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

We theoretically investigate a supersymmetric collective mode called 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 the zero momentum and the explicit breaking term. We also numerically evaluate the gap of 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 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 $^6$Li-$^7$Li, $^{40}$K-$^{41}$K, and $^{173}$Yb-$^{174}$Yb mixtures. Furthermore, we analyze the Goldstino spectral weight in a $^{173}$Yb-$^{174}$Yb 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 the fermionic single-particle excitation as well as the relationship between the GOR relation and Tan's contact.


I. INTRODUCTION
The supersymmetry is a symmetry with respect to an interchange between bosons and fermions [1][2][3]. While the existence of supersymmetry is expected in the context of particle physics, its evidence or any indications have not been observed in high-energy experiments yet [4]. However, apart from whether the supersymmetric partners such as squark exist or not in our world, it is really an interesting problem to explore the consequences of the supersymmetry using fermions and bosons that are well established in condensed matter physics.
A remarkable feature of supersymmetry in a Bose-Fermi mixture is the emergence of Nambu-Goldstone mode called Goldstino [28][29][30][31][32]. While a usual Nambu-Goldstone mode propagates as a bosonic mode, the Goldstino behaves as a fermionic mode. Such a fermionic collective excitation has also been predicted in quantum electrodynamics as well as quantum chromodynamics (QCD) [33][34][35]. Observation of this collective mode is really important to see the supersymmetric properties in a Bose-Fermi mixture that is realized in a tabletop experiment. Since the Goldstino is a fermionic collective mode associated with the broken supersymmetry, it becomes a gapless mode when the system possesses the exact supersymmetry. However, the explicit supersymmetry breaking such as mass-imbalance between fermions and bosons is unavoidable in cold atom experiments. In such a case, the Goldstino has a finite energy-gap associated with explicit breaking parameters. If one can observe the gapped Goldstino and its spectral properties agree with the result of theoretical analysis, it should be an evidence for the existence of supersymmetry in these systems. Indeed, the first example of Nambu-Goldstone bosons in particle physics was pions, which are also gapped modes due to the explicitly broken chiral symmetry associated with the current quark mass [36].
In this work, we theoretically examine the energy gap of Goldstino in a Bose-Fermi mixture with explicitly broken supersymmetry. We focus on a few candidates for nearlysupersymmetric Bose-Fermi mixtures, namely, 6 Li-7 Li, 40 K-41 K, and 173 Yb-174 Yb mixtures.
We determine thermodynamic properties of weakly interaction mixtures within the Hartree-Fock mean-field approximation above the Bose-Einstein condensation (BEC) temperature.
By developing a gap formula for the Goldstino, which corresponds to the Gell-Mann-Oakes-Renner (GOR) relation in QCD [37], based on the memory function method [38], we show how the explicit supersymmetry-breaking terms affect the Goldstino gap in these systems.
By comparing it with the numerical results of the random phase approximation (RPA), we clarify that the effects of the branch point is significant in the presence of the mass-imbalance between fermions and bosons. Furthermore, we discuss how to observe the goldstino gap from the single-particle spectral function of a Fermi atom. This paper is organized as follows: In Sec. II, we introduce our model and the formulation for thermodynamic quantities and the Goldstino gap within GOR and RPA. In Sec. III, we show our numerical results in RPA on the Goldstino gap in a few Bose-Fermi mixture systems, and discuss it. Section IV is devoted to discussion on how the Goldstino pole can affect the fermionic single-particle spectrum, in order to suggest possibility for detecting Goldstino in experiments. We summarize our studies in Sec. V. We calculate the Goldstino spectral function in the free limit in Appendix A to check the absence of numerical artifact.

