Collective modes in a system with two spin-density waves the `Ribault' phase of quasi-one-d
a r X i v :c o n d -m a t /9910052v 2 29 J u n 2000
Collective modes in a system with two spin-density waves:the ‘Ribault’phase of
quasi-one-dimensional organic conductors
N.Dupuis (1?,2)and Victor M.Yakovenko (2)
(1)Laboratoire de Physique des Solides,Universit′e Paris-Sud,91405Orsay,France (2)Department of Physics,University of Maryland,College Park,MD 20742-4111,USA
(January 21,2000)We study the long-wavelength collective modes in the magnetic-?eld-induced spin-density-wave (FISDW)phases experimentally observed in organic conductors of the Bechgaard salts family,fo-cusing on phases that exhibit a sign reversal of the quantum Hall e?ect (Ribault anomaly).We have recently proposed that two SDW’s coexist in the Ribault phase,as a result of umklapp processes.When the latter are strong enough,the two SDW’s become circularly polarized (helicoidal SDW’s).In this paper,we study the collective modes which result from the presence of two SDW’s.We ?nd two Goldstone modes,an out-of-phase sliding mode and an in-phase spin-wave mode,and two gapped modes.The sliding Goldstone mode carries only a fraction of the total optical spectral weight,which is determined by the ratio of the amplitude of the two SDW’s.In the helicoidal phase,all the spectral weight is pushed up above the SDW gap.We also point out similarities with phase modes in two-band,bilayer,or d +id ′superconductors.We expect our conclusions to hold for generic two-SDW systems.
PACS Numbers:72.15Nj,73.40Hm,75.30Fv
I.INTRODUCTION
In electron systems with broken symmetries,such as superconductors or DW systems,quasi-particle excita-tions are often gapped,and the only low-lying excitations are collective modes.The latter thus play a crucial role in various low-energy properties.
In an incommensurate SDW system,there are two (gapless)Goldstone modes:a sliding mode and a spin-wave mode,which result from the spontaneous breaking of translation symmetry in real space and rotation sym-metry in spin space,respectively.1,2Contrary to the case of superconductors,collective modes in DW systems di-rectly couple to external probes and therefore show up in various experiments.For instance,the sliding mode,which is pinned by impurities in real systems,can be de-pinned by a strong electric ?eld.This leads to non-linear conduction,observed in DW systems.1
The aim of this article is to study long-wavelength collective modes in a quasi-1D system where the low-temperature phase exhibits two SDW’s.The presence of two SDW’s gives rise to a rich structure of col-lective modes,which in principle can be observed in experiments.3
Our results are based on a particular case:the magnetic-?eld-induced spin-density-wave (FISDW)phases of the organic conductors of the Bechgaard salt family.4–7These FISDW phases share common features with standard SDW phases,but also exhibit remarkable properties like the quantization of the Hall e?ect.We have shown that umklapp processes may lead in these systems to the coexistence of two SDW’s with compara-ble amplitudes,which provides a possible explanation of the sign reversal of the quantum Hall e?ect (QHE)(the
so-called Ribault anomaly 8,9)observed in these conduc-tors.There are two motivations for studying this particu-lar case.i)The Bechgaard salts,as a possible candidate for a two-SDW system,present their own interest.ii)A conductor with two SDW’s is in general not easy to analyze,even at the mean-?eld level.10The analysis sim-pli?es when a strong magnetic ?eld quantizes the electron motion.
Nevertheless,we expect our conclusions to be quite general and to apply (at least qualitatively)to other sys-tems with two SDW’s.This hope is strongly supported by the similarities 3that exist between collective modes in two-SDW conductors and phase modes in two-band,11bilayer 12or d +id ′(Ref.13)superconductors,and,to a lesser extent,plasmon modes in semiconductor double-well structures.14These similarities suggest that collec-tive modes in two-component systems present generic fea-tures that do not depend on the particular case consid-ered.
A.Umklapp processes in quasi-1D SDW systems
Consider a quasi-1D conductor with a SDW ground state.In presence of umklapp processes transferring mo-mentum K (K being a vector of the reciproqual space),spin ?uctuations at wave vectors Q and K ?Q are cou-pled.Thus,the formation of a SDW at wave vector Q 1will automatically be accompanied by the formation of a second SDW at wave vector Q 2=K ?Q 1,provided that Q 1=Q 2.The case Q 1=Q 2=K /2corresponds to a (single)commensurate SDW.Umklapp processes pin the SDW whose position with respect to the underlying crystal lattice becomes ?xed:the sliding mode is gapped.
For two incommensurate SDW’s(Q1=Q2),15the total spin-density modulation can then be written as
S(r) = i=1,2S i cos(Q i·r+θi),(1.1)
where r=(na,mb)(with n,m integers)denotes the po-sition in real space(a and b being the lattice spacings along and across the conducting chains).
Even for Q1=Q2,the distinction between the two SDW’s may appear somewhat unjusti?ed since one can-not distinguish between cos(Q1·r)and cos(Q2·r)when r is taken as a discrete variable.However,umklapp pro-cesses do lead to the presence of two non-vanishing order
parameters, c?
k↑c k+Q
1↓
and c?
k↑
c k+Q
2↓
,in the SDW
phase.16This doubles the number of degrees of freedom of the SDW condensate,which yields for instance twice as many collective modes(as compared to the case with a single SDW).Thus,it is natural to speak of two SDW’s in the ground state of the system.Furthermore,we note that cos(Q1·r+θ1)and cos(Q2·r+θ2)are indistinguish-able only ifθ1=?θ2(with,again,r being a discrete variable).This is precisely the equilibrium condition ob-tained by minimizing the mean-?eld condensation energy (see Sec.III).Condensate?uctuations do not in general satisfy the conditionθ1=?θ2,and it is then more ap-propriate to view these?uctuations as originating from two di?erent SDW’s.
In a quasi-1D system with a single SDW,the wave vec-tor Q of the spin-density modulation is determined by the nesting properties of the Fermi surface:E(k+Q)??E(k),where E(k)is the energy with respect to the Fermi level.[For a perfectly nested Fermi surface,one would have E(k+Q)=?E(k).]Umklapp processes are important only if both Q1and Q2=K?Q1are good nesting vectors.Otherwise,one of the two SDW’s has a very small amplitude and can be ignored for any practical purpose.
For a2D(or3D)conductor,the geometry of the Fermi surface appears to be crucial.Consider the following dispersion law,which is linearized in the vicinity of the Fermi level:
Eα(k x,k y)=v F(αk x?k F)?2t b cos(k y b+ακ)+···
(1.2) where k x and k y are the electron momenta along and across the conducting chains.α=+(?)corresponds to right(left)movers with momenta close toαk F.v F and k F are the Fermi velocity and momentum for the motion along the chains,t b the interchain transfer integral,and b the interchain spacing.The ellipses in(1.2)represent small corrections that generate deviations from perfect nesting.κis a parameter which parameterizes the shape of the Fermi surface.
