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
Acknowledgments
Collin Broholm IQM @ Johns Hopkins
Wesley T. Fuhrman Johns Hopkins
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US Department
of Energy National Science Foundation
Zlatko Tešanović IQM @ Johns Hopkins
Michael Levin University of Chicago
Kondo Topological Insulators
Topological Train Tracks
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•
• Z Z
22topology 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
• 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
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Rashba S.O.C. on a TI's Boundary
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• 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
• 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
Triplet Instabilities
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• 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
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:
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:
Triplet Condensates & Vortices
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– 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
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
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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
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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”
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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
The Hamiltonian of TI Surfaces
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– 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
Correlated Lattice TI Films
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• 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
Competing Orders
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PN, arXiv 1606.02317 (2016)
Kondo Topological Insulators
Vortex Lattice Melting
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• 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
Effective Theory of Fractional TIs
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• 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
Duality
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• 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
Abelian Quantum Hall Liquids
• Low-energy dynamics:
Chern-Simons
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• Fractionalization
– Measured charge, spin quantized – Dual vorticity quantized
– Combine in quantum spin-Hall liquids Laughlin sequence, quantized
dA dC
Maxwell
Kondo Topological Insulators
Hierarchical States
• Emergent symmetries in the low-energy dynamics
– Multiple vortex “flavors”
Laughlin states:
• SU(2) hierarchy for spin-S particles
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Rashba Spin-Orbit Coupling
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• Low-energy dynamics:
– Helical modes: Sp
• 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
Kondo Insulators
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• 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
Kondo TIs (SmB 6 )
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• Is SmB
6a 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
Collective Modes in SmB 6
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• 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
Collective Mode Theory
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• 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
SmB 6 Theory + Experiment
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• 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
Kondo TI Boundary
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• 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
Hybridized Surface Regime
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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
Surface Instabilities
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• 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
Surface Local Moment Regime
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• 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
Conclusions
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• 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)
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
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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
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Localization of Hybridized Electrons
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• 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