Abstract
Recently suggested subwavelength lattices offer remarkable prospects for the observation of novel superfluids of fermionic polar molecules. It becomes realistic to obtain a topological pwave superfluid of microwavedressed polar molecules in 2D lattices at temperatures of the order of tens of nanokelvins, which is promising for topologically protected quantum information processing. Another foreseen novel phase is an interlayer pwave superfluid of polar molecules in a bilayer geometry.
Introduction
Nonconventional superconductors and superfluids attract a great deal of interest due to their nontrivial transport properties and/or topological behavior^{1,2,3,4,5,6,7,8,9,10,11}. This behavior has been actively discussed in two dimensions (2D) for the p_{x} + ip_{y} superfluid of identical fermions, where Cooper pairs have orbital angular momentum equal to unity^{12,13,14,15,16,17}. Quantized vortices in this superfluid carry zeroenergy Majorana modes on their cores^{3,18,19}. These modes cause the vortices to obey nonAbelian exchange statistics, which is a basis for topologically protected quantum information processing^{20,21}. However, the pwave topological superfluid of ultracold atoms is either collisionally unstable near a Feshbach resonance, or has a vanishingly low superfluid transition temperature far from the resonance^{22,23,24}.
Successful experiments on the creation of groundstate ultracold polar molecules^{25,26,27,28,29,30,31,32,33,34,35,36} opened fascinating prospects for obtaining nonconventional superfluids^{37,38,39}. In particular, microwavedressed polar molecules confined to 2D may acquire an attractive dipoledipole tail in the interaction potential, which ensures the emergence of collisionally stable pwave superfluid with a reachable transition temperature^{16,17}. Another interesting system concerns fermionic polar molecules in a bilayer geometry. Here they may form interlayer superfluids in which Cooper pairs consist of molecules belonging to different layers^{40,41,42,43}.
It this paper we consider novel pwave superfluids of fermionic polar molecules in 2D lattice geometries (see Fig. 1). It is shown that a collisionally stable topological p_{x} + ip_{y} superfluid of identical microwavedressed polar molecules may emerge in a 2D lattice due to a longrange character of the dipoledipole interaction. We also show how one can get a pwave interlayer superfluid of fermionic polar molecules in a bilayer geometry, which can be a quantum simulator of superconductivity in layered condensed matter systems^{7,8}. It is crucial to rely on the recently proposed subwavelength lattices^{44,45,46,47,48,49,50,51,52,53,54}, where the lattice constant (interlayer spacing in the bilayer system) can be as small as about 50 nm. An increase of energy scales in such lattices makes it realistic to obtain sizeable transition temperatures of the order of tens of nanokelvins.
General Relations and Qualitative Arguments
The superfluid pairing of identical fermions is characterized by the order parameter Δ(r, r′) = V(r − r′) , where V(r − r′) is the interaction potential, the symbol 〈...〉 denotes the statistical average, and is the field operator of fermions. For spin1/2 fermions one of the field operators in the expression for Δ(r, r′) is for spin↑ fermions, and the other one for spin↓ fermions. In free space the order parameter depends on the coordinates r and r′ only through the difference (r − r′). In 2D the transition temperature T_{c} of a Fermi gas from the normal to superfluid regime is set by the KosterlitzThouless transition. However, for a weak attractive interaction the order parameter and the superfluid transition temperature can be found in the BCS approach^{55}. For both spinless and spin1/2 fermions the renormalized gap equation for the order parameter in the momentum space, , reads (see^{16,17} and references therein):
where f(k′, k) is the offshell scattering amplitude and E_{k} =k^{2}/2m with m being the particle mass. The single particle excitation energy is given by where μ is the chemical potential, and . For weak interactions chemical potential coincides with the Fermi energy (k_{F} is the Fermi momentum). The quantity δV(k′, k) is a correction to the bare interparticle interaction due to polarization of the medium by colliding particles. The leading terms of this quantity introduced by Gor’kov and MelikBarkhudarov^{56}, are second order in the bare interaction (see Methods).
