Characterising band topology out-of-equilibrium using Chern marker - - PowerPoint PPT Presentation

Dr Gunnar Möller 
 Royal Society University Research Fellow

Conference on Signatures of Topology in Condensed Matter | (smr 3330) ICTP Trieste, Oct. 21-25, 2019

Visitor, Cavendish Laboratory, University of Cambridge School of Physical Sciences, University of Kent, Canterbury UK

Characterising band topology 


• ut-of-equilibrium 


using Chern marker currents

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics 15, 257 (2019).

• B. Andrews & G. Möller, Phys. Rev. B 97, 035159 (2018).
• L. Schoonderwoerd, F. Pollmann, G. Möller, arxiv:1908.00988

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Overview

• Topological order in the fractional quantum Hall effect
• Emulating the effects of magnetic fields without external fields.

Motivation & Background

Chern marker: • diagnosis of critical properties

• propagation of Chern marker currents
• measurement of the Chern marker

Main Part : New Probes for Topological Order - non-interacting systems

• Direct quantum Hall plateau transition between |C|>1 Chern

insulators and fractional Chern insulators

• new fractional Chern insulator state in higher Chern

bands at filling fractions

Brief appetiser : Novel Phases - Fractional Chern Insulators

ν = r r|Ck| + 1

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

2D electron gas

• a macroscopic quantum phenomenon: magnetoresistance in 2D electron gases
• different topologically ordered phases at each Hall plateau
• supporting fractionalized abelian and non-Abelian excitations

Topological Order in the Quantum Hall Effect

Where?

• in semiconductor hetero-

structures with clean two- dimensional electron gases

• at low temperatures (~0.1K)

and in strong magnetic fields

kBT ⌧ ~ωc = ~eB/me

What?

• plateaus in Hall conductance
• simultaneously: (near) zero

longitudinal resistance

σxy = ν e2 h

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Semiclassical picture of IQHE: skipping orbits

cyclotron motion produces no net current in bulk of sample at edge of sample, ‘skipping orbits’ contribute a uni-directional current picture for quantum transport: absence of backscattering ⇒ dissipationless current
 ⇒ no voltage drop along lead!

σxy = ν e2 h

J J

E several LLs filled: low-energy or ‘gapless’ excitations present near boundary

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Quantum Hall conductance as a topological invariant

kx

ky

C

S

Geometrical phase analogous to Aharonov-Bohm effect Using Stokes’ theorem:

(C) = i Z

C

hu~

k| d

d~ k |u~

kid~

k ⌘ Z

C

~ A(~ k)d~ k (C) = Z

C

~ A(~ k)d~ k = Z

S

~ rk ⇥ ~ A(~ k)d~

• C = ∂S

~ B = ~ rk ⇥ ~ A(~ k)

Berry curvature:

C =

1 2π

R

BZ d2k B(k)

Chern number: is a property of the band eigenfunctions, only! Effective ‘vector potential’ called Berry connection takes only integer values!

~ A(~ k) = i Z

UC

u~

k(~

r)∗~ rku~

k(~

r) d2r

• C determines the Hall conductance - TKNN (1981):

σxy = e2 h X

n occ.

Cn

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Berry curvature and Chern number

kx

ky

C

S

Geometrical phase analogous to Aharonov-Bohm effect Using Stokes’ theorem:

(C) = i Z

C

hu~

k| d

d~ k |u~

kid~

k ⌘ Z

C

~ A(~ k)d~ k (C) = Z

C

~ A(~ k)d~ k = Z

S

~ rk ⇥ ~ A(~ k)d~

• C = ∂S

~ B = ~ rk ⇥ ~ A(~ k)

Berry curvature:

C =

1 2π

R

BZ d2k B(k)

Chern number: is a property of the band eigenfunctions, only! Effective ‘vector potential’ called Berry connection topological: takes only integer values!

