Universal thermodynamics of a strongly interacting Fermi
gas
Hui HuHui Hu1,21,2, Peter D. Drummond, Peter D. Drummond22, and Xia-Ji Liu, and Xia-Ji Liu22
1.1. Physics Department, Physics Department, Renmin University of ChinaRenmin University of China;;2.2. ACQAOACQAO, Physics Department, , Physics Department, University of QueenslandUniversity of Queensland..
WhyWhy strongly interacting Fermi strongly interacting Fermi gases?gases?
Neutron star matter
Quark-gluon plasma of Big Bang
High Tcsuperconductors
Atomic Fermi gases with FR
Lattice field theory
Universal systems
• Fermions are the building blocks of matter.• Strongly-interacting Fermi gases are stable.
What is unitarity?What is unitarity?
LL
pFh
=
-V0
R
George Bertsch’s problem: 0→>>>>∞→ RLa
from Thomas’ talk
scattering l
0>a
0<a
B0
Magnetic-field Feshbach resonance: Magnetic-field Feshbach resonance: 66Li and Li and 4040KK
μν 2
1
0
2
−−= gVeff
magnetic detuning
Unitarity: σ(k)=4π/k22
Universality Conjecture:Universality Conjecture: One length scale: L = nOne length scale: L = n-1/3-1/3
Thermodynamics independent of Thermodynamics independent of structurestructure
Universality Conjecture:Universality Conjecture: One length scale: L = nOne length scale: L = n-1/3-1/3
Thermodynamics independent of Thermodynamics independent of structurestructure
Universal thermodynamics in a harmonic trap
Experimental investigations of Experimental investigations of thermodynamicsthermodynamics
Optical Trap LoadingOptical Trap LoadingForced EvaporationForced Evaporation High-Field ImagingHigh-Field Imaging
Problem: no reliable thermometryProblem: no reliable thermometry !!
Potential energy is obtained from axial cloud size
mask
Energy from interacting gasEnergy from interacting gas
0>a
0<aB0
Isentropic sweep to the weak coupling BCS limit
Isentropic ThermometryIsentropic Thermometry
Integrate
x
From Thomas- Fermi Fit: “true” temperature (entropy) for
non-interacting gas
mask
Entropy from noninteracting Entropy from noninteracting gasgas
Nozieres and Schmitt-Rink Nozieres and Schmitt-Rink theorytheory
Mean field thermodynamic potential
Where the free Nambu propagator:
Pair fluctuations
Where the Cooper pair propagator:
Gap and particle number equations
gap: i.e.,
num:
i.e.,
an additional term from conserving requirement
Other BCS-BEC crossover theories?
Theories T aammInter-molecule interactions (BEC)
MF T=0 aamm=2=2aaFF condensate-condensate
Original NSRT>Tc
aamm=0=0aaFF None !
Strinati et al. (NSR)
T=0 aamm=2=2aaFF ???
HLD T=0aamm=0.5=0.577aaFF
Includes pairing fluctuations
““aamm = 0.6 = 0.6aaFF”” within 2% accuracy, by Petrov within 2% accuracy, by Petrov et al.,et al., PRL, PRL, 20042004
?
Comparison to the heat capacity data by Thomas et al. (6Li)
• Cloud size at 840G gives Energy at unitarity (Virial theorem)
• Cloud size at 1200G gives Entropy at unitarity (isentropic sweep)
The measurement of entropy is a conceptual advance owing to John Thomas et al., with accuracy at 1% level
Duke, Thomas groupDuke, Thomas group: : L. Luo L. Luo et alet al., ., PRL PRL 9898, 080402 (2007)., 080402 (2007).
?
Comparison to the heat capacity data by Jin et al. (40K)
• Cloud size at 202G gives potential energy at unitarity
• Noninteracting cloud size gives non-interacting temperature
We then re-analyze the JILA data of potential energy and noninteracting temperature to obtain the energy and entropy.
