Many Fermi polarons at nonzero temperature
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Many Fermi polarons at nonzero temperature
Hiroyuki Tajima and Shun Uchino Quantum Hadron Labolatory, RIKEN Nishina Center, Wako, Saitama 351-0198,Japan Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo,169-8050, JapanE-mail: [email protected]
May 2018
Abstract.
An extremely polarized mixture of an ultracold Fermi gas is expectedto reduce to a Fermi polaron system, which consists of a single impurity immersedin the Fermi sea of majority atoms. By developing a many-body T -matrix theory,we investigate spectral properties of the polarized mixture in experimentally relevantregimes in which the system of finite impurity concentration at nonzero temperatureis concerned. We explicitly demonstrate presence of polaron physics in the polarizedlimit and discuss effects of many polarons in an intermediate regime in a self-consistentmanner. By analyzing the spectral function at finite impurity concentration, we extractthe attractive and repulsive polaron energies. We find that a renormalization ofmajority atoms via an interaction with minority atoms and a thermal depletion ofthe impurity chemical potential are of significance to depict the many-polaron regime.
1. Introduction
Understanding effects of impurities immersed in an environment is one of the key issuesin physics. In nuclear physics, heavy hadrons in nuclear matter such as charm hadronsare now discussed in context of impurity problems [1]. In condensed matter physics,a number of impurities problems have been examined for a long time, depending onconditions of impurities such as mobile or immobile and presence or absence of a spin-exchange interaction [2]. A particularly fundamental class of the problems is the polaronin which a mobile impurity interacts with an environment [3, 4]. The concept ofthe polaron appears in a variety of the materials such as metal, semiconductor, andsuperconductor systems [5, 6].Currently, there is a growing interest in an ultracold atomic gas as a quantumsimulator of polaron physics [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The Feshbachresonance available in an ultracold atomic gas allows us to control an interactionbetween impurity and bath and to investigate the strong coupling regime, which isgenerally challenging in quantum many-body physics [18]. In addition, by using radio-frequency (rf) spectroscopy, we can address spectral properties of the systems including any Fermi polarons at nonzero temperature T -matrix approximation [32, 33, 34, 35], functionalrenormalization [36, 37], and diagrammatic Monte Carlo [38, 39, 40, 41, 42] aresuccessfully applied. In the case of finite polarization, the polaron-polaron interaction isdiscussed [43, 44, 45, 46, 47]. In reality, however, none of these theoretical assumptionsare exactly satisfied in corresponding experiments; the temperature is about fromcentesimal to few tenths of the Fermi temperature [48] and impurity concentration is ofthe order of 10 percent. Thus, it is important to directly analyze such regimes in termsof many-body calculations accessible to the strong coupling regime.In this paper, we examine spectral properties in the polarized mixture of anultracold Fermi gas with a many-body T -matrix theory, which allows us directly to plugin the finite temperature and the impurity concentration effects. We demonstrate thatby shifting impurity concentration, the spectral function of impurities shows crossoverbehaviors from a single polaron to many polarons. By analyzing the spectral functionin detail, we extract the polaron energy as a function of impurity concentration. Wepoint out that a renormalization of majority atoms due to minority atoms plays acrucial role in understanding the system at a finite density, which has been overlookedin previous studies. In addition, we show that the impurity chemical potential is largelyaffected by finite temperature effects compared to other quantities. We also predicta quasiparticle-like peak in a high-energy regime of the spectral function of majorityatoms, which cannot be captured with single-impurity theories and may be measuredwith rf spectroscopy.
