Bose-Hubbard Model (BHM) at Finite Temperature

Transcription

1 Bose-Hubbard Model (BHM) at Finite Temperature - a Layman s (Sebastian Schmidt) proposal - pick up Diploma work at FU-Berlin with PD Dr. Axel Pelster (Uni Duisburg-Essen) ~ Diagrammatic techniques, high-order, resummation theory Prof. Hagen Kleinert (FU-Berlin) ~Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 4 th edition

2 Ultracold Atoms What s Hot? Quantum Information (QI) Quantum Simulation (QS) Atom Chip High Tc QCD Schmiedmayer (TU Vienna) W. Hofstetter (Frankfurt U)

3 Experiment+Theory Need to know what I simulate Quantitative agreement between Theory and Experiment Challenges: LDA Trapping homogeneous inhomogeneous Thermometer? Finite T Superfluid density, Mott phase???

4 Numerical vs Analytical DMRG, DMFT, NRG, Monte-Carlo (1D, 2D, 3D) BUT: Large-scale/flexibility? Fermions: Monte-Carlo sign Cassandra Problem Mean-Field Theory + Quantum Corrections Perturbation Theory + Resummation BHM=Benchmark Disorder, Geometries Flavour, Spinor,. Fermi-Hubbard

5 Outline Formulation of the Problem Bose Hubbard Model Finite-T Phase Diagram Status Quo Bogoliubov Strong Coupling Decoupling Monte-Carlo as a Benchmark Proposal: From Above From Below

6 Bose-Hubbard Model Lattice Pseudopotential Wannier Representation Lowest band N.N. hopping

7 Superfluid-Mott Insulator Superfluid Mott Insulator

8 Finite-T Phase Diagram Formulation of the Problem T/J NL SF MI U/J What is Tc (Uc) in 3D?

9 Bogoliubov Momentum Representation Cubic lattice Ansatz: Expand up to quadratic fluctuations & Minimize wrt. condensate density Linear order: Quadratic order:

10 Bogoliubov Quadratic fluctuations Bogoliubov Transformation No Gap Sound Modes! Quasiparticle Energies

11 Bogoliubov Phase transition via Condensate fraction van Oosten et al., PRA 63, (2001) T=0 NO Phase transition at T=0! BUT Good description of the condensate at weak coupling

12 Decoupling Ansatz in site-basis with consistent mean-field theory Mean field Hamiltonian Diagonalization in site-basis Perturbation Theory 0 0 K. Sheshardi et al., EPL 22, 257 (1993) van Oosten et al., PRA 63, (2001)

13 Decoupling Phase transition ~Landau expansion! =0 Mott Lobes Fisher et al., PRB 40, 546 (1989) Alexander Hoffmann, Diploma Thesis, FU-Berlin (2006)

14 Decoupling Finite Temperature Alexander Hoffmann, Diploma Thesis, FU-Berlin (2007)

15 Strong Coupling Unperturbed system is site-diagonal! Rayleigh-Schroedinger perturbation theory Particle (hole) excitation gap Phase Transition: 3 rd order +linear extrapolation Freericks & Monien, PRB 53, 2691 (1996)

16 Status Quo T/J NL Decoupling Reentrant classical field effect (zero Matsubara mode) H.Kleinert, S.Schmidt, A.Pelster, PRL 93, (2004) From Below Bogoliubov SF MI Strong-Coupling From Above U/J Monte-Carlo Data B. Capogrosso-Sansone, N. V. Prokofev, B. V. Svistunov, PRB 75, (2007)

17 From Above Diagrammatic Strong Coupling + T>0 Ohliger & Pelster, unpublished (2008) Decoupling MFT= Subset of strong-coupling diagrams Quantum Correction completely analytic! Phase diagram, Time-of-flight pictures, Visibility, Spin-1 Bosons,.. T=0 Finite-T Ednilson Santos, FU-Berlin, PhD Thesis (in preparation)

18 From Below Imaginary time path integral Classical fields Bosonic Action

19 From Below Motivation: Generic approach for superfluid phase? Background field Tree Level Quadratic Fluctuations Perturbation Free Energy

20 From Below One-Loop Approximation Tree Level U Quadratic Fluctuations with Kernel U +2U U +2U Fourier & Matsubara decomposition of Fluctuations and Kernel

21 From Below One-Loop Approximation Loop counter U Quasiparticle Energies U U Extremalization wrt. Bogoliubov very general approach: long-range, multi-component, disorder etc allows for systematic diagrammatic analysis Def: 1 st Bogoliubov Correction = 2-Loop Background (has never been calculated?)

22 From Below Two-Loop Approximation Taylor Expansion Ensemble Average Wick Theorem Feynman Rules Pseudopotential

23 From Below Two-Loop Approximation 2 NO Divergencies on the lattice! Gap or NO Gap? That s the Question! Diagrams are finite!!!! Automating higher orders (Resummation via VPT)