Density Matrix Renormalization Group and Reaction-Diffusion Processes

TL;DR

This paper applies the density matrix renormalization group (DMRG) to one-dimensional reaction-diffusion models, enabling accurate estimation of critical points and exponents near phase transitions using finite-size scaling.

Contribution

It introduces a method for applying DMRG to non-symmetric stochastic Hamiltonians in reaction-diffusion models, including new strategies for choosing the density matrix.

Findings

Accurate critical points and exponents obtained from finite-size scaling.

Successful diagonalization of non-symmetric matrices for reaction-diffusion models.

Analysis of relaxation times and density profiles in directed percolation models.

Abstract

The density matrix renormalization group (DMRG) is applied to some one-dimensional reaction-diffusion models in the vicinity of and at their critical point. The stochastic time evolution for these models is given in terms of a non-symmetric ``quantum Hamiltonian'', which is diagonalized using the DMRG method for open chains of moderate lengths (up to about 60 sites). The numerical diagonalization methods for non-symmetric matrices are reviewed. Different choices for an appropriate density matrix in the non-symmetric DMRG are discussed. Accurate estimates of the steady-state critical points and exponents can then be found from finite-size scaling through standard finite-lattice extrapolation methods. This is exemplified by studying the leading relaxation time and the density profiles of diffusion-annihilation and of a branching-fusing model in the directed percolation universality class.