2. Formulation
We consider the grand canonical Hamiltonian for the two-component Fermi mixtureinteracting through the broad Feshbach resonance [18] (we set ~ = k B = 1), H = X k ,σ ξ p ,σ c † p ,σ c p ,σ + g X p , q , k c † p , ↑ c † q , ↓ c q + k , ↓ c p − k , ↑ , (1)where c p ,σ represents the fermionic annihilation operator with momentum p andpseudospin σ = ↑ , ↓ . ξ p ,σ = p m − µ σ is the kinetic energy of atoms with mass m measuredfrom the chemical potential µ σ . The interatomic interaction is local and the couplingconstant g ( <
0) can be characterized with the s -wave scattering length a s [18]. Notice any Fermi polarons at nonzero temperature ↑ ( ↓ ) is the majority (minority) spin.We wish to examine the spectral function directly related to rf spectroscopyexperiments, which is defined as A σ ( p , ω ) = − π Im G σ ( p , iω n → ω + iδ ) , (2)where the one-particle thermal Green’s function is given by G σ ( p , iω n ) = 1 iω n − ξ p ,σ − Σ σ ( p , iω n ) , (3)with the self-energy Σ σ ( p , iω n ). Here ω n = (2 n + 1) πT is the fermionic Matsubarafrequency ( T is the temperature) and δ is an infinitesimally small number. We notethat the analytic continuation in Eq. (2) is numerically done by the Pad´e approximationwith δ = 10 − ε F where ε F is the Fermi energy of majority atoms (see also AppendixA). From the definitions above, it follows that the problem reduces to obtaining theself-energy that contains bare essentials of the strongly interacting Fermi mixture.To obtain the polaron energy ω qp ∈ R , we determine the pole ω pole ∈ C of G ↓ ( p , ω + iδ ) by solving a self-consistent equation ω pole = Σ ↓ ( p = 0 , ω pole + iδ ) − µ ↓ . (4)In general, ω pole locates on the complex plane of ω , and especially in the case of repulsivepolaron near the unitarity in which the s -wave scattering length diverges, the imaginarypart of the self-energy is non-negligible. Therefore, we rewrite Eq. (4) as ω pole + µ ↓ = ω qp − i Γ , (5)with the decay rate Γ ∈ R . Here, ω qp and Γ are related to the self-energy as ω qp = ReΣ ↓ ( p = 0 , ω qp − µ ↓ − i Γ + iδ ) , (6)Γ = − ImΣ ↓ ( p = 0 , ω qp − µ ↓ − i Γ + iδ ) . (7)By solving the above two equations, we can obtain ω qp and Γ, respectively.In addition, the chemical potential µ σ is obtained from the so-called numberequation n σ ( µ σ ) = T X p ,iω n G σ ( p , iω n ) , (8)where n σ represents the particle density of atoms with the state σ . In this work, wedefine the impurity concentration y as y = n ↓ / ( n ↑ + n ↓ ).To obtain a reasonable self-energy, we use many-body T -matrix theories, whichare known to reproduce fundamental properties in spin-balanced [49, 50, 51, 52] andpolaron limits [32, 33, 34, 35]. The simplest type of the T -matrix theories is thenon-selfconsistent approximation whose self-energy is composed of the bare Green’sfunction. However, such an approximation does not contain an interaction betweenimpurities, which is inevitable to discuss the finite impurity concentration case. Toovercome the drawback of the non-selfconsistent approximation, we adopt an extended any Fermi polarons at nonzero temperature Σ ↓ = t tt G ↑ G ↑ G ↑ ↓ ↓ ↓ ↓↓ ↓ (a) (b) Figure 1. (a) Diagrammatic expression for the ETMA self-energy of impurities Σ ↓ ,where the non-selfconsistent T -matrix approximation is recovered if the dressed Green’sfunction of medium G ↑ (double solid line) is replaced by the non-interacting one.This self-energy includes the induced polaron-polaron interaction diagrammaticallydescribed by the process (b). Shaded circle represents the many-body T -matrix t . T -matrix approximation (ETMA) [53, 54, 55, 56, 57], which contains the interactionbetween impurities (see Fig. 1) and therefore meets the purpose of the paper. In thisformalism, as diagrammatically shown in Fig. 1(a), the self-energy Σ σ ( p , iω n ) is givenby Σ σ ( p , iω n ) = T X q ,iν n t ( q , iν n ) G − σ ( q − p , iν n − iω n ) , (9)where t ( q , iν n ) = g gχ ( q , iν n ) , (10)is the many-body T -matrix ( ν n = 2 nπT is the bosonic Matsubara frequency). In Eq.(10), the lowest-order-pair-correlation function χ ( q , iν n ) is given by χ ( q , iν n ) = T X p ,iω j G ↑ ( p + q , iω j + iν n ) G ↓ ( − p , − iω n )= X p − f ( ξ p + q , ↑ ) − f ( ξ − p , ↓ ) ξ p + q , ↑ + ξ − p , ↓ − iν n (11)where f ( x ) = 1 / ( e x/T + 1) is the Fermi distribution function. In Eq. (11), G σ ( p , iω j ) =1 / ( iω j − ξ p ,σ ) is the bare Green’s function. Physically, t ( q , iν n ) describes superfluidfluctuations in the particle-particle channel [51]. Since the dressed Green’s function G ↑ in Eq. (9) [or Fig. 1(a)] involves the self-energy Σ ↑ , the polaron-polaron interactionprocess described by Fig. 1 (b) is automatically included in the self-energy of minorityatoms Σ ↓ . We note that Σ σ ( p , iω n ) is numerically obtained by self-consistently solvingEq. (9) with calculating µ σ from Eq. (8), as shown in Fig. A1.Recently, it was shown that the ETMA well reproduces thermodynamic propertiesin spin-balanced systems [58, 59]. In what follows, we demonstrate that the