Strong-Coupling and Finite Temperature Effects on p -wave Contacts
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Strong-Coupling and Finite Temperature E ff ects on p -wave Contacts Daisuke Inotani and Yoji Ohashi Department of Physics, Keio University,3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan (Dated: April 20, 2018)
Abstract
We theoretically investigate strong-coupling and finite temperature e ff ects on the p -wave contacts, as wellas the asymptotic behavior of the momentum distribution in large momentum region in a one-componentFermi gas with a tunable p -wave interaction. Including p -wave pairing fluctuations within a strong-couplingtheory, we calculate the p -wave contacts above the superfluid transition temperature T c from the adiabaticenergy relations. We show that while the p -wave contacts related to the scattering volume monotonicallyincreases with increasing the interaction strength, one related to the e ff ective range non-monotonically de-pends on interaction strength and its sign changes in the intermediate-coupling regime. The non-monotonicinteraction dependence of these quantities is shown to originate from the competition between the increaseof the cuto ff momentum and the decrease of the coupling constant of the p -wave interaction with increasingthe e ff ective range. We also analyze the asymptotic form of the momentum distribution in large momen-tum region. In contrast to the conventional s -wave case, we show that the asymptotic behavior cannot becompletely described by only the p -wave contacts, and the extra terms, which is not related to the thermody-namic properties, appear. Furthermore, in high temperature region, we find that the extra terms dominate thesub-leading term of the large-momentum distribution. We also directly compare our results with the recentexperimental measurement, by including the e ff ects of a harmonic trap potential within the local densityapproximation. We show that our model explains the dependence on the interaction strength of the p -wavecontacts. Since the p -wave contacts connect the macroscopic properties to the thermodynamic properties ofthe system, our results would be helpful for further understanding many-body properties of this anisotropicinteracting Fermi gas in the normal phase. PACS numbers: 03.75.Ss,05.30.Fk,67.85.-d . INTRODUCTION One of the most crucial features of ultra cold Fermi gas is a tunability of the interaction strengthbetween Fermi atoms, realized by Feshbach resonance[1, 2]. Since, by using the tunable s -waveinteraction, the BCS-BEC crossover, in which the superfluid properties change from the weak-coupling BCS-like one to the Bose Einstein condensation of strongly binding molecular bosonas increasing the interaction strength[3–7], has been realized in the ultracold Fermi gas[8, 9],strong-coupling e ff ects on various physical quantities have been extensively investigated bothexperimentally[10–17] and theoretically[18–30]. In addition to the s -wave interaction, in coldatom physics, a tunable p -wave interaction has already been realized[31–41] and the p -wave pairformation has also been detected [32, 34, 37–39]. However the p -wave superfuidity has not beenrealized yet in this system. The main experimental di ffi culty of this system is a very short life timeof the p -wave molecules, caused by three-body loss[42, 43], as well as dipolar relaxations[37]. Inview of this current experimental status in this research field, it is helpful to start from understand-ing the many-body properties of this system in the normal phase above the superfluid transitiontemperature T c .In this regard, the p -wave contacts have recently been measured in a polarized K Fermi gaswith the p -wave interaction from the weak-coupling regime to the strong-coupling regime[44].The concept of the contact was first introduced for a two-component Fermi gas with a contact-type s -wave interaction by S. Tan[45–47]. In these works, the contact is originally defined as thecoe ffi cient of the leading term of the asymptotic behavior of the momentum distribution at large p .Surprisingly, it has found that this quantity completely describes how the total energy of the systemchanges when one increases the interaction strength. Furthermore, the contact is also related notonly to the total energy but also to various thermodynamic quantities. These relations between thecontact and the thermodynamic properties are known as the Tan’s universal relations, which havebeen experimentally and theoretically verified in the cold atom Fermi gas[48–55] . In this sense,the contact connects the microscopic properties to the macroscopic (or thermodynamic) propertiesof the system.The concept of the contact has been recently extended to the case with the p -waveinteraction[56–61]. One of the most remarkable di ff erence from the s -wave contact is that in the p -wave case, the contacts consist of 6 components, in principle, because the p -wave interactionconsist of three components and each component is characterized by two parameters, the p -wave2cattering volume and the e ff ective range. In Ref. [44], by estimating the p -wave contacts fromtwo di ff erent quantities, the momentum distribution and the spectral weight of the excitations, apart of the universal relations in the p -wave interacting Fermi gas has been experimentally proved.However, the observed p -wave contact C R which is related to the e ff ective range cannot be ex-plained by a simple model. Thus, for further understanding of these experimental results, moreprecise theoretical analysis is necessary. Especially, the strong-coupling and the finite tempera-ture e ff ects might be important because the measurement of the p -wave contacts has been donein the normal phase above T c and for a wide range of the interaction strength. In addition, morerecently, it has been theoretically pointed out that, in the sub-leading term of the asymptotic formof the momentum distribution, there is an extra term, which are not related to the thermodynamicproperties but comes from the