Rashbons: Properties and their significance
RRashbons: Properties and their significance
Jayantha P. Vyasanakere ∗ and Vijay B. Shenoy † Centre for Condensed Matter Theory, Department of Physics,Indian Institute of Science, Bangalore 560 012, India (Dated: August 24, 2011)In presence of a synthetic non-Abelian gauge field that induces a Rashba like spin-orbit interaction,a collection of weakly interacting fermions undergoes a crossover from a BCS ground state to a BECground state when the strength of the gauge field is increased [Phys. Rev. B , 014512 (2011)]. TheBEC that is obtained at large gauge coupling strengths is a condensate of tightly bound bosonicfermion-pairs whose properties are solely determined by the Rashba gauge field – hence calledrashbons. In this paper, we conduct a systematic study of the properties of rashbons and theirdispersion. This study reveals a new qualitative aspect of the problem of interacting fermions innon-Abelian gauge fields, i. e, that the rashbon state induced by the gauge field for small centreof mass momenta of the fermions ceases to exist when this momentum exceeds a critical valuewhich is of the order of the gauge coupling strength. The study allows us to estimate the transitiontemperature of the rashbon BEC, and suggests a route to enhance the exponentially small transitiontemperature of the system with a fixed weak attraction to the order of the Fermi temperature bytuning the strength of the non-Abelian gauge field. The nature of the rashbon dispersion, andin particular the absence of the rashbon states at large momenta, suggests a regime of parameterspace where the normal state of the system will be a dynamical mixture of uncondensed rashbonsand unpaired helical fermions. Such a state should show many novel features including pseudogapphysics. PACS numbers: 03.75.Ss, 05.30.Fk, 67.85.Lm
I. INTRODUCTION
Cold atoms are a promising platform for quantumsimulations. Controlled generation of synthetic gaugefields has provided impetus to the realization of novelphases in cold atomic systems. The recent generationof synthetic non-Abelian gauge fields in Rb atoms is akey step forward in this regard. While a uniform Abeliangauge field is merely equivalent to a galilean transforma-tion, even a uniform non-Abelian gauge field nurturesinteresting physics. The clue that a uniform non-Abelian gauge field cru-cially influences the physics of interacting fermions camefrom the study of bound states of two spin- fermionsin its presence. The remarkable result found for spin- fermions in three spatial dimensions interacting viaa s -wave contact interaction in the singlet channel isthat high-symmetry non-Abelian gauge field configura-tions (GFCs) induce a two-body bound state for any scat-tering length however small and negative. The physicsbehind this unusual role of the non-Abelian gauge fieldthat produces a generalized Rashba spin-orbit interac-tion, was explained by its effect on the infrared densityof the states of the noninteracting two-particle spectrum.The non-Abelian gauge field drastically enhances the in-frared density of states, and this serves to “amplify theattractive interactions”. A second most remarkable fea-ture demonstrated in ref. [6] is that wave function of thebound state that emerges has a triplet content and as-sociated spin-nematic structure similar to those found inliquid He.The above study motivated the study of interactingfermions at a finite density in the presence of a non- Abelian gauge field. At a finite density ρ ( ∼ k F , k F is theFermi momentum), the physics of interacting fermions ina synthetic non-Abelian gauge field is determined by twodimensionless scales. The first scale is associated withthe size of the interactions − /k F a s where a s is the s -wave scattering length, and the second one, λk F , is de-termined by the non-Abelian gauge coupling strength λ .For small negative scattering lengths ( − /k F a s (cid:29) fixedscattering length , even if small and negative, the non-Abelian gauge field induces a crossover of the groundstate from the just discussed BCS superfluid state to anew type of BEC state. The BEC state that emerges isa condensate of a collection of bosons which are tightlybound pairs of fermions. Remarkably, at large gauge cou-plings λ/k F (cid:29)
