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Strongly Correlated Strongly Correlated Topological Insulators Topological Insulators

Predrag Nikoli ć Predrag Nikoli ć

George Mason University George Mason University

Institute for Quantum Matter @ Johns Hopkins University Institute for Quantum Matter @ Johns Hopkins University

Nordita Nordita July 13, 2016 July 13, 2016

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Acknowledgments

Collin Broholm IQM @ Johns Hopkins

Wesley T. Fuhrman Johns Hopkins

2/33

US Department

of Energy National Science Foundation

Zlatko Tešanović IQM @ Johns Hopkins

Michael Levin University of Chicago

Kondo Topological Insulators

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Topological Train Tracks

3/33

• Z Z

22

topology in 1D: topology in 1D:

– – Odd # of defectsOdd # of defects

in a loop (measurable) in a loop (measurable)

– – Defects can be createdDefects can be created or removed in pairs

or removed in pairs

– – Invariant: the presenceInvariant: the presence of a “charm” element of a “charm” element

Kondo Topological Insulators

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• Low-energy electrons live on the TI boundary...

– How would they be affected by strong interactions?

– Are there realistic ways to correlate them?

• Correlated TI ultra-thin films

– Triplet orders & SU(2) vortices

– Incompressible quantum liquids (fractional TIs)

• Correlations on a Kondo TI surface (SmB

6

)

– 2D “topological” heavy fermion system

Strongly Correlated TIs

4/33 Kondo Topological Insulators

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Rashba S.O.C. on a TI's Boundary

5/33

• Particles + static SU(2) gauge field

Yang-Mills flux matrix (“magnetic” for μ=0) SU(2) generators

(spin projection matrices)

...

– Rashba S.O.C.  Dirac spectrum

D. Hsieh, et.al, PRL 103, 146401 (2009) Y. Zhang, et.al, Nature Phys. 6, 584 (2010)

Cyclotron:

SU(2) flux:

Strongly correlated electrons with Rashba spin-orbit coupling

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• Artificially engineer correlations in a TI film

– A 2D band-insulator with pairing interactions and

tunable chemical potential  pair condensation QCP

• Cooper pair device

– TI film (say Bi2Se3)

– Pairing by proximity to a superconductor – QCP tuned by gate voltage, SC's Tc

• Exciton device

– TI film in a capacitor

– Biased capacitor creates an exciton gas – QCP tuned by capacitor bias

Correlated TIs in Heterostructures

PN, T.Duric, Z.Tesanovic PRL 110, 176804 (2013)

Seradjeh, Franz

Strongly correlated electrons with Rashba spin-orbit coupling 6/33

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Triplet Instabilities

7/33

• Pairing between electrons of opposite SU(2) charge

– Cooper pairs or excitons have spin  couple to the Rashba

PN, T.Duric, Z.Tesanovic PRL 110, 176804 (2013) PN, Z.Tesanovic

PRB 87, 104514 (2013) PN, Z.Tesanovic

PRB 87, 134511 (2013) a triplet pair mode enhanced

by Rashba S.O.C.

Kondo Topological Insulators

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Pairing Channels

8/33 Charge and spin fractionalization in strongly correlated topological insulators

• The minimal TR-invariant topological band-insulator

– 2 spin states ( ) X 2 surface states ( ) – short-range interactions

Decouple all interactions  6 Hubbard-Stratonovich fields:

Spin-singlet: , Spin-triplet:

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Effective Theory

9/33 Charge and spin fractionalization in strongly correlated topological insulators

• The most generic Cooper-pair action

– Integrate out all fermion fields near the Cooper Mott transition – Spin-triplets feel spin-orbit coupling

Constructed from the SU(2) gauge symmetry (idealized) There is SU(2) “magnetic” flux:

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Triplet Condensates & Vortices

10/33

– Rashba S.O.C.: momentum-dependent Zeeman – A large-momentum mode has low energy

• Landau-Ginzburg picture

– Helical “spin-current” T1 & T2 condensates – T1 phase can be TR-invariant

– T1 breaks rotation and translation symmetries – T1 has metastable vortex clusters & lattices

PN, PRA 90, 023623 (2014)

Kondo Topological Insulators

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Type-I Condensates

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Spin current without charge current

• Spin current densities & the Hamiltonian

Rashba S.O.C. 

TR-invariant

11/33

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Type-I Vortices

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Conservation laws:

– no sources for , and

– vortex is source/drain

• Neutrality: vortex quadruplets

– vortices carry two “charges”

– U(1) q (anti)vortex is bound to Ña vector (anti)vortex

12/33

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Type-I Vortex Structures

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Non-neutral clusters • Domain wall

• Vortex lattice

– unit cell is a quadruplet – square geometry

– a changes by np between singularities  rigid (meta)stable structure

 one (n =1) vortex per SU(2) “flux quantum”

13/33

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Stability of Vortex States

• Continuum: vortex cores are costly  uniform states

• Do vortex lattices ever win?

