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arXiv:1001.4518 · Physical Review Letters

Short-Time Operator Product Expansion for rf Spectroscopy of a Strongly Interacting Fermi Gas

Eric Braaten, Daekyoung Kang, Lucas Platter

Published Jun 4, 2010• 103 citations
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Abstract

Universal relations that hold for any state provide powerful constraints on systems consisting of fermions with two spin states interacting with a large scattering length. In radio-frequency (rf) spectroscopy, the mean shift in the rf frequency and the large-frequency tail of the rf transition rate are proportional to the contact, which measures the density of pairs with small separations. We show that these universal relations can be derived and extended by using the short-time operator product expansion of quantum field theory. This is a general method for identifying aspects of many-body physics that are controlled by few-body physics.

DOI
10.1103/physrevlett.104.223004

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Universal relations that hold for any state provide powerful constraints on systems consisting of fermions with two spin states interacting with a large scattering length. In radio-frequency (rf) spectroscopy, the mean shift in the rf frequency and the large-frequency tail of the rf transition rate are proportional to the contact, which measures the density of pairs with small separations. We show that these universal relations can be derived and extended by using the short-time operator product expansion of quantum field theory. This is a general method for identifying aspects of many-body physics that are controlled by few-body physics.
We derive universal relations for the rf spectroscopy of a two-dimensional Fermi gas consisting of two spin states interacting through an SSS-wave scattering length. The rf transition rate has a high-frequency tail that is proportional to the contact and displays logarithmic scaling violations, decreasing asymptotically like $1/({\ensuremath{\omega}}^{2}{ln}^{2}\ensuremath{\omega})$. Its coefficient is proportional to ${ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$, where a2D{a}_{2\mathrm{D}}a2D​ and ${a}_{2\mathrm{D}}^{\ensuremath{'}}$ are the two-dimensional scattering lengths associated with initial-state and final-state interactions. The clock shift is proportional to the contact and to $\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$. If $|\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})|\ensuremath{\gg}1$, the clock shift arises as a cancellation between much larger contributions proportional to ${ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$ from bound-bound and bound-free rf transitions.
Shina Tan has derived some universal relations that hold for any state of a system consisting of fermions with two spin states that have a large scattering length. These relations involve an intensive quantity called the contact that measures the number of pairs of atoms that are very close together. We show how these relations can be derived in the framework of quantum field theory using standard renormalization methods and the operator product expansion. They allow the contact density to be identified as the expectation value of a local operator constructed out of quantum fields.
In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.
A set of universal relations between various properties of any few-body or many-body system consisting of fermions with two spin states and a large but finite scattering length have been derived by Shina Tan. We derive generalizations of the Tan relations for a two-channel model for fermions near a Feshbach resonance that includes a molecular state whose detuning energy controls the scattering length. We use quantum field theory methods, including renormalization and the operator product expansion, to derive these relations. They reduce to the Tan relations as the scattering length is made increasingly large.
The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.
The behavior of fermions with two spin states that interact with a large scattering length is constrained by universal relations that hold for any state of the system. These relations involve a central property of the system called the contact, which measures the number of pairs of fermions with different spins that have small separations. The contact controls the thermodynamics of the system as well as the large-momentum and high-frequency tails of correlation functions. This review summarizes the current theoretical and experimental status of these universal relations.