Goldstino Spectrum in an Ultracold Bose-Fermi Mixture with Explicitly Broken Supersymmetry

PHYSICAL REVIEW RESEARCH 3, 013035 (2021)

Goldstino spectrum in an ultracold Bose-Fermi mixture with explicitly broken supersymmetry

Hiroyuki Tajima ,1,2,* Yoshimasa Hidaka ,2,3 and Daisuke Satow4 1Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan 2RIKEN Nishina Center, Wako, Saitama 351-0198, Japan 3RIKEN iTHEMS, Wako, Saitama 351-0198, Japan 4Arithmer Inc., Minato-Ku, Tokyo 106-6040, Japan

(Received 23 January 2020; accepted 3 December 2020; published 12 January 2021)

We theoretically investigate a supersymmetric collective mode called the Goldstino in a Bose-Fermi mixture. The explicit supersymmetry breaking, which is unavoidable in cold-atom experiments, is considered. We derive the Gell-Mann–Oakes–Renner (GOR) relation for the Goldstino, which gives the relation between the energy gap at zero momentum and the explicit breaking term. We also numerically evaluate the gap of the Goldstino above the Bose-Einstein condensation temperature within the random phase approximation (RPA). While the gap obtained from the GOR relation coincides with that in the RPA for the mass-balanced system, there is a deviation from the GOR relation in the mass-imbalanced system. We point out that the deviation becomes large when the Goldstino pole is close to the branch point, although it is parametrically a higher order with respect to the mass-imbalanced parameter. To examine the existence of the Goldstino pole in realistic cold atomic systems, we show how the mass-imbalance effect appears in 6Li - 7Li, 40K - 41K,and173Yb - 174Yb mixtures. Furthermore, we analyze the Goldstino spectral weight in a 173Yb - 174Yb mixture with realistic interactions and show a clear peak due to the Goldstino pole. As a possibility to observe the Goldstino spectrum in cold-atom experiments, we discuss the effects of the Goldstino pole on fermionic single-particle excitation as well as the relationship between the GOR relation and Tan’s contact.

DOI: 10.1103/PhysRevResearch.3.013035

I. INTRODUCTION Recently, Bose-Fermi mixtures with a small mass imbalance between bosons and fermions such as 6Li - 7Li [17–19], Supersymmetry is symmetry with respect to an interchange 39K - 40K [20], 40K - 41K [21], 84Sr - 87Sr [22], 87Rb - 87Sr between bosons and fermions [1–3]. While the existence of [23], 161Dy - 162Dy [24], and 173Yb - 174Yb [25,26] have been supersymmetry is expected in the context of particle physics, experimentally realized. The boson-boson or boson-fermion evidence or any indications of it have not yet been observed interactions in some of the mixtures can be tuned due to in high-energy experiments [4]. However, apart from whether the magnetic Feshbach resonance [18,21,23,27–29]. In this or not supersymmetric partners such as squarks exist in our sense, examining supersymmetry in such cold atomic systems world, it is a really interesting problem to explore the conse- is promising. quences of supersymmetry using fermions and bosons, which A remarkable feature of supersymmetry in a Bose-Fermi are well established in condensed matter physics. mixture is the emergence of a Nambu-Goldstone (NG) mode An ultracold atomic gas is currently one of the most called the Goldstino [30–34]. While the usual NG mode useful systems for investigating quantum many-body phe- propagates as a bosonic mode, the Goldstino behaves as a nomena, due to the controllability of physical parameters such fermionic mode. Such a fermionic collective excitation has as interaction, density, temperature, and quantum statistical also been predicted in quantum electrodynamics as well as properties of atoms by use of isotopes [5–7]. In particular, the quantum chromodynamics [35–37]. Observation of this col- Feshbach resonance [8] enables us to investigate this atomic lective mode is really important for seeing the supersymmetric system from the weak-coupling to the strong-coupling limit properties in a Bose-Fermi mixture that are realized in a table- in a systematic manner. In this regard, the supersymmetric top experiment. Since the Goldstino is a fermionic collective properties of this system have been extensively discussed mode associated with broken supersymmetry, it becomes a theoretically [9–16]. gapless mode when the system possesses exact supersymmetry. However, explicit supersymmetry breaking such as the mass imbalance between fermions and bosons is unavoidable *[email protected] in cold-atom experiments. In this case, the Goldstino has a ﬁnite energy gap associated with the explicit breaking param- Published by the American Physical Society under the terms of the eters. If one can observe the gapped Goldstino and its spectral Creative Commons Attribution 4.0 International license. Further properties agree with the results of theoretical analysis, this distribution of this work must maintain attribution to the author(s) should be evidence for the existence of supersymmetry in and the published article’s title, journal citation, and DOI. these systems. Indeed, the ﬁrst example of NG bosons in

