# Introduction to Path Integral Methods in Condensed Matter Physics

Mentor: | Yaroslav Rodionov |
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Revision: | 7 Dec 2013 |

## Course Summary

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.

## Course Format

Hours of lecture | Hours of discussion | Hours of independent study | Hours total |
---|---|---|---|

34 | 17 | 45 | 96 |

**Please note** that students are expected to study outside of class for three hours for every hour in class.

## Course Content

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.

## Textbooks

Primary textbooks:

- Alexander Altland and Ben D. Simons.
*Condensed Matter Field Theory*. Cambridge University Press, Cambridge ; New York, 2 edition edition, April 2010. - Jean Zinn-Justin.
*Quantum Field Theory and Critical Phenomena*. Oxford University Press, Oxford : New York, 4 edition edition, August 2002. - Lev D Landau and Evgenij M Lifshits.
*Course of theoretical physics. Statistical Physics*. volume 5. Pergamon Press., Oxford [u.a.], 2007.

Additional textbooks:

- Alexei M. Tsvelik.
*Quantum Field Theory in Condensed Matter Physics*. Cambridge University Press, Cambridge; New York, 2 edition edition, January 2007. - Hagen Kleinert.
*Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, and Financial Markets, Third Edition*. World Scientific Publishing Company, Singapore; River Edge, NJ, 3 edition edition, February 2004. - Michael E. Peskin and Dan V. Schroeder.
*An Introduction To Quantum Field Theory*. Westview Press, Reading, Mass, first edition edition edition, October 1995.

## Homework Assignments

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.

## Grading

Class participation | 10% |

Homework assignments | 20% |

Midterm exam | 20% |

Final exam | 50% |