Superfluid Fermi-Fermi mixture: phase diagram, stability, and soliton formation

# Superfluid Fermi-Fermi mixture: phase diagram, stability, and soliton formation

###### Abstract

We study the phase diagram for a dilute Bardeen-Cooper-Schrieffer superfluid Fermi-Fermi mixture (of distinct mass) at zero temperature using energy densities for the superfluid fermions in one (1D), two (2D), and three (3D) dimensions. We also derive the dynamical time-dependent nonlinear Euler-Lagrange equation satisfied by the mixture in one dimension using this energy density. We obtain the linear stability conditions for the mixture in terms of fermion densities of the components and the interspecies Fermi-Fermi interaction. In equilibrium there are two possibilities. The first is that of a uniform mixture of the two components, the second is that of two pure phases of two components without any overlap between them. In addition, a mixed and a pure phase, impossible in 1D and 2D, can be created in 3D. We also obtain the conditions under which the uniform mixture is stable from an energetic consideration. The same conditions are obtained from a modulational instability analysis of the dynamical equations in 1D. Finally, the 1D dynamical equations for the system are solved numerically and by variational approximation (VA) to study the bright solitons of the system for attractive interspecies Fermi-Fermi interaction in 1D. The VA is found to yield good agreement to the numerical result for the density profile and chemical potential of the bright solitons. The bright solitons are demonstrated to be dynamically stable. The experimental realization of these Fermi-Fermi bright solitons seems possible with present setups.

###### pacs:
03.75.Ss, 64.75.+g, 03.75.Kk

## I Introduction

After the experimental realization of a trapped Bose-Einstein condensate (BEC) books () there has been a great effort to trap and cool the Fermi atoms to degeneracy by sympathetic cooling in the presence of a second Bose or Fermi component. The second component is needed to facilitate evaporative cooling not possible due to lack of interaction in a single-component Fermi gas exp1 (); exp2 (). Apart from the observation of the degenerate Bose-Fermi mixtures Li exp3 (); mix-ex (), Na-Li exp4 () and Rb-K exp5 (); exp5x (), there have been studies of the following spin-polarized degenerate Fermi-Fermi mixtures K-K exp1 () and Li-Li exp2 () in different hyperfine states.

Specially challenging has been the experimental realization of vortex lattice in a Bardeen-Cooper-Schrieffer (BCS) superfluid Fermi gas exp6 (); exp7 (); bcsexp (); fw () in Bose-Fermi mixture employing a weak attractive interaction among the intra-species fermions by using a Feshbach resonance FESH (); FESH (). This attractive interaction allows the formation of BCS pairs leading to a BCS superfluid fw (); leggett-book (). In the last few years by further increasing this attraction several experimental groups have observed the crossover cross () from the paired BCS state to the BEC of molecular dimers with ultra-cold two-hyperfine-component Fermi vapors of K greiner () and Li atoms zwierlein (); chin (). Another possibility is to use two distinct Fermi atoms for this purpose as suggested in Ref. skac () in a study of collapse in a Fermi-Fermi mixture (Li-K is a possible candidate for future exploration.) The Feshbach-resonance management of the Fermi interaction could be utilized to study a superfluid Fermi-Fermi mixture in a controlled fashion FESH ().

In Bose-Fermi mixtures, there have been several studies on phase separation molmer (); pethick (); viverit (); das (); sala-toigo (), soliton-like structures Sadhan-BFsoliton (), and collapse skac2 (), recently. The phase diagram of the Bose-Fermi mixture in three dimensions (3D) has been studied by Viverit et al. viverit (), whereas the same in one dimension (1D) has been studied by Das das (). Bright solitons have been observed in BECs of Li Li-soliton () and Rb Rb-soliton () atoms and studied subsequently BECsolitons (). It has been demonstrated using microscopic BFsoliton(BBrepBFattr) () and mean-field hydrodynamic Sadhan-BFsoliton () models that the formation of stable fermionic bright solitons is possible in a degenerate Bose-Fermi mixture in the presence of a sufficiently attractive interspecies interaction which can overcome the Pauli-blocking repulsion among fermions. The formation of a soliton in these cases is related to the fact that the system can lower its energy by forming high density region (bright soliton) when the interspecies attraction is large enough to overcome the Pauli-blocking interaction in the degenerate Fermi gas (and any possible repulsion in the BEC) skbs (). There have also been studies of mixing-demixing transition in degenerate Bose-Fermi sala-sadhan2 () and Fermi-Fermi skabm () mixtures, and soliton formation in Fermi-Fermi mixtures ff () .

