The Role of the Core Energy in the Vortex Nernst Effect
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t The Role of the Core Energy in the Vortex Nernst Effect
Gideon Wachtel and Dror Orgad
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
We present an analytical study of diamagnetism and transport in a film with superconductingphase fluctuations, formulated in terms of vortex dynamics within the Debye-H¨uckle approximation.We find that the diamagnetic and Nernst signals decay strongly with temperature in a mannerwhich is dictated by the vortex core energy. Using the theory to interpret Nernst measurements ofunderdoped La − x Sr x CuO above the critical temperature regime we obtain a considerably betterfit to the data than a fit based on Gaussian order-parameter fluctuations. Our results indicate thatthe core energy in this system scales roughly with the critical temperature and is significantly smallerthan expected from BCS theory. Furthermore, it is necessary to assume that the vortex mobility ismuch larger than the Bardeen-Stephen value in order to reconcile conductivity measurements withthe same vortex picture. Therefore, either the Nernst signal is not due to superconducting phasefluctuations, or that vortices in underdoped La − x Sr x CuO have highly unconventional properties. Over the past decade the Nernst effect has become awidely used tool in the study of strongly correlated elec-tronic systems. The Nernst signal e N = E y / ( − ∂ x T ),defined by the ratio between a measured electric field E y and a transverse applied temperature gradient ∂ x T in an electrically isolated system subjected to an exter-nal magnetic field H z , is typically very small in non-magnetic normal metals. Conversely, a much strongereffect may arise in the flux-flow regime of superconduc-tors, owing to the transverse electric fields induced bythe motion of vortices down the temperature gradient.Consequently, the observation of a large Nernst signal inthe pseudogap state of the cuprates has been taken asevidence that these systems support vortex-like super-conducting fluctuations over a wide temperature rangeabove their critical temperature, T c . However, othershave attributed the large Nernst signal to the responseof quasiparticles in a symmetry-broken state competingwith superconductivity. Despite its appealing nature, the vortex based pic-ture has not been previously justified by an analyti-cal treatment. However, several studies have calculatedthe Nernst signal arising from superconducting order-parameter fluctuations. The contribution of BCS Gaus-sian fluctuations to the thermoelectric response of thenormal state near T c was obtained in Refs. 8,9. This re-sult was subsequently extended to a wider range of tem-peratures and magnetic fields , as well as to scenar-ios beyond that of BCS fluctuations. Experimentally,good agreement with the Gaussian theory was found inamorphous Nb . Si . films and in overdoped, but notunderdoped cuprates (see, however, Ref. 17).A different approach, more pertinent to the presentstudy, was taken by Podolsky et al. , who built uponthe premise that in underdoped cuprates, supercon-ductivity is destroyed at T c by strong phase fluctuations,whereas pairing correlations survive up to a considerablyhigher scale T p . Ignoring superconducting amplitudefluctuations the authors calculated the Nernst signal ina stochastic two-dimensional (2D) XY model via numer-ical simulations and a high-temperature expansion. Inaddition, they devised a simulation method to calculate the thermoelectric response based on vortex dynamics. In this Letter we aim to bridge the aforementionedtheoretical gap and present an analytical study of dia-magnetism and transport in an extreme type-II super-conducting film that is formulated directly in terms ofvortices. We focus on temperatures above T c where thereis a finite density, n f , of free, unbound vortices. Ourapproach, which treats the vortex interactions within aDebye-H¨uckle approximation, is inspired by Ambegaokar et al. who considered vortex dynamics in the contextof superfluid films. A similar route was taken in thestudy of the resistive transition of superconducting filmsby Halperin and Nelson. Our treatment identifies the vortex core energy ǫ c asan important energy scale which controls the strong tem-perature dependence of the fluctuation signals. Usingthe theory we are able to obtain a fit to the transversethermoelectric response of underdoped La − x Sr x CuO (LSCO) which is superior to the one based on Gaussianfluctuations. The available data imply that both ǫ c and T c share a similar doping dependence, with ǫ c ≈ − T c .Such values are significantly lower than the Fermi energy,which is the expected ǫ c from BCS theory. Moreover,in order to reconcile the vortex picture with conductiv-ity data, one needs to assume that the vortex mobilityis much larger than the Bardeen-Stephen value. Thus,unless the strong Nernst and diamagnetic signals in un-derdoped LSCO are of non-superconducting origin, it ap-pears that the vortex core is unconventional and plays animportant role in this system.
