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arXiv:0803.1125 · Physical Review Letters

Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion

Eric Braaten, Lucas Platter

Published May 21, 2008• 298 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.100.205301

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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.
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.
We consider a homogeneous, balanced gas of strongly interacting fermions in two spin states interacting through a large scattering length. Finite-range corrections are needed for a quantitative description of data which experiments and numerical simulations have provided. We use a perturbative field-theoretical framework and a tool called the operator product expansion (OPE), which together allow for the expression of finite-range corrections to the universal relations and momentum distribution. By using the OPE, we derive the 1/k61/{k}^{6}1/k6 part of the momentum tail, which is related to the sum of the derivative of the energy with respect to the finite range and the averaged kinetic energy of opposite spin pairs. By comparing the 1/k41/{k}^{4}1/k4 term and the 1/k61/{k}^{6}1/k6 correction in the momentum distribution to provided quantum Monte Carlo (QMC) data, we show that including the 1/k61/{k}^{6}1/k6 part offers marked improvements. Our field-theoretical approach allows for a clear understanding of the role of the scattering length and finite effective range in the universal relations and the momentum distribution.
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.
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.
In a recent work, Anderson et al. [Phys. Rev. C 82, 054001 (2010)] used the renormalization group (RG) evolution of the momentum distribution to show that, under appropriate conditions, operator expectation values exhibit factorization in the two-nucleon system. Factorization is useful because it provides a clean separation of long- and short-distance physics, and suggests a possible interpretation of the universal high-momentum dependence and scaling behavior found in nuclear momentum distributions. In the present work, we use simple decoupling and scale-separation arguments to extend the results of Anderson et al. to arbitrary low-energy AAA-body states. Using methods that are reminiscent of the operator product expansion (OPE) in quantum field theory, we find that the high-momentum tails of momentum distributions and static structure factors factorize into the product of a universal function of momentum that is fixed by two-body physics, and a state-dependent matrix element that is the same for both and is sensitive only to low-momentum structure of the many-body state. As a check, we apply our factorization relations to two well studied systems, the unitary Fermi gas and the electron gas, and reproduce known expressions for the high-momentum tails of each.
Many-body fermion systems are important in many branches of physics, including condensed matter, nuclear, and now cold atom physics. In many cases, the interactions between fermions can be approximated by a contact interaction. A recent theoretical advance in the study of these systems is the derivation of a number of exact universal relations that are predicted to be valid for all interaction strengths, temperatures, and spin compositions. These equations, referred to as the Tan relations, relate a microscopic quantity, namely, the amplitude of the high-momentum tail of the fermion momentum distribution, to the thermodynamics of the many-body system. In this work, we provide experimental verification of the Tan relations in a strongly interacting gas of fermionic atoms by measuring both the microscopic and macroscopic quantities in the same system.