Bose-Hubbard Model at Finite Temperature
Bose-Hubbard Model at Finite Temperature
Bose-Hubbard Model at Finite Temperature
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<strong>Bose</strong>-<strong>Hubbard</strong> <strong>Model</strong> (BHM)<br />
<strong>at</strong> <strong>Finite</strong> Temper<strong>at</strong>ure<br />
- a Layman’s (Sebastian Schmidt) proposal -<br />
pick up Diploma work <strong>at</strong> FU-Berlin with<br />
PD Dr. Axel Pelster (Uni Duisburg-Essen)<br />
~ Diagramm<strong>at</strong>ic techniques, high-order, resumm<strong>at</strong>ion theory<br />
Prof. Hagen Kleinert (FU-Berlin)<br />
~P<strong>at</strong>h Integrals in Quantum Mechanics, St<strong>at</strong>istics, Polymer Physics,<br />
and Financial Markets, World Scientific, 4 th edition
Ultracold Atoms – Wh<strong>at</strong>’s Hot?<br />
Quantum Inform<strong>at</strong>ion (QI)<br />
Quantum Simul<strong>at</strong>ion (QS)<br />
Atom Chip<br />
High Tc<br />
QCD<br />
Schmiedmayer (TU Vienna)<br />
W. Hofstetter (Frankfurt U)
Experiment+Theory<br />
Need to know wh<strong>at</strong> I simul<strong>at</strong>e<br />
Quantit<strong>at</strong>ive agreement<br />
between Theory and Experiment<br />
Challenges:<br />
LDA<br />
Trapping homogeneous inhomogeneous<br />
Thermometer?<br />
<strong>Finite</strong> T Superfluid density, Mott phase ???
Numerical vs Analytical<br />
DMRG, DMFT, NRG,…<br />
Monte-Carlo (1D, 2D, 3D)<br />
BUT:<br />
• Large-scale/flexibility?<br />
• Fermions: Monte-Carlo sign<br />
• “Cassandra” Problem<br />
Mean-Field Theory<br />
+ Quantum Corrections<br />
Perturb<strong>at</strong>ion Theory<br />
+ Resumm<strong>at</strong>ion<br />
BHM=Benchmark<br />
Disorder, Geometries<br />
Flavour, Spinor,….<br />
Fermi-<strong>Hubbard</strong>
Outline<br />
• Formul<strong>at</strong>ion of the Problem<br />
<strong>Bose</strong> <strong>Hubbard</strong> <strong>Model</strong><br />
<strong>Finite</strong>-T Phase Diagram<br />
• St<strong>at</strong>us Quo<br />
Bogoliubov<br />
Strong Coupling<br />
Decoupling<br />
Monte-Carlo as a Benchmark<br />
• Proposal:<br />
“From Above”<br />
“From Below”
<strong>Bose</strong>-<strong>Hubbard</strong> <strong>Model</strong><br />
L<strong>at</strong>tice<br />
Pseudopotential<br />
Wannier Represent<strong>at</strong>ion<br />
•Lowest band<br />
•N.N. hopping
Superfluid-Mott Insul<strong>at</strong>or<br />
Superfluid<br />
Mott Insul<strong>at</strong>or
<strong>Finite</strong>-T Phase Diagram<br />
Formul<strong>at</strong>ion of the Problem<br />
T/J<br />
NL<br />
SF<br />
MI<br />
U/J<br />
Wh<strong>at</strong> is Tc (Uc) in 3D ?
