Atom-dimer and dimer-dimer scatterings in a spin-orbit coupled Fermi gas
AAtom-dimer and dimer-dimer scatterings in a spin-orbit coupled Fermi gas
M. Iskin
Department of Physics, Ko¸c University, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey (Dated: February 16, 2021)Using the diagrammatic approach, here we study how spin-orbit coupling (SOC) affects thefermion-dimer and dimer-dimer scattering lengths in the Born approximation, and benchmarktheir accuracy with the higher-order approximations. We consider both isotropic and Rashba cou-plings in three dimensions, and show that the Born approximation gives accurate results in the1 / ( mαa s ) (cid:28) − m is the mass of the fermions, α is the strength of the SOC, and a s isthe s -wave scattering length between fermions. This is because the higher-loop contributions forma perturbative series in the 1 / ( mαa s ) < Z of the dimer propagator. In sharp contrast, since Z grows with the square-root of the bindingenergy of the dimer in the 1 / ( mαa s ) > I. INTRODUCTION
The diagrammatic approach has proven to be a pow-erful technique for studying few-body problems in manybranches of theoretical physics. For instance, in the con-text of short-range two-body interactions between parti-cles, it has been successfully applied to both the three-body [1–7] and four-body [2, 3, 8, 9] problems to verifythe known exact results for the fermion-dimer [10, 11]and dimer-dimer [12] scattering lengths, respectively. Inaddition, the approach have recently been generalized tothe three-body problem with arbitrary-range two-bodyinteractions, and applied to the electron-exciton scatter-ing in semiconductors, i.e., to the so-called three-bodyCoulomb problem [13].Furthermore, in the context of BCS-BECcrossover [14], the fermion-dimer and dimer-dimerscattering lengths appear in some of the many-bodyproperties of dilute Fermi gases, including their low-energy collective modes, superfluid density, etc.. Suchappearances are quite natural in those parameter regimeswhere a strongly interacting Fermi-Fermi mixture can bemapped to a weakly-interacting Bose-Fermi mixture ofpaired (i.e., bosonic dimers) and unpaired (i.e., excess)fermions [4, 15, 16]. However, it is also known that theusual treatment of the BCS-BEC crossover through aGaussian fluctuation approach yields fermion-dimer anddimer-dimer scattering lengths that are consistent withthe lowest-order Born approximation [2, 3, 8].Given the recent surge of experimental [17–20] andtheoretical [21–27] interests in spin-orbit-coupled Fermigases, here we extend the diagrammatic approach to therelevant few-body problems. In particular, we study howSOC affects the fermion-dimer and dimer-dimer scat-tering lengths in the Born approximation, and bench-mark their accuracy with the higher-order approxima-tions. Our primary findings for the isotropic and Rashbacouplings in three dimensions are as follows. We showthat the Born approximation gives accurate results in the1 / ( mαa s ) (cid:28) − / ( mαa s ) < Z of the dimer prop- agator. While Z decays to 0 in the 1 / ( mαa s ) → −∞ limit, it grows with the square-root of the binding en-ergy of the dimer in the 1 / ( mαa s ) > / ( mαa s ) < II. ONE-BODY PROBLEM
In the (cid:104)↑ | = (cid:0) (cid:1) and (cid:104)↓ | = (cid:0) (cid:1) basis of the σ z Pauli matrix, the single-particle problem is governed bythe Hamiltonian matrix h k = ε k σ + α k · σ (1)in momentum space, where k = ( k x , k y , k z ) is the wavevector, ε k = k / (2 m ) is the usual dispersion with k = (cid:113) k x + k y + k z in units of (cid:126) = 1, σ is a unit matrix, α ≥ k space, and σ = ( σ x , σ y , σ z ) is a vectorof Pauli matrices. The eigenvalues and eigenvectors of a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b h k are determined by the unitary transformation U k = 1 (cid:112) k ( k − k z ) (cid:18) k x − ik y k z − kk − k z k x + ik y (cid:19) , (2)where U † k h k U