Vortex Hamiltonian and dynamics . A 2D supercon-ductor, at temperatures well below T p where the orderparameter amplitude is frozen, can be described by an XY -type Hamiltonian density of a phase field θ coupledvia its charge, (2 e < A , and a constant superfluid density ρ s : H = (1+ ψ ) " ρ s (cid:18) ∇ θ − e ¯ hc A (cid:19) + X i ǫ c δ ( r − r i ) . (1)We assume that only vortices contribute to the other-wise uniform ∇ θ . A vortex i of vorticity n i = ± r i = ( x i , y i ) contributes ∇ θ i ( r ) = n i ˆz × ∇ ln | r − r i | r = n i ˆz × ( r − r i ) | r − r i | , (2)where r is the vortex core radius, and ˆz is a unit vectorperpendicular to the plane. The continuum model andvortex configuration, Eqs. (1,2), are valid at scales longerthan r . Thus, a region of radius r around r i is implicitlyremoved from the first term in Eq. (1). Its energy isgiven by the vortex core energy , ǫ c , which we assumeto be constant across the sample. Following Luttinger ,we have introduced a “gravitational” field ψ ( r ) in orderto study the response of the system to a temperaturegradient.For concreteness, we consider a superconducting stripof infinite extent along the y direction, and of finite width L in the x direction. When needed, a constant transversetemperature gradient is applied via ψ ( r ) = ψ ′ x , and auniform electric field E = E y ˆy is applied along the strip.Working in the extreme type-II limit we assume the pres-ence of a uniform perpendicular magnetic field B ˆz , andchoose the gauge A = A + A E , where A = Bx ˆy , and E = − ∂ t A E /c . By symmetry, the average (over vortices’positions) phase gradient h ∇ θ i is directed along the stripand is independent of the y coordinate.We approach the model given by Eq. (1) within amean-field Debye-H¨uckle approximation, in which cor-relations between vortices are ignored. This is possibleat temperatures higher than the Beresinskii-Kosterlitz-Thouless (BKT) transition temperature T BKT , for lengthscales longer that the Debye-H¨uckle screening length r s ,where vortex interactrions are screened by thermally ex-cited vortices. The effective description at such scales isstill given by Eq. (1), provided that ρ s and ǫ c assumerenormalized values, which include contributions fromthe superflow at shorter distances. Consequently, theseparameters become temperature dependent. Dynamicsis introduced into the model by assuming that the prob-ability P i ( r i , t ) to find the i th vortex at position r i andtime t obeys a mean field Fokker-Planck equation. Thecorresponding probability current density for vortrex i isgiven by J i ( r i , t ) = − µP i ( r i , t ) h ∇ i H i i − µT ∇ i P i ( r i , t ) , (3)where H = R d r H , µ is the vortex mobility, T the tem-perature (here, and throughout k B = 1), ∇ i is the gra-dient with respect to r i , and h· · ·i i denotes an averageover the position of all vortices besides r i . Near equilib-rium this reproduces the mean-field Debye-H¨uckle theory,provided one ignores fluctuations by taking h ( ∇ θ ) i ≈ ( h ∇ θ i ) . The residual effect of fluctuations is accountedfor by renormalizing ρ s and ǫ c . For convenience we define the mean field u ( x ) ≡h ∂ y θ i / π and a ( x ) ≡ A y /φ where φ = π ¯ hc/e is theflux quantum. Using these definitions we find that the x component of the probability current density of vortex i is given by J ix ( x ) = µP i ( x ) h π ρ s n i (1 + ψ )( u − a ) − ǫ c ∂ x ψ i − µT ∂ x P i ( x ) . (4)Similarly, the average vorticity current density along x is J vx ( x ) = X i n i J ix ( x )= 4 π ρ s µn f (1 + ψ )( u − a ) − µǫ c ∂ x ψ∂ x u − µT ∂ x u, (5)where ∂ x u ( x ) = n ( x ) = P i n i P i ( x ) is the mean vortic-ity, whose bulk value, as shown below, is set by B , and n f ( x ) = P i P i ( x ) is the density of free vortices. Withinthe equilibrium Debye-H¨uckle approximation it is pos-sible to show that n f ≃ q r − e − ǫ c /T + n , (6)which establishes a strong dependence of n f on T , forsmall B . The average y component of the electric currentdensity J e = − c h δ H /δ A i is given by J ey = 4 π ρ s cφ (1 + ψ )( u − a ) . (7)Thus, the first term in Eq. (4) is just the vortex drift inresponse to the Magnus force it experiences in an elec-tric current J ey . Note, that all free vortices, and not onlythose responsible for the excess vorticity, contribute tothe vorticity current, Eq. (5), via their response to theMagnus force. As a result, the strong temperature depen-dence of n f is also reflected in the transport coefficients. Equilibrium magnetization . In equilibrium ψ = 0, E y = 0, and we must have J vx = 0. We therefore need tofind u ( x ) which solves the following equation4 π ρ s n f ( u − ¯ nx ) − T ∂ x u = 0 , (8)with ¯ n defined such that a ( x ) = Bx/φ = ¯ nx . We solvethis equation, for small B , by choosing boundary con-ditions in which the vorticity n ( x ) = ∂ x u ( x ) vanishes at x = 0 and x = L . In terms of the Debye-H¨uckle screeninglength, r − s = 4 π ρ s n f /T , we find u ( x ) = ¯ n (cid:20) x + r s e − x/r s − e − ( L − x ) /r s e − L/r s (cid:21) . (9)The deviation of u from ¯ nx near the edge leads, accord-ing to Eq. (7), to edge currents. Their integral gives riseto an average magnetization density M z = 1 c A Z dy Z L dx xJ ey ≃ − T Bφ n f , (10)where A is the area of the strip. Here, and in the fol-lowing, we ignore corrections of order O ( r s /L ). Similarexpressions to Eq. (10) were obtained in several previousstudies. Electric conductivity . In order to study the lin-ear response of the system to a weak perturbing field E y ( ω ) e − iωt we need to obtain the dynamics of u ( x, t ).By employing translational invariance in the y direction one can show that ∂u∂t = − J vx . (11)This is a local version of the equation used in Refs. 21,22.Solving it using Eq. (5), we find in the bulk u ( x, t ) =¯ nx + u ( ω ) e − iωt where u ( ω ) = 11 − iωτ cE y ( ω ) iωφ , (12)and where we have introduced the relaxation time 1 /τ =4 π ρ s µn f . Eq. (7) then implies an electric conductivity σ s ( ω ) = 4 e h hµn f − iωτ . (13)This result is identical to the conductivity obtained byHalperin and Nelson for temperatures above T c . Thermoelectric coefficients . For systems with particle-hole symmetry or when superconducting fluctuationsdominate, the Nernst signal is given by e N = ρα xy = − ρα yx , where α yx is defined by J ey = α yx ( − ∂ x T ). Lut-tinger has shown that α yx can be deduced from theresponse to a “gravitational” field ψ according to therelation J ey = T α yx ( − ∂ x ψ ). Thus, we solve Eq. (56)in the presence of ψ ( x, t ) = ψ ′ ( ω ) xe − iωt . By writing u ( x, t ) = u ( x ) + u ( ω ) e − iωt , where u ( x ) is the equilib-rium solution of Eq. (8), we find that to first order in ψ ′ ( ω )¯ u ( ω ) = 1 L Z L dx u ( x, ω ) = − M z φ n f + ǫ c ¯ n − iωτ ψ ′ ( ω )4 π ρ s n f . (14)Eq. (7) leads then to the average electric current density J ey ( ω ) = 1 A Z dy Z L dx J ey ( x, ω ) ≃ − M z φ n f + ǫ c ¯ n − iωτ cψ ′ ( ω ) n f φ + cM z ψ ′ ( ω ) . (15)The response of u ( x, ω ) is given by the first term above.An additional contribution, of opposite sign, comes frommagnetization currents near the edges. Contrary to someprevious studies where this additional contributionhad to be subtracted , in our treatment its oppositeeffect is explicitly included in the second term. In theDC limit, ω →
