Ultra-cold fermions in real or fictitious magnetic fields: The BCS-BEC evolution and the type-I--type-II transition
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Ultra-cold fermions in real or fictitious magnetic fields:The BCS-BEC evolution and the type-I–type-II transition
M. Iskin and C. A. R. S´a de Melo Department of Physics, Ko¸c University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey. School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. (Dated: November 21, 2018)We study ultra-cold neutral fermion superfluids in the presence of fictitious magnetic fields, as wellas charged fermion superfluids in the presence of real magnetic fields. Charged fermion superfluidsundergo a phase transition from type-I to type-II superfluidity, where the magnetic properties ofthe superfluid change from being a perfect diamagnet without vortices to a partial diamagnet withthe emergence of the Abrikosov vortex lattice. The transition from type-I to type-II superfluidityis tunned by changing the scattering parameter (interaction) for fixed density. We also find thatneutral fermion superfluids such as Li and K are extreme type-II superfluids, and that theyare more robust to the penetration of a fictitious magnetic field in the BCS-BEC crossover regionnear unitarity, where the critical fictitious magnetic field reaches a maximum as a function of thescattering parameter (interaction).
PACS numbers: 03.75.Ss, 03.75.Hh, 05.30.Fk
A key experiment in the verification that neutralFermi superfluids can evolve from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein condensation(BEC) regime was the observation of quantized vorticesthroughout the BCS-BEC evolution [1] upon rotation ofthe atomic cloud. This observation had a very dramaticimpact beyond the atomic physics community, because itshowed that superfluidity of Cooper pairs and of tightlybound bosonic molecules for s-wave pairing are the man-ifestation of the same type of physics. The key tool thatpermitted such realization is the tunability of the inter-action between fermions through the use of Feshbach res-onances. The same kind of tunability does not exist in He, the standard condensed matter neutral superfluid,or in superconductors. The situation is even worse inneutron and proton superfluids, which are thought to ex-ist in the core of neutron stars.Very recently, a new technique was developed that per-mitted the production of fictitious magnetic fields whichcan couple to neutral bosonic atoms [2, 3]. These ficti-tious magnetic fields are produced through an all opticalRaman process, couple to a fictitious charge, but producereal effects like the creation of vortices in the superfluidstate of bosons. In principle, the same technique can beapplied to ultra-cold fermions, which coupled with thecontrol over interaction using Feshbach resonances allowsthe exploration of superfluidity not only as a function ofinteraction, but also as a function of fictitious magneticfield. It is in anticipation of similar experiments involvingultracold fermions that we address in this manuscript theeffects of fictitious magnetic fields on fermion superfluidsas a function of interaction.Unlike neutral superfluids, standard condensed mat-ter charged superfluids (superconductors) can be of twotypes [4]. Many superconductors are now known to betype-II (including heavy fermions, organics, and high- T c -30 -20 -10 -8 -4 0 4 8 n r F a s )Type-I Type-IIBCS BEC FIG. 1. Universal phase diagram of the dimensionless fermiondensity nr q versus scattering parameter 1 / ( k F a s ), where r q = q / ( mc ) is the classical radius of a fermion with mass m andcharge q ; c is the speed of light, k F is the Fermi momentumand a s is the scattering length. The dotted line separatesregions of type-I and type-II superfluidity. cuprates), where the application of an external magneticfield beyond the lower critical field H c leads to a non-uniform superfluid phase, which appears in the form ofthe Abrikosov vortex lattice, until