A. Model
We consider a non-relativistic Bose-Fermi mixture described by the Hamiltonian, where ψ b(f ) is the field operator of a boson (fermion) with a mass m b(f ) and the chemical potential µ b(f ) . U bb(bf ) is the coupling constant of a boson-boson (boson-fermion) interaction, which is assumed to be a contact-type. When m f = m b , µ b = µ f , U bb = U bf , there is a supersymmetry corresponding to interchange between bosons and fermions: ψ b → ψ f and The corresponding Noether charges are which commute with the Hamiltonian, is the local operator that creates the boson and annihilates the fermion [10]. Unlike the supersymmetry in relativistic systems, the anti-commutation relation between supercharges is not the Hamiltonian but the total particle number: In this sense, the supersymmetry in a non-relativistic Bose-Fermi mixture is a different type from that in relativistic theories. The order parameter of supersymmetry breaking is the total number density, , which is always broken in a finite density system. As in an ordinary symmetry breaking, the supersymmetry breaking leads to a gapless excitation. If the excitation can be identified as a single-mode excitation, it is called the Goldstino. In general, the excitation may be located at a branch point where two or multi-particles continuum starts. This is especially the case for the non-interacting system, where there is no Goldstino. The excitation is the particle-hole one.
The interaction plays an important role in the existence of the Goldstino. In the following analysis, we assume the existence of the Goldstino excitation, and we numerically check it in the RPA in Sec. III. Since the order parameter is expressed as the expectation value of the anti-commutation relation of the supercharge and its density, the Goldstino belongs to the type-B mode [39][40][41][42][43][44], which typically has a quadratic dispersion.
In a realistic situation, the supersymmetry is explicitly broken because all parameters cannot be exactly tuned in experiments. In particular, the mass imbalance is unavoidable.
The effect of the explicit breaking can be expressed as the commutation relation between the Hamiltonian and the supercharge: where is the twice reduced mass], and ∆U = U bf − U bb . These explicit breakings cause a finite gap of the Goldstino, whose formula is derived in the next subsection.

B. Gell-Mann-Oakes-Renner relation
Pions are the Nambu-Goldstone bosons associated with the spontaneous breaking of chiral symmetry in QCD. The Gell-Mann-Oakes-Renner (GOR) formula relates the pion mass and the current quark mass that explicitly breaks chiral symmetry [37]. We here generalize the GOR relation to that of Goldstino in a Bose-Fermi mixture. For this purpose, we employ the memory function method [38], which is a different method used in the original derivation [37]. The memory function method is a systematic and useful way to derive the dispersion relation. We consider the retarded Goldstino propagator defined as After Laplace-Fourier transformation, we obtain The energy gap is obtained from the pole of Γ R (p, z) in the complex z plane. Our purpose is to rewrite Γ R (p, z) to a useful form. Here, we introduce the Liouville operator L as Lq ≡ [H, q] such that we can express q(r, t) as q(r, t) = e iLt q(r, 0). Using L, we can rewrite Eq. (6) as where r(z) = (z − iL) −1 . Since we are interested in the zero-momentum gap of Goldstino, hereafter we take p = 0, Using the identity zr(z) = 1 + iLr(z), we obtain where is the bosonic (fermionic) one]. We introduce the memory function K(z) such that Using K(z), one can formally obtain Equation (11) indicates that the zero-momentum pole is given by z − K(z) = 0.
To derive the gap formula for the Goldstino, we decompose K(z) into the static part that is responsible for the gap and the dynamic one that is responsible for the dissipation.
Multiplying Eq. (10) by z and using Eq. (9) and zr(z) = 1 + iLr(z), we obtain From Eq. (10), the left hand side of Eq. (12) can be written as Substituting Eq. (13) into Eq. (12), we obtain Noting the following relation: we can express the dynamic part as The Eq. (17). Therefore, at the leading order in ǫ, we can neglect Φ(z). We note that we also Precisely speaking, there is the other contribution of order one, the . This vanishes in the correlation function at p = 0. At the leading order in ǫ, we find the solution, Performing the analytic continuation z → −iω G , we obtain the GOR relation for the Goldstino as We emphasize that this formula works for any supersymmetric Hamiltonian with a small explicit breaking term and local interactions because we have not employed the specific form of the Hamiltonian. The gap is linearly proportional to the explicit breaking term, whose property can be understood as the type-B breaking [39][40][41][42][43][44]. In contrast, the type-A breaking predicts that the gap is proportional to the square root of the explicit breaking term. We also note that although the dynamic part Φ(z) is of order ǫ 2 , it might not be small if there is a singularity in Φ(z). As is seen later, this is the case when the branch point is close to For the Hamiltonian (1) that we employ in the present paper, using Eq. (4), we can Therefore, the goldstino gap in the present model is given by where U ≡ (U bf + U bb )/2. Here E and V are the average kinetic and potential energy of one particle per volume: These parameters can be also expressed by the pressure P (T, µ, m f , m b , U bb , U bf ) as a func- We note that, this result is correct up to the first order in explicit symmetry breaking, and we did not use any approximations such as RPA in its derivation. We also note that the in zero-range models are known to be associated with the so-called Tan's contact C bb(bf ) [45][46][47][48] as where a bf and a bb are boson-fermion and boson-boson scattering lengths, respectively. The universal relations with respect to this quantity is expected to hold even in the weakly repulsive case [49,50]. Indeed, C bb is analytically obtained within the mean-field Bogoliubov theory at T = 0 in Refs. [49,51]. The GOR relation is therefore rewritten as Since Tan's contact can precisely be observed, this relation is also useful to address the Goldstino properties in recent experiments. However, a strong contact-type interaction beyond the present mean-field approximation generally involves an ultraviolet divergence in a three-dimensional system [52,53]. In this paper, we restrict ourselves in the weak-coupling regime and it is left for a future problem. Since we assume a homogeneous case with the translational symmetry, we can take r → 0 after acting ∇ 2 in the terms in E .