k gives the dispersion relations of the s = ± helicity bands ε s k = k m + sαk, (3)and U k | ↑ ( ↓ ) (cid:105) gives the corresponding eigenstates. Weillustrate these dispersions in Fig. 1 as a function of k ,and note that the ground state of the − -helicity bandcorresponds to a degenerate shell of k states with theradius k m = mα and energy ε − k m = − mα / s = + s = -k m ε ks FIG. 1: One-body dispersions ε s k = k / (2 m ) + sαk for the s = ± helicity bands. The minimum of the lower band cor-responds to a shell of k states with the radius k m = mα andenergy ε − k m = − mα / Given the Hamiltonian matrix in Eq. (1), the propa-gator of the single particle can be written as G ( k , k ) = 1( k + i + ) σ − h k , (4)where k is the energy, and we set the chemical potential µ to 0 for the few-body problems of interest below. In ouranalysis, we reexpress such propagators via the genericrelation 1 / ( Aσ − B · σ ) = ( Aσ + B · σ ) / ( A − B ) =(1 / (cid:80) s ( σ + s (cid:98) B · σ ) / ( A − sB ) , where (cid:98) B = B /B and B = | B | . III. TWO-BODY PROBLEM
Having in mind the atomic Fermi gases where thebosonic dimer is a result of a short-range interaction be-tween ↑ and ↓ fermions, our two-body interaction is gov-erned by the Hamiltonian density h r = − gψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) (5)in real space, where g ≥ ψ † σ ( r ) and ψ σ ( r ) are the fermionicfield operators. A convenient way to understand theaction of this term is through a Hubbard-Stratonovich transformation in the imaginary-time functional path-integral formalism [22–27]. Introducing the Hubbard-Stratonovich fields ∆ = − gψ ↓ ψ ↑ and ¯∆ = − g ¯ ψ ↑ ¯ ψ ↓ , where ¯ ψ σ and ψ σ are the corresponding Grassmann vari-ables with suppressed arguments x = ( r , τ ) for notationalsimplicity, the action that corresponds to Eq. (5) is re-placed by three terms ¯∆∆ /g + ∆ ¯ ψ ↑ ¯ ψ ↓ + ¯∆ ψ ↓ ψ ↑ . If oneinterprets ¯∆ = ∆ ∗ as the complex dimer field then thefirst term describes free dimers with a bare propagator − g , and the second and third terms describe the dimer-fermion and fermion-dimer conversion processes, respec-tively. = + q , q q , q q , q q , q k + q , k + q − k , − k Π( q , q ) FIG. 2: Diagrammatic representation of the two-body bind-ing problem. The dimer propagator (colored bars) is deter-mined by dressing its bare propagator (uncolored bars) withinfinitely-many fermion-fermion bubbles (solid and dashedlines), forming eventually a geometric series.
The diagrammatic representation of the two-bodybinding problem in k space is shown in Fig. 2, wherethe physical dimer propagator D ( q , q ) is determined bydressing its bare value, which is a constant in space,with repeated interactions between its fermionic con-stituents [2, 3]. The resultant geometric series can besummed over to yield D ( q , q ) = − g g Π( q , q ) , (6)where Π( q , q ) corresponds to the fermion-fermion bub-ble diagram that is given byΠ( q , q ) = Tr2 (cid:88) k σ y G ( k + q , k + q ) σ y G T ( − k , − k ) . (7)Here, Tr is a trace over the spin sector, and (cid:80) k represents (cid:80) k ,k = i (cid:82) d k (cid:82) dk / (2 π ) . In our diagrams, while thesolid lines correspond to the fermion propagators that aredescribed by Eq. (4), the dashed lines correspond to theirdimer partners that are described by the transpose T ofEq. (4). This is because the dimer is formed between aparticle that is governed by h k + q and a hole that is gov-erned by − h T − k in k space [22–27]. In accordance with theFeynman rules, each fermion line, dimer line and vertexcarries a factor of i . In addition, we associate each dimer-creation (-annihilation) vertex with an additional factorof ∓ iσ y to account for the fermion-dimer (dimer-fermion)conversion terms, i.e., − i ¯∆ σ y and i ∆ σ y , respectively, inthe particle-hole sectors.Noting the relation σ y B · σ T σ y = − B · σ , and inte-grating k in the upper half plane in which