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arXiv:1705.10979 · Nuclear Physics B

Density and current profiles in Uq(A2(1)) zero range process

Atsuo Kuniba, Vladimir V. Mangazeev

Published Jul 11, 2017• 3 citations
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Abstract

The stochastic R matrix for U q ( A n ( 1 ) ) introduced recently gives rise to an integrable zero range process of n classes of particles in one dimension. For n = 2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of q -bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.

DOI
10.1016/j.nuclphysb.2017.07.001

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The stochastic R matrix for U q ( A n ( 1 ) ) introduced recently gives rise to an integrable zero range process of n classes of particles in one dimension. For n = 2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of q -bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.
Abstract We obtain exact formulas of the first two cumulants of particle current in the q -boson zero range process on a ring via exact perturbative solution of the TQ-equation. The result is represented as an infinite sum of double contour integrals. We perform the asymptotic analysis of the large system size limit N → ∞ of the expressions obtained. For | q | ≠ 1 the leading terms of the second cumulant reproduce the N 3/2 scaling expected for models in the Kardar–Parisi–Zhang universality class. The scaling q ≍ exp ( − α / N ) → 1 corresponds to the crossover between the Kardar–Parisi–Zhang and Edwards–Wilkinson universality classes. Under this scaling the sum converges to an integral, resulting in the crossover scaling function derived previously for the asymmetric simple exclusion process and conjectured to be universal.
We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.
An exact and rigorous calculation of the current and density profile in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and forward-ordered sequential dynamics is presented. The method is based on a matrix product representation of the steady-state probability distribution. The main idea is to choose a suitable representation in which the scalar products describing the current and local density profile for a chain of arbitrary finite size depend only on the elements in a finite number of rows and columns. This makes possible the use of a truncated finite-dimensional representation of the matrices and vectors involved. After performing the calculations, we lift the truncation by letting its dimensionality go to infinity. In this limit the results become exact for any size of the chain. By rescaling one of the infinite-dimensional matrix representations found in the work of Derrida et al. [J. Phys. A 26, 1493 (1993)] for their algebra, we obtain a symmetric ``propagator'' matrix. Its truncated version is diagonalized by orthogonal transformation for easy calculation of the relevant scalar products. An interpretation of the phase transitions between the different phases is given in terms of eigenvalue splitting from a bounded quasicontinuous spectrum. A precise description of the local density profiles is given for all values of the parameters. It is shown that the leading-order asymptotic form of the position-dependent terms in the local density changes within the low- and high-density phases, signaling the presence of a higher-order transition.
The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.
The asymmetric exclusion process on a ring in one dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second-class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover, the variance (diffusion constant) grows as L 1/2 where L is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.