1, the nature of the bosons that make upthe condensate is determined solely by the gauge field and is not influenced by the scattering length (so long asit is non-zero), or by the density of particles. In otherwords, the BEC state that is attained in the λ/k F (cid:29) a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug FIG. 1. BCS-RBEC crossover induced by a non-Abeliangauge field. Here, a s is the s -wave scattering length, k F isthe Fermi momentum determined by the density, T F is theFermi temperature (= k F / T is the temperature λ is thestrength of the gauge coupling. The solid red line is the transi-tion temperature of the superfluid phase (shaded in light red)obtained in ref. 18 using the Nozi´eres-Schmitt-Rink theory .The solid blue curve is based on the estimate presented in thiswork. The figure reveals the qualitative features of the full“phase diagram” in the T − a s − λ space. (Figure courtesy:Sudeep Kumar Ghosh) fixed scattering length is to be contrasted with the tra-ditional BCS-BEC crossover by tuning the scatteringlength , but with no gauge field. Gong et al. haveinvestigated the crossover including the effects of a Zee-man field along with a non-Abelian gauge field. Certainproperties of rashbons in the EO gauge field (explainedlater) have been investigated in references [16] and [17].It was shown in ref. [7] that the Fermi surface of thenon-interacting system (with a s = 0) in presence of thenon-Abelian gauge field undergoes a change in topologyat a critical gauge coupling strength λ T (of order k F ).For weak attractions ( − /k F a s (cid:29) λ (cid:38) λ T was ar-gued to be primarily determined by the properties of theconstituent anisotropic rashbons (see Sect. V of ref. [7]).It is, therefore, necessary and fruitful to undertake a de-tailed study of the properties of rashbons and their dis-persion, and this is the aim of this paper.In this paper, we study the properties of rashbons andtheir dependence on the nature of the non-Abelian gaugefield, i.e., we obtain properties of rashbons for the mostinteresting gauge field configurations. This study entailsa study of the anisotropic rashbon dispersion, i. e., de-termination of its energy as a function of its momentumby the study of the two-body problem in a non-Abeliangauge field with a resonant scattering length (1 /λa s = 0).In addition to the determination of the properties of rash- bons, we report here a new qualitative result. It is shownthat when the momentum of a rashbon exceeds a criticalvalue which is of the order of the gauge coupling strength,it ceases to exist . Stated otherwise, when the center ofmass momentum of the two fermions that make up thebound pair exceeds a value of order of the gauge couplingstrength, the bound state disappears. To uncover thephysics behind this result, the two-fermion problem in agauge field is investigated in detail for a range scatter-ing of lengths and centre of mass momenta. The studyreveals a hitherto unknown feature of the non-Abeliangauge fields: while the non-Abelian gauge field acts asattractive interaction amplifiers for fermions with centreof mass momenta q much smaller than the gauge fieldstrength ( q (cid:28) λ ), the gauge field suppresses the forma-tion of bound states of fermions with large centre of massmomenta ( q (cid:38) λ ) . In fact, it is demonstrated here thatwhen q (cid:38) λ , a positive scattering length (very strong at-traction) is necessary to induce a bound state of the twofermions, quite contrary to q (cid:28) λ where a bound stateexists (essentially) for any scattering length.The results we report here have two significant out-comes. (1) A full qualitative picture of the BCS-BECcrossover scenario in the presence of a non-Abelian gaugefield is obtained (see Fig. 1) based on the results reportedhere. Most notably, it is shown that the transition tem-peratures of a system of fermions with a very weak at-traction can be enhanced to the order of the Fermi tem-perature (determined by the density) by the applicationof a non-Abelian gauge field. (2) Our two body resultsat large centre