2643-1564/2021/3(1)/013035(14) 013035-1 Published by the American Physical Society TAJIMA, HIDAKA, AND SATOW PHYSICAL REVIEW RESEARCH 3, 013035 (2021) particle physics was pions, which are also gapped modes due ∇2 = 3 ψ† − − μ ψ to the explicitly broken chiral symmetry associated with the H d r b (r) b b(r) 2mb current quark mass [38]. ∇2 In this work, we theoretically examine the energy gap + 3 ψ† − − μ ψ d r f (r) f f (r) of the Goldstino in a Bose-Fermi mixture with explic- 2m f itly broken supersymmetry. We focus on a few candidates Ubb for nearly supersymmetric Bose-Fermi mixtures, namely, + d3rψ†(r)ψ†(r)ψ (r)ψ (r) 2 b b b b 6Li - 7Li, 40K - 41K, and 173Yb - 174Yb mixtures. We determine the thermodynamic properties of weakly interacting mixtures + U d3rψ†(r)ψ (r)ψ†(r)ψ (r), (1) within the Hartree-Fock mean-field approximation above the bf b b f f Bose-Einstein condensation (BEC) temperature. By devel- where ψ is the field operator of a boson (fermion) with oping a gap formula for the Goldstino, which corresponds b( f ) mass mb( f ) and chemical potential μb( f ). Ubb(bf) is the cou- to the Gell-Mann–Oakes–Renner (GOR) relation in quantum pling constant of a boson-boson (boson-fermion) interaction, chromodynamics [39], based on the memory function formal- which is assumed to be a contact type. These coupling con- ism [40], we show how the explicit supersymmetry-breaking stants are related to the scattering length abb(bf) as Ubb = terms affect the Goldstino gap in these systems. By comparing (4πa )/m and U = (2πa )/m , respectively, where m = it with the numerical results of the random phase approxi- bb b bf bf r r 1/(1/m f + 1/mb) is the reduced mass. In this paper, we mation (RPA), we clarify that the effects of the branch point measure the interaction strength by using the dimensionless are significant in the presence of the mass imbalance between parameters k a and k a , where k = (6π 2N )1/3 is a mo- fermions and bosons. Furthermore, we discuss how to observe b bb b bf b b mentum scale for the boson density Nb. In general, there is the Goldstino gap from the single-particle spectral function a non-s-wave fermion-fermion interaction such as the dipole- of a Fermi atom. While the previous work done by two of dipole interaction given by the authors is dedicated to the two-dimensional system [14] and the three-dimensional one in the BEC phase [15] with 1 V = d3rψ†(r)ψ†(r )U (r − r )ψ (r )ψ (r). (2) ideal situations such as supersymmetric interactions, in this ff 2 f f ff f f paper we discuss the three-dimensional system with realistic 6 physical parameters above the Bose-Einstein condensation Although it is negligible in several Fermi atoms such as Li and 40K far away from higher partial-wave Feshbach reso- temperature TBEC. Furthermore, we consider the case with a mass imbalance where the fermionic mass is slightly lighter nances like a p-wave resonance [41] at low temperature, the than the bosonic one. In this case, the effects of the branch scattering in these higher partial waves would be important at high enough temperatures [42]. Moreover, the dipole-dipole point are more important. We show that, even when the mass 161 162 imbalance is small, the peak of the Goldstino disappears and interaction would become significant in a Dy - Dy mix- it is buried in the continuum spectrum if the interaction is too ture with large magnetic dipole moments [24]. In this work, weak. We also note that there is a work which considered a we consider the case in which these interactions are negligible mass-imbalanced Bose-Fermi system [13] in which a Bose- for simplicity. We note that the intercomponent interaction Fermi mixture trapped on an optical lattice was considered, Ubf involves a factor 2, in contrast to the intracomponent = μ = μ = and the species of atoms were not specified. These points are interaction Ubb [43]. When m f mb, b f , and Ubb improved in this paper. Ubf, there is a supersymmetry corresponding to interchange ψ → ψ ψ → ψ This paper is organized as follows: In Sec. II, we introduce between bosons and fermions: b f and f b.The corresponding Noether charges are our model and the formulation for thermodynamic quantities and the Goldstino gap within the GOR and RPA. In Sec. III, = 3 , † = 3 † , we show and discuss our numerical results within the RPA Q d rq(r) Q d rq (r) (3) on the Goldstino gap in a few Bose-Fermi mixture systems. † Section IV is devoted to discussion of how the Goldstino pole which commute with the Hamiltonian, [H, Q] = [H, Q ] = 0. = ψ ψ† can affect the fermionic single-particle spectrum, in order to Here, q(r) f (r) b (r) is the local operator that creates the suggest the possibility of detecting the Goldstino in experi- boson and annihilates the fermion [10]. Unlike the supersym- ments. We summarize our studies in Sec. V. In Appendix A, metry in relativistic systems, the anticommutation relation we show the detailed derivation of the GOR relation based on between supercharges is not the Hamiltonian but the total the memory function formalism. We calculate the Goldstino particle number operator: spectral function in the free limit in Appendix B to check { , †}= 3 ψ† ψ + 3 ψ† ψ . the absence of numerical artifacts. In Appendix C, we discuss Q Q d r f (r) f (r) d r b (r) b(r) (4) the cutoff dependence of the fermionic single-particle spectral function. In this sense, the supersymmetry in a nonrelativistic Bose- Fermi mixture is a different type from that in relativistic theories. The order parameter of supersymmetry breaking { , † } = ψ† ψ + II. FORMALISM is the total number density, Q q (r) b (r) b(r) ψ† ψ f (r) f (r) , which is always broken in a finite-density A. Model system. For a spontaneous breaking of bosonic continuous We consider a nonrelativistic Bose-Fermi mixture de- symmetry, if the order parameter is expressed as the expec- scribed by the Hamiltonian tation value of the commutation relation between a charge

013035-2 GOLDSTINO SPECTRUM IN AN ULTRACOLD BOSE-FERMI … PHYSICAL REVIEW RESEARCH 3, 013035 (2021) and a charge density, the breaking pattern is called type B After Fourier transformation, we obtain [44–49]. On the other hand, if no such order parameter exists, ∞ the breaking pattern is called type A. The NG modes corre- R(p,ω) = i dt d3reiωt−ip·rθ(t ){q(r, t ), q†(0, 0)}. sponding to type B typically exhibit a quadratic dispersion. −∞ (10) A typical example of a type B NG mode is the magnon in R ,ω a ferromagnet, in which the order parameter is expressed as The energy gap is obtained from the pole of (p ) in the complex ω plane. Since we are interested in the the expectation value of the commutation relation between = spins. Replacing the commutator with the anticommuta- zero-momentum gap of Goldstino, hereafter we take p 0. The memory function formalism systematically decomposes tor, we can identify the supersymmetry breaking pattern as R ω type B. As in an ordinary symmetry breaking, the supersym- ( ) into the form metry breaking leads to a gapless excitation. If the excitation −N R(ω) = , (11) can be identified as a single-mode excitation, it is called ω + + i (ω) the Goldstino. In general, the excitation may be located at a =ψ† ψ +ψ† ψ branch point where a two- or multiparticle continuum starts. where N f (r) f (r) b (r) b(r) is the total number This is especially the case for the noninteracting system, density. (ω) and ={[H, Q], q†(0, 0)}/N are called the where there is no Goldstino. The excitation is the particle-hole dynamic and static parts of the memory function. We do not one. The interaction plays an important role in the existence show the explicit form of (ω); the important point is that of the Goldstino. In the following analysis, we assume the (ω) is parametrically higher oder compared with with existence of Goldstino excitation, and we numerically check it respect to the explicit breaking term (see Appendix A for in the RPA in Sec. III. Since the order parameter is expressed more details). Therefore, at the leading order of the explicit as the expectation value of the anticommutation relation of breaking term, the energy gap ωG is expressed as the supercharge and its density, the Goldstino belongs to the 1 type B mode [44–49], which typically has a quadratic disper- ω = ωGOR ≡− { , , † , }. G G [H Q] q (0 0) (12) sion. N In a realistic situation, the supersymmetry is explicitly We emphasize that this formula works for any supersymmetric broken because all parameters cannot be exactly tuned in ex- Hamiltonian with a small explicit breaking term and local periments. The effect of the explicit breaking can be expressed interactions because we have not employed the specific form as the commutation relation between the Hamiltonian and the of the Hamiltonian. The gap is linearly proportional to the supercharge, explicit breaking term, whose property can be understood as type B breaking [44–49]. In contrast, type A breaking predicts ∇2 [H, Q] = d3rψ†(r) χ + μ ψ (r) that the gap is proportional to the square root of the explicit b 2m f ω r breaking term. We note that although the dynamic part ( ) is higher order, it might not be small if there is a singularity in − 3 ψ† ψ† ψ ψ , U d r b (r) b (r) b(r) f (r) (5) (ω). As shown later, this is the case when the branch point ωGOR is close to G . where we define For the Hamiltonian, (1), that we employ in the present paper, using Eq. (5), we obtain μ ≡ μ − μ , f b (6) χ { , , † , }= {∇2ψ† }ψ + ψ† ∇2ψ 1 1 mb − m f [H Q] q (r 0) [ b (r) b(r) f (r) f (r)] χ ≡ − m = , (7) 2mr m m r m + m f b b f + μ ψ† ψ + ψ† ψ [ b (r) b(r) f (r) f (r)] U ≡ Ubf − Ubb. (8) † † − U[ψ (r)ψ (r)ψb(r)ψb(r) These explicit breakings cause a finite gap of the Goldstino, b b + ψ† ψ ψ† ψ . whose formula is shown in the next subsection. 2 b (r) b(r) f (r) f (r)] (13) Therefore, the Goldstino gap in the present model is given by B. Gell-Mann–Oakes–Renner relation χ ωGOR =−μ − {∇2ψ† }ψ + ψ† ∇2ψ Pions are the NG bosons associated with the spontaneous G [ b (r) b(r) f (r) f (r) ] 2mrN breaking of chiral symmetry in quantum chromodynamics. U The GOR formula relates the pion mass and the current quark + ψ† ψ† ψ ψ [ b (r) b (r) b(r) b(r) mass that explicitly breaks chiral symmetry [39]. We can gen- N + ψ† ψ ψ† ψ eralize the GOR relation to that of Goldstino in a Bose-Fermi 2 b (r) b(r) f (r) f (r) ] mixture. For this purpose, we employ the memory function 2U formalism [40], which is a different formalism from the one =−μ + χE+ V , (14) used in the original derivation [39]. The derivation is slightly U technical, so that here we show the only result. For readers where U ≡ (Ubf + Ubb)/2. Here E and V are the average who are interested in the derivation, see Appendix A. kinetic and interaction energy of one particle per volume: We consider the retarded Goldstino propagator, defined as −∇2 −∇2 E= 1 ψ† ψ + ψ† ψ , R , ≡ θ { , , † , }. b (r) b(r) f (r) f (r) (15) (r t ) i (t ) q(r t ) q (0 0) (9) N 2mr 2mr