there are twosimple poles at k = − ε ± k , we find [25]Π( q , q ) = 14 (cid:88) ss (cid:48) k ss (cid:48) (cid:98) k · (cid:98) Q q − ε s k − ε s (cid:48) Q , (8)where Q = k + q . Equation (8) shows that only theintra-band processes contribute to the bubble diagramwhen the dimer is stationary, i.e., when its center-of-mass momentum q vanishes. Therefore, we can re-express the stationary bubble diagram as Π( , q ) =(1 / (cid:80) sk G s ( k , k + q ) G s ( − k , − k ) , where G s ( k , k ) = 1 k − ε s k + i + (9)is the fermion propagator in the s = ± helicity basis.In the lowest order in q and q , Eq. (6) has the genericstructure of a simple pole [22–27] D ( q , q ) = Zq − q / (2 m B ) + µ B + i + (10)where Z = 8 π ( | ε b | − mα ) / / ( m √ m | ε b | ) correspondsto the residue of the pole, 2 m/m B = 7 / − − mα / | ε b | ) / / − mα / | ε b | determines the effective massof the bosonic dimer, and µ B = 2 µ − ε b → − ε b cor-responds to its chemical potential. Noting that − mα is the two-body continuum threshold, the energy of thetwo-body bound state ε b ≤ − mα or the two-body bind-ing energy | ε b |− mα of the dimer can be simply found bylooking at the pole of D ( , ε b ), leading to the relation 1 =( g/ (cid:80) s k / (2 ε s k − ε b ) . In addition, we substitute g withthe usual t-matrix relation between two fermions in vac-uum without the SOC, 1 /g = − mV / (4 πa s )+ (cid:80) k / (2 ε k ) , where (cid:80) k = V (cid:82) d k / (2 π ) in units of V = 1. This leadsto ε b = − mα − / (2 ma s ) ± (cid:112) / (4 m a s ) + α /a s for a s ≶
0, showing that ε b ≤ − mα for all parameters. Thisexpression is analytically tractable in three limits [22, 25–27]: we find that (i) ε b = − mα − m α a s and m B = 6 m in the limit when 1 / ( mαa s ) (cid:28) −
1, (ii) ε b = − mα and m B = 3 √ / (2 √ − m ≈ . m in the unitaritylimit when 1 / ( mαa s ) = 0, and (iii) ε b = − / ( ma s ) and m B = 2 m in the limit when 1 / ( mαa s ) (cid:29)
1. These resultsare illustrated in Fig. 10 for the completeness of the pre-sentation. Note that the latter limit recovers the usualtwo-body problem with no SOC in the 1 / ( mαa s ) (cid:29) α → + . IV. THREE-BODY PROBLEM
In this section, we are interested in the scattering t-matrix t −− k (0) between the lowest-energy fermion in the − -helicity band and a stationary dimer. For this purpose,we introduce a shorthand notation T k ( p ) = (cid:20) t ↑↑ k ( p ) t ↑↓ k ( p ) t ↓↑ k ( p ) t ↓↓ k ( p ) (cid:21) , (11) where k = ( k , k ) refers collectively to the momentumand energy of the incoming fermion, and p = ( p , p )refers collectively to the momentum and energy exchangebetween the outgoing fermion and the dimer. We refer toFig. 5 for the clarity of its meaning. Once T k (0) is eval-uated, we transform it to the helicity basis via Eq. (2),and obtain U † k T k (0) U k . Using the spherical coordinateswhere (cid:98) k = (sin θ k cos φ k , sin θ k sin φ k , cos θ k ) , we find t ssk (0) = t ↑↑ k (0) 1 + s cos θ k t ↓↓ k (0) 1 − s cos θ k s Re[ t ↑↓ k (0) sin θ k (cos φ k + i sin φ k )] (12)for the diagonal elements with Re the real part. Notein particular that t −− k (0) = t ↓↓ k (0) for k that is alignedwith the z axis, i.e., when θ k = 0. In this paper, weare interested in the fermion-dimer scattering length a BF that is determined by [2, 3] a BF = m BF π Zt −− k (0) , (13)where m BF = 2 m B m F / ( m B + m F ) is twice the re-duced mass of the fermion and the dimer, and k =( mα (cid:98) k , − mα /
2) corresponds to the lowest-energy eigen-state in the − -helicity band. (i) k , k k , k , ε b , ε b − k , − k + ε b (ii) q − k , q − k + ε b q − k , q − k + ε b q , q + ε b k − q , k − q (iii) q − k , q − k + ε b q , q + ε b k − q , k − q k − Q , k − Q Q − k + q , Q − k + q + ε b Q − k , Q − k + ε b Q , Q + ε b FIG. 3: Diagrammatic representations of the (i) zero-loopBorn, (ii) one-loop and (iii) two-loop contributions to thefermion-dimer scattering t-matrix.