of mass momenta suggest that the normalstate of the fermion system in non-Abelian gauge fieldwill be a “dynamic mixture” of rashbons and interact-ing helical fermions. These could therefore show manynovel features such as pronounced pseudogap character-istics (see ref. 20 and references therein).The next section, II, contains the preliminaries whichincludes the formulation of the problem. Sec. III con-tains a report on the properties of rashbons, and thisis followed by sec. IV which discusses the bound state oftwo fermions for arbitrary centre of mass momentum andscattering length for specific high symmetry gauge fields.The importance of the results obtained here is discussedin sec. V, and the paper is concluded with a summary insec. VI. II. PRELIMINARIES
The Hamiltonian of the fermions moving in a uni-form non-Abelian gauge field that leads to a generalizedRashba spin-orbit interaction is H R = (cid:90) d r Ψ † ( r ) (cid:18) p − p λ · τ (cid:19) Ψ( r ) , (1)where Ψ( r ) = { ψ σ ( r ) } , σ = ↑ , ↓ are fermion operators, p is the momentum, is the SU(2) identity, τ µ ( µ = x, y, z )are Pauli matrices, p λ = (cid:80) i p i λ i e i , e i ’s are the unitvectors in the i -th direction, i = x, y, z . The vector λ = λ ˆ λ = (cid:80) i λ i e i describes a gauge-field configuration(GFC) space; we refer λ = | λ | as the gauge-couplingstrength. Throughout, we have set the mass of thefermions ( m F ), Planck constant ( (cid:126) ) and Boltzmann con-stant ( k B ) to unity.In this paper we specialize to λ = ( λ l , λ l , λ r ) asthis contains all the experimentally interesting high-symmetry GFCs. Moreover, it is shown in Ref. [6 and7], that this set of gauge fields captures all the qualita-tive physics of the full GFC space. Specific high sym-metry GFCs are obtained for particular values of λ r and λ l : λ r = 0 corresponds to extreme oblate (EO) GFC; λ r = λ l corresponds to spherical (S) GFC, and λ l = 0corresponds to extreme prolate (EP) GFC.The interaction between the fermions is described bya contact attraction in the singlet channel H υ = υ (cid:90) d r ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) . (2)Ultraviolet regularization of the theory described by H = H R + H υ is achieved by exchanging the bare inter-action v for the scattering length a s via υ + Λ = πa s ,where Λ is the ultraviolet momentum cutoff. Note that a s is the s -wave scattering length in free vacuum, i. e.,when the gauge field is absent ( λ = 0).The one-particle states of H R are described by thequantum numbers of momentum k and helicity α (whichtakes on values ± ) : | k α (cid:105) = | k (cid:105) ⊗ | α ˆ k λ (cid:105) . The one-particle dispersion is ε k α = k − α | k λ | , where k λ is de-fined analogously with p λ and | α ˆ k λ (cid:105) is the spin-coherentstate in the direction α ˆ k λ . The two-particle states of H can be described using the basis states | qk αβ (cid:105) = | ( q + k ) α (cid:105) ⊗ | ( q − k ) β (cid:105) where, q = k + k is the centerof mass momentum and k = ( k − k ) / k and k . Note that q is a good quantum number for the fullHamiltonian ( H ). The non-interacting two-particle dis-persion is E free qk αβ = ε ( q + k ) α + ε ( q − k ) β . In the presenceof interactions, bound states emerge as isolated poles ofthe T -matrix, and are roots of the equation14 πa s = 1 V (cid:88) k αβ (cid:32) | A q αβ ( k ) | E ( q ) − E free qk αβ + 14 k (cid:33) (3)where, A q αβ ( k ) is the singlet amplitude in | qk αβ (cid:105) , V isthe volume, E ( q ) = E th ( q ) − E b ( q ) is the energy of thebound state. Here E th ( q ) is the scattering threshold and E b ( q ) is the binding energy, both of which depend on q as indicated.In the absence of the gauge field ( λ = 0), the boundstate exists only for a s > E b ( q ) = − /a s is inde-pendent of q . The threshold is E th ( q ) = q /
4. Physi-cally, this corresponds to the fact that a critical attrac-tion in necessary in free vacuum ( λ = 0) for the formationof the two-body bound state. As shown in ref. [6], state of EO S EP η t λ r / λ (b) m a ss / m F m l m r m ef (a) FIG. 2. (Color online) Rashbon properties for different GFCs. (a)
In-plane, perpendicular and effective masses. (b)
