Insulator to Superfluid/Superconductor Transition in 2 - PowerPoint PPT Presentation

Insulator ¡to ¡Superfluid/Superconductor ¡ Transition in ¡2 ¡dimensions Nandini Trivedi Physics Department The Ohio State University trivedi.15@osu.edu http://trivediresearch.org.ohio-­‑state.edu/ Conference on Frontiers in Two-Dimensional Quantum Systems, ICTP, Trieste Nov 13-17, 2017

Can an insulator become a SC? How does SC arise when there is no no Fermi surface?

Roadmap: Semi-Metal à Topological SC ? Insulator à SC Proximate Insulator Metal à SC BCS Eliashberg

Metal: Electron Waves Instability of Fermi surface p y Binding p F energy ∆ of pair g p x ∆ ∼ E m e − 1 wave length of electron wave p = h/ λ g E F = p 2 Fermi energy F ∆ < E m < E F 2 m

BCS Prediction ∆ /T c = 1 . 8 Binding energy of Δ vs ¡Tc pair or Energy 35 gap in single 30 particle spectrum @T=0 25 Tc(K) 20 Al 1.2 Δ (meV) Pb 7.2 15 Nb 9.2 10 Nb 3 Ge 20.0 5 MgB2 38.6 0 0.0 50.0 100.0 150.0 200.0 250.0 H 3 S 203.0 Tc ¡(K) Transition temp 1 meV ~ 10K

GAP vs Tc Δ vs ¡Tc 2Δ/Tc vs ¡Tc 60 30 50 25 40 20 BCS BCS Δ (meV) Cuprate 30 2Δ/Tc Cuprate 15 FeSC FeSC BCS ¡Theory BCS ¡Theory 20 10 2Δ/kB ¡Tc ¡= ¡3.53 2Δ/kB ¡Tc ¡= ¡3.53 10 5 0 0 0.0 50.0 100.0 150.0 200.0 250.0 0.0 50.0 100.0 150.0 200.0 250.0 Tc ¡(K) Tc ¡(K)

Another way the BCS paradigm can break down if T c ⌧ E mode ⌧ E F (1) Strong coupling to glue: Fermi sphere greatly perturbed (2) non-adiabatic limit: electrons are slower than the mode E F τ e > τ mode E mode E F < E mode Bismuth: E F =25 meV E mode= 12 meV Tc= 0.5 mK~0.05 meV Prakash, Kumar, Thamizhavel, Ramakrishnan, Science 355, 52–55 (2017)

How can the BCS paradigm break down? (1) Strong coupling to glue: Fermi sphere greatly perturbed T c ∼ ∆ × (2) Non-adiabatic limit: electrons are sl er than the mode slow ower ∆ ∼ E m e − 1 × g

Strong glue: BCS-BEC Crossover weak attraction: strong attraction: pairing and coherence pairing and coherence occur at the same occur at different temperature temperatures attraction BCS limit T c = min( Δ 0 , ρ S ) • cooperative BEC limit Cooper pairing • tightly bound • pair size molecules • pair size M. Randeria and E. Taylor, Ann. Rev. Cond. Mat. Phys. 5 , 209 (2014)

Superfluid density and stiffness n s = superfluid number density 4 π n s e 2 p,s = c 2 = ω 2 r s = sf mass density λ 2 w s = sf plasma frequency scale m ∗ L London penetration depth F = 1 Z d d x | r θ | 2 2 D s directly related to the spectral weight in the delta function in the optical conductivity [ D s ] = Energy L d − 2 dim=2 Layered dim=3 D s ∼ ~ 2 n 2 / 3 D s ∼ ~ 2 D s ∼ ~ 2 n s s m ∗ n s m ∗ d m ∗

Insulator: charge cannot move Band C-Mott S-Mott Pauli Attraction Repulsion Ω | "#i� | #"i singlet + many other examples of insulators including disordered (localized) insulators