Most calculations on quasi-1D SDW systems assume κ=0,which corresponds to the Fermi surface shown in Fig.1a.There are two‘best’nesting vectors:Q1?(2k F,π/b?δ)and Q2?(2k F,?π/b+δ),where the small correctionδ(|δ|?π/b)is due to deviations from perfect nesting.17,18Since Q2=(4k F,0)?Q1,these two vectors are coupled by umklapp scattering if the system is half-?lled(4k F=2π/a).Two SDW’s with equal amplitudes will form simultaneously at low temperature. Consider now the caseκ=π/4in a half-?lled band, which corresponds to the asymmetric Fermi surface shown in Fig.1b.This Fermi surface has been pro-posed as a good approximation to the actual Fermi sur-face of the Bechgaard salts.19,20The best nesting vector Q1=(2k F,π/2b?δ)is now non-degenerate.By umk-lapp scattering,Q1couples to Q2=(2k F,?π/2b+δ), which is not a good nesting vector.At low temperature, two SDW’s will form simultaneously,but the one with wave vector Q2will have a vanishingly small amplitude.
B.The Ribault phase of the Bechgaard salts The organic conductors of the Bechgaard salts family (TMTSF)2X(where TMTSF stands for tetramethylte-traselenafulvalene)are well known to have remarkable properties in a magnetic?eld.In three members of this family(X=ClO4,PF6,ReO4),a moderate magnetic?eld of a few Tesla destroys the metallic phase and induces a series of SDW phases separated by?rst-order phase transitions.4,5
According to the so-called quantized nesting model (QNM),5the formation of the magnetic-?eld-induced spin-density waves(FISDW)results from competition between the nesting properties of the Fermi surface and the quantization of the electron motion in a magnetic ?eld.The formation of a SDW opens a gap,but leaves closed pockets of electrons and/or holes in the vicin-ity of the Fermi surface.In the absence of a mag-netic?eld,these pockets are too large(due to imper-fect nesting)for the SDW phase to be stable.In the presence of a magnetic?eld H,they become quantized into Landau levels(more precisely Landau subbands). In each FISDW phase,the SDW wave vector is quan-tized,Q N=(2k F+NG,Q y)with N integer,so that an integer number of Landau subbands are?lled.[Here G=eHb/ˉh?k F and?e is the electron charge.]As a result,the Fermi level lies in a gap between two Landau subbands,the SDW phase is stable,and the Hall conduc-tivity is quantized:σxy=?2Ne2/h per one layer of the TMTSF molecules.6,7As the magnetic?eld increases,the value of the integer N changes,which leads to a cascade of FISDW transitions.The QNM predicts the integer N to have always the same sign.While most of the Hall plateaus are indeed of the same sign,referred to as pos-itive as convention,a negative QHE is also observed at certain pressures(the so-called Ribault anomaly).8,9The most commonly observed negative phases correspond to N=?2and N=?4.
In the Bechgaard salts,a weak dimerization along the
chains leads to a half-?lled band.Umklapp processes transferring4k F=2π/a are allowed.Thus the forma-tion of a SDW at wave vector Q N=(2k F+NG,Q y) will be accompanied by a second SDW at wave vector QˉN=(4k F,0)?Q N=(2k F?NG,?Q y),i.e.there is coexistence of phases N and?N.21Note that in our notation,QˉN has both the signs of N and Q y reversed compared to Q N.As discussed in the preceding section, actual coexistence may occur only if QˉN(like Q N)is a good nesting vector.This is the case with the Fermi sur-face shown in Fig.1a(since Q y～π/b),but not with the one shown in Fig.1b(since Q y～π/2b).
In Ref.21,we have studied the e?ect of umklapp scat-tering on the FISDW phases,starting from the Fermi surface shown in Fig.1a.We have shown that for weak umklapp scattering Q y=π/b.In that case,the SDW with negative quantum number?|N|has a vanishing amplitude and can be ignored.However,for N even, there exists a critical value of the umklapp scattering strength above which the system prefers to form two transversely commensurate SDW’s(Q y=π/b).For cer-tain dispersion law,22the SDW with negative quantum number?|N|has the largest amplitude,which leads to a negative Hall plateau(Fig.2a).Since the umklapp scattering strength is sensitive to pressure,we have sug-gested that this provides a natural explanation for the negative QHE(Ribault anomaly)observed in the Bech-gaard salts.23[In the following,the‘negative’phases are referred to as‘Ribault’phases.]
It should be noted that this explanation relies on a simple Fermi surface(Fig.1a),which does not necessar-ily provide a good approximation to the actual Fermi surface of the Bechgaard salts.With the more realis-tic(according to band calculations)Fermi surface shown in Fig.1b,umklapp processes have only a small e?ect and do not lead to negative QHE.24Therefore,our ex-planation of the Ribault anomaly should be taken with caution.For it to be correct,the parameterκshould be smaller(typically not larger thanπ/10)than the value π/4predicted by band calculation.In the following,we consider only the caseκ=0.
We have shown in Ref.21that the SDW’s in the Rib-ault phase are likely to become circularly polarized(he-licoidal SDW’s)when the umklapp scattering strength is further increased(Fig.2b).The QHE vanishes in the he-licoidal phase.We will see that the circular polarization also a?ects the collective modes.
C.Outline of the paper
In the next section,we introduce the e?ective Hamilto-nian describing the FISDW phases.The partition func-tion is written as a functional integral over a bosonic aux-iliary?eld that describes spin?uctuations.In Sec.III,we perform a saddle-point approximation,thus recovering the mean-?eld results of Ref.21.We obtain the mean-?eld propagators and the mean-?eld particle-hole suscep-tibilities.In Sec.IV,we derive the low-energy e?ective action of the SDW phase by taking into account?uctua-tions of the bosonic auxiliary?eld around its saddle-point value.We consider only‘phase’?uctuations,i.e.sliding and spin-wave collective modes.We do not study ampli-tude collective modes,which are gapped and do not cou-ple to phase?uctuations in the long-wavelength limit.1 We?nd two sliding modes:a(gapless)Goldstone mode corresponding to a sliding of the two SDW’s in opposite directions(out-of-phase oscillations),and a gapped mode corresponding to in-phase oscillations(Sec.V).The real part of the conductivity exhibit two peaks,which re?ects the presence of two sliding modes.The low-energy mode carries only a fraction of the total spectral weightω2p/4 (ωp is the plasma frequency),which is determined by the ratio of the amplitudes of the two SDW’s(Sec.VI). The spin-wave modes are studied in Sec.VII.There is a Goldstone mode corresponding to in-phase oscillations of the two SDW’s,and a gapped mode corresponding to out-of-phase oscillations.The spectral function Imχret is computed in Sec.VIII(χret is the retarded transverse spin-spin correlation function).Both gapped modes are found to lie above the mean-?eld gap in the case of the Bechgaard salts.
In Sec.IX,we study the collective modes in the heli-coidal phase where the two SDW’s are circularly polar-ized.For a helicoidal structure,one cannot distinguish between a uniform spin rotation and a global transla-tion,so that there are only two phase modes.The Gold-stone mode does not contribute to the conductivity and is therefore a pure spin-wave mode.Thus all the spectral weight in the conductivityσ(ω)is pushed up above the mean-?eld gap.Both modes contribute to the spin-spin correlation function.
It should be noted that the long-wavelength modes are not the only modes of interest in the FISDW phases. There also exist magneto-rotons at?nite wave vectors (q x=G,2G,...).25,26These modes are not considered in this paper.
In the following,we takeˉh=k B=1.
II.MODEL AND EFFECTIVE HAMILTONIAN In the vicinity of the Fermi energy,the electron disper-sion law in the Bechgaard salts is approximated as E(k x,k y)=v F(|k x|?k F)+t⊥(k y b),(2.1) where k x and k y are the electron momenta along and across the one-dimensional chains of TMTSF.In Eq.(2.1),the longitudinal electron dispersion is lin-earized in k x in the vicinity of the two one-dimensional Fermi points±k F,and v F=2at a sin(k F a)is the corre-sponding Fermi velocity(t a being the transfer integral and a the lattice spacing along the chains).Whenever necessary,we will impose a ultraviolet energy cuto?E0to simulate a?nite bandwidth.The function t⊥(u),which
describes the propagation in the transverse direction,is periodic:t⊥(u)=t⊥(u+2π).It can be expanded in Fourier series as
t⊥(u)=?2t b cos(u)?2t2b cos(2u)
?2t3b cos(3u)?2t4b cos(4u)···(2.2) If we retain only the?rst harmonic(t b),we obtain a Fermi surface with perfect nesting at(2k F,π/b).The other harmonics t2b,t3b···?t b generate deviations from perfect nesting.They have been introduced in order to keep a realistic description of the Fermi surface despite the linearization around±k F.In the following,we shall retain only t b,t2b and t4b.t3b does not play an impor-tant role and can be discarded.21We do not consider the electron dispersion in the z direction,because it is not important in the following(its main e?ect is to introduce a3D threshold?eld below which the FISDW cascade is suppressed5).
The e?ect of the magnetic?eld H along the z direc-tion is taken into account via the Peierls substitution k→?i??e A.(The charge e is positive since the ac-tual carriers are holes.)In the gauge A=(0,Hx,0)we obtain the non-interacting Hamiltonian
H0= α,σ d2r?ψ?ασ(r)[v F(?iα?x?k F)
+t⊥(?ib?y?Gx)+σh]?ψασ(r).(2.3)
Here?ψασ(r)is a fermionic operator for right(α=+)and left(α=?)moving particles.σ=+(?)for up(down)
spin.We use the notation r=(x,mb)(m integer)and d2r=b m dx.G=eHb is a magnetic wave vec-tor and h=μB H is the Zeeman energy(we assume the electron gyromagnetic factor to be equal to two).Diago-nalizing the Hamiltonian(2.3)we obtain the eigenstates and eigenenergies
φαk(r)=
1
L x L y
e i k·r+i(α/ωc)T⊥(k y b?Gx),
?ασ(k)≡?ασ(k x)=v F(αk x?k F)+σh,(2.4) where L x L y is the area of the system,ωc=v F G,and
T⊥(u)= u0du′t⊥(u′).(2.5)
In the chosen gauge the energy depends only on k x,i.e. the dispersion law is one-dimensional.This re?ects the localization of the electron motion in the transverse di-rection,which is at the origin of the QNM(see Ref.21for a further discussion).Note that contrary to k y,the quan-tum number k x is not the momentum since the operator ?x does not commute with the Hamiltonian.
The interacting part of the Hamiltonian contains two terms corresponding to forward(g2)and umklapp(g3) scattering:H int=
g2
2 α,σ
d2r e?iα4k F x?ψ?ˉασ(r)?ψ?ˉαˉσ(r)?ψαˉσ(r)?ψασ(r),
(2.6) whereˉα=?αandˉσ=?σ.For repulsive interaction
g2,g3≥0.We shall assume that umklapp scattering is ‘weak’:g3 In a mean-?eld theory,the cuto?E0is of the order of the bandwidth t a.It is however well known that mean-?eld theory completely neglects?uctuations and cannot be directly applied to quasi-1D systems where the physics,at least at high temperature,is expected to be one-dimensional.The general wisdom is that there is at low temperature a dimensional crossover from a1D to a2D(or3D)regime.27,28In Bechgaard salts(at ambient pressure),this dimensional crossover occurs via coherent single-particle interchain hopping.Although the exper-imental assignment of the crossover temperature is still under debate(see for instance Refs.29–31),the success of the QNM in explaining the phase diagram of the com-pounds(TMTSF)2PF6and(TMTSF)2ClO4provides a clear evidence of the relevance of the Fermi surface at low temperature.In the3D regime,a mean-?eld theory is justi?ed provided that the parameters of the theory are understood as e?ective parameters(renormalized by1D ?uctuations).27E0is then smaller than the bare band-width(～t a)and corresponds to the renormalized trans-verse bandwidth.For the same reason,the interaction amplitudes g2and g3can be di?erent than their bare values.The Hamiltonian[Eqs.(2.3)and(2.6)]should therefore be understood as an e?ective low-energy Hamil-tonian. Since collective modes are best studied within a func-tional integral formalism,we write the partition function as Z= Dψ?Dψe?S0?S int whereψ(?)is an anticom-muting Grassmann variable.The actions S0and S int are given by S0= dτ α,σ d2rψ?ασ(r,τ)?