Higher angular momentum pairing from transverse gauge interactions
HHigher angular momentum pairing from transverse gauge interactions
Suk Bum Chung , Ipsita Mandal , Srinivas Raghu , and Sudip Chakravarty Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA Department of Physics, Stanford University, Stanford, California 94305, USA (Dated: November 6, 2018)In this paper, we study superconductivity of nonrelativistic fermions at finite-density coupled toa transverse U (1) gauge field, with the effective interaction including the Landau-damping. Thismodel, first studied by Holstein, Norton, and Pincus [Phys. Rev B, , 2649 (1973)] has been knownas an example of a non-Fermi liquid, ı.e. a metallic state in which the decay rate of a quasiparticleis large compared to the characteristic quasiparticle energy; other examples of the non-Fermi liquidincludes the 2d electron gas in a magnetic field at ν = 1 / (cid:96) ≥
2. Our results are obtained from a solution of the Dyson-Nambu equation. Note that in thisproblem there is a quantum critical point between a non-Fermi liquid state and the superconductingstate, as the critical coupling is nonzero. This is in contrast to a weakly coupled metal, whichexhibits superconductivity for infinitesimally weak interaction regardless of its sign.
I. INTRODUCTION
A classic problem of condensed matter physics is thatof a finite density of electrons coupled to transverse gaugefields. As was first noticed by Holstein, Norton, and Pin-cus (HNP) and elaborated further by Reizer , such asystem exhibits anomalous properties, since the trans-verse component of the gauge field remains unscreened.While the original motivation of HNP was to understandthe effects of the electromagnetic field coupled to a metal,the same field theory applies to the case of fermions cou-pled to an emergent gauge field, as is the case in cer-tain spin-liquids, and perhaps even the normal state ofcuprate superconductors and compressible quantumHall systems at the filling fraction of 1 / As is truefor any metal, the system possesses infinitely many gap-less excitations; however the decay rate of these “quasi-particles” is parametrically larger then their character-istic energy. A largely unsolved issue is whether or notsuch a non-Fermi liquid metal can remain stable againstthe formation of superconductivity.While the treatments of HNP , and Reizer , werebased on the random phase approximation (RPA), ananalysis based on renormalization group treatment inthe vicinity of the upper-critical dimension, (cid:15) = (3 − d ),suggested that the problem of fermions coupled to trans-verse gauge bosons leads to a non-Fermi liquid fixedpoint, also found by Nayak and Wilczek , who consid-ered a more general expansion in (cid:15) = 3 − ( d + x ), where x isthe exponent characterizing the range of the four fermioninteraction. The stability of the non-Fermi liquid fixedpoint with respect to competing ordered states, however,remains an outstanding issue. The issue has also beenaddressed in the context of color superconductivity ;in the simplest case up and down quarks with two dif-ferent flavors form s -wave superconductivity whereas thequarks of the third color remain gapless. Here we consider T = 0 and show that emergent trans-verse U (1) gauge bosons, more appropriate to condensedmatter physics, coupled to fermions in a fermi liquid canopen up superconducting gap. Once a gap is present,all non-Fermi liquid anomalies are cut off and the the-ory is self consistent. The quantum phase transition isa continuous transition at a nonzero critical gauge cou-pling. This is unlike conventional theory of superconduc-tivity where arbitrarily small attractive coupling leadsto superconductivity. It is also unlike Kohn-Luttingersuperconductivity where infinitesimally small repulsivecoupling leads to superconductivity.Our key results here are twofold. Firstly, the non-Fermi liquid metal resulting from coupling a Fermi sur-face to a transverse U (1) gauge field remains a sta-ble ground state below a critical threshold gauge cou-pling; there is no superconducting instability. Above thethreshold coupling, there are instabilities towards uncon-ventional superconducting states with angular momen-tum (cid:96) ≥
2. Our analysis is carried out via a straight-forward Hatree-Fock solution of the self-consistent gapequation. While it neglects fluctuation effects, it pro-vides a simple qualitative guideline to the possible natureof the ground states, and is likely to remain reliable closeto d = 3+1, which is the upper-critical dimension for theproblem at hand; the problem we consider correspondsto x = 0. We also briefly touch upon the case d = 2 + 1.This paper is organized as follows. In Sec. II, wepresent our system of nonrelativistic fermions at finite-density coupled to a transverse U (1) gauge field. InSec. III, we derive the Hartree-Fock superconductinggap equation from the Dyson formalism. In Sec. IVwe present our numerical solution to the gap equation.Lastly we discuss the implications and the validity of ourresults. In an Appendix we consider a similar treatmentat d = 2+1, which we believe is still above the lower criti-cal dimension, and our Hartree-Fock treatment could still a r X i v : . [ c ond - m a t . s t r- e l ] M a y rovide a qualitative guide. II. THE MODEL SYSTEM
