Daniel E. Sheehy The Hubbard model in cold atoms and in the high-Tc cuprates [email protected] “What are the key outstanding problems from condensed matter physics which ultracold atoms and molecules can address?” Aspen, June 2009
Daniel E. Sheehy
The Hubbard model in cold atomsand in the high-Tc cuprates
“What are the key outstanding problems from condensed matter physics whichultracold atoms and molecules can address?”
Aspen, June 2009
• Recently: Fermion Hubbard model in cold atom exp’ts Jordens et al Nature 2008, Schneider et al Science 2008
doping xAF SC
T
PG FL
T*
Tc
TNMFL
• Key Outstanding problem: Normal phase of cuprates
Pseudogap state ofunderdoped cuprates Optimal doping:
Marginal Fermi liquid
Quantum critical point?
Emergent particle-hole symmetry?
Meinders et al PRB 1993Honma & Hor PRB 2008Chakraborty et al 0807.2854
Outline
• Cuprate high-Tc superconductors: 2D Hubbard model
Neglects a lot of stuff!
BSCCO-2212
Bond ordered state inpseudogap state?
Kohsaka et al Nature 2008Macridin & Jarrell PRB 2008
Next: Cuprates
One e- per Cu
“half-filled”: Pauliwould allow 2 e-/site
Next: Hubbard Model
Cuprate high-Tc superconductors
YBCOCopper-oxygen planes (1,2,or 3)
other stuff
Cu CuO
Cu
O
O Cu
O
• Physics of SC: CuO planes
– Layers: copper-oxygen planes
• cuprates are layered
Hubbard model
t: Hopping matrix element between sites
U>0: Model Coulomb repulsion
!! "#><
++$=i
iiji
ijji nnUcccctH%
%%%%,,
)( † †
!!! jiiccn =
†nearest neighbor
P.W. Anderson, The theory of superconductivity in the high-Tc cuprates
• Question: What physics of the cuprates is captured by the Hubbard model?Answer: Noone knows
Cold atoms in optical lattices: Direct realizationSuperfluid-Mott transition of bosons:
Jaksch et al PRL 1998, Fisher et al 1989
Repulsive fermions:Hofstetter et al PRL 2002
Greiner et al Nature 2001; Spielman et al PRL 2007
Jordens et al Nature 2008, Schneider et al Science 2008…
Next: AF state
• One fermion per site– band theory: metal
Mott insulator
Half filling: Antiferromagnetic Mott insulator
!
U >> t– Large : No double occupancy!
• Antiferromagnetic order?Aligned spins:no virtual hops
AF alignment:virtual hops OK
Map to Heisenberg AF
!
H = J Si "< i, j>
# S j
Auerbach, “Interacting electronsand quantum magnetism”
!
J ~ t2/U
Next: cuprate phase diagram
• Quantitatively accurate at half filling (“parent compound”)– e.g., Birgeneau et al PRB 1999
Spin correlations above AF transition
X=0: AntiferromagnetWell understood!!
X~0.05: d-wave SC below Tc
++
-
-
!
px
!
py
Momentum-space Fermi surface:d-wave gap
Optimal Doping: x~0.2
Underdoped cuprates: Pseudogap behavior below T*
Hole concentration
Phase diagram of the hole-doped cuprates
Schematic!
doping xAF SC
T
PG FL
T*
Tc
TNMFL
Review: Norman and Pepin cond-mat/0302347
Marginal Fermi liquid: Unusual behaviorat optimal doping
Fermi liquid
Next: Pseudogap
Tc = 83K
Lowest T: no low-energytunneling
Cooper pair binding
!
2"
Highest T curve: tunnelingno problem
Where is the onset ofsuperconductivity?
What is the Pseudogap?• Numerous Exp’ts: Strong correlations above Tc Suppression of
low-energy states
Renner et al PRL 1998
BiSrCaCuO Tunneling: Inject electrons
into SC
SC
• Pseudogap: How to observe in cold-atom experiments?
RF spectroscopy, Photoemission (Stewart Nature 2008) Next: Scenarios
Pseudogap scenarios
doping xAF SC
T
PG FL
T*
Tc
TNMFL
Why is T* so large?
• Onset of order below T*
-d-density wave (Chakravarty et al PRB 2001)
-current loop order (Aji et al PRB 2008)
-bond order (Macridin et al PRB 2008)
– Next: Bond order
Has not been observed!
Why would an ordered phase be unstable to superdonductivity?
• Pairing above Tc - phase fluctuations (Emery + Kivelson Nature 95 , Franz and Tesanovic, PRL 01)
- BEC-BCS crossover (Maly et al PRB 96)
Nearest neighbor singlet correlations
!