τψασ(r,τ)+H0[ψ?,ψ] , S int=? 1 ?g(r)= g2g3e i4k F x g3e?i4k F x g2 .(2.9) Introducing a complex auxiliary?eld?ασ(r,τ),we de-couple S int by means of a Hubbard-Stratonovitch trans-formation.This leads to the action S=S0+ d2r dτ??↑(r,τ)?g(r)?↑(r,τ) ? σ d2r dτ??σ(r,τ)?g(r)Oσ(r,τ).(2.10) We use the spinor notation?σ=(?+σ,??σ)T and Oσ=(O+σ,O?σ)T.Since?ασcouples to the ?eld Oασ(r,τ)=O?ˉαˉσ(r,τ),it satis?es the constraint ?ασ(r,τ)=??ˉαˉσ(r,τ).Note that the action(2.10)main-tains spin-rotation invariance around the magnetic?eld axis.In the FISDW phase,because of the Zeeman term, the magnetization is perpendicular to the magnetic?eld axis,so that the only spin-wave mode corresponds to rotation around the z axis.In zero magnetic?eld,a dif-ferent approach should be used in order to maintain the full SU(2)spin symmetry(see Ref.2). III.MEAN-FIELD THEORY In this section we look for a mean-?eld solution cor-responding to a phase with two sinusoidal(i.e.linearly polarized)SDW’s: S x(r) = p=±M pN cos(φpN)cos(Q pN·r+θpN) S y(r) = p=±M pN sin(φpN)cos(Q pN·r+θpN) S z(r) =0,(3.1) where Sν(r)= ασσ′ψ?ˉασ(r)τνσσ′ψασ′(r)is the spin-density operator andτν(ν=x,y,z)the Pauli matri-ces.Because of the Zeeman coupling with the magnetic ?eld,the SDW’s are polarized in the(x,y)-plane.The variableφpN determines the polarization axis,whileθpN gives the position of the SDW’s with respect to the un-derlying crystal lattice. We assume that the external parameters(magnetic ?eld,pressure...)are such that the system is in the Rib-ault phase,characterized by a negative QHE and the coexistence of two SDW’s with comparable amplitudes. We choose the sign of N such that N refers to the SDW with the largest amplitude(M N≥MˉN).[N is even and negative in the Ribault phase.] The mean-?eld solution corresponds to a saddle-point approximation with a static auxiliary?eld32 ?ασ(r)= Oασ(r,τ) = p?(pN)ασe iαQ pN·r.(3.2) The relation between?ασand Oασ results from the stationarity condition of the saddle-point action(see Eq.(3.4)below).Because of the constraint??ασ(r)= ?ˉαˉσ(r),the order parameters satisfy?(pN)? ασ=? (pN) ˉαˉσ . Among the eight complex order parameters,only four are therefore independent and su?cient to character- ize the SDW phase.The order parameters?(pN) ασ= |?(pN) ασ|e i? (pN) ασare related to the spin-density modulation (3.1)by |?(pN) ασ |= M pN 2 ?(pN)+↑??(pN)?↑ , φpN=? 1 We denote by ω=2πT (n +1/2)(n integer)fermionic Matsubara frequencies.We emphasize that here k and k ′refer to the quantum numbers of the eigenstates φαk of H 0.We have introduced the (particle-hole)pairing amplitudes ??ασ(k ,k ′)= d 2r φα?k (r )φˉαk ′(r )??ασ(r ) =δk ′y ,k y ?π/b p (g 2?(pN ) ασ+g 3?(ˉpN )ˉασ)× ∞ n =?∞ I n e iαn (k y b ?π/2)δk ′x ,k x ?αQ (pN )x +αnG , (3.7) where I n ≡I n (q y =π/b ).The coe?cients I n (q y ),de?ned by I n (q y )= 2π du (g 22?g 23)I N I ˉN .(3.10) In the QLA,the mean-?eld action therefore reduces to a 2×2matrix (as in BCS theory): S MF =β d 2r ??↑(r )? g (r )?↑(r )? α,k ,ω (ψ?α↑(k ,ω),ψ?ˉα↓(k ?αQ 0,ω)) iω??α↑(k x )?? α↑(k y )???α↑ (k y )iω+?α↑(k x ) ψα↑(k ,ω) ψˉα↓(k ?αQ 0,ω) ,(3.11) using ?ˉα↓(k x ?α2k F )=??α↑(k x ).It is remarkable that within the QLA the mean-?eld action can still be written as a 2×2matrix.The presence of a second SDW changes the expression of the pairing amplitude (which becomes k y -dependent),but not the fact that the state (k ,?)couples only to (k +Q 0,+).This is not true in zero magnetic ?eld where the presence of a second SDW leads to complicated mean-?eld equations.10 A.Ground-state energy The mean-?eld action being Gaussian,we can integrate out the fermion ?elds to obtain the ground-state conden-sation energy (per unit area),?E =?(T ln Z )/L x L y ? E N ,where T →0and E N =?N (0)E 2 0is the normal state energy: ?E = α p ?(pN )?α↑??(pN )α↑2|??(ˉN )α↑|2? N (0) 2 +ln 2E 0 ?E=? N(0) ?(N) α↑= |??(ˉN) ˉα↑ /??(N) α↑ |?r|IˉN/I N| g2 ,ζ= IˉN ?N ,?γ= ?? ˉN 1+rγ ,γ= ?γ?rζ ω2+?2ασ(k x)+|??ασ(k y)|2 , Fασ(k,ω)= ?? ασ (k y) Eασ(k)= max k y |??ασ(k y)| = 1?|?γ| 2 I2pN ln2E02?ω2ν+v2F q2x 4 I N IˉN?γ 1?ω2ν+v2F q2x 4 I2N(1??γ2) 1?ω2ν+v2F q2x 4 I N IˉN?γ,(3.29) whereων=2 πTν(νinteger)is a bosonic Matsubara frequency.Eqs.(3.29)are valid in the low-energy limit v2F q2x,ω2ν???2N(1??γ2).We have used the QLA(see Appendix A),whose validity is discussed in detail in Ref. 25.Although the QLA does not predict accurately the value of the mean-?eld gaps??pN and?pN,it is an ex-cellent approximation of the low-energy properties(such as the long-wavelength collective modes)once the values of??pN and?pN are known.Since the absolute value of the latter do not play an important role for our purpose, the QLA turns out to be a perfect mean to compute the collective modes. IV.LOW-ENERGY EFFECTIVE ACTION In this section we derive the low-energy e?ective ac-tion determining the sliding and spin-wave modes.We do not consider amplitude modes which are gapped and do not couple to‘phase’?uctuations in the long-wavelength limit.We consider only longitudinal(i.e.with the wave vector parallel to the chains)?uctuations(q y=0).These ?uctuations are of particular interest since they couple to an electric?eld applied along the chains(which is a common experimental situation):see Sec.VI on the op-tical conductivity.Including a?nite q y would allow to obtain the e?ective mass of the collective modes in the transverse direction.Such a calculation is however much more involved and will not be attempted here. The low-energy e?ective action is derived by study- ing?uctuations of the auxiliary?eld?