013035-3 TAJIMA, HIDAKA, AND SATOW PHYSICAL REVIEW RESEARCH 3, 013035 (2021) 1 U † † C. Mean-field approximation V = ψ (r)ψ (r)ψb(r)ψb(r) N 2 b b In this paper, we employ the weak-coupling mean-field approximation to calculate the Goldstino gap by using the + ψ† ψ ψ† ψ . U b (r) b(r) f (r) f (r) (16) GOR relation, (14). At a weak coupling, the thermal average with respect to the interaction term in Eq. (14) can be Here, we take the limits of χ → 0(mb → m f ) and U → 0 approximated as (U → U ) to suppress the higher-order breaking terms be- bb bf ψ† ψ† ψ ψ 2, ing proportional to χ 2 and U 2 when we define E and V b (r) b (r) b(r) b(r) 2Nb (21) in Eqs. (15) and (16). These parameters can also be expressed † † ψ (r)ψb(r)ψ (r)ψ f (r) NbNf , (22) by the pressure P(T,μ,m f , mb,Ubb,Ubf ) as a function of T , b f μ , m f , mb, Ubb, and Ubf: where the particle number densities Nb( f ) are obtained as 1 ∂P ∂P 3 E= m2 + m2 , (17) d q b f b Nb = nb(ξ ), (23) mr ∂m f ∂mb π 3 q (2 ) ∂P ∂P 3 V =−U + . (18) d k f ∂ ∂ Nf = n f (ξ ), (24) Ubf Ubb (2π )3 k

We note that this result is correct up to first order in where nb( f )(x) = 1/(exp(x/T ) ∓ 1) is the Bose (Fermi) distri- explicit symmetry breaking, and we did not use any ξ b = 2/ − μ + bution function. Here we have defined q q (2mb) b approximations such as the RPA in its derivation. We f H and ξ = k2/(2m ) − μ + H . The Hartree shift H also note that the expectation values of the local opera- b k f f f b( f ) ψ† ψ† ψ ψ is given by tor b (r) b( f )(r) b(r) b( f )(r) in zero-range models are known to be associated with the so-called Tan’s contact H = + , b 2UbbNb UbfNf (25) Cbb(bf) [50–53]as H = U N . (26) U C C f bf b V = bf + bb . (19) 2 2 N (4πabf ) (4πabb ) Substituting Eqs. (23) and (24) into Eq. (14), one obtains ωGOR =−μ + χE + . The universal relations with respect to this quantity are ex- G HF 2Nb U (27) pected to hold even in the weakly repulsive case [54,55]. Here, we have defined Indeed, Cbb is analytically obtained within the mean-field Bo- goliubov theory at T = 0inRefs.[54] and [56]. The GOR 1 d3k k2 E ≡ n ξ b + n ξ f . (28) relation is therefore rewritten as HF π 3 b k f k N (2 ) 2mr U C C = ωGOR =−μ + χE+2 bf + bb In particular, in the mass-balanced case (mb m f ) relevant G N (4πa )2 (4πa )2 for a 87Sr - 87Rb mixture, one finds bf bb ωGOR =−μ + 2N U. (29) =−μ + χE+ 1 abf − abb Cbf + Cbb . G b π 2 2 2 N 2mr mb abf abb Our result agrees with the result in Ref. [12] obtained in a tight-binding model. We note that Eq. (29) obtained in the (20) normal phase is different from the result in Ref. [16], which Since Tan’s contact can be observed precisely, this rela- considered the BEC phase at zero temperature. We also note tion is also useful to address the Goldstino properties in that in the mean-field approximation one can obtain Tan’s = π 2 2 = π 2 2 2 recent experiments. However, a strong repulsive interaction contacts as Cbf 16 abfNbNf and Cbb 16 abbNb .One beyond the present weak-coupling mean-field approximation can reproduce Eq. (27) by substituting them into Eq. (20). generally involves an effective range correction acting as a momentum cutoff to avoid an ultraviolet divergence in a D. Random phase approximation three-dimensional system [57,58]. In this case, one has to We compare the results of the GOR relation with the RPA extend Eq. (20) to a relation with the effective ranges of the calculation to see the effects of continuum and higher-order interactions. It is particularly important to develop the relation correction in the explicit breaking term. In Ref. [13], it is between the GOR relation and the high-momentum tails of the reported that the imbalance of the hopping amplitude in the distribution functions [51], which is left for future work. In lattice model (which corresponds to the mass imbalance in this paper, we restrict ourselves to the weak-coupling regime. the present case) generates the continuum mode in addition to Since we assume a homogeneous case with translational sym- the Goldstino pole, within the RPA. In this work, we address metry, we can take r → 0 taking ∇2 in the terms in E. how the GOR relation works in the presence of such a con- We also note that the GOR relation derived in this paper is tinuum. We consider the series of fermion-boson bubble valid both below and above the Bose-Einstein condensation diagrammatically described in Fig. 1. The explicit form of the temperature T . To address the BEC phase below T , one BEC BEC Goldstino propagator R reads has to take the mean-field term associated with the condensate into account [59]. In this paper, we consider the normal phase (p,ω) R(p,ω) = , (30) above TBEC for simplicity. 1 + Ubf(p,ω)

