Asymmetric exclusion model with impurities

Abstract

An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with an exponent 3 2 , our model has a scaling exponent One-dimensional three-state quantum Hamiltonians and master equations of reaction-diffusion processes have played an important role in describing strongly correlated electrons and nonequilibrium statistical mechanics in the last decades, mainly due to their intrinsic and nontrivial many-body behavior. Remarkably, in one dimension several models in this category are exactly solvable, as the spin-1 Sutherland ͓1͔ and t-J ͓2͔ models, and the asymmetric diffusion of two types of particles ͓3͔. In its formulation in terms of particles with two global conservation laws, these models describe the dynamics of two types of particles on the lattice, where the total number of particles of each type is conserved separately. In order to ensure integrability, all known models in this class satisfy some particle-particle exchange symmetries ͓4,5͔. Recently, we introduced a new class of three-state model in the context of high-energy physics that is integrable despite it does not have particle-particle exchange symmetry ͓5͔. The quantum version is closely related to the strong regime of the t-U Hubbard model and the XXC model ͓6͔. In this work we formulate a stochastic model related to ͓5͔ that describes an asymmetric exclusion process ͑ASEP͒ with impurities. Although our model can be solved by the coordinate Bethe ansatz, we are going to formulate a matrix product ansatz ͑MPA͒ ͓4,7͔ due its simplicity and unifying implementation for arbitrary systems. This MPA introduced in ͓4,7͔ can be seen as a matrix product formulation of the coordinate Bethe ansatz and it is suited to describe all eigenstates of integrable models. We solve this model with periodic boundary condition through the MPA and we analyze the spectral gap for some special cases. Our model displays the full spectrum of the ASEP without impurities ͓8͔ plus new levels. The first excited state belongs to these new levels and has unusual scaling exponents. Although the ASEP without impurities belongs to the KPZ universality class ͓9͔ ͑dy-namic exponent 3 2 ͒, our model displays a scaling exponent 5 2 . The model we propose describes the dynamics of two types of particles ͑types 1 and 2͒ on the lattice, where the total number of particles of each type is conserved. In this model if the neighbor sites are empty, particles of type 1 can jump to the right or to the left with rates ⌫ 0 1 1 0 and ⌫ 1 0 0 1 , respectively. Particles of type 2 "impurities" do not jump to the neighbor sites if they are empty, but can change positions with neighbor particles of type 1 with rates ⌫ 2 1 1 2 and ⌫ 1 2 2 1 if particle 1 is on the left or on the ri...

Asymmetric exclusion model with several kinds of impurities

“…While particles of type 1 can jump to neighbors sites if they are empty, like in ASEP, particles of type 2 (called impurities) do not jump to empty sites but exchange positions with neighbor particles of type 1. We show that this model has a relaxation time longer than the ones for the ASEP, and displays a scaling exponent of z = 5 2 [23] (of order L 3 2 ×L = L 5 2 [23]). We obtained this result by solving the Bethe Ansatz equation for the half-filling sector and in the totally asymmetric diffusion process [23].…”

“…Recently, we introduced a new class of 3-state model that is integrable despite it do not have particle-particle exchange symmetry [22]. In [23] we extend the model [22] and formulate an one-dimensional asymmetric exclusion process with one kind of impurities (ASEPI). This model describes the dynamics of two types of particles (type 1 and 2) on a lattice of L sites, where each lattice site can be occupied by at most one particle.…”

“…We formulate an asymmetrical diffusion model of N = 1, 2, 3, ... kinds of particles with impurities (N-ASEPI), where particles of kind 1 can jump to neighboring sites if they are empty and particles of kind α = 2, 3, ..., N (called impurities) only exchange positions with particles if satisfy a well defined dynamics. Different from the ASEPI [23], our generalized model can have more than one particle on each site (multiple site occupation). Although our model can be solved by the coordinate Bethe ansatz, we are going to formulate a new matrix product ansatz (MPA) [24,21] due its simplicity and unifying implementation for arbitrary systems.…”

“…In order to check our conjecture, we solve numerically the Bethe equation with N = 3 and N = 4 for the totally asymmetric diffusion and found that the gap for the first excited state scales with L − 7 2 and L − 9 2 in these cases. Furthermore, we also generalize the model [23] to include quantum spin chain and solve the Bethe Ansatz equation for symmetric and asymmetric diffusion.…”

“…In section 2 we generalize the model [23] to include quantum spin chain and solve the Bethe Ansatz equation for the symmetric and asymmetric diffusion. The generalization for several kinds of impurities is done in section 3.…”