ασ(r,τ)around its saddle-point value?ασ(r).We write?ασ(r,τ)= ?ασ(r)+ηασ(r,τ)and calculate the e?ective action to quadratic order in the?uctuating?eldη.The fermionic action(2.10)can be rewritten as S[ψ?,ψ]=S MF[ψ?,ψ]? σ d2r dτη?σ(r,τ)?Oσ(r,τ) + d2r dτ[η?↑(r,τ)?g(r)η↑(r,τ) (r,τ)??↑(r)+c.c.)].(4.1) +(η? ↑ Integrating out the fermions,we obtain to quadratic or- der inηthe e?ective action 1 S[η?,η]= 2 ?(pN)+↑(r,τ)??(pN)?↑(r,τ) , 1 φpN(r,τ)=? θN(?q)θˉN(?q) ,(4.8) 2 ?q(θN(??q),θˉN(??q))D?1ch(?q) 1 S[φ]= δc ++ rζ ζ ?2rδc +??(1?r 2)(δc ++δc ???δc 2+?)=0. (5.2) A.Goldstone mode Eq.(5.2)admits the solution ω2 ν + v 2 F q 2 x =0.After an-alytical continuation to real frequencies (iων→ω+i 0+),we obtain a mode with a linear dispersion law (Fig.4) ω=v F q x , (5.3) θN (q x ,ω)=?θˉN (q x ,ω). (5.4) As expected from the mean-?eld analysis (Sec.III),the Goldstone mode corresponds to an out-of-phase oscilla-tion of the two SDW’s.We do not ?nd any renormaliza-tion of the mode velocity because we have not taken into account long-wavelength charge ?uctuations.The latter couple to the sliding modes and renormalize the velocity to v F (1+g 4N (0))1/2,where g 4is the forward scattering strength.1In order to obtain this velocity renormaliza-tion,it would have been necessary to include forward scattering in the interaction Hamiltonian and to intro-duce an auxiliary ?eld for the long-wavelength charge ?uctuations.2 B.Gapped mode The second mode obtained from (5.2)corresponds to a gapped mode with dispersion law (Fig.4) ω2=v 2F q 2x +ω2 0,(5.5)ω20 = 121?r 2 3+5?γ2 I N I ˉ N .(5.6) The oscillations of this mode satisfy θN r ?γ?ζ r (3+?γ2+4?γζ)2+(1?r )24ζ?γ(3+?γ2) δA 0(r ,τ) A 0=0 . (6.2) Following Sec.II,we introduce the auxiliary ?eld ?ασ(r ,τ)and integrate out the fermions.This leads to the action S [??,?,A 0]= d 2r dτ ??↑(r ,τ)? g (r )?↑(r ,τ)? α Tr ln(?G ?1α+?A 0),(6.3) where G ?1 α(r ,τ;r ′,τ′)= G (0)?1 α↑(r ,τ;r ′,τ′) δ(r ?r ′ )δ(τ?τ′ )?? α↑(r ,τ)δ(r ?r ′)δ(τ?τ′)???α↑ (r ,τ)G (0)?1ˉα↓ (r ,τ;r ′,τ′) ,(6.4) ?A 0(r ,τ;r ′,τ′)=δ(r ?r ′)δ(τ?τ′)A 0(r ,τ)?1. (6.5) We denote by ?1the 2×2unit matrix.Expanding the action with respect to A 0,we obtain S [??,?,A 0]=S [??,?]+ α Tr[G α?A 0]+O (A 20 ),(6.6)where S [??,?]is the action without the source ?eld. Here Tr denotes the trace with respect to time,space and matrix indices.From (6.2),we then obtain the fol-lowing expression of the charge operator ρ(r ,τ)= α tr G α(r ,τ;r ,τ),(6.7) where tr denotes the trace with respect to the matrix indices only (i.e.Tr ?O = d 2r dτtr ?O (r ,τ;r ,τ)for any operator ?O ).To calculate ρ,we write ??σ(r ,τ)=??σ(r )+?ησ(r ,τ),?ησ(r ,τ)=?g (r )ησ(r ,τ). (6.8) Then we have G ?1α=G MF ?1α?Σα, (6.9) Σα(r ,τ;r ′,τ′)=?δ(r ?r ′)δ(τ?τ′) × 0?ηα↑(r ,τ)?η? α↑(r ,τ) 0 ,(6.10) where G MF is the mean-?eld propagator.The charge op-erator is given by ρ(r ,τ)= α tr G MF α(r ,τ;r ,τ) + α tr[G MF αΣαG MF α](r ,τ;r ,τ)+O (?η2).(6.11) The ?rst term in the rhs of (6.11)is the uniform charge density ρ0in the mean-?eld state.The other terms corre-spond to charge ?uctuations ρDW induced by the conden-sate ?uctuations.To lowest order in phase ?uctuations,we obtain ρDW (r ,τ)= α tr[G MF αΣαG MF α](r ,τ;r ,τ).(6.12) Considering only phase ?uctuations [Eq.(4.4)],we ob-tain (Appendix B) ρDW (r ,τ)= 1 2 ??(N )+↑(r ,τ)???(N )?↑(r ,τ) , (6.14) ??(pN ) ασ(r ,τ)= g 2?pN ?(pN )ασ(r ,τ)+g 3?ˉpN ?(ˉpN ) ˉ ασ(r ,τ)πb ?τ?θ N (r ,τ).(6.17) Note that the current j DW is parallel to the chains.Con-densate ?uctuations do not induce current ?uctuations in the transverse direction. This simple result [Eqs.(6.13)and (6.17)]can be un-derstood as follows.For a system with a single SDW,the charge ?uctuation ρDW =?x θ/πb is obtained by requiring the SDW gap to remain tied to the Fermi level.Let us brie?y recall this argument.A ?uctuat-ing SDW potential ?ασ(x,τ)=?ασe iα2k F x +i?ασ(x,τ)??ασe iα2k F x +ix?x ?ασ(x,τ)couples the state (k x ,↓)to (k x +2k F +?x ?+↑(x,τ),↑),and (k x ,↑)to (k x +2k F ??x ??↑(x,τ),↓).[For simplicity,we neglect the transverse direction.]The resulting gap will open at the Fermi level only if the system modi?es its density in such a way that δk F +↑(x,τ)+δk F ?↓(x,τ)=?x ?+↑(x,τ),δk F +↓(x,τ)+δk F ?↑(x,τ)=??x ??↑(x,τ). (6.18) Here we denote by α(k F +δk F ασ(x,τ))the Fermi wave vector on the (ασ)branch of the spectrum.This Fermi wave vector is time and space dependent because of the density ?uctuations.Eqs.(6.18)imply the charge density variation δρ(x,τ)= 1 2πb ?x ?+↑(x,τ)???↑(x,τ) = 1 πb ?xθN(r,τ)?rγ?xθˉN(r,τ) πb ?τθN(r,τ)?rγ?τθˉN(r,τ)π2b2(1+γr)2 ×[D++ ch (ων)+r2γ2D?? ch (ων)?2rγD+? ch (ων)]. (6.22) where D pp′ ch (ων)= θpN(ων)θp′N(?ων) [see Eq.(4.8)]. The conductivity is de?ned byσ(ω)=i/(ω+ i0+)Πret(ω)where the retarded correlation function Πret(ω)is obtained fromΠ(ων)by analytical continu-ation to real frequency iων→ω+i0+.Using the expres-sion of c pp′(Appendix A),we obtain the dissipative part of the conductivity Re[σ(ω)]=ω2p 3+5?γ2 +δ(ω±ω0) 4?γ 8e2v F/b is the plasma frequency. Eq.