Quantitative theory of transport in vortex matter of type-II superconductors in the presence of random pinning

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Quantitative theory of transport in vortex matter of type-II superconductors

in the presence of random pinning

B. Rosenstein and V. Zhuravlev

National Center for Theoretical Sciences and Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China

共Received 27 January 2007; revised manuscript received 24 April 2007; published 17 July 2007兲

We quantitatively describe the competition between interactions, thermal fluctuations, and random quenched disorder using the dynamical Martin-Siggia-Rose approach 关Phys. Rev. A 8, 423 共1973兲兴 to the Ginzburg-Landau model of the vortex matter. The approach first used by Dorsey et al.关Phys. Rev. B 45, 523 共1992兲兴 to describe the linear response far from Hc1is generalized to include both pinning and finite voltage. It allows one

to calculate the non-Ohmic I-V curve, thereby extending the theory beyond the linear response. The static flux line lattice in type-II superconductors undergoes a transition into three disordered phases: the vortex liquid共not pinned兲, the homogeneous vortex glass 共pinned, if one disregards an exponentially small creep at finite tem-peratures兲, and the crystalline Bragg glass 共pinned兲 due to both thermal fluctuations and disorder. The location of the glass transition line in the homogeneous phase is determined and compared to experiments. The line is clearly different from both the melting line and the second peak line describing the translational and rotational symmetry breaking at high and low temperatures, respectively. Time correlation and response functions of the order parameter as functions of the time difference are calculated in both the liquid and the amorphous homogeneous phases. They determine the relaxation properties of the vortex matter due to the combined effect of pinning and thermal fluctuation. We calculate the critical current as a function of magnetic field and temperature in the homogeneous phase. The surface in the J-B-T space defined by this function separates between a dissipative moving vortex matter regime and vortex glass. A quantitative theory of the peak effect, qualitatively different from the conventional one due to Pippard关C. Tang, X. Ling, S. Bhattacharya, and P. M. Chaikin, Europhys. Lett. 35, 597共1996兲; A. B. Pippard, Philos. Mag. 19, 217 共1969兲; A. I. Larkin and Yu. N. Ovchinnikov, J. Low Temp. Phys. 34, 409共1978兲兴, is proposed.

DOI:10.1103/PhysRevB.76.014507 PACS number共s兲: 74.40.⫹k, 74.25.Ha, 74.25.Dw

I. INTRODUCTION

In type-II superconductors for which the penetration depth␭ exceeds the correlation length␰, the magnetic field penetrates the sample in the form of Abrikosov vortices which strongly interact, thereby creating an elastic “vortex matter.” Impurities always present in a sample lead to inho-mogeneities both on the microscopic scale共described by the electron’s mean free path兲 and the mesoscopic scale 共pin-ning兲, which greatly affect the thermodynamic and, espe-cially, dynamic properties of the vortex matter. When disor-der is strong enough, it pins the vortex matter, resulting in dissipationless persistent current, thereby recovering an original defining property of superconductor. In addition, thermal fluctuations also significantly influence the vortex matter, either directly by melting the vortex lattice into a vortex liquid or by changing the efficiency of the disorder 共thermal depinning兲. They are especially important in high Tc

superconductors. As a result of a delicate interplay between disorder, interactions, and thermal fluctuations, even the static H-T phase diagram of high Tcsuperconductors is very

complex and is still far from being reliably determined. Once electric current J is injected into a sample, one is faced with the problem of describing the dynamical phase diagram, which should be drawn in the three dimensional space T-H-J. This makes the analysis essentially more com-plicated, especially if one intends to study it beyond linear response. Generally, there are two phases: the pinned phase, in which the vortices are pinned and thus the resistivity

van-ishes 共perfect superconductivity exists兲, and the unpinned phase, in which vortices can move due to Lorentz force and thus a finite resistivity appears. The transition surface is de-termined by the critical current as function of magnetic field H and temperature T. When the critical current vanishes, the intersection of the surface with the H-T plane gives the “static” irreversibility line.

Theoretically, the problem of the vortex matter subject to both thermal fluctuations and disorder has a long history. Two major simplifications are generally made. In the major-ity of the works, the vortex matter is considered as an array of elastic lines.2 _This _{共London兲 approximation is generally}

valid far from the higher critical field Hc2, when the vortex

density is low. An alternative simplification to the vortex matter is valid far enough from the lower critical field Hc1.

At high vortex densities, magnetic fields of many vortices overlap and the resulting magnetic inductance is nearly ho-mogeneous. It is usually supplemented by the so-called low-est Landau level 共LLL兲 approximation.3,4 _{The original idea}

of the vortex glass5 _{and the continuous glass transition}

ex-hibiting the glass scaling of conductivity in statics appeared early in the framework of the frustrated XY model.6 _The

model was studied by the renormalization group and the variational methods, and has been extensively simulated numerically.7,8 _{In analogy to the theory of spin glasses, in}

this model the replica symmetry is broken when crossing the glass transition line. The frustrated XY model ran into several problems. For finite penetration depth, it has no transition9

and, moreover, there was a difficulty in explaining sharp

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Bragg peaks observed in the experiments at low magnetic fields. To address the last problem, another simplified model had been proven to be more convenient: the elastic medium approach to a collection of interacting linelike objects sub-ject to both the pinning potential and the thermal bath. The resulting theory was treated using the Gaussian approximation10,11 _{and renormalization group.}6_{The original}

problem of the very fast destruction of the vortex lattice by disorder was solved with the vortex matter being in the rep-lica symmetry broken phase for dimensionality 2⬍D⬍4, and it was termed “Bragg glass.” In D = 2, there exists dis-crete replica symmetry broken transition. It is possible to address the problem of dynamics in the presence of thermal fluctuation using an approach in which one directly simulates the interacting line-like objects subject to both the pinning potential and the thermal bath Langevin force.12

A vast majority of theoretical works deal with the linear response and cannot address the flux dynamics at finite cur-rents. The I-V curves of the flux flow are very nonlinear. However, one should be very cautious in interpreting the experimental results since they generally do not distinguish between the bulk and the edge effects. In several experiments in which the bulk was isolated共either by Corbino geometry13

or by varying the width of the sample14_{兲, one finds that the}

bulk I-V has a structure simpler than the commonly accepted nonlinear form. It is important to achieve a theoretical un-derstanding of both the bulk and edge contributions in the region of the magnetic phase diagram in which these experi-ments on NbSe2were performed. Very often the flux

dynam-ics is investigated in the London limit with widely separated vortices, so that one can model them as an array of line-like thin objects interacting pairwise. The motion in the presence of disorder looks like a rather chaotic advance in channels, with sudden hops between pinning centers and occasional avalanches. This picture cannot be directly applied in situa-tions when magnetic field of vortices overlap, creating a ho-mogeneous magnetic field. Still, the core regions might ex-hibit this type of behavior. Moreover, the connection between the qualitative description of the motion and the resulting I-V curves is not clear at present.

