Exact results for interface structure and wetting functional relations for sl(3) Chiral Potts model at q2=-1 facet shapes in a Wulff Crystal macroscopic description of phase separation in the 2D Ising model Peierls instability for the Holstein model Falicov-Kimball models - a partial review of the ground states problem Hubbard model with magnetic field - antiferromagnetism and paramagnetism quantum hall effect without divergence of the localization length on quantum stochastic dynamics for quantum spin systems on a lattice - low temperature problem interface states of quantum spin systems thermodynamic behaviour of the Bogoliubov Weakly imperfect Bose gas sum rules for Coulomb systems with hard cores ground state energy of the low density Bose gas universality and scaling in random matrix models and random polynomials KAM-renormalization-group for Hamiltonian systems with two degrees of freedom on the landscape of attractors in a neural network with interacting patterns potentials for one-dimensional restrictions of Gibbs measures. (Part contents)
Exact results for interface structure and wetting functional relations for sl(3) Chiral Potts model at q2=-1 facet shapes in a Wulff Crystal macroscopic description of phase separation in the 2D Ising model Peierls instability for the Holstein model Falicov-Kimball models - a partial review of the ground states problem Hubbard model with magnetic field - antiferromagnetism and paramagnetism quantum hall effect without divergence of the localization length on quantum stochastic dynamics for quantum spin systems on a lattice - low temperature problem interface states of quantum spin systems thermodynamic behaviour of the Bogoliubov Weakly imperfect Bose gas sum rules for Coulomb systems with hard cores ground state energy of the low density Bose gas universality and scaling in random matrix models and random polynomials KAM-renormalization-group for Hamiltonian systems with two degrees of freedom on the landscape of attractors in a neural network with interacting patterns potentials for one-dimensional restrictions of Gibbs measures. (Part contents)
Part 1 Satellite meeting of Statphys-19: quantum domains in ferromagnetic anisotropic Heisenberg chains, F.C. Alcaraz et al algebraic solution of the coincidence problem for crystals and quasi crystals, M. Baake reflection equations and surface critical phenomena, M.T. Batchelor quantum field theories in terms of group-valued local fields - an overview, L.L. Chau U(1)-invariant local and integrable lattice formulation of the massive Thirring model, C. Destri dilute algebras and solvable lattice models, U. Grimm quantum group and the Hofstadter problem, Y. Hatsugai et al domain walls in the spin-S quantum Ising chain, M. Henkel mutual exclusion statistics in the exactly solvable model of the Mott metal-insulator transition, M. Kohmoto et al boundary Yang-Baxter in the RSOS/SOS representation, W.M. Koo and C. Ahn probability of phase separation and two point temperature correlation functions for the Bose gas with delta interaction, V.E. Korepin and A.R. Its the generalized chiral clock model and its phase diagram, H. Au-Yang and J.H.H. Perk stochastic reaction-diffusion processes, operator algebras and integrable quantum spin chains, G.M. Schutz vertex-face correspondence in elliptic solutions of the Yang-Baxter equation, Y. Shibukawa logarithmic anomalies of susceptibility for solvable models, M. Takahashi on chiral Hubbard model at strong interaction, D.F. Wang soluble free-fermion models in d dimensions, F.Y. Wu bosonization based on Bethe Ansatz equations and proof of the conformal conjecture, Y.S. Wu and Y Yu. Part 3 The seventh Nankai workshop: one dimensional lattice models of electrons with r2 hopping and exchange, Ch. Gruber and D.F. Wang symmetry group invariants for spontaneous magnetization, J.-M. Maillard Zamolodchikov-Faddeev algebra vertex models, H.-P. Eckle scaling properties of the Ising model in a field, U. Grimm and B. Nienhuis experimental realizations of integrable reaction-diffusion processes in biological and chemical systems, G.M. Schutz and other papers. (Part contents).
