This note is concerned with gauge choices for the time-dependent Ginzburg-Landau equations of superconductivity. The requiations model the state of a superconducting sample in a magnetic field near the critical tempeature. Any two solutions related through a ``gauge transformation`` describe the same state and are physically indistinquishable. This ``gauge invariance`` can be exploited for analtyical and numerical purposes. A new gauge is proposed, which reduces the equations to a particularly attractive form

Fleckinger-Pelle, J.

Kaper, H.G.

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U S T I  The submitted manuzcript has been authored by a contractor of the U. S. Government under contract No. W-31-104ENG-38. Accordingly, the U. S. Government retains a nonexclusive. royalty-free Icense to publish or reproduce the published form of this contribution. or allow otherz to do ro, for I U. S. Government purposes. FLECKINGER-PELLE, JACQUELINE; KAPER, Hxxs G. Gauges for the  Ginzburg-Landau Equations of Superconductivity This note is concerned with gauge choices f o r  the time-dependent GINZBURG-LANDAU equations of superconductivity. The equations model the state of a superconducting sample in a magnetic field near the critical temperature. Any  two solutions related through a zgauge transformation” describe the same state and are physically indistinguishable. This ugauge invariance” can be ezploited fo r  analytical and numerical purposes. A new gauge is proposed, which reduces the equations to a particularly attractive form. 1. The Ginzburg-Landau Model of Superconductivity In the GINZBURG-LANDAU theory of phase transitions [I], the state of a superconducting material near‘ the critical temperature is described by a complex-valued order parameter $, a real-valued vector potential A, and, when the state changes with time, a real-valued scalar potential 4. The role of q5 differs from that of 7c1 and A: the latter are predictive variables, whose evolution is governed by dimerential equations; the former is more like a Lagrange multiplier. After suitable nondimensionalization, the equations and boundary conditions satisfied by + and A are usually presented in the form r )  (g + i r ) )  + = - (kv + A> ~ + (1 - I + [ ? )  $ in n x (0, m), (1) (2) (3) ( V x A - H ) x n  = 0 onaiEx(0,co). (4) - -V x v x A - -Q+*T+ - +vi*) - ~+I*A + v x H 8A -+v+ in SI x (0, m), 26 i --y+ on 852 x (O,co), (,.+:)- = n The domain 0 corresponds to  the region occupied by the superconducting material. We assume that S2 is bounded, 9 C DRD with D = 2 or  D = 3, and the boundary an of R is sufficiently regular; n denotes the outer unit normal to ail. The parameters of the model are r ) ,  a {dimensionless) friction coefficient; 6, the (dimensionless) GINZBURG- LAXDAU parameter; and y, a nonnegative parameter, which is zero if the superconducting material is surrounded by vacuum. The vector field H is a given applied magnetic f i e ld :  in practice, H is either timeindependent or time- periodic. As usual, V = grad, V x  = curl, V. z div, and V2 = V . V = A. Furthermore, i is the imaginary unit, and a superscript denotes complex conjugation. The system of Eqs. (1)-(4), with appropriate initial conditions, constitutes the tine-dependent GINZBURG-LANDAU (TDGL) model ofsuperconductrvzty. It was first proposed by SCHMID [2] and subsequently obtained as an s p p -  totic limit of the microscopic BARDEEN-COOPER-SCHRIEFFER (BCS) model of superconductivity by GoR’KOv AND ELIASHBERG [3]. Details can be found in the physics literature; standard references are ABRIKOSOV [4], DE- GENNES [SI, and TINKHAM [SI. In this article, we will work mostly with a rescaled version of the TDGL model, because the equations are somewhat simpler. Let u = l/(qn2). In the rescaled TDGL model, time is measured in units of 1/., the scalar potential in units of u/n, and the vector potential and applied magnetic field in units of l / ~ .  