In this paper, we investigate the 共bulk兲 dynamics of the vortex matter beyond linear response using the disordered Ginzburg-Landau model. In statics, the replica method of handling disorder in the framework of LLL Ginzburg-Landau 共GL兲 model was utilized in both three dimensions and two dimensions15,16 _{to obtain the irreversibility line}

along with other properties of the disordered vortex matter. The irreversibility lines of YBCO and a two-dimensional 共2D兲 organic superconductor were found to be in good agree-ment with experiagree-ment.16 Tesanovic and Herbut used super-symmetry共for columnar defects in layered materials兲.17

Dy-namics in the presence of thermal fluctuations and disorder is phenomenologically described using the time dependent Ginzburg-Landau 共TDGL兲 model, in which the coefficients have random components.2,18_{Such an attempt was made by}

Dorsey et al.1 _{in the homogeneous} _{共liquid兲 phase using a}

dynamic Martin-Siggia-Rose 共MSR兲 formalism.19,20 _They

obtained the irreversibility line and formulated the linear re-sponse theory of the vortex matter.

The main purpose of this paper is to study the dynamics of three-dimensional vortex matter beyond linear response

using the dynamical approach1 _{within the TDGL model at}

finite electric field. In strongly type-II superconductors共for which the Ginzburg-Landau ratio␬=␭/␰is large兲, magnetic and electric fields inside the superconductor are homoge-neous over a wide range of parameters. As mentioned above, high homogeneity of magnetic field in the mixed state origi-nates from superposition of many共B˜/Hc1Ⰷ1兲 vortices. The

same argument is valid for an electric field which arises due to vortex motion. Indeed, the electric field is related by Lor-entz transformation to the homogeneous magnetic field ap-pearing in the frame moving with vortices. Linear response, namely, conductivity in the limit of zero current, is typically obtained using the Kubo formula.1,21–23_{A theory with a finite}

electric field allows one to obtain the I-V curve beyond the linear response. Finite electric fields without disorder have been considered within the framework of TDGL in Refs.4,

21, and24.

Complex motion of vortex matter featuring mesoscopic avalanches, channels, and islands of pinned flux described above, which might exist, are not probed directly in our method of averaging over the white noise disorder. Analyti-cally, one preforms the disorder average of conductivity and magnetization, and does not directly see the picture above. To get a glimpse into the dynamics, time dependent tion functions are calculated. We study the dynamic correla-tion funccorrela-tion C共r,␶, r

⬘

,␶

⬘

兲 and the response functions R共r,␶, r

⬘

,␶

⬘

兲共averaged over disorder and thermal fluctua-tions兲 of the order parameter␺ within an appropriately gen-eralized Gaussian approximation in both the flux flow and the pinned phases. We consider the stationary case only,1

namely, when the correlation function depends on the time difference and is therefore characterized by the spectrum C_␻. The critical surface Tg_{共H,J兲 in the three-dimensional space}

T-H-J, separating the pinned and unpinned phases, is ob-tained as a surface at which C_␻→⬁ for ␻→0. Above this surface, the real space correlator decays exponentially, while below it is a constant at large time scales. The constant is proportional to the Edwards-Anderson order parameter char-acterizing transition to a glassy state.25_Approaching

critical-ity in the parameter space T→Tg

, various quantities diverge powerwise in共T−Tg兲, with critical exponents calculated in mean field. The static glass transition line, namely, the line at zero electric field, coincides with the one obtained using the replica method.16_{A relation between the dynamical and}

rep-lica methods, which is important for understanding the na-ture of any glass transition, is discussed.

We show that the leading contribution to conductivity near the glass line comes from the first Landau level rather than from the LLL. The conductivity diverges on the static glass line. This has important physical consequences. It is well known that within LLL both magnetization and conduc-tivity are proportional to the superfluid density兩␺兩2_.

Magne-tization is generally proportional to the superfluid density in the first order in 1 /␬2_{共see Ref.}₂₆_{兲, while for arbitrary LLL}

configuration␺, the electric current can be written as J⬀共zˆ ⫻ⵜ兲兩␺兩2_{. This relation does not hold for higher Landau}

lev-els共LLs兲. Experimentally, however, while magnetization is continuous across the line, the conductivity diverges. This is known to happen in a wide range of materials and

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param-eters. Compare, for example, a recent experiment27_on

mag-netization in BSCCO and an earlier transport measurement in the same material and the same range of parameters.28The fact that near the glass line contribution from higher LL 共HLL兲 takes over removes this difficulty of the LLL theory. The exponentially small vortex creep is not included共it cor-responds to instanton contribution in the formulation adopted兲, and therefore, is not general in this respect.

The main result of the present paper is the calculation of the I-V curves in a wide range of parameters. The I-V looks often as a shifted straight line, rather different from the smooth power-like behavior seen in many experiments. However, as mentioned above, it is consistent with recent measurements in which the edge contribution was mini-mized.13,14,29_{In addition, we provide an alternative theory of}

the peak effect in the critical current. A sharp increase of the critical current is not considered as gradual, due to softening of the vortex lattice before melting, but rather an abrupt jump into the homogeneous vortex glass state, in which we obtain large current diminishing fast while approaching the glass line.

The paper is organized as follows. The disordered TDGL model at a constant electric field is defined in Sec. II. The Martin-Siggia-Rose formalism is applied to it using the basis of Landau levels in electric field. In Sec. III, we briefly de-scribe a variant of the Gaussian approximation functional integral method30,31 _{and apply it to the MSR action for the}

LLL sector. We find the dynamical glass transition surface and calculate various critical exponents. The stationary cor-relations in the pinned phase are also found. It is shown in Sec. IV that the main contribution to the current near the glass transition line comes from higher Landau levels, and their leading contribution is calculated. The rest of the paper deals with applications of the obtained general results to ex-periments. The irreversibility line and the I-V curves are con-sidered in Sec. V, while the critical current and the theory of the peak effect are discussed in Sec. VI. A brief discussion of the applicability of the theory and conclusions are the sub-jects of Sec. VII.

II. TIME DEPENDENT GL MODEL IN THE PRESENCE OF BOTH DISORDER AND

THERMAL FLUCTUATIONS A. Basic equations and assumptions

Our starting point is the TDGL equation26_{in the presence}

of thermal fluctuations, which on the mesoscopic scale are represented by a共complex兲 white noise21,24␨:

ប2_␥

4m*D␶␺= − ␦

␦␺*F +␨, 共1兲

where m* _{is the effective mass of the Cooper pair. The}

co-variant time derivative is D_␶⬅_⳵␶⳵ +ie_ប*⌽, where ⌽ is the sca-lar electric potential describing the driving force in a purely dissipative dynamics. We assume that the charge of the Coo-per pairs is negative −e*_{, with positive e}*_{= 2e. The inverse}

diffusion constant␥/ 2 controlling the time scale of dynami-cal processes via dissipation is real, although a small

imagi-nary共Hall兲 part is generally present.32 _{The variance of the}

thermal noise␨determines the temperature T = tTc:

具␨共r,␶兲␨*_共r

_⬘

⬘

2兩␺兩 4

册

_. _共3兲

兩␺兩2␺+␨, 共5兲

where a共non-Hermitian兲 linear operator is defined by Lˆ⬅ ប 2_␥ 4m*D␶− ប2 2m*D 2₋ ប 2 2mc* ⳵z 2 −␣Tc共1 − t兲. 共6兲

We make several assumptions共identical to those made in Ref.21 and major parts of Ref.4兲 to simplify the problem.

As was discussed in the Introduction, in strongly type-II su-perconductors, ␬=␭/␰Ⰷ1; magnetic and electric fields are very homogeneous since fields of vortices overlap. Therefore the Maxwell type equations for electromagnetic field are not considered. The axes are chosen in such a way that the mag-netic and electric fields are oriented along the negative z direction and the negative y direction, respectively. The vor-tices are moving along the positive x direction due to the Lorentz force. The electric and magnetic fields E = −ⵜ⌽ −_⳵t⳵A; B =ⵜ⫻A are written in the Landau gauge, with the

vector potential A =共By,0,0兲 and the scalar potential ⌽ = Ey, respectively. Temperatures, currents, and magnetic fields should be close “enough” to the dynamical phase tran-sition line Hc2共T,E兲 in order to apply the GL approach based

on an assumption that the order parameter␺is small.

B. Martin-Siggia-Rose formalism for TDGL model A Langevin type dynamics can be formulated as a func-tional problem with the dynamical “partition function,”

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de-fined by the MSR functional integral19_{over the order}

param-eter ␺ and an additional “ghost” field33 _␾_{. The ghost field}

allows exact integration over the white noise␨. As was no-ticed by Sompolinsky and Zippelius,20 _{who considered a}

similar model in the context of the spin glass theory, this allows one to exactly average over disorder by performing a Gaussian integration over the random field U共x兲 correlated according to Eq.共4兲 without invoking the replica trick:

Z =

冕

D␺*D␺D␾*D␾exp兵− AMSR关␺,␾兴其. 共7兲

Since共a rather nontrivial兲 derivation is closely analogous to the one presented in detail in Ref. 20, with certain aspects further clarified in Ref. 34, we just presented the resulting actionAMSR关␺,␾兴. This choice is more convenient for

per-forming Gaussian approximation and does not lead to any mathematical complications.

The functional approach enables us to calculate both the dynamics correlators and response function of the system in close analogy to the calculation of the static correlators in statistical physics. For example, the dynamical correlator and the response function are

in Eq.共10兲 appears due to thermal noise averaging, while the

last one represents a functional Jacobian arising from nor-malization of the ghost field integration.19,34_{The two quartic}

parts describe the disorder and the interactions, respectively.

In the theoretical part of this paper, we will use appropri-ate units of time␶

⬘

=␶/␶GLwith characteristic GL time scale

␶GL=␥␰2 and coordinates x

⬘

= x / l, y

⬘

= y / l, z

⬘

= z /␰z, where

magnetic length is l =␰b−1/2 with b = B / Bc2. It is convenient

to combine the fields in a dimensionless two component col-umn ⌿ =

冉

␣l2␰z t

冊

冕

r␶ 关⌿r*共␶兲␴↑⌿r共␶兲兴关⌿r*共␶兲␴x⌿r共␶兲兴. 共13兲

Here, we introduced two dimensionless couplings n = np

4␲

冑

2Gi 共1 − t兲2

t , 共14兲

characterizing the relative strength of disorder compared to interactions, and

g = 8␲bt

冑

2Gi, 共15兲

characterizing the interactions compared to thermal fluctua-tions with

冑

2Gi =₄_␲␣b2_T⬘

c␰2␰z. The inverse propagator matrix is D−1=

冉

− 2g Lˆ

+

Lˆ − 1

冊

, 共16兲

where the operator Lˆ of Eq.共6兲 has the form

Lˆ =1 2共⳵␶+ 4ibvy兲 − b关共⳵x− iy兲 2₊_⳵ y 2_{兴 −}_⳵ z2−共1 − t兲, 共17兲 with velocity of fluxons

v =e

*_␥_El3

4ប 共18兲

given in units of c. We make use of Pauli matrices␴x,␴y,␴z,

and␴_↑,↓=1₂共1±␴z兲.

Since the spectrum of the operator Lˆ is discrete共the Lan-dau quantization兲, in a certain range of parameters, one can significantly simplify the problem by considering the “low energy” states only. It is therefore advantageous to reexpress the model in the Landau level basis.

C. Landau level basis in the presence of electric field The moving Landau harmonics are solutions of linearized time dependent GL equation in the presence of electric field.

