Many Fermi polarons at nonzero temperature

An extremely polarized mixture of an ultracold Fermi gas is expected to reduce to a Fermi polaron system, which consists of a single impurity immersed in the Fermi sea of majority atoms. By developing a many-body T -matrix theory, we investigate spectral properties of the polarized mixture in experimentally relevant regimes in which the system of finite impurity concentration at nonzero temperature is concerned. We explicitly demonstrate presence of polaron physics in the polarized limit and discuss effects of many polarons in an intermediate regime in a self-consistent manner. By analyzing the spectral function at finite impurity concentration, we extract the attractive and repulsive polaron energies. We find that a renormalization of majority atoms via an interaction with minority atoms and a thermal depletion of the impurity chemical potential are of significance to depict the many-polaron regime.


I. INTRODUCTION
Understanding effects of impurities immersed in an environment is one of the key issues in physics. In nuclear physics, heavy hadrons in nuclear matter such as charm hadrons are now discussed in context of impurity problems 1 . In condensed matter physics, a number of classes on impurities problems have been examined for a long time, depending on conditions of impurities such as mobile or immobile and presence or absence of a spinexchange interaction 2 . A particularly fundamental class of the problems is the polaron in which a mobile impurity interacts with an environment 3,4 . The concept of the polaron appears in a variety of the materials such as metal, semiconductor, and superconductor systems 5,6 .
Currently, there is a growing interest in an ultracold atomic gas as a quantum simulator of polaron physics [7][8][9][10][11][12][13][14][15][16] . The Feshbach resonance available in an ultracold atomic gas allows us to control an interaction between impurity and bath and to investigate the strong coupling regime, which is generally challenging in quantum many-body physics 17 . In addition, by using radiofrequency (rf) spectroscopy, we can address spectral properties of the systems including excited branches 18 . For example, rf spectroscopy experiments confirmed existence of a repulsive polaron, which is a quasiparticle associated with a repulsive interaction and is a metastable excited many-body state [10][11][12][13]15 . The repulsive polaron also receives attention in terms of the realization of repulsive many-body states such as itinerant ferromagnetism [19][20][21] .
Interpretations of polaron experiments in ultracold atomic gases are grounded on the theoretical analyses in which the system with a single impurity at zero temperature is assumed. In the case of the Fermi polaron whose bath consists of fermions, due to such assumptions, theoretical treatments such as variational methods [22][23][24][25][26][27][28] , T -matrix approximation 29-32 , functional renormalization 33,34 , and diagrammatic Monte Carlo 35-39 are successfully applied. In reality, however, none of these theoretical assumptions are exactly satisfied in corresponding experiments; the temperature is about from centesimal to few tenths of the Fermi temperature and impurity concentration is of the order of 10 percent. Thus, it is important to directly analyze such regimes in terms of many-body calculations accessible to the strong coupling regime.
In this paper, we examine spectral properties in the polarized mixture of an ultracold Fermi gas with a manybody T -matrix theory to bridge the gap between singleimpurity theories and cold-atom experiments in polaron systems. We demonstrate that by shifting impurity concentration, the spectral function of impurities shows crossover behaviors from a single polaron to many polarons. By analyzing the spectral function in detail, we extract the polaron energy as a function of impurity concentration. We point out that a renormalization of majority atoms due to minority atoms plays a crucial role in understanding the system at a finite density, which has been overlooked in previous theoretical studies. Our analyses are consistent with the recent experiment by Scazza et al. 15 . We also predict a quasiparticle-like peak in a high-energy regime of the spectral function of majority atoms, which cannot be captured with single-impurity theories and may be measured with rf spectroscopy.