center-of-mass motion of the pair molecules[61]. Unfortunately,the existence of this extra term was not considered in the experiment. Thus, in order to describethe experimental results, it is necessary to calculate not only the p -wave contacts but also the extraterm.In this paper, we theoretically clarify the strong-coupling and the finite temperature e ff ectson the p -wave contacts, defined from the adiabatic energy relations, as well as the asymptoticbehavior of the large momentum distribution of a one-component Fermi gas with a tunable p -wave interaction. Including p -wave pairing fluctuations within the strong-coupling theory devel-oped by Nozi`eres and Schmitt-Rink (NSR)[5, 62–65], we numerically evaluate the temperaturedependence of these quantities in the entire coupling regime above the superfluid transition tem-perature T c . By comparing the p -wave contacts with the asymptotic behavior we show that thecontribution from the extra terms becomes remarkable in high temperature region. Including thee ff ects of the harmonic trap potential within the local density approximation (LDA)[66], we cal-culate these quantities in the trapped system, which are compared with the recent experimentalmeasurement[44].This paper is organized as follows. In Sec. II, we explain our strong-coupling theory for anultracold Fermi gas with a p -wave interaction. In Sec. III, we show our numerical results on T c .Here we clarify how p -wave pairing fluctuations a ff ects the p -wave contacts in uniform case. InSec. IV, we determine the temperature dependence of the p -wave contacts and the asymptoticbehavior of the momentum distribution. We show that the extra terms in the asymptotic formbecomes remarkable in high temperature region. In Sec. V, we extend our theory to the trappedsystem and we directly compare our numerical results with the resent experiment on K Fermi3as[44]. Throughout this paper, we take ~ = k B =
1, and the system volume V is taken to be unity,for simplicity. II. FORMULATION
We consider a one-component Fermi gas with a tunable p -wave interaction V p ( p , p ′ ), describedby a Hamiltonian[62–65, 67–73], H = X p ξ p c † p c p + X p , p ′ , q V p ( p , p ′ ) c † p + q / c †− p + q / c − p ′ + q / c p ′ + q / . (1)Here c p is an annihilation operator of a Fermi atom with a kinetic energy ξ p = p / (2 m ) − µ measured from the chemical potential µ (where m is the mass of Fermi atom). In Eq. (1) the p -wave interaction V p ( p , p ′ ) is assumed to have a separable form as V p ( p , p ′ ) = − X i = x , y , z γ i ( p ) U i γ i ( p ′ ) , (2)with tunable interaction strength U i in the p i -wave Cooper channel ( i = x , y , z ). Here γ i ( p ) is thebasis function of the p i -wave interaction given by γ i ( p ) = p i F ( i )c ( p ) . (3)In Eq. (3), to eliminate the well-known ultraviolet divergence, we introduce a component-dependent cuto ff function F ( i ) c ( p ) = + (cid:16) p ( i ) c / p (cid:17) n , (4)where p ( i ) c is a cuto ff momentum of the p i -wave interaction. Throughout this paper, we take n = ff ected by the detail of the high-momentum structure of the p -wave interaction in the normal phase discussed in this paper[70].In cold atom physics the p -wave interaction is conveniently characterized in terms of the scat-tering volume v i and the e ff ective range R i , which are respectively related to U i and F ( i ) c ( p ) as4 π v i m = − U i − U i X p p ε p h F ( i )c ( p ) i , (5) R − i = π m X p p ε p h F ( i )c ( p ) i . (6)4 W= G + + p V ( p , p ) FIG. 1: Feynman diagrams describing the strong-coupling correction δ Ω to the thermodynamic potential Ω within the NSR theory. The solid and dashed line describe the bare single-particle Green’s function G ( p , i ω m ) = (cid:16) i ω m − ξ p (cid:17) − and the p -wave interaction V p ( p , p ′ ), respectively. In K Fermi gas, due to the magnetic dipole-dipole interaction between p -wave Feshbachmolecules, the p z -wave Feshbach resonance splits from the other degenerate p x and p y -wave Fes-hbach resonances[33] where z -axis is taken to be parallel to an external magnetic field. As a result,the realized p -wave interaction V p ( p , p ′ ) essentially has uniaxial anisotropy, leading component-dependence of v i and R i . However, for the e ff ective range R i the anisotropy, as well as the field-dependence is not remarkable as reported from the experiment on K Fermi gas[33]. Thus wesimply assume the e ff ective range is channel-independent as R x = R y = R z = R and we take R = . p − following the experimental situation in Ref. [44]. In this assumption, the anisotropyof the p -wave interaction is described by the di ff erence of the inverse scattering volume betweenthe stronger p z -wave component and the weaker p x and p y -component as v − z ≥ v − x = v − y . Thuswe introduce a parameter δ v − = v − z − v − x = v − z − v − y > p z -wave resonance. Although,near the p x and p y -wave resonances in K Fermi gas, the p z -wave interaction is expected to beweakly repulsive, this situation cannot be described in our model where all p -wave interaction isassumed to be attractive. In addition, we mention that the p -wave Feshbach resonance has alsobeen observed in Li Fermi gas[35–41], where, in contrast to K case, the splitting of the p -waveFeshbach resonances has not been reported, that means in this system the anisotropy of the p -waveinteraction is weak compared to K Fermi gas. Thus we also treat δ v − as a tunable parameter, andwe clarify how the anisotropy a ff ects the p -wave contacts. In this scale, the interaction strength isdescribed by only v − z for given δ v − , and the weak-coupling BCS regime and the strong-couplingBEC regime are characterized as ( v z k ) − < v z k ) − >
