Asymmetric Fermion Superfluid with Inter- and Intra-Species Pairings

We investigate the phase structure of an asymmetric fermion superfluid with inter- and intra-species pairings. The introduction of the intra-species pairing mechanism in canonical ensemble changes significantly the phase diagram and brings in a new state with coexisting inter- and intra-species pairings. Different from the case with only inter-species pairing, all the fermion excitations are fully gapped in the region with intra-species pairing.

where ψ σ i (x) are fermion fields with i = a, b and (pseudo-)spin σ =↑, ↓, the coupling constants g, g a and g b controlling respectively the inter-and intra-species pairings, are negative to keep the interactions attractive. The Lagrangian has the symmetry U a (1) ⊗ U b (1) with the element U (θ a , θ b ) defined as U (θ a , θ b )ψ σ a = e iθa ψ σ a and U (θ a , θ b )ψ σ b = e iθ b ψ σ b . We introduce the condensates of a-a, b-b and a-b pairs, Φ a = −g a ψ ↓ a ψ ↑ To incorporate the FFLO state in the study, we assume that the condensates are in the form of Φ a (x) = ∆ a e 2iqa·x , Φ b (x) = ∆ b e 2iq b ·x and Φ(x) = ∆e i(qa+q b )·x with ∆ a , ∆ b and ∆ being independent of x. Obviously, the translational symmetry and rotational symmetry in the FFLO state are spontaneously broken. Note that, for the sake of simplicity, the FFLO state we considered here is its simplest pattern, namely the single plane wave FFLO state or the so-called FF state.
The key quantity of a thermodynamic system is the partition function Z which can be represented by the path integral By performing a gauge transformation for the fermion fields, χ σ a = e −iqa·x ψ σ a and χ σ b = e −iq b ·x ψ σ b , the path integral over χ σ a,b in the partition function Z of the system at mean field level can be calculated easily, and we obtain the mean field thermodynamic potential in the imaginary time formulism of finite temperature field theory, where ω n = (2n+1)πT is the fermion frequency, p is the fermion momentum, and the inverse of the Nambu-Gorkov propagator can be written as (1) is non-renormalizable, a regularization scheme should be applied. We use the s-wave scattering lengthe regularization which relates the bare coupling constants g, g a and g b to the low energy limit of the corresponding two-body T-matrices in vacuum by with the s-wave scattering lengthes a a , a b and a. Such a scheme is reliable in the whole region of interacting strength, and therefore it is possible to extend our study to the BCS-BEC crossover, although we focus in this paper only on the weak coupling BCS region. For simplicity, we have taken the same particle masses m a = m b = m and will take all the condensates as real numbers. One should note that if the relative momentum q a −q b is large enough, the uniform superfluid would be unstable due to the stratification of the superfluid components characterized by Φ a (x) and Φ b (x). The case here is analogous to the multi-component BEC. To avoid such a dynamic instability [31,32,33], we choose q a = q b = q [34]. After computing the frequency summation and taking a Bogoliubov-Valatin transformation from particles a and b to quasi-particles, the thermodynamic potential can be expressed in terms of the quasi-particles, where E ∓ ± are the quasi-particle energies with ǫ ± , δǫ and ǫ ∆ defined as ∆ a = 0 and ∆ b = 0 correspond, respectively, to the spontaneous symmetry breaking patterns U a (1) ⊗ U b (1) → U b (1) and U a (1), and ∆ = 0 means the breaking pattern U a (1) The condensates and the FFLO momentum as functions of temperature and chemical potentials are determined by the gap equations, and the ground state of the system is specified by the minimum of the thermodynamic potential, namely by the second-order derivatives. To see clearly the effect of the mismatch between the two species a and b, we introduce the average chemical potential µ = (µ a + µ b )/2 and the chemical potential mismatch δµ = (µ b − µ a )/2 instead of µ a and µ b . Without loss of generality, we assume µ b > µ a . We choose p F a, p F a a , p F a b as the free parameters of the model, where p F = √ 2mµ is the average Fermi momentum, and set p F a = −0.58 and p F a a = p F a b = 0.73 p F a in the numerical calculation. We have checked that in the BCS region of 0 < p F |a|, p F |a a |, p F |a b | < 1 and 0 < |a a | = |a b | < |a| to guarantee weak interaction for a-b pairing and more weak coupling for a-a and b-b pairings, there is no qualitative change in the obtained phase diagrams. Note that, in the symmetric case with δµ = 0, from a direct integration of the gap equation, one obtains the famous result [35] for the inter-species pairing gap ∆ 0 at zero temperature.