For instance, the diagrammatic representations of thezero-loop, one-loop and two-loop contributions to thefermion-dimer scattering t-matrix are shown in Fig. 3 [1–7, 13]. The zero-loop contribution is known as the Bornapproximation, and in accordance with the Feynmanrules given above, it is given by T (0) k (0) = − σ y G T ( − k , − k + ε b ) σ y , (14)where the minus sign is due to the exchange of an iden-tical fermion. By plugging Eq. (14) into Eq. (12), wefind t ssk (0) (0) = 1 k − ε b + ε s k , (15)which is physically intuitive. This is because, since bothdimers are stationary in the Born diagram, the helicitybands are not coupled, and t ssk (0) can be directly ex-pressed as t ssk (0) (0) = − G s ( − k , − k + ε b ) . Furthermore,by plugging t −− k (0) (0) = 1 / ( | ε b | − mα ) into Eq. (13), wefind a Born BF = 2 m BF (cid:112) | ε b | − mα m √ m | ε b | (16)in the Born approximation, suggesting that the fermion-dimer interaction is repulsive for all parameters. InFig. 4, we show a Born BF as a function of 1 / ( mαa s ), whichis analytically tractable in three limits: (i) a Born BF = − a s / / ( mαa s ) (cid:28) −
1, (ii) a Born BF =6 √ / [ mα (5 √ − ≈ . / ( mα ) in the unitarity limitwhen 1 / ( mαa s ) = 0, and (iii) a Born BF = 8 a s / / ( mαa s ) (cid:29)
1. Note that the latter limit recov-ers the usual three-body problem with no SOC in the1 / ( mαa s ) (cid:29) α → + . FIG. 4: Fermion-dimer scattering length in the zero-loop Born, one-loop and two-loop approximations. Thehigher-loop contributions form a perturbative series in the1 / ( mαa s ) < / ( mαa s ) > To go beyond the Born approximation, we considerthe one-loop contribution that is shown in Fig. 3(ii). Inaccordance with the Feynman rules, this diagram is givenby T (1) k (0) = (cid:88) q σ y G T ( q − k , q − k + ε b ) σ y D ( q , q + ε b ) × G ( k − q , k − q ) σ y G T ( q − k , q − k + ε b ) σ y . (17)Noting the relation ( σ + (cid:98) A · σ )( σ + (cid:98) B · σ ) = (1 + (cid:98) A · (cid:98) B ) σ + ( (cid:98) A + (cid:98) B + i (cid:98) A × (cid:98) B ) · σ , we first integrate q in theupper half plane in which there are two simple poles at q = k − ε ± q , and reduce the t-matrix contribution to T (1) k (0) = 12 (cid:88) s q D ( q + k , k + ε b − ε s q )(2 ε s q − ε b ) ( σ − s (cid:98) q · σ ) . (18)Noting that T k (0) has a spherical symmetry in k space,we choose an incoming momentum k = mα (cid:98) k z that isaligned with the z axis, and perform the remaining in-tegrations numerically in the q space [28]. In Fig. 4,we show how the one-loop contribution affects a Born BF as afunction of 1 / ( mαa s ). In the one-loop approximation, wefind that a Born BF becomes attractive in the 1 / ( mαa s ) > T (2) k (0) = (cid:88) qQ σ y G T ( q − k , q − k + ε b ) σ y D ( q , q + ε b ) × G ( k − q , k − q ) σ y G T ( Q − k + q , Q − k + q + ε b ) × σ y G ( k − Q , k − Q ) D ( Q , Q + ε b ) σ y × G T ( Q − k , Q − k + ε b ) σ y , (19)where a minus sign is included due to the fermion ex-change. We integrate q and Q in their upper half planesin which there are two simple poles at q = k − ε ± q − k and two simple poles at Q = k − ε ± Q − k . In addition, bytaking advantage of the symmetry of the diagram withrespect to the internal variables q and Q , we reduce thet-matrix contribution to T (2) k (0) = 18 (cid:88) ss (cid:48) s (cid:48)(cid:48) qQ D ( q + k , k + ε b − ε s q ) D ( Q + k , k + ε b − ε s (cid:48) Q )(2 ε s q − ε b )(2 ε s (cid:48) Q − ε b )( ε s q + ε s (cid:48) Q + ε s (cid:48)(cid:48) K − k − ε b ) (cid:18)(cid:8) ss (cid:48)(cid:48) (cid:98) q · (cid:98) K + s (cid:48) s (cid:48)(cid:48) (cid:98) Q · (cid:98) K + ss (cid:48) (cid:98) q · (cid:98) Q (cid:9) σ + (cid:8) s (cid:98) q + s (cid:48) (cid:98) Q + s (cid:48)(cid:48) (cid:98) K − ss (cid:48) s (cid:48)(cid:48) [( (cid:98) q · (cid:98) Q ) (cid:98) K − ( (cid:98) Q · (cid:98) K ) (cid:98) q / − ( (cid:98) q · (cid:98) K ) (cid:98) Q / (cid:9) · σ (cid:19) , (20)where K = Q + q + k . We again choose an incomingmomentum k = mα (cid:98) k z that is aligned with the z axis,and perform the remaining integrations numerically inthe q and Q spaces [28]. In Fig. 4, we show how thecombination of the one-loop and two-loop contributionsaffects a Born BF as a function of 1 / ( mαa s ). While the two-loop contribution is negligible in the 1 / ( mαa s ) (cid:28) − a BF in the 1 / ( mαa s ) > =+ k T (q)kT (p) k , k , ε b p , p + ε b k − p , k − p p − k , p − k + ε b k − q , k − q q − k + p , q − k + p + ε b q , q + ε b FIG. 5: Diagrammatic representation of the three-bodyproblem. The fermion-dimer scattering t-matrix is deter-mined by repeating the fermion-exchange process infinitely-many times, forming eventually an integral equation.
By comparing the zero-loop, one-loop and two-loopapproximations in Fig. 4, we observe that while thehigher-loop contributions form a perturbative series inthe 1 / ( mαa s ) < / ( mαa s ) > Z decays to 0 in the1 / ( mαa s ) → −∞ limit, and that it increases as (cid:112) | ε b | inthe 1 / ( mαa s ) → + ∞ limit, this observation is caused bythe incremental growth of the power of Z that is comingfrom the additional dimer propagators within each loop.For this reason, a proper description of the latter regionrequires infinitely-many loop diagrams at all orders [1–7, 13]. A practical way to handle such summations ispresented in Fig. 5, where the fermion-dimer scattering t-matrix is determined by repeating the fermion-exchangeprocess infinitely-many times, forming eventually an in-tegral equation. In accordance with the Feynman rules,this diagram is given by T k ( p ) = − σ y G T ( p − k , p − k + ε b ) σ y − (cid:88) q T k ( q ) D ( q , q + ε b ) G ( k − q , k − q ) × σ y G T ( q − k + p , q − k + p + ε b ) σ y , (21) where the minus signs are due to the fermion exchanges.Integrating q in the upper half plane where T k ( q ) is an-alytic and there are two simple poles at q = k − ε ± q − k ,we reduce the t-matrix equation to T k ( p , p ) = − (cid:88) s σ − s (cid:98) k (cid:48) · σ p − k + ε b − ε s k (cid:48) (22) − (cid:88) ss (cid:48) q D ( q , k + ε b − ε s Q ) p + ε b − ε s Q − ε s (cid:48) K T k ( q , k − ε s Q ) × [(1 − ss (cid:48) (cid:98) Q · (cid:98) K ) σ + ( s (cid:98) Q − s (cid:48) (cid:98) K − iss (cid:48) (cid:98) Q × (cid:98) K ) · σ ] . Here k (cid:48) = p − k , Q = k − q and K = p − k + q areintroduced for the simplicity of the presentation.In the usual three-body problem with no SOC, t ( p , p ) is not only a real function but it is also restrictedto the so-called on-the-shell value t [ p , p = − p / (2 m )]for both the incoming and outgoing fermions [1–7, 13]. Inaddition, using the spherical symmetry of the t-matrix,the problem reduces to a simple integral equation with asingle variable for t ( | p | ), whose numerical computationconverges very fast. However, since the helicity bandsare coupled due to the non-stationary dimers, there aretwo shells contributing to Eq. (22). Furthermore, giventhat the t-matrix is a 2 × V. FOUR-BODY PROBLEM
Motivated by the overall success of the Born approx-imation in the fermion-dimer scattering problem, herewe apply the diagrammatic approach to the scatteringt-matrix t BB (0) between two stationary dimers in theone-loop Born and two-loop approximations. Despiteits simplicity, we expect a Born BB to be quite accurate inthe 1 / ( mαa s ) (cid:28) − / ( mαa s ) < / ( mαa s ) > (i) , ε b , ε b , ε b , ε b k , k + ε b k , k + ε b − k , − k − k , − k (ii) , ε b , ε b , ε b , ε b k − q , k − q − k , − k + ε b q − k , q − k + ε b k , k k , k q , q + ε b FIG. 6: Diagrammatic representation of the (i) one-loopBorn