0, respectively (where k F is the Fermimomentum).To investigate strong-coupling and finite temperature e ff ects on the p -wave contacts above the5 -wave superfluid transition temperature T c , we include fluctuations in the p -wave Cooper channelwithin a strong-coupling theory developed by Nozi`eres and Schmitt-Rink (NSR theory)[5, 62–65].In this approximation, the total thermodynamic potential is given by Ω = Ω + δ Ω , (7)where Ω = T X p ln h + e − ξ p / T i (8)is the non-interacting part and the strong-coupling correction δ Ω is diagramatically shown in Fig.1. Summing up these diagrams, we obtain δ Ω = T X q ,ν n Tr ln h − ˆ Γ − ( q , i ν n ) i . (9)Here ˆ Γ = { Γ i , j } ( i , j = x , y , z ) is a 3 × p -wave Cooperchannel given by ˆ Γ ( q , i ν n ) = − ˆ U p − ˆ U p ˆ Π ( q , i ν n ) , (10)where, ˆ U P = diag[ U x , U y , U z ], and Π i , j ( q , i ν n ) = X p γ i ( p ) tanh (cid:18) ξ p + q T (cid:19) + tanh (cid:18) ξ − p + q T (cid:19) ξ p + q + ξ − p + q − i ν n γ j ( p ) (11)is the 3 × p -wave contacts, we briefly summarize the Tan’scontact in an s -wave interacting Fermi gas. In this case, the Tan’s contact C s is originally definedfrom the asymptotic behavior of the large-momentum distribution n p as n p → C s p , ( s − wave) (12)where high momentum limit k F ≪ p is taken. C s is also proportional to the derivative of the totalenergy of the system with respect to the inverse s -wave scattering length a − s as C s = − π m ∂ E ∂ a − s . ( s − wave) (13)This equation is known as the adiabatic energy relation.6n the p -wave case, due to the momentum dependence of the p -wave interaction V p ( p , p ′ ), theasymptotic form of the momentum distribution consists of two terms as n p → C ( ˆ p ) p + C ( ˆ p ) p , ( p − wave) (14)where ˆ p = p / | p | . On the other hand, the adiabatic energy relation Eq. (13) is straightforwardlyextended to the p -wave case as C ( i ) v = − m ∂ Ω ∂ v − i ! T ,µ , (15) C ( i ) R = − m ∂ Ω ∂ R − i ! T ,µ . (16)We note that since the p -wave interaction is characterized by 6 parameters ( v i and R i for i = x , y , z ), the p -wave contacts also consist of 6 components, in principle. For the leading term inEq. (14) ( ∝ p − ), C ( i ) v defined from Eq. (15) completely capture the coe ffi cient C as C ( ˆ p ) ∝ P i C ( i ) v ˆ p i . However, as first mentioned in [61], C in the sub-leading term cannot be written byonly C ( i ) R and the extra terms appear. These extra terms also exist within our theoretical framework.To clearly see this, although in the original Tan’s work[45–47] the contact is defined from themomentum distribution, here, we first define the p -wave contacts C ( i ) v and C ( i ) R from the adiabaticenergy relations Eq. (15) and Eq. (16), respectively. Substituting the thermodynamic potential Ω Eq. (7) into Eqs. (15) and (16), and using Eqs. (5) and (6) to eliminate U i and F ( i ) c , we obtain theexpressions for the p -wave contacts C ( i ) v and C ( i ) R within the NSR theory as C ( i ) v = − m πβ X q ,ν n Γ i , i ( q , i ν n ) , (17) C ( i ) R = m πβ X q ,ν n (cid:16) ε B q − i ν n − µ (cid:17) Γ i , i ( q , i ν n ) , (18)where ε B q = q / (4 m ) is a kinetic energy of a pair molecule with center-of-mass momentum q and ν n = π nT is boson Matsubara frequency. On the other hand, the momentum distribution n p isconveniently calculated from the single-particle Green’s function G ( p , i ω m ) as n p = β X ω m G ( p , i ω m ) , (19)where ω m = (2 m + π T is fermion Matsubara frequency. G ( p , i ω m ) being consistent with thethermodynamic potential Eq. (7) is given by[63–65] G ( p , i ω m ) = G ( p , i ω m ) + G ( p , i ω m ) Σ ( p , i ω m ) G ( p , i ω m ) , (20)7here Σ ( p , i ω m ) = β X q ,ν n X i , j = x , y , z γ i (cid:18) p − q (cid:19) Γ i , j ( q , i ν n ) γ j (cid:18) p − q (cid:19) G ( − p + q , − i ω m + i ν n ) (21)is the strong-coupling self-energy correction. In the large momentum region p F ≪ p ≪ R − , n p becomes n p → π X i C ( i ) v ˆ p i p + π X i (cid:16) C ( i ) R + η i (cid:17) ˆ p i + πζ + π X i , j κ i , j ˆ p i ˆ p j p , (22)where ˆ p i = p i / | p | . Indeed, we find that the extra terms ζ , η i and κ i j appear in the sub-leading termin Eq. (22). The coe ffi cients of these extra terms, ζ , η i and κ i j are given by ζ = − m πβ X q ,ν n X i , j q i q j Γ i , j ( q , i ν n ) , (23) η i = m πβ X q ,ν n q + q i ! Γ i , i ( q , i ν n ) + q i X j , i q j (cid:16) Γ i , j ( q , i ν n ) + Γ j , i ( q , i ν n ) (cid:17) , (24) κ i , j = − m πβ X q ,ν n q i Γ i , i ( q , i ν n ) , ( i = j ) − m πβ X q ,ν n h q i Γ j , j ( q , i ν n ) + q j Γ i , i ( q , i ν n ) + q i q j (cid:16) Γ i , j ( q , i ν n ) + Γ j , i ( q , i ν n ) (cid:17)i . ( i , j )(25)From Eqs. (23)-(25), we find that ζ , η , and κ result from the center-of-mass motion of the pairsas discussed in Ref. (Note that Γ i , j ( q , i ν n ) physically describes the pair propagation.) At T = T c , the preformed Cooper pairs with finitemomentum q , ζ , η i and κ i , j to the sub-leading term of n p becomes more dominant than C ( i ) R in a high temperature regime.We will also discuss in Sec. V how strong the e ff ects of them are in the experimental situation inRef.[44]. We briefly note that, using the rotational symmetry around z -axis of V p ( p , p ′ ) in Eq. (1),we obtain C ( x ) v = C ( y ) v , C ( x ) R = C ( y ) R , η x = η y , κ x , z = κ y , z , and κ x , x = κ y , y = κ x , y = κ y , x To calculate C ( i ) v and C ( i ) R , as well as ζ , η i and κ i j , in the normal phase, we need to determinethe suprefluid transition temperature T c and the chemical potential µ ( T ). In the present uniaxiallyanisotropic case ( U z ≥ U x = U y ), the p -wave superfluid transition from the normal phase alwaysoccurs in the stronger p z -wave Cooper channel. Thus the equation for T c is obtained from the8houless criterion in the p z -wave channel Γ − zz (0 , = = U z X p γ z ( p )2 ξ p tanh ξ p T ! . (26)By solving Eq. (26) together with the particle number equation obtained from the thermodynamicpotential Ω as N = − ∂ Ω ∂µ ! T = X p n F ( ξ p ) + T X q ,ν n Tr " ˆ Γ ( q , i ν n ) ∂ ˆ Π ( q , i ν n ) ∂µ ! T , (27)we determine T c and µ ( T = T c ). After that, the chemical potential above T c is simply obtained bysolving only Eq. (27) for given T ( > T c ), v − z and δ v − .For the comparison with the recent experiment, here, we summarize how to measure the p -wavecontacts from the momentum distribution. Experimentally, the