(6.23)satis?es the conductivity sum rule ∞?∞dωRe[σ(ω)]=ω2p ?γ +δc?? r?γ A.Goldstone mode From Eq.(7.1)we deduce the existence of a mode with a linear dispersion law ω=v F q x , (7.2) φN (q x ,ω)=φˉN (q x ,ω). (7.3) The spin-wave Goldstone mode corresponds to in-phase oscillations of the two SDW’s,in agreement with the conclusion of the mean-?eld analysis (Sec.III).We do not ?nd any renormalization of its velocity because we have not taken into account the coupling with the long-wavelength spin ?uctuations.1 B.Gapped mode The other solution of (7.1)corresponds to a gapped mode with the dispersion law ω2=v 2F q 2x +ω2 1,ω21= 121?r 2 1?? γ2 I N I ˉ N .(7.4) The oscillations of this mode satisfy φN (q x ,ω)=?rγφˉN (q x ,ω). (7.5) In the gapped mode,the oscillations of the two SDW’s are out-of-phase.38As expected,the largest SDW has the smallest oscillations.Like the gapped sliding mode,this mode is found to lie in general above the quasi-particle excitation gap,and is therefore expected to be strongly damped due to coupling with quasi-particle ex-citations.Thus,the spin-wave modes are similar to the sliding modes as shown in Fig.4. VIII.SPIN-SPIN CORRELATION FUNCTION In this section we calculate the transverse spin-spin correlation function.For real order parameters ?(pN ) ασ,the mean-?eld magnetization S (r ) is parallel to the x axis.Transverse (to the magnetization)spin ?uctuations corresponds to ?uctuations of the operator S y (r ,τ)= i 4 α,α′,σ,σ′ σσ′ ?ασ(r ,τ)??α′σ′(r ′,τ′ ) ? 1 g 22N (0) ??N ??ˉN v F q x × ?4r (1?r 2)2 ?γ2+ζ2ω13(1+r 2 ) |?γζ| , (8.6) for ω,q x >0and q y =0.Both spin-wave modes con-tribute to the spectral function.The spectral weight car-ried by the Goldstone mode diverges as 1/q x as expected for a quantum antiferromagnet.39 Eqs.(6.23)and (8.6)predict that all the spectral weight is carried by the in-phase modes,i.e.the gapped sliding mode and the gapless spin-wave mode,whenever both SDW’s have the same amplitude (?γ=ζ=±1). IX.HELICOIDAL PHASE The analysis of the helicoidal phase turns out to be much simpler than the one of the sinusoidal phase.In the helicoidal phase,the mean-?eld gap does not depend on the transverse momentum k y ,which signi?cantly sim-pli?es the computation of the collective modes.In this section,we shall describe the properties of the helicoidal phase,but skipping most of the technical details of the derivation. A.Mean-?eld theory The helicoidal phase is characterized by the order parameter 21 ?ασ(r )= O ασ(r ,τ) =?ασe iαQ p (α,σ)N ·r ,(9.1) with p (+,↑)=p (?,↓)=+,p (+,↓)=p (?,↑)=?. (9.2) In the notations of the preceding sections,this corre-sponds to ?(N )+↑≡?+↑=???↓,?(ˉN )+↓≡?+↓=?? ?↑,?(N ) +↓=?(N ) ?↑=0,and ?(ˉN )+↑=?(ˉN )?↓=0.The fact that some order parameters vanish in the helicoidal phase makes the computation of the collective modes much sim-pler.The spin-density modulation is given by 21 S x (r ) =2|?+↑|cos(?Q N ·r ??+↑) +2|??↑|cos(Q ˉN ·r ???↑), S y (r ) =2|?+↑|sin(?Q N ·r ??+↑) +2|??↑|sin(Q ˉN ·r ???↑), S z (r ) =0, (9.3) which corresponds to two helicoidal SDW’s with opposite chiralities.The mean-?eld action is still given by (3.6),but the pairing amplitudes are given by (in the QLA) ?? ασ(k ,k ′)=δk ′,k ?αQ 0??ασ(k y ),?? ασ(k y )=??ασe iαp (α,σ)N (k y b ?π/2),??ασ=I p (α,σ)N (g 2?ασ+g 3?ˉασ). (9.4) The ground-state energy reads ?E = α 12 α |??α↑|2 |?? α↑| =? 1 ?+↑ ,?γ= ?? ?↑ω2+?2ασ (k x )+|??ασ|2, F ασ(k ,ω)= ?? ασe iαp (α,σ)N (k y b ?π/2)2 ? q (?+↑(??q ),??↑(??q ))D ?1(?q ) ?+↑(?q ) ??↑(?q ) , (9.11) D ?1 (?q )= 2g 2?2+↑ 1?c ++?r 2c ??γr (1?c ++?c ??) γr (1?c ++?c ??)γ2 (1?c ???r 2c ++) .(9.12) The dispersion of the collective modes is obtained from det D ?1(?q )=0.We ?nd a Goldstone mode satisfying ω=v F q x ,?+↑=??↑. (9.13) This mode corresponds to a uniform spin rotation. Equivalently,it can also be seen as a translation of the two SDW’s in opposite directions.40Thus it combines characteristics of the two Goldstone modes of the sinu-soidal phase. There is also a gapped mode with dispersion law ω2=v 2F q 2x +ω22,where ω2 2=161?r 2 ??+↑???↑??↑ = γ2(1?r 2)2+2rγ(γ+r )2+2rγ(1+γr )2 2πb ?x ??+↑(r ,τ)????↑(r ,τ) , (9.16) j DW (r ,τ)=? ie g 2?ασ+g 3?ˉ ασ(9.18) is the ?uctuating phase of the e?ective potential ?? ασ.From Eq.(9.17)we can calculate the current-current cor-relation function,which yields the conductivity Re[σ(ω)]= ω2p 2 α ?2α↑D αα(? q ),Tr q χxy =0. (9.21) This yields the spectral function ImTr qχretμμ= π I N IˉN δ(ω?v F q x)(1?r2)2+1+r2?γζ + δ(ω?ω2) (1?r2)2 + 1+r2 ?