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

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CCNU, Ji-sheng Chen Aug, 2006

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Ji-sheng Chen Phys Dep., CCNU, Wuhan 430079 [email protected]. Universal thermodynamics of Dirac fermions near the unitary limit regime and BEC-BCS crossover. Contents. 1.Motivations 2. The universal dimensionless coefficient ξ and energy gap Δ 3. Conclusions and prospects. - PowerPoint PPT Presentation

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

Page 1: Universal thermodynamics of  Dirac  fermions near the unitary limit regime and BEC-BCS crossover

CCNU, Ji-sheng Chen Aug, 2006

Page 2: Universal thermodynamics of  Dirac  fermions near the unitary limit regime and BEC-BCS crossover

Aug, 2006,

Page 3: Universal thermodynamics of  Dirac  fermions near the unitary limit regime and BEC-BCS crossover

CCNU, Ji-sheng Chen Aug, 2006

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

Ji-sheng ChenPhys Dep., CCNU, Wuhan 430079

[email protected]

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Contents1.Motivations

2. The universal dimensionless coefficient ξand energy gap Δ

3. Conclusions and prospects

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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

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Many-Body PhysicsA challenging topic:1, Strong coupled limit2, Long-range

force/correlating~thermodynamicsStatistical physics:microscopic

dynamics approach the macroscopic thermodynamics?

Clear dynamics~unclear thermodynamics

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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

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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

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Full description of ( Condensed Matter) Phase diagram

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

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Collective correlating;Ground state : Ladder diagram ressumation1 、 Binding energy:K,Kc, symmetry energy coefficient,isospin…2 、 Pairing Correlations:…

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Ultra-Cold dilute degenerate atomic fermions gas(quantum effects) BEC vs BCS: Cross-Over Near the Feshbach resonance, the bare scattering lengths between two-body particles diverge!

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

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Novel Physics

Key point:”physics”

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

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

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

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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

2fkm

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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 yearsToo many updating works

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

Bewitching in the fundamental Fermi-Dirac statisticsEven 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 superconductivity using universality and experiments with trapped ultracold atoms

MBX

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Its exact value/how to approach? MFT? No, “go beyond” MFT For example, epsilon expansion (Incorporate T?) cond-mat/0604500, Y Nishida, D T Son

Phys. Rev. Lett. 97, 050403 (2006)(ξ=0.475,Δ/μ=1.31 or Δ/Ef=0.62 )

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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?

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

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cond-mat/0608282 v1 11 Aug 2006

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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

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Motivation:Topology similar to Feshbach resonance

Key point:”physics”Landau Pole?

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

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

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Why and how? Let the fermion have an “electric” charge g Should be stabilized by a fictive opposite 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)

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

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General expressions for energy density and pressure as well as entropy

0 ,|BA m n

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Generalized Renormalizaion condition

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At T=0 Tailor

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Non-relativistic limit relativistic limit With the relativistic expression

through odd-even staggering

4 / 9

255 /1818 2

ff

km

Non-relativistic limit, Tc ≈ 0.157 Tf4 / 9 fk Relativistic limit: Tc ≈ 0.252 T

f

7 / 9

Statistical

weight factor

5/34/3

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

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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 EffectKondo Physics, Confinement

*m

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d=2, ξ =0Similar to this

diagram?Strong repulsion leads

to “attraction”

Long range correlation controls the global behaviors of the system

Quantum Many-body Effect

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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 transitiond=2, ξ =0

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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,…

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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 the implicit pairing correlation contribution to binding

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

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

Mean field theory:

the lowest order

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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 / ,1847 / 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

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D-dimensions:nucl-th/0608063

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3.Conclusions and Prospectsa.Non trivial screening effectsAnti-screened(off-shell) vector boson propagatorCoupled Dyson-Schwinger equations “instead of” the involved integral equations of Fock-like exchange

Effective interaction: Landau pole

“contribution”

Infinite Feynman DiagramsBut not conventional resummation

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B,Highlights:many-body physicsa, In-medium vector condensation formalism Lorentz violation may be an important tool within the frame of continuum field theoryb,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...

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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

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Thank You!