0, we therefore obtain α yx = − ek B h Bn f φ ǫ c k B T = ǫ c T cM z T . (16) This result should be compared with the constant ratiobetween α yx and cM z /T , which was found for high tem-peratures in Refs. 8,18 and 20.Next, we consider the linear response ratio ˜ α xy betweenan applied electric field and a transverse heat currentdensity, J Qx = ˜ α xy E y . We deduce J Q , which in our modelequals the energy current density, from the conservationequation ∂ t H + ∇ · J Q = J e · E . Its source term originatesfrom the explicit time dependence of H via A . The result J Q = − ρ s (cid:28) ∂θ∂t (cid:18) ∇ θ − e ¯ hc A (cid:19)(cid:29) + X i ǫ c J i , (17)is consistent with the form used by Ussishkin et al. ,once modified to include the energy current associatedwith the vortex cores. If we additionally assume that thelong superconducting strip is periodic in the y direction,then the x component of the first term in Eq. (17) mustvanish by symmetry, and we find that Onsager’s relation˜ α xy ( B ) = T α yx ( − B ) is obeyed. Discussion . Often (see Refs. 1,4 and referencestherein), a phenomenological quantity called the vor-tex transport entropy, s φ , is invoked in order to relatethe temperature gradient to the thermal force acting ona vortex, i.e. f = − s φ ∇ T . Based on Eq. (4) andLuttinger , we identify s φ = ǫ c /T . For low tempera-tures where there are no thermally excited vortices andthe flux-flow resistivity is the dominant form of damp-ing, one can show by neglecting vortex interactions that α yx = − cs φ /φ . When taken together with the aboveidentification of s φ , this result is consistent with Eq. (16),since at low temperatures ¯ n f φ = B .As the temperature is raised through T BKT , the den-sity of free vortices, n f , rapidly increases. Our results,Eqs. (6,10,16), indicate that both M z and α yx shouldexhibit a consequent strong reduction with temperature,much faster than the 1 /T ln( T /T c ) decay expected fromGaussian fluctuations. To look for such behavior inthe cuprates we compare Eq. (16) divided by the LSCOlayer separation, d = 6 . α yx = α yx /d with under-doped LSCO data. According to Eq. (6), n f is deter-mined by the renormalized vortex core energy ǫ c , whichreflects fluctuations at distances below r s and is temper-ature dependent. For weak magnetic fields and in thecritical regime above T BKT this renormalization leads to n F ∼ exp( − b/ √ T − T BKT ) , while at high tempera-tures n F ∼ exp[ − ǫ c / ( T − ˜ b )]. Here b and ˜ b are con-stants and ǫ c is the bare core energy. The lack of detailedknowledge about the the full temperature dependence of ǫ c allows for considerable freedom in the fitting proce-dure. In order to constrain the fit, and since we are onlyinterested in a rough estimate of ǫ c , we choose to considera constant ǫ c and also set φ / πr = 50T. Furthermore,we concentrate on the limit B → T c , where the renormalization effects areexpected to be small, but low enough so that vortices aredistinct objects, i.e. r n f ≪
1. Figure 1 depicts the mea-sured B → − α yx /B for LSCO samples with − − − − α D y x / B ( V / K ΩT m ) T (K) x = 0 . . . FIG. 1: − lim B → α yx /B = ( ν − ν n ) of underdopedLa − x Sr x CuO , where ν is the Nernst coefficient, ν n a sub-tracted background due to quasiparticles, and ρ is the in-planeresistivity. The data for x = 0 . , .
10 was extracted fromRefs. 2,3,33, and for x = 0 .
12 from Ref. 18. The data wasfitted to Eqs. (16) (solid color curves). In the regime indi-cated by the dashed curves r n f > .
35, and the theory isnot expected to be applicable. The solid black curves depictthe best fit to the Gaussian fluctuations theory. x ( T c ) = 0 .
07 (11 K) , .
10 (27 . .
12 (29 K). Thesolid color lines are the theoretical fits in the temperaturewindow 1 . T c < T < ∼ T c , with a constant ǫ c as the onlyfree fitting parameter. From these curves we find ǫ c ≈ , ,
143 K, for the different doping levels. Compara-ble, but somewhat larger values, ǫ c ≈ T c , were foundby analyzing penetration depth measurements in under-doped Y − x Ca x Ba Cu O − δ bilayer films. For compar-ison we also include the best fit to the data based on thetheory of Gaussian fluctuations.
Clearly, the data ex-hibits a faster decay than the Gaussian theory above thecritical region around T c . In addition, we fitted the datato the high- T result α yx ∝ T − of the stochastic XY model. We obtained a good fit for x = 0 .
12, but foundoverestimation of the data in the range 1 . T c < T < T c (3 T c ) for x = 0 .