γζ (9.22) forω,q x>0and q x→0.Both modes contribute to the spectral function.Although the Goldstone mode is a pure spin-wave mode,the gapped mode has character-istics of both a spin wave and a sliding mode,as can be seen from the spectral function. X.CONCLUSION We have studied the long-wavelength collective modes in the FISDW phases of quasi-1D conductors,focusing on phases that exhibit a sign reversal of the QHE(Rib-ault anomaly).We have recently proposed that two SDW’s,with wave vectors Q N=(2k F+NG,Q y)and QˉN=(2k F?NG,?Q y),coexist in the Ribault phase,as a result of umklapp scattering.When the latter is strong enough,the two SDW’s become circularly polarized(he-licoidal SDW’s).The presence of two SDW’s gives rise to a rich structure of collective modes,which strongly depends on the polarization(linear or circular)of the SDW’s. Regarding the sliding modes,we?nd that the out-of-phase oscillations are gapless in the long-wavelength limit.The fact that this Goldstone mode corresponds to out-of-phase(and not in-phase)oscillations is related to the pinning by the lattice(due to umklapp processes) that would occur for a single commensurate SDW.The other sliding mode is gapped and corresponds to in-phase oscillations.In Bechgaard salts,this mode is expected to lie above the quasi-particle excitation gap and should therefore be strongly damped due to the coupling with the quasi-particle excitations.In the helicoidal phase, there is no low-energy sliding mode,since the Goldstone mode is a pure spin-wave mode.[For a helicoidal SDW, one cannot distinguish between a uniform spin rotation and a global translation,so that we cannot classify the modes in sliding and spin-wave modes.] The dissipative part of the conductivity,Re[σ(ω)],ex-hibits two peaks:a low-energy peak corresponding to the Goldstone mode,and a(broader)peak at high energy due to the incoherent gapped mode.The low-energy spectral weight is directly related to the ratio of the amplitudes of the two SDW’s that coexist in the Ribault phase.When the umklapp scattering strength(g3)increases(experi-mentally this corresponds to a pressure decrease),spec-tral weight is transfered from the low-energy peak to high energies.Above a critical value of g3,the sinusoidal phase becomes unstable with respect to the formation of a he-licoidal phase.At the transition,the low-energy spec-tral weight suddenly drops to zero(Fig.5),since there is no low-energy optical spectral weight in the helicoidal phase.Thus,the formation of the helicoidal phase can be detected by measuring the low-energy optical spectral weight.We also note that the absence of a low-energy sliding mode means that there is no in?nite Fr¨o hlich conductivity41in this helicoidal phase,which is there-fore a true insulating phase even in an ideal system(i.e. with no impurities).In a real system(with impurities), this implies the absence of non-linear dc conductivity. The spin-wave modes exhibit a similar structure.The in-phase oscillations of the two SDW’s are gapless(Gold-stone mode),while the out-of-phase oscillations are gapped.In the helicoidal phase,the Goldstone mode is a pure spin-wave mode,but the gapped mode contributes both to the conductivity and to the transverse spin-spin correlation function. As discussed in the Introduction,these conclusions rely on a simple Fermi surface(Fig.1a),which does not neces-sarily provide a good approximation to the actual Fermi surface of the Bechgaard salts.They should therefore be taken with caution regarding their relevance to the organic conductors of the Bechgaard salts family.How-ever,we have derived a number of experimental conse-quences that should allow to test our theory.In the si-nusoidal phase,we predict a possible reentrance of the phase N=0within the cascade.21The low-energy peak in the optical conductivity Re[σ(ω)]carries only a frac-tion of the total spectral weightω2p/4.This fraction should decrease with pressure.At low pressure,the si-nusoidal phase may become helicoidal.The helicoidal phase is characterized by a vanishing QHE,21a kinetic magneto-electric e?ect,21and the absence of low-energy spectral weight in the optical conductivity as well as the absence of non-linear dc conductivity.In the alterna-tive scenario proposed by Zanchi and Montambaux,23 the Ribault phase does not exhibit any special features compared to the positive phases,apart from the unusual behavior of the magneto-roton modes,26but these modes have not been observed yet. We expect our conclusions regarding the structure of the collective modes and the associated spectral func-tions to hold for generic two-SDW systems and not only for the FISDW phases that exhibit the Ribault anomaly. It is clear that the existence of four long-wavelength col-lective modes(two spin-wave and two sliding modes)re-sults from the presence of a second SDW,which doubles the number of degrees of freedom.Two of these modes should be gapless as expected from Goldstone theorem in a system where two continuous symmetries are sponta-neously broken:the translation symmetry in real space and the rotation symmetry in spin space. This belief is supported by the striking analogy with collective modes in other systems like two-band,bi-layer,and d+id′superconductors,11–13and,to a lesser extent,plasmon modes in semiconductor double-well structures.14While phase modes are in general di?cult to observe in superconductors,since they do not directly couple to external probes(see however Ref.13),collec-tive modes in SDW systems directly show up in response functions like for instance the dc and optical conductivi-ties. Finally,we point out that we expect a similar struc-ture of collective modes in the FISDW phases of the compound(TMTSF)2ClO4.Due to anion ordering,the unit cell contains two sites,which leads to two electronic bands at the Fermi level.This doubles the number of degrees of freedom and therefore the number of collec-tive modes(without considering the possible formation of a second SDW due to umklapp scattering).As for phonon modes in a crystal with two molecules per unit sites,we expect an acoustic(Goldstone)mode and an optical mode. ACKNOWLEDGMENT This work was partially supported by the NSF un-der Grant DMR9815094and by the Packard Foundation. Laboratoire de Physique des Solides is associ′e au CNRS. APPENDIX A:MEAN-FIELD SUSCEPTIBILITIESˉχAND?