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
Conference on Frontiers in Two-Dimensional Quantum Systems, ICTP, Trieste Nov 13-17, 2017
Nandini Trivedi Physics Department The Ohio State University
trivedi.15@osu.edu http://trivediresearch.org.ohio-‑state.edu/
BCS Eliashberg Proximate Insulator
EF = p2
F
2m
p = h/λ
wave length of electron wave
Fermi energy
Binding energy
g
5 10 15 20 25 30 35 0.0 50.0 100.0 150.0 200.0 250.0
Δ (meV) Tc ¡(K)
Δ vs ¡Tc
Tc(K)
Al 1.2 Pb 7.2 Nb 9.2 Nb3Ge 20.0 MgB2 38.6 H3S 203.0
Binding energy of pair or Energy gap in single particle spectrum @T=0 Transition temp
1 meV ~ 10K
10 20 30 40 50 60 0.0 50.0 100.0 150.0 200.0 250.0
Δ (meV) Tc ¡(K)
Δ vs ¡Tc
BCS Cuprate FeSC BCS ¡Theory 2Δ/kB ¡Tc ¡= ¡3.53
5 10 15 20 25 30 0.0 50.0 100.0 150.0 200.0 250.0
2Δ/Tc Tc ¡(K)
2Δ/Tc vs ¡Tc
BCS Cuprate FeSC BCS ¡Theory 2Δ/kB ¡Tc ¡= ¡3.53
(1) Strong coupling to glue: Fermi sphere greatly perturbed (2) non-adiabatic limit: electrons are slower than the mode
τe > τmode EF < Emode
Bismuth: EF=25 meV Emode=12 meV Tc= 0.5 mK~0.05 meV
Prakash, Kumar, Thamizhavel, Ramakrishnan, Science 355, 52–55 (2017)
(1) Strong coupling to glue: Fermi sphere greatly perturbed
g
(2) Non-adiabatic limit: electrons are sl slow
er than the mode
attraction
Strong glue: BCS-BEC Crossover
BCS limit
Cooper pairing
BEC limit
molecules
weak attraction: pairing and coherence
temperature strong attraction: pairing and coherence
temperatures
Tc = min(Δ 0,ρS)
ns= superfluid number density rs= sf mass density ws= sf plasma frequency scale directly related to the spectral weight in the delta function in the optical conductivity
p,s = c2
L
F = 1 2Ds Z ddx|rθ|2 [Ds] = Energy Ld−2
dim=2 Layered dim=3
Ds ∼ ~2 m∗ ns
Ds ∼ ~2ns m∗d
Ds ∼ ~2n2/3
s
m∗
London penetration depth
+ many other examples of insulators including disordered (localized) insulators
singlet
c~rs
Tc scale set by electronic energies not necessarily by coupling to a mode g: Driven by pressure, magnetic field, doping, gating
(Heavy ¡Fermion)
Georg Knebel, Dai Aoki, Jacques Flouquet, arXiv:0911.5223
Organic Superconductors
κ-‑(ET)2Cu2(CN)3
P . Giraldo-Gallo et al. (I. Fisher), PRB 85 85, 174503 (2012),; Nature Comm. 6, 8231 (2015)
Kurosaki et al. (Saito), PRL 95 95, 177001 (2005)
SrTiO3-‑δ SrTi1-‑xNbxO3
Xiao Lin et al. (K. Behnia), PRL 112 112, 207002 (2014)
arXiv:1703.06369 A full superconducting dome of strong Ising protection in gated monolayer WS2
Fermi Insulator Bose Insulator BEC BCS
crossover crossover
SC- Insulator Transition
Loh, Randeria, Trivedi, Chen, Scalettar, : Superconductor-Insulator Transition and Fermi-Bose Crossovers”
# of states energy
# of states energy
# of states energy
Fermi Band Insulator Metal
Fermi Insulator Bose Insulator BEC BCS
crossover crossover
SC- Insulator Transition
Increase hopping t between wells
U=0
U 6= 0
2D Fermion model for band-insulator à SC transition
local attraction -|U| à no sign-problem in QMC à possible to realize in cold atoms expts.