1
2"# $ #"( )
• Scanning tunneling in pseudogap
Kohsaka et al Nature 2008
Anderson, Sachdev, …
Bond order• Exotic magnetic order: Valence bond
Broken rotational symmetry
– Next: Quantum critical point
Recent Dynamical Mean Field resultsMacridin et al PRB 2008
Similar bond-ordered phases
• Pseudogap: Subtle ordering
Quantum critical point scenarioTallon + Loram cond-mat/0005063Sachdev Science 2000Orenstein & Millis Science 2000
e.g., d-density wave, current loop order, valence bond order…
Heavy-fermionSC CePd2Si2
AF fluctuations mediate SC Mathur et al Nature 1998
Next: Marginal FL/”Strange metal”
• Strong fluctuations mediate SC?
• Terminates at a quantum critical point?
Under “dome” near optimal doping!
doping xAF SC
T
PG FL
T*
Tc
TNMFL
– Enlarged symmetry– Lack of energy scale– Marginal Fermi liquid
doping xAF SC
T
PG FL
T*
Tc
TNMFL
E.g., T-linear in-plane resistivity:
Strange metal/Marginal Fermi liquidAnomalous temperature dependencies at optimal doping
Ando et al, PRL 2004
Exactly linear at optimal doping
Deviations from linearity inunderdoped region (pseudogap)
Varma et al PRL 1989
Next: Quantum critical point
Matsuura PRB 1992
Next: Can the Hubbard model capture this?
Honma and Hor, PRB 2008
Universal vanishing of thermopower at optimal doping!
Evidence for Quantum Critical point: ThermopowerThermopower: Voltage drop under an applied temperature difference
!
S = "#V
#T
Chakraborty et al 0807.2854
Hubbard model Thermopower Beni PRB 1974Lewis PRB 1976Mukerjee PRB 1995
• Hubbard model in the atomic limit:
!! "#><
++$=i
iiji
ijji nnUcccctH%
%%%%,,
)( † †
!
t" 0 (no hopping, strong coupling)
• Thermopower:
!
S" ln1# x
2x
Vanishes at xc=1/3!
• Vanishing thermopower: Particle-hole symmetry Entropy carried by particles or holes?
• Idea: Thermopower data implies emergent particle-hole symmetry at optimal dopingChakraborty et al 0807.2854
– Away from atomic limit: – How can we test this scenario in cold-atom experiments?
xc<1/3
Next: Why p-h symmetry?
• Dynamical mean-field theory: Particle-hole symmetry at
!
x " 0.2
Vidhyadhiraja PRL 2009
• Dope 1 hole
Lower Hubbard band:Filled
Upper Hubbard band:Empty
Single-particle density of states
UN statesN states
Energy
Spectral Weight Transfer • Half filling
Meinders et al PRB 1993Chakraborty et al 0807.2854
(Still in atomic limit!)
Hole doping: Introduction of low energy excitations
UN-1 states N-1 states
2 states
!
µ
!
(",#)
Next: Dope more holes…
Energy
Spectral Weight Transfer 2 Meinders et al PRB 1993Chakraborty et al 0807.2854
• Dope xN holes? x=fraction of sites with holes
• Real space picture:
Two ways to put in a hole… Two ways to put in a particle…
UN(1-x) states N(1-x) states
2xN states
!
(",#)
!
µ
• Particle hole symmetry point:
!
1" x = 2x
!
x =1
3
U
!
µ
DeMarco: Measure thermopower & transport• How to probe in cold-atom Hubbard experiments?
Other probes sensitive to p-h symmetry?
• Phase diagram based on DMFT:
Park et al PRL 2008
Mott transition of 2-D Hubbard model• 2-D: No long-range order for continuous symmetries
– No AF order! – Can have Mott transition: Ising
Spins localized, but no LRO
• Cold-atoms: Test scenarios for the Mott transition– Role of local magnetic correlations
– Many electronic materials: Coupled to lattice
Critical point
Phase sep.
Entropy of insulator is lower than Fermi liquiddue to short-range correlations
Concluding remarks• Key Outstanding problem: Normal phase of cuprates/Hubbard model
Pseudogap state ofunderdoped cuprates Optimal doping:
Marginal Fermi liquid
doping xAF SC
T
PG FL
T*
Tc
TNMFL
– Emergent particle-hole symmetry?
• Quantum critical point? – Other evidence: Transport in phase underneath SC dome (Large B field)
Boebinger et al PRL 1996
Bond ordered state inpseudogap?
Kohsaka et al Nature 2008Macridin & Jarrell PRB 2008
• Can we probe the pseudogap in cold atom experiments?