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 University12.02.2015Philipp 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]

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

IntroductionThe 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:

S. Inouye et al., Nature 392, 151 (1998).

Scattering length of Na atoms near a Feshbach resonance

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 6Li atoms in a magnetic field.

This is a universal function applicable to all strongly interacting fermionic systems.

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?

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

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 C?

: Number of pairs

Number of atoms with wavenumber k larger than K

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

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

E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

General OPE:

Starting point: expression for momentum distribution:

OPE (non-trivial)

take the expectation value

E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).

C

After Fourier transformation:

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

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 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

How can the unitary limit be understood in a broader context?

Key parameter: kFa

well understood well understooddifficult

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).