Thetriplet content of rashbons. affairs change drastically in the presence of a non-Abeliangauge field. For q = 0, the presence of the gauge fieldalways reduces the critical attraction to form the boundstate and in particular, for special high symmetry GFCs(e.g. λ = ( λ l , λ l , λ r ) with λ r ≤ λ l ) two body bound stateforms for any scattering length. III. PROPERTIES OF RASHBONS
The bound state that emerges in the presence of thegauge field when the scattering length is set to the reso-nant value 1 /a s = 0 is the rashbon. As argued above, thebinding energy of the rashbon state for all the GFCs con-sidered here (except for the EP GFC) is positive. Theenergy of the rashbon state E R ( q = 0) determines thechemical potential of the RBEC. Other properties of theRBEC are determined by the rashbon dispersion E R ( q ),and in particular the transition temperature will be de-termined by the mass of the rashbons.The curvature of the rashbon dispersion E R ( q ) at q = defines the effective low-energy inverse mass ofrashbons. The dispersion is in general anisotropic andthe inverse mass is, in general, a tensor. However, due totheir symmetry, for the GFCs considered in this paper ( λ of the form ( λ l , λ l , λ r )), E R ( q ) = E R ( q l , q r ), where q l isthe component of q on the x − y plane, and q r is the com-ponent along e z . Thus, the inverse mass tensor is com-pletely specified by its principal elements - in-plane in-verse mass ( m − l ) and the “perpendicular” inverse mass, m − r . m − l = ∂ E R ( q l , q r ) ∂q l (cid:12)(cid:12)(cid:12)(cid:12) q =0 , m − r = ∂ E R ( q l , q r ) ∂q r (cid:12)(cid:12)(cid:12)(cid:12) q =0 (4)An effective mass m ef defined as m ef = (cid:113) ( m r m l ) (5)is useful in the discussions that follow.In addition to the anisotropy in their orbital motion,rashbons are intrinsically anisotropic particles. Theirpair-wave function has both a singlet and triplet compo-nent; the weight of the pair wave function in the tripletsector η t is the triplet content. The triplet component istime reversal symmetric, but does not have the spin rota-tional symmetry – it is therefore a spin nematic. Keepingthis interesting aspect in mind, we shall also investigateand report the triplet content of rashbons, and its depen-dence on the gauge field.Before presenting the results we make a general obser-vation. The threshold energy ( E th ) becomes increasinglyflat as a function of q in the small q /λ regime as oneapproaches spherical gauge field in the GFC space. Infact, for the spherical GFC, it is exactly constant in thesmall q /λ regime (see Fig. 3). The mass is therefore de-termined entirely by the variation of the binding energywith q (this may be contrasted in free vacuum case dis-cussed before). It is reasonable therefore to expect thatthe effective mass of rashbons is always greater than twicethe bare fermion mass and for it to be the largest for thespherical GFC.Fig. 2 (a) shows the in-plane, perpendicular, and ef-fective masses for different GFCs. Rashbons emergingfrom spherical GFCs have the highest m ef and that fromEP GFCs have the least. It is interesting to note thatapart from the spherical GFC, there is yet another GFC( λ r ≈ . λ - see fig. 2 (a) ) where the low energy dis-persion is isotropic, i. e., rashbon has a scalar mass. Thetriplet content is shown in fig. 2 (b) for different GFCs. η t is minimum (1/4) for spherical GFC and maximum(1/2) for EP GFC.A detailed study of rashbon dispersion as a functionof its momentum q (centre of mass momentum of thefermions that make up the rashbon) revealed a hithertounreported and rather unexpected feature. The full rash-bon dispersion as a function of q for the spherical (S)GFC is shown in Fig. 3. The rashbon energy increaseswith increasing q and eventually for q/λ (cid:38) .
3, there isno two body bound state! This curious result motivatedus to perform a more detailed investigation of the disper-sion of the bound fermions (bosons) at arbitrary scatter-ing lengths (away from resonance which corresponds torashbons), in order to uncover the physics behind thisphenomenon. This study, conducted for specific highsymmetry GFCs, is presented in the next section.