In order to gain insight in what is happening, we first omit the correction δV(k′, k) in Eq. (1). We then put k = k_{F}, and notice that the main contribution to the integral over k′ in Eq. (1) comes from k′ close to k_{F}. At temperatures T tending to the critical temperature T_{c} from below, we put in . For the pairing channel related to the interaction with orbital angular momentum l, this immediately leads to an estimate:
The quantity ρ(k_{F}) = m/2π in the exponent of Eq. (2) is the density of states on the Fermi surface, and f_{l}(k_{F}) is the onshell scattering amplitude.
In the lattice with a period b satisfying the condition k_{F}b ≪ 1, the superfluid paring of fermions can be considered as that of particles with effective mass m^{*} > m in free space. The density of states ρ(k_{F}) is then given by the same expression, with m replaced by m^{*}. Thus, the BCS exponent [ρ(k_{F}) f_{l}(k_{F})]^{−1} in the lattice is smaller than in free space at the same k_{F} (density) if there is no significant reduction in the scattering amplitude. Hence, although the Fermi energy decreases by the same factor m/m^{*}, the critical temperature T_{c} in the lattice can be much larger than in free space. This is the case for the swave pairing of shortrange interacting spin1/2 fermions in the tight binding model, if the extension of the particle wavefunction in the lattice site greatly exceeds the characteristic radius of the interparticle interaction. An increase of the critical temperature for the swave superfluidity by the lattice potential has been indicated in refs 57 and 58.
The situation changes for the pwave pairing of identical fermions attractively interacting via a shortrange potential. This pairing in an optical lattice at very low temperatures has been considered in ref. 59 (more sophisticated lattice models, where pwave pairing is constructed with the use of swave pairing at intermediate stages, were recently suggested in refs 60 and 61). In the tight binding model two such fermions can not be in the same lattice site unless one of them occupies a higher Bloch band. Therefore, the main contribution to the scattering amplitude comes from the interaction between two fermions sitting in neighboring sites^{59}. In particular, the fermions undergo quantum tunneling from the centers of their sites and experience the shortrange interaction in the spatial region where their wavefunctions are attenuated. This strongly suppresses the interaction amplitude and leads to a very low critical temperature. We however show below that the picture is drastically different for an attractive longrange interaction between the fermions.
Pwave Pairing of Microwavedressed Polar Molecules in a 2D Lattice
We will consider identical fermionic polar molecules in a 2D lattice of period b. Being dressed with a microwave field, they acquire an attractive dipoledipole tail in the interaction potential^{16,17,62,63}:
Here d is an effective dipole moment, and we assume that Eq. (3) is valid at intermolecular distances . This leads to superfluid pwave pairing of the molecules. In free space the emerging ground state is the topological p_{x} + ip_{y} superfluid, and the leading part of the scattering amplitude can be obtained in the first Born approximation^{16,17}. We assume the weakly interacting regime at a small filling factor in the lattice, k_{F}b ≪ 1.
The Hamiltonian of the system is , with
where , are the annihilation and creation operators of a molecule with quasimomentum q, and ε_{q} is the single particle energy. In the low momentum limit we have ε_{q} = q^{2}/2m^{*}, where m^{*} > m is the effective mass in the lowest Bloch band. The quantity describes the interaction between the molecules and is given by
where is the field operator of a particle in the lattice site j located at r_{j} in the coordinate space. At a small filling factor in the low momentum limit, the main contribution to the matrix elements of comes from intermolecular distances (see Methods). Therefore, we may replace the summation over r_{j} and by the integration over d^{2}r_{j} and . As a result the Hamiltonian of the system reduces to
where the first term in the right hand side is (4) rewritten in the coordinate space. We thus see that the problem becomes equivalent to that of particles with mass m^{*} in free space.