Preface Part I. Quantum Mechanics: 1. On the low density of Boson models: Langevin equation L. Accardi and L. Y. Gang 2. Schrodinger operators with potentials supported by null sets S. Albeverio, J. E. Fenstad, R. Hoegh-Krohn, W. Karwowski and T. Lindstrom 3. Landau Hamiltonians on symmetric spaces J. E. Avron and A. Pnueli 4. Renormalization group analysis and quasicrystals J. Bellissard 5. The measurement problem in the stochastic formulation of quantum mechanics Ph. Blanchard, M. Cini and M. Serva 6. Schrodinger operators at threshold D. Bolle 7. Spectral analysis and scattering theory for Schrodinger operators with an interaction supported by a regular curve J. F. Brasche and A. Teta 8. Lattice models of solids V. V. Evstratov and B. S. Pavlov 9. Schrodinger operators on unusual manifolds P. Exner and P. Seba 10. High energy resolvent estimates for Schrodinger operators A. Jensen 11. Point interactions as solvable models W. Kirsch 12. Stochastic model of quantum mechanics V. P. Maslov Part II. Quantum Field Theory: 13. Maximality of infinite dimensional Dirichlet forms and Hoegh-Krohn's model of quantum fields S. Albeverio and S. Kusuoka 14. Global Markov property in quantum field theory and statistical mechanics: a review on results and problems S. Albeverio and B. Zegarlinski 15. Hypercontractivity: a bibliographic review E. B. Davies, L. B. Gross and B. Simon 16. Progress and problems in algebraic quantum field theory S. Doplicher 17. Hoegh-Krohn's model of quantum fields and the absolute continuity of measures S. Kusuoka 18. Baryon states in lattice QCD R. A. Minlos and E. A. Zhinzhina 19. The energy of a Weyl soliton R. F. Streater Part III. Statistical Physics: 20. Morphology and classification of galaxies: a stochastic model S. Albeverio, Ph. Blanchard, D. Gandolfo, R. Hoegh-Krohn and M. Mebkhout 21. Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction R. L. Dobrushin and S. B. Shlosman 22. Verification of the global Markov property for strongly coupled trigonometric interactions R. Gielerak 23. Decomposition of Ising models and the Mayer expansion C. M. Newman and G. A. Baker, Jr 24. Towards time-dynamics for bosonic systems in quantum statistical mechanics A. G. Shuhov, Yu. M. Suhov and A. V. Teslenko 25. Towards time-dynamics for bosonic systems in quantum statistical mechanics, 2 A. G. Shuhov and Yu. M. Suhov.
Dynamical impurity problems, V.J. Emery and S.A. Kivelson bosons in a random potential, A.P. Young statistical mechanics of vortices in type-II superconductors, D.A. Huse and L.R. Radzihovsky correlation functions of solvable models, B. Nienhuis random matrices and two-dimensional gravity, F. David aperiodic structures - geometry, diffraction spectra, and physical properties, J.M. Luck universal aspects of interacting lines and surfaces, M. Lassig and R. Lipowsky phase transitions in colloidal dispersions, H.N.W. Lekkerkerker and G.J. Vroege polymers, both living and dead, M.E. Cates amplitude equations for pattern forming systems, M. van Hecke et al turbulence and Navier-Stokes-equations, S. Grossmann.