Bogoliubov<br />
Momentum<br />
Represent<strong>at</strong>ion<br />
Cubic l<strong>at</strong>tice<br />
Ans<strong>at</strong>z:<br />
Expand up to quadr<strong>at</strong>ic fluctu<strong>at</strong>ions<br />
& Minimize wrt. condens<strong>at</strong>e density<br />
Linear order:<br />
Quadr<strong>at</strong>ic order:
Bogoliubov<br />
Quadr<strong>at</strong>ic<br />
fluctu<strong>at</strong>ions<br />
Bogoliubov<br />
Transform<strong>at</strong>ion<br />
No Gap<br />
Sound Modes !<br />
Quasiparticle Energies
Bogoliubov<br />
Phase transition<br />
via Condens<strong>at</strong>e fraction<br />
van Oosten et al.,<br />
PRA 63, 053601 (2001)<br />
T=0<br />
NO Phase transition <strong>at</strong> T=0 !<br />
BUT Good description of the condens<strong>at</strong>e <strong>at</strong> weak coupling
Decoupling<br />
Ans<strong>at</strong>z in<br />
site-basis<br />
with<br />
consistent mean-field theory<br />
Mean field Hamiltonian<br />
Diagonaliz<strong>at</strong>ion in site-basis<br />
Perturb<strong>at</strong>ion Theory<br />
0<br />
0<br />
K. Sheshardi et al., EPL 22, 257 (1993)<br />
van Oosten et al., PRA 63, 053601 (2001)
Decoupling<br />
Phase transition<br />
~Landau expansion<br />
!<br />
=0<br />
Mott Lobes<br />
Fisher et al., PRB 40, 546 (1989)<br />
Alexander Hoffmann,<br />
Diploma Thesis, FU-Berlin (2006)
Decoupling<br />
<strong>Finite</strong> Temper<strong>at</strong>ure<br />
Alexander Hoffmann,<br />
Diploma Thesis, FU-Berlin (2007)
Strong Coupling<br />
Unperturbed<br />
system<br />
is site-diagonal!<br />
Rayleigh-Schroedinger<br />
perturb<strong>at</strong>ion theory<br />
Particle (hole)<br />
excit<strong>at</strong>ion gap<br />
Phase Transition:<br />
3 rd order +linear extrapol<strong>at</strong>ion<br />
Freericks & Monien, PRB 53, 2691 (1996)
St<strong>at</strong>us Quo<br />
T/J<br />
NL<br />
Decoupling<br />
Reentrant<br />
classical field effect<br />
(zero M<strong>at</strong>subara mode)<br />
H.Kleinert, S.Schmidt, A.Pelster,<br />
PRL 93, 160402 (2004)<br />
“From Below”<br />
Bogoliubov<br />
SF<br />
MI<br />
Strong-Coupling<br />
“From Above”<br />
U/J<br />
Monte-Carlo D<strong>at</strong>a<br />
B. Capogrosso-Sansone, N. V. Prokofev, B. V. Svistunov, PRB 75, 134302 (2007)
From Above<br />
Diagramm<strong>at</strong>ic Strong Coupling + Resumm<strong>at</strong>ion @ T>0<br />
Ohliger & Pelster, unpublished (2008)<br />
•Decoupling MFT= Subset of strong-coupling diagrams<br />
•Quantum Correction completely analytic !<br />
Phase diagram, Time-of-flight pictures, Visibility, Spin-1 Bosons,…..<br />
T=0 <strong>Finite</strong>-T<br />
Ednilson Santos, FU-Berlin, PhD Thesis (in prepar<strong>at</strong>ion)
From Below<br />
Imaginary time p<strong>at</strong>h integral<br />
Classical fields<br />
Bosonic Action
From Below<br />
Motiv<strong>at</strong>ion: Generic approach for superfluid phase?<br />
Background field<br />
Tree Level<br />
Quadr<strong>at</strong>ic<br />
Fluctu<strong>at</strong>ions<br />
Perturb<strong>at</strong>ion<br />
Free Energy
From Below<br />
One-Loop Approxim<strong>at</strong>ion<br />
Tree Level<br />
U<br />
Quadr<strong>at</strong>ic<br />
Fluctu<strong>at</strong>ions<br />
with Kernel<br />
U<br />
+2U<br />
U<br />
+2U<br />
Fourier & M<strong>at</strong>subara decomposition<br />
of Fluctu<strong>at</strong>ions and Kernel
From Below<br />
One-Loop Approxim<strong>at</strong>ion<br />
Loop counter<br />
U<br />
Quasiparticle Energies<br />
U<br />
U<br />
Extremaliz<strong>at</strong>ion wrt.<br />
Bogoliubov<br />
very general approach: long-range, multi-component, disorder etc<br />
allows for system<strong>at</strong>ic diagramm<strong>at</strong>ic analysis<br />
Def: 1 st Bogoliubov Correction = 2-Loop Background<br />
(has never been calcul<strong>at</strong>ed?)
From Below<br />
Two-Loop Approxim<strong>at</strong>ion<br />
Taylor Expansion<br />
Ensemble Average<br />
Wick Theorem<br />
Feynman Rules<br />
Pseudopotential
From Below<br />
Two-Loop Approxim<strong>at</strong>ion<br />
2<br />
NO Divergencies on the l<strong>at</strong>tice!<br />
Gap or NO Gap?<br />
Th<strong>at</strong>’s the Question!<br />
Diagrams are finite!!!!<br />
Autom<strong>at</strong>ing higher orders<br />
(Resumm<strong>at</strong>ion via VPT)
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