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