C. Mean-field approximation
In this paper, we employ the weak-coupling mean-field approximation to calculate the Goldstino gap by using the GOR relation (21). At a weak coupling, the thermal average with respect to the interaction term in Eq. (21) can be approximated as where the particle number densities N b(f ) is obtained as where Substituting Eqs. (30) and (31) into Eq. (21), one can obtain Here, we defined In particular, in the mass-balanced case (m b = m f ), one can find Here we note that, a similar result was obtained in Ref. [16], but their result differs from ours in the interaction term, while 2N b ∆U in our result agrees with the result in Ref. [12] obtained in a tight-binding model. This difference may originate from the higher order of the explicit breaking term beyond the mean-field approximation. We also note that in the mean-field approximation one can obtain Tan's contacts as C bf = 16π 2 a 2 bf N b N f and C bb = 16π 2 a 2 bb N 2 b . One can reproduce Eq.

D. Random phase approximation
We compare the results of the GOR relation with the RPA calculation to see effects of continuum and higher order correction in the explicit breaking term. We consider the series of fermion-boson bubble Π diagrammatically described by Fig. 1. The explicit form of the Goldstino propagator Γ R reads where is a bubble diagram with respect to the fermion-boson exchange. Here δ is an infinitesimally small value. In the numerical calculations, δ is taken to be 10 −3 ε b in this paper. We note that, the continuum is generated when the kinematics of 1 to 2 scattering is possible for multiple ω due to multiple k: For p = 0 and χ > 0, the branch point is located at There is the continuum spectrum for ω ≥ ω BP . Since ω BP can be written as is always in the continuum for χ > 0. In contrast, when χ < 0, the continuum spectrum exists for ω ≤ ω BP . Thus, ω = ω GOR G is in the continuum when The Goldstino gap is obtained by the zero point of the denominator of Γ R (0, ω), i.e, 1 + U bf Π(0, ω) = 0. In the mass-balanced case, one can analytically estimate the goldstino gap from and therefore We note that, the k dependence completely vanishes from Eq. (39) at p = 0, and therefore the width of the continuum becomes zero. Equation (41)  To see this, we parametrize the denominator of Γ R (0, ω) as whereΦ Φ(ω) plays a similar role of the dynamic part of the memory function defined in Eq. (17), From this expression, the correction in the χ 2 order is evaluated as We can estimate the scale ofΦ(ω GOR G ) as χ 2 E 2 HF /(U bf N), where the integral of ( E HF − k 2 /2m r ) 2 is estimated to be E 2 HF . Since χ E HF ∼ ω GOR G and ω BP ∼ U bf N for a small explicit symmetry breaking case, we obtaiñ Similarly, we can estimate n-th order in χ as of order ω GOR G |ω GOR G /ω BP | n−1 . This expansion breaks down if |ω GOR G /ω BP | is not small even though χ ≪ 1.
When χ > 0, there is a contribution from the imaginary part ofΦ(ω GOR G ) to the dispersion relation, which can be analytically evaluated as wherek = 2m r (ω GOR G − ω BP )/χ = 2m r E HF + 2m r U bf N/χ. We see that the factor χ −3/2 appears in contrast to the previous order estimateΦ ∼ χ 2 . If the ω GOR G is far from Since the mass-imbalance effect is generally unavoidable in actual cold atom experiments, in the following we therefore focus on the mass-imbalanced effect on the gap by taking µ f = µ b and U bf = U bb , unless otherwise specified. As realistic candidates, we consider 6 Li-7 Li, 40 K-41 K, and 173 Yb-174 Yb mixtures. Even in these systems, it is generally difficult to control U bf and U bb independently. However, in the case of 6 Li-7 Li and 40 K-41 K mixtures, the boson-boson scattering length a bb = (m b U bb )/(4π) can be tuned due to the magnetic Feshbach resonance [26,27], while the boson-fermion one a bf = (m r U bf )/(4π) is almost independent of the magnetic field (noting that a bf = 2.16 nm [18] and a bf = 5.13 nm [54] in 6 Li-7 Li and 40 K-41 K mixtures, respectively). In 173 Yb-174 Yb mixtures, two scattering lengths are precisely determined as a bf = 7.34 nm and a bb = 5.55 nm [55].

