QCD-like phase diagram with Efimov trimers and Cooper pairs in resonantly interacting SU(3) Fermi gases

We investigate color superfluidity and trimer formation in resonantly interacting SU(3) Fermi gases with a finite interaction range. The finite range is crucial to avoid the Thomas collapse and treat the Efimov effect occurring in this system. Using the Skorniakov-Ter-Martirosian (STM) equation with medium effects, we show the effects of the atomic Fermi distribution on the Efimov trimer energy at finite temperature. We show the critical temperature of color superfluidity within the non-selfconsistent $T$-matrix approximation (TMA). In this way, we can provide a first insight into the phase diagram as a function of the temperature $T$ and the chemical potential $\mu$. This phase diagram consists of trimer, normal, and color-superfluid phases, and is similar to that of quantum chromodynamics (QCD) at finite density and temperature, indicating a possibility to simulate the properties of such extremely dense matter in the laboratory.


I. INTRODUCTION
The concept of quantum simulation has opened new possibilities for exploring the properties of novel materials as well as exotic matter in extreme conditions [1][2][3]. Recently, ultracold atoms have been used as quantum simulators of strongly correlated systems thanks to the tunability of their physical parameters such as interparticle interaction [4][5][6][7]. For example, ultracold atomic Fermi gases loaded into an optical lattice can realize the Hubbard model, which is relevant to high-T c cuprate superconductors [3,[8][9][10][11]. The antiferromagnetic behavior which plays an important role for the superconducting mechanism of high-T c cuprates has been observed in this atomic system [3,11]. As another example, strongly interacting two-component Fermi gases can be used as quantum simulators of dilute neutron matter [12][13][14]. Indeed, the observed thermodynamic quantities of homogeneous Fermi gases near the unitarity limit [15][16][17][18] quantitatively reproduce the equation of state of neutron matter obtained by numerical simulations [19] which is crucial for understanding the interior of a neutron star [20,21].
The analog simulation of quantum chromodynamics (QCD) [22,23], where quarks with three colors strongly interact with each other, can be regarded as one of the next important challenges for cold-atom physics [24][25][26]. While numerous efforts have been made to explore the phase diagram of finite-density QCD where various phenomena such as color superconductivity [27] have been theoretically proposed, they could not be confirmed by high-energy experiments due to the extreme densities, nor by exact numerical simulations due to the sign problem. On the other hand, three-component fermion systems have been experimentally realized by using mixtures of fermionic atoms in three different internal states [26,[28][29][30][31][32][33][34][35].
These systems reach the quantum degenerate regime around T ≃ 0.3T F [28] where T F is the Fermi degenerate temperature of non-interacting atoms. In this regime, the existence of color superfluidity similar to color superconductivity in QCD has been theoretically examined in three-component Fermi gases [36][37][38][39][40][41][42][43]. Although there are several differences between these atomic systems and QCD, they constitute a good starting point to study the strong-coupling effects in three-component fermion systems.
In most of the previous works on color superfluidity [36][37][38][39][40][41][42][43], zero-range interactions have been used to describe the attraction between the three components. However, such zerorange interactions are known to lead to a collapse of the system [44,45] in connection with the existence of Efimov trimers [30,33,35,[46][47][48][49][50]. A finite range of interactions is therefore necessary to prevent the collapse and properly treat Efimov trimers. The effect of finite range and the Efimov trimers on the phase diagram of three-component fermionic system was studied in Ref. [51]. However, this study was limited to zero temperature.
In this paper, we investigate in-medium Efimov trimers and color superfluidity in resonantly interacting SU(3) Fermi gases with finite interaction range and finite temperature.
Finally, we summarize this paper in Sec. IV. For simplicity, we take = k B = 1 and the system volume V is taken to be unity.

A. Hamiltonian
We consider a symmetric three-component fermionic system. We model the interaction between two different components by a two-channel Feshbach resonance model [7,51]  system is therefore described by the following Hamiltonian, where ξ F p = p 2 2m − µ and ξ B q = q 2 4m + ν − 2µ are the kinetic energies of a fermion with mass m and the diatomic molecule, respectively, and µ is the fermionic chemical potential. c p,j and b q,ij are annihilation operators of a Fermi atom with the internal state j = 1, 2, 3 and a diatomic molecule made of i − j atomic pair. The energy of diatomic molecules ν and the atom-dimer Feshbach coupling g are related to the scattering length a and the range parameter R * as follows, We note that the effective range r e of this two-channel interaction is here negative and is associated with R * = − 1 2 r e . In this paper, we focus on the unitarity limit 1/a = 0.

