CCNU, Ji-sheng Chen Aug, 2006

Aug, 2006,

CCNU, Ji-sheng Chen Aug, 2006

Universal thermodynamics of Dirac fermions near the unitary limit regime and BEC-BCS crossover

Ji-sheng Chen

Phys Dep., CCNU, Wuhan 430079

chenjs@iopp.ccnu.edu.cn

Aug, 2006,

Contents1.Motivations

2. The universal dimensionless coefficient ξand energy gap Δ

3. Conclusions and prospects

Aug, 2006,

1. MotivationPhase transtion and phase structure

a 、 Changes of symmetry is the central topic of physics (nuclear physics, condensed physics, high energy physics etc.)

b 、 Through in-medium Lorentz violation! Many-body effects

Aug, 2006,

Many-Body Physics

A challenging topic:1, Strong coupled limit2, Long-range

force/correlating~thermodynamicsStatistical physics:microscopic

dynamics approach the macroscopic thermodynamics?

Clear dynamics~unclear thermodynamics

Aug, 2006,

Why Study Ultra-Cold Gases?Answer: Coherent Quantum

Phenomena

High Temperature:Random thermal motion dominates

Low Temperature:Underlying quantum behavior revealed

Quantum wave-like

behaviorClassical particle-like

behavior

Aug, 2006,

Quantum Coherence

Technology:Precision Measurement,

Navigation, Sensing

Direct Applications:Quantum Computing,

Quantum Information Processing

Intellectually Exciting:Counterintuitive,

Fundamental part of nature

Single particle “textbook” physics

Correlated Many-body physics-Connections to other fields

Condensed Matter, Nuclear

Aug, 2006,

Full description of （ Condensed Matter) Phase diagram

a,Astrophysicsb,Heavy ion collisionsc,Strongly correlated electronsd,Cosmology。。。

Aug, 2006,

Collective correlating;Ground state ： Ladder diagram ressumation

1 、 Binding energy:K,Kc, symmetry energy coefficient,isospin…

2 、 Pairing Correlations:…

Aug, 2006,

Aug, 2006,

Aug, 2006,

Ultra-Cold dilute degenerate atomic fermions gas(quantum effects)

BEC vs BCS: Cross-Over Near the Feshbach resonance, the bare scat

tering lengths between two-body particles diverge!

| |a 6 9(10 ), (10 )T K T nK

Aug, 2006,

Novel Physics

Key point:”physics”

Aug, 2006,

Aug, 2006,

Unitary limit, |a| diverges(main characteristic).

Short range force but long-range correlation, system details “erased”!

Dilute unitary gas: not “ideal free Fermi gas.”

Aug, 2006,

Universal property: dimensional analysis, the only dimensionful parameter is the Fermi momentum .

The corresponding energy scale is the Fermi kinetic energy

The system details do not contribute to the thermodynamics properties

fk

2

2fk

m

Aug, 2006,

Various approaches tried and results differ remarkably.

1,The “theoretical results” ξ ∼ 0.3 − 0.6. 2,Experimental results quite different, ξ ≈

0.74±0.07[5], ξ = 0.51±0.04[6], ξ ≈ 0.7[7], ξ = 0.27+0.12−0.09[8].

New result is about ξ=0.46 ±0.05, Science 311, 503 (2006)

3, The lattice result ξ = 0.25 ± 0.03 of Lee Dean et al.

This ξ attracts much attention in recent years

Too many updating works

Aug, 2006,

A challenging topic in contemporary physics: Related to many realistic problems

Bewitching in the fundamental Fermi-Dirac statistics

Even closely related with the SU(Nc) physics, e.g.,1. nucl-th/0606019, T Schaefer,From Trapped Atoms to Liberated Quarks1. nucl-th/0606046, E.V. Shuryak, Locating strongly coupled color superconduct

ivity using universality and experiments with trapped ultracold atoms

MBX

Aug, 2006,

Its exact value/how to approach?

MFT? No, “go beyond” MFT For example, epsilon expansion (Incor

porate T?) cond-mat/0604500, Y Nishida, D T SonPhys. Rev. Lett. 97, 050403 (2006)(ξ=0.475,Δ/μ=1.31 or Δ/Ef=0.62 )

Aug, 2006,

1, 20-40 particles extending to infinite particles system, eliable? Quantum Monte Carlo simulation, for example

Carlson et al., PRL, 91, 050401(0.44) (2003), “More accurate” 0.42, Δ/μ=1.2 PRL(2005)

PRL 95, 030404 (2005) (0.42)

PRL 96, 090404 (2006)(0.42)…Tc=0.23 Tf;

Phys. Rev. Lett. 96, 160402 (2006): 0.493, Tc =0.15 Tf.

New result “More exact” 0.44, Tc=0.25 Tf, cond-mat/0608154

2, Local density functional theory? At finite T?

Aug, 2006,

More challenging topic: the superfluid phase transition temperature Tc/energy gap0.05-1.5

At the unitary cross-over point, the superfluid transition temperature is also of the order of the Fermi kinetic energy and thus the weak-coupling

theories such as the BCS- or the

Bogoliubov-type are not applicable.

