Lecture 3: Berry connection and topological invariantsLecture 3: Berry connection and topological...
Transcript of Lecture 3: Berry connection and topological invariantsLecture 3: Berry connection and topological...
[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
Su-Schrieffer-Heeger (SSH) model: staggered hopping
Intracell hopping Intercell hopping
Sublattice symmetry: leads to
PBC:
D = 1 and chiral symmetry
satisfies
trivialtopological
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’s
area
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
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 (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
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
Geometric phases irrelevant (go to parallel transport gauge)
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