In this paper we investigate the phase diagram of a BCS superfluid Fermi-Fermi mixture of fermion components of distinct mass at zero temperature using energy densities for the superfluid Fermi components in one, two (2D), and three dimensions. We derive the conditions of stability of the mixture in terms of the densities of the components and the strength of interspecies interaction. The two possible phases of the mixture are a uniformly mixed configuration and a totally separated pure-phase configuration. Unlike in a Bose-Fermi mixture viverit (); das (), no complicated mixed phases are allowed in a superfluid Fermi-Fermi mixture in 1D and 2D. However, a mixed and a pure phase is allowed in 3D. In 1D, two pure and separated phases of the fermion components appear for low fermion densities, whereas the opposite is found in 3D. In 1D, a uniform mixture appears for large fermion densities with the opposite taking place in 3D. In 2D, the condition for uniform mixture and phase separation is independent of density of the components. In 1D, we find the uniform mixture to be unstable for small fermion densities, whereas In 3D, the uniform mixture is unstable for large fermion densities.

The 1D configuration is of special interest due to soliton formation by modulational instability of a uniform mixture. To study this phenomenon we derive a set of dynamical equations of the system as the Euler-Lagrange equation of an appropriate Lagrangian. The condition of stability of the uniform mixture and the formation of soliton for attractive interspecies Fermi-Fermi interaction were studied from an energetic consideration as well as with a linear stability analysis of the constant-amplitude solution of the above dynamical equations. We solved the 1D dynamical equations numerically and variationally to study some features of the bright solitons. The numerical results for the density of the fermion components as well as their chemical potentials are found to be in good agreement with the variational findings. These bright solitons are found to be stable numerically when they are subjected to a perturbation.

The dependence of Fermi energy densities in 1D and 2D on atomic densities has counterparts in Bose systems and the analysis presented here is also applicable to these Bose systems. The 2D Fermi energy density has a quadratic dependence on atomic density as in a dilute BEC obeying the Gross-Pitaevskii equation, thus allowing the present results to be applicable to such a BEC books (). The 1D Fermi energy density, on the other hand, has a cubic dependence on atomic density as in a Tonks-Girardeau Tonks () (TG) Bose gas observed recently tg (), thus making the present results applicable to this system.

The paper is organized as follows. In Sec. II we consider the stability condition of a uniform BCS superfluid Fermi-Fermi mixture from an energetic consideration. In Sec. III we consider a two-phase BCS superfluid Fermi-Fermi mixture in 1D, 2D and 3D and study the possibility of the formation of two phases from a consideration of pressure, energy and chemical potential of the system. We can have two pure phases or a uniformly mixed phase in all dimensions. In addition, in 3D, we can have a pure and a mixed phase. In Sec. IV we consider the Euler-Lagrange nonlinear dynamical equations for the system in 1D and study the modulational instability of the constant-amplitude solution representing the uniform mixture. The condition of modulational instability for attractive Fermi-Fermi interaction is found to be consistent with the condition of stability of the uniform mixture obtained from an energetic consideration in Sec. II. We further solve these dynamical equations numerically and variationally to analyze the properties of the Fermi-Fermi solitons. Finally, in Sec. V we present a summary of our study.

## Ii Uniform Superfluid Fermi-Fermi Mixture

### ii.1 Energy Density of a Component

We consider a single-component dilute BCS superfluid of spin-half Fermi atoms of mass and density with a weak attraction between fermions with opposite spin orientations. In 3D, the energy density of this system is given by yang1 (); yang2 (); heis (); salasnich (),

 E3D=(3/5)n3εF (1)

where is the Fermi energy, is the Fermi momentum (this expression was first obtained by Lee and Yang yang2 () in the weak-coupling BCS limit). Modifications to this expression for a description of the BCS-BEC crossover, for stronger attraction between fermions, have also been considered salasnich (). The total density of the fermions in a 3D box is obtained by filling the quantum states up to the Fermi energy and is given by . (The factor of 2 in the expression for accounts for BCS pairing in each level.) Hence the energy density in (1) becomes

 E3D=3(3π2)2/3ℏ210mn5/33=35A3n5/33, (2)

with .

