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

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 theory: s ymmetry breaking classification of matter phases key concept: local order parameter Magnetism Liquid-gas “Superconductivity” magnetization M density difference n(L)-n(G) pair amplitude ψ Experimental detectability!

Haldane PRL88 Introduction. Kane,Mele PRL05 Bernevig et al Science 2006 …. many more TOPOLOGICAL INSULATORS quantum materials eluding the G-L paradigm! bulk (band) insulator + with gapless edge modes. Dirac semi-metal + Spin-Orbit Coupling + = States classified in terms of the Topological Properties of the Hilbert space of Bloch functions : Trivial Non-Trivial key concept: global topological invariant

The quantum spin-Hall insulator Initial focus on graphene but small SOC ( gap ~ 10 -3 meV ) Kane,Mele PRL 2005 Bernevig et al Science 2006 Idea : look for systems with a larger SOC. Konig et al Science 2007 BHZ model : 2 QHI + Time Reversal Symmetry. CdTe/HgTe quantum wells . P 3/2 S h ( k ) = d ( k ) · τ Orbital pseudo-spin structure S P 3/2 P 1/2 P 1/2 d ( k )=[ λ sin k x , λ sin k y , M − ε ( k )]

Topological QPT 𝒟 1 =0 BHZ description of topological transition: h ( k ) = d ( k ) · τ spin texture M> 2 d ( k ) Continuous Topological Quantum Phase Transition band structure evolves smoothly with control parameters… 6 4 2 trivial 0 band insulator M< 2 M=2 M> 2 -2 -4 -6 M X

Topological QPT 𝒟 1 =0 BHZ description of topological transition: h ( k ) = d ( k ) · τ spin texture M> 2 d ( k ) Continuous Topological Quantum Phase Transition band structure evolves smoothly with control parameters… 4 3 2 Dirac cone 1 semi-metal 0 -1 M< 2 M=2 M> 2 -2 -3 -4 M X

Topological QPT 𝒟 1 =1 𝒟 1 =0 BHZ description of topological transition: h ( k ) = d ( k ) · τ spin texture M< 2 M> 2 d ( k ) Continuous Topological Quantum Phase Transition band structure evolves smoothly with control parameters… 3 2 1 band inversion 0 QSH insulator M< 2 M=2 M> 2 -1 -2 -3 M X

What about the interaction? Quest for larger SOC…heavy elements compounds ( 5d/4,5f ) Hexaborides Sm/Pu B 6 , Magnetic Order Ir-based pyrochlores: Sr 2 Ir 2 O 7 , etc.. Mott Dzero et al. PRL 2010 U D. Pesin, L. Balents, NP 2010 Hohenadler , Assad. Journal of Phys. 2013 Topological Mott Ins. Deng et al PRL 2013 “Simple” Topological Band Ins. New materials? Materials SOC Engineering correlated TI: Transition Metal Oxides Heterostructures D. Xiao et al. Nat. Comm. 2011 LaAuO 3

DMFT solution Dynamical Mean-Field Theory non-perturbative solution of the interacting problem Idea: Reduce the interacting lattice problem to a self-consistent impurity problem Impurity solver Advantages: + local quantum physics (beyond Hartree-Fock) . Imaginary time + non-perturbative in the interaction + access to topological invariant DMFT Drawbacks: - neglects spatial fluctuations - computational demanding… Lattice problem Impurity problem Self-consistency solve using Exact Diagonalization & CTQMC Obtain dynamical (non-scalar) self-energy. Describes the effects of interaction.

BHZ - Interaction AA et al PRL 2015 AA et al PRB 2016 BHZ effective minimal model + multi-orbital interactions LOW-SPIN HIGH-SPIN ✓ N 2 4 + S 2 ◆ H I = ( U − J H ) N ( N − 1) z 2 − 2 T 2 − J H z Vs 2 M M eff M e ff = M + Tr[ τ z ˆ Σ (0)] / 2 Effective reduction of the Mass term: 0 M > 2 M=2.5 M-2W -0.2 Hartree-Fock Interaction driven TI CORRELATED TI M eff < 2 Top. Ins. Re Σ (i ω ) -0.4 M eff = M c U=3.30 Trivial -0.6 U=3.40 U=3.50 Mott Ins. U=3.60 Mott phase at large U U U=3.70 -0.8 QSHI U=3.80 U=3.90 Mott Ins. U=4.50 Mott U=7.24 -1 0 10 20 30 ω n

