Condensed Matter Seminars

Second-order differential equations are ubiquitous in physics, but every once in a while there are special cases where the order of these equations may be reduced. One example of such a reduction is the quantum harmonic oscillator as solved by P. Dirac. Dirac introduced the annihilation and creation operators, which permit solving of the second-order problem presented by the linear Schrödinger equation by reduction to a first-order linear equation. In the context of superconductivity, a system of non-linear second-order differential equations describing vortex lines in superconductors may be reduced to first-order at a particular value of the Ginzburg-Landau parameter. This reduction can be achieved by two procedures. One approach uses an ingenious minimization of the Ginzburg-Landau free energy functional tailored to the presence of a topological defect (the Bogomol’nyi procedure). The other uses the Ginzburg-Landau equations and operators analogous to Dirac’s annihilation and creation operators (the Sarma procedure). The Bogomol’nyi procedure leads to the famous Bogomol’nyi equations, published in 1976. The Sarma procedure was never published in a journal but is hinted at in P. G. de Gennes’s 1966 classic Superconductivity of Metals and Alloys.  We will show that, in the particular case of the vortex line, the Sarma procedure recovers the Bogomol’nyi equations and that the Sarma procedure is in fact more general, as to be applicable it does not rely upon the presence of a topological defect.