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Operator Product Expansion and Sum Rule Approach to Unitary Fermi Gas

Explore the operator product expansion technique applied to the unitary Fermi gas system, analyzing sum rules for the single-particle spectral density using maximum entropy methods. Extract whole spectrum with Bertsch parameter and Contact as input. Generalize to finite temperature, study pseudo-gap existence, consider higher-dimensional operators. Role of high-dimensional operators, channels, and beyond the unitary limit. Collaborators: N. Yamamoto, T. Hatsuda, Y. Nishida.

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Operator Product Expansion and Sum Rule Approach to Unitary Fermi Gas

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  1. Operator product expansion and sum rule approach to the unitary Fermi gas arXiv:1501:06053 [cond.mat.quant-gas] Schladming Winter School “Intersection Between QCD and Condensed Matter” 05.03.2015 Philipp Gubler (ECT*) Collaborators: N. Yamamoto (Keio University), T. Hatsuda (RIKEN, Nishina Center), Y. Nishida (Tokyo Institute of Technology)

  2. 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

  3. Introduction The Unitary Fermi Gas A dilute gas of non-relativistic particles with two species. Unitary limit: only one relevant scale Universality

  4. 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

  5. Parameters charactarizing the unitary fermi gas (2) The “Contact” C interaction energy Tan relations kinetic energy S. Tan, Ann. Phys. 323, 2952 (2008); 323, 2971 (2008); 323, 2987 (2008).

  6. What is the value of the “Contact”? Using Quantum Monte-Carlo simulation: S. Gandolfi, K.E. Schmidt and J. Carlson, Phys. Rev. A 83, 041601 (2011). 3.40(1) From experiment: J.T. Stewart, J.P. Gaebler, T.E. Drake, D.S. Jin, Phys. Rev. Lett. 104, 235301 (2010).

  7. A new development: Use of the operator product expansion (OPE) General OPE: works well for small r! Applied to the momentum distribution nσ(k): C E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

  8. 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).

  9. Comparison with Monte-Carlo simulations OPE results Quantum Monte Carlo simulation P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011). Y. Nishida, Phys. Rev. A 85, 053643 (2012).

  10. The sum rule approach in QCD (“QCD sum rules”) Kramers-Kronig relation is calculated using the operator product expansion (OPE) Borel transform input M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).

  11. Our approach PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].

  12. MEM PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].

  13. The sum rules in practice: 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].

  14. Results Re Σ Im Σ A |k|=0.0 |k|=0.6 kF |k|=1.2 kF In a mean-field treatment, only these peaks are generated PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].

  15. Results pairing gap PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].

  16. Comparison with other methods Monte-Carlo simulations Luttinger-Ward approach P. Magierski, G. Wlazłowski and A. Bulgac, Phys. Rev. Lett. 107, 145304 (2011). R. Haussmann, M. Punk and W. Zwerger, Phys. Rev. A 80, 063612 (2009). Qualitatively consistent with our results, except position of Fermi surface.

  17. 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)

  18. Outlook • 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 • …

  19. Backup slides

  20. What is C? : Number of pairs Number of atoms with wavenumber k larger than K

  21. The basic problem to be solved given ? “Kernel” (but only incomplete and with error) 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:

  22. The Maximum Entropy Method prior probability likelihood function (Shannon-Jaynes entropy) Corresponds to ordinary χ2-fitting. “default model” M. Jarrel and J.E. Gubernatis, Phys. Rep. 269, 133 (1996). M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).

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