and (ii) two-loop contributions to the dimer-dimer scat-tering t-matrix. rules, it is given by t BB (0) = − Tr2 (cid:88) k (cid:2) σ y G ( k , k + ε b ) σ y G T ( − k , − k ) (cid:3) , (23)where the minus sign is due to the fermion exchange.Noting the relations ( σ ± (cid:98) A · σ )( σ ± (cid:98) A · σ ) = 2( σ ± (cid:98) A · σ ) , and ( σ ± (cid:98) A · σ )( σ ∓ (cid:98) A · σ ) = 0 , we first integrate k inthe upper half plane in which there are two double polesat k = − ε ± k , and reduce the t-matrix contribution to t BB (0) = 12 (cid:88) s k ε s k − ε b ) . (24)This is a physically intuitive result because, since alldimers are stationary in the Born diagram, the helic-ity bands are not coupled, and the diagram can be di-rectly expressed as t BB (0) = ( − / (cid:80) sk (cid:2) G s ( k , k + ε b ) G s ( − k , − k ) (cid:3) . In this paper, we are interested in the dimer-dimerscattering length a BB that is determined by [2, 3, 8, 9] a BB = m B π Z t BB (0) . (25)Plugging t BB (0) = m √ m ( | ε b | + 2 mα ) / [16 π ( | ε b | − mα ) / ] above, we find [25–27] a Born BB = m B ( | ε b | + 2 mα ) (cid:112) | ε b | − mα m √ m | ε b | , (26)in the Born approximation, which suggests that thedimer-dimer interaction is repulsive for all parameters.In Fig. 7, we show a Born BB as a function of 1 / ( mαa s ),which is analytically tractable in three limits: (i) a Born BB = − a s in the limit when 1 / ( mαa s ) (cid:28) −
1, (ii) a Born BB =3 √ / [ mα (2 √ − ≈ . / ( mα ) in the unitarity limitwhen 1 / ( mαa s ) = 0, and (iii) a Born BB = 2 a s in the limitwhen 1 / ( mαa s ) (cid:29)
1. Note that the latter limit recov-ers the usual four-body problem with no SOC in the1 / ( mαa s ) (cid:29) α → + . FIG. 7: Dimer-dimer scattering length in the one-loop Bornand two-loop approximations. The higher-loop contributionsform a perturbative series in the 1 / ( mαa s ) < / ( mαa s ) > To go beyond the Born approximation, we considerthe two-loop contribution that is shown in Fig. 6(ii). Inaccordance with the Feynman rules, this diagram is givenby t BB (0) = − Tr2 (cid:88) kq σ y G T ( q − k , q − k + ε b ) σ y G ( k , k ) D ( q , q + ε b ) σ y G T ( − k , − k + ε b ) σ y G ( k , k ) σ y G T ( q − k , q − k + ε b ) σ y G ( k − q , k − q ) . (27)We first integrate q in the upper half plane in whichthere are two simple poles at q = k − ε ± q − k , and thenintegrate k in the upper half plane in which there are twosimple poles at k = ε b − ε ± k . This reduces the t-matrixcontribution to t BB (0) = 14 (cid:88) ss (cid:48) kq D ( q , ε b − ε s k − ε s (cid:48) Q )(2 ε s k − ε b ) (2 ε s (cid:48) Q − ε b ) (1 + ss (cid:48) (cid:98) k · (cid:98) Q ) , (28)where Q = k + q , and the remaining integrations areperformed numerically in the k and q spaces [28]. InFig. 7, we show how the two-loop contribution affects a Born BB as a function of 1 / ( mαa s ). While the two-loopcontribution is negligible in the 1 / ( mαa s ) (cid:28) − a Born BB in the 1 / ( mαa s ) > VI. MANY-BODY PROBLEM
The fermion-dimer and dimer-dimer scattering lengthsoffer valuable insights for some of the many-body prop-erties of Fermi gases. For instance, in the case ofpopulation-imbalanced Fermi gases, a BF and a BB can beused to map the strongly-interacting Fermi-Fermi mix-ture of ↑ and ↓ fermions to a weakly-interacting Bose-Fermi mixture of paired fermions (dimers) and unpaired(excess) ones [4, 15, 16]. In the parameter regime wherethis effective description holds, the existing literature ontrue Bose-Fermi mixtures can be easily utilized to char-acterize the imbalanced Fermi gases. FIG. 8: Critical boundary between the uniform superfluid(U) and phase separation (PS) that is determined by the ef-fective weakly-interacting Bose-Fermi mixture description ofa population-imbalanced Fermi gas in the Born approxima-tion.
