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This seminar discusses the application of the operator product expansion and sum rules to study the single-particle spectral density of the unitary Fermi gas. Topics covered include the unitary Fermi gas, Bertsch parameter, Tan's Contact, recent theoretical developments, formulation of sum rules, results of the single-particle spectral density, and conclusions.
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Application of the operator product expansion and sum rules to the study of the single-particle spectral density of the unitary Fermi gas arXiv:1501:06053 [cond.mat.qunat-gas] Seminar atYonsei University 12.02.2015 Philipp Gubler (ECT*) Collaborators: N. Yamamoto (Keio University), T. Hatsuda (RIKEN, Nishina Center), Y. Nishida (Tokyo Institute of Technology)
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
Introduction The Unitary Fermi Gas A dilute gas of non-relativistic particles with two species. Unitary limit: only one relevant scale Universality
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: Scattering length of Na atoms near a Feshbach resonance S. Inouye et al., Nature 392, 151 (1998).
Example of universality M. Horikoshi et al., Science, 327, 442 (2010). Measured using the two lowest spin states of 6Li atoms in a magnetic field. Ideal Fermi Gas This is a universal function applicable to all strongly interacting fermionic systems. Unitary Fermi Gas
Parameters charactarizing the unitary fermi gas (1) 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? Bertsch parameter
Values of ξ during the last few years ~ 0.37 M.G. Endres, D.B. Kaplan, J.-W. Lee and A.N. Nicholson, Phys. Rev. A 87, 023615 (2013).
Parameters charactarizing the unitary fermi gas (2) The “Contact” C interaction energy kinetic energy Tan relations S. Tan, Ann. Phys. 323, 2952 (2008); 323, 2971 (2008); 323, 2987 (2008).
What is C? : Number of pairs Number of atoms with wavenumber k larger than K
What is the value of C? Experiment Theory J.T. Stewart, J.P. Gaebler, T.E. Drake, D.S. Jin, Phys. Rev. Lett. 104, 235301 (2010). S. Gandolfi, K.E. Schmidt and J. Carlson, Phys. Rev. A 83, 041601 (2011). Using Quantum Monte-Carlo simulation: 3.40(1)
Deriving the Tan-relations with the help of the operator product expansion Zero-Range model: Operator corresponding to the contact density: E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).
General OPE: Starting point: expression for momentum distribution: OPE (non-trivial) E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).
take the expectation value After Fourier transformation: C E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).
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).
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).
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].
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].
Results pairing gap PG, N. Yamamoto, T. Hatsuda and Y. Nishida, arXiv:1501.06053 [cond-mat.quant-gas].
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.
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)
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 • …
Taken from: M. Randeria, Nature Phys. 6, 561 (2010). How can the unitary limit be understood in a broader context? Key parameter: kFa well understood well understood difficult C.A.R. Sá de Melo, M. Randeria and J.R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993).