Similarly, the energy density of a dilute 1D superfluid of atom density is given by yang3 (); recati ()

 E1D=(1/3)n1εF. (3)

This was obtained using the Gaudin-Yang (GY) model yang3 () of fermions weakly interacting via zero-range (-function) potential, and was later extended to the description of the BCS-to-unitarity crossover recati (). (For repulsive interaction the GY model gives xt () a Tomonaga-Luttinger liquid TL (), while for attractive interaction it leads to a Luther-Emery liquid LE (). For weak attraction the ground state of the system is a BCS superfluid fw (); KO (). With the increase of attraction, the strong-coupling regime of tightly bound dimers is attained, which behaves like a hard core Bose gas, or like a 1D noninteracting Fermi gas, known as the TG gas Tonks (); ad-sa ().) The general solution for the ground-state energy in the GY model has been obtained by solving the Bethe ansatz bethe () equations for all strengths of interaction connecting the weak-attraction regime of BCS condensate to the strong-attraction regime of of tightly bound dimers described by the Lieb-Liniger model ll () of repulsive bosons. This solution can be presented as an expansion series in limits of weak or strong interactions. The limiting value of this solution in the weak interaction BCS limit is given by Eq. (3) recati (); xt ().

The fermion density of the BCS superfluid in a 1D box is , hence, in this case, , and energy density (3) becomes recati2 ()

 E1D=π2ℏ224mn31=13A1n31, (4)

with . The energy density of a TG gas Tonks () is given by ad-sa () and is very similar to that given by Eq. (4). The difference in numerical factors between the two expressions is due to pairing in the present Fermi superfluid allowing two fermions (spin up and down) in the same quantum level. Hence the 1D results of the present study is also applicable to a TG gas.

Finally, a counterpart of relations (1) and (4) for the 2D superfluid is luca2 () , the 2D density being , with . Thus, the energy density of the 2D superfluid can be written as luca2 ()

 E2D=πℏ22mn22=12A2n22, (5)

with .

Here we specify the criteria of applicability of Eqs. (2), (4), and (5) for different dimensionalities. These results are valid for a dilute BCS superfluid. In 3D, at low densities, with the Fermi-Fermi scattering length, gaps are small and have little effect on the total energy of the system heis (). The total energy density of the ground state can then be expanded in powers of the small parameter . At low densities Eq. (2) includes the lowest order term in this expansion yang2 (). The condition of validity of Eq. (2) can be related to the gas parameter in 3D: , as the density . In 1D, for a interaction of strength the dimensionless coupling constant and the condition of validity of Eq. (4) is recati (). In two dimensions an attractive interaction leads to a bound state of energy and the condition of diluteness for the validity of Eq. (5) can be expressed as

### ii.2 Stability Condition of the Uniform Mixture

We consider a uniform mixture of two types of fermions, containing , atoms (of mass and ) , in a box of size (in 1D the size is a length, in 2D an area, and in 3D a volume) with distinct mass at zero temperature. The energy density of the uniform mixture is given by

 E1D = 13A1n31(1)+g12n1n2+13λA1n31(2), (6) E2D = 12A2n22(1)+g12n1n2+12λA2n22(2), (7) E3D = 35A3n5/33(1)+g12n1n2+35λA3n5/33(2), (8)

respectively, for 1D, 2D and 3D systems, where denotes the density of each component in D, . The nonlinear terms involving in above equations represent the interaction between two types of atoms arising solely from the atomic scattering length , where is the reduced mass of atoms. The terms involving in the above equations, although is similar to the interaction term for bosons (with representing the self interaction of a dilute boson gas with the Bose-Bose scattering length and the mass of an atom), have a different origin as we have seen. These terms originating from the energy of the fermions occupying the lowest quantum levels at zero temperature obeying Pauli principle generate an effective repulsion between the fermions and is usually called Pauli-blocking interaction.

The chemical potentials for species in 1D, 2D and 3D, are given, respectively, by

 μ1 = A1n21(1)+g12n1(2),μ2=g12n1(1)+λA1n21(2), (9) μ1 = A2n2(1)+g12n2(2),μ2=g12n2(1)+λA2n2(2), (10) μ1 = A3n2/33(1)+g12n3(2),μ2=g12n3(1)+λA3n2/33(2).