II. FORMULATION
We consider the grand canonical Hamiltonian for the two-component Fermi mixture interacting through the broad Feshbach resonance (we set = k B = 1), lated to rf spectroscopy experiments, which is defined as where the one-particle thermal Green's function is given by with the self-energy Σ σ (p, iω n ). Here ω n = (2n + 1)πT is the fermionic Matsubara frequency and δ is a infinitesimally small number. We note that the analytical continuation in Eq. (2) is numerically done by the Pàde approximation with δ = 10 −3 ε F (ε F is the Fermi energy of majority atoms). From the definitions above, it follows that the problem reduces to obtaining the self-energy that contains bare essentials of the strongly interacting Fermi mixture. The polaron energy ω qp is obtained by solving a self-consistent equation 40 To see a renormalization of majority atoms, we also calculate the chemical potential µ σ from the so-called number equation where n σ represents the particle density of atoms with the state σ.
To obtain a meaningful self-energy, we use many-body T -matrix theories, which are known to reproduce fundamental properties in spin-balanced 41-44 and polaron limits [29][30][31][32] . The simplest type of the T -matrix theories is the non-selfconsistent approximation whose self-energy is composed of the bare Green's function. However, such an approximation does not contain an interaction between impurities, which is inevitable to discuss the finite impurity concentration case. To overcome the drawback of the non-selfconsistent approximation, we adopt an extended T -matrix approximation (ETMA) [45][46][47][48][49] , which contains the interaction between impurities (please, see Fig. 1) and therefore meets the purpose of the paper. In this formalism, as diagrammatically shown in Fig. 1(a), the self-energy Σ σ (p, iω n ) is given by is the many-body T -matrix (ν n = 2nπT is the bosonic Matsubara frequency). In Eq. (7), the lowest-order-paircorrelation function χ(q, iν n ) is given by where f (x) = 1/(e x/T + 1) is the Fermi distribution function. Physically, t(q, iν n ) describes superfluid fluctuations in the particle-particle channel 43 . Since the dressed Green's function G ↑ in Eq. (6) [or Fig. 1(a)] involves the self-energy Σ ↑ , the polaron-polaron interaction process described by Fig. 1 (b) is automatically included in the self-energy of minority atoms Σ ↓ . We note that Σ σ (p, iω n ) is numerically obtained by self-consistently solving Eq. (6) with calculating µ σ from Eq. (5).
Recently, it was shown that the ETMA well reproduces thermodynamic properties in spin-balanced systems 50,51 .
In what follows, we demonstrate that the ETMA also provides reasonable results on spectral properties in the polarized system such as the polarons. In this paper, we focus on the low temperature regime which is relevant for recent experiments. We point out that calculations at the higher temperature with our formulation are numerically cheap, since convergence of the Matsubara sum at high temperature is fast. This is in sharp contrast to theoretical analyses based on a single impurity problem at T = 0, where it is difficult to estimate temperature effects in a systematic manner.