The scattering amplitude at k = k_{F}, which enters the exponential factor in Eq. (2), is obtained from the solution of the scattering problem in the lattice. For particles that have mass m^{*} (see Methods), the amplitude is written as follows
where , and B is a numerical coefficient coming from shortrange physics. Since for weak interactions two fermions practically do not get to the same lattice site, for calculating B we may introduce a perfectly reflecting wall at intermolecular distances r ~ b (see Methods). For the superfluid pairing the most important are particle momenta ~k_{F}. Therefore, the lowmomentum limit requires the inequality k_{F}b ≪ 1.
The solution of the gap equation (1) then leads to the p_{x} + ip_{y} superfluid with the critical temperature (see Methods):
where the coefficient κ is related to B and depends on the ratio (see Methods). There are two important differences of equation (8) from a similar equation in free space obtained in ref. 16. First, the Fermi energy E_{F} is smaller by a factor of m/m^{*}, and the effective dipoledipole distance is larger than the dipoledipole distance in free space by m^{*}/m. Second, the coefficient B and, hence, κ in free space is obtained from the solution of the Schrödinger equation in the full microwaveinduced potential of interaction between two molecules, whereas here B follows from the fact that the relative wavefunction is zero for r ≤ b (perfectly reflecting wall).
It is clear that for the same 2D density n (and k_{F}) the critical temperature in the lattice is larger than in free space because the BCS exponent in Eq. (8) is smaller. However, in ordinary optical lattices one has the lattice constant nm. In this case, for m^{*}/m ≈ 2 (still the tight binding case with b/ξ_{0} ≈ 3, where ξ_{0} is the extension of the particle wavefunction in the lattice site) and at a fairly small filling factor (let say, k_{F}b = 0.35) the Fermi energy for the lightest alkaline polar molecules NaLi is about 10 nK (n ≈ 2 × 10^{7} cm^{−2}). Then, even for approaching unity the critical temperature is only of the order of a nanokelvin (for k_{F}b = 0.35 and Fig. 2 in Methods gives κ ~ 1).
The picture is quite different in recently introduced subwavelength lattices^{44,45,46,47,48,49,50,51,52}, where the lattice constant can be as small as nm. This strongly increases all energy scales, and even for a small filling factor the Fermi energy may become of the order of hundreds of nanokelvins. Subwavelength lattices can be designed using adiabatic dressing of statedependent lattices^{44}, multiphoton optical transitions^{45,46}, spindependent optical lattices with timedependent modulations^{47}, as well as nanoplasmonic systems^{48}, vortex arrays in superconducting films^{49}, periodically patterned graphene monolayers^{50}, magneticfilm atom chips^{51}, and photonic crystals^{52,53,54}. These interesting proposals already stimulated studies related to manybody physics in such lattices, in particular the analysis of the Hubbard model and engineering of spinspin Hamiltonians^{52}.
In the considered case of p_{x} + ip_{y} pairing in the 2D lattice, putting b = 50 nm, for the same k_{F}b as above the Fermi energy for NaLi molecules exceeds 200 nK (n ≈ 4 × 10^{8} cm^{−2}). Then, for the same κ ~ 1 and approaching unity we have T_{c} ~ 20 nK, which is twice as high as in free space. An additional advantage of the lattice system is the foreseen quantum information processing, since addressing qubits in the lattice is much easier than in free space.
Note that there is a (secondorder) process, in which the interaction between two identical fermions belonging to the lowest Bloch band provides a virtual transfer of one of them to a higher band. Then, the two fermions may get to the same lattice site and undergo the inelastic process of collisional relaxation. The rate constant of this secondorder process is roughly equal to the rate constant in free space, multiplied by the ratio of the scattering amplitude (divided by the elementary cell area) to the frequency of the potential well in a given lattice site (the difference in the energies of the Bloch bands). This ratio originates from the virtual transfer of one of the fermions to a higher band and does not exceed (ξ/b)^{2}. Even in not a deep lattice, where m^{*}/m is 2 or 3, we have (ξ/b)^{2} < 0.1. Typical values of the rate constant of inelastic relaxation in free space are ~10^{−8}–10^{−9} cm^{2}/s^{16}, and hence in the lattice it will be lower than 10^{−9} or even 10^{−10} cm^{2}/s. Thus, the rate of this process is rather low and for densities approaching 10^{9} cm^{−2} the decay time will be on the level of seconds or even tens of seconds.