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Mathematically, we define them as the right eigenfunctions of the operator Lˆ, Eq. 共17兲, with eigenvalues L_Nk,k

z:

Lˆ共e−i␻␶_␸

Nk,k_z兲=LNk,k_z共e−i␻␶␸Nk,k_z兲, where

␸Nk,k_z= 1

冑

2NN!␲1/2e −v2/2_H N共y − k + iv兲exp关ikzz兴 ⫻exp关ikx兴exp

冋

−1 2共y − k + iv兲 2

册

_, _共19兲

and HN共y兲 are Hermit polynomials. The basis functions are

normalized as

冕

r

␸

¯Nk⬘共r兲␸Mk共r兲 = 共2␲兲2␦k⬘k␦N,M 共20兲

and␦k⬘k is the 2D␦ function with k =兵k,kz其. Since the

op-erator is not Hermitian, they are different from the left eigen-functions ␸¯_Nk,k

z defined as Lˆ

†_共ei␻␶_␸_¯

Nk,k_z兲=LNk,k* _z共ei␻␶␸¯Nk,k_z兲,

see Ref.24.

The order parameter and the ghost fields are expanded, therefore, via the moving Landau harmonics as

⌿r共␶兲 =

1 共2␲兲3/2

兺

N=0

冕

␻k

e−i␻␶␸Nk共r兲⌿Nk,␻. 共21兲

We will use in Sec. III the LLL subspace, although higher levels will be necessary for the calculation of supercurrent and will be considered in Sec. IV. It will be convenient to separate the presumably more important LLL subspace from the rest, the HLL part:

⌿r共␶兲 = ⌽r共␶兲 + ⌰r共␶兲, 共22兲

where⌽_k,_␻⬅⌿_0k,␻ and ⌰_k,_␻⬅兺_N=1⌿_Nk,_␻. As will become clear in Sec. IV, it will be sufficient for our purposes to consider effects up to first order in⌰r共␶兲, and now, we

con-centrate on the LLL sector.

In the LLL sector, the quadratic part of the action takes the form Afree=

冕

k,␻ ⌽k,*␻Dk−1␻⌽k,␻, Dk−1_␻=

冢

− 2g i␻

⬘

2 + kz 2 + 2ah − i␻

⬘

2 + kz 2_{+ 2a} h − 1

冣

, 共23兲

where ␻

⬘

=␻− 4bvk and ah= −共1−t−b−bv2兲/2 is the

“dis-tance” on the dynamical phase diagram from the mean field normal-superconductor transition line. The quartic terms are

Adis= − ng 共2␲兲5/2

冕

关k兴,␻,␯␦k1−k2+k3−k4 f0共关k兴兲共⌽k₁,␻ * _␴ x⌽k₂,␻兲 ⫻共⌽k₃,␯ * _␴ x⌽k₄,␯兲, 共24兲 Aint= g 2共2␲兲7/2

冕

关k兴,关␻兴␦␻1−␻2+␻3−␻4 k₁−k2+k3−k4 _f 0共关k兴兲共⌽k₁,␻₁ * _␴ ↑⌽k₂,␻2兲 ⫻共⌽k₃,␻₃ * _␴ x⌽k₄,␻₄兲, 共25兲

where关k兴 and 关␻兴 denote a set of all variables 兵k1, k2, k3, k4其

and 兵␻1,␻2,␻3,␻4其, respectively, and ␦_␻₁−␻2+␻3−␻4

k1−k2+k3−k4 _{is a}

product of Dirac ␦ functions ␦共␻1−␻2+␻3−␻4兲␦共k1− k2

+ k3− k4兲. The Gaussian damping factor f0共关k兴兲

= exp

关

−共k2−k1兲

2_+共k 4−k1兲2

2

兴

arises from matrix elements of four

LLL functions.

III. CONSISTENT GAUSSIAN APPROXIMATION FOR THE MSR ACTION

Since the model is highly nontrivial even in the simplest cases, one has to use an approximation scheme. We utilize a method which evolved from the Gaussian variational ap-proach to quantum mechanics, referred to here as Gaussian approximation.31_{Generally, it captures the basic physical}

as-pects, although its precision might not be very high共perhaps similar to a dynamical mean field theory in the band struc-ture calculations兲.

Since we will calculate higher correlators using well known, sometimes inconsistently, Gaussian approximation, we start with a brief description of the general method intro-duced in detail in Ref. 30, utilizing the so-called Gaussian effective action and “truncated” Dyson-Schwinger equations. Naively using the Hartree-Fock procedure 共Wick contrac-tions兲 to calculate correlators leads to several important in-consistencies. For example, approximate correlators of Gold-stone bosons are not massless, contrary to the GoldGold-stone theorem. This is crucial in calculating effects dependent on massless modes like the divergence of conductivity on the glass line. An additional advantage of this approach over the resummation of the diagram technique used in Refs.1and23

共borrowed from the physics of weak localization兲 is that it is systematic and unambiguous, without any reference to the “large number of components” limit.

A. Gaussian effective action and correlators

The exact effective action A共⌿¯ ,G兲 introduced by Corn-wall et al.35 _{is defined as the functional of two variational}

parameters, a “classical”共or shift兲 field ⌿¯

A=具⌿A典 = Z−1

冕

⌿,⌿*⌿Aexp兵− AMSR关⌿,⌿

*_{兴其共26兲}

and a two-field connected correlator GAB=具共⌿A−⌿¯A兲共⌿B * −⌿¯B *_兲典 = Z−1

冕

⌿,⌿*共⌿A−⌿ ¯ A兲共⌿B * −⌿¯B *_{兲exp兵− A} MSR关⌿,⌿*兴其. 共27兲 Here, indices of the field⌿ stand for the full set of variables and parameters: A =兵␻, k , i其, where i=1,2 is the ⌿ column

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index. Minimization of the effective action yields the “shift” and the “gap” equations

␦A共⌿¯ ,G兲 ␦⌿¯ A = 0, ␦A共⌿¯ ,G兲 ␦GAB = 0, 共28兲 respectively.

In the Gaussian approximation,A共⌿¯ ,G兲 is approximated 共up to an unimportant constant兲 by

A共⌿¯ ,G兲 ⯝ AG共⌿¯ ,G兲 = Afree关⌿¯ 兴 − Tr log G + Tr兵D−1G其

+具Adis典G+具Aint典G, 共29兲

whereAfree关⌿¯ 兴 is the quadratic part of AMSR关⌿¯ 兴 action in

Eq.共10兲 and 具Adis典G, 具Aint典Gare calculated as the Gaussian

averages with the field shifted by⌿¯ :

具O典G=

冕

⌿,⌿* O关⌿,⌿*_{兴exp关− 共⌿}*₋_⌿¯*_兲G−1_{共⌿ − ⌿}_{¯ 兲兴}

冕

⌿,⌿* exp关− 共⌿ *₋_⌿_¯*_兲G−1_{共⌿ − ⌿}_{¯ 兲兴} .

This allows calculation of correlators of the fields via func-tional differentiation with respect to the shift fields. In the rest of this section, we will solve the gap equations for the MSR action, while correlators共including the four-point func-tions兲 are considered in Sec. IV.

B. Application to the LLL model: The gap equation The Gaussian approximation has been applied to the LLL time dependent Ginzburg-Landau model in the absence of electric field using diagram resummation in Ref.1. Here, we use the Gaussian effective action approach leading to an identical gap equation. Assuming that the gauge 共electric charge兲 U共1兲 symmetry is unbroken, the one-field averages 具␺典, 具␾典 and the “charged” two-field cor-relators like 具␺␺典 should vanish. The invariant 共“neutral”兲 two-point Green functions include the correlator C共r,␶,

_⬘

_{兲典, and the auxiliary field correlator}

冉

C_␻k R_␻k

R_␻k* B_␻k

冊

, 共30兲 where ␦_␻␻_⬘⬅␦_␻−␻_⬘ and ␦kk⬘=␦k−k⬘ are the one- and

two-dimensional Dirac delta functions ␦共␻−␻

⬘

兲 and ␦共k − k

⬘

兲␦共kz− kz

⬘

兲, respectively.

Making use of the above relation, the quartic terms in Eq. 共29兲 become 具Adis典g= − ng 共2␲兲5/2␦␻␦k

冕

关k兴,关k⬘兴,␻ fR共k

⬘

− k兲Tr共␴xG_␻k␴xG_␻k⬘兲, 共31兲 具Aint典g= g␦_␻␦k 2共2␲兲7/2

冕

_关k兴,关k_⬘_兴,␻,␯fR共k

⬘

− k兲关Tr共␴↑G␻k␴xG␯k⬘兲 + Tr共␴↑G␻k兲Tr共␴xG␯k⬘兲兴, 共32兲 where ␦_␻=_2␲1兰_␶ and ␦k= 1

共2␲兲2兰r are infinite constants, and

fR共k兲=exp

关

− k2

2

兴

is a reduced version of the damping factor

f0共关k兴兲 due to the momentum conservation. In the disorder

term, we omitted the Wick contractions within the curly brackets in Eq.共24兲, since they are canceled exactly by the

disorder “counterterms” in the MSR action, see Ref.34. Cal-culating the functional derivative ofAg共⌽¯ ,G兲, the gap

equa-tion is written in a well known form GAB −1 = DAB −1 + MAB, MAB= ␦ ␦GAB

共具Adis典gauss+具Aint典gauss兲.

Note that GABis a symmetric function with respect to

trans-position of all its indices. Substituting the matrix MAB as a

function of GABinto the gap equation, we arrive at

G_␻k−1= D˜_␻k−1− 2ng 共2␲兲5/2

冕

关k⬘兴 fR共k

⬘

− k兲␴xG␻k⬘␴x 共33兲 with D˜_␻k−1= D_␻k−1+ g 共2␲兲7/2

冕

关k⬘兴,␯ fR共k

⬘

− k兲 ⫻关␴↑Tr共␴xG␯k⬘兲 +␴xTr共␴↑G␯k⬘兲兴. 共34兲

Here, one clearly sees the difference between the disorder and the interaction contributions in the Gaussian approxima-tion. The interaction term just renormalizes the quadratic term in the action, while the disorder term is both frequency and wave vector dependent. We make use of this observation to simplify the gap equation.

1. Simplification of the gap equation for response function and correlator

To allow a solution of the gap equation in a closed form, we expand correlation functions共of k

⬘

兲 in Eqs. 共33兲 and 共34兲,

␴xG_␻k⬘␴xand ␴_↑Tr共␴xG_␯k⬘兲+␴xTr共␴_↑G_␯k⬘兲, near the point

k

⬘

= k and retain only the first two terms. The correlation function depends analytically on electric field, and the lead-ing correction ⳵

2_G ␯k

⳵k2 is proportional to the square of the

elec-tric field. Furthermore, one can neglect it due to the typical smallness of the velocity parameter关defined in Eq. 共18兲兴, v

Ⰶ1. This approximation does not mean that we will not be interested in important nonanalytic nonlinearities later on. For justification of the above statement, one observes that, according to Eq. 共23兲, the inverse propagator D_␻k−1 depends only on the combination␻

⬘

=␻− 4bvk of␻and k. Since in the zero order of the expansion all the dependencies on ␻

(7)

and k come from D_␻k, the exact Green function, G_␻k, also depends on␻

⬘

only and, therefore, ⳵

2_G ␻k ⳵k2 ⯝v2 ⳵2_G ␻_⬘kz ⳵␻⬘2 . Here, we

have used the fact that G_␻_⬘k_zand its second derivative are of

the same order inv共see the solution obtained below兲 and are analytic. We also have solved the full equation numerically by iterations and verified that the errors are exponentially negligible. In this approximation, the gap equation is trans-formed into the following form 共in what follows, we drop “primes” in␻

⬘

using notation G_␻k

zfor G␻k= G␻−4bvk,kz兲: G_␻k z −1 _{= D}_˜ ␻kz −1 ₋ 2ng 共2␲兲2

冕

k_z⬘ ␴xG␻k z ⬘␴x, 共35兲 D˜_␻k z −1 = D_␻k z −1 + g 共2␲兲3

冕

k_z⬘,␯ 关␴↑Tr共␴xG␯k z ⬘兲 +␴xTr共␴_↑G␯k z ⬘兲兴. 共36兲 Equation 共35兲 is satisfied for vanishing correlator of the

ghost fields, B_␻= 0, because the Eq.共11兲 matrix elements in

both G_␻k

z

−1 _and_␴

xG␻k_z⬘␴xare proportional to B␻as it should be

generally.34 _{The correspondent matrix element of D}˜

␻kz

−1 _is

equal to zero if the following condition is valid: 1 共2␲兲3

冕

␻kz Tr共␴xG_␻k_z兲 = 1 共2␲兲3

冕

␻kz 共R_␻k_z+ R_␻k z * _{兲 = 2.} 共37兲 The above self-consistency condition should be checked for a solution of the gap equation. The dependence on momen-tum along the magnetic field direction z is rather trivial and will be treated first.

