CONCEPTS OF THEORETICAL SOLID STATE PHYSICS
CONTENTS FROM PARTICLES TO ?ELDS CLASSICAL HARMONIC CHAIN: PHONONS FUNCTIONAL ANALYSIS AND VARIATIONAL PRINCIPLES MAXWELL’S EQUATIONS AS A VARIATIONAL PRINCIPLE QUANTUM CHAIN QUANTUM ELECTRODYNAMICS NOETHER’S THEOREM SUMMARY AND OUTLOOK PROBLEMS SECOND QUANTIZATION INTRODUCTION TO SECOND QUANTIZATION APPLICATIONS OF SECOND QUANTIZATION FEYNMAN PATH INTEGRAL THE PATH INTEGRAL: GENERAL FORMALISM CONSTRUCTION OF THE PATH INTEGRAL APPLICATIONS OF THE FEYNMAN PATH INTEGRAL FUNCTIONAL ?ELD INTEGRAL CONSTRUCTION OF THE MANY-BODY PATH INTEGRAL FIELD INTEGRAL FOR THE QUANTUM PARTITION FUNCTION FIELD THEORETICAL BOSONIZATION: A CASE STUDY PERTURBATION THEORY GENERAL STRUCTURES AND LOW-ORDER EXPANSIONS GROUND STATE ENERGY OF THE INTERACTING ELECTRON GAS IN?NITE-ORDER EXPANSIONS BROKEN SYMMETRY AND COLLECTIVE PHENOMENA MEAN-?ELD THEORY PLASMA THEORY OF THE INTERACTING ELECTRON GAS BOSE–EINSTEIN CONDENSATION AND SUPER?UIDITY SUPERCONDUCTIVITY FIELD THEORY OF THE DISORDERED ELECTRON GAS RESPONSE FUNCTIONS CRASH COURSE IN MODERN EXPERIMENTAL TECHNIQUES LINEAR RESPONSE THEORY ANALYTIC STRUCTURE OF CORRELATION FUNCTIONS ELECTROMAGNETIC LINEAR RESPONSE THE RENORMALIZATION GROUP THE ONE-DIMENSIONAL ISING MODEL DISSIPATIVE QUANTUM TUNNELING RENORMALIZATION GROUP: GENERAL THEORY RG ANALYSIS OF THE FERROMAGNETIC TRANSITION RG ANALYSIS OF THE NONLINEAR ?-MODEL BEREZINSKII–KOSTERLITZ–THOULESS TRANSITION TOPOLOGY EXAMPLE: PARTICLE ON A RING HOMOTOPY ?-0TERMS WESS–ZUMINO TERMS CHERN–SIMONS TERMS I 2M + I<J VII(RI ?RJ), ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? HEI = II VEI(RI ?RI) (1 1) HERE, RI (RI) DENOTE THE COORDINATES OF THE VALENCE ELECTRONS (ION CORES) AND HE, HI, AND HEI DESCRIBE THE DYNAMICS OF ELECTRONS, IONS AND THE INTERACTION OF ELECTRONS AND IONS, RESPEC- TIVELY (SEE FIG 1 1) OF COURSE, THE HAMILTONIAN EQ (1 1) CAN BE MADE “MORE REALISTIC,” FOR EXAMPLE BY REMEMBERING THAT ELECTRONS AND IONS CARRY SPIN, ADDING DISORDER, OR INTRODUC- ING HOST LATTICES WITH MULTI-ATOMIC UNIT-CELLS HOWEVER, FOR DEVELOPING OUR PRESENT LINE OF THOUGHT THE PROTOTYPE H WILL SU?CE THE FACT THAT A SEEMINGLY INNOCUOUS HAMILTONIAN LIKE EQ (1 1) IS CAPABLE OF GENERATING THE VAST PANOPTICON OF METALLIC PHENOMENOLOGY CAN BE READ IN REVERSE ORDER: ONE WILL NORMALLY NOT BE ABLE TO MAKE THEORETICAL PROGRESS BY APPROACHING THE PROBLEM IN AN “AB INITIO” MANNER, I E BY AN APPROACH THAT TREATS ALL MICROSCOPIC CONSTITUENTS AS EQUALLY RELEVANT DEGREES OF FREEDOM HOW THEN CAN SUCCESSFUL ANALYTICAL APPROACHES BE DEVELOPED? THE ANSWER TO THIS QUESTION LIES IN A NUMBER OF BASIC PRINCIPLES INHERENT IN GENERIC CONDENSED MATTER SYSTEMS 4 FINALLY, CONDENSED MATTER SYSTEMS TYPICALLY POSSESS A NUMBER OF INTRINSIC SYMMETRIES FOR EXAMPLE, OUR PROTOTYPE HAMILTONIAN ABOVE IS INVARIANT UNDER SIMULTANEOUS TRANS- LATION AND ROTATION OF ALL COORDINATES, WHICH EXPRESSES THE GLOBAL GALILEAN INVARIANCE OF THE SYSTEM (A CONTINUOUS SET OF SYMMETRIES) SPIN ROTATION INVARIANCE (CONTINUOUS) AND 1 THE IMPORTANCE OF THIS POINT IS ILLUSTRATED BY THE EMPIRICAL OBSERVATION THAT THE MOST CHALLENGING SYSTEMS IN PHYSICAL SCIENCES ARE OF MEDIUM (AND NOT LARGE) SCALE, E G , METALLIC CLUSTERS, MEDIUM-SIZED NUCLEI OR LARGE ATOMS CONSIST OF O(101–102) FUNDAMENTAL CONSTITUENTS SUCH PROBLEMS ARE WELL