The model consists of the equations (8, + ib) r ~ ,  = - (iv + A)* 4 + n2 (I - \ $ I 2 )  in Q x (0, m), (5) - (’7) u ( & A + V + )  = -V x V x A +  J, + V  x H in SZ x (O,m), (6) (Vx A-H)xn = 0 onaRx(O,W) ,  (8) J, = (2i)-’ ($*V+ - $V+*) - 1$12A = -Re [4* (iV + A) $1. - ( i V + A ) $ - n  = -i-p+!~ on 8S2 x (O,OO), together with the appropriate initial conditions, Here, we have introduced the abbreviation at = a/%. Furthermore, (9) The quantity 3, is the so-called svpercurrent or, more correctly, supercurrent density. The supercurrent is a phe- s (or COOPER nomenological quantity, which is thought of as a flux of moving ‘‘superelectrons.n The s pairs), whose density is ns = ] $ I 2 ,  are responsible for the superconducting properties of the material. For example, the supercurrent prevents a magnetic field from penetrating a superconducting region. Note that E = -dA/dt - Vd is the electric f ield and B = V x A the magnetic induction. Therefore, Eq. ( 6 )  may be viewed as Faraday’s law, V x B = J,  where the total current J is the sum of the supercurrent J,, a “normal” current J, = aE, and the transport current Jt = V x H. The normal current obeys OHM’S law with a “normal conductivity’’ coefficient u. In the steady state, the TDGL model admits the trivial solution $ = 0, V x A = H. (Recall that the scalar potential vanishes in the steady state.) This solution represents the superconductor in the normal stale, where the magnetic field penetrates the sample uniformly and the material has lost all superconducting properties. 2. Gauge Invariance Equations (5) and (6) require initial conditions for the order parameter and the vector potential. Here, the concept of gauge invariance enters. Because the physical state of the system at t = 0 is completely determined by the magnetic induction B, the superelectron density n,, and the supercurrent J,, we have a significant degree of freedom in the choice of initial data for $ and A. In fact, if the pair ($0, Ao) properly specifies the physical initial state, then so does any other pair ($&, Ah) that is related to ($0, Ao) by a transformation Gxo:  ($0, Ao) - ($4, Ah) = ($oe’xo, A0 + VXO) . (10) Here, xo can be any (sufficiently smooth) real-valued function of position. Equation (10) is the gauge transfornation for the stationary GINZBURG-LANDAU model. There is a similar, though more complicated, gauge transformation for the time-dependent GINZBURG-LANDAU model. In terms of the rescaled variables, it is G x :  ($ ,A,  4) - (#, A’, 4’) = ($eix ,  A + Vx, d - dtx) . (11) Here, x can be any (sufficiently smooth) real-valued function of position and time. Mathematically, gauge invariance reflects a lack of uniqueness. The TDGL model defines only an equivalence class of solutions, and by choosing a par- ticular gauge x we select a representative from this class. The physical relevance of gauge invariance is the following. At each instant, the macroscopic state of the superconductor is entirely specified in terms of the electromagnetic variables E and B, the superelectron density n,, and the supercurrent J,. These quantities are invariant under the gauge transformation (ll), so the states ($, A, 4) and ( $ I ,  A’, 4‘) are macroscopically indistinguishable. The choice of a particular gauge x does not affect the specification of the physical state of the system. 3. Gauge Choices The most common gauge is the “COULOMB gauge,” where the vector potential is divergence-free at all times. If ($0, Ao) are the initial data given with Eqs. (5) and ( 6 ) ,  one determines the initial gauge xo by solving the boundary value problem ~ Ax0 = -V . A0 in Q, V X O .  n = -Ao. n on dQ, (12) and changes to the image ($;,Ah) of ($0, A*) under the gauge transformation Gx0. At any time t > 0, one takes the solution (111, A, 4) of the rescaled TDGL model, solves the boundary value problem A x = - V - A  i n R ,  V x - n = - A . n  ondR,  (13) u A ~ = V * J , ,  V - A = O  inRx(O,co) ,  (14) V $ - n  = --y$, V 4 - n  = 0, A .  