We first consider systems of grand canonical ensemble with fixed chemical potentials. The phase diagram in T − δµ plane is shown in Fig.1. The left panel is the familiar case without intra-species pairing [36]. Both thermal fluctuations and large chemical potential mismatch can break the Cooper pairs. The homogeneous BCS state can exist at low temperature and low mismatch, and the inhomogeneous FFLO state survives only in a narrow mismatch window. The phase transition from the superfluid to normal phase is of second order, and the transition from the homogeneous to inhomogeneous superfluid at zero temperature is of first order and happens at δµ c = ∆ 0 / √ 2 [36]. When the intra-species pairing is included as well, see the right panel of Fig.1, the inhomogeneous FFLO state of a-b pairing is eaten up by the homogeneous superfluid of a-a and b-b pairings at low temperature, just as we expected, and survives only in a small triangle at high temperature. The phase transition from the a-b pairing superfluid to the a-a and b-b pairing superfluid is of first order. From the assumption of µ b > µ a , the temperature to melt the condensate ∆ b is higher than that to melt ∆ a , which leads to a phase with only b-b pairing. Since µ b increases and µ a decreases with mismatch δµ, the region with only b-b pairing becomes more and more wide when δµ increases. Note that, for systems with fixed chemical potentials there is no mixed phase of inter-and intra-species pairings, and the situation is similar to a three-component fermion system [37]. The phase diagram of an asymmetric fermion superfluid in grand canonical ensemble. δµ is the chemical potential mismatch between the two species, and the average chemical potential is fixed to be µ = 50∆0 with ∆0 being the corresponding symmetric gap at δµ = 0. The left panel is for the familiar case with only inter-species pairing, and in the right panel the intra-species pairing is included as well.
For many physical systems, the fixed quantities are not chemical potentials µ a and µ b but particle number densities n a = −∂Ω/∂µ a and n b = −∂Ω/∂µ b , or equivalently speaking the total number density n = n b + n a and the number density asymmetry α = δn/n with δn = n b − n a . Such systems are normally described by the canonical ensemble and the essential quantity is the free energy which is related to the thermodynamic potential by a Legendre transformation, F = Ω + µ a n a + µ b n b = Ω + µn + δµδn.
It is easy to prove that the gap equations in the canonical ensemble are equivalent to (9) in the grand canonical ensemble and give the same solutions. While the candidates of the ground state of the system are the same in both ensembles, the stability conditions for the two ensembles are very different and lead to much more rich phase structure in systems with fixed number densities. Our method to obtain the phase diagram at fixed total number density n is as the following. We first calculate the gap equations (9) and obtain all the possible homogeneous phases and inhomogeneous FFLO phase, then compare their free energies to extract the lowest one at fixed T and α, and finally investigate the stability of the system against number fluctuations by computing the number susceptibility matrix where Y i are susceptibility vectors with elements (Y i ) m = ∂ 2 Ω/(∂µ i ∂x m ) and R is a susceptibility matrix with elements R mn = ∂ 2 Ω/(∂x m ∂x n ) in the order parameter space constructed by x = (∆ a , ∆ b , ∆, q). The state with non positive-definite χ (denoted by χ < 0) is unstable against number fluctuations and may be a phase separation [38]. The phase diagram without intra-species pairing can be found in [12,29,30]. As a comparison we recalculate it and show it in the left panel of Fig.2. By computing the gap equations (9) and then finding the minimum of the free energy F , we find that the superfluid is in homogeneous state at high temperature and FFLO state at low temperature. However, the number susceptibility in the region of low temperature and low number asymmetry is not positive-definite, the FFLO state in this region is therefore unstable against the number fluctuations, and the ground state is probably an inhomogeneous mixture of the BCS superfluid and normal fermion fluid. The shadowed region is the gapless superfluid with δµ > ∆ where the energy gap to excite qusi-particles is zero and the system may be sensitive to the thermal and quantum fluctuations. The phase diagram with both inter-and intra-species pairings is shown in the right panel of Fig.2. Besides the familiar phase with only inter-species pairing (∆ = 0, ∆ a = ∆ b = 0) and the expected phases with only intra-species pairing (∆ a , ∆ b = 0, ∆ = 0 and ∆ b = 0, ∆ = ∆ a = 0), there appears a new phase where the two kinds of pairings coexist (∆, ∆ a , ∆ b = 0). In this new phase the FFLO momentum is zero and the number susceptibility is negative, χ < 0. Therefore, the homogeneous superfluid in this region is unstable against number fluctuations, and the ground state is probably a inhomogeneous mixture of these three superfluid components. In the familiar phase with only inter-species pairing, there remain a stable FFLO region and an unstable FFLO triangle where the number susceptibility is negative and the system may be in the state of phase separation. As we expected in the introduction, the gapless state appears only in the inter-species pairing superfluid, and in the region with intra-species pairing all the fermions are fully gapped. In summary, we have investigated the phase structure of an asymmetric two-species fermion superfluid with both inter-and intra-species pairings. Since the attractive interaction for the intra-species pairing is relatively weaker, its introduction changes significantly the conventional phase diagram with only inter-species pairing at low temperature. For systems with fixed chemical potentials, the inhomogeneous superfluid with inter-species pairing at low temperature is replaced by the homogeneous superfluid with intra-species pairing, while for systems with fixed species numbers, the two kinds of pairings can coexist at low temperature and low number asymmetry. In any region with intra-species pairing, the interesting gapless superfluid is washed out and all fermion excitations are fully gapped. To finally determine the exact state of the system in the regions with negative number susceptibility (χ < 0) in Fig.2, further investigation, especially considering other possible inhomogeneous superfluid phases, is needed.
Since the intra-species pairing is significant at low temperature, our result including both inter-and intra-species pairings is expected to influence the characteristics of those low temperature fermion systems. For instance, it will change the equation of state of the nuclear superfluid (where neutron-proton and neutron-neutron, proton-proton pairings play the roles of inter-and intra-species pairings) or the color superconductor of quark matter (where up-down and up-up, down-down quark pairings play the roles of inter-and intra-species pairings in two flavor case) in compact stars. On the other hand, systems involving two different alkali atoms can be used to study the result we obtained.