✗
Checkerboard
✓
Triangular lattice bilayer
Fix T=0, t⊥ = 1 t/t⊥ |U|/t⊥
attraction band structure
hni = 1
Diagrams & Mean-field theory Strong coupling Boson regime Fermion Determinantal Quantum Monte Carlo (QMC) “atomic” limit Changing band structure attraction
2 9 1.5 2
t t U t
FI BI BEC BCS (Fermion) Band Insulator Metal for |U|=0
Non-interacting Limit
Compensated (semi)metal with Electron & Hole FS’s 2 9 1.5 2
t t U t
FI BI BEC BCS (Fermion) Band Insulator Superconductor for |U| > 0
LIMITS: t=0 ATOMIC 2/9 2 FI Metal BI
Atomic Limit: tk = 0
N=0,1,2,3,4
µ = 0
Egap < ωpair
Egap > ωpair
pair susceptibility
Imχ(q = 0, ω)
ωpair
Pa Pairing ng Ins nstability in n a Band nd Ins nsul ulator
2/9 2
FI Metal BI
¼ χ0ðωÞ ¼ 1 N X
k
1 − 2fk 2εk − ω − i0þ
in Insulator = 2-particle gap à 0 at SIT
t t
à transition from insulator to SC Note: near the SIT
Pa Pairing ng Ins nstability in n a Band nd Ins nsul ulator
SIT
Single-particle (band) gap in Insulator, Finite at SIT Insulator
2/9 2
FI Metal BI
0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0
FI BI BEC BCS
t t energy scales t
1
pair
Eg Eg0 Ds
Mean Field Theory for SC state
SIT
Insulator Superconductor
Single-particle Energy Gap
Band gap in Insulator SC gap in BCS regime BEC regime near SIT BEC BCS ¡à
Superfluid Stiffness even outside the BEC regime à phase fluctuation dominate
2 9 2
t t U t
FI BI BEC BCS
(a) Diagrams & MFT
Strong coupling Bosons
QMC “atomic” limit
band structure attraction
Determinental Quantum Monte Carlo
Attractive Hubbard -- Free of fermion sign-problem at all fillings
2 9 1.5 2
t t U t
FI BI BEC BCS
Diagrams & MFT Actual Phase Diagram
2 9 2
t t U t
FI BI BEC BCS
(a)
ed jUj=t⊥ ¼ 4. (not shown).
across the SIT for at T ¼ 0.0803t⊥. text) and (b) superflu
12 × 12 × 2 bilayer (b) N for specific
SIT Insulator SC Single-Particle Density of States From QMC + Maximum Entropy Persistence of single-particle gap across the SC-Insulator transition
Can see gap directly from imaginary time QMC data without analytic continuation
2 9 2
t t U t
FI BI BEC BCS
(a)
T/t⊥ = 0.08 U/t⊥ = 4
QMC: Pairing structure factor QMC: Superfluid density SF INS SF INS
QMC
Λxx(ri, rj, t) = hjx(ri, t)j†
x(rj, 0)i
Ps = 1 N X
i,j
hc†
i↑c† i↓cj↓cj↑i
2 Predictions:
ARPES (angle resolved photoemission spectroscopy)
How to identify the BCS & BEC regimes in the crossover?
11/19/17
Fermion Spectral function
Angle resolved
= probability to make an excitation at momentum k and energy w
Single particle Green function Spectral function
Minimum gap at k=kF
k
u2
k
A(k, ω) = u2
kδ(ω − Ek) + v2 kδ(ω + Ek)
Ek = ± q ξ2
k + |∆k|2
u2
k + v2 k = 1
v2
k = 1
2 ✓ 1 − ξk Ek ◆
kF
Mi Minimum gap loc
space
BEC regime (“strong pairing”)
point
✏k = 0 k = 0
BCS regime (“weak pairing”)
contour
✏k = µ k = “kF ”
* Topology of “Minimum Gap Locus” * Gap-edge singularity in DOS
Crossover from BCS to BEC regime
ß BCS ß BEC BCS BEC BEC
Min gap locus is contour Min gap locus is point 1/(square-root) In DOS DOS has jump discontinuity (2D)
* Topology of “Minimum Gap Locus” * Gap
ap-ed edge e sing ngul ularity in n DOS How to identify the BCS & BEC regimes in the crossover?
