Strong correlation effects in 2D topological quantum phase - - PowerPoint PPT Presentation

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Strong correlation effects in 2D topological quantum phase - - PowerPoint PPT Presentation

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Strong correlation effects in 2D topological quantum phase transitions Adriano Amaricci IOM Frontiers in 2D Quantum Systems, ICTP Trieste Wuerzburg Dresden G.Sangiovanni B.Trauzettel J. Budich M.Capone Introduction. Ginzburg-Landau

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SLIDE 1

Strong correlation effects in 2D topological quantum phase transitions

Frontiers in 2D Quantum Systems, ICTP

IOM

Adriano Amaricci

M.Capone G.Sangiovanni B.Trauzettel

  • J. Budich

Trieste Wuerzburg Dresden

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SLIDE 2

Introduction.

Ginzburg-Landau theory: symmetry breaking classification of matter phases

Magnetism “Superconductivity”

Experimental detectability! key concept: local order parameter

Liquid-gas

magnetization M density difference n(L)-n(G) pair amplitude ψ

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SLIDE 3

key concept: global topological invariant

Introduction.

TOPOLOGICAL INSULATORS

quantum materials eluding the G-L paradigm!

States classified in terms of the Topological Properties

  • f the Hilbert space of Bloch functions:

bulk (band) insulator + with gapless edge modes.

Dirac semi-metal + Spin-Orbit Coupling

Haldane PRL88 Kane,Mele PRL05 Bernevig et al Science 2006 …. many more

+ =

Trivial Non-Trivial

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SLIDE 4

The quantum spin-Hall insulator

Idea: look for systems with a larger SOC.

BHZ model: 2 QHI + Time Reversal Symmetry.

Initial focus on graphene but small SOC (gap ~ 10-3meV)

P3/2

P1/2

P1/2

S S

P3/2

Bernevig et al Science 2006

Konig et al Science 2007

h(k) = d(k) · τ

Kane,Mele PRL 2005

Orbital pseudo-spin structure

CdTe/HgTe quantum wells.

d(k)=[λ sin kx, λ sin ky, M −ε(k)]

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SLIDE 5

Continuous Topological Quantum Phase Transition

  • 6
  • 4
  • 2

2 4 6 M X

trivial band insulator

M=2 M< 2 M> 2

band structure evolves smoothly with control parameters…

Topological QPT

h(k) = d(k) · τ

BHZ description of topological transition:

𝒟1=0 d(k)

M> 2

spin texture

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SLIDE 6
  • 4
  • 3
  • 2
  • 1

1 2 3 4 M X

Dirac cone semi-metal

M=2 M< 2 M> 2

Continuous Topological Quantum Phase Transition

band structure evolves smoothly with control parameters…

h(k) = d(k) · τ

BHZ description of topological transition:

𝒟1=0 d(k)

M> 2

spin texture

Topological QPT

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SLIDE 7

band inversion QSH insulator

  • 3
  • 2
  • 1

1 2 3 M X

M=2 M< 2 M> 2

Continuous Topological Quantum Phase Transition

band structure evolves smoothly with control parameters…

h(k) = d(k) · τ

BHZ description of topological transition:

𝒟1=0 𝒟1=1 d(k)

M< 2 M> 2

spin texture

Topological QPT

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SLIDE 8

Quest for larger SOC…heavy elements compounds (5d/4,5f)

What about the interaction?

Hexaborides Sm/PuB6, Ir-based pyrochlores: Sr2Ir2O7 , etc..

  • D. Pesin, L. Balents, NP 2010

Dzero et al. PRL 2010 Hohenadler , Assad. Journal of Phys. 2013 Deng et al PRL 2013

U

SOC

Mott

Topological Band Ins.

“Simple” Materials

Topological Mott Ins.

Magnetic Order

Engineering correlated TI: Transition Metal Oxides Heterostructures

  • D. Xiao et al. Nat. Comm. 2011

LaAuO3

New materials?

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SLIDE 9

Advantages:

+ local quantum physics (beyond Hartree-Fock). + non-perturbative in the interaction + access to topological invariant

Drawbacks:

  • neglects spatial fluctuations
  • computational demanding…

Dynamical Mean-Field Theory non-perturbative solution of the interacting problem

Idea: Reduce the interacting lattice problem to a self-consistent impurity problem

DMFT solution

Self-consistency

DMFT

Impurity solver

Imaginary time Lattice problem Impurity problem

solve using Exact Diagonalization & CTQMC

Obtain dynamical (non-scalar) self-energy. Describes the effects of interaction.

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SLIDE 10

BHZ effective minimal model + multi-orbital interactions

BHZ - Interaction

AA et al PRL 2015

M

LOW-SPIN

Meff

HIGH-SPIN

AA et al PRB 2016

Vs

Meff = M + Tr[τz ˆ Σ(0)]/2

Effective reduction of the Mass term:

10 20 30

ωn

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

ReΣ(iω)

U=3.30 U=3.40 U=3.50 U=3.60 U=3.70 U=3.80 U=3.90 U=4.50 U=7.24 Trivial QSHI Mott

Meff = Mc

Hartree-Fock

CORRELATED TI

Mott Ins.

