Nuclear Physics B344 (1990) 627—645 North-Holland ON THE REMARKABLE STRUCTURE OF THE SUPERCONDUCTING INTERMEDIATE STATE David J.E. CALLAWAY* Department of Physics, The Rockefeller Unicersity, 1230 York A venue, New York, NY 10021, USA Received 19 December 1989 If a large square plate of a type I superconductor is placed in a perpendicular magnetic field, the field will penetrate it in a pattern of domains. The intermediate-state problem is to predict this pattern. In the critical Ginzburg—Landau theory, the horizontal and vertical directions of the plate are essentially Fourier conjugate coordinates, like position and momen- tum in quantum mechanics. Thus, the intermediate state allows us a rare direct glimpse of quantum phase space. It is also demonstrated that, although Landau’s (1937) textbook model is inconsistent with the Ginzburg—Landau equations, its qualitative nature is correct. Elongated structures consistent with a complete spontaneous breakdown of discrete rotational invariance are predicted. Comments with regard to lattice Higgs simulations are also made. 1. Prolegomena The problem of the magnetic intermediate state in superconductors is both extremely fascinating and extraordinarily difficult. Its solution involves the analysis of highly degenerate ground states and intricate fractal-like patterns on many length scales (cf. sect. 7) and utilizes arcane mathematical structures that are most familiar perhaps to the string theorist. Yet, for all the complexity of its solution, the problem itself can be stated with deceptive ease: Consider a block of type I superconducting material placed in a weak magnetic field. Because of the Meissner effect the magnetic field is expelled. As the field strength is increased the magnetic field penetrates the superconductor. The problem is to predict this pattern of penetration. 2. Background and synopsis The simplest illustration of the magnetic intermediate state is a problem originally addressed by Landau [1] and considered in detail here. Landau’s prob- * Work supported in part by the US Department of Energy under contract no. DE-ACO2- 87ER40325-Task B 1. 0550-3213/90/$03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland)