7855:
3887:
3577:
1349:
636:
3882:{\displaystyle {\begin{aligned}F=D\circ D=dA+A\wedge A=\left({\frac {\partial A_{\nu }}{\partial x^{\mu }}}+A_{\mu }A_{\nu }\right)dx^{\mu }\wedge dx^{\nu }={\frac {1}{2}}\left({\frac {\partial A_{\nu }}{\partial x^{\mu }}}-{\frac {\partial A_{\mu }}{\partial x^{\nu }}}+\right)dx^{\mu }\wedge dx^{\nu }\\\end{aligned}}}
1185:
399:
6146:
6313:
argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov–Witten theory of Calabi–Yau orbifolds to FJRW theory an analogous Landau–Ginzburg "FJRW" theory. Witten's sigma
5566:
1179:
4506:
converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices. The sum over vortices, with multiplicity, just equals the degree of the line bundle; as a result, one may write a line bundle on a
Riemann surface as a flat bundle, with
2245:
5220:
2319:
from the normal state is of second order for Type II superconductors, taking into account fluctuations, as demonstrated by
Dasgupta and Halperin, while for Type I superconductors it is of first order, as demonstrated by Halperin, Lubensky and Ma.
2398:
in 1957. He used
Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux
2751:
2343:. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In
4140:
1044:
6028:
6199:
5895:
1344:{\displaystyle \nabla \times \mathbf {B} ={\frac {4\pi }{c}}\mathbf {J} \;\;;\;\;\mathbf {J} ={\frac {e^{*}}{m^{*}}}\operatorname {Re} \left\{\psi ^{*}\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)\psi \right\},}
631:{\displaystyle f_{s}=f_{n}+\alpha (T)|\psi |^{2}+{\frac {1}{2}}\beta (T)|\psi |^{4}+{\frac {1}{2m^{*}}}\left|\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)\psi \right|^{2}+{\frac {\mathbf {B} ^{2}}{8\pi }},}
91:, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of
2257:
is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor.
5069:
5340:
4206:
4042:
2422:. This is the same functional as given above, transposed to the notation commonly used in Riemannian geometry. In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including
1860:
3035:
2080:
1993:
2869:
5697:
4723:
1670:
5639:
2106:
4951:
1590:
919:
1481:
6033:
5987:
4631:
3582:
235:
5003:
2328:
In the original paper
Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states. The
3474:
2818:
6447:
Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and have not managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century".
5132:
2945:
1711:
5770:
6239:
3530:
3430:
4405:
972:
4898:
4455:
2783:
2496:
6151:
Note that these are both first-order differential equations, manifestly self-dual. Integrating the second of these, one quickly finds that a non-trivial solution must obey
5275:
4803:
6282:
3949:
for additional articulation of this specific notation.) To emphasize this, note that the first term of the
Ginzburg–Landau functional, involving the field-strength only, is
281:
4840:
4756:
4296:
5332:
5120:
1034:
877:
4658:
2552:
1534:
699:
6205:
Roughly speaking, this can be interpreted as an upper limit to the density of the
Abrikosov vortecies. One can also show that the solutions are bounded; one must have
4585:
3943:
728:
3917:
350:
317:
6020:
5810:
5720:
2889:
2361:
penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength
748:
380:
4481:
3292:
2909:
768:
4425:
4236:
3324:
3163:
2516:
1411:
1012:
830:
797:
668:
352:
is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper. Assuming smallness of
152:
5098:
4362:
3108:
4504:
853:
5790:
5740:
5295:
4776:
4556:
4332:
4260:
3553:
3494:
3384:
3364:
3344:
3258:
3231:
3211:
3187:
3128:
3078:
3058:
2540:
2467:
2447:
1371:
992:
4064:
6141:{\displaystyle {\begin{aligned}{\overline {\partial }}_{A}\psi &=0\\*(iF)&={\frac {1}{2}}\left(\sigma -\vert \psi \vert ^{2}\right)\\\end{aligned}}}
1778:
6157:
5818:
2332:
breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In
1600:(remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of
7285:
8219:
6360:
5008:
5561:{\displaystyle {\mathcal {L}}\left(\psi ,A\right)=2\pi \sigma \operatorname {deg} L+\int _{\Sigma }{\frac {i}{2}}dz\wedge d{\overline {z}}\left}
1746:
is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only
7708:
7518:
4151:
1539:
1174:{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m^{*}}}\left(-i\hbar \nabla -{\frac {e^{*}}{c}}\mathbf {A} \right)^{2}\psi =0}
17:
3955:
1428:
2085:
It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value
6314:
models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions.
7889:
7600:
7559:
4522:, which may be analyzed in a similar fashion, and which possesses many similar properties, including self-duality. When such systems are
2956:
2015:
1928:
7952:
6598:
2823:
1903:
The
Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor. The first characteristic length was termed
6425:
5651:
2240:{\displaystyle \lambda ={\sqrt {\frac {m^{*}}{\mu _{0}e^{*2}\psi _{0}^{2}}}}={\sqrt {\frac {m^{*}\beta }{\mu _{0}e^{*2}|\alpha |}}},}
7338:
4663:
1603:
5580:
4903:
2371:, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called
7363:
1492:. This corresponds to the normal conducting state, that is for temperatures above the superconducting transition temperature,
7698:
7467:
6659:
6541:
6405:
888:
8311:
8049:
7927:
7590:
7278:
800:
7328:
5903:
4597:
1510:
Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is
8159:
7446:
4299:
164:
8247:
7858:
6631:
4956:
84:
4239:
2092:. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth,
7968:
7788:
7595:
7538:
7358:
7248:
4660:
of one-forms over a
Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic:
3447:
2791:
1904:
7258:
8164:
7882:
7748:
7651:
7631:
7391:
7271:
7254:
5215:{\displaystyle F=-\left(\partial _{A}{\overline {\partial }}_{A}+{\overline {\partial }}_{A}\partial _{A}\right)}
7209:, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546
6912:
García-Prada, Oscar (1994). "A Direct
Existence Proof for the Vortex Equations Over a Compact Riemann Surface".
7947:
7917:
7554:
7313:
7308:
7147:
6995:
Greene, B.R.; Vafa, C.; Warner, N.P. (September 1989). "Calabi-Yau manifolds and renormalization group flows".
3556:
2336:, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value
8347:
8288:
8140:
8084:
8059:
6960:
Vafa, Cumrun; Warner, Nicholas (February 1989). "Catastrophes and the classification of conformal theories".
6533:
6329:
2922:
2916:
1675:
4587:, the functional can be re-written so as to explicitly show self-duality. One achieves this by writing the
8190:
8119:
7827:
7703:
7353:
6382:
5749:
2395:
38:
8135:
8054:
7822:
7616:
6208:
4055:
3499:
3389:
4375:
4263:
928:
8342:
8242:
8237:
7875:
6339:
6294:
4845:
4434:
6557:
David J. E. Callaway (1990). "On the remarkable structure of the superconducting intermediate state".
2759:
2472:
8195:
7942:
7564:
6266:
5228:
4781:
4519:
2543:
1885:
In
Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to form a
1421:
Consider a homogeneous superconductor where there is no superconducting current and the equation for
880:
387:
124:
6594:
2746:{\displaystyle {\mathcal {L}}(\psi ,A)=\int _{M}{\sqrt {|g|}}dx^{1}\wedge \dotsm \wedge dx^{m}\left}
240:
7978:
7800:
6623:
6483:
4812:
4728:
4268:
3265:
1378:
5307:
5103:
1017:
860:
8321:
8180:
8112:
8029:
7672:
7528:
7497:
7472:
7411:
6302:
4636:
2786:
1513:
1769:, the right hand side of the equation above is positive and there is a non-trivial solution for
1390:
675:
8232:
8205:
8185:
8107:
7839:
7763:
7758:
7693:
7621:
7523:
7426:
7416:
7401:
7318:
7227:
6344:
4564:
3922:
2519:
2344:
2293:
704:
391:
3895:
326:
286:
8102:
7812:
7807:
7646:
7641:
7574:
7477:
7441:
7431:
7386:
7244:
5999:
5795:
5705:
4427:
vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to
4048:
3237:
2912:
2874:
2412:
2333:
2278:
733:
355:
69:
6615:
4463:
3271:
2894:
753:
8306:
8262:
7753:
7487:
7323:
7294:
7166:
7049:
7004:
6969:
6870:
6774:
6719:
6568:
6495:
6254:
4410:
4214:
3297:
3136:
2501:
1396:
997:
808:
775:
646:
130:
96:
68:. In its initial form, it was postulated as a phenomenological model which could describe
2391:, are Type I, while almost all impure and compound superconductors are Type II.
2277:
is presently known as the Ginzburg–Landau parameter. It has been proposed by Landau that
72:
without examining their microscopic properties. One GL-type superconductor is the famous
8:
8352:
7817:
7783:
7718:
7636:
7626:
7421:
6616:
5077:
4588:
4457:
4341:
4135:{\displaystyle D^{*}D\psi ={\frac {1}{2}}\left(\sigma -\vert \psi \vert ^{2}\right)\psi }
3294:, and is normalized differently. In physics, one conventionally writes the connection as
3083:
2419:
2357:
leads to a mixed state (also known as the vortex state) in which an increasing amount of
1393:, but is principally different due to a nonlinear term — determines the order parameter,
95:, where in many cases exact solutions can be given. This general setting then extends to
92:
7170:
7053:
7008:
6973:
6946:
M.C. Hong, J, Jost, M Struwe, "Asymptotic limits of a Ginzberg-Landau type functional",
6874:
6778:
6723:
6572:
6499:
4486:
835:
8293:
8044:
8004:
7451:
7182:
7156:
7116:
7096:
7065:
7039:
6894:
6798:
6743:
5775:
5725:
5280:
4761:
4592:
4541:
4369:
4317:
4245:
3538:
3479:
3369:
3349:
3329:
3243:
3216:
3196:
3172:
3113:
3063:
3043:
2525:
2452:
2432:
2423:
2400:
1594:
When the right hand side of this equation is positive, there is a nonzero solution for
1356:
977:
154:
of a superconductor near the superconducting transition can be expressed in terms of a
4428:
8069:
7898:
7502:
7186:
7120:
7061:
7016:
6981:
6929:
6898:
6886:
6839:
6790:
6747:
6735:
6688:
6655:
6627:
6580:
6559:
6537:
6464:
6401:
4523:
3946:
3564:
3166:
65:
7069:
6802:
103:, again owing to its solvability, and its close relation to other, similar systems.
8278:
8252:
8024:
7999:
7932:
7773:
7174:
7106:
7057:
7012:
6977:
6921:
6878:
6829:
6782:
6727:
6680:
6576:
6503:
6456:
6393:
6262:
6250:
3261:
3190:
2380:
2316:
116:
6194:{\displaystyle 4\pi \operatorname {deg} L\leq \sigma \operatorname {Area} \Sigma }
5890:{\displaystyle \operatorname {deg} L=c_{1}(L)={\frac {1}{2\pi }}\int _{\Sigma }iF}
8301:
8034:
7795:
7533:
7482:
6858:
6762:
6707:
6397:
4559:
4335:
3234:
2388:
158:
120:
57:
7178:
7111:
7084:
6484:"First-Order Phase Transitions in Superconductors and Smectic-A Liquid Crystals"
730:
are phenomenological parameters that are functions of T (and often written just
8039:
7983:
7973:
7937:
7738:
7713:
7492:
7142:
7134:
7030:
Witten, Edward (16 August 1993). "Phases of N = 2 theories in two dimensions".