BEYOND THE REACH OF FEW-BODY QUANTUM MECHANICS WHILE NOT YET ACCESSIBLE TO RELIABLE STATISTICAL MODELING OFTEN THE ONLY VIABLE PATH TO APPROACHING SYSTEMS OF THIS TYPE IS MASSIVE USE OF PHENOMENOLOGY CONCEPTS OF THEORETICAL SOLID STATE PHYSICS , (1 2) WHERE THE COE?CIENT KS DETERMINES THE STEEPNESS OF THE LATTICE POTENTIAL NOTICE THAT H CAN BE INTERPRETED AS THE HAMILTONIAN OF N POINT-LIKE PARTICLES OF MASS M ELASTICALLY CONNECTED BY SPRINGS WITH SPRING CONSTANT KS (SEE FIG 1 2) 2 IN THIS TEXT, WE WILL FOCUS ON THE DYNAMICAL BEHAVIOR OF LARGE SYSTEMS, AS OPPOSED TO THEIR STATIC STRUCTURAL PROPERTIES IN PARTICULAR, WE WILL NOT ADDRESS QUESTIONS RELATED TO THE FORMATION OF DE?NITE CRYSTALLOGRAPHIC STRUCTURES IN SOLID STATE SYSTEMS CONCEPTS OF THEORETICAL SOLID STATE PHYSICS KS (I – 1)A IA (I + 1)A RI – ?I M FIGURE 1 2 TOY MODEL OF A ONE-DIMENSIONAL SOLID; A CHAIN OF ELASTICALLY BOUND MASSIVE POINT PARTI- CLES LAGRANGIAN FORMULATION AND EQUATIONS OF MOTION JOSEPH–LOUIS LAGRANGE 1736– , (1 3) WHERE T AND U DENOTE RESPECTIVELY THE KINETIC AND POTENTIAL ENERGY SINCE WE ARE INTERESTED IN THE PROPERTIES OF THE LARGE-N SYSTEM, WE CAN EXPECT BOUNDARY E?ECTS TO BE NEGLIGIBLE THIS BEING SO, WE ARE AT LIBERTY TO IMPOSE ON OUR ATOMIC CHAIN THE TOPOLOGY OF A CIRCLE, I E WE ADOPT PERIODIC BOUNDARY CONDITIONS IDENTIFYING RN+1 = R1 FURTHER, ANTICIPATING THAT THE E?ECT OF LATTICE VIBRATIONS ON THE SOLID IS WEAK (I E LONG- RANGE ATOMIC ORDER IS MAINTAINED) WE MAY ASSUME THAT THE DEVIATION OF THE IONS FROM THEIR CONCEPTS OF THEORETICAL SOLID STATE PHYSICS A L DX, WHERE L = NA NOTE THAT, AS DE?NED, THE FUNCTIONS ?(X, T) HAVE DIMENSIONALITY [LENGTH]1/2 EXPRESSED IN TERMS OF THE NEW DEGREES OF FREEDOM, THE CONTINUUM LIMIT OF THE LAGRANGIAN THEN READS L[?] = L DX L(?, ?X?, ??), L(?, ?X?, ??) = M (?X?)2, (1 4) 3 INDEED, FOR REASONS THAT WILL BECOME CLEAR, HIGHER-ORDER CONTRIBUTIONS TO THE TAYLOR EXPANSION ARE IMMATERIAL IN THE LONG-RANGE CONTINUUM LIMIT CONCEPTS OF THEORETICAL SOLID STATE PHYSICS T ? S S[?] M FIGURE 1 3 SCHEMATIC VISUALIZATION OF A ?ELD: A MAPPING ? FROM A BASE MANIFOLD M INTO A TARGET SPACE T (IN THIS CASE, T ARE THE REAL NUMBERS; BUT, IN GENERAL, T CAN BE MORE COMPLICATED) A FUNCTIONAL ASSIGNS TO EACH ? A REAL NUMBER S[?] THE GRID EMBEDDED INTO M INDICATES THAT ?ELDS IN CONDENSED MATTER PHYSICS ARISE AS CONTINUUM LIMITS OF DISCRETE MAPPINGS WHERE THE LAGRANGIAN DENSITY L HAS DIMENSIONALITY [ENERGY]/[LENGTH] AND WE HAVE DESIG- NATED THE PARTICLE MASS BY THE MORE COMMON SYMBOL M ?M SIMILARLY, THE CLASSICAL ACTION ASSUMES THE CONTINUUM FORM S[?] = DT L[?] = DT L 4 IF YOU ARE UNFAMILIAR WITH THE NOTION OF MANIFOLDS (FOR A CRASH COURSE, SEE PAGE 537), THINK OF M AND T AS SUBSETS OF SOME VECTOR SPACE FOR THE MOMENT, THIS LIMITATION WON’T DO ANY HARM 5 IN SOME (RARE) CASES IT BECOMES NECESSARY TO DE?NE ?ELDS IN A MORE GENERAL SENSE (E G AS SECTIONS OF MATHEMATICAL OBJECTS KNOWN AS ?BER BUNDLES) HOWEVER, IN PRACTICALLY ALL CONDENSED MATTER APPLICATIONS THE MORE RESTRICTIVE DE?NITION ABOVE WILL SU?CE 6 BY CONTRAST, THE CONDITION OF LORENTZ INVARIANCE IMPLIES THE ABSENCE OF SUCH FACTORIZATIONS IN RELATIVISTIC ?ELD THEORY IN CLASSICAL STATISTICAL ?ELD THEORIES, I E THEORIES PROBING THE THERMODYNAMIC BEHAVIOR OF LARGE SYSTEMS, M IS JUST SPACE-LIKE CONCEPTS OF THEORETICAL SOLID STATE PHYSICS