ETMAalso provides reasonable results on spectral properties in the polarized system such asthe polarons. In this work, we focus on the relevant parameter regimes to the recentexperiments. After discussing the comparison between our results and the previous any Fermi polarons at nonzero temperature -3-2-10120 0.2 0.4 0.6 0.8 1 1.2 Attractive polaron ω qp / F Repulsive polaron -0.500.51 0 0.25 0.5 0.75 1 T / T F m / m * m / m * ( k F a s ) -1 ( k F a s ) -1 =0.6 ( k F a s ) -1 Figure 2. (Left panel) Interaction dependence of the polaron energy near the zeroimpurity density limit ( y = n ↓ / ( n ↑ + n ↓ ) . − ) at T = 0 . T F . Solid lines showattractive (lower) and repulsive (upper) polaron energies calculated by the ETMA.The dots represent the experimental results in Li Fermi gases [16]. (Right panel)The effective mass of repulsive polarons m ∗ near the zero impurity limit. The solidline shows our result with the ETMA. The dashed line is the result in the previouswork [11]. The black dots are observed effective masses in Ref. [16]. The inset showsthe calculated temperature dependence of m ∗ at ( k F a s ) − = 0 . works of experiments as well as theories at the low temperature and impurity densityregime, we clarify effects of finite temperature and impurity density.
3. Result
We first show how our many-body T -matrix theory works well even in the zero-impuritydensity limit at the low temperature through a comparison between our numericalresults and the recent experimental measurements [16] as well as previous theoreticalstudies. In our formalism, the zero-impurity density limit is achieved by puttingthe large chemical potential difference µ ↑ − µ ↓ such that the impurity concentration y = n ↑ / ( n ↑ + n ↓ ) . − is enough small. The left panel of Fig. 2 shows the attractiveor repulsive polaron energy ω qp as a function of inverse scattering length ( k F a s ) − withthe Fermi momentum of majority atoms k F . In our calculation, the temperature is fixedat T = 0 . T F (where T F is the Fermi temperature of majority atoms). Our resultsshow good agreements with recent experimental results in Li Fermi gases [16]. Wenote that while the experiment [16] has been done at a bit higher impurity density andhigher temperature compared with our theoretical input, the differences do not leadto significant consequences on the polaron energy 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 tonon-interacting one G ↑ ( p , iω n ) = 1 / ( iω n − ξ p ,σ ) [47] in the zero impurity density any Fermi polarons at nonzero temperature Z ( k F a s ) -1 Attractive polaronRepulsive polaron
ETMAΓ PF Γ PP k F a s ) -1 Γ / ε F Figure 3.
The left panel shows residue Z of each polaron calculated by the ETMA(solid line) in the zero-impurity limit at T = 0 . T F and the functional renormalizationgroup (FRG) in Ref. [36] (dashed line). The right panel is the interaction dependenceof the decay rate of repulsive polarons Γ at T = 0 . T F . In this figure, the black dotsare experimental results [16]. Γ PF and Γ PP are the decay rate at T = 0 of polaron-to-bare-atom and polaron-to-polaron processes, respectively [11]. limit. Thus, our approach based on the ETMA turns out to be a natural extension ofthe non-selfconsistent T -matrix approximation with a single impurity to discuss finitetemperature and density in the Fermi polaron system.Our result of the effective mass m ∗ subtracted from G ↓ ( p , ω + iδ ) near the singleimpurity limit is shown in the right panel of Fig. 2 and is consistent with the previouswork [11]. The small difference between the previous and our works comes from thefinite temperature effects as shown in the inset of the right panel of Fig. 2. It is quitenatural that m ∗ decreases with increasing the temperature since the temperature effectsgradually suppress the interaction effects. This is the reason why our calculated m ∗ at T = 0 . T F is larger than that of the previous work obtained at T = 0. On the otherhand, the experimental results [16] show heavier effective masses than our evaluation inspite of the fact that the experimental temperature T = 0 . T F is higher than our case.We also numerically checked that the effect of a finite-impurity density does not leadsuch significant difference. The large mass renormalization in the recent experiment [16]cannot be explained by finite temperature or impurity density effect by means of theETMA.The left panel of Fig. 3 shows the residue Z of minority Green’s function at ω = ω pole , which is calculated as [36] Z − = − ∂∂ω G − ↓ ( p = 0 , ω + iδ ) (cid:12)(cid:12)(cid:12)(cid:12) ω = ω pole . (12)Our results of Z for attractive and repulsive polarons show good agreement with thetheoretical study based on the functional renormalization group at T = 0 [36], whichnon-perturbatively involves higher order corrections such as three-body process. From any Fermi polarons at nonzero temperature y y -1.5-1-0.500.5 0 0.1 0.2 0.3 ↑ / ε F (cid:1) ↓ / ε F = (cid:0)(k F a s ) -1 (cid:2) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:8)(cid:9)(cid:10) Figure 4.