IV. DISPERSION OF BOSONS AT ARBITRARYSCATTERING LENGTHS FOR SPECIFIC GFCS
In this section we investigate the dispersion of thebosonic bound state of two fermions at arbitrary scat-tering length. Results of the boson dispersion obtainedby solving eqn. (3) will be presented for the S and EOGFCs.
A. Spherical GFC
Spherical (S) GFC corresponds to λ r = λ l and henceproduces an isotropic boson dispersion as discussed be-fore. The boson dispersion depends only on q = | q | .Solving eqn. (3), the boson dispersion obtained for var-ious scattering lengths is as shown in fig. 3(a). Thekey features of this spectrum are the following. For anyscattering length, however large and positive , there existsa critical center of mass momentum q c such that when q > q c the bound state ceases to exist.This is best understood by fixing attention on a par-ticular momentum q . When the momentum is “small”,there is a bound state for any attraction . This is in factthe case for all q < q o , where q o = 2 λ √ . For q > q o , a crit-ical attraction described by a nonzero scattering length a sc is necessary for the formation of a bound state. For q = q + o , the critical scattering length is a sc = − √ λ . Onincreasing q , a stronger attractive interaction is requiredto produce a bound state and when q reaches ∼ λ , a res-onant attraction is necessary to produce a bound state.For q (cid:38) λ , a very strong attractive interaction describedby a small positive scattering length is necessary to pro-duce a bound state. In fact, for q (cid:29) λ , the critical scat-tering length scales as a sc ∼ (cid:113) λq . The dependence of a sc on the centre of mass momentum is shown in Fig. 3(b).How do we understand these results? Here the ε − γ model introduced in Ref. 6 comes to our rescue. Themodel states that if the infrared density of states g s ( ε ) ∼ ε γ for 0 ≤ ε ≤ ε , where ε is the energy measuredfrom the scattering threshold, then the critical scatteringlength is given by √ ε a sc ∝ γ Θ( γ ) / (2 γ − γ <
0, the criticalscattering length vanishes.It is evident that there is a drastic change in the in-frared density of states at q = q o . In fact, this special mo-mentum q o is such that the threshold energy correspondsto that state where the relative momentum k betweenthe pair of fermions vanishes. Clearly, for q < q o , thereare many degenerate k states that produce a nonzerodensity of states at the threshold. In fact, when q = 0,the density of states diverges as 1 / √ ε , i.e., γ = − / q < q o , there is still a finite density of states at thethreshold with an effective γ <
0. Thus the critical scat-tering length, as given by the (cid:15) − γ model, vanishes. Letus turn our attention to what happens for q (cid:29) λ . Fromeqn. (3) it is evident that the density of states g s ( ε ) has -0.7-0.5-0.3 0 0.3 0.6 0.9 1.2 1.5 E ( q ) / λ q / λ E th λ a s / √ -1/2-1-2-4rashbon+4 (a) -15-10-5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 λ a s c q / λ (b) FIG. 3. (Color online) (a)
The boson dispersion for various scattering lengths in spherical GFC. Note that for any givenscattering length, the bound state disappears after some critical momentum. (b)
Critical scattering length ( a sc ) as a functionof momentum. a sc goes as 1 / √ q in the large q/λ limit. the contributions from the ++, −− , + − and − + chan-nels. It can be shown that in the regime q (cid:29) λ , of the ++channel has a density of states that has ε / behaviour.The + − and − + channels have a higher threshold whichis λq larger than the threshold of the ++ channel; thedensity of states of the + − / − + channels goes as √ ε formthis higher threshold. These arguments provide an esti-mate of ε ≈ qλ . The result on the critical scatteringlength is then a sc ∼ √ q , precisely as obtained from thefull numerical solution shown in Fig. 3(b).As a by product of the analysis of the boson dispersion,we were able to obtain an analytical expression for themass of bosons (which is isotropic in this case) m B m F = 6 E ( ) / E ( ) / + 2 (cid:112) E ( ) λ − (cid:0) E ( ) + λ (cid:1) / , (6)where, E ( ) = − λ − (cid:32) a s + (cid:115) a s + 4 λ (cid:33) (7)At a given λ , as expected, mass for a small positivescattering length a s > (4 + √ m F ≈ . m F . Interestingly, the valueof m B /m F for a small negative scattering length limit is(integer) 6. B. Extreme oblate GFC