The uniformly mixed phase is energetically stable if its energy is a minimum with respect to small variations of the densities, while the total number of fermions and bosons are held fixed. The conditions of stability [are the conditions of a minimum of as a function of two variables and and] are given by

 ∂2E∂n2(1)≡∂μ1∂n(1)≥0,∂2E∂n2(2)≡∂μ2∂n(2)≥0, (12) ∂2E∂n2(1)∂2E∂n2(2)−(∂2E∂n(1)∂n(2))2≡∂μ1∂n(1)∂μ2∂n(2)−∂μ1∂n(2)∂μ2∂n(1)≥0,

where we have dropped the dimension suffix. The solution of these inequalities gives the region in the parameters’ space where the uniformly mixed phase is energetically stable. Using Eqs. (12) and (II.2) the condition of stability of the uniform mixture in 1D, 2D, and 3D are given, respectively, by viverit (); ad-sa ()

 4A21λn(1)n(2)≥g212, (14) A22λ≥g212, (15) 4A23λ≥9g212n1/3(1)n1/3(2). (16)

These conditions are determined by and not the sign of .

In 1D, we find from Eq. (14) with a finite , that at small fermionic densities (small and ) the uniform mixture is unstable: the ground-state of the system displays demixing if and becomes a localized Fermi-Fermi bright soliton if ad-sa (). The mixture is stable at large fermionic densities. In 2D, Eq. (15) reveals that the condition for stability is independent of density. In 3D, Eq. (16) predicts that for a finite , the mixture is unstable at large fermionic densities, leading to collapse for and to demixing for , and stable at small fermionic densities. It is realized that as we move from 1D to 3D through 2D, the condition of stability of the uniform mixture changes from large fermion densities to small fermion densities. This result is quite similar to that in a Bose-Fermi mixture viverit (); das (), where the condition of stability of the uniform mixture is independent of the bosonic density and has a similar dependence on fermion density, e.g., during the passage from 1D to 3D through 2D, the condition of stability changes from large fermion density to small fermion density.

From inequality (12) the stability condition of a single component uniform gas can be represented as , which, using Eqs. (9), (10) and (II.2), is realized for denoting a repulsive system. In the presence of a second component, inequality (II.2) can be written as viverit ()

 ∂μ1∂n(1)−(∂μ2∂n(1))2∂n(2)∂μ2≥0, (17)

as . The first term in inequality (17) represents the effective repulsion among fermions of type 1. The second term, representing an induced interaction due to the presence of component 2, reduces the repulsion and tries to destabilize the uniform mixture. The uniform mixture becomes unstable when the second term in inequality (17) becomes larger than the first term. This happens for both attractive and repulsive interspecies interaction .

The inequality (II.2) can be written as

 c21c22≥4g212n(1)n(2), (18)

where represent sound velocities in the two superfluid components, . The sound velocity of the 1D Fermi-Fermi mixture can be obtained following a procedure suggested by Alexandrov and Kabanov kabanov (); ad-sa () for a two-component BEC:

 c12=1√2√c21+c22±√(c21−c22)2+16g212n(1)n(2). (19)

The homogeneous mixture becomes unstable when the sound velocity becomes imaginary, e.g., when inequality (18) is violated.

## Iii Two-Phase Superfluid Fermi-Fermi Mixture

In the last section we considered a uniform mixture of two components in equilibrium. Here we explore the more interesting case of two types of fermions with different possible densities in different regions of a box of size The components may mix uniformly or form separate phases depending on the initial conditions mass, density, interspecies interaction etc.

The conservation of the number of particles, and , of the two species can be expressed as viverit (); das ()

 Ni=Sni=S2∑j=1ni,jfj,2∑j=1fj=1, (20)

where represent the species and represent the phases (different region with distinct density of gas), represent the overall density of the two species, is the density of species in phase , and represent the size of each phase with the fraction of size in phase . For a two-component system one can have only two distinct phases, , as the inclusion of more phases leads to inconsistency viverit (). Here we have dropped the dimension label and also removed the parentheses () from the component label .

The total energy of the system is given by

 E = 2∑j=1Ej≡S2∑j=1fjEj (21)

where denotes the energy density of phase and its total energy. The pressure of phase is given by . The chemical potential of component in phase is defined by .