Calculated chemical potential of majority atoms as a function of impurityconcentration y at T = 0 . T F . Inset shows results of minority atoms. In each figure,we use the same line-style at each impurity concentration.The circle represents theattractive polaron energy ω aqp at the single-impurity limit. this comparison, one can find that the reside Z is essentially described well by theladder-approximation scheme at the single-impurity limit.However, the decay rate of repulsive polarons Γ obtained from Eqs. (6) and (7) isgenerally smaller compared to FRG results [36] since our calculation does not incorporatethe effect of three-body decay associated with atom-dimer scatterings [60] as well as thedecay to attractive polarons, which can be considered by replacing G ↓ in χ ( q , iν n ) withdressed one G ↓ [11]. Since G ↓ is concerned, the ETMA may reproduce the decay rate ofpolaron-to-bare-atom transition Γ PF rather than that of polaron-to-polaron transitionΓ PP calculated in the previous work at the single-impurity limit with exactly T = 0 [11].Although the correct physical process may be the latter, the former is closer to theexperimental result. In addition, our result involves finite temperature effects whichenhance the decay of the quasi-particles [47], which is visible in the weak repulsiveinteracting regime where the collisional effects are relatively small.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 inFig. 4, µ ↑ deviates from the Fermi energy and decreases with increasing y due to theself-energy shift ReΣ ↑ ( p , ω + iδ ) associated with the strong pairing interaction. Thisrenormalization effect on majority atoms becomes more remarkable when the pairinginteraction gets stronger. Furthermore, the shifts of µ σ are not explained by the simplemean-field shift Σ MF σ = πa s m n − σ , since the scattering length a s diverges near the unitaritylimit. However, the shift of µ ↑ is proportional to n ↓ even at ( k F a s ) − = 0 at small y .By using the linear fitting with respect to n ↓ /n ↑ in the small impurity-density regime( y < . µ ↑ = ε F (cid:20) − . n ↓ n ↑ (cid:21) ≡ ε F (cid:20) − . y − y (cid:21) . (13) any Fermi polarons at nonzero temperature a = 1 /k F given byΣ MF ( a = 1 /k F ) = 4 πmk F n ↓ ≃ . n ↓ n ↑ ε F . (14)Since the chemical potential plays a crucial role in the thermodynamics of a unitaryFermi gas in which µ ↑ /ε F in the unpolarized case takes a universal constant calledBertsch parameter [62], we expect that the origin of pre-factor 0 .
526 in the second termof the right hand side of Eq. (13) would be important in terms of the thermodynamics ofthe many polarons. We emphasize that these renormalization effects cannot be capturedwith single-impurity theories. The renormalization is of the order of a tenth of theFermi energy in the typical cold-atom experiments whose impurity concentration is 0 . .