a second critical field H c is reached, when the system becomes normal. Othercharged superfluids are known to be type-I and do notallow the magnetic field to penetrate the sample. Thesesystems are perfect diamagnets until the critical field H c is reached, where the charged superfluid becomes normal.The parameter that characterizes the type of charged su-perfluid is the Ginzburg-Landau parameter κ = λ/ξ cor-responding to the ratio between the penetration depth λ of the magnetic field into the sample and the coher-ence length ξ of the charged superfluid, such that type-Isuperfluids have κ < / √ κ > / √ Li and Kare extreme type-II superfluids, and for charged super-fluids we find a phase transition from type-I to type-IIsuperfluidity for fermions of density n = k F / (3 π ) inter-acting via a contact potential characterized by the inter-action parameter 1 / ( k F a s ). As shown in Fig. 1, the phaseboundary in the density n versus interaction parameter1 / ( k F a s ) occurs when the critical value κ c = 1 / √ only in the BCS limit [5–8]. In contrast, here we showthat, microscopically, a clean (no disorder) charged su-perfluid can exhibit a type-I to type-II transition inducedby interactions. The phase diagram shown in Fig. 1 hasa wider applicability to include standard charged super-fluids (like superconductors of condensed matter physics)and even proton superfluidity in nuclei or neutral stars,as long as the interactions can be described by a con-tact potential with corresponding scattering length a s .In addition, we indicate that neutral (charged) superflu-ids are more robust to the penetration of ficitious (real)magnetic fields near unitarity, where the critical fictitious(real) magnetic fields reach a maximum as a function ofthe scattering parameter.To describe the transition from type-I to type-II super-fluidity as a function of the interaction parameter and theproperties of neutral (charged) superfluids in the presenceof fictitious (real) magnetic fields during the BCS-BECevolution for s-wave superfluids in three dimensions, westart with the Hamiltonian density¯ H ( r ) = X σ ψ † σ ( r ) (cid:18) − ¯ h ∇ m − µ (cid:19) ψ σ ( r ) + ˆ U ( r ) , (1)where ˆ U ( r ) = R d r ′ V ( r , r ′ ) ψ †↑ ( r ) ψ †↓ ( r ′ ) ψ ↓ ( r ′ ) ψ ↑ ( r ) con-tains the attractive contact interaction potential V ( r , r ′ ) = − gδ ( r − r ′ ), and ψ † σ ( r ) is the creation operatorof fermions with mass m and spin σ . Notice that g has di-mensions of energy times volume. To make progress, werewrite the Hamiltonian H = R d r ¯ H ( r ) from real spaceto momentum space H = X k ,σ ξ k ψ † k ,σ ψ k ,σ − g X k , k ′ , q b † k , q b k ′ , q , (2)where b † k , q = ψ † k + q / , ↑ ψ †− k + q / , ↓ creates a fermion pairwith center of mass momentum q and relative momentum2 k , ξ k = ǫ k − µ is the kinetic energy term with ǫ k =¯ h k / (2 m ) and µ is the chemical potential.Integration over the fermion fields [9] leads to the orderparameter equation1 g = 1 L X k tanh [ ξ k / (2 T c )]2 ξ k (3) at the critical temperature T c , where the order parametervanishes. Here L is the sample volume. The interaction g can be written in terms of the scattering length a s lead-ing to 1 /g = − m/ (cid:0) π ¯ h a s (cid:1) + (1 /L ) P k [1 / (2 ǫ k )]. Thesecond self-consistency relation is the number equation N = X k ,σ f( ξ k ) + T c X q ∂ (cid:2) ln( L K /T c ) (cid:3) ∂µ (4)where f( ξ k ) is the Fermi function, and K − = 1 g − L X k − f( ξ k + q / ) − f( ξ − k + q / ) ξ k + q / + ξ − k + q / − i ¯ hω (5)is the pair propagator, and ω is the Matsubara frequencyfor bosons.The effective action is T S eff / ¯ h = P q K − ( q ) | ∆( q ) | + b L P q ,q ,q ∆( q )∆ ∗ ( q )∆( q )∆ ∗ ( q − q + q ) in termsof the order parameter ∆( q ), where q = ( q , iω ). Tostudy thermodynamic properties, we take i ¯ hω = 0, orequivalently, ∆( r , τ ) ≡ ∆( r ), leading to the effective La-grangian density L eff = a | ∆ | + X i,j ¯ h c ij m ∇ i ¯∆ ∇ j ∆ + b | ∆ | . Using the notation X k = tanh[ ξ k / (2 T )] and