Equation of state of the unitary Fermi gassmp/Delta/... · Equation of state of the unitary Fermi...

Equation of state of the unitary Fermi gasIgor Boettcher

Institute for Theoretical Physics,University of Heidelberg

with S. Diehl, J. M. Pawlowski, and C. Wetterich

Δ13, 11. 1. 2013Cold atoms Functional

RG

The many-body problem

The many-body problempossibility of a statistical

description

collective degrees of freedom

The many-body problem1st step: Find the right Hamiltonian H

2nd step: Determine the partition function Z



H is knownfor coldatoms andQCD!



path integral

Euclidean quantum field theory

H is knownfor coldatoms andQCD!

Shopping listWhat are the generic features of quantum many-body systems?

What are reliable theoretical methodsto describe such systems?

What observables reveal advancementsand short-comings of theory?

Shopping listWhat are the generic features of quantum many-body systems?

What are reliable theoretical methodsto describe such systems?

What observables reveal advancementsand short-comings of theory?

cold atoms

neutron stars

nuclear matterheavy ion collisions

quark gluon plasma

high-Tc superconductors early universe

Shopping listTheory Experiments

with cold atomsPhase diagram andEquation of state

Momentum distribution

Transport coefficients

Density images

Collective modefrequencies anddamping constants

Expansion afterrelease from trap

Response functions...

The equation of stateClassical ideal gas:

Virial expansion for interacting gas:

Van-der-Waals equation of state:

Pressure P(μ,T)Bose gas

Density n=(∂P/∂μ)TBose gas

Bose gas

Isothermal compressibility (∂2P/∂μ2)T

Isothermal compressibility (∂2P/∂μ2)TBose gas Position of critical line:

phase diagram

Superfluid phase transition

The BCS-BEC CrossoverTwo cornerstones of quantum condensation:

Cooper pairingof weakly attractivefermions

Bose condensationof weakly repulsivebosons

BCS BEC


BCS BEC


BCS BEC

Unitary Fermi gas

The BCS-BEC Crossover

3D BCS-BEC crossover

(results from FunctionalRenormalization Group)

The BCS-BEC Crossover

2D BCS-BEC crossover

(results from FunctionalRenormalization Group)

BCSBKT

High Tc super-conductors!?

The BCS-BEC CrossoverKey observables

Dimer-dimer scattering length

Tan contact

Equation of state

Bertsch parameter Critical temperature

The BCS-BEC CrossoverKey observables

Dimer-dimer scattering length

Tan contact

Equation of state

Bertsch parameter Critical temperature

One problem:

Thermodynamics from density profiles

S. Nascimbène et al.

Ho, Zhou

local density approximation


M. J. H. Ku et al.,Science 335, 563-567 (2012)

Unitary Fermi gas at MIT by Zwierlein group


M. J. H. Ku et al.,Science 335, 563-567 (2012)


Tc/TF=0.167(13)



Bertsch parameter ξ:EoS at T=0



Bertsch parameter ξ:EoS at T=0 ξ=0.376(5)



Bertsch parameter ξ:EoS at T=0 ξ=0.376(5)

ξ=0.370(5)(8)

Equation of state from Functional RGExperiment:

Latest FRG:(Floerchinger, Scherer, Wetterich)



sacrifice ξ and Tc/TF




Most likely error source:

Frequency-independent cutoffs

First estimate: ξ=0.41 with 4D-fermioncutoff in the simplest truncation




Most likely error source:

Frequency-independent cutoffs

First estimate: ξ=0.41 with 4D-fermioncutoff in the simplest truncation

→ contact me during the workshopif you are interested in that

Tan contactMomentumdistribution

Tan relation

Asymptotic fermion self-energy

Tan contact

FRG: IB, S. Diehl, J. M. Pawlowski, C. Wetterich

Tan contact

Thank you for yourattention!

FRG: IB, S. Diehl, J. M. Pawlowski, C. Wetterich

Additional slides

Microscopic ModelMany-body Hamiltonian

Microscopic ModelMany-body Hamiltonian

Microscopic action

Macroscopic physicsHow to compute the partition function?

Integration

Macroscopic physicsHow to compute the partition function?

scale dependent partition function

Solve flow equation

Wetterich equationeffective action

Microphysics Macrophysics

fluctuations

Contact in the BCS-BEC Crossover

Momentum distributionIdeal Fermi gas: Fermi-Dirac distribution

Interactions

Momentum distributionTan contact C

Several exact relations, e.g.:

Contact from the FRGfull macroscopic propagator

Contact from the FRGFactorization of the RG flow for large p:

Contact from the FRGFactorization of the RG flow for large p:

Flowing contact

Contact from the FRGUniversal regime is enhanced for the Unitary

Fermi gas

Contact from the FRGTemperature dependent contact of the Unitary Fermi gas

Contact from the FRGContact at T=0 in the BCS-BEC crossover

Contact from the FRGMomentum distribution of the Unitary FermiGas at the critical temperature

without contact term

with contact term

Increase of densityContribution from high energetic particles

to the density

Substantial effect on

at Tc

Two-dimensional BCS-BEC Crossover

Two-dimensional BCS-BEC CrossoverWhy two dimensions?

● Enhanced effects of quantum fluctuations → test and improve elaborate methods

● Understand pairing in two dimensions → high temperature superconductors

How?

Highly anisotropic traps!

What is different?Scattering physics in two dimensions

Scatteringamplitude

Crossover parameter

No scale invariance, but strong correlations for

Equation of state at T=0

for

Equation of state at T=0

for

BKT BCS


for


for

Damping ofn-th mode:Thank you for your attention

and enjoy lunch!

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Equation of state of the unitary Fermi gassmp/Delta/... · Equation of state of the unitary Fermi...

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