We first present our action for the fermions coupledto the transverse gauge bosons at d = 3. Our action is Wick-rotated at T = 0, which gives us the form most con-venient for investigating the quantum phase transitionfrom a non-Fermi liquid to a superconducting state. Thefull action consists of three main terms - the free fermionaction denoted by ψ ’s, the free U (1) gauge field, denotedby A ’s, and the fermion-boson coupling ( i, j = 1 , , S F = (cid:90) dω π (cid:90) d k (2 π ) ψ † ( k , ω ) [ iω − ( ε k − µ )] ψ ( k , ω ) , (1) S G = (cid:90) dω π (cid:90) d q (2 π ) A † i ( q , ν ) (cid:2) q + ν (cid:3) (cid:0) δ ij − q i q j /q (cid:1) A j ( q , ν ) , (2) S int = gm (cid:90) dω π (cid:90) dν π (cid:90) d p (2 π ) (cid:90) d k (2 π ) ψ † ( k , ν ) { A i ( k − p , ν − ω ) p i } ψ ( p , ω ) , (3)where repeated indices are summed over, m is the mass ofthe fermion, ε k = k / m and g is the gauge coupling. Wework in the Coulomb gauge such that ∂ i A i = 0. To sim-plify notation we have set (cid:126) and the gauge boson velocity c g to unity. We have subsumed all four fermion interac-tions into S F in the spirit of Landau theory. The gaugegroup is assumed to be noncompact, which is not entirelya trivial assumption. In (2 + 1)-dimensions, Polyakov has shown that the gauge field is massive for the com-pact case due to instanton effects, at least in the absenceof the matter field. In the presence of massless fermionsand in (3 + 1) dimensions, such topological excitationsare assumed not to play any important role. Since we are investigating the possibility of supercon-ductivity cutting off the non-Fermi liquid behavior, weuse for our effective transverse gauge boson mediated in-teraction the same effective interaction that is knownto give rise to the non-Fermi liquid behavior in thissystem . It was shown previously that this effective in-teraction can be obtained simply by taking the one-loopcorrected gauge boson propagator as the vertex does notreceive any anomalous dimension to one-loop order fromthe Ward identity a conclusion that was also reached byPolchinski. In other words, our problem is to find outwhether the interaction mediated by the gauge bosonswith the one-loop corrected effective action S G = (cid:90) dν π (cid:90) d q (2 π ) A † i ( q , ν ) (cid:2) q + ν + γ | ν | /qv F (cid:3) (cid:0) δ ij − q i q j /q (cid:1) A j ( q , ν ) , (4)can lead to superconductivity in (3 + 1) dimensions. Thecorresponding gauge propagator D ij ( k , ν ) = (cid:0) δ ij − k i k j k (cid:1) k + γ | ν | kv F (5)is shown graphically in Fig. 1. where v F is the Fermivelocity, γ = αk F , and α = g v F π is our analog of the finestructure constant; note that we have dropped the ν term, keeping only the most relevant low frequency termthat arises as the result of the Landau damping. It isimportant to note that our fine structure constant α canbe much larger than that of quantum electrodynamics,given that v F can be much larger than the gauge velocity,set typically by a strongly correlated many body system,and of course g itself is not known in general.Our problem is analogous to the Kohn-Luttinger prob-lem, albeit with some qualifications. To see how this is so, note that the transverse gauge bosons mediate current-current interaction. The bare current-current interactionin the Cooper channel is repulsive for the same reason astwo parallel wires with anti-parallel currents repel eachother. Thus, our problem shares the most essential char-acteristic of the Kohn-Luttinger problem - the Cooperpairing out of a repulsive bare interaction. However,whereas in the Kohn-Luttinger problem the infinitesimalbare interaction is sufficient to destabilize the Fermi liq-uid through Cooper pairing, in our problem a finite inter-action strength - that is to say, a finite coupling constant- is required for Cooper pairing to destabilize the non-Fermi liquid . This is because the screening of our in-teraction is dynamical, meaning that we recover the bareinteraction in the zero frequency limit. Together with thefact that at the zero frequency limit we will have equallydivergent repulsion in all angular momentum channels,this makes the perturbative method used in the Kohn-2 q q qpp − q FIG. 1. One-loop corrected gauge boson propagator. Herefor compactness we use Euclidean four vector notation: p =( p , p ) ≡ ( ω, p ). Luttinger problem quite unsuitable for our problem.
