The idea of the course is to get students acquainted with path integral approach to problems of contemporary condensed matter physics. The aim is to give students firm command of this approach via carefully selected examples and problems. The course contains mathematical digression into complex calculus, the basics of second quantization, field quantization, path integral description of quantum statistical mechanics, finite temperature perturbation theory, theory of linear response, basics of renormalization group analysis and effective field theory. The final project consists of the theoretical description of single electron transistor via effective Ambegaokar-Eckern-Schoen action.
The plan is to work through the following topics:
Selected topics from complex calculus
Integrals in the complex plane, computation of infinite sums and products,
Rouche's theorem. Saddle point methods in contour integration. Method of contour integral in differential equations. The construction of solution asymptotes. Typical solutions of Schroedinger equation, Laguerre and Hermite polynomials.
Second quantization of many particle systems and quantized fields
General scheme of second quantization, gases of free bosons and fermions, the second quantization of oscillator, Baker-Campbell-Housdorff –theorem, quantization of scalar and vector fields, spontaneous symmetry breaking.
Introduction into path integral formalism
Computation of propagation amplitude in quantum mechanics, general idea of a path integral, Gaussian integrals, coherent state basis, grassmann variables and fermion path integral, computation of partition function for free particle and oscillator.
Linear response and perturbation theory
Linear response, calculation of physical observables from linear response theory, simplest field correlators, perturbation theory.
Renormalization group, Hubbard-Stratonovich transformation
The renormalization idea, renormalization semi-group, Hubbard-Stratonovich transformation, BSC model, Gross-Neveu model. Ambegaokar-Eckern-Schoen effective action for single electron transistor.
Weekly, 1 problem set in total, due at the beginning of the lecture. You may also submit via e-mail before the due date/time. It is of outmost importance that you invest your own effort into solving problems. Should you consult any sources, please provide references. Homework assignments should be typed. Legible handwritten assignments are acceptable.