The Ruelle-Araki transfer operator for one-dimensional classical systems

2005 ◽  
pp. 40-65
Keyword(s):  
Transfer Operator ◽  
One Dimensional ◽  
Classical Systems

scholarly journals Efficient Numerical Evaluation of Thermodynamic Quantities on Infinite (Semi-)classical Chains

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Christian B. Mendl ◽  
Folkmar Bornemann
Keyword(s):  
Transfer Operator ◽  
Classical System ◽  
Free Energy Density ◽  
Classical Particle ◽  
Thermodynamic Quantities ◽  
Nls Equation ◽  
Classical Systems ◽  
Particle Chain ◽  
Dynamics Simulations ◽  
Good Agreement

AbstractThis work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger (NLS) equation and to a classical system on a cylindrical lattice. A comparison with molecular dynamics simulations performed for the NLS model shows very good agreement.

scholarly journals A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

2018 ◽  
Vol 40 (3) ◽  
pp. 612-662
Author(s):  
ALEXANDER ADAM ◽  
ANKE POHL
Keyword(s):  
Transfer Operator ◽  
Zeta Functions ◽  
Cusp Forms ◽  
Dimensional Representation ◽  
Triangle Group ◽  
Selberg Zeta Function ◽  
One Dimensional ◽  
Transfer Operators ◽  
Finite Dimensional ◽  
Automorphic Functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

scholarly journals On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-18 ◽  
Author(s):  
J. Hutchinson ◽  
J. P. Keating ◽  
F. Mezzadri
Keyword(s):  
Classical System ◽  
Companion Paper ◽  
Quantum Systems ◽  
Spin Systems ◽  
Two Dimensional ◽  
Critical Properties ◽  
One Dimensional ◽  
Classical Systems ◽  
Long Range Interactions ◽  
Quantum Hamiltonian

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.

scholarly journals QUANTUM CORNER — TRANSFER MATRIX DMRG

2008 ◽  
Vol 19 (08) ◽  
pp. 1145-1161 ◽  
Author(s):  
ERIK BARTEL ◽  
ANDREAS SCHADSCHNEIDER
Keyword(s):  
Thermal Properties ◽  
Transfer Matrix ◽  
Heisenberg Model ◽  
Quantum Systems ◽  
Test Cases ◽  
The Novel ◽  
Simple Test ◽  
Ising Chain ◽  
One Dimensional ◽  
Classical Systems

We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.

scholarly journals Factorizations of one-dimensional classical systems

2008 ◽  
Vol 323 (2) ◽  
pp. 413-431 ◽  
Author(s):  
Şengül Kuru ◽  
Javier Negro
Keyword(s):  
One Dimensional ◽  
Classical Systems

KOLMOGOROV SYSTEMS AND ANOSOV SYSTEMS IN QUANTUM THEORY

2001 ◽  
Vol 04 (01) ◽  
pp. 85-119 ◽  
Author(s):  
H. NARNHOFER
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Theory ◽  
Quantum Field ◽  
Modular Systems ◽  
One Dimensional ◽  
Classical Systems ◽  
Anosov Systems

In analogy to classical systems, quantum K-systems and quantum Anosov systems are defined. Their relation especially for modular systems is discussed as well as the consequences on clustering properties. Examples for such systems in the framework of quantum field theory and one-dimensional theories are offered.

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