III. GAP EQUATION FROM DYSONFORMALISM
We obtain the pairing gap self-consistently throughDyson formalism, as this technique can be ap-plied straightforwardly to a frequency-dependentinteraction . In this formalism, we treat the pairinggap as the self-energy of the fermion propagator in theNambu basis, as seen in Fig. 2. In our Dyson equation, ≈ k − k k q − q − kk − q FIG. 2. Single-particle-exchange contribution to the off-diagonal component of the Dyson equation. The black blobdenotes the proper self-energy, and the grey blob (along withthe associated fermion lines) denotes the exact propagator ofthe fermions. As in Fig.1, we have used the Euclidean fourvector notation for simplicity. Σ( k , ω ) = (cid:90) d qdν (2 π ) V eff ( k , ω ; q , ν ) G ( q , ν ) , (6)the self-energy Σ( k , ω ) is due to the gauge boson medi-ated interaction in the Cooper channel V eff ( k , ω ; q , ν ) = − g m ( k i ) D ij ( k − q , ω − ν )( − k j )= g m k − [ k · ( k − q )] ( k − q ) ( k − q ) + γ | ω − ν | v F | k − q | . (7)In order to derive the gap equation from this Dyson equa-tion, we note that the self-energy by definition is the dif- ference between the free and interacting propagator: − Σ( k , ω ) = G − ( k , ω ) − G − ( k , ω ) . (8)In the Nambu basis, the inverse non-interacting fermionpropagator is G − ( k , ω ) = (cid:18) iω − ξ k iω + ξ k (cid:19) , (9)where ξ k ≡ v F ( k − k F ). Due to our assumption that thesole consequence of the interaction is the Cooper pairing,the inverse of the interacting fermion propagator thentakes the following form in the Nambu basis: G − ( k , ω ) = (cid:18) iω − ξ k ∆ ∗ ( k , ω )∆( k , ω ) iω + ξ k (cid:19) . (10)Then, G ( k , ω ) = − ω + ξ k + | ∆( k , ω ) | (cid:18) iω + ξ k − ∆ ∗ ( k , ω ) − ∆( k , ω ) iω − ξ k (cid:19) . (11)We can see now that the gap equation can be derivedby inserting the above propagators into the Dyson equa-tion. On the Fermi surface, the gap equation comes outto be∆(ˆ k , ω ) = − (cid:90) d q dν (2 π ) V eff ( ˆk − ˆq , ω − ν ) ∆(ˆ q , ν ) ν + ξ q + | ∆(ˆ q , ν ) | , (12)where V eff ( ˆk − ˆq , ω − ω (cid:48) ) = ( gv F ) θ k F (1 − cos θ ) + γ | ω − ω (cid:48) | /v F √ k F √ − cos θ . (13) V eff ( k , ω ; q , ν ) is evaluated on the Fermi surface with θ being the scattering angle: cos θ = ˆk · ˆq . Our gap equa-tion does not take into account the fermion wave func-tion renormalization factor Z ; it has also been droppedin Refs. 17, 23–25. The infrared anomaly on the Fermisurface ∝ ( α/ π ) ln (Λ / | ω | ), shown previously, is cutoffat the scale of the pairing gap. Thus, setting Z ( ω ) ≈ q and decompose the gap intoall angular momentum channels (cid:96) ’s. We can trans-form the integral by introducing the density of states atthe Fermi energy, (cid:82) d q/ (2 π ) → N (0) (cid:82) dξ q (cid:82) d (cos θ ) / N (0) = k F /π v F ). We obtain at T = 03(ˆ k, ω ) = − α (cid:90) dν π (cid:90) d (cos θ )2 1 + cos θ (1 − cos θ ) + α √ | ω − ν | E F √ − cos θ ∆(cos θ, ν ) (cid:112) ω (cid:48) + | ∆(cos θ, ν ) | . (14)Being a nonlinear integral equation involving the gap,we cannot strictly speaking decouple the various angularmomentum channels. However, we shall assume approx-imate decoupling. Such decoupled gap equations for dif-ferent channels can be considered as local minima of thefree energy. Therefore the self-consistency condition forthe pairing gap for each angular momentum (cid:96) is, usingthe Fermi energy, E F , as the unit energy,˜∆ (cid:96) (˜ ω ) = − (cid:90) d ˜ ν ˜ V (cid:96) (˜ ω − ˜ ν ) ˜∆ (cid:96) (˜ ν ) (cid:113) ˜ ν + | ˜∆ (cid:96) (˜ ν ) | , (15)where˜ V (cid:96) (˜ ω ) = ˜ α (cid:90) d (cos θ ) P (cid:96) (cos θ )(1 + cos θ )(1 − cos θ ) + π ˜ α √ | ˜ ω |√ − cos θ , (16)with ˜ ω = ω/E F , ˜ ν = ν/E F , ˜∆ (cid:96) = ∆ (cid:96) /E F and ˜ α = α/ π .We see here that ˜ α can be treated as the dimensionlesseffective interaction strength. We plot V (cid:96) ( ω ) in Fig.3. l=0l=1l=2l=3l=4l=5l=6 Π Α(cid:142) Ω(cid:142) (cid:45) (cid:45) V l (cid:142) (cid:72) Ω(cid:142) (cid:76)
Α(cid:142)
FIG. 3. Angular momentum decomposition of the interaction˜ V (cid:96) (˜ ω ) / ˜ α versus π ˜ α ˜ ω/ √ (cid:96) = 0 , , , , , , . Note that the x -axis is logarithmic in ˜ ω . It does not require us to solve the gap equation to con-clude that not only s -wave but also p -wave pairing is im-possible from our Dyson formalism. As with the conven-tional BCS self-consistency condition, for an interaction˜ V (cid:96) (˜ ω ) that is always repulsive, our gap equation Eq.(15)does not have a nontrivial solution. In other words, anon-trivial solution requires V (cid:96) ( ω ) < (cid:96) = 2 , , , , (cid:96) = 0 ,
1. In general, we can derive from Eq.(16) an identical loga-rithmically divergent repulsion˜ V (cid:96) (˜ ω ) ∼ −
23 ˜ α log | ˜ ω | , (17)in the low frequency limit for all (cid:96) by making use of thefact that the forward scatter cos θ ≈ | ˜ ω | (cid:28) | ˜ ω | (cid:29)
1, the Landau damping dominates, hence˜ V (cid:96) (˜ ω ) ≈ (cid:32) √ π (cid:33) C l | ˜ ω | (18)where C (cid:96) = (cid:82) − P (cid:96) (cos θ )(1 + cos θ ) √ − cos θ . We findthat C (cid:96) < (cid:96) ≥ C , C >
0. It is easy to seethat C (cid:96) ’s fall off rapidly with (cid:96) .A representative plot for ˜ V (cid:96) (˜ ω ) is shown in Fig. 3. The s -and p -wave channels are repulsive at all frequencies,while channels for (cid:96) ≥ ω ( (cid:96), α ) for (cid:96) ≥ ω ( (cid:96), α ) ≈ (cid:32) √ π ˜ α (cid:33) . (cid:96) . . (19)which indicates attractive interaction at higher frequen-cies for (cid:96) ≥ Z -factor vanishes for the fermions, as in a non-Fermi liquid, for which only the one-loop result is avail-able. If this is the case, non-Fermi liquid effects mustbe directly included within the Dyson formalism, perhapsrestricting the validity of the higher angular momentumpairing.That p -wave pairing is not possible is in contrast tothe d = 2 result derived from the large- N approxima-tion. This is a robust result and