013035-4 GOLDSTINO SPECTRUM IN AN ULTRACOLD BOSE-FERMI … PHYSICAL REVIEW RESEARCH 3, 013035 (2021)

ωGOR the branch point is close to G . To see this, we parametrize the denominator of R(0,ω)as 1 + ,ω = ω − ωGOR − ˜ ω , 1 Ubf (0 ) G ( ) (35) UbfN FIG. 1. Feynman diagrams for the Goldstino propagator con- where sisting of RPA series of boson-fermion bubble . The solid (dashed) 1 d3k line and the black dot represent the fermion (boson) propagator and ˜ (ω) = n ξ f + n ξ b 3 f k b k the boson-fermion interaction Ubf , respectively. N (2π ) [ω + μ − χk2/(2m ) − 2UN ]2 × r b . (36) where ω + iδ − χk2/(2m ) − ω r BP 3 ξ f + ξ b d k n f k nb k−p ˜ ω (p,ω) =− (31) ( ) plays a role similar to that of the dynamic part of (2π )3 ω + δ + ξ b − ξ f i k−p k the memory function defined in Eq. (A12), although the ω = ωGOR ˜ ωGOR definition is different. At G , ( G ) is explicitly is a bubble diagram with respect to the fermion-boson ex- proportional to χ 2: change. Here δ is an infinitesimally small value. In the δ −3ε 1 d3k numerical calculations, is taken to be 10 b in this paper. ˜ ωGOR = χ 2 ξ f + ξ b 2 2/3 G n f k nb k Here εb = (6π Nb) /(2mb) is the energy scale associated N (2π )3 with the boson density Nb. We note that the continuum is [k2/(2m ) −E ]2 generated when the kinematics of 1 to 2 scattering is possible × r HF . (37) ωGOR − ω − χ 2/ + δ for multiple ω due to multiple k: G BP k (2mr ) i ω + ξ b = ξ f . From this expression, the correction in the χ 2 order is k−p k (32) evaluated as For p = 0 and χ>0, the branch point is located at χ 2 3 ω = ξ f − ξ b =−μ + − GOR 1 d k f b BP 0 2 UNb UbfN. There is a con- ˜ ω n ξ + n ξ 0 G π 3 f k b k tinuum spectrum for ω ωBP. Since ωBP can be written as UbfN N (2 ) ω = ωGOR − χE − ω = ωGOR 2 BP G HF UbfN, G is always in the k 2 χ> χ< × E − . (38) continuum for 0. In contrast, when 0, the contin- HF 2m ω ω ω = ωGOR r uum spectrum exists for BP. Thus, G is in the ˜ ωGOR χ 2E2 / continuum when UbfN −χEHF. We can estimate the scale of ( G )as HF (UbfN), 2 2 The Goldstino gap is obtained by the zero point of the where the integral of (EHF − k /2mr ) is estimated to be R ,ω + ,ω = E2 χE ∼ ωGOR ω ∼ denominator of (0 ), i.e., 1 Ubf (0 ) 0. In the HF. Since HF G and BP UbfN for a small mass-balanced case, one can analytically estimate the Gold- explicit symmetry-breaking case, we obtain stino gap from ωGOR ˜ ωGOR ∼ ωGOR G . = + ,ω ( G ) G (39) 0 1 Ubf (0 G) ωBP 3 ξ f + ξ b χ d k n f k nb k Similarly, we can estimate the nth order in as of or- = 1 − U GOR GOR n−1 bf 3 der ω |ω /ω | . This expansion breaks down if (2π ) ωG + iδ + μ − 2UNb + UbfN G G BP |ωGOR/ω | is not small even though χ 1. ω + μ − 2UN G BP = G b , (33) When χ>0, there is a contribution from the imaginary ω + iδ + μ − 2UN + U N ˜ ωGOR G b bf part of ( G ) to the dispersion relation, which can be and therefore analytically evaluated as π 3 ω =−μ + . d k f G 2Nb U (34) −Im ˜ ωGOR = χ 2 n ξ + n ξ b G N (2π )3 f k b k We note that the k dependence completely vanishes from 2 Eq. (32)atp = 0, and therefore the width of the continuum k2 × −EHF becomes 0. Equation (34) coincides with the GOR relation 2mr given by Eq. (29)[13–15]. Beyond the RPA, there will be k2 corrections coming from interactions between quasiparticles. × δ ωGOR − ω − χ G BP 2m Since the differences in chemical potentials and interactions r simply induces a shift of the Goldstino pole, the supersym- 3 U 2m U N 2 E metric collective mode can be confirmed experimentally by = bf r bf + χ HF π χ 1 checking the interaction and chemical potential dependences 4 UbfN of the gap in a weakly interacting mass-balanced mixture. × n ξ f + n ξ b , (40) f k˜ b k˜ On the other hand, in the presence of the mass imbalance between bosons and fermions, there is a correction to the GOR where k˜ = 2m (ωGOR − ω )/χ = relation, which is parametrically higher order in the explicit r G BP −3/2 breaking term. However, the correction may not be small if 2mrEHF + 2mrUbfN/χ. We see that the factor χ

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the fermionic number density Nf and the chemical potential μb = μ f for three cases with a ﬁxed bosonic number density Nb at which the two conditions above are realized. The Bose- Einstein condensation temperature TBEC is identiﬁed by the Hugenholtz-Pines relation [61]

μb − b = μb − 2UbbNb(T = TBEC)

− UbfNf (T = TBEC) = 0. (41)