momentum distribution is observedby taking absorption image after time-of-flight with an imaging beam. The imaging beam used inRef. [44] propagates along z -direction parallel to the Feshbach magnetic field. Then one obtainsthe 2-dimensional momentum distribution ¯ n ( p x , p y ) which is the averaged n p over the p z directionas ¯ n (cid:16) p x , p y (cid:17) = R dp z π n p . When we ignore the extra terms from Eq. (22), we obtain a formula¯ n (cid:16) p x , p y (cid:17) → π ¯ C v ρ + π ¯ C R ρ , (28)where ρ = q p x + p y and ¯ C v = C ( z ) v + X i = x , y C ( i ) v , (29)¯ C R = C ( z ) R + X i = x , y C ( i ) R . (30)We briefly note that the coe ffi cients 1 / / p x - and p y -component of the p -wave contacts are due to the anisotropy of the momentum distribution. Theyalso assume that C xv = C yv and C xR = C yR can be negligible near the p z -wave resonance field becausethe p z -wave resonance and the p x and p y -wave resonance are well-separated in K Fermi gas.Finally, they obtain C ( z ) v and C ( z ) R near the p z -wave resonance by fitting.9he e ff ects from the extra terms do not considered in Eq. (28), but they actually modify Eq.(28) as ¯ n ( ρ ) → π ¯ C v ρ + π (cid:16) ¯ C R + δ (cid:17) ρ . (31)where δ = ζ + η z + X i = x , y η i + κ z , z + X i = x , j (cid:0) κ i , z + κ z , i (cid:1) + X i , j = x , y κ i , j (32)is the correction, coming from the extra terms, to the coe ffi cient of the sub-leading term ( ∝ ρ − ) of¯ n (cid:16) p x , p y (cid:17) . Thus, the experimental result for C ( z ) R should be compared not with ¯ C R but with ¯ C R + δ Since the measurement has been done under the trap potential for confining the atoms, themeasured ¯ n ( p x , p y ) is also spatially averaged over the gas cloud. Thus we also need to considerthe e ff ects of the trap potential to directly compare with the experiment. Using the local densityapproximation (LDA)[66], this is simply achieved by replacing µ by the position dependent chem-ical potential µ ( r ) = µ − V ( r ) where V ( r ) = m ω r / r measured from the trap center ( r =
0) with the trap frequency ω . Then, Eqs. (7)-(11) becomeposition dependent. From the total thermodynamic potential R d r Ω ( r ), the LDA particle numberequation is given by N = − Z d r ∂ Ω ( r ) ∂µ ( r ) ! T , (33)and the equation for T c is derived from the Thouless criterion at the trap center r = C ( i ) v and C ( i ) R ,as well as ζ , η i and κ i , j in the trapped system are obtained by replacing µ by µ ( r ) and carrying outthe r integral. For example, C ( i ) v = − m Z d r ∂ Ω ( r ) ∂ v − i ! T ,µ , (34) C ( i ) R = − m Z d r ∂ Ω ( r ) ∂ R − i ! T ,µ . (35)In Sec. III and IV, we first focus on the p -wave contacts and the extra terms in the uniformsystem and we will discuss how strong pairing fluctuations and finite temperature a ff ect thesequantities. After that, including the e ff ects of the harmonic trap potential, we directly compare ourresults with the recent measurement in Sec. V.10 C v ( i ) k F / N C v ( z ) (a) -0.04-0.03-0.02-0.01 C R / ( N k F ) ( i ) C R ( z ) (c) C v ( x ) (b) C v ( y ) = C v ( i ) k F / N -25 -20 -15 -10 -5 0 5 10 15 20 25 ( v z k F3 ) - C R ( x ) (d) C R ( y ) = ( d v k F3 ) - = = = -0.04-0.03-0.02-0.01 C R / ( N k F ) ( i ) FIG. 2: (Color online) calculated p -wave contacts C ( i ) v and C ( i ) R ( i = x , y , z ) at the superfluid transitiontemperature T c as a function of the strength of the p z -wave interaction ( v z k ) − for three typical anisotropicparameter [( δ vk ) − = , , C ( i ) v and C ( i ) R in the BEC limit in theisotropic case and the anisotropic limit. III. p -WAVE CONTACTS AT THE SUPERFLUID TRANSITION TEMPERATURE IN UNI-FORM SYSTEM Figure 2 shows the p -wave contacts (a), (b) C ( i ) v and (c), (d) C ( i ) R for i = x , y , z in a uniformone-component Fermi gas with a tunable p -wave interaction with a uniaxial anisotropy at T = T c as a function of the interaction strength ( v z k ) − for ( δ vk ) − = , ,
8. As shown in Fig. 2 (a) and11b), C ( i ) v monotonically increases as a function of ( v z k ) − from the weak-coupling regime to thestrong-coupling regime. On the other hand, from Fig. 2 (c) and (d), we find that although C ( i ) R firstincreases in the weak-coupling regime [( v z k ) − < ∼ v z k ) − , C ( i ) R decreasesand vanishes around ( v z k ) − =
0. Eventually, C ( i ) R becomes negative in the strong-coupling regime( v z k ) − > ff erent interaction-dependence between C ( i ) v and C ( i ) R , we start from the strong-coupling BEC limit. Noting that, in this limit, ˆ Γ ( q , i ν n ) is reduced to the single-particle BoseGreen’s function as Γ i , j ( q , i ν n ) = α ( µ ) h i ν n − (cid:16) ε Bq − µ ( i ) B (cid:17)i δ i , j . (36)Here α ( µ ) = X p F ( p )12 ξ p ≃ m π R − − p m | µ | ! , (37)and µ ( z ) B = , (38) µ ( x ) B = µ ( y ) B = − m δ v − πα ( µ ) , (39)is the chemical potential of the p i -wave molecular Boson. In deriving Eq. (36), we use the fact that | µ | ≫ T c in this limit. Substituting Eq. (36) into Eqs. (17) and (18), we obtain the strong-couplingexpressions for C ( i ) v and C ( i ) R as C ( i ) v ≃ R − − p m | µ | / N ( i )B ≃ − m ∂ E ( i )bind ∂ v − i N ( i )B , (40) C ( i ) R ≃ m µ NR − − p m | µ | / N ( i )B ≃ − m ∂ E ( i )bind ∂ R − i N ( i )B , (41)where N ( i )B = P q n B (cid:16) ε B q − µ ( i )B (cid:17) and E ( i )bind = − R / ( mv i ) are, respectively, the number and the two-body binding energy of the p i -wave molecular Boson [ n B ( ε ) is the Bose distribution function].In figure 2, Eqs. (40) and (41) in the isotropic limit [( δ vk ) − =
