Preface -- CONTENTS -- CHAPTER 1 MATHEMATICAL THEORY OF THE ISING MODEL AND ITS GENERALIZATIONS: AN INTRODUCTION Yuri Kozitsky -- 1 Introduction -- 2 Classical Models -- 2.1 Local Hamiltonians and Gibbs States -- 2.2 Analytic Properties of Local Gibbs States -- 2.3 Correlation Inequalities -- 2.4 Infinite-Volume Gibbs States -- 2.5 Phase Transitions and Critical Points -- 2.6 Uniqueness of Gibbs States for the Ising Model -- 2.7 Self-Similarity, One-Dimensional and Hierarchical Models -- 3 Quantum Models -- 3.1 Local Gibbs States -- 3.2 Green and Matsubara Functions -- 3.3 Euclidean Approach -- 3.4 Phase Transitions and Critical Points -- Acknowledgments -- References -- CHAPTER 2 RELAXATION IN QUANTUM SPIN CHAINS: FREE FERMIONIC MODELS Dragi Karevski -- 1 Introduction -- 2 Quantum Spin Chains -- 2.1 Free Fermionic Models -- 2.2 Canonical Diagonalization -- 2.3 Excitation Spectrum and Eigenvectors -- 2.3.1 XY -Chain -- 2.3.2 Ising Chain -- 3 Equilibrium Behaviour -- 3.1 Critical Behaviour -- 3.1.1 Surface Magnetization -- 3.1.2 Bulk Magnetization -- 3.2 Time-Dependent Correlation Functions -- 4 Non-Equilibrium Behaviour -- 4.1 Heisenberg Equations of Motion -- 4.2 Time-Dependent Behaviour -- 4.2.1 Transverse Magnetization -- 4.2.2 Boundary Effects -- 4.2.3 Two-Time Functions -- 4.2.4 Critical Ising Chain -- 4.2.5 XX-Chain -- 5 Discussion and Summary -- Acknowledgements -- References -- CHAPTER 3 QUANTUM PHASE TRANSITIONS IN ALTERNATING TRANSVERSE ISING CHAINS Oleg Derzhko -- 1 Classical and Quantum Phase Transitions -- 2 Spin-1/2 Ising Chain in a Transverse Field as the Simplest Model for the Quantum Phase Transition Theory -- 3 Spin-1/2 Ising Chain in a Transverse Field with Regularly Alternating Hamiltonian Parameters: Continued Fraction Approach -- 4 Effects of Regularly Alternating Bonds/Fields on the Quantum Phase Transition.

5 Related Models -- Acknowledgments -- References -- CHAPTER 4 PHASE TRANSITIONS IN TWO-DIMENSIONAL RANDOM POTTS MODELS Bertrand Berche and Christophe Chatelain -- 1 Introduction -- 2 Perturbative Approach in 2D -- 2.1 Replicas and Relevance Criterion -- 2.2 Perturbation Techniques -- 2.2.1 Average Correlation Functions -- 2.2.2 Multiscaling and Higher Order Moments of the Correlators -- 2.2.3 Are these Effects Measurable? -- 3 Numerical Techniques in 2D -- 3.1 Monte Carlo Simulations -- 3.1.1 Cluster Algorithms -- 3.1.2 Definition of the Physical Quantities -- 3.2 Transfer Matrix Technique -- 4 Analysis of Numerical Data in 2D -- 4.1 Location of the Random Fixed Point -- 4.2 Temperature Dependence -- 4.3 Finite-Size Scaling -- 4.4 Short-Time Dynamics Scaling -- 4.5 Conformal Mappings -- 4.6 Impact of Rare Events and Non-Self-Averaging -- 5 Numerical Results and Comparison with Perturbative Expansions in 2D -- 5.1 Regime q > 4 -- 5.1.1 Randomness Induces a Second-Order Regime -- 5.1.2 Comparison between Finite-Size Scaling and Conformal Mappings -- 5.2 Regime q < 4 -- 5.2.1 Tests of Replica Symmetry -- 5.2.2 Multiscaling -- 5.2.3 Probability Distribution of Correlation Functions -- 6 Conclusion and Summary of the Main Results -- Acknowledgements -- References -- CHAPTER 5 SCALING OF MIKTOARM STAR POLYMERS Christian von Ferber -- 1 Introduction -- 1.1 Star Polymers and Polymer Networks -- 1.2 Miktoarm Star Polymers and Multifractal Spectra -- 1.3 Interactions of Star Polymers -- 1.4 Comparison with Exact Results in 2D -- 2 Miktoarm Star Polymers -- 2.1 Model and Notations -- 2.2 Renormalization -- 2.3 Renormalization Group Flow and the Fixed Points -- 2.4 Results for the Exponents -- 2.5 Resummation -- 2.6 Numerical Results -- 3 Multifractal Spectra for the Polymer Absorber Model -- 3.1 Multifractal Spectrum -- 3.2 Resummation and Results.

4 Comparison with 2D Exact Results -- 4.1 Multifractal Spectrum in 2D -- 5 Scaling Theory of Forces between Star Polymers -- 5.1 Two-Star Polymers -- 5.2 Three Stars -- 5.3 Computer Simulation -- 6 Summary -- 6.1 Miktoarm Star Polymers -- 6.2 Multifractality -- 6.3 2D Exact Results and Extensions -- Acknowledgments -- References -- CHAPTER 6 FIELD THEORETIC APPROACHES TO THE SUPERCONDUCTING PHASE TRANSITION Flavio S. Nogueira and Hagen Kleinert -- 1 Introduction -- 2 Review of the HLM Theory -- 2.1 HLM Mean-Field Theory -- 2.2 Renormalization Group in d = 4 - Dimensions -- 2.3 Critical Exponents -- 2.4 1/N Expansion -- 3 Existence of the Charged Fixed Point -- 3.1 Scaling Near the Charged Fixed Point -- 3.2 Duality -- 3.2.1 Duality in the Lattice Ginzburg-Landau Model -- 3.2.2 The Disorder Field Theory -- 4 The Physical Meaning of the Critical Exponent η -- 5 Renormalization Group Calculation at Fixed Dimension and Below Tc -- 6 Concluding Remarks -- Acknowledgements -- References -- Index.