III. RESULT
We first show how our many-body T -matrix theory works well even in a low impurity density regime at the low temperature limit through a comparison between our numerical results and the recent experimental measurements 15 . Figure 2 shows the attractive or repulsive polaron energy ω qp as a function of inverse scattering length (k F a s ) −1 with the Fermi momentum of majority atoms k F . In our calculation, the temperature is fixed at T = 0.03T F (where T F is the Fermi temperature of majority atoms) and the impurity concentration is y = We note that while the experiment 15 has been done at a bit higher impurity density and higher temperature compared with our theoretical input, the differences do not lead to significant consequences as discussed below. In addition, in the zero impurity density limit, the ETMA reduces to the non-selfconsistent T -matrix approximation, which is known to describe polaron properties quantitatively, since the majority one-particle Green's function G ↑ (p, iω n ) in the ETMA reduces to non-interacting one G 0 ↑ (p, iω n ) = 1/(iω n − ξ p,σ ) 52 in the zero impurity density limit. Thus, our approach based on the ETMA is a natural extension of the non-selfconsistent T -matrix approximation with a single impurity to discuss finite temperature and density in the polaron system.
We next look at how impurity concentration y affects the chemical potential µ σ . We note that µ ↑ = ε F in the single impurity case at T = 0. However, as shown in Fig. 3, µ ↑ deviates from the Fermi energy and decreases with increasing y due to the self-energy shift ReΣ ↑ (p, ω + iδ) associated with the strong pairing interaction. This renormalization effect on majority atoms becomes more remarkable when the pairing interaction gets stronger. In addition, the chemical potential of minority atoms µ ↓ increases with y in each interaction strength.
We note that µ ↓ shows a discontinuity at y = 0 since µ ↓ → −∞ in the absence of impurities (y = 0), while µ ↓ is finite for more than or equal to a single impurity (y → 0) 24 . Furthermore, the shifts of µ σ are not explained by the simple mean-field shift Σ MF σ = 4πas m n −σ , since the scattering length a s diverges near the unitarity limit. We emphasize that these renormalization effects  cannot be captured with single-impurity theories. The renormalization is of the order of a tenth of the Fermi energy in the typical cold-atom experiments whose impurity concentration is 0.1 to 0.3. We expect that such a significant shift can be measured with the state-of-the-art precision thermodynamic measurement 51 . A renormalization of majority atoms is also visible in the spectral function A ↑ (p = 0, ω). In Fig. 4, we show the spectral function at y = 10 −3 , 0.18 and 0.26 at (k F a s ) −1 = 0.2. It turns out that the stable pole position shifts toward the lower energy with increasing y due to the shift of µ ↑ . From Eq. (3), the shift of the peak in Fig. 4 is directly related to the change of the self-energy of majority atoms as given by ω +µ ↑ = Σ ↑ (p = 0, ω +iδ). This is nothing but the renormalization effect of majority atoms. In addition, we find that a metastable peak associated with the upper branch appears at finite impurity concentration even in the spectral function of majority atoms. The presence of such a peak originates from the upper peak of the minority Green's function that is explicitly contained in the self-energy of majority atoms. We also confirm that the metastable-peak structure is enhanced in the vicinity of the strong coupling limit. By considering that the intensity of such an upper peak in the majority spectral function is comparable to that in minority spectral function, its experimental validation with rf spectroscopy is promising.
On the other hand, in contrast to majority atoms, the shift of the spectral function A ↓ (p = 0, ω) of minority atoms shown in inset of Fig. 4 is weak against impurity concentration. In Fig. 5 (a), we show impurity concentration dependence of the attractive polaron energy ω a qp obtained from Eq. (4). We find that ω a qp measured from −µ ↓ is almost independent of y from the weak coupling region to unitary region. However, in the strong coupling region [(k F a s ) −1 = 0.4 in Fig. 5(a)], the polaron energy turns to slightly increase with increasing y. We argue that this indicates the presence of the polaron-polaron interaction, which is indeed known to be positive by means of the Fermi liquid theory 52-54 .
In Fig. 5 (b), we show the calculated repulsive polaron energy ω r qp measured from −µ ↓ as a function of y in the strong coupling region [(k F a s ) −1 = 0.4, 0.8 and 1.2]. Although the repulsive polaron energy does not represent any noteworthy behavior related to the polaron-polaron interaction, the result itself is consistent with the recent experiment 15 . The inset of Fig. 5 (b) shows the comparison between y-dependence of attractive and repulsive polaron energies at (k F a s ) −1 = 0.8, where we set an offset (= 2.5ε F ) on the attractive polaron energy. Seemingly, this behavior may be surprising, since there is the polaron-polaron interaction in the attractive polarons and is the extreme region where strong pairing fluctuations appear due to divergent many-body T -matrix. However, our estimation shows that the decay time of the repulsive polaron in the strong coupling regime is of the order of the Fermi time [≃ O(1)ε −1 F ] in contrast to the stable attractive polaron. By considering that it takes about 2.7 collisions for ↑ and ↓ particles to thermalize 55,56 and the collision time in the strong coupling regime is again about the Fermi time, we interpret the behavior of the repulsive polaron as meaning that there is no enough time to have the polaron-polaron interaction in an equilibrium way.
We note that we stop the calculations of ω r qp at the superfluid instability point, which can be identified by the so-called Thouless criterion 57 , [t(q = 0, iν n = 0)] −1 = 0.
At the fixed temperature, the Thouless criterion is more likely to be satisfied in the regime (k F a s ) −1 0, where the transition temperature of the superfluid is higher and increases with increasing y. To correctly describe the superfluid phase transition in a strongly interacting spin-imbalanced Fermi gas, we have to consider the existence of the first order phase transition and the phase separation 58,59 . In this paper, we avoid such a regime by focusing on lower impurity concentration. We also note that although the realization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state 60,61 has been predicted in a uniform polarized Fermi gas 58 , such an exotic superfluid state is known to be unstable against superfluid fluctuations 62,63 . Figure 6 shows the temperature dependence of the attractive polaron energy ω a qp + µ ↓ (a) and the chemical potential of majority atoms µ ↑ (b) at the unitarity limit [(k F a s ) −1 = 0]. From this result, we find that the temperature dependence of polaron properties is not so strong in the very low temperature regime (T 0.1T F ). While µ ↑ shows a monotonic decrease due to the ther-  6. (a) The attractive polaron energy ω a qp + µ ↓ and (b) the chemical potential of majority atoms µ ↑ as a function of y at T = 0.03TF, 0.10TF, and 0.20TF at the unitarity limit (1/kFas = 0). mal depletion with increasing T , the attractive polaron energy at T = 0.20T F exhibits a slight suppression in the case of the large impurity concentration (y 0.2), which might be originated from a non-Fermi liquid behavior due to many-body effects with thermal fluctuations 52 .

IV. CONCLUSION
We have theoretically investigated Fermi polarons at finite impurity concentration and finite temperature within the framework of the many-body T -matrix theory, which can also describe polaron properties in the zero impurity density and zero temperature limits. We have pointed out that majority atoms are affected by the strong pairing interaction with impurities. In particular, we have showed the renormalization effects on the chemical potential as well as quasi-particle spectral function of majority atoms. In addition, we have predicted the appearance of the metastable quasiparticle spectrum in majority atoms. A detailed study on such a metastable many-body state is an interesting future work. The renormalization of the chemical potential and metastable peak structure in the spectral function of majority atoms can be observed by thermodynamic measurements and rf spectrum measurements, respectively.
We have also extracted the polaron energy as a function of impurity concentration to discuss the polaronpolaron interaction, which is consistent with the experimental observation 15 . We have found that in the strong coupling region [(k F a s ) −1 0.4], although the polaronpolaron interaction is visible in the lower branch, this effect is weaker in the upper branch, reflecting the difference between the decay times of attractive and repulsive polarons.
In this paper, we have emphasized that these manybody effects in the polaron problem at finite temperature and finite impurity density are beyond previous single impurity theories. It is also interesting to extend our analyses to mass-imbalanced 10 and two-dimensional systems 11 already realized in ultracold Fermi gases. The effects of a harmonic trap is also important and our present work can include such effects by employing the local density approximation 49 .