverses a loop, the Bloch states acquire a Berry phase:

• The Berry curvature is an important local property of a Bloch band that affects electron dynamics

• What is the meaning of the band Berry phase? Why would the value of change?

• Time-dependent Hamiltonians• Motion of electrons in external fields

• Semiclassical picture of electron dynamics: electrons are wave packets composed of Bloch wave functions with momenta close to a given value

• These move with the group velocity which is usually written as

• In applied electric field , Berry curvature changes this to

• Extra term is normal to , leads to a Hall current! Come back to this later

Band theory and topology

Part 3Band topology in one dimension

Adiabatic charge transfer in 1D

• Consider a 1D band insulator under a slow periodic perturbation

• In the -picture and are defined on a torus: both and are periodic and can be treated on the same footing

• Recall the adiabatic evolution of wave functions. Apart from phase factors

• Find the number current in this state using

• Velocity operator (in the Heisenberg picture)

• In the -picture

D. J. Thouless, Quantization of particle transport, Phys. Rev. B27, 6083 (1983)

Adiabatic charge transfer in 1D• Now find to first order

• Can insert the (identically zero) term with and use

• Integrate over the 1D BZ: the first term gives zero. Sum over the filled bands

• Total number of electrons transferred over one time cycle in one band is

• This is the first Chern class, an integer! Let us prove it for this case

Adiabatic charge transfer in 1D• Rescale and

use the Stokes theorem

• Each integral connects two equivalent points, so gives a Berry phase

• Denote

• Then

• Now, the phase of can be obtained in two different ways

• This gives : the total flux of the Berry curvature through the torus is an integer multiple of

Adiabatic charge transfer in 1D

• Intuition: sliding potential

• Can slide by several lattice period in one time cycle

• Or can return to the original position, and may be zero. When is ?

• Adiabatic pumping. Vary parameters with frequency

• DC current

• Time dependence through parameters

• There must be at least two parameters

• Extend their values from the surface of the torus to its interior

• Chern number counts

the number of monopoles inside the torus

• To have there must be band crossings (degeneracies) for some values of parameters inside the torus, so that the total monopole charge is nonzero

Adiabatic charge transfer in 1D

Interactions and disorder

• Can extend the previous arguments to a more realistic system

• One-body potential is slowly varying and periodic in time, but not necessarily in space, so can include a disorder potential

• Assume that the ground state is non-degenerate and is separated by a finite gap from excited states

• Consider twisted boundary conditions for many-body wave functions

• By analogy with the one-particle case perform a unitary transformation

• The eigenstates of this Hamiltonian should be periodic

Q. Nui and D. J. Thouless, J. Phys. A 17, 2453 (1984)

Interactions and disorder

• This describes a ring with a uniform vector potential

which creates a magnetic flux through the ring

• Gauge potential allows to express the current as a derivative

• Current operator: general definition

• In one dimension this becomes

• In our case this gives

Interactions and disorder

• Now we repeat our previous calculation with replaced by the ground-state many-body wave function :

• The key point: as long as the Fermi level is in the gap, the current is insensitive to the boundary conditions specified by , so we can average over it

• Also , so is defined on a 2-torus. Total charge transferred over the time period is given by the quantized Chern number!

Polarization of band insulators in 1D

• Problems with definition of polarization as the dipole moment of a unit cell divided by its volume: electric dipole moment is not well defined for a periodic charge distribution

• Textbook picture (Clausius-Mossotti)

• Then, naively

• However, dipole moment depends on the choice of the unit cell

Modern theory of polarization

• Maxwell’s equation

• No external field

• Continuity equation

• Up to a divergence-free part, can integrate in time

• “Unquantized” adiabatic particle transport

• Substitute here the adiabatic current

• Adiabatic change may be due to some parameter (e.g. deformation)

Modern theory of polarization

• The adiabatic parameter may be cyclic

• Then polarization is defined modulo a quantum

• In 3D the quantum is related to a Bravais lattice vector and the unit cell volume

• In 1D the quantum is

• Polarization is related to the surface (end charge)

• In 3D

• In 1D