Interlayer pwave Superfluid of Fermionic Polar Molecules in a Bilayer System
Another interesting novel superfluid of fermionic polar molecules is expected in a bilayer system, where dipoles are oriented perpendicularly to the layers and in opposite directions in different layers.
Such a bilayer configuration, but with all dipoles oriented in the same direction, has been considered in refs 40, 41, 42, 43. As found, it should form an interlayer swave superfluid, where Cooper pairs are formed by dipoles of different layers due to the swave dipolar interaction between them.
For the dipoles of one layer that are opposite to the dipoles of the other one, the picture of interlayer pairing is different. The swave pairing is practically impossible, and the system may form pwave and higher partial wave superfluids. This type of bilayer systems can be created by putting polar molecules with rotational moment J = 0 in one layer, and molecules with J = 1 in the other. Then, applying an electric field (perpendicular to the layers) one gets a fieldinduced average dipole moment of J = 0 molecules parallel to the field, and the dipole moment of J = 1 molecules oriented in the opposite direction. One should also prevent a flipflop process in which the dipoledipole interaction between given J = 1 and J = 0 molecules reverses their dipoles, thus inducing a rapid threebody decay in collisions of a dipolereversed molecule with two original ones. This can be done by making the electric field inhomogeneous, so that it is larger in the layer with J = 0 molecules and the flipflop process requires an increase in the Stark energy. This process will be suppressed if the difference in the Stark energies of molecules in the layers significantly exceeds the Fermi energy, which is a typical kinetic energy of the molecules (~100 nK for the example considered below). This is realistic for present facilities.
For the dipole moment close to 1 Debye and the interlayer spacing of 50 nm, one thus should have the field gradient (perpendicularly to the layers) significantly exceeding 0.5 kV/cm^{2}. This could be done by using electrodes consisting of four rods, and even a higher gradient ~30 kV/cm^{2} should be achievable^{64,65}. By changing the positions of the rods one can obtain the field gradient exceeding 0.5 kV/cm^{2} in the direction perpendicular to the layers of the bilayer system. The field itself will not be exactly perpendicular to the layers and there will also be the field gradient parallel to the layers. This, however, does not essentially influence the physics.
The potential of interaction between two molecules belonging to different layers has the form:
where L is the interlayer spacing, r is the inlayer separation between the molecules, and −d^{2} is the scalar product of the average dipole moments of these molecules. The potential V_{L}(r) is repulsive for and attractive at larger r. The potential well is much more shallow than in the case of all dipoles oriented in the same direction, which was considered in refs 40, 41, 42. We have checked that swave interlayer dimers, which exist at any r_{*}/L, are weakly bound even for r_{*}/L ≈ 3. Their binding energy at is much smaller than the Fermi energy at least for k_{F}L > 0.1. For such r_{*}/L, interlayer dimers with orbital angular momenta l ≥ 1 do not exist. We thus are dealing with purely fermionic physics.
For the analysis of the superfluid pairing we are interested in particle momenta k ~ k_{F}. As well as in the case of all dipoles oriented in the same direction^{40,41,42,43}, under the condition k_{F}r^{*} ≪ 1 (where r^{*} = md^{2}/) the amplitude of interlayer interaction is obtained in the Born approximation. The Fourier transform of the potential (9) is
and in the first Born approximation the onshell amplitude of the lwave scattering at k = k_{F} reads (see Methods):
The swave amplitude is positive, i.e. the swave channel corresponds to repulsion. Note that for extremely low collision energies comparable with the dimer binding energy, where the Born approximation is not accurate, the swave scattering amplitude can be negative. This, however, does not lead to superfluid swave pairing.