BCS Min gap locus is contour 1/(square-root) In DOS
37
38
Tamaghna Hazra
Minimum ¡gap ¡locus ¡ is ¡a ¡contour ¡at
✏k = µ
Minimum ¡gap ¡ locus ¡is ¡a ¡point at k=0
40
Stewart, Gaebler & Jin, Nature 454, 744 (2008)
1/(kF a) = −∞
non-interacting
1/(kF a) = 0
unitarity
1/(kF a) ≈ 1
(BEC)
unpaired fermions
molecules
BEC regime
k-resolved RF Spectroscopy (~ ARPES)
interacting Fermi Gases Example from Cold Atoms Experiment
Fermi Insulator Bose Insulator BEC BCS
Summary of Insulator-Superconductor Transition:
crossover crossover
SC- Insulator Transition Lowest energy excitation is a charge e fermion Lowest energy excitation has charge 2e
àgoes soft at SIT Minimum Gap locus is a Point Minimum Gap locus is a Contour * Topology of minimum gap locus * Gap-edge singularity in DOS
Disorder-‑driven ¡Insulator-‑Superconductor ¡Transition
– Quantum phase transition – Is the SIT “fermionic” or “bosonic”?
Wh What is is the nature of the in insula lator? R → ∞ R → 0 IN S SC SIT T
Conductor-Insulator Quantum Phase Transition (ed. V. Dobrosavljevic, N. Trivedi, and J. M. Valles) Oxford (2012)
disorder
Haviland, Liu, Goldman, PRL 62,2180 (1989).
Egap
Egap
T Tc
wpair
Phase ¡Diagram Disorder ¡tuned ¡SIT
BCS
P''( )/ω ω 1 2 3 4 5 ω ¬1 1 ω V = 0.1 c P''( )/ω ω 2 pair ω 1 ¬1 1 2 ω V = 1.8 bTc = min(Egap, ρs) T ∗ ∼ Egap
2ωpair
pair susceptibility
Lo Local gap and Lo Local superfluid density
G(V), normalized 50 100 150 240 250 260 270 280 275 265 255 245 50 100 150 x [nm] [µV] y [nm] ∆ ¬1 1 V (mV) 1.0 0.5 G, normalized 1.5 a b ¬1 1 V (mV) 1.0 0.5 G, normalized 1.5 a b¬2 ¬1 1 2 0.1 0.2 0.3 ω V = 0.5 T = 0.1 T = 0.125 T = 0.2 T = 0.25 T = 0.5
b d
N(ω) ω
DOS ¡gapped ¡on ¡both sides ¡of ¡transition ¡ Disorder ¡leads ¡to ¡‘’SC ¡puddles’’
a b
0.4
¬2 ¬1 1 2 0.1 0.2 0.3 ω V = 3 T = 0.1 T = 0.25 T = 0.5 T = 1.5 T = 2.5
f
N(ω) ω
Theory: ¡Ghosal et ¡al, ¡ ¡PRB ¡65 ¡ ¡ 014501 ¡(2001) ¡PRL ¡81 ¡3940 ¡ (1998) Bouadim et ¡al, ¡Nat. ¡Phys. ¡7, ¡884 ¡ (2011) Experiment: ¡Sacepe, ¡et ¡ al, ¡Nat. ¡Phys 7 239 ¡ (2011), ¡PRL ¡101, ¡ 157006 ¡(2008)
Scan-tunneling Scan-‑squid
BCS Eliashberg Proximate Insulator
Proximate Insulator