For equilibrium, the pressure in one phase has to be equal to that in the other. If two phases are occupied by atoms of the same type, the chemical potential for that type of atoms in two phases should also be equal so that the equilibrium can be energetically maintained. If the atom density of one type of atom in a phase is zero then the chemical potential of that type of atom in this phase should be larger than that in the other, so that the atoms do not flow to the phase with no atoms of this type viverit ().

In the following we consider a system composed of two phases comprising of fractions and of size . There are three following possibilities to be analyzed in 1D, 2D, and 3D, although some of them may not materialize in a particular case:

(i) Two pure and separated phases with one type of atom occupying a distinct phase.

(ii) A mixed and a pure phase where the density of one type of atoms is zero in one phase.

(iii) Two mixed phases where both phases are occupied by both type of atoms.

In the following we deal with the three possibilities in 1D, 2D and 3D. First, we consider the 2D case as the algebra is significantly simpler in this case.

### iii.1 Two-Dimensional (2D) Mixture

From Eqs. (7) we find that the expressions for total energy and pressure in this case are

 Ej = SjEj≡Sj[12A2(n21,j+λn22,j)+g12n1,jn2,j], (22) Pj ≡ −∂Ej∂Sj=12A2n21,j+g12n1,jn2,j+12A2λn22,j. (23)

In deriving Eq. (23) we recall that . From Eq. (22) the chemical potentials are given by

 μ1,j = A2n1,j+g12n2,j (24) μ2,j = A2λn2,j+g12n1,j (25)

#### iii.1.1 Two Pure Phases

In case of two pure and separated phases one should have, for example corresponding to the type one atoms occupying phase 1 only () and type 2 atoms occupying phase 2 only ().

Equality of pressure in the two phases yields

 n21,1=λn22,2. (26)

As the number of atoms is zero in one of the phases, one has the inequalities and on the chemical potential, which, using Eqs. (24) and (25), become

 A2λn2,2 ≤ g12n1,1, (27) A2n1,1 ≤ g12n2,2. (28)

Eliminating and among Eqs. (26), (27), and (28) we get

 A22λ≤g212, (29)

consistent with inequality (15). We have the uniform mixture for inequality (15); for the opposite inequality (29) we have the separated phases in equilibrium. These inequalities are independent of the atomic densities.

In the present case the overall densities of the two species are given by

 n1=f1n1,1=fn1,1,n2=f2n2,2=(1−f)n2,2. (30)

Let us now consolidate these findings using energetic considerations comparing the total energy of a phase-separated configuration with that of a uniform mixture. The energy of the mixture is given by

 Emix = S[12A2n21+g12n1n2+12A2λn22], = S[12A2f2n21,1+g12n1,1n2,2f(1−f) + 12A2λn22,2(1−f)2], (32)

where we have used Eqs. (30). The energy of the phase-separated system with the same number of atoms is

 Esep = S[12A2fn21,1+12A2λ(1−f)n22,2]. (33)

Using Eq. (26), one has for the difference

 Emix−Esep=Sf(1−f)n22,2√λ[g12−A2√λ]. (34)

When the system naturally moves to the separated phase and this happens for , consistent with inequality (29), leading to a stable separated phase. In the opposite limit, when , the energetic consideration favors the uniform mixture and this happens for , consistent with inequality (15).

#### iii.1.2 A mixed and a pure phase

Here we consider one mixed phase (phase 1) and one pure phase (phase 2) consistent with , which means that the type 1 atoms occupy only phase 1, whereas type 2 atoms occupy both phases 1 and 2. Using Eq. (23) the equality of pressure in two phases leads to

 12A2n21,1+g12n1,1n2,1+12A2λn22,1=12A2λn22,2. (35)

From Eq. (25) the equality of the chemical potential of type 2 atoms in two phases () leads to

 n1,1=A2λ(n2,2−n2,1)/g12. (36)

From Eq. (24) the inequality of the chemical potential of type 1 atoms in two phases () leads to

 A2n1,1

which using Eq. (36) yields

 A22λ

Substituting Eq. (36) into Eq. (35) and after some straightforward algebra we obtain

 (A22λ−g212)(n2,1−n2,2)2=0, (39)

which allows two possibilities. For , the only solution is the trivial, nevertheless unacceptable, one , which means that the type 2 atoms form a uniform configuration and not a mixed phase. However, if , one can have a mixed phase with . Nevertheless, this condition enters in contradiction with inequality (38), showing that one cannot have one mixed and one pure phase in this case.