Y k =sech [ ξ k / (2 T )], the coefficients of the Lagrangian densityare a ( µ, T ) = g − L P k X k ξ k for the constant term, L c ij ( µ c , T c ) = X k (cid:20)(cid:18) X k ξ k − Y k ξ k T c (cid:19) δ ij + X k Y k T c ¯ h k i k j mξ k (cid:21) for the coefficient of the gradient terms ∇ i ¯∆ ∇ j ∆ , and L b ( µ c , T c ) = P k (cid:16) X k ξ k − Y k ξ k T c (cid:17) for the coefficient of thenon-linear quartic term. Notice that c ij = cδ ij for s-wavesuperfluids.In general, near T c , a ( µ, T ) = − a ǫ ( T ) , where a = T c [ ∂a/∂T ] T c and ǫ ( T ) = (1 − T /T c ). In theBCS limit of 1 / ( k F a s ) → −∞ , L a = D F , where D F = mk F L / (2 π ¯ h ) is the density of single parti-cle states per spin channel at the Fermi energy ǫ F .Also the coefficient of the quartic term is L b = (cid:2) ζ (3) / (8 π T c ) (cid:3) D F , while the coefficient of the gradi-ent term is L c = (cid:2) ζ (3) / (12 π T c ) (cid:3) D F ǫ F . Here, thezeta function ζ (3) = 1 . T c = (cid:0) e γ − /π (cid:1) ǫ F exp [ − π/ (2 k F | a s | )] , with e γ ≈ . µ = ǫ F . However, in theBEC limit of 1 / ( k F a s ) → + ∞ , L a = D F ǫ F / (4 | µ | ) . Correspondingly the coefficient of the quartic term is L b = ( π/ D F / ( | µ | p ǫ F | µ | ) , and the coefficient of thegradient term is L c = ( π/ D F / p ǫ F | µ | . In this case, T c ≈ . ǫ F and µ = E b / , where E b = − ¯ h / ( ma s ) isthe two-particle binding energy in vacuum.Next, we scale the order parameter to ψ ( r ) = √ c ∆( r )and introduce an external (real or fictitious) magneticfield via the vector potential A ( r ), using the substitution ∇ i → ∇ i − iqA i / (¯ hc ), where q is the real or fictitiousparticle charge and c is the speed of light. The differencein free energy density between the charged superfluid andits normal state in the presence of magnetic fields takesthe Ginzbug-Landau form F GL = α | ψ | + β | ψ | + ¯ h m (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i ∇ − q ¯ hc A (cid:19) ψ (cid:12)(cid:12)(cid:12)(cid:12) + | H | π where H = ∇ × A is the real or fictitious microscopicmagnetic field. The parameter α = a/c changes sign at T = T c , however β = b/c is always positive guaranteeingthe stability of the theory. It is also useful to define theflux quantum Φ q = π ¯ hc /q , which will be used below.Minimization of F GL with respect to ψ and A lead theorder parameter equation αψ + β | ψ | ψ + ¯ h m (cid:18) − i ∇ − q ¯ hc A (cid:19) ψ = 0 (6)and to the current density j = − ¯ hqim ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) − q mc | ψ | A . (7)Using the relation ∇ × H = 4 π j /c and taking the curl ofthe current density leads to the London equation λ ∇ × ( ∇ × H ) + H = 0 where λ = p mc / (16 πq | ψ | ) is themagnetic penetration depth. Since | ψ | = | α | /β = | a | c/b in weak magnetic fields, the penetration depth becomes λ ( T ) = λ GL | ǫ ( T ) | − / , where λ GL = p b/ (16 πr q a c ) . Here, r q = q / ( mc ) is the classical radius of a fermionwith mass m and charge q in CGS units. Since | ψ | plays the role of the superfluid density n s , we maywrite | ψ | = | ψ | | ǫ ( T ) | = n s = n s, | ǫ ( T ) | , where n s, = | ψ | = a c/b is the temperature independentprefactor. This observation allows us to write k F λ GL = p [3 π/ (16 k F r q )] ( n/n s, ) . The prefactor n s, reflects azero temperature extrapolation of the superfluid density n s , however, in a Galilean invariant system we must have n s, ≈ n/