~ A(~ k) = i Z

UC

u~

k(~

r)∗~ rku~

k(~

r) d2r

• Chern number provides universal classification of all possible single-particle bands (cl. A)

integral over real-space, so it does not matter in what physical space the system ‘lives’ in reciprocal space, only the change of the scalar product on that space matters

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Quantum Hall effect without magnetic fields

Many different opportunities for emulating magnetic fields:

The fractional quantum Hall effect is observed under extreme conditions

• strong magnetic fields of several Tesla
• very low temperatures
• clean / high mobility semiconductor samples
• 1. Cold Atomic Gases
• 2. Solid State
• 3. Photons

Fe3Sn2

(proposed)

spin orbit
 coupling strain rotation laser assisted hopping Si waveguides

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Landau-levels vs lattice models with effective magnetic flux

kx ky B = r ⇥ A

Berry curvature in the Hofstadter model

kx ky B

Berry curvature in a Landau-Level: flat

continuum lattice

H = −J X

hα,βi

h ˆ b†

αˆ

bβeiAαβ + h.c. i + X Vijˆ niˆ nj

H = 1 2m|p − eA|2 + ˆ V

A = Bzxey

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

C = 1 2π Z

BZ

d2k (∂kxAky − ∂kyAkx)

ˆ H = −t1 X

hi,ji

⇣ ˆ c†

i ˆ

cj + H.c. ⌘ − t2 X

h hi,ji i

⇣ eiϕij ˆ c†

i ˆ

cj + H.c. ⌘ + M X

i2A

ˆ ni − M X

i2B

ˆ ni

Part I: New Probes for Topological Order (free fermions)

• x

y

a

+ t2 +M

L3a

a t1

3aL/2

0.5

ϕ/2π

−2 −1 1 2

M

b

−1 1

c

Take a simple target model: Haldane model Phase diagram with trivial and topological phases: Chern index: 
 Global topological quantum number

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

The topological Chern marker

C = 1 2π Z

BZ

d2k (∂kxAky − ∂kyAkx)

From the global Chern index:

trace in momentum space

Bianco, R. & Resta, R. Phys. Rev. B 84, 241106 (2011)

To the local Chern marker

hun0,k|∂kun,ki = ihψn0,k|r|ψn,ki

• rewrite in terms of position operator
• note trace can be rewritten in position space

c(rα) = 4π Ac Im X

s=A,B

hrαs| ˆ P ˆ x ˆ Qˆ y ˆ P|rαsi

0.5

ϕ/2π

−2 −1 1 2

M

b

−1 1

c

ˆ P = X

q∈occ

|ψqihψq|, ˆ Q = ˆ 1 ˆ P

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

• x

y

a

• +

L3a 3aL/2

Geometry considered for our analysis of Chern marker

• Finite-size system, LxL unit cells of honeycomb lattice
• Open boundary conditions

Initially, consider C(r) in single unit cell in centre of flake later, we also study values for different unit cells along one of the lattice directions

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

0.5

ϕ/2π

−2 −1 1 2

M

b

−1 1

c

Critical properties from the local Chern marker

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

1.4 1.6 Mc 1.8 2.0 M 0.0 0.5 1.0 c

Increasing L

a

−10 −5 5 10 (M − Mc)L1/ν 0.0 0.5 1.0 c

b

2.7 3.0 3.3

logL

−1.2 −1.4 −1.6

log∆M

Trace the behaviour of the local Chern marker at centre across phase transition

ϕ = π/2

tune M through transition at constant

Perfect scaling collapse is achieved for ν = 1, in line with the low-energy Dirac theory at the transition

identify width of transition and assume scaling of correlation length

ξ ∼ (∆M)−ν

we find scaling collapse of the form

c ∼ f(ξ/L) = f((M − Mc)−ν/L) = ˜ f((M − Mc)L1/ν)