B. non-selfconsistent T -matrix approximation
We first employ the non-selfconsistent T -matrix approximation within the Matsubara formalism [54][55][56][57][58][59][60][61][62][63][64] to determine thermodynamic properties such as critical temperature T c of color superfluidity. In this framework, the thermal Green's function of dressed atoms is given by where we used the four-momentum notation p = (p, iω n ) and ω n = (2n + 1)πT is the fermionic Matsubara frequency. Σ(p) is the fermionic self-energy. In the case of just two fermions in vacuum, the self-energy is given by the diagram shown in Fig. 1 where Q = (Q, iν n ′ ) and ν n ′ = 2πn ′ T is the bosonic Matsubara frequency. G 0 (p) = 1/ iω n − ξ F p and D(Q) are the in-medium Green's functions of a non-interacting fermion and a dressed molecule, respectively. We note that although this equation (5) as the same form as that in vacuum, here G 0 and D contain the medium effects. We also note that the factor 2 in Eq. (5) comes from the degree of freedom with respect to internal states.
Similarly, the thermal Green's function of dressed molecules with the ultraviolet renormalization is given by where Ξ(Q) is the bosonic self-energy diagrammatically shown in Fig. 1(b). Here again, we take the vacuum form where f (ξ) = 1/(e ξ/T + 1) is the Fermi-Dirac distribution function. The chemical potential is the Fermi energy of an ideal Fermi gas] is obtained by solving the number equation, In addition, we obtain the critical temperature T c from the Thouless criterion D −1 (Q = 0, iν n = 0) = 0 [73], which gives (1/a = 0)

C. in-medium Skorniakov-Ter-Martirosian equation
To determine the trimer energy E M 3 in the medium, we consider the three-body T -matrix equation [74,75] which is diagrammatically shown in Fig. 2. In the vacuum case, it gives the so-called Skorniakov-Ter-Martirosian (STM) equation which exactly describes Efimov physics in three-body problems. Using the thermal Green's functions within the Matsubara formalism, we obtain the in-medium three-body T -matrix We note that p, p ′ , and P are the four-momenta of incoming fermion, outcoming fermion, and the center-of-the-mass, respectively (we suppress the index of internal states for simplicity).
The factor 2 in the second term of r. h. s. of Eq. (10) comes from the degree of freedom with respect to internal states. G 0 (q) with q = (q, iΩ n ) indicates intermediate states of fermions which are integrated. By including Pauli-blocking effects on G 0 (p) (see Appendix A), we obtain the in-medium STM equation in the unitarity limit (1/a → 0), where κ(q) 2 = 3 4 q 2 − mE M 3 , L(p) is the function defined by Eq. (A9) in Appendix A, and is the statistical factor associated with the Fermi-Dirac distribution of atoms given by f (ξ F p ) = 1/ e ξ F p /T + 1 . One can find that the ordinary STM equation in a three-body problem is recovered by setting F (p, q) = 1. We note that Eq. (12) is qualitatively consistent with the previous works [76,77]. We note however that Ref. [76] is restricted to T = 0 with the choice We briefly note that similar in-medium three-body equations were employed in nuclear physics [78][79][80][81][82][83][84][85].
III. RESULTS Figure 3 shows the ground-state trimer energy E M 3 with the medium effects, which can be obtained numerically from Eq. (11) as a dimensionless function where λ T = 2π mT is the thermal de Broglie wavelength. Physically, the range parameter R * gives the typical size of the Efimov trimer [86]. In this regard, the ratio between R * and λ T represents how trimer states are affected by finite temperature effects. In the case of R * ≪ λ T and µ/T < ∼ 0, E M 3 approaches the vacuum limit given by E V 3 = −0.01385/(mR 2 * ) which is close to that of a universal trimer [87], since the trimer size is small enough compared to the typical thermal length scale. In addition, the ratio µ/T is associated with the fugacity z = e µ/T and represents Pauli-blocking effects due to the atomic Fermi distribution, which plays a significant role when µ > 0. The absolute value of E M 3 is greatly reduced in the Fermi degenerate region. Finally, E M 3 disappears in the region where µ/T and R * /λ T are relatively large. The physical interpretation of these effects is that Fermi atoms from the medium weaken the Efimov attraction between three atoms forming a trimer state. This phenomenon is somewhat similar to the Gor'kov-Melik-Barkhudarov (GMB) correction in weak-coupling superconductors for which the size of Cooper pairs is large [88] and the pairing interaction is screened by the medium [89][90][91][92][93].
To see how these effects would appear in actual experiments at given temperatures and densities, we plot in Fig. 4 (a) the typical temperature dependence of trimer energy E M 3 at different range parameters according to the density equation of state obtained from the nonselfconsistent T -matrix approximation (TMA). By numerically solving the number equation Eq. (8), we obtain the temperature-dependent chemical potential µ as shown in Fig. 4 (b).