6507:
6350:
6286:
4527:
4515:
3560:
3440:
corresponding to the symmetry group of the fiber. Here, the symmetry group is
2329:
155:
88:
6684:
6647:
6611:
4314:, it is conventional to study the Ginzburg–Landau functional for the manifold
8336:
7834:
7768:
7743:
7343:
6933:
6890:
6843:
6834:
6817:
6794:
6739:
6692:
6334:
6310:
6274:
4311:
3568:
2416:
2411:
The Ginzburg–Landau functional can be formulated in the general setting of a
2358:
2097:
1895:| indicates the fraction of electrons that have condensed into a superfluid.
1389:. The first equation — which bears some similarities to the time-independent
320:
100:
31:
6818:"Special metrics and stability for holomorphic bundles with global sections"
8074:
8064:
8019:
8014:
7778:
7688:
7406:
7348:
7333:
7138:
6925:
6708:"Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations"
6468:
6460:
6355:
6324:
6290:
6258:
5064:{\displaystyle {\overline {\partial }}_{A}={\overline {\partial }}+A^{0,1}}
1881:
from below. Such a behavior is typical for a second order phase transition.
6675:
Hitchin, N. J. (1987). "The Self-Duality Equations on a Riemann Surface".
8200:
8009:
7436:
7263:
6298:
6278:
5990:
5572:
5123:
4365:
3437:
2948:
1374:
6948:
Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt
7912:
7044:
6882:
6786:
6731:
5743:
4201:{\displaystyle D^{*}F=-\operatorname {Re} \langle D\psi ,\psi \rangle }
1886:
112:
61:
7085:"The Witten equation, mirror symmetry, and quantum singularity theory"
4037:{\displaystyle {\mathcal {L}}(A)=YM(A)=\int _{M}*(1)\vert F\vert ^{2}}
2100:. Expressed in terms of the parameters of Ginzburg–Landau model it is
7569:
7867:
6289:
possess a degenerate critical point. The same month, together with
1855:{\displaystyle |\psi |^{2}=-{\frac {\alpha _{0}(T-T_{c})}{\beta }},}
1536:). Under this assumption the equation above can be rearranged into:
8227:
7667:
6763:"Vortices in holomorphic line bundles over closed Kähler manifolds"
5334:
so derivatives are purely imaginary). The functional then becomes
3433:
3030:{\displaystyle *(1)={\sqrt {|g|}}dx^{1}\wedge \dotsm \wedge dx^{m}}
2394:
The most important finding from Ginzburg–Landau theory was made by
2075:{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{4m^{*}|\alpha |}}}.}
2009:(superconducting phase), where it is more relevant, it is given by
1988:{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m^{*}|\alpha |}}}.}
383:
7226:
1174 (1957)].) Abrikosov's original paper on vortex structure of
7161:
7101:
2911:
have been absorbed so that the potential energy term is a quartic
4058:
for the Ginzburg–Landau functional are the Yang–Mills equations
2864:{\displaystyle \vert \psi \vert ^{2}=\langle \psi ,\psi \rangle }
2384:
2376:
1413:. The second equation then provides the superconducting current.
83:
Later, a version of Ginzburg–Landau theory was derived from the
77:
45:
6527:
5692:{\displaystyle \operatorname {Area} \Sigma =\int _{\Sigma }*(1)}
2372:
2096:. It was previously introduced by the London brothers in their
283:
is a measure of the local density of superconducting electrons
6285:
in November 1988; in this generalization one imposes that the
8283:
8257:
4718:{\displaystyle \Omega ^{1}=\Omega ^{1,0}\oplus \Omega ^{0,1}}
3441:
1665:{\displaystyle \alpha :\alpha (T)=\alpha _{0}(T-T_{\rm {c}})}
5634:{\displaystyle *(1)={\frac {i}{2}}dz\wedge d{\overline {z}}}
3366:; in Riemannian geometry, it is more convenient to drop the
8316:
6269:
is called a Landau–Ginzburg theory. The generalization to
5301:
4946:{\displaystyle D=\partial _{A}+{\overline {\partial }}_{A}}
73:
7083:
Fan, Huijun; Jarvis, Tyler; Ruan, Yongbin (1 July 2013).
1898:
6869:(3). Springer Science and Business Media LLC: 527–546.
6718:(3). Springer Science and Business Media LLC: 277–292.
5225:
Note that in the sign-convention being used here, both
4407:, where one can specify any finite set of points where
1585:{\displaystyle |\psi |^{2}=-{\frac {\alpha }{\beta }}.}
914:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
7230:
derived as a solution of G–L equations for κ > 1/√2
6390:
Springer Handbook of Electronic and Photonic Materials
4431:. In the limit of weak coupling, it can be shown that
1476:{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi =0.}
6654:(Fifth ed.). Springer-Verlag. pp. 521–522.
6211:
6160:
6031:
6002:
5906:
5821:
5798:
5778:
5752:
5728:
5708:
5654:
5583:
5343:
5310:
5283:
5231:
5135:
5106:
5080:
5011:
4959:
4906:
4848:
4815:
4784:
4764:
4731:
4666:
4639:
4600:
4567:
4544:
4489:
4466:
4437:
4413:
4378:
4344:
4320:
4271:
4248:
4217:
4154:
4067:
3958:
3925:
3898:
3580:
3541:
3502:
3482:
3450:
3392:
3372:
3352:
3332:
3300:
3274:
3246:
3219:
3199:
3175:
3139:
3116:
3086:
3066:
3046:
2959:
2925:
2897:
2877:
2826:
2794:
2762:
2555:
2528:
2504:
2475:
2455:
2435:
2109:
2018:
1931:
1781:
1678:
1606:
1542:
1516:
1431:
1399:
1359:
1188:
1047:
1020:
1000:
980:
931:
891:
863:
838:
811:
778:
756:
736:
707:
678:
649:
402:
358:
329:
289:
243:
167:
133:
64:, is a mathematical physical theory used to describe
6773:(1). Springer Science and Business Media LLC: 1–17.
6556:
6482:
Halperin, B; Lubensky, T; Ma, S (11 February 1974).
6433:. IBM Thomas J. Watson Research Center. p. 970.
4842:. This allows the vector potential to be written as
4518:, then one may write a very similar functional, the
4514:
When the manifold is four-dimensional, possessing a
4511:
singular points and a covariantly constant section.
2323:
7133:
5982:{\displaystyle c_{1}(L)=c_{1}(L)\in H^{2}(\Sigma )}
4626:{\displaystyle d=\partial +{\overline {\partial }}}
6481:
6233:
6193:
6140:
6014:
5981:
5889:
5804:
5784:
5764:
5734:
5714:
5691:
5633:
5560:
5326:
5289:
5269:
5214:
5114:
5092:
5063:
4997:
4945:
4892:
4834:
4797:
4770:
4750:
4717:
4652:
4625:
4579:
4550:
4498:
4475:
4449:
4419:
4399:
4356:
4326:
4290:
4254:
4230:
4200:
4134:
4036:
3937:
3911:
3881:
3547:
3524:
3488:
3468:
3424:
3378:
3358:
3338:
3318:
3286:
3252:
3225:
3205:
3181:
3157:
3122:
3102:
3072:
3052:
3029:
2939:
2903:
2883:
2863:
2812:
2777:
2745:
2534:
2510:
2490:
2461:
2441:
2347:, raising the applied field past a critical value
2239:
2074:
1987:
1854:
1756:Below the superconducting transition temperature,
1717:Above the superconducting transition temperature,
1705:
1664:
1584:
1528:
1475:
1405:
1365:
1343:
1173:
1028:
1006:
994:with respect to variations in the order parameter
986:
966:
913:
871:
847:
824:
791:
762:
742:
722:
693:
662:
630:
374:
344:
311:
275:
229:
146:
6595:The magnetic properties of superconducting alloys
6293:they argued that these theories are related by a
5126:, the field strength can similarly be written as
2756:The notation used here is as follows. The fibers
115:'s previously established theory of second-order
8334:
6994:
3945:skew-symmetric matrix. (See the article on the
230:{\displaystyle \psi (r)=|\psi (r)|e^{i\phi (r)}}
6950:(1996) International press (Boston) pp. 99-123.
4998:{\displaystyle \partial _{A}=\partial +A^{1,0}}
670:is the free energy density of the normal phase,
6677:Proceedings of the London Mathematical Society
6536:. Vol. 8. Oxford: Butterworth-Heinemann.
6442:
6440:
5996:The Lagrangian is minimized (stationary) when
4298:. Note that these are closely related to the
2375:because the flux carried by these vortices is
7883:
7279:
7082:
6828:(1). International Press of Boston: 169–213.
3469:{\displaystyle \langle \cdot ,\cdot \rangle }
2820:so that the square of the norm is written as
2813:{\displaystyle \langle \cdot ,\cdot \rangle }
6911:
6856:
6622:(Third ed.). Springer-Verlag. pp.
6309: = 2 theories in two-dimensions",
6120:
6113:
5533:
5526:
5468:
5444:
4444:
4438:
4195:
4180:
4115:
4108:
4025:
4018:
3463:
3451:
2858:
2846:
2834:
2827:
2807:
2795:
2718:
2711:
2677:
2667:
2655:
2648:
6914:Bulletin of the London Mathematical Society
6528:Lev D. Landau; Evgeny M. Lifschitz (1984).
6446:
6437:
6427:Pairing symmetry in cuprate superconductors
4372:persists in these general cases, including
2542:. The Ginzburg–Landau functional is then a
7890:
7876:
7293:
7286:
7272:
7145:(2013), "Surface Defects and Resolvents",
6959:
6652:Riemannian Geometry and Geometric Analysis
6650:(2008). "The Ginzburg–Landau Functional".
6618:Riemannian Geometry and Geometric Analysis
6614:(2002). "The Ginzburg–Landau Functional".
6599:Journal of Physics and Chemistry of Solids
6277:in 2 spacetime dimensions was proposed by
5571:The integral is understood to be over the
1229:
1228:
1224:
1223:
7160:
7110:
7100:
7043:
6833:
5108:
4387:
3193:(this is not the same as the free energy
2933:
2765:
2478:
6423:
3386:(and all other physical units) and take
2406:
1416:
855:, where e is the charge of an electron),
6815:
6760:
6674:
3496:is a form taking values in the algebra
2940:{\displaystyle \sigma \in \mathbb {R} }
1706:{\displaystyle \alpha _{0}/\beta >0}
1485:This equation has a trivial solution:
14:
8335:
7364:Two-dimensional conformal field theory
7029:
6863:Communications in Mathematical Physics
6767:Communications in Mathematical Physics
6712:Communications in Mathematical Physics
6705:
6380:
2947:. The integral is explicitly over the
1899:Coherence length and penetration depth
7897:
7871:
7267:
7243:A.A. Abrikosov's 2003 Nobel lecture:
5765:{\displaystyle \operatorname {deg} L}
7253:V.L. Ginzburg's 2003 Nobel Lecture:
7219:, 1442 (1957) (English translation:
6859:"Invariant connections and vortices"
6646:
6610:
6419:
6417:
6392:. Springer Handbooks. p. 1233.
6361:Bogomol'nyi–Prasad–Sommerfield bound
6022:solve the Ginzberg–Landau equations
2919:, with a minimum at some real value
1753:solves the Ginzburg–Landau equation.
6530:Electrodynamics of Continuous Media
6244:
6234:{\displaystyle |\psi |\leq \sigma }
4305:
3559:to the non-Abelian setting, as the
3525:{\displaystyle {\mathfrak {su}}(n)}
3508:
3505:
3444:, as that leaves the inner product
3425:{\displaystyle A=A_{\mu }dx^{\mu }}
2261:The original idea on the parameter
37:For the nonlinear instability, see
24:
6383:"High-Temperature Superconductors"
6188:
6039:
5973:
5951:
5876:
5799:
5709:
5672:
5661:
5450:
5399:
5346:
5198:
5183:
5163:
5151:
5034:
5015:
4973:
4961:
4929:
4914:
4817:
4733:
4700:
4681:
4668:
4641:
4615:
4607:
4574:
4400:{\displaystyle M=\mathbb {R} ^{2}}
4051:on a compact Riemannian manifold.
3961:
3793:
3778:
3756:
3741:
3647:
3632:
3571:. It is conventionally written as
2871:. The phenomenological parameters
2785:are assumed to be equipped with a
2558:
1653:
1297:
1189:
1123:
967:{\displaystyle F=\int f_{s}d^{3}r}
925:The total free energy is given by
900:
551:
394:and exhibits U(1) gauge symmetry:
319:analogous to a quantum mechanical
25:
8364:
7859:Template:Quantum mechanics topics
6414:
5702:is the total area of the surface
4893:{\displaystyle A=A^{1,0}+A^{0,1}}
4450:{\displaystyle \vert \psi \vert }
2324:Classification of superconductors
1294:
1120:
548:
7854:
7853:
6822:Journal of Differential Geometry
6305:. In his 1993 paper "Phases of
2778:{\displaystyle \mathbb {C} ^{n}}
2491:{\displaystyle \mathbb {C} ^{n}}
1321:
1231:
1219:
1196:
1147:
1022:
907:
893:
865:
832:is an effective charge (usually
605:
575:
7202:V.L. Ginzburg and L.D. Landau,
7127:
7076:
7023:
6988:
6953:
6940:
6905:
6850:
6809:
6754:
6706:Taubes, Clifford Henry (1980).
6699:
6668:
5270:{\displaystyle A^{1,0},A^{0,1}}
4798:{\displaystyle {\overline {z}}}
4533:
2310:
1922:(normal phase), it is given by
106:
7148:Journal of High Energy Physics
6640:
6604:
6587:
6550:
6521:
6475:
6374:
6221:
6213:
6082:
6073:
5976:
5970:
5954:
5948:
5945:
5939:
5923:
5917:
5850:
5844:
5686:
5680:
5593:
5587:
5517:
5501:
5489:
4015:
4009:
3990:
3984:
3972:
3966:
3838:
3812:
3557:electromagnetic field strength
3519:
3513:
3096:
3088:
2986:
2978:
2969:
2963:
2602:
2594:
2575:
2563:
2226:
2218:
2061:
2053:
1974:
1966:
1840:
1821:
1792:
1783:
1659:
1638:
1622:
1616:
1553:
1544:
1454:
1445:
1070:
1061:
717:
711:
688:
682:
496:
487:
483:
477:
451:
442:
438:
432:
368:
360:
339:
333:
306:
300:
276:{\displaystyle |\psi (r)|^{2}}
263:
258:
252:
245:
222:
216:
201:
197:
191:
184:
177:
171:
13:
1:
6534:Course of Theoretical Physics
6424:Tsuei, C. C.; Kirtley, J. R.
6367:
4835:{\displaystyle \Omega ^{0,1}}
4751:{\displaystyle \Omega ^{1,0}}
4291:{\displaystyle \delta =d^{*}}
2917:spontaneous symmetry breaking
2265:belongs to Landau. The ratio
7062:10.1016/0550-3213(93)90033-L
7017:10.1016/0550-3213(89)90471-9
6982:10.1016/0370-2693(89)90473-5
6857:García-Prada, Oscar (1993).
6581:10.1016/0550-3213(90)90672-Z
6398:10.1007/978-3-319-48933-9_50
6042:
5626:
5453:
5431:
5327:{\displaystyle e^{i\theta }}
5186:
5166:
5115:{\displaystyle \mathbb {C} }
5037:
5018:
4932:
4790:
4618:
2429:For a complex vector bundle
1029:{\displaystyle \mathbf {A} }
872:{\displaystyle \mathbf {A} }
7:
7823:Quantum information science
7112:10.4007/annals.2013.178.1.1
6816:Bradlow, Steven B. (1991).
6761:Bradlow, Steven B. (1990).
6317:
4653:{\displaystyle \Omega ^{1}}
2449:over a Riemannian manifold
1889:. In this interpretation, |
1529:{\displaystyle \psi \neq 0}
123:and Landau argued that the
10:
8369:
8220:Technological applications
6508:10.1103/PhysRevLett.32.292
6295:renormalization group flow
4778:and have no dependence on
4300:Yang–Mills–Higgs equations
694:{\displaystyle \alpha (T)}
390:density has the form of a
36:
29:
8271:
8218:
8173:
8149:
8128:
8092:
8083:
7992:
7962:Characteristic parameters
7961:
7905:
7848:
7731:
7681:
7660:
7609:
7583:
7547:
7511:
7460:
7379:
7372:
7301:
7196:
6593:Abrikosov, A. A. (1957).
6330:Gross–Pitaevskii equation
6267:degenerate critical point
5746:, as before. The degree
4580:{\displaystyle M=\Sigma }
4520:Seiberg–Witten functional
3938:{\displaystyle n\times n}
1038:Ginzburg–Landau equations
1014:and the vector potential
881:magnetic vector potential
723:{\displaystyle \beta (T)}
85:Bardeen–Cooper–Schrieffer
18:Ginzburg–Landau parameter
7979:London penetration depth
6257:with a unique classical
5122:so that the bundle is a
4056:Euler–Lagrange equations
3912:{\displaystyle A_{\mu }}
3326:for the electric charge
2426:(see discussion below).
2383:superconductors, except
1379:electric current density
345:{\displaystyle \psi (r)}
312:{\displaystyle n_{s}(r)}
39:Ginzburg–Landau equation
30:Not to be confused with
27:Superconductivity theory
8272:List of superconductors
8150:By critical temperature
7519:2D free massless scalar
7412:Quantum electrodynamics
7339:QFT in curved spacetime
7228:Type-II superconductors
7179:10.1007/JHEP09(2013)070
6685:10.1112/plms/s3-55.1.59
6488:Physical Review Letters
6381:Wesche, Rainer (2017).
6275:supersymmetric theories
6015:{\displaystyle \psi ,A}
5805:{\displaystyle \Sigma }
5715:{\displaystyle \Sigma }
2884:{\displaystyle \alpha }
2787:Hermitian inner product
2345:Type II superconductors
2294:Type II superconductors
743:{\displaystyle \alpha }
375:{\displaystyle |\psi |}
7840:Quantum thermodynamics
7764:On shell and off shell
7759:Loop quantum cosmology
7601:N = 4 super Yang–Mills
7560:N = 1 super Yang–Mills
7427:Scalar electrodynamics
7417:Quantum chromodynamics
7319:Conformal field theory
7295:Quantum field theories
6835:10.4310/jdg/1214446034
6461:10.1002/cphc.200400182
6340:Stuart–Landau equation
6235:
6195:
6142:
6016:
5983:
5891:
5806:
5786:
5766:
5736:
5716:
5693:
5635:
5562:
5328:
5297:are purely imaginary (
5291:
5271:
5216:
5116:
5094:
5065:
4999:
4947:
4894:
4836:
4799:
4772:
4752:
4719:
4654:
4633:. Likewise, the space
4627:
4581:
4552:
4526:, they are studied as
4500:
4477:
4476:{\displaystyle D\psi }
4451:
4421:
4401:
4358:
4328:
4292:
4256:
4232:
4202:
4136:
4038:
3939:
3913:
3883:
3549:
3526:
3490:
3470:
3426:
3380:
3360:
3340:
3320:
3288:
3287:{\displaystyle n>1}
3254:
3227:
3207:
3183:
3159:
3124:
3104:
3074:
3060:-dimensional manifold
3054:
3031:
2941:
2905:
2904:{\displaystyle \beta }
2885:
2865:
2814:
2779:
2747:
2536:
2512:
2498:, the order parameter
2492:
2463:
2443:
2334:Type I superconductors
2281:are those with 0 <
2279:Type I superconductors
2241:
2076:
1989:
1856:
1707:
1666:
1586:
1530:
1477:
1407:
1367:
1345:
1175:
1030:
1008:
988:
968:
921:is the magnetic field.
915:
873:
849:
826:
793:
764:
763:{\displaystyle \beta }
744:
724:
695:
664:
632:
376:
346:
313:
277:
231:
148:
87:microscopic theory by
70:type-I superconductors
54:Landau–Ginzburg theory
50:Ginzburg–Landau theory
7918:Bean's critical state
7813:Quantum hydrodynamics
7808:Quantum hadrodynamics
7432:Scalar chromodynamics
7089:Annals of Mathematics
6679:. s3-55 (1): 59–126.
6236:
6196:
6143:
6017:
5984:
5892:
5807:
5787:
5767:
5737:
5717:
5694:
5636:
5563:
5329:
5292:
5272:
5217:
5117:
5100:, where the fiber is
5095:
5066:
5000:
4948:
4895:
4837:
4800:
4773:
4753:
4720:
4655:
4628:
4582:
4553:
4501:
4478:
4452:
4422:
4420:{\displaystyle \psi }
4402:
4359:
4329:
4293:
4257:
4233:
4231:{\displaystyle D^{*}}
4203:
4137:
4039:
3940:
3914:
3884:
3550:
3527:
3491:
3471:
3436:taking values in the
3427:
3381:
3361:
3346:and vector potential
3341:
3321:
3319:{\displaystyle d-ieA}
3289:
3255:
3238:field strength tensor
3228:
3208:
3189:is the corresponding
3184:
3160:
3158:{\displaystyle D=d+A}
3125:
3110:of the metric tensor
3105:
3075:
3055:
3032:
2942:
2913:mexican hat potential
2906:
2886:
2866:
2815:
2780:
2748:
2537:
2522:of the vector bundle
2513:
2511:{\displaystyle \psi }
2493:
2464:
2444:
2413:complex vector bundle
2407:Geometric formulation
2242:
2077:
1990:
1857:
1708:
1667:
1587:
1531:
1478:
1417:Simple interpretation
1408:
1406:{\displaystyle \psi }
1368:
1346:
1176:
1036:, one arrives at the
1031:
1009:
1007:{\displaystyle \psi }
989:
969:
916:
874:
850:
827:
825:{\displaystyle e^{*}}
794:
792:{\displaystyle m^{*}}
765:
745:
725:
696:
665:
663:{\displaystyle f_{n}}
633:
382:and smallness of its
377:
347:
314:
278:
237:, where the quantity
232:
149:
147:{\displaystyle f_{s}}
8348:Quantum field theory
8093:By magnetic response
7784:Quantum fluctuations
7754:Loop quantum gravity
7324:Lattice field theory
7214:Zh. Eksp. Teor. Fiz.
7204:Zh. Eksp. Teor. Fiz.
6926:10.1112/blms/26.1.88
6303:Calabi–Yau manifolds
6255:quantum field theory
6209:
6158:
6029:
6000:
5904:
5819:
5796:
5776:
5750:
5726:
5706:
5652:
5581:
5341:
5308:
5281:
5229:
5133:
5104:
5078:
5009:
4957:
4904:
4846:
4813:
4782:
4762:
4729:
4664:
4637:
4598:
4565:
4542:
4487:
4464:
4435:
4411:
4376:
4368:. The phenomenon of
4342:
4318:
4269:
4246:
4215:
4152:
4065:
3956:
3923:
3896:
3578:
3539:
3500:
3480:
3476:invariant; so here,
3448:
3390:
3370:
3350:
3330:
3298:
3272:
3264:, but is in general
3244:
3217:
3213:given up top; here,
3197:
3173:
3137:
3114:
3084:
3064:
3044:
2957:
2923:
2895:
2875:
2824:
2792:
2760:
2553:
2526:
2502:
2473:
2453:
2433:
2107:
2016:
1929:
1779:
1676:
1604:
1540:
1514:
1429:
1397:
1391:Schrödinger equation
1357:
1186:
1045:
1018:
998:
978:
929:
889:
861:
836:
809:
776:
754:
734:
705:
676:
647:
400:
356:
327:
287:
241:
165:
131:
97:quantum field theory
76:, and generally all
8045:persistent currents
8030:Little–Parks effect
7818:Quantum information
7422:Quartic interaction
7171:2013JHEP...09..070G
7054:1993NuPhB.403..159W
7009:1989NuPhB.324..371G
6974:1989PhLB..218...51V
6920:(1). Wiley: 88–96.
6875:1993CMaPh.156..527G
6779:1990CMaPh.135....1B
6724:1980CMaPh..72..277T
6573:1990NuPhB.344..627C
6500:1974PhRvL..32..292H
6273: = (2,2)
5772:of the line bundle
5093:{\displaystyle n=1}
4758:are holomorphic in
4725:, so that forms in
4593:Dolbeault operators
4589:exterior derivative
4458:converges uniformly
4357:{\displaystyle n=1}
4262:, analogous to the
3260:corresponds to the
3233:corresponds to the
3167:connection one-form
3103:{\displaystyle |g|}
2915:; i.e., exhibiting
2518:is understood as a
2420:Riemannian manifold
2167:
1868:approaches zero as
93:Riemannian geometry
8005:Andreev reflection
8000:Abrikosov vortices
7704:Nambu–Jona-Lasinio
7632:Higher dimensional
7539:Wess–Zumino–Witten
7329:Noncommutative QFT
6883:10.1007/bf02096862
6787:10.1007/bf02097654
6732:10.1007/bf01197552
6345:Reaction–diffusion
6231:
6191:
6138:
6136:
6012:
5979:
5887:
5802:
5782:
5762:
5732:
5712:
5689:
5631:
5558:
5324:
5287:
5267:
5212:
5112:
5090:
5061:
4995:
4943:
4890:
4832:
4795:
4768:
4748:
4715:
4650:
4623:
4577:
4548:
4538:When the manifold
4499:{\displaystyle dA}
4496:
4473:
4447:
4417:
4397:
4370:Abrikosov vortices
4354:
4324:
4288:
4252:
4228:
4198:
4132:
4047:which is just the
4034:
3935:
3909:
3879:
3877:
3545:
3522:
3486:
3466:
3422:
3376:
3356:
3336:
3316:
3284:
3250:
3223:
3203:
3179:
3155:
3120:
3100:
3070:
3050:
3027:
2937:
2901:
2881:
2861:
2810:
2775:
2743:
2546:for that section:
2532:
2508:
2488:
2459:
2439:
2424:Abrikosov vortices
2237:
2153:
2072:
1985:
1852:
1703:
1662:
1582:
1526:
1473:
1403:
1363:
1341:
1171:
1026:
1004:
984:
964:
911:
869:
848:{\displaystyle 2e}
845:
822:
789:
760:
740:
720:
691:
660:
628:
372:
342:
309:
273:
227:
144:
8343:Superconductivity
8330:
8329:
8248:quantum computing
8214:
8213:
8070:superdiamagnetism
7899:Superconductivity
7865:
7864:
7727:
7726:
7032:Nuclear Physics B
6997:Nuclear Physics B
6962:Physics Letters B
6661:978-3-540-77340-5
6560:Nuclear Physics B
6543:978-0-7506-2634-7
6407:978-3-319-48931-5
6100:
6045:
5869:
5792:over the surface
5785:{\displaystyle L}
5735:{\displaystyle *}
5629:
5607:
5515:
5456:
5434:
5412:
5290:{\displaystyle F}
5189:
5169:
5040:
5021:
4935:
4793:
4771:{\displaystyle z}
4621:
4551:{\displaystyle M}
4327:{\displaystyle M}
4255:{\displaystyle D}
4095:
4049:Yang–Mills action
3947:metric connection
3807:
3770:
3731:
3661:
3565:affine connection
3548:{\displaystyle F}
3489:{\displaystyle A}
3379:{\displaystyle e}
3359:{\displaystyle A}
3339:{\displaystyle e}
3253:{\displaystyle A}
3226:{\displaystyle F}
3206:{\displaystyle F}
3182:{\displaystyle F}
3123:{\displaystyle g}
3080:with determinant
3073:{\displaystyle M}
3053:{\displaystyle m}
2990:
2697:
2606:
2535:{\displaystyle E}
2462:{\displaystyle M}
2442:{\displaystyle E}
2232:
2231:
2170:
2169:
2067:
2066:
1980:
1979:
1847:
1730:, the expression
1577:
1366:{\displaystyle J}
1318:
1260:
1216:
1144:
1106:
987:{\displaystyle F}
623:
572:
529:
472:
117:phase transitions
66:superconductivity
16:(Redirected from
8360:
8279:bilayer graphene
8253:Rutherford cable
8165:room temperature
8160:high temperature
8090:
8089:
8050:proximity effect
8025:Josephson effect
7969:coherence length
7892:
7885:
7878:
7869:
7868:
7857:
7856:
7774:Quantum dynamics
7447:Yang–Mills–Higgs
7402:Non-linear sigma
7392:Euler–Heisenberg
7377:
7376:
7288:
7281:
7274:
7265:
7264:
7212:A.A. Abrikosov,
7190:
7189:
7164:
7131:
7125:
7124:
7114:
7104:
7080:
7074:
7073:
7047:
7027:
7021:
7020:
6992:
6986:
6985:
6957:
6951:
6944:
6938:
6937:
6909:
6903:
6902:
6854:
6848:
6847:
6837:
6813:
6807:
6806:
6758:
6752:
6751:
6703:
6697:
6696:
6672:
6666:
6665:
6644:
6638:
6637:
6621:
6608:
6602:
6601:, 2(3), 199–208.
6591:
6585:
6584:
6554:
6548:
6547:
6525:
6519:
6518:
6516:
6514:
6479:
6473:
6472:
6444:
6435:
6434:
6432:
6421:
6412:
6411:
6387:
6378:
6263:potential energy
6251:particle physics
6245:In string theory
6240:
6238:
6237:
6232:
6224:
6216:
6200:
6198:
6197:
6192:
6147:
6145:
6144:
6139:
6137:
6133:
6129:
6128:
6127:
6101:
6093:
6052:
6051:
6046:
6038:
6021:
6019:
6018:
6013:
5988:
5986:
5985:
5980:
5969:
5968:
5938:
5937:
5916:
5915:
5896:
5894:
5893:
5888:
5880:
5879:
5870:
5868:
5857:
5843:
5842:
5811:
5809:
5808:
5803:
5791:
5789:
5788:
5783:
5771:
5769:
5768:
5763:
5741:
5739:
5738:
5733:
5721:
5719:
5718:
5713:
5698:
5696:
5695:
5690:
5676:
5675:
5640:
5638:
5637:
5632:
5630:
5622:
5608:
5600:
5567:
5565:
5564:
5559:
5557:
5553:
5552:
5551:
5546:
5542:
5541:
5540:
5516:
5508:
5476:
5475:
5463:
5462:
5457:
5449:
5435:
5427:
5413:
5405:
5403:
5402:
5369:
5365:
5350:
5349:
5333:
5331:
5330:
5325:
5323:
5322:
5304:is generated by
5296:
5294:
5293:
5288:
5276:
5274:
5273:
5268:
5266:
5265:
5247:
5246:
5221:
5219:
5218:
5213:
5211:
5207:
5206:
5205:
5196:
5195:
5190:
5182:
5176:
5175:
5170:
5162:
5159:
5158:
5121:
5119:
5118:
5113:
5111:
5099:
5097:
5096:
5091:
5074:For the case of
5070:
5068:
5067:
5062:
5060:
5059:
5041:
5033:
5028:
5027:
5022:
5014:
5004:
5002:
5001:
4996:
4994:
4993:
4969:
4968:
4952:
4950:
4949:
4944:
4942:
4941:
4936:
4928:
4922:
4921:
4899:
4897:
4896:
4891:
4889:
4888:
4870:
4869:
4841:
4839:
4838:
4833:
4831:
4830:
4804:
4802:
4801:
4796:
4794:
4786:
4777:
4775:
4774:
4769:
4757:
4755:
4754:
4749:
4747:
4746:
4724:
4722:
4721:
4716:
4714:
4713:
4695:
4694:
4676:
4675:
4659:
4657:
4656:
4651:
4649:
4648:
4632:
4630:
4629:
4624:
4622:
4614:
4586:
4584:
4583:
4578:
4557:
4555:
4554:
4549:
4505:
4503:
4502:
4497:
4482:
4480:
4479:
4474:
4456:
4454:
4453:
4448:
4429:Kähler manifolds
4426:
4424:
4423:
4418:
4406:
4404:
4403:
4398:
4396:
4395:
4390:
4363:
4361:
4360:
4355:
4333:
4331:
4330:
4325:
4306:Specific results
4297:
4295:
4294:
4289:
4287:
4286:
4261:
4259:
4258:
4253:
4237:
4235:
4234:
4229:
4227:
4226:
4207:
4205:
4204:
4199:
4164:
4163:
4141:
4139:
4138:
4133:
4128:
4124:
4123:
4122:
4096:
4088:
4077:
4076:
4043:
4041:
4040:
4035:
4033:
4032:
4005:
4004:
3965:
3964:
3944:
3942:
3941:
3936:
3918:
3916:
3915:
3910:
3908:
3907:
3888:
3886:
3885:
3880:
3878:
3874:
3873:
3858:
3857:
3845:
3841:
3837:
3836:
3824:
3823:
3808:
3806:
3805:
3804:
3791:
3790:
3789:
3776:
3771:
3769:
3768:
3767:
3754:
3753:
3752:
3739:
3732:
3724:
3719:
3718:
3703:
3702:
3690:
3686:
3685:
3684:
3675:
3674:
3662:
3660:
3659:
3658:
3645:
3644:
3643:
3630:
3555:generalizes the
3554:
3552:
3551:
3546:
3531:
3529:
3528:
3523:
3512:
3511:
3495:
3493:
3492:
3487:
3475:
3473:
3472:
3467:
3431:
3429:
3428:
3423:
3421:
3420:
3408:
3407:
3385:
3383:
3382:
3377:
3365:
3363:
3362:
3357:
3345:
3343:
3342:
3337:
3325:
3323:
3322:
3317:
3293:
3291:
3290:
3285:
3262:vector potential
3259:
3257:
3256:
3251:
3232:
3230:
3229:
3224:
3212:
3210:
3209:
3204:
3191:curvature 2-form
3188:
3186:
3185:
3180:
3164:
3162:
3161:
3156:
3129:
3127:
3126:
3121:
3109:
3107:
3106:
3101:
3099:
3091:
3079:
3077:
3076:
3071:
3059:
3057:
3056:
3051:
3036:
3034:
3033:
3028:
3026:
3025:
3004:
3003:
2991:
2989:
2981:
2976:
2946:
2944:
2943:
2938:
2936:
2910:
2908:
2907:
2902:
2890:
2888:
2887:
2882:
2870:
2868:
2867:
2862:
2842:
2841:
2819:
2817:
2816:
2811:
2784:
2782:
2781:
2776:
2774:
2773:
2768:
2752:
2750:
2749:
2744:
2742:
2738:
2737:
2736:
2731:
2727:
2726:
2725:
2698:
2690:
2685:
2684:
2663:
2662:
2642:
2641:
2620:
2619:
2607:
2605:
2597:
2592:
2590:
2589:
2562:
2561:
2541:
2539:
2538:
2533:
2517:
2515:
2514:
2509:
2497:
2495:
2494:
2489:
2487:
2486:
2481:
2468:
2466:
2465:
2460:
2448:
2446:
2445:
2440:
2396:Alexei Abrikosov
2389:carbon nanotubes
2317:phase transition
2306:
2305:
2291:
2290:
2246:
2244:
2243:
2238:
2233:
2230:
2229:
2221:
2216:
2215:
2203:
2202:
2192:
2188:
2187:
2177:
2176:
2171:
2168:
2166:
2161:
2152:
2151:
2139:
2138:
2128:
2127:
2118:
2117:
2081:
2079:
2078:
2073:
2068:
2065:
2064:
2056:
2051:
2050:
2037:
2036:
2027:
2026:
1994:
1992:
1991:
1986:
1981:
1978:
1977:
1969:
1964:
1963:
1950:
1949:
1940:
1939:
1905:coherence length
1894:
1867:
1861:
1859:
1858:
1853:
1848:
1843:
1839:
1838:
1820:
1819:
1809:
1801:
1800:
1795:
1786:
1774:
1752:
1745:
1735:
1712:
1710:
1709:
1704:
1693:
1688:
1687:
1671:
1669:
1668:
1663:
1658:
1657:
1656:
1637:
1636:
1599:
1591:
1589:
1588:
1583:
1578:
1570:
1562:
1561:
1556:
1547:
1535:
1533:
1532:
1527:
1506:
1491:
1482:
1480:
1479:
1474:
1463:
1462:
1457:
1448:
1412:
1410:
1409:
1404:
1372:
1370:
1369:
1364:
1350:
1348:
1347:
1342:
1337:
1333:
1329:
1325:
1324:
1319:
1314:
1313:
1304:
1282:
1281:
1261:
1259:
1258:
1249:
1248:
1239:
1234:
1222:
1217:
1212:
1204:
1199:
1180:
1178:
1177:
1172:
1161:
1160:
1155:
1151:
1150:
1145:
1140:
1139:
1130:
1107:
1105:
1104:
1103:
1087:
1079:
1078:
1073:
1064:
1035:
1033:
1032:
1027:
1025:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
974:. By minimizing
973:
971:
970:
965:
960:
959:
950:
949:
920:
918:
917:
912:
910:
896:
878:
876:
875:
870:
868:
854:
852:
851:
846:
831:
829:
828:
823:
821:
820:
798:
796:
795:
790:
788:
787:
769:
767:
766:
761:
749:
747:
746:
741:
729:
727:
726:
721:
700:
698:
697:
692:
669:
667:
666:
661:
659:
658:
637:
635:
634:
629:
624:
622:
614:
613:
608:
602:
597:
596:
591:
587:
583:
579:
578:
573:
568:
567:
558:
530:
528:
527:
526:
510:
505:
504:
499:
490:
473:
465:
460:
459:
454:
445:
425:
424:
412:
411:
381:
379:
378:
373:
371:
363:
351:
349:
348:
343:
318:
316:
315:
310:
299:
298:
282:
280:
279:
274:
272:
271:
266:
248:
236:
234:
233:
228:
226:
225:
204:
187:
153:
151:
150:
145:
143:
142:
21:
8368:
8367:
8363:
8362:
8361:
8359:
8358:
8357:
8333:
8332:
8331:
8326:
8297:
8267:
8210:
8169:
8156:low temperature
8145:
8124:
8079:
8035:Meissner effect
7988:
7984:Silsbee current
7957:
7923:Ginzburg–Landau
7901:
7896:
7866:
7861:
7844:
7796:Quantum gravity
7723:
7682:Particle theory
7677:
7656:
7605:
7579:
7543:
7507:
7461:Low dimensional
7456:
7397:Ginzburg–Landau
7368:
7359:Topological QFT
7297:
7292:
7235:Sov. Phys. JETP
7221:Sov. Phys. JETP
7199:
7194:
7193:
7143:Seiberg, Nathan
7135:Gaiotto, Davide
7132:
7128:
7081:
7077:
7028:
7024:
6993:
6989:
6958:
6954:
6945:
6941:
6910:
6906:
6855:
6851:
6814:
6810:
6759:
6755:
6704:
6700:
6673:
6669:
6662:
6645:
6641:
6634:
6609:
6605:
6592:
6588:
6555:
6551:
6544:
6526:
6522:
6512:
6510:
6480:
6476:
6445:
6438:
6430:
6422:
6415:
6408:
6385:
6379:
6375:
6370:
6365:
6320:
6283:Nicholas Warner
6247:
6220:
6212:
6210:
6207:
6206:
6159:
6156:
6155:
6135:
6134:
6123:
6119:
6106:
6102:
6092:
6085:
6067:
6066:
6056:
6047:
6037:
6036:
6032:
6030:
6027:
6026:
6001:
5998:
5997:
5964:
5960:
5933:
5929:
5911:
5907:
5905:
5902:
5901:
5875:
5871:
5861:
5856:
5838:
5834:
5820:
5817:
5816:
5797:
5794:
5793:
5777:
5774:
5773:
5751:
5748:
5747:
5727:
5724:
5723:
5707:
5704:
5703:
5671:
5667:
5653:
5650:
5649:
5621:
5599:
5582:
5579:
5578:
5547:
5536:
5532:
5507:
5485:
5481:
5480:
5471:
5467:
5458:
5448:
5447:
5440:
5436:
5426:
5404:
5398:
5394:
5355:
5351:
5345:
5344:
5342:
5339:
5338:
5315:
5311:
5309:
5306:
5305:
5282:
5279:
5278:
5255:
5251:
5236:
5232:
5230:
5227:
5226:
5201:
5197:
5191:
5181:
5180:
5171:
5161:
5160:
5154:
5150:
5149:
5145:
5134:
5131:
5130:
5107:
5105:
5102:
5101:
5079:
5076:
5075:
5049:
5045:
5032:
5023:
5013:
5012:
5010:
5007:
5006:
4983:
4979:
4964:
4960:
4958:
4955:
4954:
4937:
4927:
4926:
4917:
4913:
4905:
4902:
4901:
4878:
4874:
4859:
4855:
4847:
4844:
4843:
4820:
4816:
4814:
4811:
4810:
4785:
4783:
4780:
4779:
4763:
4760:
4759:
4736:
4732:
4730:
4727:
4726:
4703:
4699:
4684:
4680:
4671:
4667:
4665:
4662:
4661:
4644:
4640:
4638:
4635:
4634:
4613:
4599:
4596:
4595:
4566:
4563:
4562:
4560:Riemann surface
4543:
4540:
4539:
4536:
4528:Hitchin systems
4488:
4485:
4484:
4465:
4462:
4461:
4436:
4433:
4432:
4412:
4409:
4408:
4391:
4386:
4385:
4377:
4374:
4373:
4343:
4340:
4339:
4336:Riemann surface
4319:
4316:
4315:
4308:
4282:
4278:
4270:
4267:
4266:
4247:
4244:
4243:
4222:
4218:
4216:
4213:
4212:
4159:
4155:
4153:
4150:
4149:
4118:
4114:
4101:
4097:
4087:
4072:
4068:
4066:
4063:
4062:
4028:
4024:
4000:
3996:
3960:
3959:
3957:
3954:
3953:
3924:
3921:
3920:
3903:
3899:
3897:
3894:
3893:
3876:
3875:
3869:
3865:
3853:
3849:
3832:
3828:
3819:
3815:
3800:
3796:
3792:
3785:
3781:
3777:
3775:
3763:
3759:
3755:
3748:
3744:
3740:
3738:
3737:
3733:
3723:
3714:
3710:
3698:
3694:
3680:
3676:
3670:
3666:
3654:
3650:
3646:
3639:
3635:
3631:
3629:
3628:
3624:
3581:
3579:
3576:
3575:
3540:
3537:
3536:
3504:
3503:
3501:
3498:
3497:
3481:
3478:
3477:
3449:
3446:
3445:
3416:
3412:
3403:
3399:
3391:
3388:
3387:
3371:
3368:
3367:
3351:
3348:
3347:
3331:
3328:
3327:
3299:
3296:
3295:
3273:
3270:
3269:
3245:
3242:
3241:
3235:electromagnetic
3218:
3215:
3214:
3198:
3195:
3194:
3174:
3171:
3170:
3138:
3135:
3134:
3115:
3112:
3111:
3095:
3087:
3085:
3082:
3081:
3065:
3062:
3061:
3045:
3042:
3041:
3021:
3017:
2999:
2995:
2985:
2977:
2975:
2958:
2955:
2954:
2932:
2924:
2921:
2920:
2896:
2893:
2892:
2876:
2873:
2872:
2837:
2833:
2825:
2822:
2821:
2793:
2790:
2789:
2769:
2764:
2763:
2761:
2758:
2757:
2732:
2721:
2717:
2704:
2700:
2699:
2689:
2680:
2676:
2658:
2654:
2647:
2643:
2637:
2633:
2615:
2611:
2601:
2593:
2591:
2585:
2581:
2557:
2556:
2554:
2551:
2550:
2527:
2524:
2523:
2503:
2500:
2499:
2482:
2477:
2476:
2474:
2471:
2470:
2454:
2451:
2450:
2434:
2431:
2430:
2409:
2370:
2356:
2341:
2326:
2313:
2303:
2301:
2288:
2286:
2256:
2225:
2217:
2208:
2204:
2198:
2194:
2193:
2183:
2179:
2178:
2175:
2162:
2157:
2144:
2140:
2134:
2130:
2129:
2123:
2119:
2116:
2108:
2105:
2104:
2091:
2060:
2052:
2046:
2042:
2038:
2032:
2028:
2025:
2017:
2014:
2013:
2007:
1973:
1965:
1959:
1955:
1951:
1945:
1941:
1938:
1930:
1927:
1926:
1920:
1901:
1890:
1880:
1872:gets closer to
1863:
1834:
1830:
1815:
1811:
1810:
1808:
1796:
1791:
1790:
1782:
1780:
1777:
1776:
1775:. Furthermore,
1770:
1768:
1747:
1741:
1731:
1729:
1689:
1683:
1679:
1677:
1674:
1673:
1652:
1651:
1647:
1632:
1628:
1605:
1602:
1601:
1595:
1569:
1557:
1552:
1551:
1543:
1541:
1538:
1537:
1515:
1512:
1511:
1505:
1493:
1486:
1458:
1453:
1452:
1444:
1430:
1427:
1426:
1425:simplifies to:
1419:
1398:
1395:
1394:
1358:
1355:
1354:
1320:
1309:
1305:
1303:
1287:
1283:
1277:
1273:
1272:
1268:
1254:
1250:
1244:
1240:
1238:
1230:
1218:
1205:
1203:
1195:
1187:
1184:
1183:
1156:
1146:
1135:
1131:
1129:
1113:
1109:
1108:
1099:
1095:
1091:
1086:
1074:
1069:
1068:
1060:
1046:
1043:
1042:
1021:
1019:
1016:
1015:
999:
996:
995:
979:
976:
975:
955:
951:
945:
941:
930:
927:
926:
906:
892:
890:
887:
886:
864:
862:
859:
858:
837:
834:
833:
816:
812:
810:
807:
806:
783:
779:
777:
774:
773:
755:
752:
751:
735:
732:
731:
706:
703:
702:
677:
674:
673:
654:
650:
648:
645:
644:
615:
609:
604:
603:
601:
592:
574:
563:
559:
557:
541:
537:
536:
532:
531:
522:
518:
514:
509:
500:
495:
494:
486:
464:
455:
450:
449:
441:
420:
416:
407:
403:
401:
398:
397:
367:
359:
357:
354:
353:
328:
325:
324:
294:
290:
288:
285:
284:
267:
262:
261:
244:
242:
239:
238:
209:
205:
200:
183:
166:
163:
162:
159:order parameter
138:
134:
132:
129:
128:
109:
58:Vitaly Ginzburg
52:, often called
42:
35:
28:
23:
22:
15:
12:
11:
5:
8366:
8356:
8355:
8350:
8345:
8328:
8327:
8325:
8324:
8319:
8314:
8309:
8304:
8299:
8295:
8291:
8286:
8281:
8275:
8273:
8269:
8268:
8266:
8265:
8260:
8255:
8250:
8245:
8240:
8235:
8233:electromagnets
8230:
8224:
8222:
8216:
8215:
8212:
8211:
8209:
8208:
8203:
8198:
8193:
8188:
8183:
8177:
8175:
8174:By composition
8171:
8170:
8168:
8167:
8162:
8157:
8153:
8151:
8147:
8146:
8144:
8143:
8141:unconventional
8138:
8132:
8130:
8129:By explanation
8126:
8125:
8123:
8122:
8117:
8116:
8115:
8110:
8105:
8096:
8094:
8087:
8085:Classification
8081:
8080:
8078:
8077:
8072:
8067:
8062:
8057:
8052:
8047:
8042:
8037:
8032:
8027:
8022:
8017:
8012:
8007:
8002:
7996:
7994:
7990:
7989:
7987:
7986:
7981:
7976:
7974:critical field
7971:
7965:
7963:
7959:
7958:
7956:
7955:
7950:
7945:
7943:Mattis–Bardeen
7940:
7935:
7930:
7928:Kohn–Luttinger
7925:
7920:
7915:
7909:
7907:
7903:
7902:
7895:
7894:
7887:
7880:
7872:
7863:
7862:
7849:
7846:
7845:
7843:
7842:
7837:
7832:
7831:
7830:
7820:
7815:
7810:
7805:
7804:
7803:
7793:
7792:
7791:
7781:
7776:
7771:
7766:
7761:
7756:
7751:
7746:
7741:
7739:Casimir effect
7735:
7733:
7729:
7728:
7725:
7724:
7722:
7721:
7716:
7714:Standard Model
7711:
7706:
7701:
7696:
7691:
7685:
7683:
7679:
7678:
7676:
7675:
7670:
7664:
7662:
7658:
7657:
7655:
7654:
7649:
7644:
7639:
7634:
7629:
7624:
7619:
7613:
7611:
7607:
7606:
7604:
7603:
7598:
7593:
7587:
7585:
7584:Superconformal
7581:
7580:
7578:
7577:
7572:
7567:
7565:Seiberg–Witten
7562:
7557:
7551:
7549:
7548:Supersymmetric
7545:
7544:
7542:
7541:
7536:
7531:
7526:
7521:
7515:
7513:
7509:
7508:
7506:
7505:
7500:
7495:
7490:
7485:
7480:
7475:
7470:
7464:
7462:
7458:
7457:
7455:
7454:
7449:
7444:
7439:
7434:
7429:
7424:
7419:
7414:
7409:
7404:
7399:
7394:
7389:
7383:
7381:
7374:
7370:
7369:
7367:
7366:
7361:
7356:
7351:
7346:
7341:
7336:
7331:
7326:
7321:
7316:
7311:
7305:
7303:
7299:
7298:
7291:
7290:
7283:
7276:
7268:
7262:
7261:
7251:
7241:
7233:L.P. Gor'kov,
7231:
7210:
7198:
7195:
7192:
7191:
7126:
7075:
7045:hep-th/9301042
7038:(1): 159–222.
7022:
7003:(2): 371–390.
6987:
6952:
6939:
6904:
6849:
6808:
6753:
6698:
6667:
6660:
6639:
6632:
6603:
6586:
6567:(3): 627–645.
6549:
6542:
6520:
6494:(6): 292–295.
6474:
6455:(7): 930–945.
6436:
6413:
6406:
6372:
6371:
6369:
6366:
6364:
6363:
6358:
6353:
6351:Quantum vortex
6348:
6342:
6337:
6332:
6327:
6321:
6319:
6316:
6287:superpotential
6246:
6243:
6230:
6227:
6223:
6219:
6215:
6203:
6202:
6190:
6187:
6184:
6181:
6178:
6175:
6172:
6169:
6166:
6163:
6149:
6148:
6132:
6126:
6122:
6118:
6115:
6112:
6109:
6105:
6099:
6096:
6091:
6088:
6086:
6084:
6081:
6078:
6075:
6072:
6069:
6068:
6065:
6062:
6059:
6057:
6055:
6050:
6044:
6041:
6035:
6034:
6011:
6008:
6005:
5978:
5975:
5972:
5967:
5963:
5959:
5956:
5953:
5950:
5947:
5944:
5941:
5936:
5932:
5928:
5925:
5922:
5919:
5914:
5910:
5898:
5897:
5886:
5883:
5878:
5874:
5867:
5864:
5860:
5855:
5852:
5849:
5846:
5841:
5837:
5833:
5830:
5827:
5824:
5801:
5781:
5761:
5758:
5755:
5731:
5711:
5700:
5699:
5688:
5685:
5682:
5679:
5674:
5670:
5666:
5663:
5660:
5657:
5643:
5642:
5628:
5625:
5620:
5617:
5614:
5611:
5606:
5603:
5598:
5595:
5592:
5589:
5586:
5569:
5568:
5556:
5550:
5545:
5539:
5535:
5531:
5528:
5525:
5522:
5519:
5514:
5511:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5484:
5479:
5474:
5470:
5466:
5461:
5455:
5452:
5446:
5443:
5439:
5433:
5430:
5425:
5422:
5419:
5416:
5411:
5408:
5401:
5397:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5368:
5364:
5361:
5358:
5354:
5348:
5321:
5318:
5314:
5286:
5264:
5261:
5258:
5254:
5250:
5245:
5242:
5239:
5235:
5223:
5222:
5210:
5204:
5200:
5194:
5188:
5185:
5179:
5174:
5168:
5165:
5157:
5153:
5148:
5144:
5141:
5138:
5110:
5089:
5086:
5083:
5058:
5055:
5052:
5048:
5044:
5039:
5036:
5031:
5026:
5020:
5017:
4992:
4989:
4986:
4982:
4978:
4975:
4972:
4967:
4963:
4940:
4934:
4931:
4925:
4920:
4916:
4912:
4909:
4887:
4884:
4881:
4877:
4873:
4868:
4865:
4862:
4858:
4854:
4851:
4829:
4826:
4823:
4819:
4792:
4789:
4767:
4745:
4742:
4739:
4735:
4712:
4709:
4706:
4702:
4698:
4693:
4690:
4687:
4683:
4679:
4674:
4670:
4647:
4643:
4620:
4617:
4612:
4609:
4606:
4603:
4576:
4573:
4570:
4547:
4535:
4532:
4516:spin structure
4495:
4492:
4472:
4469:
4446:
4443:
4440:
4416:
4394:
4389:
4384:
4381:
4353:
4350:
4347:
4323:
4307:
4304:
4285:
4281:
4277:
4274:
4264:codifferential
4251:
4225:
4221:
4209:
4208:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4162:
4158:
4143:
4142:
4131:
4127:
4121:
4117:
4113:
4110:
4107:
4104:
4100:
4094:
4091:
4086:
4083:
4080:
4075:
4071:
4045:
4044:
4031:
4027:
4023:
4020:
4017:
4014:
4011:
4008:
4003:
3999:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3963:
3934:
3931:
3928:
3906:
3902:
3892:That is, each
3890:
3889:
3872:
3868:
3864:
3861:
3856:
3852:
3848:
3844:
3840:
3835:
3831:
3827:
3822:
3818:
3814:
3811:
3803:
3799:
3795:
3788:
3784:
3780:
3774:
3766:
3762:
3758:
3751:
3747:
3743:
3736:
3730:
3727:
3722:
3717:
3713:
3709:
3706:
3701:
3697:
3693:
3689:
3683:
3679:
3673:
3669:
3665:
3657:
3653:
3649:
3642:
3638:
3634:
3627:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3583:
3561:curvature form
3544:
3535:The curvature
3521:
3518:
3515:
3510:
3507:
3485:
3465:
3462:
3459:
3456:
3453:
3419:
3415:
3411:
3406:
3402:
3398:
3395:
3375:
3355:
3335:
3315:
3312:
3309:
3306:
3303:
3283:
3280:
3277:
3249:
3222:
3202:
3178:
3154:
3151:
3148:
3145:
3142:
3119:
3098:
3094:
3090:
3069:
3049:
3038:
3037:
3024:
3020:
3016:
3013:
3010:
3007:
3002:
2998:
2994:
2988:
2984:
2980:
2974:
2971:
2968:
2965:
2962:
2935:
2931:
2928:
2900:
2880:
2860:
2857:
2854:
2851:
2848:
2845:
2840:
2836:
2832:
2829:
2809:
2806:
2803:
2800:
2797:
2772:
2767:
2754:
2753:
2741:
2735:
2730:
2724:
2720:
2716:
2713:
2710:
2707:
2703:
2696:
2693:
2688:
2683:
2679:
2675:
2672:
2669:
2666:
2661:
2657:
2653:
2650:
2646:
2640:
2636:
2632:
2629:
2626:
2623:
2618:
2614:
2610:
2604:
2600:
2596:
2588:
2584:
2580:
2577:
2574:
2571:
2568:
2565:
2560:
2531:
2507:
2485:
2480:
2458:
2438:
2408:
2405:
2365:
2351:
2339:
2330:Meissner state
2325:
2322:
2312:
2309:
2254:
2248:
2247:
2236:
2228:
2224:
2220:
2214:
2211:
2207:
2201:
2197:
2191:
2186:
2182:
2174:
2165:
2160:
2156:
2150:
2147:
2143:
2137:
2133:
2126:
2122:
2115:
2112:
2089:
2083:
2082:
2071:
2063:
2059:
2055:
2049:
2045:
2041:
2035:
2031:
2024:
2021:
2005:
1996:
1995:
1984:
1976:
1972:
1968:
1962:
1958:
1954:
1948:
1944:
1937:
1934:
1918:
1900:
1897:
1883:
1882:
1876:
1851:
1846:
1842:
1837:
1833:
1829:
1826:
1823:
1818:
1814:
1807:
1804:
1799:
1794:
1789:
1785:
1764:
1754:
1725:
1702:
1699:
1696:
1692:
1686:
1682:
1661:
1655:
1650:
1646:
1643:
1640:
1635:
1631:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1581:
1576:
1573:
1568:
1565:
1560:
1555:
1550:
1546:
1525:
1522:
1519:
1501:
1472:
1469:
1466:
1461:
1456:
1451:
1447:
1443:
1440:
1437:
1434:
1418:
1415:
1402:
1362:
1340:
1336:
1332:
1328:
1323:
1317:
1312:
1308:
1302:
1299:
1296:
1293:
1290:
1286:
1280:
1276:
1271:
1267:
1264:
1257:
1253:
1247:
1243:
1237:
1233:
1227:
1221:
1215:
1211:
1208:
1202:
1198:
1194:
1191:
1170:
1167:
1164:
1159:
1154:
1149:
1143:
1138:
1134:
1128:
1125:
1122:
1119:
1116:
1112:
1102:
1098:
1094:
1090:
1085:
1082:
1077:
1072:
1067:
1063:
1059:
1056:
1053:
1050:
1024:
1003:
983:
963:
958:
954:
948:
944:
940:
937:
934:
923:
922:
909:
905:
902:
899:
895:
884:
867:
856:
844:
841:
819:
815:
804:
801:effective mass
786:
782:
771:
759:
739:
719:
716:
713:
710:
690:
687:
684:
681:
671:
657:
653:
627:
621:
618:
612:
607:
600:
595:
590:
586:
582:
577:
571:
566:
562:
556:
553:
550:
547:
544:
540:
535:
525:
521:
517:
513:
508:
503:
498:
493:
489:
485:
482:
479:
476:
471:
468:
463:
458:
453:
448:
444:
440:
437:
434:
431:
428:
423:
419:
415:
410:
406:
370:
366:
362:
341:
338:
335:
332:
308:
305:
302:
297:
293:
270:
265:
260:
257:
254:
251:
247:
224:
221:
218:
215:
212:
208:
203:
199:
196:
193:
190:
186:
182:
179:
176:
173:
170:
141:
137:
108:
105:
56:, named after
26:
9:
6:
4:
3:
2:
8365:
8354:
8351:
8349:
8346:
8344:
8341:
8340:
8338:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8292:
8290:
8287:
8285:
8282:
8280:
8277:
8276:
8274:
8270:
8264:
8261:
8259:
8256:
8254:
8251:
8249:
8246:
8244:
8241:
8239:
8236:
8234:
8231:
8229:
8226:
8225:
8223:
8221:
8217:
8207:
8204:
8202:
8199:
8197:
8194:
8192:
8191:heavy fermion
8189:
8187:
8184:
8182:
8179:
8178:
8176:
8172:
8166:
8163:
8161:
8158:
8155:
8154:
8152:
8148:
8142:
8139:
8137:
8134:
8133:
8131:
8127:
8121:
8120:ferromagnetic
8118:
8114:
8111:
8109:
8106:
8104:
8101:
8100:
8098:
8097:
8095:
8091:
8088:
8086:
8082:
8076:
8073:
8071:
8068:
8066:
8065:supercurrents
8063:
8061:
8058:
8056:
8053:
8051:
8048:
8046:
8043:
8041:
8038:
8036:
8033:
8031:
8028:
8026:
8023:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7997:
7995:
7991:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7967:
7966:
7964:
7960:
7954:
7951:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7926:
7924:
7921:
7919:
7916:
7914:
7911:
7910:
7908:
7904:
7900:
7893:
7888:
7886:
7881:
7879:
7874:
7873:
7870:
7860:
7852:
7847:
7841:
7838:
7836:
7835:Quantum logic
7833:
7829:
7826:
7825:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7802:
7799:
7798:
7797:
7794:
7790:
7787:
7786:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7769:Quantum chaos
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7744:Cosmic string
7742:
7740:
7737:
7736:
7734:
7730:
7720:
7717:
7715:
7712:
7710:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7690:
7687:
7686:
7684:
7680:
7674:
7671:
7669:
7666:
7665:
7663:
7659:
7653:
7650:
7648:
7645:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7623:
7620:
7618:
7617:Pure 4D N = 1
7615:
7614:
7612:
7608:
7602:
7599:
7597:
7594:
7592:
7589:
7588:
7586:
7582:
7576:
7573:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7552:
7550:
7546:
7540:
7537:
7535:
7532:
7530:
7527:
7525:
7522:
7520:
7517:
7516:
7514:
7510:
7504:
7501:
7499:
7498:Thirring–Wess
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7476:
7474:
7473:Bullough–Dodd
7471:
7469:
7468:2D Yang–Mills
7466:
7465:
7463:
7459:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7384:
7382:
7378:
7375:
7371:
7365:
7362:
7360:
7357:
7355:
7352:
7350:
7347:
7345:
7344:String theory
7342:
7340:
7337:
7335:
7332:
7330:
7327:
7325:
7322:
7320:
7317:
7315:
7314:Axiomatic QFT
7312:
7310:
7309:Algebraic QFT
7307:
7306:
7304:
7300:
7296:
7289:
7284:
7282:
7277:
7275:
7270:
7269:
7266:
7260:
7256:
7252:
7250:
7246:
7242:
7240:, 1364 (1959)
7239:
7236:
7232:
7229:
7225:
7222:
7218:
7215:
7211:
7208:
7205:
7201:
7200:
7188:
7184:
7180:
7176:
7172:
7168:
7163:
7158:
7154:
7150:
7149:
7144:
7140:
7139:Gukov, Sergei
7136:
7130:
7122:
7118:
7113:
7108:
7103:
7098:
7094:
7090:
7086:
7079:
7071:
7067:
7063:
7059:
7055:
7051:
7046:
7041:
7037:
7033:
7026:
7018:
7014:
7010:
7006:
7002:
6998:
6991:
6983:
6979:
6975:
6971:
6967:
6963:
6956:
6949:
6943:
6935:
6931:
6927:
6923:
6919:
6915:
6908:
6900:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6853:
6845:
6841:
6836:
6831:
6827:
6823:
6819:
6812:
6804:
6800:
6796:
6792:
6788:
6784:
6780:
6776:
6772:
6768:
6764:
6757:
6749:
6745:
6741:
6737:
6733:
6729:
6725:
6721:
6717:
6713:
6709:
6702:
6694:
6690:
6686:
6682:
6678:
6671:
6663:
6657:
6653:
6649:
6643:
6635:
6633:3-540-42627-2
6629:
6625:
6620:
6619:
6613:
6607:
6600:
6596:
6590:
6582:
6578:
6574:
6570:
6566:
6562:
6561:
6553:
6545:
6539:
6535:
6531:
6524:
6509:
6505:
6501:
6497:
6493:
6489:
6485:
6478:
6470:
6466:
6462:
6458:
6454:
6450:
6443:
6441:
6429:
6428:
6420:
6418:
6409:
6403:
6399:
6395:
6391:
6384:
6377:
6373:
6362:
6359:
6357:
6354:
6352:
6349:
6346:
6343:
6341:
6338:
6336:
6335:Landau theory
6333:
6331:
6328:
6326:
6323:
6322:
6315:
6312:
6311:Edward Witten
6308:
6304:
6300:
6296:
6292:
6288:
6284:
6280:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6242:
6228:
6225:
6217:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6154:
6153:
6152:
6130:
6124:
6116:
6110:
6107:
6103:
6097:
6094:
6089:
6087:
6079:
6076:
6070:
6063:
6060:
6058:
6053:
6048:
6025:
6024:
6023:
6009:
6006:
6003:
5994:
5992:
5989:is the first
5965:
5961:
5957:
5942:
5934:
5930:
5926:
5920:
5912:
5908:
5884:
5881:
5872:
5865:
5862:
5858:
5853:
5847:
5839:
5835:
5831:
5828:
5825:
5822:
5815:
5814:
5813:
5779:
5759:
5756:
5753:
5745:
5729:
5683:
5677:
5668:
5664:
5658:
5655:
5648:
5647:
5646:
5623:
5618:
5615:
5612:
5609:
5604:
5601:
5596:
5590:
5584:
5577:
5576:
5575:
5574:
5554:
5548:
5543:
5537:
5529:
5523:
5520:
5512:
5509:
5504:
5498:
5495:
5492:
5486:
5482:
5477:
5472:
5464:
5459:
5441:
5437:
5428:
5423:
5420:
5417:
5414:
5409:
5406:
5395:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5366:
5362:
5359:
5356:
5352:
5337:
5336:
5335:
5319:
5316:
5312:
5303:
5300:
5284:
5262:
5259:
5256:
5252:
5248:
5243:
5240:
5237:
5233:
5208:
5202:
5192:
5177:
5172:
5155:
5146:
5142:
5139:
5136:
5129:
5128:
5127:
5125:
5087:
5084:
5081:
5072:
5056:
5053:
5050:
5046:
5042:
5029:
5024:
4990:
4987:
4984:
4980:
4976:
4970:
4965:
4938:
4923:
4918:
4910:
4907:
4900:and likewise
4885:
4882:
4879:
4875:
4871:
4866:
4863:
4860:
4856:
4852:
4849:
4827:
4824:
4821:
4808:
4787:
4765:
4743:
4740:
4737:
4710:
4707:
4704:
4696:
4691:
4688:
4685:
4677:
4672:
4645:
4610:
4604:
4601:
4594:
4590:
4571:
4568:
4561:
4545:
4531:
4529:
4525:
4521:
4517:
4512:
4510:
4493:
4490:
4470:
4467:
4459:
4441:
4430:
4414:
4392:
4382:
4379:
4371:
4367:
4351:
4348:
4345:
4338:, and taking
4337:
4321:
4313:
4312:string theory
4303:
4301:
4283:
4279:
4275:
4272:
4265:
4249:
4241:
4223:
4219:
4192:
4189:
4186:
4183:
4177:
4174:
4171:
4168:
4165:
4160:
4156:
4148:
4147:
4146:
4129:
4125:
4119:
4111:
4105:
4102:
4098:
4092:
4089:
4084:
4081:
4078:
4073:
4069:
4061:
4060:
4059:
4057:
4052:
4050:
4029:
4021:
4012:
4006:
4001:
3997:
3993:
3987:
3981:
3978:
3975:
3969:
3952:
3951:
3950:
3948:
3932:
3929:
3926:
3904:
3900:
3870:
3866:
3862:
3859:
3854:
3850:
3846:
3842:
3833:
3829:
3825:
3820:
3816:
3809:
3801:
3797:
3786:
3782:
3772:
3764:
3760:
3749:
3745:
3734:
3728:
3725:
3720:
3715:
3711:
3707:
3704:
3699:
3695:
3691:
3687:
3681:
3677:
3671:
3667:
3663:
3655:
3651:
3640:
3636:
3625:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3574:
3573:
3572:
3570:
3569:vector bundle
3566:
3562:
3558:
3542:
3533:
3516:
3483:
3460:
3457:
3454:
3443:
3439:
3435:
3417:
3413:
3409:
3404:
3400:
3396:
3393:
3373:
3353:
3333:
3313:
3310:
3307:
3304:
3301:
3281:
3278:
3275:
3267:
3263:
3247:
3239:
3236:
3220:
3200:
3192:
3176:
3168:
3152:
3149:
3146:
3143:
3140:
3131:
3117:
3092:
3067:
3047:
3022:
3018:
3014:
3011:
3008:
3005:
3000:
2996:
2992:
2982:
2972:
2966:
2960:
2953:
2952:
2951:
2950:
2929:
2926:
2918:
2914:
2898:
2878:
2855:
2852:
2849:
2843:
2838:
2830:
2804:
2801:
2798:
2788:
2770:
2739:
2733:
2728:
2722:
2714:
2708:
2705:
2701:
2694:
2691:
2686:
2681:
2673:
2670:
2664:
2659:
2651:
2644:
2638:
2634:
2630:
2627:
2624:
2621:
2616:
2612:
2608:
2598:
2586:
2582:
2578:
2572:
2569:
2566:
2549:
2548:
2547:
2545:
2529:
2521:
2505:
2483:
2456:
2436:
2427:
2425:
2421:
2418:
2414:
2404:
2402:
2397:
2392:
2390:
2386:
2382:
2378:
2374:
2368:
2364:
2360:
2359:magnetic flux
2354:
2350:
2346:
2342:
2335:
2331:
2321:
2318:
2308:
2299:
2295:
2284:
2280:
2276:
2272:
2268:
2264:
2259:
2253:
2234:
2222:
2212:
2209:
2205:
2199:
2195:
2189:
2184:
2180:
2172:
2163:
2158:
2154:
2148:
2145:
2141:
2135:
2131:
2124:
2120:
2113:
2110:
2103:
2102:
2101:
2099:
2098:London theory
2095:
2088:
2069:
2057:
2047:
2043:
2039:
2033:
2029:
2022:
2019:
2012:
2011:
2010:
2008:
2001:
1982:
1970:
1960:
1956:
1952:
1946:
1942:
1935:
1932:
1925:
1924:
1923:
1921:
1914:
1910:
1906:
1896:
1893:
1888:
1879:
1875:
1871:
1866:
1849:
1844:
1835:
1831:
1827:
1824:
1816:
1812:
1805:
1802:
1797:
1787:
1773:
1767:
1763:
