Large momentum part of a strongly correlated Fermi gas

doi:10.1016/j.aop.2008.03.005

Annals of Physics

Volume 323, Issue 12, December 2008, Pages 2971-2986

https://doi.org/10.1016/j.aop.2008.03.005 Get rights and content

Abstract

It is well known that the momentum distribution of the two-component Fermi gas with large scattering length has a tail proportional to $1 / k^{4}$ at large k. We show that the magnitude of this tail is equal to the adiabatic derivative of the energy with respect to the reciprocal of the scattering length, multiplied by a simple constant. This result holds at any temperature (as long as the effective interaction radius is negligible) and any large scattering length; it also applies to few-body cases. We then show some more connections between the $1 / k^{4}$ tail and various physical quantities, including the pressure at thermal equilibrium and the rate of change of energy in a dynamic sweep of the inverse scattering length.

Section snippets

The theorem

Consider an arbitrary number of fermions with mass m in two spin states $↑$ and $↓$ , having an s-wave contact interaction with large scattering length a ( $∣ a ∣ ≫ r_{0}$ [1]) between opposite spin states, and confined by any external potential $V_{ext} (r)$ . To discuss their momentum distribution, it is convenient to use a box with volume $Ω$ and impose a periodic boundary condition. For a large uniform gas, $Ω$ is its actual volume; but for a gas confined in a trap, it is better to use (conceptually) a very large box

The proof

Although the theorem can be proved with conventional means as well, we shall use the formalism developed in [4]. This formalism, as we shall see below, allows us to prove this universal theorem in a universal (and simple) way.

The basis of the formalism developed in [4] is a pair of linearly independent generalized functions $Λ (k)$ and $L (k)$ , satisfying $Λ (k) = 1 (∣ k ∣ < \infty), \int \frac{d^{3} k}{(2 π)^{3}} \frac{Λ (k)}{k^{2}} = 0,$ $L (k) = 0 (∣ k ∣ < \infty), \int \frac{d^{3} k}{(2 π)^{3}} \frac{L (k)}{k^{2}} = 1,$ $Λ (- k) = Λ (k), L (- k) = L (k) .$ There is no contradiction at all, because the integrals of

Two-body case: a conventional proof

It is cumbersome to prove the above theorem with conventional means. Here we just demonstrate the two-body case, and then sketch the extension to the N-body cases. This shall help to convince the readers that the adiabatic sweep theorem is indeed correct, even if they are not used to the succinct (and “unconventional”) proof in the last section.

Now consider two fermions in opposite spin states with an external potential $V_{ext} (r)$ . We consider an energy eigenstate described by a wave function $ϕ (r_{1},$

Physical meaning

The meaning of Eq. (1) is intuitively clear. If we tune the inverse scattering length slightly and slowly, quantum mechanical first-order perturbation determines the amount by which the energy level shifts. In the limit of zero effective interaction radius, two fermions (in opposite spin states) do not interact unless they appear at the same position, so the energy shift should be proportional to the probability that this occurs. Meanwhile, $Ω C$ characterizes this probability [4]; recall that

Acknowledgments

The author thanks T.L. Ho for a stimulating suggestion, and thanks E. Braaten for comments. He also thanks K. Levin for the introduction to ultracold Fermi gases, and S. Giorgini for communication.

References (50)

V.A. Belyakov
Sov. Phys. JETP
(1961)
L. Viverit
Phys. Rev. A
(2004)
S. Tan,...
M. Greiner
Phys. Rev. Lett.
(2005)
C.A. Regal, et al.,...

M. RanderiaE. Timmermans

Phys. Lett. A

(2001)

M. Holland

Phys. Rev. Lett.

(2001)

Q. Chen

Phys. Rep.

(2005)

K. Huang et al.

Phys. Rev.

(1957)

T.D. Lee et al.

Phys. Rev.

(1957)

G.V. Skorniakov et al.

Sov. Phys. JETP

(1957)

D.S. Petrov et al.

Phys. Rev. A

(2005)

D.S. Petrov et al.

Phys. Rev. Lett.

(2004)

S. Tan,...

T.D. Lee et al.

Phys. Rev.

(1957)

T.T. Wu

Phys. Rev.

(1959)

E. Braaten et al.

Phys. Rev. Lett.

(2002)

S. Stringari

Europhys. Lett.

(2004)

Cited by (711)

Exploring Short-Range Correlations in symmetric nuclei: Insights into contacts and entanglement entropy
2024, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
The Short-Range Correlations between nucleons in nuclei are regarded as a complex system. We investigate the relationship between the orbital entanglement entropy of SRCs $S_{i j}$ in nuclear structures and Tan contact $c_{i j}$ , and find that the orbital entanglement entropies and Tan contacts corresponding to proton-proton SRC pairs and neutron-proton SRC pairs in nuclei demonstrate a scaling relation. More specifically, the proportionality of entanglement entropy between proton-proton pairs and neutron-proton pairs is directly related to the ratio of nuclear contacts within the atomic nucleus, demonstrating an approximate ratio of 2.0. Our research suggests that this scaling relationship should hold true for all symmetric nuclei, furthermore, we offer a possible explanation for this phenomenon.
Few-body Bose gases in low dimensions—A laboratory for quantum dynamics
2023, Physics Reports
Cold atomic gases have become a paradigmatic system for exploring fundamental physics, which at the same time allows for applications in quantum technologies. The accelerating developments in the field have led to a highly advanced set of engineering techniques that, for example, can tune interactions, shape the external geometry, select among a large set of atomic species with different properties, or control the number of atoms. In particular, it is possible to operate in lower dimensions and drive atomic systems into the strongly correlated regime. In this review, we discuss recent advances in few-body cold atom systems confined in low dimensions from a theoretical viewpoint. We mainly focus on bosonic systems in one dimension and provide an introduction to the static properties before we review the state-of-the-art research into quantum dynamical processes stimulated by the presence of correlations. Besides discussing the fundamental physical phenomena arising in these systems, we also provide an overview of the calculational and numerical tools and methods that are commonly used, thus delivering a balanced and comprehensive overview of the field. We conclude by giving an outlook on possible future directions that are interesting to explore in these correlated systems.
Universal many-body properties of one-dimensional mass-imbalanced highly polarized Fermi gases
2023, Physics Letters, Section A: General, Atomic and Solid State Physics
We investigate the universal properties of mass-imbalanced highly polarized Fermi gases in one-dimensional scenarios. By treating the mass ratio as a parameter, we delve into the binding energy, quasiparticle residue, and correlation functions of the systems. We provide a thorough momentum distribution for impurities and spin-up atoms within the Fermi sea. When the impurity mass is infinite, we discover that the quasiparticle residue approaches to zero and the systems exhibit the signature of the Anderson orthogonality catastrophe in strong coupling regime. We observe an increase in Tan contact under large mass ratios and study density correlations by applying Fourier transform. Crucially, we demonstrate the existence of the molecular state in one-dimensional polaron system with strong attractive interactions, unlike previous conclusions.
Thermal-contact capacity of one-dimensional attractive Gaudin-Yang model
2024, Chinese Physics B
Universal dynamic scaling and Contact dynamics in quenched quantum gases
2024, Frontiers of Physics
Thermal fading of the 1/k4 tail of the momentum distribution induced by the hole anomaly
2024, Physical Review A

View all citing articles on Scopus

View full text

Published by Elsevier Inc.