M=2.5

Interaction driven TI Mott phase at large U

M > 2

M-2W

Meff < 2

Mott Ins.

  • Top. Ins.

U

HI = (U − JH)N(N − 1) 2 − JH ✓N 2 4 + S2

z

2 − 2T 2

z

◆

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SLIDE 11

Correlated QSHI

Strong coupling: 1st order TQPT correlated many-body character Weak coupling: Continuous ~ U=0

0.1 0.2 0.3 0.4

1/U

0.2 0.25 0.3 0.35 0.4 0.45 0.5

1/M

Topological Insulator

Band Insulator M

  • t

t I n s u l a t

  • r

Quantum Critical Point Triple point

(a)

Uncorrelated topological transition

[HgTe/CdTe quantum wells]

10 1 0.1

Tr[τz ˆ ΣHF−τz ˆ Σ(0)]/2

Phase diagram M-U (flipped view).

AA et al PRL 2015

correlation strength.

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SLIDE 12

Correlated QSHI

AA et al PRL 2015

1.6 1.8 2.0 2.2 0.32 0.34 0.36 0.38 0.40

4.0 5.0 5.5 5.7 5.9 6.1 6.3 6.5 6.7 7.0 U

A clear picture from the iso-U curves

∆Meff =Meff(BI)−Meff(QSH)

h Hi

Metastable states hallmark of 1st transition.

2.7 2.8 2.9 3 3.1

M

0.8 0.9 1 U=5.5 U=5.9 U=6.0 U=6.1 U=6.2 U=6.3 U=6.8 (a)

κ = ∂hTzi/∂M Diverging orbital compressibility at U=Uc Experimental accessible quantities marking the TQPT.

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SLIDE 13

U<Uc

The transition to a topological state

  • ccurs thru band-gap closing.

Dirac cone formation.

U>Uc

No gap-closing No suppression of any symmetries protecting the topological state.

Absence of gap closure

Γ

k

0.05 0.1

P(k)

M=3.22 M=3.23 M=3.24 M=3.251 M=3.26 M=3.27 M=3.28

U=2

Γ

k

0.02 0.04 0.06 0.08

P(k)

M=4.560 M=4.561 M=4.562 M=4.563 M=4.564 M=4.565 M=4.566 M=4.567 M=4.568 M=4.569

U=11

Breakdown of the gap-less TQPT paradigm…

Weak Coupling Strong Coupling

AA et al PRB 2016 AA et al PRL 2015

Dirac Cone Gap!

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SLIDE 14

Correlated edge states

AA et al. PRB 2017

H = X

kxyy0

Ψ+

kxyM(kx)δyy0Ψkxy0 +

X

kxyy0

⇣ Ψ+

kxyTδy + 1y0Ψkxy0 +H.c.

⌘

M=[M −2t cos kx]Γ5+λ sin kxΓx

T=−tΓ5 + iλ 2 Γy

Consider a 2D stripe.

  • π

π

Ny kx

  • 3
  • 2
  • 1

1 2 3 −π π E(kx) kx

Helical gapless states localized at the edges. What’s the effects of strong correlation on the 2D stripe?

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SLIDE 15

Correlated edge states

Sequence of transitions to reach the Mott state.

Uc1 Uc2…Uc

2.8 3.0 3.2 3.4 3.6 3.8 4.0

U

0.0 0.2 0.4 0.6

Z

y=1 y=2 y=3 y=4 y=5

kx Ny

AA et al. PRB 2017

Y=1

U

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SLIDE 16

U<Uc1 Uc1<U<Uc2 Uc2<U<Uc3

50 40 30 20 10 1

  • 10

10

U=4.42 U=3.98 U=3.62

yi xi

  • 10

1 10 C

GAPPED EDGE

Z2 =(C↑−C↓)/2

Cσ(r)=2πihr|ˆ xσ

P ˆ

yσ

Q ˆ

yσ

P ˆ

xσ

Q|ri

Topological properties with OBC: Local Chern Marker

Gapped Edge

M

  • t

t ( t r i v i a l )

U<Uc1

Uc1<U<Uc2 U>Uc

Bulk compression —> Edge state reconstruction

Correlated edge states

What’s the fate of the edge states?

AA et al. PRB 2017

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SLIDE 17

Conclusions.

  • Topological States can be favoured by strong interaction.
  • Emergent thermodynamic character: 1st order transition.
  • New paradigm for TQPT : no gap closing but no symmetry breaking!
  • Correlation driven edge states reconstruction.

Outlook…

  • Break TRS or IS: correlation effects in Weyl SM.
  • Interplay of strong interaction and SOC: from models to real materials.
  • Topological Mott Insulators.
  • Condensed matter realization of excitations beyond “standard model”

AA et al. PRL 2015 AA et al. PRB 2016 AA et al. PRB 2017 AA et al. in preparation 2017