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Many-body physics with atomic gases in optical lattices Sebastiano Pilati “Abdus Salam” International Centre for Theoretical Physics Trieste- Italy Theory: Ping Nang Ma ETH – Zurich Prof. Matthias Troyer ETH – Zurich Prof. Stefano Giorgini INO-CNR “BEC” Center – University of Trento Prof. Xi Dai Chinese Academy of Science – Beijing Barcelona – 19 January 2012 Experiments: Prof. Salomon group ENS - Paris Prof. Nägerl group University of Innsbruck Main idea: use ultra-cold gases to test/further develop computational techniques for many-fermion systems Big challenge: strongly correlated electronic systems quantum magnetism (ferromagnetism, antif., spin liquids) giant magnetoresistance High-TC superconductors Motivation: ultracold gases are accessible and tunable Measuring the Equation of State: -) Exp: Nascimbene, Navon, Jiang, Chevy, Salomon, Nature 2010 (ENS - Paris) -) Theory: Ho, Zhou, Nature Phys (2009) Tuning the interaction strength SPIN UP: SPIN DOWN: s-wave scattering length Feshbach resonance “metastable” repulsive branch magnetic field attractive branch BEC-BCS crossov. BEC unitarity BCS Theory: Legget, Nozier-Schmitt-Rink Exp: ENS(Paris), MIT etc. EOS of attractive Fermi gas: BEC-BCS crossover Quantum MC —— experiment weakly attractive Fermi gas weakly repulsive Bose gas of molecules interaction parameter BEC unitarity BCS QMC: Astrakharchik at al. PRL (2004) SP, Giorgini PRL (2008) Exp: Nascimbene at al. Nature(2010) (ENS-Paris) Spin-imbalanced Fermi gases: N↑ > N↓ EF↑ EF↓ ? Polarized superfluid Phase-separated state magnetic susceptibility normalized pressure ▬▬ superfluid ▬▬ normal exp. ▬▬ QMC (normal) ▬▬ QMC (SF) interaction parameter b = (μ↑ - μ ↓) / μ BCS Energy difference EN-ES [3/5NEF] normalized pressure Equation of state: normal phase / 2m a BEC ▬▬ QMC 3 2 EC E N ES N 8 EF Nascimbène, Navon, Pilati, Chevy, Giorgini, Georges, Salomon, PRL 106 215303 (2011) MIT 2008 Tuning the interaction strength SPIN UP: SPIN DOWN: s-wave scattering length Feshbach resonance “metastable” repulsive branch magnetic field attractive branch BEC-BCS crossov. BEC unitarity BCS Theory: Legget, Nozier-Schmitt-Rink Exp: ENS(Paris), MIT etc. Two component repulsive Fermi gases STONER INSTABILITY → paradigm for itinerant ferromagnetism SPIN UP: SPIN DOWN: N↑ and N↓ fixed interaction strength PERTURBATIVE THEORIES: Mean-field theory (Stoner model): 2nd order phase t. at kFa = 1.57 2nd order perturbation theory (Duine and MacDonald,PRL2009): 1st order phase t. at kFa = 1.054 EXPERIMENT@MIT: Repulsive Fermi gases: Equation of State with Fixed-Node DMC “Fully polarized” ferromagnet Hard-Sphere ▲ Soft-Sphere Square-Well —— 2nd order p. th. Eint = 0 Ekinetic = 22/3 EIFG Small non-universal effects at kFa > 0.6 Homogeneous phase unstable at kFa ≈ 0.9 → but repulsive gas is unstable! (see exp. @ MIT : Sanner et al. arXiv:1108.2017) Inverse magnetic susceptibility SW HS Ferromagnetic transition kFa = 0.82(1) (HS) kFa = 0.86(1) (SW) Periodic potentials: optical lattices V0 d /2 V ( x) V0 sin 2 (x / d ) harmonic terms V0 : laser intensity λ : laser wavelength Bosons: single-band Hubbard model Wannier w0 phase diagram Fisher at al. (1989) Single-atom (and single-site) microscope I. Bloch’s group –Munich (2011) How to describe inter-atomic interactions: dilute systems → only s-wave scattering length matters V(r) V(r) r r 1. “BEST” approx. : Fermi-Huang Contact pseudopotential 4 2 V r a r r r m 2. (regularized δ-function interaction) gas-like state is out of equilibrium r0 << a needs huge number of Bloch bands effective (low-energy) Hubbard parameter U H.P. Büchler PRL (2010) δ-function interaction → 1st Born approximation 4 2 V r a r m 3. Hard-core potential (purely repulsive) V r if r a, 0 otherwise used to determine Hubbard parameter U Jaksch et al. PRL (1998) gas-like state is at equilibrium r0 >> a r0 = a possible non-universal effects at very large a Hard Spheres in OL (cont. space) vs. single-band Bose-Hubbard S.P., Troyer, arXiv: 1108.1408 optical lattice intensity n=1 Insulator —— ♦ —— hard-spheres in OL (cont. space) Single-band Bose-Hubbard Exp. (Innsbruck) Multi-orbital