We see that Nf is smaller than Nb at low T . The qualitative temperature dependence of these quantities is unchanged among 6Li - 7Li, 40K - 41K, and 173Yb - 174Yb mixtures. While the chemical potential can be precisely obtained through the density profile measurement in recent cold atomic experiments [62–64], it is not in every case. These results on number densities would be helpful for one trying to realize a nearly supersymmetric mixture. This behavior can be understood as follows: In the noninteracting case, μb = 0atTBEC and μb is negative above TBEC. In the presence of interactions, the chemical potential is effec- μ ≡ μ − H tively shifted to ¯ f (b) f (b) f (b) due to the Hartree shift. Asμ ¯ b is fixed from Nb and T , which is negative, μb becomes larger as the interaction strength increases and eventually becomes positive. On the other hand, μ f is positive in the FIG. 2. Ratio between fermionic and bosonic number densities low-temperature regime even in the absence of repulsions due Nf /Nb with different interactions as functions of the temperature T to the Fermi-Dirac statistics. Therefore, in the weak-coupling 6 7 40 41 173 174 in (a1) Li - Li, (b1) K - K, and (c1) Yb - Yb Bose-Fermi case, μ f = μb would take a positive and small value. As Nf is μ = = mixtures with U 0. (a2), (b2), and (c2) show the bosonic an increasing function ofμ ¯ f , which is proportional to μ f , Nf μ chemical potential b in systems corresponding to (a1), (b1), and needs to be much smaller than N in order to realize μ = μ . μ ε b f b (c1), respectively. In these plots, b is divided by the energy scale b This situation is similar to that of so-called Bose polarons 1 ε = 2/ = π 2 3 characterizing Nb as b kb (2mb), where kb (6 Nb) . TBEC is [65–71], where impurity atoms (which correspond to fermions the BEC temperature. in the present case) are immersed in a bosonic medium. If we increase the interaction, Nf becomes larger and finally appears in contrast to the previous order estimate exceeds Nb. ˜ ∼ χ 2 ωGOR ω We note that at stronger coupling, the system may be un- .Ifthe G is far from BP, more precisely, if ωGOR − ω χ stable against phase separation [72]. At T = 0, the mixture ( G BP ) T , the imaginary part is exponentially 1 − ωGOR − ω / χ π mb Nb 3 small by the factor exp[ ( G BP ) ( T )], so the order is expected to become unstable when kbabb m +m ( N ) ˜ ∼ χ 2 b f f estimate is still valid. at Ubb = Ubf in Ref. [72]. Since the parameter regimes we < < Since the mass-imbalance effect is generally unavoidable consider in this paper are 0.3 ∼ Nf /Nb ∼ 1.2(seeFig.2) and in actual cold-atom experiments, in the following we focus 7 mb 174 + , this stability condition can be estimated on the mass-imbalanced effect on the gap by taking μ = μ 13 m f mb 347 f b as k a < 1.6. Furthermore, as usual, such an instability is and U = U , unless otherwise specified. As realistic can- b bb ∼ bf bb weakened at finite temperature due to thermal fluctuations. didates, we consider 6Li - 7Li, 40 - 41 , and 173Yb - 174Yb K K Therefore, although we do not explicitly address this condi- mixtures. Even in these systems, it is generally difficult to tion, we assume that the homogeneous phase is realized. control Ubf and Ubb independently. However, in the case of 6Li - 7Li and 40K - 41K mixtures, the boson-boson scattering B. Spectral properties of the Goldstino length abb = (mbUbb)/(4π ) can be tuned due to the magnetic Feshbach resonance [28,29], while the boson-fermion one Using the thermodynamic quantities shown in Fig. 2 and abf = (mrUbf )/(4π ) is almost independent of the magnetic the GOR relation given by Eq. (27) and the RPA equation, = . = . field (noting that abf 2 16 nm [18] and abf 5 13 nm (30), we calculate the Goldstino gap above TBEC as shown in 6 7 40 41 ωRPA [20]in Li - Li and K - K mixtures, respectively). In Fig. 3. In the RPA calculation, we have defined G as the 173 174 Yb - Yb mixtures, two scattering lengths are precisely energy where the Goldstino spectral weight AG(p = 0,ω) has determined as abf = 7.34 nm and abb = 5.55 nm [60]. a maximum, where R AG(p,ω) = Im (p,ω), (42) III. NUMERICAL RESULTS so that it can be defined in the case where the pole has a finite A. Thermodynamic quantities imaginary part. In the case of 40K - 41K and 173Yb - 174Yb First, we discuss when the system explicitly breaks the mixtures at kbabb = 0.2, one can find that the GOR pre- supersymmetry with respect to only the mass imbalance, dictions show good agreement with the RPA calculation. namely, Ubf = Ubb and μ f = μb but mb = m f . Figure 2 shows In fact, one can check that the values of the expansion

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FIG. 3. The Goldstino gap ωG calculated by the GOR (solid lines) and RPA (dashed lines) with U = μ = 0. For 6Li - 7Li mixtures, we plot ωG at different interaction strengths, kbabb = 0.2, 40 41 173 174 0.4, and 0.8. We also show ωG in K - K and Yb - Yb mixtures at kbabb = 0.2.

|ωGOR/ω |= parameter estimated in Eq. (39) are small: G BP . × −2 |ωGOR/ω |= . × −2 40 41 9 6 10 and G BP 2 2 10 in K - K and 173 174 Yb - Yb mixtures, respectively, at T = TBEC.Also,we ωGOR/ε see that G b is quite small. This behavior can be qualitatively understood in the following way: The second term in ,ω ε / Eq. (27) gives the term which is proportional to χEHF/εb, FIG. 4. Calculated Goldstino spectral weight AG(p ) b Nb ωGOR/ε E /ε within the RPA in (a) 6Li - 7Li, (b) 40 - 41 ,and(c)173Yb - 174Yb in G b. Assuming that the factor HF b is not far from K K ωGOR/ε Bose-Fermi mixtures at T = TBEC. Parameters are set at kbabb = 0.2 unity, one can make an order estimate of G b by checking − − = μ = χ. Indeed, the values χ = 1.2 × 10 2 and χ = 2.9 × 10 3 and U 0. in 40K - 41K and 173Yb - 174Yb mixtures explain the order of ωGOR/ε magnitude for G b. Although we do not show the numerical results explicitly finite momenta even in 40K - 41K and 173Yb - 174Yb mixtures. at stronger supersymmetric couplings, this agreement is un- Figure 5(a) shows AG(p = 0,ω) at zero momentum in a 6 7 changed in these mixtures. In this regard, we conclude that Li - Li mixture at T = 1.16TBEC. With increasing supersym- mb the mass-imbalance effect in these systems is negligibly small metric interaction kbabb (= abf since we take Ubb = Ubf), 2mr in this temperature region. To confirm the existence of the one can see the crossover from the regime where the singular- Goldstino, exploring the interaction and density dependences ity associated with ωBP is dominant to the coexistence of the of the Goldstino gap given by Eq. (29) is suitable. We note sharp Goldstino pole and continuum plateau. It is possible to that thermodynamic quantities such as the chemical potential check the sharp Goldstino peak also at finite momentum, from were observed precisely within a relative error of less than 4% the spectral function at kbabb = 0.8 plotted in Fig. 5(b).While in a recent cold-atom experiment [62–64]. in the noninteracting case (kbabb = 0) a kink structure can be On the other hand, the mass-imbalance effect on the Gold- found around ω = 0, it originates mainly from the tip of the stino gap in 6Li - 7Li mixtures with χ = 1/13 is not so small continuum, as one can see in Appendix B. We note that the compared to the other two systems. In fact, the values of contribution at ω<0 in the noninteracting case is an artifact |ωGOR/ω |= . δ = −3ε the expansion parameter, G BP 0 621, 0.308, and associated with the small imaginary part i 10 bi.We 0.151, for kbabb = 0.2, 0.4, and 0.8 at T = TBEC are not small also note that some of the analysis above in a tight-binding compared with those in 40K - 41K and 173Yb - 174Yb mixtures. model was done in Ref. [13], but this is the first time that We note that the difference between the GOR and the RPA we got results for gases of realistic Bose-Fermi mixtures. is accidentally small at kbabb = 0.2. This is just a coinci- Figure 6(a) shows a comparison of the Goldstino gap from the dence caused by the singular behavior of the branch point. GOR relation vs the RPA calculation (corresponding to Fig. 5) 6 7 Figure 4 shows the RPA spectral weight AG(p,ω)oftheGold- in a Li - Li mixture at T = 1.16TBEC. If we increase kbabb stino at T = TBEC, kbabb = 0.2, and finite momentum. While (and simultaneously kbabf such that Ubf = Ubb), one finds 40 41 173 174 > AG(p,ω)in K - K and Yb - Yb mixtures exhibits that the two results approach each other around kbabb ∼ 0.6. a sharp peak associated with the supersymmetric collective In this regime, as shown in Fig. 5, a sharp Goldstino peak mode around ω = 0, this peak in a 6Li - 7Li mixture is greatly appears since the branch point is separated from the pole. We ˜ ωGOR /ωGOR broadened due to the branch point at weak coupling. We note also plot ( G ) G given by Eq. (37)inFig.6(b). Since ˜ ωGOR that the Goldstino spectrum merges with the continuum at ( G ) represents the higher-order corrections included in