2. Indeed, in the BCS limit n s, = n/ k F λ GL = p π/ (8 k F r q ), while in the BEC limit n s, = 3 n/ k F λ GL = p π/ (2 k F r q ). The BCSvalue of k F λ GL is slightly smaller than its BEC value,however throughout the BCS-BEC evolution, k F λ GL does not change substantially.The coherence length can be extracted from Eq. (6) as ξ ( T ) = ¯ h/ p m | α ( T ) | leading to ξ ( T ) = ξ GL | ǫ ( T ) | − / where ξ GL = ¯ h p c/ (2 ma ) . Unlike the penetration depth,the coherence length ξ GL changes substantially duringthe BCS to BEC evolution. In the BCS regime, k F ξ GL = p ζ (3) / (12 π )( ǫ F /T c ) is very large, and in terms of k F a s becomes k F ξ GL = A exp[ π/ (2 k F | a s | )] , where A = p ζ (3) / (12 π )( πe − γ / . In the BEC regime, k F ξ GL = p π/ | µ | /ǫ F ) / is also very large, and in terms of k F a s becomes k F ξ GL = p π/ / √ k F a s . However, k F ξ GL passesthrough a minimun in the intermediate regime, where k F ξ GL ≈ O (1).As discovered by Abrikosov [4], the parameter κ = λ ( T ) /ξ ( T ) is of fundamental importance in the charac-terization of the magnetic properties of charged super-fluids. When κ < κ c = 1 / √ κ > κ c , the chargedsuperfluid allows the penetration of magnetic field in thesuperfluid state in the form of vortices (type-II). Sincethe temperature dependence of λ ( T ) and ξ ( T ) is exactlythe same, the parameter κ = λ ( T ) ξ ( T ) = λ GL ξ GL = s mb πr q ¯ h c (8)is independent of temperature. Notice that κ p k F r q = p s/ (16 π ), where s = k F b/ ( ǫ F c ) is a dimension-less parameter which is a function of 1 / ( k F a s ) only.In Fig. 2, we show the evolution of κ as a func-tion of the scattering parameter 1 / ( k F a s ). In theBCS limit, κ p k F r q = p π / [14 ζ (3)]( T c /ǫ F ) cor-responds to κ p k F r q = B exp[ − π/ (2 k F | a s | )] , where B = p π / [14 ζ (3)](8 e γ − /π ) . While in the BEC limit, κ p k F r q = √ ǫ F / | µ | ) / corresponds to κ p k F r q = √ √ k F a s . Notice the maximum of κ in the vicinity ofunitarity and µ = 0 (1 / [ k F a s ] = 0 . qH is con-trolled, instead of H alone [2]. So it is useful to thinkof ultra-cold superfluids like Li or K as having charge q →
0, but with qH finite. In this sense, these neutralsuperfluids are extreme type-II with κ → ∞ throughoutthe BCS to BEC evolution [10]. -4 -2 -8 -4 0 4 8 κ ( k F r q ) / F a s )BCS C r o ss o v e r BEC
FIG. 2. Universal plot of the Ginzburg-Landau parameter κ versus scattering parameter 1 / ( k F a s ), where r q is the classicalradius of a fermion with mass m and charge q . To obtain the phase diagram shown in Fig. 1, we set κ = κ c in Eq. (8) and extract the fermion density n as afunction of 1 / ( k F a s ), which leads to nr q = s / (1536 π ).In Fig. 1, κ is higher (lower) than κ c below (above)the critical line indicating a type-II (type-I) charged su-perfluid phase. For fixed density n , a phase transi-tion from type-I to type-II charged superfluid occurs,as the interaction parameter 1 / ( k F a s ) increases. Elec-tron superfluids with 10 cm − ≤ n ≤ cm − have 2 . × − < nr q < . × − , and thetransition between type-I and type-II occurs in the in-terval − < / ( k F a s ) < −
4, while proton superfluidsin nuclear matter, with r q ≈ . × − cm − and10 cm − ≤ n ≤ cm − , have a type-I to type-IItransition in the range − < / ( k F a s ) < H c ( T ) determined by the condition H c ( T ) / (8 π ) = F n − F s , where F n ( F s ) is the Helmholtzfree energy for the normal (superfluid) state. For a uni-form superfluid state the energy difference is F n − F s = α / (2 β ) = a / (2 b ) leading to H c ( T ) = H c, | ǫ ( T ) | = | a ( T ) | p π/b , where H c, = a p π/b is the temperatureindependent prefactor. We define the dimensionless ther-modynamic critical field e H c, = H c, /H k F , where H k F =Φ q k F . Notice that e H c, = ¯ hω c, / (2 πǫ F ), where ω c, = | q | H c, / ( mc ) is the cyclotron frequency at H c, . Usingthe asymptotic expressions for a and b , we obtain e H c, = p k F r q p / [7 πζ (3)]( T c /ǫ F ) in the BCS regime, whichcan be rewritten as e H c, = p k F r q C exp [ − π/ (2 k F | a s | )] , with C = p / [7 πζ (3)](8 e γ − /π ) . While we obtain e H c, = p k F r q (1 /π )( ǫ F / | µ | ) / in the BEC regime,which can be rewritten as e H c, = p k F r q (1 /π ) √ k F a s . The field e H c, reaches a maximum near unitary and µ = 0, thus indicating that type-I superfluids are mostrobust to the penetration of magnetic fields in that sameregion.For type-II superfluids there are two critical fields.The first is called H c ( T ) and separates the perfect-diamagnet Meissner phase