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Unified approach to numerical transfer matrix methods for disordered systems : applications to mixed crystals and to elasticity percolation

M.A. Lemieux, Pierre Breton, A.–M. S. Tremblay

It is shown that the Negative Eigenvalue Theorem and transfer matrix methods may be considered within a unified framework and generalized to compute projected densities of states or, more generally, any linear combination of matrix elements of the inverse of large symmetric random matrices. As examples of applications, extensive simulations for one- and two-mode behaviour in the Raman spectrum of one-dimensional mixed crystals and a finite-size scaling analysis of critical exponents for the central force percolation universality class are presented.It is shown that the Negative Eigenvalue Theorem and transfer matrix methods may be considered within a unified framework and generalized to compute projected densities of states or, more generally, any linear combination of matrix elements of the inverse of large symmetric random matrices. As examples of applications, extensive simulations for one- and two-mode behaviour in the Raman spectrum of one-dimensional mixed crystals and a finite-size scaling analysis of critical exponents for the central force percolation universality class are presented.

Journal de Physique LettresJan 1, 198589 citations

Lack of self-averaging in weakly disordered one-dimensional systems

A. Crisanti, G. Paladin, Maurizio Serva +1 more

We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.

arXiv:cond-mat/9304025Journal de Physique IOct 1, 19932 citations

Matrix product states-properties and extensions(New Development of Numerical Simulations in Low-Dimensional Quantum Systems: From Density Matrix Renormalization Group to Tensor Network Formulations)

Anders W. Sandvik

物性研究Mar 5, 20110 citations

Universal distributions of overlaps from generic dynamics in quantum many-body systems

Alexios Christopoulos, Amos Chan, Andrea De Luca

We study the distribution of overlaps with the computational basis of a quantum state generated under generic quantum many-body chaotic dynamics, without conserved quantities, for a finite time t . We argue that, scaling time logarithmically with the system size t ∝ log L , the overlap distribution converges to a universal form in the thermodynamic limit, forming a one-parameter family that generalizes the celebrated Porter-Thomas distribution. The form of the overlap distribution only depends on the spatial dimensionality and, remarkably, on the boundary conditions. This picture is justified in general by a mapping to Ginibre ensemble of random matrices and corroborated by the exact solution of a random quantum circuit. Our results derive from an analysis of arbitrary overlap moments, enabling the reconstruction of the distribution. Our predictions also apply to Floquet circuits, i.e., in the presence of mild quenched disorder. Finally, numerical simulations of two distinct random circuits show excellent agreement, thereby demonstrating universality.We study the distribution of overlaps with the computational basis of a quantum state generated under generic quantum many-body chaotic dynamics, without conserved quantities, for a finite time t . We argue that, scaling time logarithmically with the system size t ∝ log L , the overlap distribution converges to a universal form in the thermodynamic limit, forming a one-parameter family that generalizes the celebrated Porter-Thomas distribution. The form of the overlap distribution only depends on the spatial dimensionality and, remarkably, on the boundary conditions. This picture is justified in general by a mapping to Ginibre ensemble of random matrices and corroborated by the exact solution of a random quantum circuit. Our results derive from an analysis of arbitrary overlap moments, enabling the reconstruction of the distribution. Our predictions also apply to Floquet circuits, i.e., in the presence of mild quenched disorder. Finally, numerical simulations of two distinct random circuits show excellent agreement, thereby demonstrating universality.

arXiv:2404.10057Physical Review ResearchJul 2, 20255 citations

The Rigorous Stochastic Matrix Product Method for the Calculations of Reduced Equilibrium Density Matrices of Multilevel Open Quantum Systems

Xin Chen

Oct 28, 20130 citations

Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

Andris Ambainis, Aram W. Harrow, Matthew B. Hastings

arXiv:0910.0472Communications in Mathematical PhysicsJan 5, 201230 citations

Products of random matrices for disordered systems

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Products of random matrices for disordered systems

Abstract

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