Operator product expansion and sum rule approach to the unitary Fermi gas Schladming Winter School “Intersection Between QCD and Condensed Matter” Philipp 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

Introduction The 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, (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, (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, (2010). From experiment:

A new development: Use of the operator product expansion (OPE) General OPE: Applied to the momentum distribution n σ (k): C E. Braaten and L. Platter, Phys. Rev. Lett. 100, (2008). works well for small r!

Y. Nishida, Phys. Rev. A 85, (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, (2012). OPE results Quantum Monte Carlo simulation P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, (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: [cond-mat.quant-gas].

MEM PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv: [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: [cond-mat.quant-gas]. in practice:

Results Im Σ Re Σ A |k|=0.0 |k|=0.6 k F |k|=1.2 k F PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv: [cond-mat.quant-gas]. In a mean-field treatment, only these peaks are generated

Results PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv: [cond-mat.quant-gas]. pairing gap

Comparison with other methods Monte-Carlo simulations P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, (2011). Luttinger-Ward approach R. Haussmann, M. Punk and W. Zwerger, Phys. Rev. A 80, (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 χ 2 -fitting. (Shannon-Jaynes entropy) “default model” The Maximum Entropy Method