3. We expect that such a significant shift can be measured with the state-of-the-artprecision thermodynamic measurement [59].The inset of Fig. 4 shows the impurity chemical potential µ ↓ , which monotonicallyincreases with increasing y and decreases with increasing the interaction strength. Atthe zero temperature, µ ↓ is equivalent to the attractive polaron energy ω aqp at y → µ ↓ = E N ↓ =1 − E N ↓ =0 is defined as the energy needed to add an impurity withzero momentum to the system where E N ↓ ( N ↓ ∈ Z ) is the energy in the presence of N ↓ impurities. Indeed, this definition is equivalent to µ ↓ = (cid:16) ∂E∂n ↓ (cid:17) S at the thermodynamiclimit, where E and S are the internal energy and entropy, respectively. At a finitetemperature, however, we have to carefully notice the difference between µ ↓ and ω aqp .An important point is that at a finite temperature there is the contribution from thermalexcited states with nonzero momenta in addition to one from the ground state with thezero momentum. Figure 5 (a) shows the impurity chemical potential µ ↓ and ω aqp of theunitarity limit as a function of y at several temperatures. In general, µ ↓ is smallerthan ω aqp in the small impurity density region ( y ≃ µ ↓ decreaseswith increasing the temperature, whereas ω aqp slightly shifts due to the temperatureeffects. Except for the strong-coupling regime beyond polaron-molecule (or polaron-BEC) transition, the number equation of impurities for µ ↓ can approximately be givenby n ↓ ≃ X p Z a f (cid:18) p m ∗ a − µ ↓ + ω aqp (cid:19) , (15)where Z a and m ∗ a are the residue and effective mass of an attractive polaron, respectively.For simplicity, we neglect the decay rate of an attractive polaron as well as the repulsivebranch. At T = 0, the solution of Eq. (15) for the low impurity density limit ( n ↓ →
0) isapparently µ ↓ = ω aqp since the Fermi distribution function f ( x ) becomes a step function θ ( − x ). On the other hand, at finite temperature, such solution have to be µ ↓ → −∞ because the summation over momenta in Eq. (15) involves the contribution from highmomentum region associated with the finite temperature. This large negative µ ↓ reflectsthe fact that a few polarons at finite temperature behave as a classical Boltzmannensemble. Indeed, if one measures the temperature by using the Fermi temperature of any Fermi polarons at nonzero temperature -3-2-10 μ ↓ ε F ω a q(cid:11) / ε F T (cid:12)(cid:13)(cid:14) T F T (cid:15) (cid:16)(cid:17)(cid:18) T F T (cid:19) (cid:20)(cid:21)(cid:22) T F ( k F a s ) -1 y μ (cid:28) / ε F T =0.03 T F T =0.10 T F T =0.20 T F (b) Figure 5.
Impurity concentration dependence of (a) µ ↓ (solid line) and ω aqp (dashedline) at T = 0 . T F , 0 . T F and 0 . T F , and (b) µ ↑ at T = 0 . T F , 0 . T F and 0 . T F .In each figure, the interaction strength is set at ( k F a s ) − = 0. impurities T F , ↓ , one can obtain TT F , ↓ = (cid:18) n ↑ n ↓ (cid:19) TT F , (16)which diverges in the limit of n ↓ → T /T F . In contrast, the region where µ ↓ > ω aqp at the large impurity density can be regarded as the Fermi degenerate regimeof attractive polarons. In this case, they make a soft Fermi surface with the effectiveFermi energy ε pF = µ ↓ − ω aqp . To access such a regime, the temperature must be muchsmaller than T F , ↓ = ( n ↓ /n ↑ ) T F . In Fig. 6, we summarize the different regimes in theFermi polaron system. We also note that the curves shown in Fig. 6 are shifted belowif the effective mass is considered, since T F , ↓ is generally in inverse proportion to theeffective mass.We note that in contrast to µ ↓ , the spectral property of the attractive polaronat the single-impurity limit is relatively robust against the finite temperature effects,since it is related to the thermodynamic property of majority atoms. At n ↓ → G ↑ ( p , ω + iδ ) ≃ δ ( ω − ξ p , ↑ ), the self-energy of impurities after the analytic continuation any Fermi polarons at nonzero temperature (cid:29)(cid:30)(cid:31) T / T F y T T F ↓ T T F ↓ T T F ↓ Figure 6.