0] and in the anisotropic limit[( δ vk ) − = ∞ ] are also shown (see the chain lines in each panel) and we find these expressionswell-describe the behavior of C ( i ) v and C ( i ) R in the strong-coupling regime [( v z k ) − > ∼ P i E ( i )bind N ( i )B . Furthermore, the number of the molecules12oes not change even when v i and R i are varied as far as | E bind | ≪ T . Thus, the last expression ofEqs. (40) and (41) are, indeed, proportional to the derivative of the total energy with respect to v − i and R − i , respectively. That is consistent with the adiabatic energy relations.From the relation between v i , R i and U i , p ( i ) c [Eqs. (5) and (6)], when one increases v − i withfixing R , only U i increases, but p ( i ) c does not change. Thus the increase of v − i simply means theincrease of the coupling constant U i . Thus E bind < v − i , leading thepositive value of C ( i ) v in the whole interaction region. On the other hand, when one increases R − with fixing v − i , we find that p ( i ) c increases, but U i decreases at the same time to fix v − i [see Eq.(5)]. Thus there is a competition between the increase of p ( i ) c which enhances the interaction andthe decrease of U i which suppresses the interaction with increasing R − . In the BEC limit, thelatter is more dominant than the former. As a result, C ( i ) R has negative value in this regime.To consider the binding energy E ( i )bind of the p i -wave pair state in the weak-coupling BCS regime,we need to include the e ff ects of the Fermi surface, which support the pairing phenomenon. Here,we estimate, E ( i )bind in the BCS regime by applying the Cooper problem to the p -wave case, wherethe equation for E ( i )bind is given as 1 U i = X | p |≥ p F γ i ( p )2 (cid:16) ε p − ε F (cid:17) − E ( i )bind . (42)Using ε F ≫ E ( i )bind in the weak-coupling limit, we obtain E ( i )bind = − ε F exp " π k F R + k v i ! − . (43)We find that, in contrast to the strong-coupling regime, E ( i )bind is a decreasing function of not only v − i but also R − . That means the increase of p ( i ) c is more significant for E ( i )bind than the decreasesof U i with increasing R − and C ( i ) R becomes positive. We briefly comment on the e ff ects of theanisotropy. As shown in Fig. 2, C ( z ) v and C ( z ) R become remarkable when the anisotropy becomesstrong, as expected. However, the ( v z k ) − -dependence of C ( i ) v and C ( i ) R are not significantly a ff ectedby the anisotropy.The coe ffi cients of the leading term ( ∝ ρ − ) ¯ C v and the sub-leading term ( ∝ ρ − ) ¯ C R + δ of¯ n (cid:16) p x , p y (cid:17) [Eq. (31)] at T c and ( δ vk ) − = C v and ¯ C R + δ seems to be similar to one of C ( i ) v and C ( i ) R shown in Fig. 2. We also note that, at T = T c , the contribution from the extra terms δ does not remarkably a ff ects the results over theentire interaction region [see the dotted line in Fig. 3 (b)]. This means the thermal excitations ofthe pair molecules are su ffi ciently suppressed, at least at T c .13 C - v k F / N C - v -0.04-0.03-0.02-0.01 0 0.01-25-20-15-10 -5 0 5 10 15 20 25 C - R / ( N k F ) , d / ( N k F ) ( v z k F3 ) - C - R + d C - R d (a) (b) FIG. 3: (Color online) coe ffi cient of (a) the leading term ¯ C v and (b) the sub-leading term ¯ C R + δ of theasymptotic form of ¯ n ( p x , p y ) at T c and ( δ vk ) − =
8, as a function of the interaction strength ( v z k ) − . Thedashed line and the dotted line in panel (b) show ¯ C R and δ , respectively. IV. TEMPERATURE DEPENDENCE OF THE p -WAVE CONTACTS IN UNIFORM SYSTEM Figure 4 and 5 show the temperature dependence of the p -wave contacts C ( i ) v , ¯ C v (in Fig. 4)and C ( i ) R , ¯ C R (in Fig. 5) in the weak-coupling regime ( v z k ) − = −
20 [Fig. 4 (a) and 5 (a)] andin the strong-coupling region ( v z k ) − =
20 [Fig. 4(b) and 5(b)] with ( δ vk ) − =
8. For ¯ C v and¯ C R , while in the strong-coupling regime, both ¯ C v and ¯ C R gradually vanish as increasing T , in theweak-coupling regime, we find a non-monotonic temperature dependence of ¯ C v and ¯ C R . The keyto understand this non-monotonic behavior is a competition between two finite temperature e ff ectson the p -wave contacts. To more clearly see this, we rewrite Eqs. (17) and (18) by using thespectral representation as C ( i ) v = − m π X q Z dz Im (cid:2) Γ i , i ( q , i ν n → z + i δ ) (cid:3) n B ( z ) (44) C ( i ) R = m π X q Z dz (cid:16) ε B q − z − µ (cid:17) Im (cid:2) Γ i , i ( q , i ν n → z + i δ ) (cid:3) n B ( z ) (45)Here, Γ i , i ( q , i ν n → z + i δ ) is the analytic continued particle-particle scattering matrix, of whichthe imaginary part physically describes the spectrum of the collective mode. One of the im-14 C v k F / N
0 0.5 1 1.5 2 C v k F / N T / T F C v ( z ) C v ( x ) = C v ( y ) C - v (a)(b) FIG. 4: (Color online) Temperature dependence of the p -wave contacts related to the scattering volume ¯ C v in Eq. (29), C ( z ) v and C ( x ) v = C ( y ) v in (a) the weak-coupling regime ( v z k ) − = −
20 and in (b) the strong-coupling regime ( v z k ) − = −
20. In this figure we take ( δ vk ) − = portant finite temperature e ff ects is the change of the structure of Im (cid:2) Γ i , i ( q , i ν n → z + i δ ) (cid:3) withincreasing T . In Fig. 6(a), (c), and (e), the temperature dependence of Im (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) in the weak-coupling regime is shown. At T = T c , because of the Thouless criterion Eq. (26)the collective excitation is gapless. However, as increasing T from T c , the peak structure inIm (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) is gradually shifted to higher energy region, that suppresses ¯ C v and ¯ C R .The other finite temperature e ff ect is included n B ( z ) in Eqs. (44) and (45), which describes the oc-cupancy of the collective excitations with energy z . As increasing T , since the thermal excitationsbecome remarkable and the collective mode with high energy ( z < ∼ T ) starts to be occupied, lead-ing the enhancement of ¯ C v and ¯ C R . Fig. 6 (b), (d) and (f) show q Im (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) n B ( z ).From this figure, we find that the latter e ff ect is more remarkable than the former in the weak-coupling regime near T c . Thus ¯ C v and ¯ C R first increase with increasing T from T c . Of course, inthe high temperature limit, the e ff ects of the interaction become negligible and ¯ C v and ¯ C R vanishas expected.On the other hand, as shown in Fig. 7, in the strong-coupling regime, the change of the structure15 C R / ( N k F ) -0.04-0.03 -0.02-0.01 0 0.01 0.02