The channels with l ≥ 1 correspond to attraction. A straightforward calculation shows that for the largest is the pwave amplitude and, hence, at sufficiently low temperatures the system will be an interlayer pwave superfluid. As for dwave and higher partial wave superfluids, they are possible only at extremely low temperatures. Thus, we confine ourselves to the pwave pairing and employ the BCS approach.
A detailed analysis of the gap equation (1), which includes first and second order contributions to the scattering amplitude and Gor’kovMelikBarkhudarov corrections, is given in Methods. The critical temperature for the pwave superfluidity proves to be (see Methods):
and for not very small k_{F}r^{*} the validity of the perturbative treatment of the Gor’kovMelikBarkhudarov corrections requires (see Methods). The functions F(k_{F}L) and β(k_{F}L) are given in Methods. For k_{F}L ranging from 0.15 to 0.3 the function F increases from 3.4 to 5, and the coefficient β is fairly large, being about 80 at k_{F}L = 0.15 (see Fig. 3 and Methods).
Creating the bilayer system by using a 1D subwavelength lattice we may have L ≈ 50 nm. In this case, for k_{F}L = 0.15 the Fermi energy of NaLi molecules is close to 100 nK, and the critical temperature for k_{F}r^{*} approaching 0.5 is about 10 nK.
For completeness, we also consider the regime of strong interactions within a single layer. Assuming that the coupling between the layers is still fairly weak, we have superfluid (interlayer) pairing between quasiparticles. Related problems have been discussed for coupled 2D Fermi liquids as models for layered superconductors^{8}. In this case, we replace the bare mass m by the effective mass m^{*} and account for renormalization of the fermionic Green functions by a factor Z < 1^{66}. Then, the expression for the transition temperature takes the form:
where we can not determine the preexponential coefficient. Therefore, Eq. (13) only gives an order of magnitude of T_{c}. For k_{F}L = 0.3 and L ≈ 50 nm the Fermi energy of NaLi molecules is about 400 nK, and for, let say, k_{F}r_{*} ≈ 2 the dimer physics is still not important. Then, using the effective mass and factor Z from the Monte Carlo calculations^{67} one may think of superfluid transition temperatures of the order of several tens of nanokelvins.
Conclusions
We have demonstrated the emergence of the topological p_{x} + ip_{y} superfluid for identical microwavedressed fermionic polar molecules in a 2D lattice. Another novel pwave superfluid is found to emerge for fermionic molecules in a bilayer system, with dipoles of one layer opposite to the dipoles of the other one. In both cases the use of subwavelength lattices with a period nm (creation of the bilayer system with the interlayer spacing nm) allows one to obtain superfluid transition temperature of the order of tens of nanokelvins. This opens interesting prospects for topologically protected quantum information processing with p_{x} + ip_{y} superfluids in 2D lattices. The interlayer pwave superfluid in bilayer systems, together with the earlier proposed swave interlayer superfluid^{40,41,42,43} and superfluids in multilayer fermionic systems^{68}, can be a starting point for the creation of more sophisticated layered structures.
Superfluidity itself can be detected in the same way as in the case of swave superfluids^{69,70}. Rotating the p_{x} + ip_{y} superfluid and inducing the appearance of vortices one can find signatures of Majorana modes on the vortex cores in the RF absorption spectrum^{71}. Eventually, one can think of revealing the structure of the order parameter by visualizing vortexrelated dips in the density profile on the approach to the strongly interacting regime, where these dips should be pronounced at least in timeofflight experiments.
Methods
Scattering problem and superfluid pairing of microwavedressed polar molecules in a 2D lattice
As we concluded in the main text, in the low momentum limit at a small filling factor the system of lattice polar molecules is equivalent to that of molecules with effective mass m^{*} in free space. We now demonstrate this explicitly by the calculation of the offshell scattering amplitude f(k′, k). For our problem the main part of the scattering amplitude can be obtained in the Born approximation^{16}.