2. Integrating out the direction along the magnetic field

As a next step, we factor out the kzdependence. Note that

the last term in Eq.共35兲 and the interaction correction in D˜_␻k

z

−1

are both kzindependent, so that the dependence of G_␻k

z

−1

on kz

is determined, according to Eq.共23兲, solely by an additive kz

dependent term in D_␻k

z

−1_{. Therefore, the solution has the form}

D˜_␻k z −1 _{= D}_˜ ␻ −1_{+ k} z 2_␴ x, G_␻k z −1 _{= G} ␻ −1_{+ k} z 2_␴ x, 共38兲

in which a matrix valued function of one variable␻, G_␻= G_␻k

z=0⬅

冉

C_␻ R_␻

R_␻* 0

冊

, 共39兲

remains to be determined. In terms of elements of G_␻, the correlators integrated over kzare given by

1 ␲

冕

k_z G_␻k z=␴x共␴xG␻兲 1/2₌

冢

C_␻ R_␻1/2+ R_␻*1/2 R␻ 1/2 R_␻*1/2 0

冣

. 共40兲

Substituting Eq.共38兲 into Eq. 共35兲, we arrive, after

inte-gration over kz,

G_␻−1= D˜_␻−1− ng 2␲共␴xG␻兲

1/2_␴

x, 共41兲

at a cubic equation for a matrix variable共␴xG␻兲1/2:

ng 2␲共␴xG␻兲

3/2_{− A}

␻共␴xG␻兲 + 1 = 0. 共42兲

Here, the matrix

A_␻= D˜_␻−1␴x=

冢

i␻ 2 + 2aint 0 − 1 − i␻ 2 + 2aint

冣

, aint= ah+ g 共2␲兲3

冕

␻kz C_␻k z, 共43兲

as well as ␴xG␻ and its powers in Eq. 共42兲 are triangular

matrices with zero关12兴 component. The 关22兴 component of Eq.共42兲, ng 2␲R␻ 3/2 −

冉

− i␻ 2 + 2aint

冊

R␻+ 1 = 0, 共44兲 determines R_␻ in terms of aint, while the 关21兴 component

expresses C_␻ via R_␻: C_␻= R␻R␻ *_共R ␻ 1/2_{+ R} ␻ *1/2_兲 R_␻1/2+ R_␻*1/2− ng 2␲R␻R␻ * . 共45兲

The remaining关11兴 component yields the complex conjugate equation to the first one, Eq.共44兲. The constant aint, in turn,

depends exclusively on C_␻, so the loop closes. It is important to emphasize that the only effect of interaction in the gap equations共44兲 and 共45兲 is to renormalize ah upward to aint.

This allows one to consider the excitations of the vortex liquid for negative ahsince aint is always positive.3

2/3 R_␻, 共46兲

and define a new parameter, the conventional LLL scaled temperature3,17 _a

T= 2共8␲/ g兲2/3ah. In this variable, the gap

equation for the correlation functions is reduced to 4r␳_␻3/2−

冋

aT+ 4 2␲

冕

_␯ c_␯ ␳␯1/2+␳␯*1/2 − i 2

冉

8␲ g

冊

2/3 ␻

册

␳␻+ 1 = 0, 共47兲 c_␻=

冉

8␲ g

冊

2/3 _␳ ␻ *_␳ ␻共␳␻1/2+␳␻*1/2兲 ␳␻1/2+␳␻*1/2− 4n␳␻*␳␻ , 共48兲

where g was defined in Eq.共15兲. In a certain range of the

(8)

does not alter the vacuum structure significantly, the solution can be found assuming the validity of the dissipation-fluctuation theorem 共DFT兲, which subsequently can be checked by substitution back into the gap equation. In a more complicated “glass” phase, the DFT is violated and a solu-tion is found later using a different method.

C. Solution of the gap equations, the glass line, and critical exponents

1. Unpinned phase

Let us assume the validity of the DFT,

冕

v

Cq_z_␯= 2␲Rq_z_␻=0,

which, after integration over qz, yields

1 2␲

冕

_␯

c_␯ ␳␯1/2+␳␯*1/2

=␳₀1/2, 共49兲

with␳0 defined as ␳0⬅␳␻=0. Combining the above relation

with Eq. 共47兲 at␻= 0, one obtains a cubic equation for ␳0

共which is a function of external parameters r and aT兲:

− 4共1 − n兲␳03/2− aT␳0+ 1 = 0. 共50兲 The solution is ␳0共aT兲 = − aT+ aT 2 d−1/3+ d1/3 12共1 − n兲4/3 , d = aT 3_{+ 12共1 − n兲}

冑

_{324共1 − n兲}2_{− 3a} T 3_{− 216共1 − n兲}2_. 共51兲 In what follows, we will need its asymptotic form for large negative aTin the clean limit, n→0,

␳01/2= −

aT

4 . 共52兲

Further, solving Eq.共50兲, one can substitute the solution

into Eq.共47兲 to obtain a closed form cubic equation for the

frequency dependent response function␳_␻, 4n␳_␻3/2− 4␳₀1/2␳_␻− aT

冉

1 −

i␻ 4ah

冊

␳␻+ 1 = 0. 共53兲

It should be noted that, according to Eq.共48兲,␳_␻determines the correlator c_␻completing the solution of the set of the gap equations. Obviously, both the response function and the cor-relator can be written in an explicit form. A characteristic shape of the correlators is shown in Fig.1. We also made use of these explicit functions for a numerical check of DFT. However, below we are mainly interested in analytic proper-ties of the correlation functions, which can be investigated without using the explicit expressions.

The result for the integral over frequency of the correlator coincides with the static correlator calculated using the rep-lica formalism in Gaussian approximation.15,16_{As was}

dis-cussed in these papers, n⬎1 corresponds to a case in which

disorder overpowers “repulsive” interaction and destabilizes the system.

2. Glass line and critical behavior

The critical surface in the “space” of dimensionless scaled parameters共t,b,v兲 is defined as a set of values of the param-eters for which the correlator C_␻at␻= 0 diverges. The static glass line is a line on this surface for zero electric field, ␯ = 0. We will argue later that below this line the supercon-ductor acquires certain “glassy” properties.