JILA, Jin groupJILA, Jin group: : J. T. Stewart J. T. Stewart et alet al., ., PRL PRL 9797, 220406 (2006)., 220406 (2006).
Evidence for UniversalityEvidence for Universality
QuickTime™ and a decompressor
are needed to see this picture.
Perturbation vs Monte Carlo
Homogeneous
Case
Energy vs T
Entropy vs T
Comparative Studies in a trap
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
What about many Fermi species?Color Superconductors!
Exactly soluble problem in one-dimension - use LDA to treat a trap
QuickTime™ and a decompressor
are needed to see this picture.
Deconfining transition - evidence from breathing mode frequencies
BCS-BEC crossover with unequal spin
populations !
Like a dance floor with more men than women, the state contains paired and unpaired entities
ExptExpt.:.: M. W. Zwierlein M. W. Zwierlein et alet al. . ScienceScience 311311, 492 (2006)., 492 (2006). ExptExpt.:.: G. B. Partidge G. B. Partidge et alet al. . ScienceScience 311311, 503 (2006)., 503 (2006).
Quantum Phase TransitionQuantum Phase Transition
Experiments: Polarized Fermi gases
Phase SeparationPhase Separation
74.0=P
Questions• What is the ground state of a polarized Fermi What is the ground state of a polarized Fermi gas?gas?
• Does the FFLO state exist? Does the FFLO state exist?
• What is the structure of FFLO state? What is the structure of FFLO state?
• What is the intrinsic reason for the phase What is the intrinsic reason for the phase separation state? separation state?
• Is there any uniform superfluid state with a finite Is there any uniform superfluid state with a finite
spin population imbalance (i.e., spin population imbalance (i.e., ppcc~ 0.10 in the ~ 0.10 in the
unitary limit)? unitary limit)?
• c c = 0.74 in the unitary limit?= 0.74 in the unitary limit?
Polarized Fermi Gas in One Dimension
Method
Mean-field Exact solution
SPWA SCBdG
Phase Diagram of a Uniform System
Spin polarization
dimensionless parameter
Integrable Gaudin Equations (Bethe Ansatz):
nc )2/1('=
Phase Diagram of a Uniform System
SF : BCS-like superfluid state with zero polarization
SFP : Superfluid state with finite polarization
N : Fully polarized normal state
Structure of SFp Phase
P=0.03
P=0.16
The SFp is of FFLO Character
????
PHASE DIAGRAM IN A ONE-DIMENSIONAL TRAP
Conclusions and references• We We analyzeanalyze recent experimental recent experimental measurements on the finite temperature measurements on the finite temperature properties of a strongly interacting atomic properties of a strongly interacting atomic Fermi gas and Fermi gas and provideprovide evidence of evidence of universal universal thermodynamicsthermodynamics. . •Extensions: exact solutions for: 1D color Extensions: exact solutions for: 1D color superconductors, polarized Fermi gassuperconductors, polarized Fermi gas
• J. T. Stewart J. T. Stewart et alet al., ., Phys. Rev. Lett.Phys. Rev. Lett. 9797, 220406 , 220406 (2006).(2006).• L. LuoL. Luo et al., Phys. Rev. Lett. et al., Phys. Rev. Lett. 9898, , 080402080402 (2007).(2007).• G. Partridge G. Partridge et alet al., ., ScienceScience 311311, 503 (2006)., 503 (2006).
• Hu, Liu & Drummond, Hu, Liu & Drummond, Europhys. LettEurophys. Lett. . 7474, 574 , 574 (2006).(2006).• Hu, Liu & Drummond, Hu, Liu & Drummond, Phys. Rev. APhys. Rev. A 7373, 023617 , 023617 (2006).(2006).• Hu, Drummond & Liu, Hu, Drummond & Liu, 3 3 469, Nature Physics (2007).469, Nature Physics (2007).
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