Operator product expansion and sum rule approach to the unitary
Fermi gas
Schladming Winter School “Intersection Between QCD and Condensed Matter”05.03.2015Philipp Gubler (ECT*)
Collaborators: N. Yamamoto (Keio University), T. Hatsuda (RIKEN, Nishina Center), Y. Nishida (Tokyo Institute of Technology)
arXiv:1501:06053 [cond.mat.quant-gas]
Contents
Introduction The unitary Fermi gas
Bertsch parameter, Tan’s contact
The method The operator product expansion Formulation of sum rules
Results: the single-particle spectral density Conclusions and outlook
IntroductionThe Unitary Fermi Gas
A dilute gas of non-relativistic particles with two species.
Unitary limit:
only one relevant scale
Universality
Parameters charactarizing the unitary fermi gas (1)The Bertsch parameter ξ
M.G. Endres, D.B. Kaplan, J.-W. Lee and A.N. Nicholson, Phys. Rev. A 87, 023615 (2013).
~ 0.37
Parameters charactarizing the unitary fermi gas (2)
The “Contact” C
Tan relations
interaction energy
S. Tan, Ann. Phys. 323, 2952 (2008); 323, 2971 (2008); 323, 2987 (2008).
kinetic energy
What is the value of the “Contact”?
S. Gandolfi, K.E. Schmidt and J. Carlson, Phys. Rev. A 83, 041601 (2011).
Using Quantum Monte-Carlo simulation:
3.40(1)
J.T. Stewart, J.P. Gaebler, T.E. Drake, D.S. Jin, Phys. Rev. Lett. 104, 235301 (2010).
From experiment:
A new development: Use of the operator product expansion (OPE)General OPE:
Applied to the momentum distribution nσ(k):
C
E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).
works well for small r!
Y. Nishida, Phys. Rev. A 85, 053643 (2012).
Further application of the OPE: the single-particle Green’s function
location of pole in the large momentum limit
Y. Nishida, Phys. Rev. A 85, 053643 (2012).
OPE resultsQuantum Monte Carlo simulation
P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011).
Comparison with Monte-Carlo simulations
The sum rule approach in QCD (“QCD sum rules”)
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).
Kramers-Kronig relation
Borel transform
is calculated using the operator product
expansion (OPE)
input
Our approach
PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
MEM
PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
The sum rules
The only input on the OPE side: Contact ζ
Bertsch parameter ξ
PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
in practice:
Results Im Σ Re Σ A
|k|=0.0
|k|=0.6 kF
|k|=1.2 kF
PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
In a mean-field treatment, only these peaks are generated
Results
PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
pairing gap
Comparison with other methodsMonte-Carlo simulations
P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011).
Luttinger-Ward approach
R. Haussmann, M. Punk and W. Zwerger, Phys. Rev. A 80, 063612 (2009).
Qualitatively consistent with our results, except position of Fermi surface.
Conclusions
Unitary Fermi Gas is a strongly coupled system that can be studied experimentally Test + challenge for theory
Operator product expansion techniques have been applied to this system recently
We have formulated sum rules for the single particle self energy and have analyzed them by using MEM
Using this approach, we can extract the whole single-particle spectrum with just two inputparameters (Bertsch parameter and Contact)
Generalization to finite temperature Study possible existence of pseudo-gap
Role of so far ignored higher-dimensional operators
Spectral density off the unitary limit Study other channels …
Outlook
Backup slides
What is C?
: Number of pairs
Number of atoms with wavenumber k larger than K
The basic problem to be solved
given (but only incomplete and
with error)
?“Kernel”
This is an ill-posed problem.
But, one may have additional information on ρ(ω), which can help to constrain the problem:
- Positivity:
- Asymptotic values:
likelihood function prior probability
M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).
M. Jarrel and J.E. Gubernatis, Phys. Rep. 269, 133 (1996).
Corresponds to ordinary χ2-fitting.
(Shannon-Jaynes entropy)
“default model”
The Maximum Entropy Method