– good candidates: metastable type-I structures

– tight-binding lattice systems: vortex cores are cheap (if small) – entropy favors vortices (order by disorder, or vortex liquids)

• Microscopic lattice model

– triplet pairing of fermions with Rashba S.O.C. on a square lattice bilayer (triplet superconductivity in a TI film)

Vortices and vortex states of Rashba spin-orbit coupled condensates 14/33

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The Hamiltonian of TI Surfaces

15/33

– SU(2) charge:

g = τ z = ±1 helicity of spin-momentum locking

2D tight-binding model SU(2) gauge field on lattice links

Continuum limit

Rashba spin-orbit coupling

Yang-Mills (magnetic) flux

– Singularity (“defect”)

Dirac point in E(k) at some energy – Dirac points can be gapped

only in pairs (TR symmetry) – SmB6 (100 surface):

M is gapped (bulk TI)

Kondo Topological Insulators

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Correlated Lattice TI Films

16/33

• Inter-surface triplet pairing

– Enhanced at large momenta by S.O.C.

PN, T.Duric, Z.Tesanovic PRL 110, 176804 (2013)

• Finite-momentum condensate:

– SU(2) vortex lattices

Similar to:

W.S.Cole, S.Zhang,

A.Paramekanti, N. Trivedi, PRL 109, 085302 (2012) PN, PRA 90, 023623 (2014)

Kondo Topological Insulators

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Competing Orders

17/33

PN, arXiv 1606.02317 (2016)

Kondo Topological Insulators

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Vortex Lattice Melting

18/33

• Quantum fluctuations

– Positional fluctuations of

SU(2)

vortices

– Grow when the condensate is weakened (by tuning the gate voltage)

– Eventually, 1st order phase transition (preempts the 2nd order one)

• Vortex liquid

– Particles per flux quantum ~ 1 – Fractional TI

Cyclotron:

SU(2) flux:

PN, T.Duric, Z.Tesanovic, PRL 110, 176804 (2013) PN, PRB 87, 245120 (2013)

Kondo Topological Insulators

• Fractionalization by vortex lattice melting?

– Numerical evidence: N. Cooper, etc. … U(1) bosonic quantum Hall – Field theory indications: a generalization of Chern-Simons

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Effective Theory of Fractional TIs

19/33

• Landau-Ginzburg theory (dual Lagrangian of vortices)

• Topological term

– equation of motion  “drift currents” in U(1)xSU(2) E&M field

PN, PRB 87, 245120 (2013)

P.N, J.Phys: Cond.Mat.

25, 025602 (2013).

dual gauge field matrix

Maxwell term (density fluctuations) spinor vortex field

Levi-Civita tensor anti-commutators flux matrix

Kondo Topological Insulators

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Duality

20/33

• Particle Lagrangian

• Boson-vortex duality in (2+1)D

Exact if:

– particles are bosons – Sz is conserved

Conjecture:

– generalize by symmetry – fermionic particles?

Kondo Topological Insulators

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Abelian Quantum Hall Liquids

• Low-energy dynamics:

Chern-Simons

21/33

• Fractionalization

– Measured charge, spin  quantized – Dual vorticity  quantized

– Combine in quantum spin-Hall liquids  Laughlin sequence, quantized

dA dC

Maxwell

Kondo Topological Insulators

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Hierarchical States

• Emergent symmetries in the low-energy dynamics

– Multiple vortex “flavors”

Laughlin states:

• SU(2) hierarchy for spin-S particles

22/33 Kondo Topological Insulators

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Rashba Spin-Orbit Coupling

23/33

• Low-energy dynamics:

– Helical modes: Sp

• Incompressible states:

– Local density constraint: one free local parameter  (r) per mode

quantized “densities”

effective Chern-Simons theory w.

constrained non-Abelian gauge field

Kondo Topological Insulators

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Kondo Insulators

24/33

• Electron spectrum

– Hybridized broad d and narrow f orbitals of the rare earth atom (e.g. Sm)

– Heavy fermion insulators:

Ef is inside the hybridization bandgap – Coulomb interactions ≫ f-bandwidth

• Correlations

– T-dependent gap – Collective modes

• Materials:

– SmB6, YbB12, Ce3Bi4Pt3, Ce3Pt3Sb3, CeNiSn, CeRhSb...

Kondo Topological Insulators

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Kondo TIs (SmB 6 )

25/33

• Is SmB

6

a topological insulator?