Next we consider the possibility of two mixed phases. The equality of pressure and chemical potential of each species in two phases leads to the following conditions

 12A2(n21,1−n21,2) + 12A2λ(n22,1−n22,2) (40) = g12(n1,2n2,2−n1,1n2,1), A2n1,1+g12n2,1 = A2n1,2+g12n2,2, (41) A2λn2,1+g12n1,1 = A2λn2,2+g12n1,2. (42)

This set of equations have only the trivial solutions and corresponding to uniform mixture. Hence two mixed phases cannot be in equilibrium.

### iii.2 One-Dimensional (1D) Mixture

From Eqs. (6), we find that the expression for total energy and pressure in this case are

 Ej = SjEj≡Sj[13A1(n31,j+λn32,j)+g12n1,jn2,j], (43) Pj ≡ −∂Ej∂Sj=23A1n31,j+g12n1,jn2,j+23A1λn32,j.

From Eq. (22) the chemical potentials are given by

 μ1,j = A1n21,j+g12n2,j (45) μ2,j = A1λn22,j+g12n1,j. (46)

#### iii.2.1 Two pure phases

In case of two pure and separated phases one should have, for example, for phase 1 and for phase 2. The condition of equal pressure then yields

 n31,1=λn32,2. (47)

For equal-mass fermions , and one obviously have have the trivial solution or the densities of the two species are equal. Of course, for the densities of the two species could be different. Chemical potential condition yields

 A1λn22,2≤g1,2n1,1. (48)

Chemical potential condition yields

 A1n21,1≤g1,2n2,2. (49)

Eliminating between Eqs. (47) and (48) or between Eqs. (47) and (49) we get

 n2,2≤B1,B1=g12/(A1λ2/3). (50)

From Eqs. (47) and (50) we obtain the following restriction on

 n1,1≤C1,C1=g12/(A1λ1/3). (51)

In this case a phase diagram showing the total densities of type 1 and 2 fermions for which the system can completely separate, can be obtained from Eq. (30) if we allow to vary from 0 to 1 and use conditions (50) and (51). This is illustrated in Fig. 1. The light gray area represents pure phases and the dark gray area represents the stable uniform mixture. The uniform mixture is unstable in the clear area below the curve given by inequality (14). For attractive interaction, one has the formation of bright solitons by modulational instability, (discussed in Sec. IV). For repulsive interaction one can have a partially demixed configuration in the clear region in Fig. 1.

Now let us see if the system spontaneously move into the phase-separated configuration from an energetic consideration. The energy of the mixed system is

 Emix = S[13A1n31+n1n2g12+13λA1n32] = S[13A1n31,1f3+n1,1n2,2f(1−f)g1,2 + 13A1n32,2(1−f)3λ]. (53)

Equation (53) is obtained with the use of Eq. (30). The energy of the separated phase system with the same number of atoms is

 Esep=13A1S[n31,1f+n32,2λ(1−f)]. (54)

Using Eq. (47), and after some straightforward algebra, the difference is given by

 Δ=n22,2Sλ1/3[g12−A1n2,2λ2/3]f(1−f) (55)

Considering the restriction (50) in the separated phase, Eq. (55) yields the following inequality

 Δ=n22,2f(1−f)Sλ1/3[1−n2,2/B1]g12≥0. (56)

For density ranges where equilibrium is possible and , is always less than . Hence, energetically the two species of fermions can separate.

#### iii.2.2 A mixed and a pure phase

Now let us consider a mixed phase (phase 1) and a pure phase (phase 2) and consider the case . The equality of pressure now leads to

 23A1n31,1+23A1λn32,1+g12n1,1n2,1=23A1λn32,2. (57)

The equality of chemical potential of species 2 in two phases () yields

 n1,1=A1λ(n22,2−n22,1)/g12. (58)

Eliminating between equations (57) and (58) (after some straightforward algebra) we get

 2Λ(1−x2)3=x3−3x+2 (59)

where , . After cancelling the trivial factor from both sides of Eq. (59), we get

 2Λ(1+x)3(1−x)=x+2 (60)

From Eq. (60) we find that the solution is obtained for corresponding to , and . The densities of the first component are and This is the special case considered in Sec. IIIB1 [see, Eqs. (50) and (51)]. The solution is a solution of two pure phases corresponding to . The domain of solution of mixed phase corresponds to corresponding to (recall that the fraction cannot be negative.) Hence for the present mixed phase to exist Eq. (60) should have the solution for . However, we find from Eq. (60) as is made slightly greater than 1, the solution turns negative (unphysical). [Please note that for Eq. (60) has two real roots: ; the latter (spurious) root is not of present physical interest.] Hence we conclude that a mixed and a pure phase cannot be realized in the present mixture.