Extreme oblate (EO) GFC corresponds to λ r = 0 with λ l = λ √ . It can be easily shown that for this GFC, E ( q l , q r ) = E ( q l ,
0) + q r . Thus, the two-body dispersion -1.1-0.9-0.7-0.5 0 0.4 0.8 1.2 1.6 E ( q l , ) / λ q l / λ E th λ a s -1-2rashbon+2 FIG. 4. (Color online) The boson dispersion for various scat-tering lengths in extreme oblate GFC. Just as in S GFC (seeFig. 3), for any given scattering length, the bound state dis-appears after some critical q l . as a function of q l provide all the nontrivial features ofthe two-body problem arising from this gauge field.Fig. 4 shows the boson dispersion for various scatteringlengths. Remarkably, we find that the dispersion has verysimilar features as found for the spherical GFC, i.e., forany given scattering length there is a q c such that for q > q c , the two-body bound state ceases to exist. Clearly,this is a generic feature of the boson (bound fermion-pair)dispersion in a gauge field.For this GFC, m r is just twice the fermion mass. Thein-plane mass ( m l ) extracted from the two-body disper-sion is shown in fig. 5. m l for small positive scatteringlength is again twice the fermion mass. The resonancevalue which corresponds to rashbon is m l (cid:39) . m F . Thisresult agrees with refs. [16 and 17]. It is again interest-ing to note that value of m l /m F in the deep BCS limit m l / m F -1 / ( λ a s ) FIG. 5. (Color online) In-plane mass of tightly bound fermionpairs in the rashbon BEC side in presence of an extreme oblateGFC. is (integer) 4.
C. Discussion
The analysis of the dispersion of the boson (bound-state of two fermions obtained in a gauge field) revealsthat the boson ceases to exist when the momentum ofthe boson exceeds a critical value. For the case of rash-bons (bosons obtained at resonance scatteirng length),the critical momentum is of the order of the strength ofthe gauge field.The analysis presented here shows that this is againbecause of the influence of the gauge field in alteringof the infrared density of states. When the momen-tum is smaller than the magnitude of the gauge coupling,the gauge field works to enhance the infrared density ofstates. On the other hand, for large momenta, the gaugefield has the opposite effect, i. e., it depletes the infrareddensity of states.
V. SIGNIFICANCE OF THE RESULTS
The above results allow us to infer many key aspectsof the physics of interacting fermions in the presence ofa non-Abelian gauge field.First, these results allow us to estimate the transitiontemperature. For large gauge couplings, the transitiontemperature as noted above will be determined by themass of the rashbons. We have argued (and demon-strated) that the mass of the rashbons is always greaterthan twice the fermion mass. Thus the transition temper-ature of RBEC will always be less than that of the usualBEC of bound pairs of fermions obtained in the absenceof the gauge field by tuning the scattering length to smallpositive values. -3 -2 -1
0 1 2 3 4 5 T c / T F λ / k F k F a s = -1/4 EO - GFC
MFTTwo Body (b) -3 -2 -1 T c / T F S - GFC (a)
FIG. 6. (Color online) Estimate of transition temperaturein spherical and extreme oblate GFCs as a function of thegauge coupling strength which takes the regular BCS state toa rashbon BEC. T c in the small λ/k F limit is obtained frommean field theory (analytical approximation is shown in thetext). T c in the large λ/k F limit is obtained from the con-densation temperature of the tightly bound pairs of fermions(analytical form for the S GFC can be obtained from eqn. (6)and eqn. (8)). Horizontal dashed line corresponds to rashbon T c . Vertical line indicates the gauge coupling correspondingto the Fermi surface topology transition . However, there is something remarkable that a syn-thetic non Abelian gauge field can achieve. Consider asystem with a weak attraction (small negative scatteringlength). In the