In the lattice the scattering amplitude is, strictly speaking, the function of both incoming quasimomenta q_{1}, q_{2} and outgoing quasimomenta . However, in the lowmomentum limit where qb ≪ 1, taking into account the momentum conservation law the amplitude becomes the function of only relative momenta k = (q_{1} − q_{2})/2 and . For the offshell scattering amplitude the first Born approximation gives:
where V(r_{1} − r_{2}) is given by Eq. (3) of the main text, and S is the surface area. The last line of Eq. (14) is obtained assuming the tightbinding regime, where the single particle wavefunction is
Here, the index j labels the lattice sites located at the points r_{j}, and N = S/b^{2} is the total number of the sites. The particle wavefunction in a given site j has extension ξ_{0} and is expressed as . In the lowmomentum limit we may replace the summation over j and j′ by the integration over d^{2}r_{j} and taking into account that b^{2} ∑_{j} transforms into ∫d^{2}r_{j}. This immediately yields
and the pwave part of the scattering amplitude is obtained multiplying Eq. (16) by exp(−iϕ) and integrating over dϕ/2π, where ϕ is the angle between the vectors k and k′. This is the same result as in free space (see^{16}). The onshell amplitude (k = k′) can be written as , where is the effective dipoledipole distance in the lattice. The applicability of the Born approximation assumes that , which is clearly seen by calculating the second order correction to the scattering amplitude.
Up to the terms , the onshell scattering amplitude following from the solution of the scattering problem for particles with mass m^{*}, is given by^{16}:
where the numerical coefficient B comes from shortrange physics. For calculating B we introduce a perfectly reflecting wall at intermolecular distances r ~ b, which takes into account that two fermions practically can not get to one and the same lattice site. The coefficient B depends on the ratio , and we show this dependence in Fig. 2a.
The treatment of the superfluid pairing is the same as in ref. 16, including the GorkovMelikBarkhudarov correction. We should only replace the mass m with m^{*}. The expression for the critical temperature then becomes:
where , and it is displayed in Fig. 2b as a function of .
Superfluid pairing of fermionic polar molecules in a bilayer system
For the interlayer interaction potential V_{L}(r) given by equation (9) in the main text, the scattering amplitude for k_{F}r^{*} ≪ 1 can be calculated in the Born approximation^{40}. The pwave part of the first order contribution to the offshell amplitude is
where
and J_{1} is the Bessel function. Regarding the second order contribution, for the solution of the gap equation we only need the onshell pwave part, which is given by
where
In fact, the true pwave scattering amplitude follows from the exact relation
where ψ(k, r) is the true wavefunction of the pwave relative motion with momentum k, normalized such that for r → ∞ we have with being the Hankel function. This amplitude is complex and it is related to the real amplitude given by equations (19)–(22), as
In order to calculate the superfluid transition temperature we use the BCS approach along the lines of ref. 16. We consider temperature T tending to T_{c} from below and rely on the renormalized gap equation (1). For the pwave pairing the order parameter is , and we then multiply Eq. (1) by and integrate over dφ_{k′} and dφ_{k}. As a result, we obtain the same equation (1) in which Δ_{k} and Δ_{k′} are replaced with Δ(k) and Δ(k′), the offshell scattering amplitude f(k′, k) with its pwave part, and δV(k′, k) with its pwave part . Calculating the contribution of the pole in the second term in square brackets and using Eq. (24) we obtain
where the symbol P stands for the principal value of the integral. In the first term in the right hand side of Eq. (25) we divide the region of integration into two parts: E_{k′} − E_{F} < ω and E_{k′} − E_{F} > ω, where Δ(k_{F}), T_{c} ≪ ω ≪ E_{F}. The contribution to the pwave order parameter from the first region we denote as Δ^{(1)}(k), and the contribution from the second region as Δ^{(2)}(k). The contribution of the second term in right hand side of equation (25) is denoted as Δ^{(3)}(k).