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

3 t* 6 9 12

t

−1 1

c

a

M = −1.0 M = −0.5 M = 0.0 M = 0.5 M = 1.0

b

Dynamics of the Chern marker

• x

y

a

• +

L3a 3aL/2

y0 10 20 30 40

y

−5

c

b

t = 0 t = 2.5 t = 5

c

Chern marker at sample centre (L=31) Distribution of Chern marker along grey cut

• Chern marker seen to propagate from edge into bulk after a quench

Time evolution after a quench in M to M=5 (from topological to trivial)

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

ϕ/2π

M

−1 1

c

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Dynamics of the Chern marker: velocity

• x

y

a

• +

L3a 3aL/2

3 t* 6 9 12

t

−1 1

c

a

M = −1.0 M = −0.5 M = 0.0 M = 0.5 M = 1.0

y0 10 20 30 40

y

−5

c

b

t = 0 t = 2.5 t = 5

y0 10 20

y

2 4 6

t*

c

0.0 0.1 0.2 0.3 0.4 0.5

ϕÕ/2π

2 3 4

v

Chern marker at sample centre (L=31) Distribution of Chern marker Regression for velocity

• Chern marker propagates from

edge into bulk

• Examination of different size

samples yields propagation velocity

• Surprisingly, the velocity

exceeds that of individual carriers in either of the bands!

Time evolution after a quench in M to M=5

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

ϕ/2π

M

−1 1

c

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

• x

y

a

• +

L3a 3aL/2

Spacetime view of the propagation

• Visual impression confirms well-formed shock wave propagating at

constant velocity

Time evolution of Chern marker shown along the grey-shaded cut

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Chern marker at centre of edge:

Steady state and finite-size effects

• Long-time evolution in finite size ultimately returns towards initial values

(unitary time evolution with free Hamiltonian)

Steady-state is close to, but not equal to ground state of trivial target Hamiltonian

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

• x

y

a

• +

L3a 3aL/2

• At intermediate times the Chern marker plateaus, indicating a steady state
• This steady state is close but not equal to value of Chern marker in non-

topological phase

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Chern marker currents

Fc := I

∂Ac

Jc · dl = − Z

Ac

∂c ∂t d2r

The Chern marker is globally conserved, so a current can be defined from continuity equation

∂c ∂t + r · Jc = 0

We measure the total flux through a unit cell:

Time evolution of flux at different locations regression for velocity

Chern marker current reveals how topological quench propagates from edge into bulk

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

y

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Measuring the Chern marker

Pαsβs0 :=hrαs| ˆ P|rβs0 i = X

Ek<EF

hrαs|ψkihψk|rβs0 i

Note: matrix elements of projector P are
 precisely those of the single particle density matrix measuring density matrix:

• bserve precession of Bloch vector from

coupling of distant eigenstates

Ardila, Heyl & Eckardt, PRL 121 (2018), arxiv: 1806.08171

c(rα) = 4π Ac Im X

s,t,u,v,w

hrαs| ˆ P|rαti hrαt|ˆ x|rαuihrαu| ˆ Q|rαvi hrαv|ˆ y|rαwihrαw| ˆ P|rαsi

the Chern marker follows by direct evaluation:

in eigenbasis

M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics (2019)

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Summary / Open questions:

• The Chern marker emulates a local order

parameter for topological phase transitions

• It acts like a conserved charge with a

corresponding current obeying a continuity equation. Disorder: How does disorder affect the propagation of the Chern marker? Inhomogeneous geometries: Can we guide the flow of Chern currents and build practical devices exploiting non-equilibrium topological states? Interacting systems: Is there a generalisation of the Chern marker to interacting many-body systems?

Main conclusions Further directions

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Brief advert: Fractional Quantum Hall Effect in Periodic Potentials

• quantized Hall response in partially filled bands?
• THEORY: Kol & Read (1993)
• Confirmations for such states?

