ID: cond-mat/9411022

Directed Paths in Random Media

October 31, 1994

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Diagonalization of replicated transfer matrices for disordered Ising spin systems

May 12, 2004

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T. Nikoletopoulos, A. C. C. Coolen
Disordered Systems and Neura...

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with ran...

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Solving the one-dimensional Ising chain via mathematical induction: An intuitive approach to the transfer matrix

July 25, 2019

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Wenlong Wang, Rogelio Díaz-Méndez, Raudys Capdevila
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The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straigh...

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From Classical to quantum stochastic process

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Gustavo Montes, Soham Biswas, Thomas Gorin
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In this paper for the first time, we construct quantum analogs starting from classical stochastic processes, by replacing random which path decisions with superpositions of all paths. This procedure typically leads to non-unitary quantum evolution, where coherences are continuously generated and destroyed. In spite of their transient nature, these coherences can change the scaling behavior of classical observables. Using the zero temperature Glauber dynamics in a linear Ising...

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Anomalous diffusion in disordered media and random quantum spin chains

April 29, 1998

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F. Igloi, H. Rieger
Statistical Mechanics
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Using exact expressions for the persistence probability and for the leading eigenvalue of the Focker-Planck operator of a random walk in a random environment we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many new exact results are obtained from this correspondence including the space and time correlations of surviving random walks and the distribution of t...

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Density matrix renormalization on random graphs and the quantum spin-glass transition

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Javier Rodriguez-Laguna
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The density matrix renormalization group (DMRG) has been extended to study quantum phase transitions on random graphs of fixed connectivity. As a relevant example, we have analysed the random Ising model in a transverse field. If the couplings are random, the number of retained states remains reasonably low even for large sizes. The resulting quantum spin-glass transition has been traced down for a few disorder realizations, through the careful measurement of selected observa...

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Density Matrix Renormalization Group Method for the Random Quantum One-Dimensional Systems - Application to the Random Spin-1/2 Antiferromagnetic Heisenberg Chain -

February 6, 1996

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Kazuo Dept. Physics, Saitama University Hida
Condensed Matter

The density matrix renormalization group method is generalized to one dimensional random systems. Using this method, the energy gap distribution of the spin-1/2 random antiferromagnetic Heisenberg chain is calculated. The results are consistent with the predictions of the renormalization group theory demonstrating the effectiveness of the present method in random systems. The possible application of the present method to other random systems is discussed.

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Directed paths on hierarchical lattices with random sign weights

February 13, 1999

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Eduardo Aponte, Ernesto Medina
Statistical Mechanics
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We study sums of directed paths on a hierarchical lattice where each bond has either a positive or negative sign with a probability $p$. Such path sums $J$ have been used to model interference effects by hopping electrons in the strongly localized regime. The advantage of hierarchical lattices is that they include path crossings, ignored by mean field approaches, while still permitting analytical treatment. Here, we perform a scaling analysis of the controversial ``sign trans...

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Analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

December 7, 2004

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Massimo Ostilli, Carlo Presilla
Other Condensed Matter
Probability

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy, as well as the correlation functions in the ground state, are exactly determined as a series expansion in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our appro...

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Statistical properties of Random Matrices and the replica method

January 7, 1997

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Giorgio Parisi
Disordered Systems and Neura...
Soft Condensed Matter

I present here some results on the statistical behaviour of large random matrices in an ensemble where the probability distribution is not a function of the eigenvalues only. The perturbative expansion can be cast in a closed form and the limits of validity of this expansion are carefully analyzed. A comparison is done with a similar model with quenched disorder, where the solution can be found by using the replica method. Finally I will apply these results to a model which s...

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Universality in chaotic quantum transport: The concordance between random matrix and semiclassical theories

November 21, 2011

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Gregory Berkolaiko, Jack Kuipers
Chaotic Dynamics
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Electronic transport through chaotic quantum dots exhibits universal, system independent, properties, consistent with random matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semicla...

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