Transcript Thomson Problem approach to the ground state of Dirac

CCNU, Ji-sheng Chen
Aug, 2006
Aug, 2006,
Universal thermodynamics of Dirac
fermions near the unitary limit
regime and BEC-BCS crossover
Ji-sheng Chen
Phys Dep., CCNU, Wuhan 430079
[email protected]
CCNU, Ji-sheng Chen
Aug, 2006
Contents
1.Motivations
2. The universal dimensionless coefficient ξand
energy gap Δ
3. Conclusions and prospects
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! Manybody effects
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
Aug, 2006,
Why Study Ultra-Cold Gases?
Answer: Coherent Quantum Phenomena
High Temperature:
Random thermal motion
dominates
Aug, 2006,
Low Temperature:
Underlying quantum
behavior revealed
Quantum Coherence
Intellectually Exciting:
Counterintuitive,
Fundamental part of nature
Single particle “textbook” physics
Correlated Many-body physics
-Connections to other fields
Condensed Matter, Nuclear
Technology:
Precision Measurement,
Navigation, Sensing
Aug, 2006,
Direct Applications:
Quantum Computing,
Quantum Information Processing
Full description of （Condensed Matter)
Phase diagram
a,Astrophysics
b,Heavy ion collisions
c,Strongly correlated electrons
d,Cosmology
。。。
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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Ultra-Cold dilute degenerate atomic fermions
gas(quantum effects)
T   K (106 ), T  nK (109 )


| a | 
BEC vs BCS: Cross-Over
Near the Feshbach resonance, the bare
scattering lengths between two-body particles
diverge!
Aug, 2006,
Novel Physics
Key point:”physics”
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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 .k f
The corresponding energy scale is the Fermi
kinetic energy k
2
f
2m
The system details do not contribute to the
thermodynamics properties

Aug, 2006,
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 result ξ = 0.25 ± 0.03 of
Lee Dean et al.
Aug, 2006,

MBX
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 Quarks
1.
nucl-th/0606046, E.V. Shuryak,
Locating strongly coupled color
superconductivity using universality and
experiments with trapped ultracold
atoms
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 )
Aug, 2006,
1, 20-40 particles extending to infinite particles system,

eliable?
Quantum Monte Carlo simulation, for
example
Carlson et al., PRL, 91, 050401(
Δ/μ=1.2 PRL(2005)
0.44) (2003), “More accurate” 0.42,
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, condPRL 95, 030404 (2005) (
mat/0608154
2, Local density functional theory? At finite T?
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 weakcoupling 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 limit
Make a detour
Aug, 2006,
Motivation:Topology similar to Feshbach
resonance
Anti-screened “vector boson” propagator
with a negative Debye mass squared m=1
Key point:”physics”
Aug, 2006,
Landau Pole?
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
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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 transversality
condition with HLS:
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
 A  0
General expressions for energy density and
pressure as well as entropy

| A0 ,mB  n

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Generalized
Renormalizaion
condition
Aug, 2006,
At T=0

Tailor
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Reasonablely consistent with the BCS theory
but with an effective scattering length



  4/9
Non-relativistic limit
relativistic limit   7 / 9
With the relativistic expression through
odd-even staggering
2
5 kf
  5 /18 f 
18 2m
  4 / 9k f
Aug, 2006,
Non-relativistic limit, Tc
≈ 0.157 Tf
Relativistic limit: Tc ≈ 0.252 Tf
Statistical
weight
factor
5/3
4/3
Main result for two-dimensions

Can even approach the extreme occasion
 0

m 
*
S/V=P=E/V=0 for fermions at unitary,
Surprisingly similar to Bose-Einstein
Condensation of 3-dimensional for ideal Bose
Fractional Quantum Hall Effect
gas
Aug, 2006,
Kondo Physics, Confinement
ong range correlation controls the global behaviors of the system
d=2, ξ =0
Quantum Many-body Effect
Aug, 2006,
Similar to this diagram?
Strong repulsion leads to
“attraction”
Ising universal class
controversial: 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
d
Aug, 2006,
or
4/9
A new type of fermions
superfluity for d=3


Stability: sound speed squared still
positive
Rough work


Aug, 2006,
Specific heat capacity, bulk and shear viscosity
of fermions, …
Polarized fermion gas,…
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.
Aug,
The2006,
sound
speed detection can judge this dilemma.
Extending to finite a
Unitary limit regime with finite scattering length
at both T and density
Mean field
theory:
the lowest
order
Aug, 2006,
Repulsive approaches to effective
attraction
Main results of nucl-th/0602065

Exactly approach some of the experimental
and quantum Monte Carlo simulation results
5
E f , Tc  0.157T f , P  1/ 4 E / V , non  relativistic
18
4
  7 / 9,   E f , Tc  0.252T f , P  1/ 7 E / V , Ultra  relativistic  lim it
9
  4 / 9,  

Same analytical result with power counting, James V.
Steele, nucl-th/0010066
  4/9
non-relativistic framework and T=0
Facilitates the comparison of non-relativistic and
Aug, 2006,
relativistic approaches to thermodynamics
D-dimensions:
nucl-th/0608063
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
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!
Aug, 2006,