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

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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

    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

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