does not depend onthe Landau damping as the source of dynamic screening.A characteristic of the transverse gauge field, seen fromEqs. (7) and (13), is that the interaction is always sup-pressed for backscattering. This general feature of thetransverse gauge field makes it more difficult to obtainattraction in the p -wave channel; obtaining attraction inthe p -wave channel out of an interaction that is repulsivefor all scattering angles – which is exactly the case inour interaction in Eq. (7) – requires the repulsion to bestronger for backscattering than for forward scattering.Moreover, generating attraction in the p -wave channelthrough the higher order effects is not really possible inour case, unlike the Kohn-Luttinger problem.4e note that the physics of the possible pairing inthe (cid:96) ≥ . In the BCS case, the repulsion athigher frequency cannot break the Cooper pairs as long asthere is any attraction at the zero frequency. By contrast,in this problem, V (cid:96) for (cid:96) ≥ d -wave channel to give us the most ro-bust pairing from Fig. 3 due to a number of differentreasons. First, the d -wave channel has the strongest at-traction in the high frequency limit, as the | C (cid:96) | of Eq.(18)falls off rapidly with (cid:96) for (cid:96) ≥
2. Secondly, when one takesinto account the logarithmic scale of the frequency axis ofFig. 3, we can see that the attractive region gets narroweras (cid:96) increases. These two factors lead to the frequencyintegral of − ˜ V (cid:96) (˜ ω ) decreasing rapidly and monotonicallywith (cid:96) for (cid:96) ≥ (cid:96) = 2 and then consider (cid:96) > IV. NUMERICAL RESULTS
The solution of Eq. (15) yields a non-vanishing gapfor α > α c . To solve this equation we employed a bruteforce iterative procedure with a high frequency cutoff ω = E F , consistent with the validity of the Landau damping.This is not always straightforward because roundoff errorbuilds up quickly, but mostly when the gaps are small.The frequency grid also plays an important role: we havefound that number of points 5 × is typically sufficientto achieve satisfactory accuracy. The results for (cid:96) = 2are shown in Fig. 4. The gap at ω = 0 is clearly theexcitation gap, which follows from the Green function inEq. (11) despite the Wick rotation. Obtaining gaps atreal frequencies requires inverse Wick rotation.Most phase transitions take place at a finite couplingwhich is nonuniversal. In fact, the critical coupling typi-cally depends on the ultraviolet cutoff. A simple exampleto check is the large- N limit for O ( N ) nonlinear sigmamodel in d = 3 where the divergence is linear in theultraviolet cutoff. There are some special cases involv-ing lower critical dimensions that are different. In onedimensional classical Ising model the critical point is ex-actly at T = 0, forced to be the case because of the finitetemperature fluctuations. Classical O (3) sigma model atthe lower critical dimension, d = 2, behaves similarlyas T →