Superconductivity domes Tc scale set by electronic energies not necessarily by coupling to a mode T c ~ r s g g: Driven by pressure, magnetic field, doping, gating

CeRhIn 5 Cuprates Pnictides BaPb 1-x Bi x O 3 (Heavy ¡Fermion) P . Giraldo-Gallo et al. (I. Fisher), PRB 85 85 , 174503 D. N. Basov & Andrey V. Chubukov, Nature Phys. 7 , 272 (2011) (2012),; Nature Comm. 6 , 8231 (2015) Organic Superconductors SrTiO 3 Georg Knebel, Dai Aoki, Jacques κ -­‑(ET) 2 Cu 2 (CN) 3 Flouquet, arXiv:0911.5223 SrTiO 3-­‑δ SrTi 1-­‑x Nb x O 3 Kurosaki et al. (Saito), PRL Xiao Lin et al. (K. Behnia), PRL 112 112 , 207002 (2014) 95 95 , 177001 (2005)

arXiv:1703.06369 A full superconducting dome of strong Ising protection in gated monolayer WS 2 J. M. Lu, O. Zheliuk, Q. H. Chen, I. Leermakers, N. E. Hussey, U. Zeitler, J. T. Ye

Band Insulator à SC Ω Fermi Bose BEC BCS Insulator Insulator crossover crossover SC- Insulator Transition Loh, Randeria, Trivedi, Chen, Scalettar, : Superconductor-Insulator Transition and Fermi-Bose Crossovers” Phys. Rev. X 6, 021029 (2016)

t Ω # of states energy Ω µ µ µ # of states # of states Fermi Metal Band Insulator energy energy t/ Ω

Fixed attraction U Fermi Bose BEC BCS Insulator Insulator crossover crossover SC- Insulator Transition U=0 U 6 = 0 Ω E 1 p = Ω E 2 p = 2 Ω Ω t/ Ω Increase hopping t between wells

The ¡Model 2D Fermion model for band-insulator à SC transition • Translationally invariant; no disorder • (at least) 2 sites/orbitals per unit cell à 2 bands in insulator • Attractive interactions à SC local attraction -|U| à no sign-problem in QMC à possible to realize in cold atoms expts. • Non-bipartite lattice à suppress CDW order Triangular lattice bilayer Checkerboard ✗ ✓

The ¡Model Triangular ¡bilayer ¡attractive ¡Hubbard ¡model | U | /t ⊥ Fix T =0, t ⊥ = 1 h n i = 1 attraction t/t ⊥ band structure

Methods attraction Strong coupling Boson regime “atomic” limit Fermion Determinantal Quantum Monte Carlo (QMC) Diagrams & Mean-field theory Changing band structure

Non-­‑interacting Non-interacting Limit BEC BEC U t 2 U t 2 BI BI (Fermion) 1.5 BCS 1.5 BCS (Fermion) Band Superconductor Band Metal Insulator for |U| > 0 FI FI Insulator for |U|=0 0 0 0 2 9 0 2 9 t t t t Compensated (semi)metal with Electron & Hole FS’s

Atomic LIMITS: t=0 ATOMIC BI 2 FI Metal 0 0 2/9

Atomic Atomic Limit: Insulators Atomic Limit: t k = 0 µ = 0 N=0,1,2,3,4 FI BI E gap < ω pair E gap > ω pair

Weakly ¡interacting Pa Pairing ng Ins nstability in n a Band nd Ins nsul ulator ¼ 1 − 2 f k χ 0 ð ω Þ ¼ 1 X BI 2 ε k − ω − i 0 þ N k 2 pair susceptibility Im χ ( q = 0 , ω ) FI Metal 0 0 2/9 ω pair o Pole in à Gap to pair excitations in Insulator = 2-particle gap à 0 at SIT