χ Using the expression(3.21)of the mean-?eld propagators G and F,we obtain ˉχασ,ασ(q+αQ pN,q′+αQ p′N,ων)=?δq,q′ T L x L y k,ω∞ n=?∞Fασ(k,ω)Fασ(k?q+αnG,ω?ων) ×I pN+n(π/b+q y)I p′N?n(π/b+q y)e?iα(p+p′)N(k y b?q y b/2?π/2).(A2) Here we have assumed that|q x|,|q′x|?G,which is the case of interest when studying the low-energy?uctuations around the mean-?eld solution. In the QLA,we retain only the term n=0in the above equations,since the other terms are strongly suppressed whenωc?T,|??pN|in the limit of long-wavelength?uctuations.25Restricting ourselves to q y=0,we then obtain ˉχασ,ασ(q+αQ pN,q+αQ p′N,ων)=? T L x L y k,ωFασ(k,ω)Fασ(k?q,ω?ων)I pN I p′N e?iα(p+p′)N(k y b?π/2).(A4) Performing the sum over k x,we obtain at T=0and in the limit|v F q x|,|ων|?|??ασ(k y)| ˉχασ,ασ(q+αQ pN,q+αQ p′N,ων)=I pN I p′N N(0) |??σα(k y)| ? 1 6|??ασ(k y)|2 ,(A5) ˉχασ,ˉαˉσ(q+αQ pN,q?αQ p′N,ων)=I pN I p′N N(0) 2 + v2F q2x+ω2ν It is useful to introduce the notations c pp′=g2[ˉχ+↑,+↑(q+Q pN,q+Q p′N,ων)?ˉχ+↑,?↓(q+Q pN,q?Q p′N,ων)]=ˉc pp′+δc pp′,(A7) whereˉc pp′=c pp′|q=ων=0andδc pp′are deduce d from Eqs.(3.29). The gap equation can be written as a function of the static(ων=0)susceptibilitiesˉχ(orˉc).From?ασ(r)= Oασ(r) =T ωFασ(r,r,ω),we deduce ?pN=I pN T ω2+?2 +↑ (k x)+|??+↑(k y)|2 ?? p′N .(A8) This can be rewritten in terms of the mean-?eld propagators ?pN=?I pN T I p′N ˉχ+↑,+↑(Q pN,Q p′N,ων=0)?ˉχ+↑,?↓(Q pN,?Q p′N,ων=0) ??p′N.(A9) Using the relation between?pN and??pN,we obtain ?? N (1?r2)IˉN = ˉc++??N (1?r2)IˉN ? r??N IˉN .(A10) From(A10),we deduce the relations ˉc++=1 γ+r ,(A11) which gives,by eliminatingγ, (1?ˉc++)(1?ˉc??)?r2ˉc++c??=0.(A12) Eq.(A12)is nothing but the gap equation rewritten in terms of the static mean-?eld susceptibilitiesˉc. In the study of the collective modes,the natural quantity to consider is notˉχbut the susceptibility?χde?ned by ?χασ,α′σ′(r,τ;r′,τ′)= ?Oασ(r,τ)?O?α′σ′(r′,τ′) ? ?Oασ(r,τ) ?O?α′σ′(r′,τ′) ,(A13) where?Oσ(r,τ)=?g(r)Oσ(r,τ).?χis related toˉχby ?χασ,α′σ′(q,q′,ων)=g22ˉχασ,α′σ′(q,q′,ων)+g23ˉχˉασ,ˉα′σ′(q?α4k F,q′?α′4k F,ων) +g2g3[ˉχˉασ,α′σ′(q?α4k F,q′,ων)+ˉχασ,ˉα′σ′(q,q′?α′4k F,ων)](A14) and satis?es ?χα↑,α′↑(q+αQ pN,q+α′Q p′N,ων)??χα↑,ˉα′↓(q+αQ pN,q?α′Q p′N,ων)=g2δα,α′(c pp′+r2cˉpˉp′) +g3δα,ˉα′(cˉpp′+c pˉp′).(A15) Eq.(A15)is used in Sec.IV. APPENDIX B:CHARGE OPERATORρDW Using G MF α(r,τ;r′,τ′)= Gα↑(r,τ;r′,τ′)Fα↑(r,τ;r′,τ′) Fˉα↓(r,τ;r′,τ′)Gˉα↓(r,τ;r′,τ′) ,(B1) where G and F are the mean-?eld propagators(see Sec.III B),we rewrite Eq.(6.12)as ρDW (r ,τ)=? α d 2r ′dτ ′ G α↑(r ,τ;r ′,τ′)?ηα↑(r ′,τ′)F ˉα↓(r ′,τ′;r ,τ)+F α↑(r ,τ;r ′,τ′)? η? α↑(r ′,τ′)G α↑(r ′,τ′;r ,τ)+F ˉα↓(r ,τ;r ′ ,τ′ )?ηα↑(r ′ ,τ′ )G ˉα↓(r ′ ,τ′ ;r ,τ)+G ˉα↓(r ,τ;r ′ ,τ ′ )?η?α↑(r ′,τ′)F α↑(r ′,τ′ ;r ,τ) . (B2) If we consider phase ?uctuations only,ηασis given by (4.4),which gives using (6.8) ?ηασ(r ,τ)=i p e iαQ pN ·r [g 2?pN ?(pN ) ασ(r ,τ)+g 3?ˉpN ?(ˉpN )ˉασ(r ,τ)].(B3) De?ning the ?uctuating phases ??(pN ) ασ of the e?ective potential ?? ασby ?? ασ(r ,τ)= p ??pN I pN e iαQ pN ·r ??(pN ) ασ(r ,τ). (B5) The relation between ??(pN ) ασand ?(pN ) ασis given by (6.15)to lowest order in phase ?uctuations. From Eqs.(B2)and (B5),we deduce ρDW (?q )=?i T I pN α,?q ′ ??(pN )α↑(?q ′) d 2r dτ d 2r ′dτ′e ?i (q ·r ?ωντ)+i (q ′·r ′?ω′ντ′)× e iαQ pN ·r ′ [G α↑(r ,τ;r ′,τ′)F ˉα↓(r ′,τ′;r ,τ)+F ˉα↓(r ,τ;r ′,τ′)G ˉα↓(r ′,τ′ ;r ,τ)] ?e ?iαQ pN ·r ′[F α↑(r ,τ;r ′,τ′)G α↑(r ′,τ′;r ,τ)+G ˉα↓(r ,τ;r ′,τ′)F α↑(r ′,τ′;r ,τ)] ,(B6) where q =(q x ,q y =0)and ?q =(q x ,ων).Using d 2r dτ d 2r ′dτ′ e ?i (q ·r ?ωντ)+i (q ′·r ′?ω′ντ′)+iαQ pN ·r ′ G α↑(r ,τ;r ′,τ′)F ˉα↓(r ′,τ′ ;r ,τ)= δ?q ,?q ′I pN k ,ω G α↑(k ,ω)F ˉα↓(k ?αQ 0?q ,ω?ων)e iαpN (k y b ?π/2) ,(B7) d 2r dτ d 2r ′dτ′ e ?i (q ·r ?ωντ)+i (q ′ ·r ′ ?ω′ ντ′ )+iαQ pN ·r ′ F ˉα↓(r ,τ;r ′,τ′) G ˉα↓(r ′,τ′ ;r ,τ)= δ?q ,?q ′I pN k ,ω F ˉα↓(k ,ω) G ˉα↓(k ?q ,ω?ων)e iαpN (k y b ?π/2) ,(B8) d 2r dτ d 2r ′dτ′ e ?i (q ·r ?ωντ)+i (q ′·r ′?ω′ντ′)?iαQ pN ·r ′ F α↑(r ,τ;r ′,τ′) G α↑(r ′,τ′;r ,τ)=δ?q ,?q ′I pN k ,ω F α↑(k ,ω) G α↑(k ?q ,ω?ων)e ?iαpN (k y b ?π/2),(B9) d 2r dτ d 2r ′dτ′ e ?i (q ·r ?ωντ)+i (q ′·r ′ ?ω′ντ′)?iαQ pN ·r ′ G ˉα↓(r ,τ;r ′,τ′)F α↑(r ′,τ′ ;r ,τ)= δ?q ,?q ′I pN k ,ω G ˉα↓(k ,ω)F α↑(k +αQ 0?q ,ω?ων)e ?iαpN (k y b ?π/2) ,(B10) we obtain ρDW (?q )=?i α,p ??pN ??(pN )α↑(?q ) T To lowest order in v F q x and ων,we have (for q y =0) T 8|?? α↑(k y )|2(iων+αv F q x ),(B12)T 8|?? α↑(k y )|2(?iων+αv F q x ). (B13) This leads to ρDW (?q )= iN (0) |?? α↑(k y )|2 ??(pN ) α↑(?q ) ? ??(pN ) ˉα↑(? q ) .(B14) Performing the sum over k y ,we ?nally obtain ρDW (?q )=i q x 2 ln 2E 0 2 ? ω2ν+v 2F q 2x 2 ?1 12??2ασ ,(C2) for q y =0and |ων|,v F |q x |?|?? ασ|.Introducing the notations c ++=g 2[ˉχ+↑,+↑(q +Q N ,q +Q N ,ων)?ˉχ+↑,?↓(q +Q N ,q ?Q N ,ων)], c ??=g 2[ˉχ?↑,?↑(q ?Q ˉN ,q ?Q ˉN ,ων)?ˉχ?↑,+↓(q ?Q ˉN ,q +Q ˉN ,ων)], (C3) the gap equation reads (1?ˉc ++)(1?ˉc ??)?r 2ˉc ++ˉc ??=0, (C4) where ˉc ++=1/(1+γr )and ˉc ??=γ/(γ+r )(ˉc pp =c pp |?q =0).