A Survey by the Editors.- I.1 Violation of the Noncrossing Rule: The Hubbard Hamiltonian for Benzene.- I.2 Two Theorems on the Hubbard Model.- I.3 The Hubbard Model: Some Rigorous Results and Open Problems.- I.4 Flux Phase of the Half-Filled Band.- II.1 Proof of the Peierls Instability in One Dimension.- II.2 Stability of the Peierls Instability for Ring Shaped Molecules.- III.1 A Model for Crystallization: A Variation on the Hubbard Model.- III.2 Phase Separation due to Quantum Mechanical Correlations.- IV.1 Theory of Ferromagnetism and the Ordering of Electronic Energy Levels.- IV.2 Ordering Energy Levels of Interacting Spin Systems.- IV.3 Magnetic Properties of Some Itinerant-Electron Systems at T > 0.- V.1 Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors.- V.2 Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field.- V.3 Constructive Macroscopic Quantum Electrodynamics.- V.4 The Laser: A Reversible Quantum Dynamical System with Irreversible Classical Macroscopic Motion.- VI.1 A Proof of Part of Haldane's Conjecture on Spin Chains.- VI.2 Rigorous Results on Valence-Bond Ground States in Antiferromagnets.- VI.3 Valence Bond Ground States in Isotropic Quantum Antiferromagnets.- VI.4 Ground State Properties of a Fully Frustrated Quantum Spin System.- VII.1 Exact Ground State Energy of the Strong-Coupling Polaron.- VIII.1 Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas.- VIII.2 Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.- VIII.3 The Quantum-Mechanical Many-Body Problem: The Bose Gas.- Some of the Early History of Exactly Soluble Models.- IX.1 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.- IX.2 Exact Solution of the F Model of an Antiferroelectric.- IX.3 Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.- IX.4 Residual Entropy of Square Ice.- IX.5 Two-Dimensional Ferroelectric Models.- IX.6 Relations Between the 'Percolation' and 'Colouring' Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the 'Percolation' Problem.- IX.7 Analytic Properties of the Free Energy for the Ice Models.- X.1 Two-Dimensional Ising Model as a Soluble Problem of Many Fermions.- X.2 Solution of the Dimer Problem by the Transfer Matrix Method.- XI.1 Two Soluble Models of an Antiferromagnetic Chain.- XII.1 Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State.- XII.2 Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum.- XIII.1 Delta-Function Fermi Gas with Two-Spin Deviates.- XIII.2 Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model on One Dimension.- XIV.1 Exact Solution of a Many-Fermion System and Its Associated Boson Field.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.
The thesis contains theoretical study of number of systems in low dimension, which are related to quantum Ising model. The main results are emergence of Majorana fermions in condensed matter system. We discuss an array of cavity-QED system, describing atom-photon (spin-boson) interacting problem, namely Rabi lattice model, and calculated its phase diagram. More importantly we showed that in any dimension, the system has Ising dynamics in strong atom-photon coupling limit. For 1D, a special relation between end-to-end spin-spin correlation is presented, which is very much different form the bulk spin-spin correlation. The entanglement between cavities is calculated and shown to have maximum value at the phase transition point. Other problems we studied were the frustrated quantum Ising model and its phase diagram as well as existence of Majorana like edge modes in different phases of the model. The fermionic version of the frustrated quantum Ising problem is also presented. On a different but related note, we present some beautiful results on Falicov Kimball (FK) model, a fermionic model similar to Hubbard model, on a square lattice. Using a canonical representation for electrons (invented by Brijesh Kumar), the FK model is mapped to classical Ising model (at infinite Hubbard interaction limit) on 2D lattice. We performed classical Monte Carlo simulation to study magnetization as a function of Hubbard interaction and also as a function of temperature. The motivation was to study (quantum) charge fluctuation (thus, quantum fluctuation of top of classical Ising thermodynamics) due to finite Hubbard interaction term. We also devote a chapter for basic numerical techniques, such as Density matrix Renormalization Group method, Exact diagonalization for spin system and classical Monte Carlo simulation (for classical 2D Ising model), which we have used in this thesis.
Some of the early history of exactly soluble models, E.H. Lieb the many faces of the chiral Potts model, H. Au-Yang and J.H.H. Perk star-triangle and star-star relations in statistical mechanics, R.J. Baxter knot theory and statistical mechanics, L.H. Kauffman star-star relation for the Potts model, C. King and F.Y. Wu finite temperature correlations of the Ising chain in a transverse field, S. Sachdev exact solutions of low-dimensional reaction-diffusion systems, A.M.R. Cadilhe et al the infinite-state Potts model and solid partitions of an integer, H.Y. Huang and F.Y. Wu exact evaluation of thermodynamic quantities for the integrable T-J chain, A. Klumper multilayer extension of two-dimensional solvable statistical models to three dimensions, V. Popkov a higher-level Bailey lemma, A. Schilling and S.O. Warnaar the Heisenberg chain as a dynamical model for protein synthesis - some theoretical and experimental results, G.M. Schutz free fermion conditions and the symmetries of integrability, C.-M. Viallet. (Part contents).