from the non-uniform phaseexhibiting vortices. The second is called H c ( T ) and sep-arates the non-uniform phase exhibiting vortices fromthe normal state. Since Li and K are extremetype-II superfluids with κ → ∞ , then H c ( T ) → H c ( T ).The calculation of H c ( T ) is performed by lineariz-ing Eq. (6) − ¯ h ( ∇ − i π A / Φ q ) ψ + 2 mα ( T ) ψ = 0 . Using the Landau gauge A = Hx ˆ y , the momen-tum components k y and k z are good quantum num-bers and the solution for ψ becomes ψ n,k y ,k z ( x, y, z ) = e ( ik y y + ik z z ) u n ( x ), which substituted in the previousequation leads to the one-dimensional Schr¨odinger equation (cid:2) − ¯ h / (2 m ) d /dx + mω s ( x − x ) / (cid:3) u n ( x ) = ǫ n u n ( x ) , where x = Φ q k y / (2 πH ) is the equilibrium po-sition of the harmonic potential, ω s = 2 | q | H/ ( mc ) isthe harmonic potential frequency, and ǫ n = | α ( T ) | − ¯ h k z / (2 m ) = ¯ hω s ( n + 1 /
2) is the eigenvalue. Thehighest magnetic field at which superconductivity nu- cleates occurs for n = 0 and k z = 0 leading tothe condition | α ( T ) | = ¯ hω s /
2. Isolating the mag-netic field from the harmonic potential frequency leadsto H c ( T ) = [Φ q / (2 π )]2 m | α ( T ) | / ¯ h , which can be fi-nally expressed in terms of the coherence length ξ ( T )as H c ( T ) = Φ q / (cid:2) πξ ( T ) (cid:3) . Substituting ξ ( T ) = ξ GL | ǫ ( T ) | − / , we write H c ( T ) = H c , | ǫ ( T ) | , where H c , = Φ q / (2 πξ GL ). Using again the reference field H k F = Φ q k F , we obtain the dimensionless upper criticalfield e H c , = H c , /H k F = 1 / (2 πk F ξ GL ) . This expressionis equivalent to the ratio ¯ hω c , / (2 πǫ F ) , where ω c , = qH c , / ( mc ) is the cyclotron frequency at H c , . In theBCS limit, e H c , = [6 π/ ζ (3)] ( T c /ǫ F ) , which in termsof the scattering parameter 1 / ( k F a s ) becomes e H c , = D exp[ − π/ ( k F | a s | )] with D = 256 e γ − / [7 πζ (3)] . In theBEC limit, e H c , = (cid:0) /π (cid:1) p ǫ F / | µ | which in termsof the scattering parameter 1 / ( k F a s ) becomes e H c , =(2 /π ) k F a s . Since k F ξ GL reaches a minimum in the re-gion near unitarity and µ = 0, it is clear that e H c , hasa maximum there, where type-II superfluids are most ro-bust to the presence of real or fictitious magnetic fields.Before concluding, we note that the quantum regime¯ hω c ≥ πT (where Landau level quantization is impor-tant) can be reached experimentally for Li and K whilepreserving superfluidity.In conclusion, we have analyzed the effects of real orfictitious magnetic fields during the BCS to BEC evo-lution of s-wave superfluids with direct application toultra-cold fermionic atoms. We have shown that a transi-tion from type-I to type-II charged superfluidity occurs asthe Ginzburg-Landau paramater crosses its critical value κ c = 1 / √ q and mass m . We haveshown that Li and K in fictitious magnetic fields areextreme type-II superfluids. Finally, we have indicatedthat the critical magnetic fields (real or fictitious) dependstrongly on the scattering parameter 1 /k F a s and reach amaximum in a region near unitarity, where superfluidityis more robust to their penetration.CSdM and MI would like to thank NSF (DMR-0709584) and ARO (W911NF-09-1-0220), and MarieCurie IRG (FP7-PEOPLE-IRG-2010-268239) andT ¨UB˙ITAK, respectively, for support. [1] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.Schunck, and W. Ketterle, Nature , 1047 (2005).[2] Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.Porto, and I. B. Spielman, Nature , 628 (2009).[3] I. B. Spielman, Phys. Rev. A , 063613 (2009).[4] A. A. Abrikosov, Sov. Phys. JETP , 1174 (1957).[5] A. A. Abrikosov, and L. P. Gorkov Sov. Phys. JETP ,1090 (1958).[6] L. P. Gorkov, Sov. Phys. JETP , 1364 (1959); Sov. Phys. JETP , 998 (1960).[7] E. Helfand, and N. R. Werthamer, Phys. Rev. , 288(1966).[8] A. A. Abrikosov, in “Fundamentals of the theory of met-als”, p. 368, North Holland, (1988). [9] C. A. R. S´a de Melo, M. Randeria, J. Engelbrecht, Phys.Rev. Lett.71