Finally, one can consider the possibility of two mixed phases. The equality of pressure and chemical potential of each species in two phases leads to

 23A1(n31,1−n31,2) + 23A1λ(n32,1−n32,2) (61) = g12(n1,2n2,2−n1,1n2,1), A1n21,1+g12n2,1 = A1n21,2+g12n2,2, (62) Aλn22,1+g12n1,1 = Aλn22,2+g12n1,2. (63)

This set of equations have only the trivial solutions and corresponding to uniform mixture and that is also possible when the condition of uniform mixture is satisfied. Hence two mixed phases cannot be in equilibrium.

### iii.3 Three-dimensional (3D) Mixture

From Eqs. (8), we find that the expression for total energy and pressure in this case are

 Ej = SjEj≡Sj[35A3n5/31,j+g12n1,jn2,j+35A3λn5/32,j], (64) Pj ≡ −∂Ej∂Sj=25A3n5/31,j+g12n1,jn2,j+25A3λn5/32,j.

From Eq. (22) the chemical potentials are given by

 μ1,j = A3n2/31,j+g12n2,j (66) μ2,j = A3λn2/32,j+g12n1,j. (67)

#### iii.3.1 Two pure phases

Again for two pure and separated phases we take . The condition of equal pressure in two phases then leads to

 n5/31,1=λn5/32,2. (68)

The chemical potential condition yields

 n1,1≥λA3n2/32,2/g12. (69)

The chemical potential condition yields

 n2,2≥A3n2/31,1/g12. (70)

Eliminating between Eqs. (68) and (69) or between Eqs. (68) and (70) we obtain

 n2,2≥B3,B3=(λ2/5A3/g12)3. (71)

Similarly, eliminating between Eqs. (68) and (69) we get

 n1,1≥C3,C3=(λ3/5A3/g12)3. (72)

In this case a phase diagram showing the total densities of type 1 and 2 fermions for which the system can completely separate, can be obtained from Eq. (30) if we allow to vary from 0 to 1 and use conditions (71) and (72). This is illustrated in Fig. 2.

To see the separation of the two types of fermions from an energetic consideration, we calculate the energies of the mixed and separated configurations. The energy of the mixed phase is viverit ()

 Emix = S[35A3n5/31+g12n1n2+35A3λn5/32] = S[35A3f5/3n5/31,1+g12n1,1n2,2f(1−f) + 35A3λn5/32,2(1−f)5/3]. (74)

The energy of the separated phase is

 Esep = S[35A3n5/311f+35A3λn5/32,2(1−f)]. (75)

Using Eq. (68) the difference can be written as

 Δ = S[3A3λn5/32,2f(f2/3−1)/5+g12n222λ3/5f(1−f) (76) + 3A3λn5/32,2(1−f){(1−f)2/3−1}/5].

Using inequality (70), Eq. (76) yields

 Δ ≥ A63λ3g512[−35f(1−f2/3) (77) + f(1−f)−35(1−f){1−(1−f)2/3}].

For , the quantity given by (77) is always positive. Hence the separated phase has less energy than the mixed phase and the system will spontaneously move into the phase separated configuration.

In this case also two mixed phases cannot be in equilibrium as in 1D.

#### iii.3.2 A mixed and a pure phase

Again we consider a mixed (species 2) and a pure (species 1) phase and consider the case . The equality of pressure now leads to

 25A3n5/31,1+g12n1,1n2,1+25A3λn5/32,1=25A3λn5/32,2. (78)

The equality of chemical potential of species 2 in two phases () yields

 n1,1=A3λ(n2/32,2−n2/32,1)/g12. (79)

Eliminating between Eqs. (78) and (79) and after some straightforward algebra we get

 2Λ(1+x)5/3−(1−x)1/3(3x3+6x2+4x+2)=0, (80)

where , and . From Eq. (80) we find that the solution is obtained for corresponding to , ,