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ω FIG. 5. (a) Goldstino spectral weight AG(p = 0,ω) at zero mo- FIG. 6. (a) Comparison of the Goldstino gap G between the 6 7 ˜ ωGOR /ωGOR mentum in a Li - Li mixture at T = 1.16TBEC and μ = U = 0 GOR and the RPA and (b) ( G ) G given by Eq. (37) in a 6 7 = . μ = = with kbabb = 0, 0.2, 0.4, 0.6, and 0.8. (b) Contour plot of AG(p,ω)at Li - Li mixture at T 1 16TBEC and U 0. kbabb = 0.8.

energy region. This result indicates that observation of the the RPA, the GOR relation is expected to be valid in the region Goldstino gap in 173Yb - 174Yb mixtures is quite promising. ˜ ωGOR /ωGOR where ( G ) G is small. In this sense, one can obtain the strong intensity of the Goldstino pole even in 6Li - 7Li mixtures in the presence of relatively strong interactions. Such a condition in this case can be expressed as χE UbfN. ωGOR From Fig. 6(b), one can confirm that the coincidence of G ωRPA . and G at kbabb 0 2inFigs.3 and 6(a) is accidental due ˜ ωGOR = to Re ( G ) 0. We note that the Lee-Huang-Yang corrections may become important in the relatively strong repulsive regime [73]. Indeed, this leading quantum correction on μb/εb 5 is known to be proportional to (kbabb ) 2 in a single-component Bose gas at T = 0. A self-consistent framework beyond the present mean-field approximation would be required to obtain > the Goldstino gap quantitatively at kbabb ∼ 1. 173 174 Furthermore, in Fig. 7 we plot AG(0,ω)ina Yb - Yb mixture with realistic interactions given by abf/abb = 7.34/5.55 [60], at T = TBEC and μ f = μb. A sharp peak in the Goldstino pole emerges at a positive energy, whereas there is a small peak in the continuum in the negative-energy region. The continuum is quite small compared to the case of the 6Li - 7Li mixture shown in Fig. 5. We have checked that FIG. 7. Calculated Goldstino spectral weight AG(p = 0,ω)εb/Nb the GOR relation shows excellent agreement with the pole 173 174 within the RPA in a Yb - Yb Bose-Fermi mixture at T = TBEC position in this case. If we increase Nb [namely, the coupling and μ = 0. Interaction parameters are chosen to reproduce the 1 = π 2 3 parameter kbabb (6 Nb) abb with fixed abb], the Goldstino experimental ratio abf /abb = 7.34/5.55 [60]. The sharp peaks at pole becomes distinct since the continuum goes to the lower- positive energy are the Goldstino poles.

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FIG. 8. Self-energy diagram for supersymmetric ﬂuctuations associated with the RPA Goldstino propagator . Solid (dashed) lines H represent fermion (boson) propagators Gb( f ) with the Hartree shift H f (b). Black dots show the boson-fermion coupling Ubf .

IV. FERMIONIC SINGLE-PARTICLE SPECTRUM In this section, in addition to the Tan’s contacts shown in Sec. II, we discuss how to detect the Goldstino gap in cold-atom experiments. One promising way is via the single- → particle excitation spectrum of a fermion as discussed in the FIG. 9. Fermionic single-particle spectral function A f (p ,ω 173 174 BEC phase [15]. In the normal phase, we consider the self- 0 ) at the zero-momentum limit in a Yb - Yb mixture at T = T with k a = 0.3anda /a = 7.34/5.55 [60]. In the energy f (p, iω) diagrammatically drawn in Fig. 8, where BEC b bb bf bb = k ω = (2 + 1)πT is the Matsubara frequency of fermions. numerical calculation, we take the momentum cutoff 2 b.Dot- ted and dashed vertical lines represent the fermionic pole position The explicit form of (p, iω)isgivenby f ξ f =−μ − p→0 ( f Ubf Nb) within the mean-field theory and the Gold- ∞ ωRPA 3 stino gap G obtained from the RPA analysis. 2 d k (p, iω) =−U T (k, iω ) f bf (2π )3 n n=−∞ f (p, iω)as H × G (p − k, iω − iω ), (43) b n 3 − + ξ b 2 d k 1 n f (Ep−k ) nb k (p, iω) = U Z . (46) H , ω = / ω − ξ b f bf G π 3 ω − − ξ b where Gb (p i ˜ k ) 1 (i ˜ k p ) is the Hartree Green’s (2 ) i Ep−k k function of a boson. Here,ω ˜ k = 2kπT is the Matsubara Furthermore, near T = TBEC it was suggested that one can use frequency for bosons. We obtain the single-particle Green’s ξ b , ω the so-called static approximation where nb( k ) has a dom- function G f (p i )as ξ b = inant contribution at k 0[74]. For qualitative illustrative 1 purposes, we use this approximation and obtain , ω = . G f (p i ) f (44) 2 iω − ξp − f (p, iω) UbfZGNb f (p, iω) . (47) iω − E The fermionic single-particle spectral function is obtained as p ,ω =−1 , ω → ω + δ → ,ω A f (p ) π ImG f (p i i ). Finally, the fermionic spectral function A f (p 0 )atthe Figure 9 shows the calculated A f (p → 0,ω)ina zero-momentum limit reads 173 174 Yb - Yb mixture at T = TBEC with realistic interactions, A f (p → 0,ω) = α+δ(ω − E+) + α−δ(ω − E−), (48) that is, kbabb = 0.3 and abf/abb = 7.34/5.55 [60]. One finds that A f (p → 0,ω) contains a double-peak structure due to where the self-energy correction. In the absence of such a correc- ωG − μ f + UbfNb ω = ξ f ≡ E± = tion, only a single fermionic pole is located at p→0 2 − μ − ( f UbfNb). In the numerical calculation, we take a finite = ω + μ − U N 2 momentum cutoff, 2kb,inEq.(43). We note that the ± G f bf b + 2 UbfZGNb (49) double-peak structure in A f (p → 0,ω) is left qualitatively 2 unchanged by the value of (see also Appendix C). We and examine the qualitative structure of A f (p,ω) by focusing on ⎛ ⎞ the Goldstino pole and using an approximate form of as ω + μ − 1⎝ G f UbfNb ⎠ α± = 1∓ . (50) 2 2 2 ZG (ωG + μ f − UbfNb) + 4U ZGNb (k, iωn ) , (45) bf iωn − Ek This double-peak structure is due to the level repulsion be- 2 where Ek = k /(2mG) + ωG is the Goldstino dispersion. ZG tween the one-particle fermion excitation and the Goldstino and mG are the wave-function renormalization and the ef- pole in A(p → 0,ω). This level repulsion enlarges the sep- fective mass of the Goldstino, respectively. Their analytical aration between the fermionic pole −(μ f − UbfNb) and the = ωRPA ω α expressions are obtained in the supersymmetric case at T Goldstino pole G . One can estimate G from E± and ±. 0[15]. By using this expression, we can analytically per- Indeed, in cold-atom experiments, radio-frequency spectro- form the summation of the fermion Matsubara frequency in scopies are employed to observe single-particle excitations