0 0.5 1 1.5 2 C R / ( N k F ) T / T F C R ( z ) C R ( x ) = C R ( y ) C - R (a)(b) FIG. 5: (Color online) Temperature dependence of the p -wave contacts related to the e ff ective range ¯ C R inEq. (30), C ( z ) R and C ( x ) R = C ( y ) R in (a) the weak-coupling regime ( v z k ) − = −
20 and in (b) the strong-couplingregime ( v z k ) − =
20. In this figure we take ( δ vk ) − = of Im (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) is always dominant in the whole temperature regime. In other words,the thermally dissociation of the preformed Cooper pairs is the most crucial finite temperaturee ff ects in the BEC regime. Thus, ¯ C v and ¯ C R monotonically decreases as decreasing the number ofthe molecular boson with increasing T .In Fig. 5, we also plot C ( i ) v and C ( i ) R . We find that the weaker-coupling components ( i = x , y ) aregradually enhanced with increasing T . When we simply consider the strong-coupling regime, thedi ff erence of the binding energy δ E bind between the stronger-coupling p z -wave molecules and theweaker-coupling p x and p y -wave molecules is given by δ E bind = R / ( m δ v ). In the low temperatureregime where T ≪ δ E bind , since most of Fermi atoms form the p z -wave pairs, C xv = C yv and C xR = C yR are negligibly small. However, as increasing T , the thermal transfer from the p z -wavemolecule to the p x and p y -wave molecule occurs. Then the di ff erence between C ( z ) v and C ( x ) v = C ( y ) v (or C ( z ) R and C ( x ) R = C ( y ) R ) gradually becomes small, and eventually vanishes in the high temperatureregion where T ≫ δ E bind . Although this e ff ects are clearly seen when the δ E bind is comparableto the Fermi energy, unfortunately, in K Fermi gas δ E bind is much larger than the Fermi energy.16 IG. 6: (Color online) (a), (c), (e) calculated Im (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) and (b), (d), (f) q n B ( z )Im (cid:2) Γ z , z ( q , i ν n → z + i δ ) (cid:3) in the weak-coupling regime ( v z k ) − = −
20. In this figure we take q = (0 , , q ) and ( δ vk ) − = v z k ) − = Thus it is di ffi cult to observe this phenomenon, which essentially comes from the anisotropy ofthe p -wave interaction, in this system.In Fig. 8, we compare ¯ C R with δ in Eq. (31) as a function of T . As mentioned above, δ comesfrom the center-of-mass (c.m) motion of the pair molecules. Indeed, in high temperature regionsince the pairs are easily excited to the states with finite c.m momentum q , δ becomes remarkable,17 T / T F C - R + d C - R d -0.04 0 0.04 C - R / ( N k F ) , d / ( N k F ) C - R / ( N k F ) , d / ( N k F ) (a)(b) FIG. 8: (Color online) Comparison between the temperature dependence of the coe ffi cient ¯ C R and δ of thesub-leading term of the asymptotic form of the momentum distribution in Eq. (31). In panel (a) and (b),the results in the weak-coupling regime ( v z k ) − = −
20 and the strong-coupling regime ( v z k ) − =
20 areshown, respectively [( δ vk ) − = and when T > ∼ T F the coe ffi cient of the sub-leading term of ¯ n ( p x , p y ) is eventually dominated by δ , compared to ¯ C R . This results implies that in order to measure the p -wave contact ¯ C R , it isneeded to su ffi ciently decrease the temperature ( T ≃ T c ) or C ( i ) R should be directly estimated fromthe thermodynamic properties of the system. V. COMPARISON WITH THE EXPERIMENTAL VALUE OF THE p -WAVE CONTACTS Before showing the results in the trapped system, we first summarize our detailed numericalparameter setting in the calculations. Following Ref. [44], we take the total atomic number N = ω = v i is estimated from the formula v i = v bg i − ∆ i δ B i ! (46)where δ B i = B − B ( i )res ( B ( i )res is the resonance field in the p i -wave channel). For v bg i , ∆ i and B ( i )res ,we use the values in Ref. [33]. In the following, the results are shown as a function of δ B z . For18 T c / T F -1-0.5 0 0.5-0.2 -0.1 0 0.1 0.2 0.3 0.4 m / e F d B (G)0.2 T F T F FIG. 9: (Color online) phase diagram of a p -wave interacting Fermi gas with a harmonic trap potential andchemical potential at the trap center on T c , calculated within the local density approximation. As shownfrom this figure, at T = . T F the superfluid transition occurs in the weak-coupling regime ( δ B ≃ . simplicity, we ignore the magnetic field dependence of the e ff ective range. However, we mentionthat, in the region where the calculation is done, R changes only about 2 percent.Figure 9 shows the phase diagram of a one-component Fermi gas with a p -wave interactionand the chemical potential at T c calculated within the LDA. Although the validity of the LDA for T c and µ , as well as the p -wave contacts is still unclear in the p -wave case, in the s -wave case, theexperimental results on the Tan’s contact is well-described within the LDA. In Ref. [44], it wasmentioned that the experiment has been done at T = . T F and the system is in the normal phase.However, within the LDA, at T = . T F , the superfluid transition occurs at δ B ≃ . C v and ¯ C R + δ with the experimental results. In additionto the result at the experimental temperature T = . T F , we also plot the results at T = . T F , inorder to compare in the intermediate and the strong-coupling region. Experimentally, in the regionwhere δ B <
