1. Application of the operator product expansion and sum rules to the study of the single-particle spectral density of the unitary Fermi gas Seminar at Yonsei University 12.02.2015 Philipp Gubler (ECT*) Collaborators: N. Yamamoto (Keio University), T. Hatsuda (RIKEN, Nishina Center), Y. Nishida (Tokyo Institute of Technology) arXiv:1501:06053 [cond.mat.qunat-gas]

2. Contents Introduction  The unitary Fermi gas Bertsch parameter, Tan’s Contact  Recent theoretical developments 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. Any relevance for the real world? It has only recently become possible to reach the unitary limit experimentally by tuning the scattering length by making use of a Feshbach resonance: S. Inouye et al., Nature 392, 151 (1998). Scattering length of Na atoms near a Feshbach resonance

5. Example of universality M. Horikoshi et al., Science, 327, 442 (2010). Ideal Fermi Gas Unitary Fermi Gas Measured using the two lowest spin states of 6 Li atoms in a magnetic field. This is a universal function applicable to all strongly interacting fermionic systems.

6. Parameters charactarizing the unitary fermi gas (1) Bertsch parameter The “Bertsch problem” What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, infinite scattering length contact interaction. (posed at the Many-Body X conference, Seattle, 1999) Or, more specifically: what is the value of ξ?

7. M.G. Endres, D.B. Kaplan, J.-W. Lee and A.N. Nicholson, Phys. Rev. A 87, 023615 (2013). ~ 0.37 Values of ξ during the last few years

8. 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

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

10. What is the value of C? 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). Theory Experiment

11. Zero-Range model: Operator corresponding to the contact density: E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008). Deriving the Tan-relations with the help of the operator product expansion

12. E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008). General OPE: Starting point: expression for momentum distribution: OPE (non-trivial)

13. take the expectation value E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008). C After Fourier transformation:

14. 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

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

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

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

18. 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].

19. 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:1501.06053 [cond-mat.quant-gas]. In a mean-field treatment, only these peaks are generated

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

21. Comparison with other methods Monte-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.

22. 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)

23. 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

24. Backup slides

25. How can the unitary limit be understood in a broader context? Key parameter: k F a well understood difficult C.A.R. Sá de Melo, M. Randeria and J.R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993). Taken from: M. Randeria, Nature Phys. 6, 561 (2010).