0. In these problems fluctuations at the lowercritical dimension enforces the value of the critical point.
FIG. 4. The gap versus the imaginary frequency for the (cid:96) = 2 channel. From top to bottom: ˜ α ≡ α/ π =11 . , . , . , . , . , . , . , . , . The pairing gap has a very strong dependence on theimaginary frequency. The superconducting phase tran-sition, as signified by the vanishing of the ω = 0 gap isa continuous transition. At the edge of the supercon-ducting phase α = α c , the zero frequency gap vanish,while the finite imaginary frequency the pairing gap re-main finite. We speculate that when analytically contin-ued to real frequencies, there should be fluctuating gapseven when the superconductivity has disappeared. FromFig. 5 we obtain the following scaling relation for the zerofrequency gap in the (cid:96) = 2 channel:∆ E F = A (cid:16) α − α c α c (cid:17) . if α > α c ,0 if α ≤ α c . (20)It is unusual that the critical exponent ≈ /
4, which isdifferent from the naive mean field value of 1 /
2. It ap-pears that the gap equation went beyond the mean-fieldformalism in the sense that our solution is non-uniformin time. As noted in the Introduction, α c is cutoff depen-dent, while neither the gap exponent nor the nature ofthe quantum phase transition is. For our present choiceof the cutoff, α c = 7 . have suggested that this transi-tion is discontinuous based on various approximations tothe gap equation. For example, in an insightful paper,Sch¨afer correctly pointed out that there is no supercon-ductivity for couplings smaller than a critical coupling ˜ α c from an equation, which ignores the vector character ofthe vertex, the frequency dependence of the gap and as-sumes that the gap for all angular momentum channelsare roughly the same except for (cid:96) = 0 ,
1. We have solved5 Α Π (cid:68)(cid:69) F FIG. 5. Plot of the gap ∆ ≡ ∆( ω = 0) for the (cid:96) = 2 as afunction of ˜ α ≡ α/ π . this gap equation to find that it leads to a jump in thegap at ˜ α c although the derivative at ˜ α c + is singular. Thejump does not imply a first order transition. There are atleast two such examples: jump in the superfluid densityin 2 D − XY model and the jump in the magnetizationin the 1 D Inverse square Ising model. While the conclu-sion that a critical ˜ α c is necessary for superconductivity,a more rigorous treatment presented here shows a gapcollapsing to zero, as discussed above.We have further solved the gap equation for (cid:96) =3 , , , ω = E F , we obtain α c ( (cid:96) = 2) > α c ( (cid:96) = 3) >α c ( (cid:96) = 4) > α c ( (cid:96) = 5) > α c ( (cid:96) = 6). Fig. 6 shows plots ofthe zero frequency gaps for these channels as function of˜ α . It appears that that the lower bound for ˜ α c ≈ .
5; notethat the gap equation gives us the continuous transitionin ∆( ω = 0) for higher (cid:96) as well. Such a cascade of higherangular momentum gaps with an accumulation point israther unusual. Since our Hartree-Fock equations do notinclude fluctuation effects, it is difficult speculate aboutthe fate of the cascade of gaps. It is not unreasonableto expect, however, that the higher angular momentumgaps will probably be eliminated from a variety of effectsnot included in our theory, but not (cid:96) = 2. V. CONCLUSION
We have shown using the Dyson formalism that a Lan-dau damped gauge boson in d = 3 + 1 induces supercon-ductivity with higher angular momentum pairing. No s -wave and p -wave solutions are possible, whereas de-pending on the magnitude of the gauge coupling, super-conducting states with (cid:96) ≥ d = 3 + 1, since l=2l=3l=3l=3l=4l=4l=5l=6 Α(cid:142) (cid:68)(cid:69) F FIG. 6. Plot of the gap ∆ ≡ ∆( ω = 0) for the (cid:96) = 2 , , , , α ≡ α/ π . the fermion-gauge boson coupling is marginal, and alsosince the superconducting gap preempts the formation ofa non-Fermi liquid. The existence of a finite critical cou-pling is also present in the case of finite-density quarkscoupled to a non-abelian gauge field, though the start-ing action is quite different. We have also carried out ouranalysis in the much more controversial case of d = 2 + 1,and our analysis suggests again that 1) superconductivityrequires a finite critical interaction, and 2) superconduct-ing solutions have (cid:96) ≥
2. However, we have not analyzedthe Chern-Simons gauge field, which will be discussedelsewhere.In this paper, we have concentrated on superconduc-tivity mediated by transverse gauge interactions. In thiscase the gauge boson is a distinct emergent field thanthe fermions to which it couples. A related class of fieldtheories involve metals in the vicinity of a Pomeranchukinstability (eg. ferromagnetism). In this case, the bosoncorresponds to a fermion bilinear, and unlike the gaugefield, it is generically massive except when tuned to criti-cality. While superconducting instabilities in the vicinityof such quantum critical points have been studied exten-sively in the literature , many issues remain open.We have not addressed the issue of the competing or-ders in this work. Therefore our results do not rule outCooper pairing being pre-empted by density wave order-ing in the high- α limit. We plan to investigate this issuethrough an analogous Hartree-Fock analysis. The resultsof this analysis will be presented elsewhere. ACKNOWLEDGMENTS
This work was supported by the funds from David. S.Saxon Presidential Term Chair at UCLA (IM), the Al-fred P. Sloan Foundation (SR), DOE under No. AC02-76SF00515 (SR) and NSF under Grant No. DMR-1004520 (SC). We would like to thank Elihu Abrahams,6ndrey Chubukov, Chetan Nayak, Joe Polchinski, andRahul Roy for sharing their insights.