For instance, it is well-known that a weakly-interactingBose-Fermi mixture is unstable against phase separa-tion with a negative compressibility when the densityof fermions n F satisfies [22] n F ≥ π U BB / (3 m F U BF )in three dimensions, where U BB = 4 πa BB /m B is the repulsive interaction between bosons, and U BF = 4 πa BF /m BF is the repulsive interaction betweenfermions and bosons. Thus, the Bose-Fermi mixturephase separates when n F ≥ π m m B ( m B + m ) a BB a BF , (29)and is otherwise uniform. By plugging the Born approx-imations Eqs. (16) and (26) into Eq. (29), we obtain thecorresponding relation for the stability of a population-imbalanced Fermi gas with SOC. The critical boundarybetween the uniform superfluid and phase separation isshown in Fig. 8.Here we remark in passing that one can study BCS-BEC evolution for any given a s by tuning the strength α of the SOC, no matter how small or large the value of a s is and independently of its sign. Its physical mecha-nism is the SOC-induced enhancement of ε b through theincrease of the single particle density of states. In par-ticular, when α is large, the nature of the bosons that make up the BEC is determined solely by αa s . For thisreason, these bosons are sometimes called rashbons in therecent literature since their properties are determined bySOC alone. See Refs. [21–27] for further discussion, in-cluding the effective Gross-Pitaevskii description of theweakly-interacting dimers in the BEC limit. VII. ANISOTROPIC (RASHBA) SPIN-ORBITCOUPLING
Our results can be easily generalized to anisotropicSOC fields. For instance, in the presence of a RashbaSOC, the one-body Hamiltonian is governed by h k = ε k σ + α k ⊥ · σ , where k = ( k ⊥ , k z ) and k ⊥ = ( k x , k y ),leading to ε s k = ( k ⊥ + k z ) / (2 m ) + sαk ⊥ . Therefore, theground state of the − -helicity band corresponds to a de-generate ring of k ⊥ states with the radius k m = mα at k z = 0 and energy ε − k m = − mα / q = ( q ⊥ , q z ) and q , Eq. (6) hasthe generic structure of a simple pole [22–27] D ( q , q ) = Zq − q ⊥ / (2 m B, ⊥ ) − q z / (2 m B,z ) + µ B + i + (30)where Z = 8 π ( | ε b | − mα ) / ( m (cid:112) m | ε b | ) correspondsto the residue of the pole, 2 m/m B, ⊥ = (2 | ε b | − mα ) / (2 | ε b | ) − [( | ε b | − mα ) / (2 | ε b | )] log(1 − mα / | ε b | )and m B,z = 2 m determine the anisotropic effective massof the dimer. In addition, µ B = | ε b | is determined by1 / ( mαa s ) = (cid:112) | ε b | / ( mα ) − log[ (cid:112) | ε b | / ( | ε b | − mα ) + (cid:112) mα / ( | ε b | − mα )] . This expression is analyticallytractable in three limits [21–23]: we find that (i) ε b = − mα − mα e / ( mαa s ) − and m B, ⊥ = 4 m in the limitwhen 1 / ( mαa s ) (cid:28) −
1, (ii) ε b ≈ − . mα and m B, ⊥ ≈ . m in the unitarity limit when 1 / ( mαa s ) = 0, and(iii) ε b = − / ( ma s ) and m B, ⊥ = 2 m in the limitwhen 1 / ( mαa s ) (cid:29)
1. Note that the latter limit re-covers the usual two-body problem with no SOC in the1 / ( mαa s ) (cid:29) α → + .Since k = ( mα (cid:99) k ⊥ , − mα /
2) corresponds to the lowest-energy eigenstate in the − -helicity band, Eq. (15) gives t −− k (0) (0) = 1 / ( | ε b | − mα ) . By plugging it into Eq. (13),we find a Born BF = 2 m BF m (cid:112) m | ε b | (31)in the Born approximation, where m B refers to the geo-metric mean ( m B, ⊥ m B,z ) / of the anisotropic effectivemass [27]. In Fig. 9, we show a Born BF as a function of1 / ( mαa s ), which is analytically tractable in three limits:(i) a Born BF ≈ . / ( mα ) in the limit when 1 / ( mαa s ) (cid:28)−
1, (ii) a Born BF ≈ . / ( mα ) in the unitarity limit when1 / ( mαa s ) = 0, and (iii) a Born BF = 8 a s / / ( mαa s ) (cid:29)
1. Note again the latter limit re-covers the usual three-body problem with no SOC in the1 / ( mαa s ) (cid:29) α → + . FIG. 9: Dimer-dimer (red) and fermion-dimer (blue) scat-tering lengths for the isotropic SOC (left: same as in Fig. 4)versus Rashba SOC (right) in the Born approximations.
