Field Theories in Condensed Matter Physics

by Vij; D. R.

• ISBN13: 9780750308762
• ISBN10: 0750308761
• Format: Nonspecific Binding
• Copyright: 2002-05-30
• Publisher: CRC Press

Summary

The application of field theoretic techniques to problems in condensed matter physics has generated an array of concepts and mathematical techniques to attack a range of problems such as the theory of quantum phase transitions, the quantum Hall effect, and quantum wires. While concepts such as the renormalization group, topology, and bosonization have become necessary tools for the condensed matter physicist, enough open problems and interesting applications remain to drive much activity in this area in the coming years. Field Theories in Condensed Matter Physics presents a comprehensive survey of the concepts, techniques, and applications of the field. Written by experts and carefully edited, the book provides the necessary background for graduate students entering the area of modern condensed matter physics. It also supplies field theorists with a valuable introduction to the areas in condensed matter physics where field theoretic concepts can be fruitfully applied.

Table of Contents

Preface	xiii
Introduction	1	(6)
Quantum Many Particle Physics	7	(62)
Pinaki Majumdar
Preamble	8	(1)
Introduction	8	(2)
Introduction to many particle physics	10	(12)
Phases of many particle systems	10	(2)
Quantities of physical interest	12	(2)
Fermi and Bose liquids	14	(8)
Phase transitions and broken symmetry	22	(9)
Phase transitions and symmetry breaking	22	(3)
Symmetry breaking and interactions in BEC	25	(6)
Normal Fermi systems: model problems	31	(15)
Neutral fermions: dilute hardcore Fermi gas	34	(5)
Charged fermions: the electron gas	39	(7)
Electrons and phonons: Migdal-Eliashberg theory	46	(17)
Weak coupling theory: BCS	48	(4)
The normal state: Migdal theory	52	(4)
BCS theory: Greens function approach	56	(2)
Superconductivity: Eliashberg theory	58	(5)
Conclusion: `field theory' and many particle physics	63	(6)
Critical Phenomena	69	(50)
Somendra M. Bhattacharjee
Preamble	70	(2)
Large system: Thermodynamic limit	72	(1)
Where is the problem?	72	(2)
Recapitulation - A few formal stuff	74	(4)
Extensivity	74	(2)
Convexity: Stability	76	(2)
Consequences of divergence	78	(3)
Generalized scaling	81	(12)
One variable: Temperature	81	(6)
Solidarity with thermodynamics	87	(1)
More variables: Temperature and field	88	(4)
On exponent relations	92	(1)
Relevance, irrelevance and universality	93	(2)
Digression	95	(4)
A first-order transition: α=1	95	(2)
Example: Polymers: no ``ordering''	97	(2)
Exponents and correlations	99	(6)
Correlation function	99	(2)
Relations among the exponents	101	(2)
Length-scale dependent parameters	103	(2)
Models as examples: Gaussian and φ4	105	(7)
Specific heat for the Gaussian model	106	(1)
Cut-off and anomalous dimensions	107	(3)
Through correlations	110	(2)
Epilogue	112	(7)
Phase Transitions and Critical Phenomena	119	(70)
Deepak Kumar
Introduction	120	(1)
Thermodynamic stability	121	(5)
Lattice gas: mean field approximation	126	(8)
Landau theory	134	(4)
Spatial correlations	138	(3)
Breakdown of mean field theory	141	(2)
Ginzburg-Landau free energy functional	143	(1)
Renormalisation group (RG)	144	(2)
RG for a one dimensional Ising chain	146	(4)
RG for a two-dimensional Ising model	150	(8)
General features of RG	158	(6)
Irrelevant variables	163	(1)
RG scaling for correlation functions	164	(3)
RG for Ginzburg-Landau model	167	(9)
Tree-level approximation	170	(2)
Critical exponents for d > 4	172	(3)
Anomalous dimensions	175	(1)
Perturbation series for d < 4	176	(7)
Generalisation to a n-component model	183	(6)
Topological Defects	189	(50)
Ajit M. Srivastava
The subject of topological defect	191	(2)
What is a topological defect?	193	(5)
Meaning of order parameter	194	(1)
Spontaneous symmetry breakdown(SSB)	195	(2)
SSB in particle physics	197	(1)
Order parameter space	197	(1)
The domain wall	198	(5)
Why defect?	200	(1)
Why topological?	201	(1)
Energy considerations	202	(1)
Examples of topological defects	203	(6)
Condensed matter versus particle physics	209	(4)
Detailed understanding of a topological defect	213	(6)
Free homotopy of maps	216	(1)
Based homotopy and the fundamental group	217	(2)
Classification of defects using homotopy groups	219	(8)
Defect structure in liquid crystals	227	(4)
Defects in nematics	228	(2)
Non abelian π1 - biaxial nematics	230	(1)
Formation of topological defects	231	(8)
Introduction to Bosonization	239	(96)
Sumathi Rao
Diptiman Sen
Fermi and Luttinger liquids	240	(7)
Bosonization	247	(21)
Bosonization of a fermion with one chirality	248	(9)
Bosonisation with two chiralities	257	(8)
Field theory near the Fermi momenta	265	(3)
Correlation functions and dimensions of operators	268	(4)
RG analysis of perturbed models	272	(9)
Applications of bosonization	281	(1)
Quantum antiferromagnetic spin 1/2 chain	282	(18)
Hubbard model	300	(9)
Transport in a Luttinger liquid - clean wire	309	(10)
Transport in the presence of isolated impurities	319	(9)
Concluding remarks	328	(7)
Quantum Hall Effect	335	(24)
R. Rajaraman
Classical Hall effect	336	(1)
Quantized Hall effect	337	(1)
Landau problem	338	(2)
Degeneracy counting	340	(1)
Laughlin wavefunction	341	(1)
Plasma analogy	342	(2)
Quasi-holes and their Laughlin wavefunction	344	(1)
Localization physics and the QH plateaux	345	(3)
Chern-Simons theory	348	(6)
Vortices in the CS field and quasiholes	354	(1)
Jain's theory of composite fermions	355	(4)
Low-dimensional Quantum Spin Systems	359	(28)
Indrani Bose
Introduction	360	(5)
Ground and excited states	365	(4)
Theorems and rigorous results for antiferromagnets	369	(7)
Lieb-Mattis theorem	369	(1)
Marshall's sign rule	370	(2)
Lieb, Schultz and Mattis theorem	372	(4)
Mermin-Wagner theorem	376	(1)
Possible ground states and excitation spectra	376	(11)
The Bethe Ansatz	387