Weakly ¡interacting Pa Pairing ng Ins nstability in n a Band nd Ins nsul ulator BI Single-particle 2 (band) gap in Insulator, Finite at SIT FI Metal 0 0 2/9 Insulator o Divergence of pairing à transition from insulator to SC Note: near the SIT t t SIT

MFT Mean Field Theory for SC state Single-particle Energy Gap FI BI BEC BCS Band gap in Insulator 1.0 E g BCS ¡ à BEC BEC regime 0.8 near SIT energy scales t SC gap in BCS regime 0.6 pair 0.4 Superfluid Stiffness 1 0.2 E g0 D s even outside the BEC regime 0.0 à phase fluctuation dominate 0.0 0.1 0.2 0.3 0.4 t t Superconductor Insulator SIT

attraction Strong coupling Bosons Actual Phase Diagram Diagrams & MFT “ atomic ” limit (a) QMC BEC BI BEC U Diagrams & MFT U t 2 BI t 1.5 BCS 2 band structure BCS FI FI 0 0 2 9 0 t t 0 2 9 t t Determinental Quantum Monte Carlo Attractive Hubbard -- Free of fermion sign-problem at all fillings

(a) BI BEC U t 2 BCS FI 0 0 2 9 t t

Single-Particle Density of States Intermediate ¡coupling: ¡QMC From QMC + Maximum Entropy (a) SC BI BEC U t SIT 2 BCS Insulator FI 0 0 2 9 t t Persistence of single-particle gap across the SC-Insulator transition Can see gap directly from imaginary time QMC data without analytic continuation onset of both across the SIT for 12 × 12 × 2 bilayer at T ¼ 0 . 0803 t ⊥ . ed j U j =t ⊥ ¼ 4 . (b) N for specific text) and (b) superflu (not shown).

QMC: Pairing structure factor QMC: Superfluid density P s = 1 h c † i ↑ c † X i ↓ c j ↓ c j ↑ i N i,j Λ xx ( r i , r j , t ) = h j x ( r i , t ) j † x ( r j , 0) i off diagonal long range order T/t ⊥ = 0 . 08 U/t ⊥ = 4 INS SF INS SF QMC

How to identify the BCS & BEC regimes in the crossover? 2 Predictions: • Topology of “Minimum Gap Locus”– ARPES (angle resolved photoemission spectroscopy) • Gap-edge singularity in DOS – tunneling

Fermion Spectral function Angle resolved photoelectron spectroscopy • = probability to make an excitation rf spectroscopy • at momentum k and energy w Single particle Green function Spectral function 11/19/17

BCS: minimum gap locus at k F A ( k, ω ) = u 2 k δ ( ω − E k ) + v 2 k δ ( ω + E k ) q ξ 2 E k = ± k + | ∆ k | 2 ✓ ◆ k = 1 1 − ξ k u 2 k + v 2 v 2 k = 1 2 E k v 2 u 2 k k k F Minimum gap at k=k F

BEC regime (“strong pairing”) BCS regime (“weak pairing”) Mi Minimum gap loc ocus in k-sp space point contour ✏ k = 0 ✏ k = µ or or k = 0 k = “ k F ”

Crossover from BCS to BEC regime * Topology of “Minimum Gap Locus” * Gap-edge singularity in DOS ß BCS ß BEC BEC BEC BCS 1/(square-root) Min gap locus is DOS has jump Min gap locus is In DOS point discontinuity (2D) contour

How to identify the BCS & BEC regimes in the crossover? * Topology of “Minimum Gap Locus” * Gap ap-ed edge e sing ngul ularity in n DOS BCS 1/(square-root) Min gap locus is In DOS contour

Single ¡particle ¡gap 37

Tamaghna Hazra 38

BCS-­‑BEC ¡Crossover ¡: ¡Topology ¡of ¡Minimum ¡gap ¡locus Minimum ¡gap ¡locus ¡ Minimum ¡gap ¡ is ¡a ¡contour ¡at locus ¡is ¡a ¡point at ✏ k = µ k=0