According to the gap equation, Eq. 共48兲, the correlator

c_␻=0becomes infinite when ␳0=

1

共2n兲2/3. 共54兲

Here, ␳0 as a function of the external parameters 共r,aT兲 is

given by Eq. 共50兲. Excluding ␳0, one arrives at a simple

formula for the glass surface: aT

g

=共2n兲2/3

冉

3 −2

n

冊

, 共55兲

which, in terms of scaled parameters共t,b,v兲, is equivalent to bvg 2 = 1 − t − b +

冉

ng 4␲

冊

2/3

冉

3 −2 n

冊

. 共56兲

The critical velocity vg determines the critical electric field

Eg, which destroys the glassy state, forcing vortices to move

in a direction perpendicular to the field. The static glass line is described by Eq. 共56兲 for zero electric field, vg= 0. This

result is consistent with the one obtained using the replica method16in a similar Gaussian approximation.

Note that for n =2₃, the glass line coincides with the dy-namic superconductor-normal metal transition line4 _a

h= 0,

whereas for a stronger disorder, 2₃⬍n⬍1, it lies in the nor-mal state, and for n艌1, the Gaussian effective action be-comes unstable, signaling breakdown of the approximation. Similar phenomena occur in the␾4 _theory.30,31 _{In the static}

共B-T兲 phase diagram, the glass line, Bg共T兲, begins from the

point共B=0, T=Tc兲, with zero derivative, dBg共Tc兲

dT = 0共see Fig. FIG. 1.共Color online兲 A typical form of the dimensionless cor-relator共the brown solid line兲, and the real 共the blue dash line兲 and imaginary共green dot line兲 parts of the response function above the glass transition.⌬⬎0 are shown as a function of frequency␻.

(9)

2兲, and crosses Bc2共T兲 line at some intermediate temperature.

The expansion of the correlation functions near the glass line in a small parameter

⌬T= aT− aT g

=共8␲/g兲2/3⌬ 共57兲 and a small frequency leads to the following critical behav-ior: ␳␻=_共2n兲12/3

冋

1 − 共2n兲1/3 2

冑

1 − i␶rel␻⌬T

册

, 共58兲 c_␻=

_冑

␶rel 2共2n兲1/3兵1 + 关1 + 共␶rel␻兲 2_兴1/2_其−1/2_⌬ T, 共59兲

where the characteristic time, ␶rel= 8 3⌬T2

冉

2␲ n2g

冊

2/3 , 共60兲

determines a long scale decay of the correlator.

On the critical surface, where⌬T= 0 and␶reldiverges, the

correlator and response function, ␳_␻g = 1 共2n兲2/3

冋

1 − 2e−i␲/4

冑

6

冉

4␲ ng

冊

1/3 ␻1/2

册

_, _共61兲 c_␻g=

冑

1 3

冉

8␲ g

冊

1/3₁ n␻ −1/2_, _共62兲

both have a fractional power dependence on␻. One, there-fore, observes criticality with exponent 1₂. In the static limit near transition, the response function

The expression for␳_␻is given by a solution of Eq.共53兲 with

␳0=共2n兲−2/3, 4n␳_␻3/2− 4共2n兲−1/3_␳ ␻− aT

冉

1 − i␻ 4ah

冊

␳␻+ 1 = 0. 共67兲

The regular component of the correlator in the␶space, there-fore, decreases as a power C共␶兲⬀共␶/␶GL兲−1/2.

Having completed the solution of the gap equation, we now use it to calculate physical quantities other than the correlator.

IV. dc SUPERCURRENT A. LLL contribution

In the homogeneous phase, the current density after dis-order averaging is independent of both␶and r. The average supercurrent density, therefore, is 共in the geometry consid-ered in this paper, the current flows in the y direction兲

jLLL⬅ JLLL J

⬘

= − i 2 共2␲兲3 VT

冕

_␶r具⌽r *_共_␶_兲_␴ +⳵y⌽r共␶兲 −关⳵y⌽r *_共_␶_兲兴_␴ +⌽r共␶兲典 = v

冕

␻kz C_␻k z, 共68兲

where具¯典 denotes both thermal and disorder averages, and J

⬘

=_共2␲兲បe3*_m*_l t ␣l2_␰ z= 2b1/2_g 共8␲兲4J0 with J0= e* 冑_2Gi 16Tc␥a

ប␰2 , using the

solu-tions of the gap equation and integrating over kz. In the liquid

and glass phases, the supercurrent is given by jLLL liq = 2␲

冉

8␲ g

冊

1/3 ␳0 1/2 v, 共69兲 jLLL glass = 2␲

冉

8␲ g

冊

1/3

冋

1 共2n兲1/3+ aT g − aT 4

册

v, 共70兲

respectively. Here,␳0 is a solution to Eq.共50兲 given in Eq.

共51兲.

Therefore, within the LLL approximation, the supercur-rent is proportional to velocity times an expression analytic in velocity. Consequently, even at criticality, there exists fi-nite conductivity jLLL

liq

/v = 2␲

共

4_ng␲

兲

1/3, which is determined solely by disorder on the mesoscopic scale ng. The Ohmic behavior implies that there apparently is flux flow in both liquid and glass phases. On the glass transition, the LLL conductivity is continuous, although not a smooth function. In the glass state, we clearly see that there is no expected

vortex pinning and the conductivity is finite. On the liquid side, despite the critical divergencies in correlation function discussed in Sec. III C, the conductivity does not diverge, again in contrast to experiment. The only piece of physics that the LLL approximation is able to capture is the Bardeen-Stephen flux flow conductivity far from the glass transition line and the fluctuation conductivity in normal phase.18_{In the}

clean limit32 _{共no disorder on the mesoscopic scale兲, n→0}

with␳₀1/2= −aT

4 关see Eq. 共52兲兴 and one obtains

␴LLL=

JLLL

E =␴n

1 − t − b

b , 共71兲

where ␴n is conductivity in the normal state. The full

for-mula provides a disorder correction.

As was mentioned in the Introduction, it is quite clear physically why the LLL approximation contains the Ohmic contribution only. The current generally is proportional to a gradient of the superfluid density, which, in turn, is propor-tional to the electric field. The pinning effects, therefore, ap-pear due to higher Landau levels only. We, therefore, gener-alize the discussion to the lowest states at which the pinning appears.