n = 0, (T x A - H) x n = 0 on dQ x (0, OO), (15) and changes to the image (+’,A’, 9’) of ($,A, 4 )  under the gauge transformation Gx. As a result, Ab and A‘ are divergence-free. The procedure amounts effectively to integrating Eqs, (5) and (6), together with the equations - starting from initial data ($0, A*) with V . A0 = 0. The equations of the TDGL(C) model (“C” for COULOMB gauge) are formally similar to the NAVIER-STOKES equations for an incompressible fluid. This similarity was exploited in recent work by TANG and WANG [7], who proved the existence of strong (D = 2 , 3 )  and weak (D = 2) solutions and of a global attractor. I Because the COULOMB gauge requires the solution of an elliptic boundary value problem for the scalar potential at each time step, it is less suitable for numerical purposes. Here, one would like to eliminate the scalar potential altogether. This is, in fact, possible. By choosing an arbitrary gauge x o  for the initial data and determining x as the solution of the initial-value problem at# = 4 in R x (O,CQ) ,  x l t = ~  = xo in R, (16) one obtains the “zero-electric potential gauge.” (The scalar 4 is also known as the electric potential.) In this gauge, the rescaled TDGL model reduces to a,$ = - ( i ~  + A ) ~  4 + (1 - 1$12) + in R x (0, m), (17) u&A = - V x V x A + J , + V x H  i n Q x ( O , m ) ,  (18) (iV + A ) +  . n  = 474 on 8S2 x (O,w), (19) ( V x A - H ) x n  = 0 onaRx(0 ,co ) .  (20) The TDGL(Z) model (“Z” for “zero-electric potential gauge”) was used by Du [8] to prove the existence and uniqueness of strong solutions. It is the kernel of the TDGL code used at ARGONNE [9, 101 for the numerical simulation of vortex dynamics in type-I1 superconductors. It is not possible to combine the COULOMB gauge and the zero-electric potential gauge. Even when the initial data are divergence-free, the solution obtained in the zero-electric potential gauge does not remain on the divergence-free manifold. To see this, take the divergence of Eq. (18): if div A is zero, it must be the case that div J, = 0. But from Eq. (17) we obtain the expression div J, = ( 2 i ) - 1 ( v & $  - $at$*), which is zero if and only if + = 0 or the phase of $ is constant in time. It is, however, possible to  couple div A to 4. The standard gauge is the “4 = -div A gauge,” which maintains the identity 4 = -div A between the (unscaled) potentials or, equivalently, u4 = -div A between the rescaled potentials at all times. The gauge x is the solution of the boundary value problem u&x - Ax = div A + a4 in R x (0, CQ), Vx n = -A . n on 8 2  x (0, m). The initial condition x l t = 0  = xo can be chosen arbitrarily. Usually, one takes xo so the initial data are divergence-free, cf. DU [SI, but this choice is in no way necessary; in fact, there is a distinct advantage in leaving x o  undetermined. The rescaled TDGL model reduces to the following TDGL(S) model (“S” for “standard 4 = -div A gauge”): (21) &$ = - (iV + A)2 $ + tc2 (1 - !$I2) $4- ia-’(div A)$ in f2 x (0, CQ), (22) u&A = AA + J, f V x H in R x (0, co), (23) V$-n = --y$, A . n = O ,  (V x A - H) x n = 0 on 8Q x (0, oo). (24) Here, the initial data ($o,Ao) must be consistent with the boundary conditions (24). The TDGL(S) model was analyzed by TAKAC 1111, who showed that it generates a dynamical process in a Cartesian product of fractional Sobolev spaces. There are, of course, many other ways to couple div A to 4. In fact, we claim that the TDGL model reduces to a more tractable form if the constraint 4 = -div A is applied to the rescaled, rather than the original, potentials. The gauge x which accomplishes this reduction is the solution of the boundary value problem (25) &x - Ax = div A + 4 in R x (0, oo), Vx . n = -A . n on aQ x (0, CQ), ~ --. with Xlt=o = xo arbitrary. In this gauge, the rescaled TDGL model reduces to  at$ = - (iv + A ) ~  $ + s2 (1 - /$12)  11 u&A = -V x V x A+aV(div  A ) +  J, + V  x H in R x (o,cQ), in 52 x (O,CQ) ,  V $ . n  = --y$, A + n = O ,  ( V x A - H ) x n = O  onaQx(0 ,oo) .  