1759:
1755:
1750:
1744:
1739:
1734:
1728:
1724:
1720:
1716:
1715:
1714:
1700:
1697:
1694:
1690:
1684:
1680:
1648:
1644:
1641:
1633:
1629:
1625:
1619:
1613:
1610:
1607:
1598:
1592:
1579:
1574:
1571:
1566:
1563:
1558:
1548:
1523:
1520:
1517:
1508:
1504:
1500:
1496:
1489:
1483:
1470:
1467:
1464:
1459:
1449:
1441:
1438:
1435:
1432:
1424:
1414:
1400:
1392:
1388:
1384:
1380:
1376:
1360:
1351:
1338:
1334:
1330:
1326:
1315:
1310:
1306:
1300:
1291:
1288:
1284:
1278:
1274:
1269:
1265:
1262:
1255:
1251:
1245:
1241:
1235:
1225:
1213:
1209:
1206:
1200:
1192:
1181:
1168:
1165:
1162:
1157:
1152:
1141:
1136:
1132:
1126:
1117:
1114:
1110:
1100:
1096:
1092:
1088:
1083:
1080:
1075:
1065:
1057:
1054:
1051:
1048:
1040:
1039:
1001:
981:
961:
956:
952:
946:
942:
938:
935:
932:
903:
897:
885:
882:
857:
842:
839:
817:
813:
805:
802:
784:
780:
772:
757:
737:
714:
708:
685:
679:
672:
655:
651:
643:
642:
641:
638:
625:
619:
616:
610:
598:
593:
588:
584:
580:
569:
564:
560:
554:
545:
542:
538:
533:
523:
519:
515:
511:
506:
501:
491:
480:
474:
469:
466:
461:
456:
446:
435:
429:
426:
421:
417:
413:
408:
404:
395:
393:
389:
385:
364:
336:
330:
322:
321:wave function
303:
295:
291:
268:
255:
249:
219:
213:
210:
206:
194:
188:
180:
174:
168:
160:
157:
139:
135:
126:
122:
118:
114:
104:
102:
101:string theory
98:
94:
90:
86:
81:
79:
75:
71:
67:
63:
59:
55:
51:
47:
40:
33:
32:Landau theory
19:
8201:oxypnictides
8136:conventional
8075:superstripes
8020:flux pumping
8015:flux pinning
8010:Cooper pairs
7922:
7850:
7779:Quantum foam
7719:Stueckelberg
7673:Chern–Simons
7610:Supergravity
7396:
7349:Supergravity
7334:Gauge theory
7237:
7234:
7223:
7220:
7216:
7213:
7206:
7203:
7152:
7146:
7129:
7095:(1): 1–106.
7092:
7088:
7078:
7035:
7031:
7025:
7000:
6996:
6990:
6968:(1): 51–58.
6965:
6961:
6955:
6947:
6942:
6917:
6913:
6907:
6866:
6862:
6852:
6825:
6821:
6811:
6770:
6766:
6756:
6715:
6711:
6701:
6676:
6670:
6651:
6648:Jost, Jürgen
6642:
6617:
6612:Jost, Jürgen
6606:
6589:
6564:
6558:
6552:
6529:
6523:
6511:. Retrieved
6491:
6487:
6477:
6452:
6449:ChemPhysChem
6448:
6426:
6389:
6376:
6356:Higgs bundle
6325:Flux pinning
6306:
6299:sigma models
6291:Brian Greene
6270:
6259:vacuum state
6248:
6204:
6150:
5995:
5899:
5701:
5644:
5570:
5298:
5224:
5073:
4806:
4591:as a sum of
4537:
4534:Self-duality
4513:
4508:
4460:to 1, while
4309:
4210:
4144:
4053:
4046:
3891:
3534:
3132:
3039:
2755:
2428:
2410:
2393:
2379:. Most pure
2366:
2362:
2352:
2348:
2337:
2327:
2314:
2311:Fluctuations
2297:
2282:
2274:
2270:
2266:
2262:
2260:
2251:
2249:
2093:
2086:
2084:
2003:
1999:
1997:
1916:
1912:
1908:
1902:
1891:
1884:
1877:
1873:
1869:
1864:
1771:
1765:
1761:
1757:
1748:
1742:
1737:
1732:
1726:
1722:
1718:
1596:
1593:
1509:
1502:
1498:
1494:
1487:
1484:
1422:
1420:
1386:
1382:
1373:denotes the
1352:
1182:
1041:
1037:
924:
639:
396:
392:field theory
110:
107:Introduction
82:
53:
49:
43:
8060:SU(2) color
8040:Homes's law
7661:Topological
7575:Wess–Zumino
7488:Sine-Gordon
7478:Gross–Neveu
7387:Born–Infeld
7354:Thermal QFT
6279:Cumrun Vafa
5991:Chern class
5573:volume form
5124:line bundle
4366:line bundle
3438:Lie algebra
3266:non-Abelian
2949:volume form
2469:with fiber
2296:those with
1375:dissipation
388:free energy
125:free energy
89:Lev Gor'kov
8353:Lev Landau
8337:Categories
8196:iron-based
8055:reentrance
7442:Yang–Mills
6368:References
5744:Hodge star
4807:vice-versa
4524:integrable
4364:; i.e., a
2544:Lagrangian
1998:while for
1887:superfluid
62:Lev Landau
7993:Phenomena
7851:See also:
7570:Super QCD
7524:Liouville
7512:Conformal
7483:Schwinger
7187:118498045
7162:1307.2578
7155:(9): 70,
7121:115154206
7102:0712.4021
6934:0024-6093
6899:122906366
6891:0010-3616
6844:0022-040X
6795:0010-3616
6748:122086974
6740:0010-3616
6693:0024-6115
6229:σ
6226:≤
6218:ψ
6189:Σ
6186:
6180:σ
6177:≤
6171:
6165:π
6117:ψ
6111:−
6108:σ
6071:∗
6054:ψ
6043:¯
6040:∂
6004:ψ
5974:Σ
5958:∈
5952:Σ
5877:Σ
5873:∫
5866:π
5826:
5800:Σ
5757:
5730:∗
5710:Σ
5678:∗
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5669:∫
5662:Σ
5659:
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5616:∧
5585:∗
5530:ψ
5524:−
5521:σ
5505:−
5493:−
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5432:¯
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5400:Σ
5396:∫
5386:
5380:σ
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4682:Ω
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4273:δ
4224:∗
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4074:∗
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3930:×
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3871:ν
3860:∧
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3802:ν
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3641:ν
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3595:∘
3464:⟩
3461:⋅
3455:⋅
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3009:⋯
3006:∧
2961:∗
2930:∈
2927:σ
2899:β
2879:α
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2856:ψ
2850:ψ
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2808:⟩
2805:⋅
2799:⋅
2796:⟨
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2709:−
2706:σ
2674:ψ
2628:∧
2625:⋯
2622:∧
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2567:ψ
2506:ψ
2381:elemental
2377:quantized
2223:α
2210:∗
2196:μ
2190:β
2185:∗
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2125:∗
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2058:α
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2030:ℏ
2020:ξ
1971:α
1961:∗
1943:ℏ
1933:ξ
1845:β
1828:−
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1433:α
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1387:real part
1331:ψ
1311:∗
1301:−
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1266:
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1193:×
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620:π
585:ψ
565:∗
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549:ℏ
543:−
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492:ψ
475:β
447:ψ
430:α
384:gradients
365:ψ
331:ψ
250:ψ
214:ϕ
189:ψ
169:ψ
111:Based on
8228:cryotron
8186:cuprates
8181:covalent
7938:Matthias
7906:Theories
7647:Type IIB
7642:Type IIA
7627:4D N = 8
7622:4D N = 1
7591:6D (2,0)
7555:4D N = 1
7534:Polyakov
7493:Thirring
7302:Theories
7255:pdf file
7245:pdf file
7070:16122549
6803:59456762
6513:April 7,
6469:15298379
6318:See also
5645:so that
4334:being a
3434:one-form
3432:to be a
2401:vortices
1862:that is
323:. While
127:density
121:Ginzburg
78:cuprates
8322:more...
8206:organic
7749:History
7732:Related
7529:Minimal
7380:Regular
7167:Bibcode
7050:Bibcode
7005:Bibcode
6970:Bibcode
6871:Bibcode
6775:Bibcode
6720:Bibcode
6569:Bibcode
6496:Bibcode
6347:systems
6265:with a
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879:is the
156:complex
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8099:Types
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7637:Type I
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7197:Papers
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386:, the
161:field
113:Landau
8312:TBCCO
8284:BSCCO
8263:wires
8258:SQUID
7828:links
7801:links
7789:links
7709:NMSSM
7694:Fermi
7437:Soler
7407:Proca
7259:video
7249:video
7183:S2CID
7157:arXiv
7117:S2CID
7097:arXiv
7066:S2CID
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4953:with
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883:, and
8317:YBCO
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8302:NbSn
8289:LBCO
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7596:ABJM
7503:Toda
7153:2013
6930:ISSN
6887:ISSN
6840:ISSN
6791:ISSN
6736:ISSN
6689:ISSN
6656:ISBN
6628:ISBN
6538:ISBN
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5277:and
5005:and
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60:and
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8243:NMR
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7652:11D
7257:or
7247:or
7175:doi
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