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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