Daniel E. Sheehy

The Hubbard model in cold atomsand in the high-Tc cuprates

Sheehy@LSU.EDU

“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?