We then use µ as an input for in-medium STM equation given by Eq. (11). This gives E M 3 showing in Fig. 4 (a). E M 3 has a peak structure which can be understood from two effects, that is, the evolution of the Fermi chemical potential and the decrease of λ T compared to the trimer size. In the high-temperature limit, we can neglect the interaction effects and µ is given by the number equation of ideal gases n = 3 p e −ξ F p /T . This gives approximately µ ≃ − 3 2 T lnT and for large temperatures E M 3 reproduces the vacuum result E V 3 due to the large negative µ. Decreasing the temperature makes the density and temperature effects more visible which weakens the trimer (E M 3 increases). However, at very low temperature µ becomes almost constant while λ T increases as 1/ √ T with decreasing temperature. As a result, it becomes larger and larger compared to trimer size R * (which is fixed in this figure), which suppresses the temperature effects and the trimer strengthens (E M 3 decreases). This decrease becomes sharper with increasing R * as shown in Fig. 4 (a). We note that our calculation is stopped at T = T c , where Eq. where all three kinds of pairs are condensed, and a normal phase (NP) where the atoms form neither trimers nor condensed pairs. The boundary between TP and NP is estimated by the curve where E M 3 = 0. Threshold curve E M 3 = 0 does not mean any phase transition between trimer and normal phases but it should be a good indication of how the trimer character disappears in the high-density region of this system. CSF is defined by the region below T c . We note that T c approaches 0 in low-density (zero-range) limit at µmR 2 * = 0 [77]. We also note that the high density limit of the critical temperature T HDL molecules in the presence of thermal-excited fermions and µ approaches 0 in this limit [77].
This indicates that the system undergoes a crossover from unitary Cooper pairs to BEC of closed channel molecules. This behavior is specific to the narrow-resonance two-channel model used in this work. Interestingly, Fig. 6 is similar to the phenomenological phase diagram of QCD consisting of the hadron phase (analogue of TP), the color superconducting phase (analogue of CSF), and the deconfined quark phase (analogue of NP) [23]. However, while the phase transition at T = T c between CSF and NP is of the second order, that of color superconductivity is of the first order due to the gauge coupling [27]. Moreover, the conjectured BEC-BCS crossover [94] in QCD with increasing the chemical potential is opposite to the BCS-BEC crossover found in this model at 1/a = 0. We stress again this is a particularity of the narrow-resonance two-channel model. the magnification around µmR 2 * = 0, where the two curves we calculated cross each other. In reality, the region near this point is expected to be dominated by strong multi-body correlation due to the competition between trimer formation and color superfluidity [80], which cannot be captured by our treatment.
Finally, we look at pairing fluctuations above T c . Indeed, it is known that pairing fluctuations become strong near T c in two-component Fermi gases near the unitarity limit. Figure 6 shows the single-particle spectral function obtained from the analytic continuation of G(p) given by Eq. (4) (δ is an infinitesimally small positive number), at µmR 2 * = 0.0298 and T mR 2 * = 0.0302 which is just above T c indicated in the inset of Fig. 5. One can see that the atomic dispersion has a gap structure near ω = 0 even in the absence of the superfluid gap. This excitation gap in the normal phase originates from strong pairing fluctuations (preformed Cooper pair) and is called pseudogap, which has been extensively discussed for various strongly correlated quantum systems such as high-T c superconductors [95,96], ultracold Fermi gases [54][55][56][57][58][59][60][61][62][63][64][97][98][99][100][101][102][103][104], color superconductivity [105,106], and nuclear matter [107][108][109]. This single-particle excitation property is accessible by photo-emission spectrum measurement in cold atom systems [98][99][100][101]. In principle, such experiments could also observe many-body effects associated with in-medium Efimov trimers. However, treating such many-body effects theoretically would require a self-consistent approach including both two-body and three-body correlations in the self-energy.
The pseudogap can also be seen in the single-particle density of states ρ(ω), which is defined by It is shown in Fig. 7 (a) at T = T c . This quantity clearly shows the pseudogap effect as a dip structure around ω = 0. The pseudogap disappears away from µ = T = 0, that is, in the high-density (large range parameter) limit as found in the case of two-component Fermi gases with negative effective range [64]. To characterize this many-body phenomenon, we introduce the pseudogap size ∆ pg defined as the half width of the dip [59] [see the inset of Fig. 7 (b)]. One can find that ∆ pg grows when µmR 2 * approaches 0 and reaches a maximum value ∆ pg ≃ 0.55ε F at µmR 2 * = 0. This enhancement of ∆ pg indicates that many-body effects associated with pairing fluctuations are important in the region around µ = T = 0 in the phase diagram of Fig. 5. This confirms the expected competition between formation of Cooper pairs and that of Efimov trimers. This competition would occur around E M 3 = 0, which is shown as the vertical dashed line in Fig. 7(b).