0, the p -wave molecules are unstable. Thus the p -wave contacts rapidly decreasearound δ B = p -wave molecules does not19 C v k F / N -0.02-0.01 0 0.01 0.02 0.03-0.2 -0.1 0 0.1 0.2 0.3 0.4 / ( k F N ) d B (G) T / T F = = C - R + d () - FIG. 10: (Color online) calculated ¯ C v and ¯ C R + δ at T = . T F (solid line) and 0 . T F (dashed line) as afunction of the external magnetic field. The filled square with the error bar shows experimental results fromRef. [44]. considered, even in this regime ¯ C v and ¯ C R + δ have finite value. Our calculation describes thecharacteristic behavior of ¯ C v and ¯ C R + δ . For example, the increase of ¯ C v as a function of δ B anda peak structure appearing in δ B dependence of ¯ C R + δ . However, our results cannot qualitativelydescribe the experimental results. In Fig. 11, we also compare ¯ C R with δ in the trapped system.We find that even at T = . T F , the contribution from δ cannot be ignored in the strong-couplingregime.Finally, we mention that the p -wave contacts explicitly depend on the atomic number N . Fig-ure 12 shows the p -wave contacts as a function of δ B for three di ff erent particle number. The N -dependence of ¯ C v and ¯ C R + δ originates from the fact that the p i -wave interaction is essentiallydescribed by two parameters, i.e. ( v i k ) − and Rk F . Experimentally v i and R are tuned with varyingthe external magnetic field and the Fermi momentum k F is determined from the total atomic num-ber N . Thus, in order to fix both ( v i k ) − and Rk F , we have to tune not only the external magneticfield and but also the atomic number. As a result, even when ¯ C v and ¯ C R + δ is renormalized by theatomic number, the value of these quantities as a function of the magnetic field still depends on the20 C R / ( k F N ) d B (G) C - R + d C - R d FIG. 11: (Color online) Comparison between ¯ C R and δ in K Fermi gas with the harmonic trap potential at T = . T F . -0.02-0.01 0 0.01-0.2 -0.1 0 0.1 0.2 0.3 0.4 d B (G) N =
N =
N = / ( k F N ) C - R + d () C v k F / N - FIG. 12: (Color online) particle number dependence of calculated ¯ C v and ¯ C R + δ at T = . T F . We showthe results for N = atomic number. Thus, to precisely measure the p -wave contacts, one needs to do the measurementwith fixing the atomic number in addition to tuning of the external magnetic field.21 I. SUMMARY
To summarize, including the e ff ects of the p -wave pairing fluctuations within the NSR theory,we theoretically investigated strong-coupling and finite temperature e ff ects on the p -wave con-tacts. We found that while the p -wave contact related to the scattering volume simply increases asincreasing the interaction strength, one related to the e ff ective range non-monotonically dependson the interaction strength and its sign changes in the intermediate regime. We clarified that thisnon-monotonic behavior originate from the competition between the increase of the cuto ff mo-mentum and the decrease of the coupling constant with increasing the e ff ective range. We alsoinvestigate the temperature dependence of the p -wave contacts, as well as the contribution fromthe extra terms which is not related to the thermodynamic properties. We showed that in hightemperature regime the sub-leading behavior of the large-momentum distribution is dominatedby the extra terms. This results imply that in order to measure the p -wave contacts, one needto decrease the temperature near the superfluid transition temperature T c or directly observe thethermodynamic quantities of the system.By including the e ff ects of the trap potential within the local density approximation, we directlycompared our results with the recent experiment. Although in quantitative level, the experimentalresults can not be described by our model, we obtained similar magnetic field dependence of the p -wave contacts. We also mentioned that the p -wave contacts explicitly depends on the atomicnumber due to the fact that the p -wave interaction is characterized by two parameters. Thus weconcluded that in order to precisely measure the p -wave contacts one needs not only to tune theexternal magnetic field but also to control the particle number of the system. In the current stage ofcold atom physics, it is one of the most challenging issues to understand how the p -wave pairingfluctuations a ff ects the many-body properties of the system. Since the p -wave contacts connectthe microscopic properties with the thermodynamic properties, our results are useful for furtherunderstanding the physical properties in the p -wave interacting Fermi gas in the normal phase. Acknowledgments
This work was supported by KiPAS project in Keio University. DI was supported by Grant-in-aid for Scientific Research from JSPS in Japan (No.JP16K17773). YO was supported by Grant-in-aid for Scientific Research from MEXT and JSPS in Japan (No.JP15H00840, No.JP15K00178,22o.JP16K05503). [1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. , 1225, (2010).[2] E. Timmermans, K. Furuya, P. W. Milonni, and A. K. Kerman, Phys. Lett. A , 228 (2001).[3] D. M. Eagles, Phys. Rev. , 456 (1969).[4] A. J. Leggett, in Modern Trends in the Theory of Condensed Matter (Springer, Berlin, 1960).[5] P. Nozi`eres, and S. Schmitt-Rink, J. Low Temp. Phys. , 195 (1985).[6] C. A. R. Sa de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett, , 3202 (1993).[7] Y. Ohashi and A. Gri ffi n, Phys. Rev. Lett , 130402 (2002).[8] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. , 040403 (2004).[9] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys.Rev. Lett. , 120403 (2004).[10] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature , 744 (2008).[11] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Nature Phys. , 569 (2010).[12] Y. Sagi, T. E. Drake, R. Paudel, R. Chapurin, and D. S. Jin, Phys. Rev. Lett. , 075301 (2015).[13] C. Sanner, E. J. Su, A. Keshet, W. Huang, J. Gillen, R. Gommers, and W. Ketterle, Phys. Rev. Lett. , 010402 (2011).[14] S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy and C. Salomon, Nature , 1057 (2010).[15] N. Navon, S. Nascimb`ene, F. Chevy, and C. Salomon, Science , 729 (2010).[16] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science , 563 (2012).[17] M. Horikoshi, M. Koashi, H. Tajima, Y. Ohashi, and M. Kuwata-Gonokami, Phys. Rev. X , 041004(2017).[18] Y. Ohashi, and A. Gri ffi n, Phys. Rev. A , 063612 (2003).[19] H. Hu, X. J. Liu, and P. D. Drummond, Phys. Rev. A , 023617 (2006).[20] R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, Phys. Rev. A , 023610 (2007).[21] N. Fukushima, Y. Ohashi, E. Taylor, and A. Gri ffi n, Phys. Rev. A , 033609 (2007).[22] H. Hu, X. J. Liu, and P. D. Drummond, Phys. Rev. A , 061605 (2008).[23] S. Tsuchiya, R. Watanabe, and Y. Ohashi, Phys. Rev. A , 033613 (2009).[24] Q. Chen, and K. Levin, Phys. Rev. Lett. , 190402 (2009).