B. Higher Landau level contribution to current The argument above leading to an Ohmic dependence of the current on electric field in LLL共confirmed by the direct calculation in the previous section兲 is not valid already for the first LL, N = 1: the current is no longer a curl of the superfluid density. It is natural to assume that the HLL cor-rections to fields in Eq. 共22兲 can be considered as a

pertur-bation. Furthermore, to leading order in the perturbation, a nonzero contribution to current comes solely from the first LL共1LL兲: j = jLLL+ jHLL, jHLL= − i 2 共2␲兲3 VT

冕

r␶ 具⌽r *_共_␶_兲_␴ +⳵y⌰r共␶兲 −关⳵y⌰r *_共_␶_兲兴_␴ +⌽r共␶兲 + c.c.典. 共72兲

This is due to the fact that the operator⳵ycontains just one,

raising or lowering the Landau number operator共in addition to a constant proportional to v兲. Thus one needs the wave function up to the first LL only,

⌿r共␶兲 = ⌽r共␶兲 + ⌰r共␶兲,

and considers it to first order in⌰r共␶兲.

In this section, we use a simplified model compared to that used in the derivation of the gap equation in the previous section. The difference consists in the application of the steepest descent approximation 共similar to that which was utilized and discussed in Sec. III B兲 already in the MSR ac-tion. As in Sec. III B, due to the Gaussian damping factor in quartic terms 关see Eq. 共24兲兴, we neglect all the terms

off-diagonal in momenta k both in the LLL and the HLL contri-butions:

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Afree=

冕

k,␻ 关⌽k,␻ * Dk␻ −1_⌽ k,␻+⌰k,␻ * D_1k␻−1 ⌰k,␻兴, Adis= − rg 共2␲兲2␦␻␦k

冕

k,关kz兴,␻,␯ ␦k_1z−k_2z+k_3z−k_4z共⌽k,k_1z,␻ * _␴ x⌽k,k_2z,␻兲 ⫻共⌽k,k* _3z,␯␴x⌽k,k_4z,␯兲 + ⌬Adis, ⌬Adis= − 23/2iv rg 共2␲兲2␦␻␦k

冕

k,关kz兴,␻,␯ ␦k_1z−k_2z+k_3z−k_4z ⫻关共⌽k,k* _1z,␻␴x⌽k,k2z,␻兲共⌽k,k3z,␯ * _␴ x⌰k,k4z,␯兲 − c.c.兴. 共73兲 Here, the inverse propagator for 1LL is

D_1k␻−1 =

冢

0 i␻

⬘

2 + kz 2_{+ 2a} h 1 − i␻

⬘

2 + kz 2 + 2ah 1 − 1

冣

, 共74兲 where ah 1

= ah+ b is the large “mass” of the first LL

excita-tions. The coefficient in the disorder term is chosen in such a way that the LLL gap equation obtained from this action coincides with Eq. 共35兲. Physically, this assumption is

equivalent to replacing an approximate Landau degeneracy by an exact one. In addition, we ignore the interaction term since, as was noted in Sec. III C, the interaction effects can be accounted for by renormalization of the free part of LLL action, where ah is replaced by aint.

The correction term⌬Adis is already proportional to the

electric field via the factorv due to the fact that the integral of a product of three LLL Landau harmonics and one 1LL harmonic, Eq. 共19兲, vanishes at zero electric field. Since

Afreedoes not “mix” LLL with 1LL, the leading contribution

to jHLL, Eq.共72兲, will be of the second order in HLL ⌰ and

will come from all the connected diagrams proportional to ⌬Adis.

The HLL contribution to the current from the disorder part is jHLL= 2ng共1 + 2v2_兲共2␲兲2_a h 1 v

冕

k_关k_z兴␻␯ ␦k_1z−k_2z+k_3z−k_4z具共⌽kk_1z␻ * _␴ x⌽kk_2z␻兲 ⫻共⌽kk_3z␯ * _␴ ↑⌽kk_2z␯兲典. 共75兲

One, therefore, faces a problem of a consistent calculation of the four-point Green’s function. Diagrammatically, within the linear response theory, it was approximated by a resum-mation of the “diffusion” and “Cooperon” chains.1 _We

cal-culate this function using a systematic approach by differen-tiating four times the Gaussian effective action defined in Sec. II.

Factorization of the action and of the current with respect to index k leads to proportionality of all physical quantities to the Landau degeneracy. Dropping below the index k and denoting kzsimply by k, one can express the HLL current as

jHLL=

2ng共1 + 2v2_兲

共2␲兲2_a h

1 v␴HLL, 共76兲

with the HLL contribution to the conductivity, ␴HLL=

冕

兵prqs␻␯其 ␦p+q−r−q2␴x u₁,v1␴ ↑ u₂,v2具⌽ p_␻ *u1⌽ q_␯ *u2⌽ r_␻ v1⌽ s_␯ v2典 c, 共77兲 proportional to the connected part of the four-point correla-tion funccorrela-tion.

C. Four-point correlators

The calculation of the four-point correlators is quite lengthy, so we first describe the general structure of the terms appearing in it. The gap equation derived in Sec. III B for Green’s function GAB=具⌽A⌽B

*_{典共where indices A and B}

de-note the full index set of the order parameter兲 can be gener-alized to include the rest of Green’s functions FAB=具⌽A⌽B典

and FAB *

=具⌽A *_⌽

B

*_{典. In U共1兲 symmetric theories, like the one}

we consider, the last propagators are identically equal to zero due to the symmetry. However, their functional derivatives do contribute to equations for the higher vertex functions and will be necessary for our discussion.

G regarded as independent of the shift fields⌽¯ . Summation over repeated indices is assumed. The equations for a four-point vertex function are derived from Eq.共78兲 by

differen-tiating it twice with respect to the shift field ⌽¯ . The field dependence of the propagators, G关⌽¯ 兴 or F关⌽¯ 兴, determined by the gap equation共with external “source” present兲, should be taken into account. It provides the “chain” parts propor-tional to ␦G

␦⌽¯, ␦F

␦⌽¯, ␦2_G

␦⌽¯_␦_⌽¯ , . . .. From the U共1兲 symmetry, one can

infer that there are two nontrivial vertex contributions: the diffuson and the Cooperon parts defined by the ␦G

␦⌽¯_␦_⌽¯* and

␦F

␦⌽¯_␦_⌽¯ second derivatives, respectively.

The diffusion and the Cooperon equations can be obtained from Eq.共78兲, differentiating its 共11兲 component with respect

to⌽¯C₂ and⌽¯C₁ *

and differentiating the共21兲 component with