Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring

Phase diagram of generalized totally asymmetric simple exclusion process on an open chain: Liggett-like boundary conditions

The totally asymmetric simple exclusion process with generalized update is a version of the discrete time totally asymmetric exclusion process with an additional interparticle interaction that controls the degree of particle clustering. Though the model was shown to be integrable on the ring and on the infinite lattice, on the open chain it was studied mainly numerically, while no analytic results existed even for its phase diagram. In this paper, we introduce boundary conditions associated with the infinite translation invariant stationary states of the model, which allow us to obtain the exact phase diagram analytically. We discuss the phase diagram in detail and confirm the analytic predictions by extensive numerical simulations.

One-dimensional discrete aggregation-fragmentation model

We study here one-dimensional model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time kinetics of the generalized totally asymmetric simple exclusion process (gTASEP) on open chains. The gTASEP is essentially the ordinary TASEP with backward-ordered sequential update (BSU), however, equipped with two hopping probabilities: p and p_{m}. The second modified probability p_{m} models a special kinematic interaction between the particles of a cluster in addition to the simple hard-core exclusion interaction, existing in the ordinary TASEP. We focus on the nonequilibrium stationary properties of the gTASEP in the generic case of attraction between the particles of a cluster. In this case the particles of a cluster have higher chance to stay together than to split, thus producing higher throughput in the system. We explain how the topology of the phase diagram in the case of irreversible aggregation, occurring when the modified probability equals unity, changes sharply to the one, corresponding to the ordinary TASEP with BSU, as soon as the modified probability becomes less than unity and aggregation-fragmentation of clusters appears. We estimate various physical quantities in the system and determine the parameter-dependent injection and ejection critical values by extensive computer simulations. With the aid of random walk theory, supported by the Monte Carlo simulations, the properties of the phase transitions between the three stationary phases are assessed.

One-dimensional irreversible aggregation with dynamics of a totally asymmetric simple exclusion process

We define and study one-dimensional model of irreversible aggregation of particles obeying a discrete-time kinetics, which is a special limit of the generalized Totally Asymmetric Simple Exclusion Process (gTASEP) on open chains. The model allows for clusters of particles to translate as a whole entity one site to the right with the same probability as single particles do. A particle and a cluster, as well as two clusters, irreversibly aggregate whenever they become nearest neighbors. Nonequilibrium stationary phases appear under the balance of injection and ejection of particles. By extensive Monte Carlo simulations it is established that the phase diagram in the plane of the injection-ejection probabilities consists of three stationary phases: a multiparticle (MP) one, a completely filled (CF) phase, and a "mixed" (MP+CF) one. The transitions between these phases are: an unusual transition between MP and CF with jump discontinuity in both the bulk density and the current, a conventional first-order transition with a jump in the bulk density between MP and MP+CF, and a continuous clustering-type transition from MP to CF, which takes place throughout the MP+CF phase between them. By the data collapse method a finite-size scaling function for the current and bulk density is obtained near the unusual phase transition line. A diverging correlation length, associated with that transition, is identified and interpreted as the size of the largest cluster. The model allows for a future extension to account for possible cluster fragmentation.

Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions

Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.