absence of the gauge field, the transitiontemperature in the BCS superfluid state is exponentiallysmall in the scattering length. Interestingly, the transi-tion temperature can be brought to the order of Fermitemperature by increasing the magnitude of the gaugefield strength (keeping the weak attraction, small nega-tive scattering length, fixed).While T c in the BCS regime is determined by thepairing amplitude (∆), in the BEC regime it is deter-mined by the condensation temperature of the emergentrashbons. The mean field estimate of the former (i.e. for small k F | a s | , a s < λ/k F ) is obtained by simultane-ously solving − / (4 πa s ) = V (cid:80) k α (cid:18) tanh ξ k α Tc ξ k α − k (cid:19) andthe number equation, ρ = V (cid:80) k α / (cid:16) exp (cid:16) ξ k α T c (cid:17) + 1 (cid:17) , where ξ k α = ε k α − µ . In this limit, the chemical poten-tial at T c is almost equal to that of the noninteractingone at zero temperature. i.e., µ ( T c , a s , λ ) ≈ µ (0 , − , λ ),and ∆ ( T =0) /T c ≈ π/e γ where ∆ ( T =0) is the pairing am-plitude at zero temperature and γ is Euler’s constant ( ≈ T c on the RBEC side can be extracted from theeffective mass ( m ef ) as condensation temperature of thebosonic pairs : T c T F = (cid:18) π ( ζ (3 / (cid:19) / m ef , (rashbon BEC) (8)where we recall that m ef = ( m r m l ) / . Using the infor-mation of mass given earlier (eqn. (6) for S GFC and fig. 5for EO GFC) one can obtain T c in this regime as a func-tion of λa s in S and EO GFCs. In particular, rashbon T c in S case is ≈ . T F and in EO case it is ≈ . T F .The rashbon T c can be obtained for various GFCs, us-ing m ef shown in fig. 2 ( a ). Since, among all GFCs, therashbon mass corresponding to S GFC is the largest, italso corresponds to condensate with the smallest T c .The results obtained in both BCS and RBEC limits for k F a s = − / T c has increased by two ordersof magnitude with increasing gauge coupling strength λ .These considerations also allow us to infer an overall qual-itative “phase diagram” in the T − a s − λ space as shownin Fig. 1.What is the nature of the system above T c ? There is aregime in the parameter space shown in Fig. 1, where thenormal state can be quite interesting. Consider for exam-ple λ ≈ . k F . The ground state will be “very bosonic”i. e., a condensate of rashbons in the zero momentumstate. On heating the system above the transition tem-perature (cid:46) T F , the system becomes normal. Rashbonsare excited to higher momenta states, and eventuallybreak up into the constituent fermions since there is nobound state at higher momenta. There, should, thereforebe a temperature range where the sytem is a dynamicalmixture of uncondensed rashbons and high energy helicalfermions – a state that should show many novel featuressuch as, among other things, a pseudogap. VI. SUMMARY
The new results of this paper are:1. A systematic enumeration of the properties of rash-bons, including closed form anlytical formulae, forvarious gauge field configurations.2. A detailed study of the rashbon (boson) disper-sion, which results in a new qualitative observa-tion. Although a zero centre of mass momentumbound state exists for any scattering length formany GFCs, the bound state vanishes when thecentre of mass momentum exceeds a critical value.Thus, although the gauge field acts to promotebound state formation for small momenta, it actsoppositely, i. e., inhibits bound state formation forlarge momenta. We provide a detailed explanationof the physics behind the phonemon.These results allow us to make two important infer-ences.1. For a fixed weak attractive interaction, the expo-nentially small transition temperature of a BCS su-perfluid can be enhanced by orders of magnitude tothe order the Fermi temperature of the system byincreasing the magnitude of the gauge coupling.2. There is a regime of T − a s − λ parameter spacewhere the normal phase of the system will havenovel features.We hope that these results will stimulate further ex-perimental and theoretical studies on this topic. Acknowledgements
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