Different regimes in the Fermi polaron system obtained from Eq. (16). Theregion above the red curve is approximated as a classical Boltzmann gas. On the otherhand, the region below the blue curve is described with a theory at T = 0. In between,there exists a soft Fermi surface in which the finite temperature effect is significant. is given by Σ ↓ ( p , ω + iδ ) = X q Z ∞−∞ dν A t ( q , ν ) b ( ν ) + f ( ξ q − p , ↑ ) ω + iδ + ξ q − p , ↑ − ν (17)where A t ( q , ν ) = − π Im t ( q , iν n → ν + iδ ) is the spectral function of a diatomic pair and b ( x ) = 1 / ( e x/T −
1) is the Bose distribution function. The finite temperature effects inEq. (17) originate from mainly f ( ξ q − p , ↑ ) and µ ↑ ≃ ε F (cid:20) − π (cid:16) TT F (cid:17) (cid:21) [4] [see Fig. 5(b)] far away from the BEC critical point of molecules. In this way, one can find thatspectral polaron properties such as ω qp determined by Eq. (4) is deeply related to howmajority fermions are affected by the temperature. We also note that the large negative µ ↓ does not notably affect Σ ↓ ( p = 0 , ω + iδ ) since µ ↓ in Eq. (17) is included in only themolecular branch A t ( q , ν ).A renormalization of majority atoms is also visible in the spectral function A ↑ ( p =0 , ω ). In Fig. 7, we show the spectral function at y = 4 × − , .
18 and 0 .
26 at( k F a s ) − = 0 .
2. It turns out that the stable pole position shifts toward the lower energywith increasing y due to the shift of µ ↑ . From Eq. (3), the shift of the peak in Fig.7 is directly related to the change of the self-energy of majority atoms as given byReΣ ↑ ( 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 appearsat finite impurity concentration even in the spectral function of majority atoms. Thepresence of such a peak originates from the upper peak of the minority Green’s functionthat is explicitly contained in the self-energy of majority atoms. We also confirm thatthe 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 any Fermi polarons at nonzero temperature !" $%& ( ω μ ↑ )/ ε F A ↑ ( p ) ε F ( ’+ μ ↓ )/ ε F A ↓ ( p )*,- ) ε F y . /12345 y y ; <>?@ Figure 7.
Spectral function A ↑ ( p = 0 , ω ) of majority atoms at ( k F a s ) − = 0 .
2. withdifferent impurity concentrations. Inset shows that of minority atoms. The solid,dot-dashed, and dashed lines represent the result of y = 4 × − , 0 .
18, and 0 . T = 0 . T F . In each figure, we use the sameline style in each impurity concentration. is comparable to that in minority spectral function, its experimental validation with rfspectroscopy is promising.On the other hand, in contrast to majority atoms, the shift of the spectral function A ↓ ( p = 0 , ω ) of minority atoms by the finite density is small as shown in the insetof Fig. 7. In Fig. 8 (a), we show impurity concentration dependence of the attractivepolaron energy ω aqp obtained from Eq. (4) at several interaction strength. We findthat ω aqp is almost independent of y from the weak coupling region to unitary region.However, in the strong coupling region [( k F a s ) − = 0 . y . We argue that this indicates the presenceof the polaron-polaron interaction, which is indeed known to be positive by meansof the Fermi liquid theory [43, 44, 47]. One can interpret that the polaron-polaroninteraction effect is visible due to the increase of pairing interaction that overcomes thefinite temperature effect. Indeed, the increase of ω aqp at ( k F a s ) − = 0 . y ≃ .
2, where
T /T F , ↓ . . µ ↓ isvery important even from such viewpoint for the polaron-polaron interaction.In Fig. 8 (b), we show the calculated repulsive polaron energy ω rqp as a function of y in the strong coupling region [( k F a s ) − = 0 . , . . y -dependence of attractive and repulsive polaronenergies at ( k F a s ) − = 0 .
8, where we set an offset (= 2 . ε F ) on the attractive polaronenergy. These results indicate that the repulsive polaron energy does not representany noteworthy behavior related to the polaron-polaron interaction, which is consistentwith the recent experiment [16]. While solely from our numerical data it is difficult to any Fermi polarons at nonzero temperature -1-0.8 ABCDEFGH -0.200.2
IJKLMN
OPQRST
UVWXYZ[\]^_‘bcdefgh ( k F a s ) -1 k F a s ) -1 p ) y ω / ε F ω / ε F ω r ij / ε F y (b) Figure 8.
Impurity concentration dependence of (a) attractive and (b) repulsivepolaron energies at T = 0 . T F . In the panel (b), the dashed line shows the repulsivepolaron energy ω rqp (¯ p ) with the initial state momentum ¯ p at ( k F a s ) − = 0 .
4. The insetof (b) shows a comparison between the attractive (solid line) and repulsive (dotted line)polaron energies at ( k F a s ) − = 0 .
