1. Aug, 2006 CCNU, Ji-sheng Chen

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3. Aug, 2006 CCNU, Ji-sheng Chen 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

4. Aug, 2006, Contents 1.Motivations 2. The universal dimensionless coefficient ξ and energy gap Δ 3. Conclusions and prospects

5. Aug, 2006, 1. Motivation Phase 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

6. Aug, 2006, Many-Body Physics A challenging topic: 1, Strong coupled limit 2, Long-range force/correlating~thermodynamics Statistical physics:microscopic dynamics approach the macroscopic thermodynamics? Clear dynamics~unclear thermodynamics

7. 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 behavior Classical particle-like behavior

8. 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

9. Aug, 2006, Full description of （ Condensed Matter) Phase diagram a,Astrophysics b,Heavy ion collisions c,Strongly correlated electrons d,Cosmology 。。。

10. Aug, 2006, Collective correlating; Ground state ： Ladder diagram ressumation 1 、 Binding energy:K,Kc, symmetry energy coefficient,isospin… 2 、 Pairing Correlations:…

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13. Aug, 2006, 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!

14. Aug, 2006, Novel Physics Key point:”physics”

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16. 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. ”

17. 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

18. 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

19. 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.,  nucl-th/0606019, T Schaefer,T Schaefer From Trapped Atoms to Liberated Quarks  nucl-th/0606046, E.V. Shuryak,E.V. Shuryak Locating strongly coupled color superconductivity using universality and experiments with trapped ultracold atoms MBX

20. Aug, 2006, 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 )Y NishidaD T Son

21. 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?

22. Aug, 2006, More challenging topic: the superfluid phase transition temperature Tc/energy gap 0.05-1.5 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

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24. Aug, 2006, cond-mat/0608282 v1 11 Aug 2006

25. 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 limit Make a detour

26. 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

27. Aug, 2006, To address this topic from the fundamental “ gauge ” theory A,Construct a simple Model: “QED” ; B, Thomson Problem as a arm to attack this problem

28. Aug, 2006, 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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31. Aug, 2006, Gauge invariance ensured by the Lorentz transversality condition with HLS:

32. Aug, 2006, General expressions for energy density and pressure as well as entropy

33. Aug, 2006, Generalized Renormalizaion condition

34. Aug, 2006, At T=0 Tailor

35. Aug, 2006, Non-relativistic limit relativistic limit With the relativistic expression through odd-even staggering Non-relativistic limit, T c ≈ 0.157 T f Relativistic limit: T c ≈ 0.252 T f Statistical weight factor 5/3 4/3 Reasonablely consistent with the BCS theory but with an effective scattering length

36. 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 Fractional Quantum Hall Effect Kondo Physics, Confinement

37. 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

38. Aug, 2006, Ising universal class controversial: 2-D ξ =1??? Relativistic limit, ξ =7/9 Non-relativistic limit, ξ=0.44 or 4/9 d<2, Unstable, no phase transition d=2, ξ =0

39. 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, …

40. 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 the implicit pairing correlation contribution to binding energy. Communications with many active experts. The sound speed detection can judge this dilemma.

41. Aug, 2006, Extending to finite a Unitary limit regime with finite scattering length at both T and density Mean field theory: the lowest order

42. 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/0010066James V. Steele non-relativistic framework and T=0 Facilitates the comparison of non-relativistic and relativistic approaches to thermodynamics Main results of nucl-th/0602065 Repulsive approaches to effective attraction

43. Aug, 2006, D-dimensions: nucl-th/0608063

44. Aug, 2006, 3.Conclusions and Prospects a.Non trivial screening effects Anti-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 resummation

45. 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...

46. 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

47. Aug, 2006, Thank You!