10 (0 . T onset , is de-fined as the temperature for which the Nernst coefficient ν = e N /B goes below a threshold value, typically around ν = 4 nV / K T. Such levels can be reached using Eq. (16)only if one takes r n f ∼
1. This, however, is beyond thevalidity of our theory. Indeed, we find that the experi-mental data begin to deviate from the theoretical curvesat temperatures where r n f > .
35, indicated by dashed lines in Fig. 1. Thus, although our theory agrees with theNernst measurements up to T ≈ T c , it cannot accountfor T onset , which is probably controlled by a combinationof lattice effects and amplitude fluctuations. The Nernst signal in the cuprate pseudogap regimeexhibits a maximum as a function of the magneticfield, which shifts to higher fields with increasingtemperature.
While we do not have a theory for themaximum we note that Eqs. (6,16) imply a crossover, setby the condition
B/φ ∼ n f ( T, B = 0), from a linear- B dependence of α yx at weak fields towards saturation athigher fields. Across this scale magnetic field-inducedvortices dominate, screening is reduced and correlationeffects are enhanced, leading potentially to the suppres-sion of α yx .In conclusion, we showed that within the vortex pic-ture of phase fluctuating superconductors, ǫ c plays anessential role in the thermoelectric response. The vortexcore energy was also found to be important in determin-ing T c of layered superconductors. Uncovering the roleplayed by ǫ c in other phenomena may help in identifyingthe physics underlying the different temperature scalesobserved in the cuprates. Equally pertinent is gaining anunderstanding of the factors which determine ǫ c itself.Here we briefly mention the need for a model of “cheapvortices”, in which vortices support a state close in en-ergy to the superconducting phase. It seems to us thatthe checkerboard state observed around vortex cores isa natural candidate.Nevertheless, if the Nernst signal in underdopedcuprates is, in fact, due to thermally excited vortices,one must also understand why experiments do not showsignatures of fluctuation enhanced conductivity over asimilar temperature range. More specifically, if the vor-tex mobility is given by the Bardeen-Stephen result , µ ≈ πe r /h σ n , then Eq. (13) gives a fluctuationcontribution σ s = σ n / πr n f , where σ n is the normalstate conductivity. This would imply, using our estimate ǫ c ≈ − T c , from fitting the LSCO Nernst data, and Eq.(6), that σ s > σ n for T < T c , in contradiction to exper-iments. To avoid such a contradiction within our model,we must therefore assume that µ is much larger than theBardeen-Stephen value, thereby reducing σ s while notaffecting M z and α yx . A similar conclusion regarding µ was reached based on THz time-domain spectroscopy inLSCO. 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I. DEBYE-H ¨UCKLE APPROXIMATION IN EQUILIBRIUM
At high temperatures, it is possible to study the vortex Hamiltonian within the Debye-H¨uckle approximation, whichis best formulated using a variational mean-field approach. Assume that the state of the system is defined by thevorticity at each lattice site, n r = 0 , − , +1. In the variational mean-field ansatz the density matrix is factored intoa product of local probabilities, ρ = Y r ρ r ( n r ) , (18)with the effect that the entropy is given by S = − Tr ρ ln ρ = − X r X n r ρ r ( n r ) ln ρ r ( n r ) . (19)Additionally, one