Correlated QSHI AA et al PRL 2015 Phase diagram M-U ( flipped view). Weak coupling: Continuous ~ U=0 0.5 10 (a) Topological Insulator 0.45 Uncorrelated Σ (0)] / 2 topological transition r o [HgTe/CdTe quantum wells] t correlation strength. a 0.4 l u Σ HF − τ z ˆ s n 1 I 1/M t t 0.35 o M Tr[ τ z ˆ Quantum Critical Point 0.3 Band Insulator 0.25 0.1 Triple point 0.2 0.1 0.2 0.3 0.4 1/U Strong coupling: 1 st order TQPT correlated many-body character

Correlated QSHI AA et al PRL 2015 Metastable states hallmark of 1 st transition. h H i U 0.40 4.0 5.0 0.38 5.5 5.7 5.9 0.36 6.1 A clear picture from the iso- U curves 6.3 0.34 ∆ M e ff = M e ff ( BI ) − M e ff ( QSH ) 6.5 6.7 0.32 7.0 1 (a) 1.8 2.0 2.2 1.6 0.9 Diverging orbital compressibility at U=U c U=5.5 U=5.9 κ = ∂ h T z i / ∂ M U=6.0 U=6.1 0.8 U=6.2 Experimental accessible quantities marking U=6.3 U=6.8 the TQPT. 2.7 2.8 2.9 3 3.1 M

Absence of gap closure AA et al PRL 2015 AA et al PRB 2016 Breakdown of the gap-less TQPT paradigm… Weak Coupling Strong Coupling M=4.560 M=3.22 0.08 U=2 U=11 M=4.561 M=3.23 M=4.562 M=4.563 M=3.24 M=4.564 M=4.565 M=3.251 0.06 M=4.566 0.1 M=3.26 M=4.567 M=4.568 M=3.27 P(k) P(k) M=4.569 M=3.28 0.04 0.05 0.02 Dirac Cone Gap! 0 0 Γ Γ k k U>U c U<U c No gap-closing The transition to a topological state occurs thru band-gap closing. No suppression of any symmetries Dirac cone formation. protecting the topological state.

Correlated edge states AA et al. PRB 2017 Consider a 2D stripe. ⇣ ⌘ X X Ψ + Ψ + k x y M ( k x ) δ yy 0 Ψ k x y 0 + 1 y 0 Ψ k x y 0 + H.c. H = k x y T δ y + k x yy 0 k x yy 0 M =[ M − 2 t cos k x ] Γ 5 + λ sin k x Γ x π T = − t Γ 5 + i λ 2 Γ y 0 k x - π N y 3 2 Helical gapless states localized 1 at the edges. E(k x ) 0 What’s the effects of strong -1 correlation on the 2D stripe? -2 -3 −π 0 π k x

Correlated edge states AA et al. PRB 2017 Sequence of transitions to reach the Mott state. y=1 y=2 0.6 y=3 y=4 y=5 0.4 Z 0.2 k x N y 0.0 2.8 3.0 3.2 3.4 3.8 4.0 3.6 U U c1 U c2 …U c U Y=1

Correlated edge states AA et al. PRB 2017 What’s the fate of the edge states? Bulk compression —> Edge state reconstruction U<U c 1 U c 1 <U<U c 2 U c 2 <U<U c 3 10 1 Topological properties with OBC: U<U c1 U=3.62 0 Local Chern Marker C -10 U c 1 <U<U c 2 U=3.98 C σ ( r )=2 π i h r | ˆ P ˆ Q � ˆ P ˆ Q | r i x σ y σ y σ x σ Gapped Edge GAPPED EDGE 10 Z 2 =( C ↑ − C ↓ ) / 2 U>U c U=4.42 0 x i M o t t ( t r i v i a l ) 50 40 30 20 -10 10 y i 1

Conclusions. AA et al. PRL 2015 • Topological States can be favoured by strong interaction. AA et al. PRB 2016 • Emergent thermodynamic character : 1 st order transition. • New paradigm for TQPT : no gap closing but no symmetry breaking! AA et al. PRB 2017 • Correlation driven edge states reconstruction. AA et al. in preparation 2017 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”