8, where we set an offset (= 2 . ε F ) on the attractivepolaron energy. pinpoint the reason of the difference from the prediction of the Fermi liquid theory, thefollowings could be conceivable: (i) smallness of the polaron-polaron interaction due tothe Pauli blocking, (ii) short lifetime of the repulsive polaron (typically of the order ofthe Fermi time), (iii) finite temperature effect as is the case with attractive polaron.We note that we stop the calculations of ω rqp at the superfluid instability point,which can be identified by the so-called Thouless criterion [63],[ t ( q = 0 , iν n = 0)] − = 0 . (18)At the fixed temperature, the Thouless criterion is more likely to be satisfied in theregime ( k F a s ) − &
0, where the transition temperature of the superfluid is higher andincreases with increasing y . To correctly describe the superfluid phase transition in astrongly interacting spin-imbalanced Fermi gas, we have to consider the existence of thefirst order phase transition and the phase separation [64, 65]. In this paper, we avoidsuch a regime by focusing on lower impurity concentration. We also note that although any Fermi polarons at nonzero temperature lmn = 0.10 T F T = 0.03 T F = 0.20 T F y / F F Figure 9.
Estimated thermal average of the impurity energy ¯ ε in the initial state.The dotted line indicate the internal energy density of an ideal Fermi gas at T = 0. the realization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [66, 67] has beenpredicted in a uniform polarized Fermi gas [64], such an exotic superfluid state is knownto be unstable against superfluid fluctuations [68, 69] (note however Ref. [70]).Furthermore, to address the more detailed experimental situation, we consider theeffect of initial-state momentum of impurities in the rf spectrum measurement [16]. Wefirst estimate the averaged momentum of impurities ¯ p by assuming that the initial stateis a non-interacting uniform Fermi gas. The thermal average of the impurity energy ¯ ε is defined as ¯ ε = 1 n ↓ X p p m f (cid:18) p m − µ i (cid:19) , (19)where µ i is the chemical potential of the initial state impurities, obtained by the solving n ↓ = X p f (cid:18) p m − µ i (cid:19) . (20)From the above equations, we can obtain ¯ p = √ m ¯ ε . Figure 9 shows the impurityconcentration dependence of ¯ ε . In the relevant region of the experimental impuritydensity (0 . . y . .
3) and temperature ( T ≃ . T F ), it is quite small compared to thetrapped case reported in the Supplemental Material of Ref. [16]. In the presence of ¯ p ,the repulsive polaron energy is obtained from ω rqp (¯ p ) − i Γ = Σ ↓ ( ¯p , ω rqp (¯ p ) − µ ↓ − i Γ + iδ ) . (21)The dashed line in Fig. (8) shows calculated ω rqp (¯ p ) at ( k F a s ) − = 0 .
4. As expected,the fiinite ¯ p leads to the negative shift of ω rqp (¯ p ) compared to ω rqp (¯ p = 0). In theexperimental paper, it is estimated that this negative shift is given by − (cid:0) − mm ∗ (cid:1) ¯ ε with ¯ ε = O (10 − ε F ) [16]. However, in our case, ¯ ε is smaller than 10 − ε F in the relevantregion, and the estimated shift is also smaller than O (10 − ε F ). This result indicates any Fermi polarons at nonzero temperature y -dependence of polaron energies. Since the harmonic trap enhance thefinite temperature effects due to the inhomogeneous density profile [57], it may also berelated to the suppression of effects of polaron-polaron interaction in the experiment.