A. Thermodynamic quantities
First, we discuss when the system explicitly breaks the supersymmetry with respect to only the mass-imbalance, namely U bf = U bb and µ f = µ b but m b = m f .
shows the fermionic number density N f and the chemical potential µ b = µ f for three cases with fixed bosonic number density N b , at which the two conditions above are realized.
Here ε b = (6π 2 N b ) 2/3 /(2m b ) is the energy scale associated with the boson density N b , and the Bose-Einstein condensation temperature T BEC is identified by the Hugenholtz-Pines relation [56] We see that N f is smaller than N b at low T . The qualitative temperature dependence of these quantities are unchanged among 6 Li-7 Li, 40 K-41 K, and 173 Yb-174 Yb mixtures.
These behavior can be understood as follows: In the non-interacting case, µ b = 0 at T BEC and µ b is negative above T BEC . In the presence of the interactions, the chemical potential is due to the Hartree shift. Asμ b is fixed from N b and T , which is negative, µ b becomes larger as the interaction strength increases, and eventually becomes positive. On the other hand, µ f is positive in the low-temperature regime even in the absence of repulsions due to the Fermi-Dirac statistics. Therefore, at a weak coupling case, µ f = µ b would take a positive and small value. As N f is an increasing function of µ f , which is proportional to µ f , N f needs to be much smaller than N b , in order to realize This situation is similar to so-called Bose polarons [57][58][59][60][61][62][63] where impurity atoms (which corresponds to a fermion in the present case) are immersed in a the bosonic medium.
If we increase the interaction, N f becomes larger and finally excess N b .
We note that at stronger coupling, the system may be unstable against the phase separation [64]. At T = 0, the mixture is expected to become unstable when [64]. Since the parameter regimes we consider in this paper are Fig. 2) and 7 13 ≤ m b m f +m b ≤ 174 347 , this stability condition can be estimated as k b a bb < ∼ 1.6. Furthermore, as usual, such an instability is weakened at finite temperature due to thermal fluctuations. Therefore, although we do not explicitly address this condition, we assume that the homogeneous phase is realized.

B. Spectral properties of Goldstino
Using the thermodynamic quantities shown in Fig. 2

and the GOR relation given by
Eq. (34) and RPA equation (37), we calculate the goldstino gap above T BEC as shown in so that it can be defined in the case that the pole has a finite imaginary part. In the case Also, we see that ω GOR G /ε b is quite small. This behavior can be qualitatively understood in the following way: The second term in Eq. (34) gives the term which is proportional to Assuming that the factor E HF /ε b is not far from unity, one can make an order estimate of ω GOR Although we do not show explicitly the numerical results at stronger supersymmetric couplings, this agreement is unchanged in these mixtures. In this regard, we conclude that the mass-imbalance effect in these system is negligibly small in this temperature region. To confirm the existence of Goldstino, exploring the interaction and density dependences of Goldstino gap given by Eq. (36) is suitable. We note that thermodynamic quantities such as chemical potential can precisely be observed within relative error of less than 4% in a recent cold atom experiment [65][66][67].
On the other hand, the mass-imbalance effect on the Goldstino gap in 6  at T = T BEC are not small compared with those in 40 K-41 K and 173 Yb-174 Yb mixtures. We note that the difference between GOR and RPA is accidentally small at k b a bb = 0.2. This is just a coincidence caused by the singular behavior of the branch point.  With increasing the supersymmetric interaction k b a bb (= m b mr a bf since we take U bb = U bf ), one can see the crossover from the regime where the singularity associated with ω BP is dominant, to the coexistence of the sharp Goldstino pole and continuum plateau. It is possible to check the sharp Goldstino peak also at finite momentum, from the spectral function at k b a bb = 0.8 plotted in Fig. 5(b). While in the non-interacting case (k b a bb = 0) a kink structure can be found around ω = 0, it originates from mainly the tip of the continuum, as one can see from Appendix A. We note that the contribution at ω < 0 in the non-interacting case is an artifact associated with the small imaginary part iδ = 10 −3 i. We also note that, some of the analysis above in a tight-binding model were done in Ref. [13], but this is the first time that we got results for gases of realistic Bose-Fermi mixtures.