approximate the average Hamiltonian by h H i ≈ ρ s Z d r (1 + ψ ) (cid:18) h ∇ θ i − e ¯ hc A (cid:19) + ǫ c X r (1 + ψ ( r )) h| n r |i , (20)while ignoring the contribution coming from fluctuations in ∇ θ , h H fluc . i = 12 ρ s Z d r (1 + ψ ) (cid:16) h ( ∇ θ ) i − h ∇ θ i (cid:17) . (21) h ∇ θ i is given by h ∇ θ ( r ) i = ∇ θ + X r ′ h n r ′ i ˆz × ( r − r ′ )( r − r ′ ) , (22)where ∇ θ is the uniform part of ∇ θ ( r ), which does not rise from vortices, h n r i = X n r ρ r ( n r ) n r , (23)and h| n r |i = X n r ρ r ( n r ) | n r | . (24) ρ r ( n r ) itself is determined by minimizing the free energy F = h H i − T S , with the constraint X n r ρ r ( n r ) = 1 , (25) ∂F∂ρ r ( n r ) = ρ s Z d r ′ (cid:18) h ∇ ′ θ ( r ′ ) i − e ¯ hc A ( r ′ ) (cid:19) · ˆz × ( r ′ − r )( r ′ − r ) n r + ǫ c (1 + ψ ( r )) | n r | + T ln ρ r ( n r ) = α. (26)Solving for ρ r we find ρ r ( n r ) = 1 z r e − βǫ c | n r |− βϕ ( r ) n r , (27)where ϕ ( r ) = ρ s Z d r ′ (cid:18) h ∇ ′ θ ( r ′ ) i − e ¯ hc A ( r ′ ) (cid:19) · ˆz × ( r ′ − r )( r ′ − r ) , (28)and z r = 1 + e − βǫ c βϕ ( r ) . (29)For small e − βǫ c we find h| n r |i ≈ e − βǫ c βϕ ( r ) , (30)and h n r i ≈ − e − βǫ c βϕ ( r ) . (31)Eliminating ϕ gives h| n r |i = q e − βǫ c + h n r i , (32)which, after dividing through by r , reads n f = q r − e − βǫ c + n . (33) II. VORTEX DYNAMICSA. Mean-Field Fokker-Planck equations
In order to formulate dynamics of the vortices in our model, we assume that the number of vortices is the same asin equilibrium, and that their vorticity is fixed. Events of vortex-anti-vortex creation and annihilation are importantfor non-linear response at T c , but have a negligible effect on linear response, and are therefore ignored. Thus, it ispossible to formulate vortex dynamics using a Fokker-Planck equation for the positions of all vortices, { r i } , each witha given vorticity { n i = ± } : ∂P ( { r i } , t ) ∂t = X i n µ ∇ i · [ P ( { r i } , t ) ∇ i H ] + µT ∇ i P ( { r i } , t ) o , (34)where µ is the vortex mobility, ∇ i is the gradient with respect to r i , and k B = 1 is used throughout. This isa complicated equation to solve, but it can be treated approximately, in a manner similar to the Debye-H¨uckleapproximation in equilibrium, by factoring the probability density into a product of single vortex probabilities, P ( { r i } , t ) = Y i P i ( r i , t ) . (35)Integrating the left side of Eq. (34) over the positions of all vortices aside from the position of the i th gives Y j = i Z d r j ∂P ( { r i } ) ∂t = Y j = i Z d r j X k Y l = k P l ( r l , t ) ∂P k ( r k , t ) ∂t = P i ( r i , t ) X k = i Y j = i,k (cid:18)Z d r j P j ( r j , t ) (cid:19) Z d r k ∂P k ( r k ) ∂t + ∂P i ( r i , t ) ∂t Y j = i (cid:18)Z d r j P j ( r j , t ) (cid:19) = ∂P i ( r i , t ) ∂t , (36)where we demand that the single vortex probabilities are normalized, Z d r j P j ( r j , t ) = 1 . (37)Preforming the same integral on the right side of the Fokker-Planck equation gives ∂P i ( r i , t ) ∂t = Y j = i Z d r j µ X k ∇ k · h P ( { r k } , t ) ∇ k H ( { r k } ) + T ∇ k P ( { r k } ) i = P i ( r i , t ) µ X k = i Z d r k ∇ k · h P k ( r k , t ) h ∇ k H i ik + T ∇ k P k ( r k ) i + µ ∇ i · h P i ( r i , t ) h ∇ i H i i + T ∇ i P i ( r i ) i , (38)where h ∇ i H i i = Y j = i (cid:18)Z d r j P j ( r j , t ) (cid:19) ∇ i H, (39)and h ∇ k H i ik = Y j = i,k (cid:18)Z d r j P j ( r i , t ) (cid:19) ∇ k H. (40) h ∇ k H i ik is similar to h ∇ k H i k except for an interaction term H ik between vortex k and vortex i : h ∇ k H i ik = h ∇ k H i k − h ∇ k H ik i k + ∇ k H ik (41)Substituting Eq. 41 into Eq. 38 we find that the single vortex Fokker-Planck equation is ∂P i ( r i ) ∂t = µ ∇ i · h P i ( r i , t ) h ∇ i H i i + T ∇ i P i ( r i ) i , (42)provided that X k = i Z d r k ∇ k · h P k ( r k , t ) ∇ k H ik − P k ( r k , t ) h ∇ k H ik i k i = 0 . (43)This can be shown to be the case on our strip where there is translational invariance in the y direction. B. Derivation of the vorticity current
As shown above, the Fokker-Planck equation can be separated into single vortex equations, ∂P i ( r i , t ) ∂t = µ ∇ i · [ P i ( r i , t ) h ∇ i H i i ] + µT ∇ i P i ( r i , t ) , (44)where h− ∇ i H i i is the force on vortex i , averaged over the position of all other vortices h ∇ i H i i = Y j = i (cid:18)Z d r j P j ( r j , t ) (cid:19) ∇ i H ( { r } ) = ∇ i δ h H i δP i ( r i ) . (45)Various average quantities can be calculated using the single vortex probability density P i ( r , t ) = h δ ( r − r i ( t )) i , (46)and the probability current density J i ( r , t ) = h δ ( r − r i ( t ))˙ r i ( t ) i . (47)Interpreting the single vortex Fokker-Planck equation as a probability conservation condition, it is evident that J i ( r i , t ) = − µP i ( r i , t ) h ∇ i H i i − µT ∇ i P i ( r i , t ) . (48)Translational invariance in the y direction (along the strip) requires that P i and J i are independent of the y coordinate.For example, the vorticity can be wrriten as ∂ x u ( x, t ) = n ( x, t ) = X i h n i δ ( r − r i ( t )) i = X i n i P i ( x, t ) , (49)the free vortex density is n f ( x, t ) = X i h δ ( r − r i ( t )) i = X i P i ( x, t ) , (50)and the vorticity current is given by J vx ( x, t ) = X i h n i δ ( r − r i ( t )) ˙ x i i = X i n i J i,x ( x, t ) . (51)Ignoring the same fluctuation term in h H i as in Eq. 20, we find ∂ h H i i ∂x i = ∂∂x i δ h H i δP i ( r i ) ≈ n i ρ s ∂∂x i Z d r ′ [1 + ψ ( x ′ )] (cid:18) h ∇ θ ( r ′ ) i − e ¯ hc A ( r ′ ) (cid:19) · ˆz × ( r ′ − r i )( r ′ − r i ) + ǫ c ∂∂x i ψ ( r i )= n i ρ s ∂∂x i Z dx ′ π [1 + ψ ( x ′ )][ u ( x ′ ) − a ( x ′ )] Z dy ′ x ′ − x i ( x ′ − x i ) + ( y ′ − y i ) + ǫ c ∂∂x i ψ ( x i )= n i ρ s ∂∂x i Z dx ′ π [1 + ψ ( x ′ )][ u ( x ′ ) − a ( x ′ )] π sign( x ′ − x i ) + ǫ c ∂∂x i ψ ( x i )= − n i π ρ s [1 + ψ ( x i )][ u ( x i ) − a ( x i )] + ǫ c ∂∂x i ψ ( x i ) . (52)Therefore, the vorticity current density is J vx ( x, t ) = X i n i J i,x ( x, t )= X i n i (cid:20) − µP i ( x i , t ) ∂ h H i i ∂x i − µT ∂P i ( x i , t ) ∂x i (cid:21) x i = x = X i n i (cid:20) µP i ( x i , t ) n i π ρ s [1 + ψ ( x i )][ u ( x i ) − a ( x i )] − µP i ( x i , t ) ǫ c ∂∂x i ψ ( x i ) − µT ∂P i ( x i , t ) ∂x i (cid:21) x i = x = X i h π ρ s µP i ( x, t )[1 + ψ ( x )][ u ( x ) − a ( x )] − µǫ c ∂ x ψ ( x ) n i P i ( x, t ) − µT n i ∂ x P i ( x, t ) i , (53)which finally gives J vx = 4 π ρ s µn f (1 + ψ )( u − a ) − µǫ c ∂ x ψ∂ x u − µT ∂ x u. (54) III. DYNAMIC EQUATION FOR u In order to study the linear response of the system to weak, time dependent, perturbing fields E and ∇ ψ , we mustobtain the dynamics of the field u ( x, t ). ∂u∂t = 12 π *X i ˙ x i ∂∂x i ∂ y θ +Z dy ∂u∂t = 12 π *X i ˙ x i ∂∂x i Z dy ∂ y θ + = 12 π *X i ˙ x i ∂∂x i n i π sign( x − x i ) + = − *X i ˙ x i n i δ ( x − x i ) + = − Z dy J vx . (55)By translational invariance in the y direction we find ∂u∂t = − J vx ..