We first notice that the main contribution to Δ(k) comes from k′ close to k_{F}. Retaining only f_{1}, which is proportional to kr^{*}, in the offshell scattering amplitude and omitting the second term in the right hand side of Eq. (25) (which is proportional to (kr^{*})^{2}) we obtain
Putting k = k_{F} and performing the integration in the first region in the first term of Eq (25), where E_{F} − ω < E_{k′} < E_{F} + ω, we may put Δ(k′) = Δ(k_{F}) and . Then, putting in and taking into account that the contribution of the second term in square brackets is zero, we obtain:
with C = 0.577 being the Euler constant, and ρ(k_{F}) = m/2π the density of states.
In the second region, where E_{k′} > E_{F} + ω or E_{k′} < E_{F} − ω, we put and keep only f_{1} in the scattering amplitude. For k = k_{F} the integral over E_{k′} from E_{F} + ω to ∞ vanishes. In the integral from 0 to E_{F} − ω we use Δ(k′) from Eq. (26) and find
where
and .
Then, we consider the Gor’kovMelikBarkhudarov corrections to the bare interaction of the molecules in the bilayer. These manybody corrections are second order in (k_{F}r^{*}) and are described by four diagrams (for details, see^{16,41,56}). For the case of pwave superfluidity of identical fermionic polar molecules they have been considered in ref. 16. They have been also studied for the interlayer swave superfluidity of dipoles oriented in the same direction in ref. 41.
We are interested in the case of sufficiently small k_{F}L. Following the same treatment as in refs 16 and 41, in the limit of k_{F}L → 0 we obtain:
where . The dominant contribution to this result comes from the diagram containing a bubble in the interaction line (diagram a) in refs 16 and 41). This contribution strongly decreases with increasing k_{F}L. In particular, for we have , and when increasing k_{F}L to 0.2. Comparing δV with the scattering amplitude f_{1}(k_{F}) we see that for not very small k_{F}r^{*} the perturbative treatment of the Gor’kovMelikBarkhudarov corrections is adequate for . We therefore confine ourselves to these values of k_{F}L.
Performing the integration in the second term of Eq. (25) we obtain the contribution of the Gor’kovMelikBarkhudarov corrections to the order parameter:
the sum of Eqs (27), (28) and (31) yields
where we put ω ~ E_{F} in the terms proportional to (k_{F}r^{*})^{2}. We should also recall that the bare mass m should be replaced with the effective mass m^{*} = m[1 − (4/3π)k_{F}r^{*}] which has been found in refs 41 and 72. Since the relative difference between m^{*} and m is small as k_{F}r^{*}, it is sufficient to replace m with m^{*} only in the multiple r^{*} ~ m in the first term of Eq. (32). This leads to the appearance of a new term
in the right hand side of equation (32). Then, dividing both sides of Eq. (32) by Δ(k_{F}) we obtain for the critical temperature:
where
and
The dependence of F and β on k_{F}L is shown in Fig. 3. We stop at k_{F}L = 0.3 because for larger values of this parameter the function F is so large that the critical temperature will be negligible.
Additional Information
How to cite this article: Fedorov, A. K. et al. Novel pwave superfluids of fermionic polar molecules. Sci. Rep. 6, 27448; doi: 10.1038/srep27448 (2016).
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Acknowledgements
We acknowledge fruitful discussions with Jun Ye, Ignacio Cirac, and Martin Zwierlein. The research leading to these results has received funding from the European Research Council under European Community’s Seventh Framework Programme (FR7/20072013 Grant Agreement no. 341197).
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A.K.F., S.I.M. and V.I.Y. equally contributed to the obtained results. G.V.S. supervised the project, and A.K.F. and G.V.S. have written the text.
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Fedorov, A., Matveenko, S., Yudson, V. et al. Novel pwave superfluids of fermionic polar molecules. Sci Rep 6, 27448 (2016). https://doi.org/10.1038/srep27448
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DOI: https://doi.org/10.1038/srep27448
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