IV. FERMIONIC SINGLE-PARTICLE SPECTRUM
In this section, in addition to Tan's contacts shown in Sec. II, we discuss how to detect the Goldstino gap in cold atom experiments. One of promising ways is the single-particle excitation spectrum of a fermion as discussed in the BEC phase [15]. In the normal phase, we consider the self-energy Σ f (p, iω ℓ ) diagrammatically drawn in Fig. 8, where ω ℓ = (2ℓ + 1)πT is the Matsubara frequency for fermions. The explicit form of Σ f (p, iω ℓ ) is given by where G H b (p, iω k ) = 1/(iω k − ξ b p ) is the Hartree Green's function of a boson. Here,ω k = 2kπT is the Matsubara frequency for bosons. We obtain the single-particle Green's function The fermionic single-particle spectral function is obtained as We examine a qualitative structure of A f (p, ω) by focusing on the Goldstino pole and using an approximate form of Γ as where E k = k 2 /(2m G ) + ω G is the Goldstino dispersion. Z G and m G are the wave-function renormalization and the effective mass of the Goldstino, respectively. Their analytical expressions are obtained in the supersymmetric case at T = 0 [15]. By using this expression, we can analytically perform the summation of fermion Matsubara frequency in Σ f (p, iω ℓ ) Furthermore, near T = T BEC it was suggested that one can use the so-called static approximation where n b (ξ b k ) has a dominant contribution at ξ b k = 0 [68]. For qualitative illustration purpose, we use this approximation and obtain Finally, the fermionic spectral function A f (p → 0, ω) at the zero-momentum limit reads where and This double-peak structure is due to the level repulsion between the one-particle fermion excitation and the Goldstino pole in A(p → 0, ω). One can estimate ω G from E ± and α ± .
Indeed, in cold atom experiments, radio-frequency spectroscopies are employed to observe single-particle excitations [69]. If the interaction and chemical potential dependences of the low-momentum excitation spectra are observed, one can estimate ω G from them.

V. SUMMARY
To summarize, we have theoretically investigated the gapped Goldstino mode in an ultracold atomic Bose-Fermi mixture with explicitly broken supersymmetry. We develop the gap formula for the Goldstino (GOR relation) by using the memory function method. Using this relation, we calculate the Goldstino gap at the first order of the explicit symmetry breaking, and compare it with the numerical results obtained within the RPA. We have confirmed that in the absence of the mass-imbalance between fermions and bosons, the Goldstino gap obtained by the GOR relation coincides with that in the RPA. We have also discussed the relationship between the derived GOR relation and Tan's contact. At the current stage, a 173 Yb-174 Yb mixture is the strongest candidate to detect the Goldstino. Indeed, using experimental values of scattering lengths and mass-ratio, we show that the Goldstino pole has a strong intensity in this mixture. While the mass-imbalance effect in 40 K-41 K and 173 Yb-174 Yb mixtures is negligibly small even in the weak-coupling regime, that in a 6 Li-7 Li mixture induces broadening of the Goldstino pole due to the singularity around the branch point in fermion-boson bubbles through the infinite sum of bubbles in the RPA. However, if we increase the interactions, the Goldstino pole becomes sharp even in this case since the branch point is well separated from the Goldstino pole. Finally, we have discussed a possibility to observe the Goldstino gap from the single-particle excitation of the Fermi atom. We show a qualitative structure of the spectral function near T = T BEC and predict the modification of the dispersion due to the coupling between the free branch and the Goldstino pole at low-momenta.
To further address realistic experimental situations, it is an important problem to investigate the radio-frequency spectra and the momentum-resolved photoemission spectra of the Fermi atom in the present mixtures. In such a case, we have to consider the inhomogeneity due to the trap potential. In addition, exploring other approaches to see the Goldstino gap, such as nonequilibrium dynamics [70,71], are also interesting future directions. of δ is small for the maximum of A G,0 (0, ω). While the finite contribution at ω < 0 in the numerical calculation originates from finite δ, we confirmed that this also does not affect our main results.