Advances in Condensed Matter Physics

Volume 2015, Article ID 582734, 9 pages

http://dx.doi.org/10.1155/2015/582734

## Theoretical Approach to the Gauge Invariant Linear Response Theories for Ultracold Fermi Gases with Pseudogap

^{1}Department of Physics, Southeast University, Nanjing 211189, China^{2}College of Physical Science and Technology, Sichuan University, Chengdu, Sichuan 610064, China

Received 22 June 2015; Accepted 2 August 2015

Academic Editor: Artur P. Durajski

Copyright © 2015 Hao Guo and Yan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperature . We adapt a particular -matrix approach which is close to the formalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities and -limit Ward identity are indeed satisfied.

#### 1. Introduction

Recently there is a broad literature on the subjects of response functions in superconductors and atomic Fermi gas superfluids [1–11], where interparticle interaction is strong enough such that the classical BCS theory is not adequate. Related experiments include the studies of the thermodynamic response functions and dynamical response [12–17]. Theoretically, linear response theories have been an important tool for studying the transport and dynamic properties of Fermi gases. Hence, it is important to assess the self-consistency of the linear response theories as well as comparing with the experimental results. There must be some general rules that the theory must follow. In [18, 19], several fundamental constraints associated with the conservation laws/Ward identities and the sum rules were addressed. Since the conservation laws are generically related to certain (gauge) symmetry of the theory, then in the broken-symmetry phase or ordered phase it is particularly difficult for many-body theories to satisfy all these constraints. It was also pointed out that the strict weakly interacting BCS mean field theory does pass all these types of testing both below and above even when the pairing population is unbalanced [18, 19]. In other words, the linear response theories of BCS superfluids can be formulated into a fully gauge invariant theory. In the normal state, the simplest Nozieres-Schmitt-Rink (NSR) is also compatible with these gauge invariance conditions [20].

In this paper, we try to build an ideal linear response theory for strongly correlated superfluids undergoing BCS-BEC crossover by a diagrammatic approach such that the fundamental constraints mentioned above can be satisfied. Our selected diagrams bear on those associated with the Goldstone modes due to the symmetry-breaking via the consistent-fluctuation-of-the-order-parameter (CFOP) approach and the conventional contributions, namely, the Maki-Thompson (MT) and Aslamazov-Larkin (AL) diagrams. As a price, we have to adapt a slightly modified formalism to incorporate the pairing fluctuation effect. We emphasize that this approach is purely a theoretical attempt until now. However, it might be a necessary step to fully understand the transport properties of strongly correlated Fermi gases.

The linear response theories must be consistent with several fundamental constraints [18, 19, 21] imposed by the Ward identities and -limit Ward identity [22]. It is well known that the Ward identities guarantee the gauge invariance of the theory. However, the -limit Ward identity is quite strange to most researchers in the condensed matter community. It in fact leads to the sum rules of compressibility and spin susceptibilities which further build the consistent connection between the single-particle thermodynamics and two-particle correlation functions. For spin response theory the -limit Ward identity is only meaningful to polarized Fermi superfluids [19]. In this paper, we focus on the unpolarized Fermi superfluids. The central difficulty of formulating the consistent linear response theory is to maintain the gauge invariance when the pseudogap self-energy is introduced by the pairing fluctuation effect. This is obviously beyond the CFOP approach since the total energy gap is now different from the order parameter.

In the following sections, we first briefly review the CFOP linear response theories in both the density and the spin channels for BCS mean field theory; then we introduce the pairing fluctuation effects via a particular -matrix formalism. We further carry on extra diagrammatic corrections in the two channels and verify that the new theories do maintain the gauge symmetry, respectively. Throughout this paper, we follow the convention .

#### 2. BCS Mean Field Theory Approach

By using the to denote the spin or pseudospin , , the Hamiltonian for a two-component Fermi gas interacting via the attractive contact interaction iswhere and are the annihilation and creation operators of fermions, is the chemical potential, and is the fermion mass. There is an implicit summation over the pseudospin indices . The Hamiltonian has a symmetry [23]where and is the phase parameter of those transformations. The first symmetry is well known for relating to the electromagnetism (EM). If the particle is charged, this symmetry naturally becomes a gauge symmetry. For a charge neutral system, the symmetry is still associated with the mass current conservation. The second symmetry is the spin rotational symmetry which is associated with the spin current conservation. Our linear response theories in the density and spin channels must respect these two symmetries, respectively. The central idea is to “gauge” the symmetries by introducing two types of weak external fields. In the density channel, it is the weak EM field , while in the spin channel it is , where is the component of the magnetic field (assuming is the axis of spin rotation) and is the magnetization.

After taking BCS mean field approximation, the order parameter or superconducting gap function is introduced . The first symmetry is spontaneously broken while the second is not. As can be seen in [18], this brings significant difference between the linear response theories in the two channels. For a homogeneous system, the BCS Hamiltonian can be expressed aswhere . As a familiar result, the BCS Green and anomalous Green functions in the momentum space arewhere is the Fermion Matsubara frequency and is the quasiparticle energy dispersion. Hereinafter we use the subscript “sc” to emphasize that these discussions are only under the BCS mean field approximation. Define ; the number and gap equations are determined by and . These identities giveThe bare Green function is . The Dyson equation gives where is the BCS self-energy.

##### 2.1. Density Channel

In the density channel, the system is perturbed by an effective external EM field , and the Hamiltonian becomes withwhere is the bare EM interaction vertex. Here is the external four momentum, where is the boson Matsubara frequency. The bare vertex satisfies the “bare” Ward identityIn a gauge invariant EM linear response theory, a full EM interaction vertex (the bare and full EM vertices are shown in Figure 1) which satisfies the full Ward identitymust be found so that the perturbed current can be expressed as with determined by the Kubo formalismwhere with being the metric tensor. By using the Ward identity (8), it is easy to show that , which further leads to the conservation of perturbed current . Hence the linear response theory is indeed gauge invariant. Under the framework of BCS mean field theory, such full vertex can be obtained either by Nambu’s integral-equation approach [23, 24] or by the CFOP approach. However, the -limit Ward identity provides an independent consistency check of the theory. It is not known if Nambu’s approach can pass this check, while the vertex given by the latter approach is proved to satisfy this conditionDetails can be found in [18]. This identity not only builds a consistent connection between the one-particle thermodynamics and two-particle response functions but also acts as the sufficient and necessary condition for the compressibility sum rule