4. Conclusion
We have theoretically investigated Fermi polarons at finite impurity concentration andfinite temperature within the framework of the many-body T -matrix theory, which canalso describe polaron properties in the zero impurity density and zero temperature limits.Our results show quantitative or semi-quantitative agreement with current experimentsas well as previous works based on single polaron theories at zero temperature.We have pointed out that majority atoms are affected by the strong pairinginteraction with impurities. In particular, we have showed the renormalization effectson the chemical potential as well as quasi-particle spectral function of majority atoms.In the case of minority atoms, the finite temperature effects play a crucial role intheir thermodynamic properties such as chemical potential. It is also related to thequantum degeneracy of attractive polarons, which leads to the competition betweenfinite temperature effects and the polaron-polaron interaction. The renormalizationof the majority chemical potential and the thermal depletion of minority chemicalpotential can be observed by recent precise thermodynamic measurements. In addition,we have predicted the appearance of the metastable peak in the high-energy region ofmajority spectral function. A detailed study on such a metastable many-body state isan interesting future work. Also, metastable peak structure in the spectral function ofmajority atoms can be detected by rf spectrum measurements.We have also extracted the polaron energy as a function of impurity concentrationto discuss the polaron-polaron interaction. We have found that in the strong couplingregion at a low temperature, although the polaron-polaron interaction is visible in thelower branch, this effect is much weaker in the upper branch. In addition, we also haveclarified that the mass-renormalization effect on the polaron energy in the uniform caseis smaller compared to the case of trapped gas clouds, by considering the initial-statemomentum of impurities.In this paper, we have emphasized that these many-body effects in the polaronproblem at finite temperature and finite impurity density are beyond previous singleimpurity theories. While our result successfully reproduces experimental results inseveral regimes and predict the polaron properties which no one has reported, we foundthat there are still differences between theories and experiments with respect to theeffective mass as well as the decay rate somehow beyond finite temperature and impuritydensity effects, which remain as our important future problem. In particular, an effectof a harmonic trap is important to compare our results with the observed rf-spectra [16]in detail, and our present work can include such effects by employing the local densityapproximation [57]. It is also interesting to extend our analyses to mass-imbalanced [10] any Fermi polarons at nonzero temperature -0.8 opst -0.4-0.20 0 5 10 15 20 25 30 Σ ↓ ( p i ω n ) / ε F uv Σ wxyz{|}~ Σ (cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134) Σ (cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146) Σ (cid:147)(cid:148)(cid:149)(cid:150)(cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156) '' ω n / ε F Figure A1.
Calculated impurity self-energy Σ ↓ ( p = 0 , iω n ) at T = 0 . T F , y = 0 . k F a s ) − = 0 and the comparison with the interpolated result obtained by the Pad´eapproximation ,where original is the self-energy from Eq. (9) without the interpolation. and two-dimensional systems [12] already realized in ultracold Fermi gases. Acknowledgments
We thank F. Scazza, A. Recati, S. Giorgini, M. Horikoshi and Y. Nishida for usefuldiscussions. H. T. is supported by a Grant-in-Aid for JSPS fellows (No. 17J03975).S. U. is supported by JSPS KAKENHI Grant Number JP17K14366. This work waspartially supported by RIKEN iTHEMS Program.
Appendix A. Analytic continuation
In general, the analytic continuation is sensitive to noises, and theoretical approacheswith statistical errors such as Monte-Carlo methods suffers from this procedure from theimaginary time τ to the real frequency ω [42, 71]. On the other hand, the ETMA used inthis work is free from statistical errors, and therefore we can implement the conventionalnumerical continuation methods. In this work, we adopt the Pad´e approximation toexamine the spectral structure in the Fermi polaron system.In our case, the self-energy has been already calculated in the complex energy planein terms of the Matsubara frequency located at imaginary energy axis. It is known thatthe Pad´e approximation is applicable to reproduce the pole structure in this plane [73].In fact, the photoemission spectra obtained from the many-body T -matrix theory withthe Pad´e approximation well reproduce the experimental result in a strongly interactingunpolarized Fermi gas [74]. In addition, the Pad´e approximation has been successfullyapplied to the Fermi polaron system with the functional renormalization group [36],which is also free from statistical errors. To double-check how the Pad´e approximation any Fermi polarons at nonzero temperature ↓ ( p = 0 , iω n ) (from n = 0 to n = 40 andfrom n = 90 to n = 100) and the interpolated results of them by means of the Pad´eapproximation, where the data between n = 40 and n = 90 are interpolated. One cansee that the Pad´e approximation smoothly interpolate the original ETMA self-energy. [1] Hosaka A, Hyodo T, Sudoh K, Yamaguchi Y, and Yasui S 2017 Progress in Particle and NuclearPhysics Solid State Physics (Thomson Press)[3] Landau L D 1933 Phys. Z. Sowjetunion Quantum Gas Experiments: Exploring Many-Body States , Vol. 3(World Scientic)[20] Jo G-B, Lee Y-R, Choi J-H, Christensen C A, Kim T H, Thywissen J H, Pritchard D E andKetterle W 2009 Science any Fermi polarons at nonzero temperature [33] Bruun G M and Massignan P 2010 Phys. Rev. Lett. , 224509[50] Perali A, Pieri P, Strinati G C and Castellani C 2002 Phys. Rev. B A550[67] Larkin A and Ovchinnikov Y 1964 Zh.Eksp.Teor.Fiz.95