IV. SUMMARY
In this paper, we have investigated some of the strong-coupling effects occurring in resonantly interacting SU(3) Fermi gases with a finite interaction range, namely, the in-medium Efimov trimer and the critical temperature T c of color superfluidity.
The trimer formation is weakened by the medium effects, which consist of thermal agitation and Fermi pressure due to Pauli exclusion. The trimer is affected by thermal agitation when the thermal de Broglie wavelength is comparable to the trimer size given by the range of interaction. The Pauli-blocking effects are significant when the chemical potential µ be- Finally, we have investigated the phase diagram with respect to the chemical potential µ and temperature T . Our calculations indicate the existence of three phases: trimers phase, normal phase, and color superfluidity. Interestingly, the obtained phase diagram is analogous to the phenomenological QCD phase diagram which consists of hadron, deconfined quark, and color superconducting phases. We emphasize that the finite interaction range plays an important role to obtain such a QCD-like phase diagram in this atomic system. Near µ = T = 0 in our phase diagram, the system is expected to be dominated by strong two-body and three-body correlations resulting from the competition between trimer formation and color superfluidity. This idea is supported by our calculation of the inmedium trimer energy and the single-particle spectral function which exhibits strong pairing fluctuations near the color superfluid phase transition. A self-consistent treatment of twobody and three-body correlations is required to understand this interesting regime, which is left as a future problem. Our analysis does not exclude the possibility of trimer superfluidity due to a possible residual attraction between the trimers [110]. Such a state would be similar to the p-wave superfluidity in a Bose-Fermi mixture [111]. Although we consider a resonant interaction in this paper, the phase diagram would quantitatively change when tuning the scattering length. In particular, the color superfluid phase (or molecular BEC) would start at a lower chemical potential in the case of a finite scattering length [51]. Since we calculate the ground-state trimer energy E M 3 with the medium effects, we set P = (0, iζ l ) in the three-body T -matrix equation given by Eq. (10) [74,75] [where iζ l = (2l + 1)πT is the fermionic Matsubara frequency]. We obtain where we ignore the first term of r. h. s. of Eq. (10) which is negligible near the pole of since Resf (x = iΩ n ) = −T . We note that C can be deformed to a clockwise path C ′ which encloses the poles of G 0 , D, and T M 3 , which give medium effects associated with the momentum distributions of atoms, molecules, and trimers, respectively. For simplicity, we consider only the pole of G 0 to incorporate the effects of the atomic Fermi-Dirac distribution.
This approximation is justified in the high temperature regime where the fugacity z = e µ/T is small. In this regime, the atomic Fermi distribution function is approximately given 2mT z, whereas the molecular and trimer distribution functions Einstein distribution function and ξ T P = P 2 /(6m) − 3µ + E M 3 is the kinetic energy of a trimer]. Using this approximation, we can analytically perform the energy integration as where q 1 = (q, iζ l − iω n − ξ F p+q ) and q 2 = (q, ξ F q ). To obtain the in-medium STM equation, we perform the analytic continuations iω n → ξ F p and iζ l → E M 3 − 3µ in Eq. (A3). In this way, we obtain T M 3 (p, p ′ ; P ) = 2g 2 Furthermore, the first integrand in Eq. (A4) gives dominant contributions near its pole, We can then approximate the arguments of D and T M 3 as and By substituting Eqs. (A5) and (A6) into Eq. (A4), we obtain T M 3 (p, p ′ ; P ) = 2g 2 where is obtained from the analytic continuation of Eq. (6). We note that κ(q) 2 = 3 4 q 2 − mE M 3 and F (k, q) is the statistical factor defined by Eq. (12). We introduce We note that although L(q) implicitly depends on p ′ and P , they do not change the equation of E M 3 . By using L(q), Eq. (A7) can be rewritten by Finally, we obtain the in-medium STM equation, that is, Eq. (11) by making the substitutions q → q + p/2 and p → −p in Eq. (A10). We note that by taking the vacuum limit µ → −∞ where f (ξ F p ) → 0, Eq. (11) reproduces the ordinary STM equation of a three-body problem at 1/a = 0 [51] given by −R * κ(p) 2 + 4π which gives the ground-state trimer energy E V 3 = −0.01385/(mR 2 * ) in vacuum.