– Theoretical proposal: Dzero, Sun, Galtski, Coleman – Residual T→0 resistivity & surface conduction

– Quantum Hall effect

– Latest neutron scattering

– Spin-sensitive ARPES D.J.Kim, J.Xia, Z.Fisk arXiv:1307.0448

A.Kebede, et al.

Physica B 223, 256 (1996)

G.Li, et al.

arXiv:1306.5221

Kondo Topological Insulators

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Collective Modes in SmB 6

26/33

• Sharp peak in inelastic neutron scattering

– Nearly flat dispersion at 14 meV

– Protected from decay (inside the 19 meV bandgap) – Theory: perturbative slave boson model

Neutron scattering experiment: Broholm group at IQM/JHU Theory: Fuhrman, PN

Kondo Topological Insulators

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Collective Mode Theory

27/33

• Perturbative slave boson theory

– Applied to the Anderson model adapted by Dzero, et al.

– Input: renormalized band-structure + 3 phenomenological param.

– Output: mode dispersion & spectral weight in the 1st B.Z.

• Slave bosons

– no double-occupancy of lattice sites by f electrons

Kondo Topological Insulators

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SmB 6 Theory + Experiment

28/33

• Match the theoretical & experimental mode spectra

– Fit the band-structure and phenomenological param.

– deduce the renormalized fermion spectrum implied band inversion at X  a TI

experiment theory

Implied qualitative FS of d electrons

Kondo Topological Insulators

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Kondo TI Boundary

29/33

• A 2D Dirac heavy fermion system

Bands “bend” near the boundary Hybridized regime Local moment regime

Bulk bands

SmB6 metallic surface electron dispersion

“heavy” surface electrons? light surface electrons?

PN, PRB 90, 235107 (2014)

In this cartoon:

Dirac surface states are made from the near-Ef portions of the bent bulk bands.

Kondo Topological Insulators

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Hybridized Surface Regime

30/33

Basic process:

f-d Hybridization assisted by a slave boson evolves into Kondo singlet upon localization

• 2D perturbative slave boson theory

– Slave bosons mediate forces among electrons – attractive in exciton, repulsive in Cooper ch.

slave boson renormalization

exciton Cooper

• Instabilities by nesting

– Magnetic (SDW)  Dirac points perish

– Superconductivity (s± or d-wave)  Dirac points live

(π,π)

(π,0)

Recall: we have a collective mode right at these wavevectors

Kondo Topological Insulators

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Surface Instabilities

31/33

• Weak-coupling orders

– Cooper:

– Exciton:

exciton Cooper

Kondo Topological Insulators

TR-invariant:

TR-broken:

Vortex lattice

• Stronger coupling at

– The same triplets as in ultra-thin TI films

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Surface Local Moment Regime

32/33

• Kondo v.s. RKKY

– Kondo wins: singlets frustrate the RKKY orders as doped “holes”  • spin liquid of localized f electrons

• … charged uncondensed slave bosons + neutral f spinons • … fluctuating vortices in the slave boson phase

2D gauge flux for d electrons (“QED3”) – RKKY wins 

• AF metal or insulator

• VBS, spin liquid, etc. Dirac metal

• The charge of f electrons is localized

– The f orbital is half-filled  Kondo lattice model – Light surface electrons?

PN, PRB 90, 235107 (2014)

Alexandrov, Coleman, Erten ; Thomson, Sachdev

Kondo Topological Insulators

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Conclusions

33/33

• Possible realizations in:

– Ultra-thin TI films – Kondo TI surfaces

Kondo Topological Insulators

• Inter-orbital pairing in TIs

– Spin-triplet pairing at large momenta due to S.O.C.

– TR-invariant vortex lattice

– Incompressible quantum (vortex) liquid (TR-invariant, non-Abelian)

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Type-II Condensates

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Spin current by charge current + spin texture

• Current densities & the Hamiltonian

Rashba S.O.C.  – charge current + spin texture

– TR broken

34/33

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Type-II Vortices

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Conservation laws:

• Vortex quadruplet

– not classically (meta)stable – charge singularities bound to

spin vortices (not antivortices)

– no sources for , and – no sources for

35/33

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Localization of Hybridized Electrons

36/33

• Mott insulators in doped lattice models

– Localization at p/q particles per lattice site – Broken translation symmetry

• Coulomb: may localize charge

– Can't localize spin so easily on a TI surface!

– S.O.C.  No spin back-scattering

• Mott insulators in doped lattice models

– Localization at

– Broken translation symmetry

A “flat” band!

(partially populated)

• Compromise: spin liquid

– Algebraic: Dirac points of spinons – Non-Abelian: fully gapped

Kondo Topological Insulators

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