Similarly, Eq. (24) gives t BB (0) = m √ m ( | ε b | + mα ) / [16 π ( | ε b | − mα ) (cid:112) | ε b | ] , and by plugging it inEq. (25), we find a Born BB = m B ( | ε b | + mα ) m √ m (cid:112) | ε b | , (32)in the Born approximation. In Fig. 9, we show a Born BB asa function of 1 / ( mαa s ), which is analytically tractablein three limits: (i) a Born BB ≈ . / ( mα ) in the limit when1 / ( mαa s ) (cid:28) −
1, (ii) a Born BB ≈ . / ( mα ) in the unitaritylimit when 1 / ( mαa s ) = 0, and (iii) a Born BB = 2 a s in thelimit when 1 / ( mαa s ) (cid:29)
1. Note again that the latterlimit recovers the usual four-body problem with no SOCin the 1 / ( mαa s ) (cid:29) α → + .In contrast to the isotropic SOC case where a BF and a BB are non-monotonous functions of 1 / ( mαa s ), theyevolve monotonously in the Rashba SOC. Their satura-tions in the 1 / ( mαa s ) (cid:28) − Z with the divergence of thet-matrices. The decay is faster in the isotropic case, caus-ing the peak in the intermediate region. Despite this ma-jor difference, the isotropic and Rashba SOC cases sharesome common properties. For instance, the decrease, in-crease and saturation of a BF are in full coordination withthose of a BB . In addition, we note that a BF is greater(smaller) than a BB in approximately the 1 / ( mαa s ) ≷ VIII. CONCLUSION
In summary, we studied how SOC affects the fermion-dimer and dimer-dimer scattering lengths in the Bornapproximation, and benchmarked their accuracy with the higher-order approximations. We considered bothisotropic and Rashba couplings in three dimensions, andfound that the Born approximation gives accurate re-sults for both a BF and a BB in the 1 / ( mαa s ) (cid:28) − / ( mαa s ) < / ( mαa s ) > Z of the dimer propagator, which decays to 0 inthe 1 / ( mαa s ) → −∞ limit, and increases as (cid:112) | ε b | in the1 / ( mαa s ) > / ( mαa s ) < / ( mαa s ) > Acknowledgments
The author acknowledges funding from T ¨UB˙ITAKGrant No. 11001-118F359.
Appendix A: Binding energy and effective mass ofthe dimer
For the sake of completeness, we present the bindingenergy | ε b | − mα and effective mass m B of the dimerin Fig. 10, where 3D SOC field refers to α k with k =( k x , k y , k z ), 2D one to α k ⊥ with k ⊥ = ( k x , k y , k = ( k x , , k -space integrations, it is equivalent to the usual two-bodyproblem with no SOC [22]. Therefore, a two-body boundstate exists only when a s > m B = 2 m that is isotropic in space.In contrast to the 1D case, a two-body bound stateexists for all a s in both 3D (isotropic) and 2D (Rashba)SOC fields, which is caused by the increase in the low-energy density of one-body states [21–27]. In addition,while the effective mass of the dimer is isotropic in the3D case where m B = m B,x = m B,y = m B,z is shownin the figure, it is anisotropic in the 2D case where only m B, ⊥ = m B,x = m B,y is shown in the figure, and m B,z =2 m for all a s . FIG. 10: Binding energy | ε b | − mα and effective mass m B,x of the dimer. 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