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[75]. If the interaction and chemical potential dependences of ACKNOWLEDGMENTS the low-momentum excitation spectra are observed, one can We would like to thank M. Horikoshi for useful discus- estimate ω from them. G sion. H.T. was supported by a Grant-in-Aid for JSPS fellows We note that, while we have employed the RPA diagram (Grant No. 17J03975) and for Scientific Research from the associated with the fermion-boson bubble resummation given JSPS (Grant No. 18H05406). Y.H. was supported in part by a in Fig. 8, fermion-boson ladder diagrams with a many-body Japan Society of Promotion of Science (JSPS) Grant-in-Aid T matrix may also appear at the same order [76]. Indeed, the for Scientific Research (KAKENHI Grant Nos. 16K17716, latter diagrams play a crucial role in Bose polarons with at- 17H06462, and 18H01211). tractive interspecies interactions [67] and show an ultraviolet divergence (see Appendix C). This beyond-mean-field correction would be important to predict more quantitative features APPENDIX A: MEMORY FUNCTION FORMALISM of fermionic single-particle excitations in the presence of the Goldstino. However, since we consider the repulsive interac- In this Appendix, we show the detailed derivation of tions, the ladder diagram in our model does not involve a sharp Eq. (11) based on the memory function formalism [40]. This pole, which is associated with two-body bound molecules in formalism is useful to describe the Langevin dynamics of slow variables such as the hydrodynamic degrees of freedom. We the strongly attractive case, as shown in Refs. [57] and [58]for R ω repulsively interacting two-component Fermi gases. Thus, we would like to obtain the equation of ( ) with the form expect that our results would correctly capture the qualitative (ω + + i (ω))R(ω) =−N, (A1) structure of A f (p → 0,ω). where and N are constants, and (ω)isafunctionofω. V. SUMMARY In the language of the memory function formalism, K(ω) = i − (ω) is called the memory function; i and (ω)are To summarize, we have theoretically investigated the the static and dynamical parts of the memory function. The gapped Goldstino mode in an ultracold atomic Bose-Fermi corresponding generalized Langevin equation reads mixture with explicitly broken supersymmetry. We have shown the gap formula for the Goldstino (GOR relation) by t ∂ − + − = , using the memory function formalism. Using this relation, we ( t i )Q(t ) dt (t t )Q(t ) R(t ) (A2) calculate the Goldstino gap at the first order of the explicit 0 symmetry breaking and compare it with the numerical re- where we have introduced the noise R(t ) that satisfies sults obtained within the RPA. We have confirmed that in the Q(t )R(t )=0. We do not give the explicit relation between absence of a mass imbalance between fermions and bosons, Eq. (A1) and Eq. (A2), which can be shown by using the the Goldstino gap obtained by the GOR relation coincides projection operator method [79,80]. Equation (A1) has a sim- with that in the RPA. We have also discussed the relationship ilar form to the Schwinger-Dyson equation in quantum field between the derived GOR relation and the Tan’s contact. At theory. Roughly speaking, + i (ω) corresponds to the self- the current stage, a 173Yb - 174Yb mixture is the strongest can- energy, and and give the gap and dissipation, respectively. didate for detecting the Goldstino. Indeed, using experimental The purpose here is to express , N, and (ω) by correla- values of scattering lengths and mass ratio, we show that the tion functions. For this purpose, we introduce the Liouville Goldstino pole has a strong intensity in this mixture. While the operator L as Lq ≡ [H, q] such that we can express q(r, t ) L mass-imbalance effect in 40K - 41K and 173Yb - 174Yb mixtures as q(r, t ) = ei t q(r, 0). This enables us to rewrite Eq. (10) is negligibly small even in the weak-coupling regime, that in at p = 0 as a 6Li - 7Li mixture induces broadening of the Goldstino pole R ω = { ω , † , }, due to the singularity around the branch point in fermion- ( ) i r( )Q q (0 0) (A3) boson bubbles through the infinite sum of bubbles in the where r(ω) =−i/(ω + L). Using the identity −iωr(ω) = RPA. However, if we increase the interactions, the Goldstino + L ω pole becomes sharp even in this case since the branch point 1 i r( ), we obtain is well separated from the Goldstino pole. Finally, we have −iωR(0,ω) = iN + i{r(ω)iLQ, q†(0, 0)}, (A4) discussed the possibility of observing the Goldstino gap from the single-particle excitation of a Fermi atom. We show the where qualitative structure of the spectral function near T = TBEC and predict the modification of the dispersion due to the cou- † † N =ψ (r)ψ f (r)+ψ (r)ψb(r) (A5) pling between the free branch and the Goldstino pole at low f b momenta. is the total number density. We now introduce the memory To further address realistic experimental situations, it is an function K(z) such that important problem to investigate the radio-frequency spectra and the momentum-resolved photoemission spectra of the K(ω)R(0,ω) = i{r(ω)iLQ, q†(0, 0)}. (A6) Fermi atom in the present mixtures. In this case, we have to consider the inhomogeneity due to the trap potential. In By construction, this satisfies (ω − iK(ω))R(ω) =−N.We addition, exploring other approaches to seeing the Goldstino would further like to decompose K(ω) into the static part i , gap, such as nonequilibrium dynamics [77,78], is also an which is responsible for the gap, and the dynamic one (ω), interesting future direction. which is responsible for the dissipation. Multiplying Eq. (A6)