The differences for energy gap Δ can be as large as several times even with Monte Carlo

Aug, 2006,

Aug, 2006,

cond-mat/0608282 v1 11 Aug 2006

Aug, 2006,

Try to obtain the analytical results with a novel approach!

Analogism between the ultra-cold atoms and infrared singularity in gauge theory

Consider it from another point of view

Return to non-relativistic limitMake a detour

Aug, 2006,

Motivation:Topology similar to Feshbach resonance

Key point:”physics”

Landau Pole?

Anti-screened “vector boson” propagator with a negative Debye mass squared m=1

Aug, 2006,

To address this topic from the fundamental “gauge” theoryA,Construct a simple Model: “QED” ; B, Thomson Problem as a arm to attack this problem

Aug, 2006,

Why and how? Let the fermion have an “electric” ch

arge g Should be stabilized by a fictive opposit

e charged Thomson background in the meantime

Simultaneously with other internal global U(1)(“hypercharge”) symmetry quantum numbers(Similar to the lepton number of electric charged electrons)

Aug, 2006,

Aug, 2006,

Aug, 2006,

Gauge invariance ensured by the Lorentz transversalitycondition with HLS: 0A

Aug, 2006,

General expressions for energy density and pressure as well as entropy

0 ,|BA m n

Aug, 2006,

Generalized Renormalizaion condition

Aug, 2006,

At T=0

Tailor

Aug, 2006,

Non-relativistic limit relativistic limit With the relativistic expression

through odd-even staggering

4 / 9

25

5 /1818 2

ff

k

m Non-relativistic limit, Tc ≈ 0.157 Tf

4 / 9 fk Relativistic limit: Tc ≈ 0.252 Tf

7 / 9

Statistical

weight factor

5/3

4/3

Reasonablely consistent with the BCS theory but with an effective scattering length

Aug, 2006,

Main result for two-dimensions Can even approach the extreme occasion

S/V=P=E/V=0 for fermions at unitary, Surprisingly similar to Bose-Einstein Condensation of 3-dimensional for ideal Bose gas

0

Fractional Quantum Hall Effect

Kondo Physics, Confinement

*m

Aug, 2006,

d=2, ξ =0

Similar to this diagram?

Strong repulsion leads to “attraction”

Long range correlation controls the global behaviors of the system

Quantum Many-body Effect

Aug, 2006,

Ising universal classcontroversial: 2-D ξ =1???

d

Relativistic limit, ξ =7/9

Non-relativistic limit,

ξ=0.44 or 4/9

d<2, Unstable, no phase transition

d=2, ξ =0

Aug, 2006,

A new type of fermions superfluity for d=3

Stability: sound speed squared still positive

Rough work Specific heat capacity, bulk and shear

viscosity of fermions, … Polarized fermion gas,…

Aug, 2006,

A Dilemma Thermodynamics university hypothesis Problem, d=3, T=0

P=2/3 E/V for ideal fermion/bose gas P<2/3 E/V for non-ideal gas Can be found in any statistical physics text books. At unitary, P=2/3 E/V??? Many arguments in the literature: due to the scaling property, similar

to ideal gas?We find P=1/4 E/V, different from that for ideal fermion gas due to th

e implicit pairing correlation contribution to binding energy.

Communications with many active experts.The sound speed detection can judge this dilemma.

Aug, 2006,

Extending to finite aUnitary limit regime with finite scattering length at both T and density

Mean field theory:

the lowest order

Aug, 2006,

Exactly approach some of the experimental and quantum Monte Carlo simulation results

Same analytical result with power counting, James V. Steele, nucl-th/0010066

non-relativistic framework and T=0

Facilitates the comparison of non-relativistic and relativistic approaches to thermodynamics

4 / 9

54 / 9, , 0.157 , 1/ 4 / ,

184

7 / 9, , 0.252 , 1/ 7 / , lim9

f c f

f c f

E T T P E V non relativistic

E T T P E V Ultra relativistic it

Main results of nucl-th/0602065

Repulsive approaches to effective attraction

Aug, 2006,

D-dimensions:nucl-th/0608063

Aug, 2006,

3.Conclusions and Prospectsa.Non trivial screening effectsAnti-screened(off-shell) vector boson propagator

Coupled Dyson-Schwinger equations “instead of” the involved integral equations of Fock-like exchange

Effective interaction: Landau pole

“contribution”

Infinite Feynman Diagrams

But not conventional resummatio

n

Aug, 2006,

B,Highlights:many-body physics

a, In-medium vector condensation formalism Lorentz violation may be an important tool within the frame of continuum field theory

b,Classical Thomson Problem(Newton third law) may be a potential non-perturbative tool to address the long range universal fluctuations and correlations. Critical phenomena:MFT?

Rich phase structure for hot and dense system~quantum Hall effects, Landau levels...

Aug, 2006,

1,To boldly approach the unitary topic with the exact “QED”

2,Classical Thomson Problem/Newton third law as a tool to approach the quantum phase transition physics(classical universal thermodynamics)

3,With the unknown side to solve the other unknown side

Aug, 2006,

Thank You!