BH (Luehmann et al.) Superfluid freezing density Interaction strength • single-band Bose-Hubbard model: Capogrosso-Sansone et al. PRB 75 134302 (2007) • Hard-sphere freezing: Kalos et al. PRA 9 2178 (1974) Fermi gases in OL: \quantum magnetism ferromagnetic phase anti-ferromagnetic phase → (strong) repulsive interactions + periodic potential Feshbach resonances Optical lattices Tuning the optical lattice intensity Unexplored regime: Kohn-Sham density functional theory Homogeneous system QMC predicts itinerant ferromagnetism at very large interaction strength (kFa ≈0.9) Chang et al. PNAS (2011), SP,Bertaina et al. PRL(2010), Conduit et al. PRL(2009) Molecule formation is too fast! Pekker et al. PRL(2011) – see D. Pekker’s talk Sanner, arXiv:1108.2017 (MIT) Single-band lattice models • Metal - Mott insulator transition observed Esslinger’s group (ETH) : Jördens et al. Nature (2008) Bloch’s group: Schneider et al. Science (2008) • anti-ferromagnetic order not observed yet! Kohn-Sham Density Functional Theory the standard computational method in material science 1998 Nobel prize in chemistry Walter Kohn Es.: band structure of silicon John A. Pople successful in calculating properties of many metals, insulators, semiconductors wrong gap in transition metal compounds → simplified models (Hubbard) Chelikowsky and Cohen, PRB (1974) Use of DFT in quantum chemistry and materials science DFT: reduces the N-body problem to a 1-body problem Energy-density functional: E EKIN drV r r drdrVINT r r r r EXC kinetic energy external potential mean-field interaction Basic approx. → Local Density Approximation: note: not the AMO LDA!! eXchange and Correlation (unknown!) homo (r) EXC dr XC • 3D electron gas : Ceperley, Alder 1980 • 3D repulsive Fermi gas: this work From Fixed-Node Quantum Monte Carlo Beyond LDA: GGA, meta GGA, hybrid functionals, LDA + U, GW, LDA + DMFT, LDA +QMC... IDEA: → use KS-DFT in shallow optical lattices → test new energy-density functionals using ultracold gases in deep OL: clean and tunable KS-DFT with LDA of a repulsive Fermi gas in OL Fixed-Node Diffusion MC simulation of the homogeneous Fermi gas with short-range interaction Accurate parametrization of the EOS: Energy vs. P E = E(kFa, P) Energy vs. kFa and magnetic susc. Polaron energy and eff. mass • Non universal corrections: small for kFa < 1 • Hartree term is local: non self-interaction error • No core electrons – no pseudopotentials KS-DFT with LDA for a repulsive Fermi gas in OL GROUND-STATE PHASE DIAGRAM part. pol. ferrom. fully pol.ferrom. param. param. increasing OL intensity TECHNICAL DETAILS: exchange-interaction functional from QMC solve self-consistent Kohn-Sham eq. Bloch Ansatz (203 k-points) expanded in plane wave basis (113 states) n=1 ferrom. anti-ferrom. staggered polarization – uniform polarization Phase diagram at half-filling Band-structure spin-density wave gap Fermi gases in OL: QMC vs. KF-DFT (preliminary) n = 0.5 a/d = 0.04 SUMMARY: KS – DFT → extensive toolbox to simulate Fermi gases in OL experiments with OL → test bed to develop new energy density functionals (maybe more important that solving Hubbard model!) KS-DFT for repulsive Fermi gases in OL: • adding an OL favours ferromagnetism • KS-LDA predicts anti-ferromagnetism at n = 1 and T = 0 OUTLOOK: • finite temperature DFT • attractive Fermi gases in OL (BEC-BCS crossover) → superconducting DFT (Gross et al. PRL 1988, A. Bulgac PRA 2007) • time-dependent DFT (see TD-DFT for unitary Fermi gas Bulgac et al. 2011) • beyond LDA: GGA, hybrid functionals, LDA +U, LDA+GW, LDA + DMFT, LDA + QMC... ADDITIONAL MATERIAL → Two-body wave functions Kohn-Sham Density Functional Theory (with LDA and beyond): the standard computational method in material science very accurate for weakly/moderately correlated materials wrong gap in transition metal compounds → simplified models (Hubbard) IDEA: → use KS-DFT in shallow optical lattices → develop energy-density functionals for the strongly correlated regime NOTE: we can tune independently interaction strength and external potential Band Structure a = 0.08d E [ER] E [ER] a = 0.16d E [ER] E [ER] V0 = 4ER E [ER] a = 0.04d E [ER] V0 = 2ER