We refer to this model as the TDGL(R) model (“R” for “rescaled 4 = -div A gauge”). The TDGL(S) and TDGL(R) models are similar, but not identical. Because- -V x V x A = AA - V(div A), they differ in the way the divergence of A is accounted for. In the TDGL(S) model, div A appears in the equation for +, in the TDGL(R) model in the equation for A. The difference has an important consequence: the TDGL(R) model describes the gradient flow of an energy functional, the TDGL(S) model does not. In fact, the GINZBURG-LANDAU energy for the TDGL(R) model is I I 1 E[$, A] = / (I(iV + A)$I2 + p 2 ( 1  - 1 $ 1 2 ) 2  + 10 x A - HI2 + a(div A)2 -y1$I2 d~(z).(29) n One readily verifies that the first variations of E with respect to +* and A are b,pE = ~ A E  =(iV + A)2 7L - tc2 (1 - 1 $ 1 2 )  $, 2 ( V  x V x A - aV(div A) - J, - ‘7 x H). The natural boundary conditions associated with E are VZC, . n  = -74, (V x A - H) x n = 0 ,  provided n . A = 0 on dR. Hence, the TDGL(R) model corresponds to the dynamical system at$ = -6*. E ,  adtA = - 3 b ~ E .  A detailed study of this dynamical system will appear in our forthcoming article [12]. (33) In conclusion, we note that the 4 = -div A gauge, in either of the forms discussed here, appears to be the most natural gauge for the TDGL model, as every stationary solution ($,A) of TDGL(S) or TDGL(R) automatically satisfies the COULOMB gauge div A = 0. Acknowledgements The work reported here was done while the second author (H. G. K.) was Visiting Professor at the Universiti des Sciences Sociales in Toulouse, France, May/June ’95. The authors thank Professor PETER TAKLC for his valuable comments and for making the manuscript (111 available prior to publication. The work of H.G.K. is supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Ofice of Computational and Technology Research, US. Department of Energy, under Contract W-31-109-Eng-38. 4. References 1 GINZBURG, V.L., LANDAU, L.D.: On the theory of superconductivity; Zh. Eksperim. i Teor. Fiz. (USSR) 20 (1950), 2 SCHMID, A.: Phys. Kondens. Mater. 5 (1966), 302. 3 GOR’KOV, L.P., ELIASHBERG, G.M.: Generalizations of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities; Zh. Eksp. Teor. Fiz. 54 (1968), 612-626; Soviet Phys.- JETP 27 1064-1082; End. transl. in TER HAAR, D.: L.D. Landau; Men of Physics, Vol. I, Pergamon Press (1965), 138-167. (1968), 328-334. 4 ABRIKOSOV, A.A.: Fundamentals of the Theory of Metals; North-Holland Publ. Co., Amsterdam 1988. 5 DEGENNES, P.: Superconductivity in Metals and Alloys; Benjamin, New York 1966. 6 TINKHAM, .: Introduction to Superconductivity; McGraw-Hill, Inc., New York 1975. 7 TANG, &., WANG, S.: Time dependent Ginzburg-Landau equations of superconductivity; preprint (1995). 8 D U ,  &.: Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity; 9 GALBREATH, N., ET AL.: Parallel solution of the three-dimensional time-dependent Ginzburg-Landau equation; in: Proc. 10 GROPP, W.D., ET AL.: Numerical simulation of vortex dynamics in type-I1 superconductors; J.  Comp. Phys. (1995) 11 TAKLC, P.: On the dynamical process generated by a superconductivity model; Proc. ICIAM 95. 12 FLECKINGER-PELL~, J., KAPER, H.G.,  TAK~C,  P.: Unpublished information, Argonne National Laboratory, 1995. Applicable Analysis (to appear). SIAM Conf. on Parallel Processing for Scientific Computing (1993); SIAM, Philadelphia. (to appear) Address: PROF. JACQUELINE FLECKINGER-PELLE, CEREMATH, Universith des Sciences Sociales, Manufacture HANS G. KAPER, Mathematics and Computer Science Division, Argonne National Laboratory, des Tabacs, 21 alMes de Brienne, 31000 Toulouse, France DR. Argonne, IL 60439, USA DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 

Gauges for the Ginzburg-Landau equations of superconductivity

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Gauges for the Ginzburg-Landau equations of superconductivity

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