Appendix: (2 + 1) -dimensions m=0m=1m=2m=3m=4m=5m=6 p aé wé- - - V é H wé L aé FIG. 7. Angular momentum decomposition of the interaction˜ V m (˜ ω ) / ˜ α versus 2 √ π ˜ α ˜ ω for m = 0 , , , , , , α = 11 . , . , . , . , . , . , The non-Fermi liquid behavior in (2 + 1) dimensionsis still not well understood due to complications of thelarge N limit . Here we consider superconductivityfrom a Hartree-Fock approximation, with the assumptionthat gap might cut off the low frequency anomalies andthe superconducting state may be qualitatively correctmodulo fluctuation effects at the transition. Of course,our calculation in d = 3 is on a much firmer groundbecause (3 + 1) is the upper critical dimension. a p D E F FIG. 9. Plot of the gap in two dimensions ∆ ≡ ∆(0) as afunction of ˜ α . Because transverse gauge field canot be realized in(1+1) dimensions, superconductivity mediated by trans-verse gauge bosons must have a larger lower critical di-mension. The gap equation for the transverse gauge bo-son mediated pairing interaction in d = 2, decomposedin terms of Fourier coefficients of cos( mφ ), is˜∆ m (˜ ω ) = − (cid:90) d ˜ ω (cid:48) ˜ V m (˜ ω − ˜ ω (cid:48) ) ˜∆ m (˜ ω (cid:48) ) (cid:113) ˜ ω (cid:48) + | ˜∆ m (˜ ω (cid:48) ) | , (A.1)where˜ V m (˜ ω ) = ˜ α (cid:90) π dφ cos( mφ )(1 + cos φ )(1 − cos φ ) + α √ | ˜ ω |√ − cos φ , (A.2)and the dimensionless coupling constant ˜ α = α / π : α = g v F πk F . (A.3)In the high frequency limit | ˜ ω | (cid:29)
1, the Landau damp-ing part in the denominator dominates and we have˜ V m (˜ ω ) ≈ √ π C m | ˜ ω | , (A.4)where C m = (cid:90) π dφ cos( mφ )(1 + cos φ ) (cid:112) − cos φ, (A.5)falls off, albeit less rapidly than d = 3, as m increases,and, once again, m = 0 and m = 1 are repulsive. Asbefore, the largest gap occurs for m = 2. Explicit solutionof the gap equation in two dimensions is shown in Fig. 8.The excitation gap as a function of ˜ α is shown in Fig. 9where the fit yields∆ E F = A (cid:16) ˜ α − α c α c (cid:17) . if ˜ α > α c ,0 if ˜ α ≤ α c . (A.6)7here α c = 1 .
89 is of course cutoff dependent and A isa dimensionless constant.We have also solved the gap equation for m = 3 , , , α c is non-vanishingas for d = 3, and the picture of cascade higher angularmomentum states is identical. The results are shown inFig. 10. m=2m=3m=2m=4m=2m=5m=6 aé D E F FIG. 10. Plot of the gap in two dimensions ∆ ≡ ∆(0) asa function of ˜ α for m = 2 , , , ,
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