25] H. Hu, X.-J. Liu, P. D. Drummond, and H. Dong, Phys. Rev. Lett. , 240407 (2010).[26] S. Tsuchiya, R. Watanabe, and Y. Ohashi, Phys. Rev. A , 043647 (2011).[27] A. Perali, F. Palestini, P. Pieri, G. C. Strinati, J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin,Phys. Rev. Lett. , 060402 (2011).[28] T. Kashimura, R. Watanabe, and Y. Ohashi, Phys. Rev. A , 043622 (2012).[29] H. Tajima, T. Kashimura, R. Hanai, R. Watanabe, and Y. Ohashi, Phys. Rev. A , 033617 (2014)[30] P. van Wyk, H. Tajima, R. Hanai, Y. Ohashi, Phys. Rev. A , 013621 (2016).[31] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. , 053201 (2003).[32] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature (London) , 47 (2003).[33] C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Phys. Rev. A , 042712 (2004).[34] J. Zhang, E. G. M. van Kempen, T. Bourdel, L. Khaykovich, J. Cubizolles, F. Chevy, M. Teichmann,L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. A , 030702(R) (2004).[35] C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A. Simoni, E. Tiesinga,C. J. Williams, and P. S. Julienne, Phys. Rev. A , 045601 (2005).[36] K. G ¨unter, T. St¨oferle, H. Moritz, M. K ¨ohl, and T. Esslinger, Phys. Rev. Lett. , 230401 (2005).[37] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. , 200403 (2007).[38] Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda, and T. Mukaiyama, Phys. Rev.Lett. , 100401 (2008).[39] J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands, P. Hannaford, and C. J. Vale ,Phys. Rev. A , 053616 (2008).[40] T. Nakasuji, J. Yoshida, and T. Mukaiyama, Phys. Rev. A , 012710 (2013).[41] R. A. W. Maier, C. Marzok, and C. Zimmermann, and Ph. W. Courteille, Phys. Rev. A , 064701(2010).[42] M. Jona-Lasinio, L. Pricoupenko, and Y. Castin, Phys. Rev. A , 043611 (2008).[43] J. Levinsen, N. R. Cooper, and V. Gurarie, Phys. Rev. A , 063616 (2008).[44] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thywissen, Nature Phys. , 599 (2016).[45] S. Tan, Ann. Phys. , 2952 (2008).[46] S. Tan, Ann. Phys. , 2971 (2008).[47] S. Tan, Ann. Phys. , 2987 (2008).[48] S. Zhang and A. J. Leggett, Phys. Rev. A , 023601 (2009).[49] E. D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford and C. J. Vale, Phys. Rev. Lett. , 170402 , 021605 (2010).[51] W. Schneider and M. Randeria, Phys. Rev. A , 021601 (2010).[52] J. T. Stewart, J. P. Gaebler, T. E. Drake and D. S. Jin, Phys. Rev. Lett. , 235301 (2010).[53] F. Werner and Y. Castin, Phys. Rev. A , 013626 (2012).[54] Y. Sagi, T. E. Drake, R. Paudel and D. S. Jin, Phys. Rev. Lett. , 220402 (2012).[55] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut, and S. Gandolfi, Phys. Rev. Lett. , 055305 (2013).[56] Z. Yu, J. H. Thywissen, and S. Z. Zhang, Phys. Rev. Lett. , 135304 (2015).[57] S. M. Yoshida and M. Ueda, Phys. Rev. Lett. , 135303 (2015).[58] S. M. Yoshida and M. Ueda, Phys. Rev. A , 033611 (2016).[59] M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Phys. Rev. Lett. , 045301 (2016).[60] S. L. Zhang, M. Y. He, and Q. Zhou, Phys. Rev. A , 062702 (2017)[61] S. G. Peng, X. Liu, H. Hu, Phys. Rev. A , 063651 (2016)[62] Y. Ohashi, Phys. Rev. Lett. , 050403 (2005).[63] D. Inotani, P. van Wyk, and Y, Ohashi, J. Phys. Soc. Jpn. , 123301 (2016).[64] D. Inotani, P. van Wyk, and Y, Ohashi, J. Phys. Soc. Jpn. , 024302 (2017)[65] D. Inotani, P. van Wyk, and Y, Ohashi, J. Phys. Soc. Jpn. , 044301, (2017)[66] C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press,Cambridge, United Kingdom, 2002).[67] V. Gurarie, L. Radzihovsky, and A. V. Andreev, Phys. Rev. Lett. , 230403 (2005).[68] V. Gurarie, L. Radzihovsky, Ann. Phys. , 2 (2007).[69] D. Inotani, R. Watanabe, M. Sigrist, Y. Ohashi, Phys. Rev. A. , 053628 (2012).[70] D. Inotani, and Y. Ohashi, Phys. Rev. A , 063638 (2015).[71] M. Iskin and C. A. R. S´a de Melo, Phys. Rev B , 224513 (2005).[72] M. Iskin and C. A. R. S´a de Melo, Phys. Rev. Lett. , 040402 (2006).[73] C.-H. Cheng and S.-K. Yip, Phys. Rev. Lett. , 070404 (2005), Phys. Rev. B , 064517 (2006)., 064517 (2006).