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

3. Ji-sheng Chen Phys Dep., CCNU, Wuhan 430079 chenjs@iopp.ccnu.edu.cn Universal thermodynamics of Dirac fermions near the unitary limit regime and BEC-BCS crossover

4. Contents 1.Motivations 2. The universal dimensionless coefficient ξand energy gap Δ 3. Conclusions and prospects

5. 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. 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. Why Study Ultra-ColdGases? Answer: Coherent Quantum Phenomena High Temperature: Random thermal motion dominates Low Temperature: Underlying quantum behavior revealed Classical particle-like behavior Quantum wave-like behavior

8. Quantum Coherence Single particle “textbook” physics Intellectually Exciting: Counterintuitive, Fundamental part of nature Correlated Many-body physics -Connections to other fields Condensed Matter, Nuclear Technology: Precision Measurement, Navigation, Sensing Direct Applications: Quantum Computing, Quantum Information Processing

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

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

13. 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. Novel Physics Key point:”physics”

16. Unitary limit, |a| diverges(main characteristic). Short range force butlong-range correlation, system details “erased”! • Dilute unitary gas: not “ideal free Fermi gas.”

17. 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. This ξ attracts much attention in recent years Too many updating works 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 resultξ = 0.25 ± 0.03 of Lee Dean et al.

19. A challenging topic in contemporary physics: Related to many realistic problems MBX Bewitching in the fundamental Fermi-Dirac statistics Even closely related with the SU(Nc) physics, e.g., • nucl-th/0606019,T Schaefer, From Trapped Atoms to Liberated Quarks • nucl-th/0606046, E.V. Shuryak, Locating strongly coupled color superconductivity using universality and experiments with trapped ultracold atoms

20. Its exact value/how to approach? • MFT? No, “go beyond” MFT • For example, epsilon expansion (Incorporate 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 )

21. 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. 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

24. cond-mat/0608282 v1 11 Aug 2006

25. 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. Anti-screened “vector boson” propagator with a negative Debye mass squared m=1 Motivation:Topology similar to Feshbach resonance Key point:”physics” Landau Pole?

27. 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. 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)

31. Gauge invariance ensured by the Lorentz transversality condition with HLS:

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

33. Generalized Renormalizaion condition

34. At T=0 • Tailor

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

36. 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. Long range correlation controls the global behaviors of the system Quantum Many-body Effect d=2, ξ =0 Similar to this diagram? Strong repulsion leads to “attraction”

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

39. 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. A DilemmaThermodynamics 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. Extending to finite aUnitary limit regime with finite scattering length at both T and density Mean field theory: the lowest order

42. Repulsive approaches to effective attraction Main results of nucl-th/0602065 • 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

43. D-dimensions:nucl-th/0608063

44. 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. 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. 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. Thank You!