013035-10 GOLDSTINO SPECTRUM IN AN ULTRACOLD BOSE-FERMI … PHYSICAL REVIEW RESEARCH 3, 013035 (2021) by −iω and using Eq. (A4) and −iωr(ω) = 1 + iLr(ω), we obtain K(ω)[iN + i{r(ω)iLQ, q†(0, 0)}] = i{iLQ, q†(0, 0)} + i{r(ω)(iL)2Q, q†(0, 0)}. (A7) From Eq. (A6), the left-hand side of Eq. (A7) can be written as i iNK(ω) + i {r(z)iLQ, q†(0, 0)}2. (A8) R(0,ω) Substituting Eq. (A8) into Eq. (A7), we obtain K(ω) = i − (ω) with 1 i = {iLQ, q†(0, 0)}, (A9) N 1 (ω) =− {r(ω)(iL)2Q, q†(0, 0)} N FIG. 10. Comparison of AG,0(0,ω) in a noninteracting 1 i † 2 6 7 = . μ = μ + {r(ω)iLQ, q (0, 0)} . (A10) Li - Li mixture at T 1 38TBEC with b f obtained from R ,ω −3 N (0 ) Eq. (B2) (dashed curve) and the numerical result with δ = 10 εb Noting the relation (solid curve). i {iLA, B} = tre−βH {[H, A], B} tre−βH case, the spectral weight AG,0(p = 0,ω) reads −i = tre−βH {A, [H, B]}=−{A, iLB}, −βH tre 3 ξ b + ξ f (A11) d k nb k n f k A , (0,ω) =−Im G 0 (2π )3 ω + μ + iδ − χk2/2 we can express the dynamic part as ∞ k2dk 1 = n ξ b + n ξ f (ω) = {r(ω)iLQ, iLq†(0, 0)} 2π b k f k N 0 2 1 × δ ω + μ − χ k δ → . + {r(ω)iLQ, q†(0, 0)} ( 0) (B1) N 2 i × {r(ω)Q, iLq†(0, 0)}. (A12) R(0,ω) By performing momentum integration, we can obtain the analytical expression of AG(0,ω)as In summary, the retarded Green function satisfies Eq. (A1), which corresponds to the generalized Langevin equation, 1 √ ,ω = √ θ ω ω (A2). The coefficients and function are given by Eqs. (A5), AG,0(0 ) 3 ( ) 2πχ 2 (A9), and (A12). ω Let us check the order of and ( ) with respect to the 1 1 × + explicit breaking term. Since both static and dynamic parts are ω ω χ −μb /T −μ / ( mb ) − χ f T proportional to iLQ = i[H, Q], K(ω) vanishes if the super- e 1 e m f + 1 symmetry is exact. Therefore, ω = 0 becomes the pole. When (B2) the supersymmetry is explicitly broken by a small parameter, [H, Q] ∼ , the static part is i ∼ , while the dynamic part μ = χ → ,ω ω = is (ω) ∼ 2 as shown in Eq. (A12). Therefore, at leading at 0. If we take 0, AG,0(0 ) diverges at only order in , we can neglect (ω). We note that we also as- 0, which indicates that the continuum has vanishing width at signed [H, q†(x, t )] ∼ . Precisely speaking, there is the other zero momentum. contribution of order 1, the divergence of the supersymmetric It can be understood also from the sum rule [14,15], ω ,ω /π = ω χ → current ∇·j in [H, q†(x, t )]. This vanishes in the correlation d AG(p ) N: At finite , AG,0 vanishes at 0, function at p = 0. At leading order in , we find the pole, (12). due to the exponential factor. Therefore, to satisfy the sum rule, the existence of divergence at ω = 0 is implied. In Fig. 10 we show a comparison between Eq. (B2) and the APPENDIX B: ANALYTICAL RESULTS IN A numerical result with δ = 10−3 in a noninteracting 6Li - 7Li NONINTERACTING MIXTURE mixture at T = 1.38TBEC. From this, we find that the effect To check that the numerical procedure to include the in- of δ is small for the maximum of AG,0(0,ω). While the finite finitesimal imaginary part iδ in the analytic continuation does contribution at ω<0 in the numerical calculation originates not cause a serious numerical artifact, we investigate the Gold- from the finite δ, we confirmed that this also does not affect stino spectral function in the free limit. In the noninteracting our main results.

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Figure 11 shows the calculated A f (p → 0,ω) with different cutoffs . One can see that the lower sharp peak is quantitatively shifted toward the lower-energy side with increasing . We emphasize, however, that the double- peak structure is qualitatively unchanged by the value of . On the other hand, the particle-particle ladder diagram, which is not considered in this work, is known to exhibit an ultraviolet divergence. The many-body T matrix responsible for this scattering process is given by

Ubf TMB(k, iωn ) = , (C2) 1 + UbfL(k, iωn ) where |p| 3 − ξ f + ξ b d p 1 n f p+k/2 nb −p+k/2 L(k, iωn ) =− (2π )3 ω − ξ f − ξ b i n p+k/2 −p+k/2 FIG. 11. Fermionic single-particle spectral function A f (p → 0, ω) at different momentum cutoffs, /kb = 1.5, 2, 2.5, and 3. Other (C3) parameters are the same as in Fig. 9. is the lowest-order particle-particle correlation function. TMB(k, iωn ) and mixing of the ladder and bubble diagrams APPENDIX C: ULTRAVIOLET CUTOFF DEPENDENCE OF may appear in the self-energy corrections in addition to THE FERMIONIC SINGLE-PARTICLE SPECTRAL Eq. (43). One can see that Eq. (C3) involves an ultraviolet FUNCTION divergence. In this case, the relation between Ubf and abf is replaced by The fermionic self-energy (p, iω → ω + iδ) analyti- f n m 1 m cally continued to the real frequency reads r = + r . (C4) π π 2 2 abf Ubf |k| d3k ∞ ,ω+ δ = 2 , Here, the cutoff , which is proportional to an inverse effec- f (p i ) Ubf dzAG(k z) π 3 −∞ (2 ) tive range [58], should be finite to reproduce the finite abf. b These ladder and finite-range corrections remain as future 1 − n f (z) + nb(ξ − ) × p k . (C1) work for more quantitative descriptions of A (p,ω). ω + δ − − ξ b f i z p−k

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