H = −J X

hα,βi

h ˆ b†

αˆ

bβeiAαβ + h.c. i + X Vijˆ niˆ nj

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

E nφ

1 2

• 1
• 2

Hofstadter model: Flat band limit of general Chern bands

near Chern # C bands with flat dispersion and band geometry

H = −J X

hα,βi

h ˆ b†

αˆ

bβeiAαβ + h.c. i

nφ → 1 |C|

C=2 C=3

Kol & Read PRB (1993)
 Möller & Cooper, PRL (2009), PRL (2015)

ν = r r|Ck| + 1

candidates for quantum liquids in higher Chern bands: Abelian composite fermion series candidates for non-Abelian states

ν = k |C| + 1

for k-body interactions

Sterdyniak et al. PRB (2013)

= Quasi Continuum Limit of |C|>1 bands

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Abelian States in Quasi Continuum Limit

|C| = 2 Band

Bosons - Correlation Function

21 42 63 0 21 42 63 0.47 0.94 1.4 x y g(r)

(0, 0) (0, 1) (1, 0) (1, 1)

|C|2 sheets

ν = 1/3 N = 7 (x mod |C|, y mod |C|) Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

• B. Andrews, G. Möller, Phys. Rev. B 97, 035159 (2018)

ν = r r|Ck| + 1

stability of Abelian series

0.5 1 1.5 2 2.5 3 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/3 ν = 2/5 ν = 3/7 ν = 1 ν = 2/3 ν = 3/5

ν = 1/3: Laughlin-like state ν = 1: BIQHE (Ster

e.g. C=2 bosons: many-body gap: consistently non-zero GS degeneracies correctly predicted by CF theory correlation functions: C2 sheets

for more data on C>2, bosons & fermions, see our publication:

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Interaction-driven competition of CI and FCI states

# bands in total = q

nφ = p q

# bands in LLL = p > 1

• Chern insulators states

LLL filling factors

• multiple magnetic sub-bands
• Fractional Chern insulators:

fractional filling of low-energy manifold integer filling of sub-bands

σxy = e2 h

m

X

i=1

Ci

(C=1 for LLL)

σxy = e2 h ν

?

U W

U ⌧ W

U

• consider states with

νCI = n/nφ = m p

νF CI = r kCr + 1

νF CI = νCI

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Interaction-driven quantum Hall plateau transition

• First signature: Hall conductance (from flux insertion)
• e.g., choose:

?

U

, thus asymptotically expect

nφ = 3 11, ν = 1 3 ⇔ n = 1 11

C=4 Chern Insulator Laughlin state (?)

• σxy jumps from CI to Laughlin:

− consider 6π flux insertion for both − Anomalous points near transition 


violate adiabatic evolution

∆Φ

• L. Schoonderwoerd, F. Pollmann, G. Möller, arxiv:1908.00988

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

More signatures of the plateau transition

• more signatures for nφ = 3

11, ν = 1 3 ⇔ n = 1 11

• L. Schoonderwoerd, F. Pollmann, G. Möller, arxiv:1908.00988
• Correlation length
• Entanglement spectrum
• Entanglement gap

?

U

(Ly = 6)

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

ν = r r|Ck| + 1

[ k odd (even) for bosons (fermions) ]

• New FTI states exist in |C|>1 Chern bands: we predicted the series

Conclusions / Summary

New states & new topological materials

G.Moller@kent.ac.uk

https://research.kent.ac.uk/theoryandsimulation/ New probes for topological order

Integer CI and fractional CI in Hofstadter lattice can exist at same n

• ur DMRG shows how interactions can drive a direct transition
• The Chern marker gives novel insights into topological matter in and
• ut-of-equilibrium - it supports Chern marker currents propagating

faster than the fastest carriers

Looking for research fellows, faculty jobs @ Kent expected in 2020

Gunnar Möller

Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Group & Support

Chris Winterowd PDRA, U. Kent

Support and funders (past and present):

Leon Schoonderwoerd PhD student, U. Kent Gunnar Möller PI, RS-URF John Pooley PhD student, U. Kent