Abstract
The hopping motion of classical particles on a chain coupled to reservoirs at both ends is studied for parallel dynamics with arbitrary probabilities. The stationary state is obtained in the form of an alternating matrix product. The properties of one- and two-dimensional representations are studied in detail and a general relation of the matrix algebra to that of the sequential limit is found. In this way the general phase diagram of the model is obtained. The mechanism of the sequential limit, the formulation as a vertex model, and other aspects are discussed.
Similar content being viewed by others
References
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries,J. Stat. Phys. 69:667–687 (1992).
G. Schütz and E. Domany, Phase transitions in an exactly soluble one-dimensional exclusion process,J. Stat. Phys. 72:277–296 (1993).
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation,J. Phys. A: Math. Gen. 26:1493–1517 (1993).
B. Derrida and M. R. Evans, Exact steady state properties of the one-dimensional asymmetric exclusion model, inProbability and Phase Transition, G. Grimmett, ed. (Kluwer Academic Publishers, 1994), pp. 1–16.
V. Hakim and J. P. Nadal, Exact results for 2D directed animals on a strip of finite width,J. Phys. A: Math. Gen. 16:L213-L218 (1983).
M. Fannes, B. Nachtergaele and R. F. Werner, Exact antiferromagnetic ground states of quantum spin chains,Europhys. Lett. 10:633–637 (1989).
A. Klümper, A. Schadschneider and J. Zittartz, Equivalence and solution of antisotropic spin-1 models and generalizedt−J fermion models in one dimension,J. Phys. A: Math. Gen. 24:L955-L959 (1991).
A. Klümper, A. Schadschneider and J. Zittartz, Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets,Europhys. Lett. 24:293–297 (1993).
H. Hinrichsen, S. Sandow and I. Peschel, On matrix product ground states for reaction-diffusion models,J. Phys. A: Math. Gen. 29:2643–2649 (1996).
H. Hinrichsen, Matrix product ground states for exclusion processes with parallel dynamics,J. Phys. A: Math. Gen. 29:3659–3667 (1996).
I. Peschel and F. Rys, New solvable cases for the eight-vertex model,Phys. Lett. A91:187–189 (1982).
P. Ruján, Order and disorder lines in systems with competing interactions: II. The IRF model,J. Stat. Phys. 29:247–262 (1982).
R. J. Baxter, Disorder points of the IRF and checkerboard Potts models,J. Phys. A: Math. Gen. 17:L911-L917 (1984).
M. T. Batchelor and J. M. J. van Leeuwen, Disorder solutions of lattice spin models,Physica A154:365–383 (1989).
N. Rajewsky, A. Schadschneider and M. Schreckenberg, The asymmetric exclusion model with sequential update,J. Phys. A: Math. Gen. 29:L305-L309 (1996).
D. Kandel, E. Domany and B. Nienhuis, A six-vertex model as a diffusion problem: Derivation of correlation functions,J. Phys. A: Math. Gen. 23:L755-L762 (1990).
G. Schütz, Time-dependent correlation functions in a one-dimensional asymmetric exclusion process,Phys. Rev. E47:4265–4277 (1993).
A. L. Owczarek and R. J. Baxter, Surface free energy of the critical six-vertex model with free boundaries,J. Phys. A: Math. Gen. 22:1141–1165 (1989).
S. Sandow, Partially asymmetric exclusion process with open boundaries,Phys. Rev. E50:2660–2667 (1994).
F. H. L. Eßler and V. Rittenberg, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries,J. Phys. A: Math. Gen. 29:3375–3407 (1996).
F. C. Alcaraz, M. Droz, M. Henkel, and V. Rittenberg, Reaction-diffusion processes, critical dynamics, and quantum chains,Ann. Phys. 230:250–302 (1994).
N. Rajewsky, L. Santen, A. Schadschneider, and M. Schreckenberg, in preparation.
Z.-Q. Ma,Yang-Baxter Equation and Quantum Enveloping Algebras (World Scientific, Singapore, 1993).
H. J. de Vega and A. González-Ruiz, BoundaryK-matrices for theXYZ, XXZ andXXX spin chains,J. Phys. A: Math. Gen. 27:6129–6137 (1994).
T. Inami and H. Konno, IntegrableXYZ spin chain with boundaries,J. Phys. A: Math. Gen. 27:L913-L918 (1994).
C. M. Yung and M. T. Batchelor, Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices,Nucl. Phys. B435:430–462 (1995).
H. J. Giacomini, Disorder solutions and the star-triangle relation,J. Phys. A: Math. Gen. 19:L537-L541 (1986).
H. Hinrichsen, K. Krebs, and I. Peschel, Solution of a one-dimensional diffusion-reaction model with spatial asymmetry,Z. Phys. B100:105–114 (1996).
G. Schütz, Generalized Bethe ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage,J. Stat. Phys. 71:471–505 (1993).
H. Hinrichsen, private communication.
R. B. Stinchcombe and G. M. Schütz, Operator algebra for stochastic dynamics and the Heisenberg chain,Europhys. Lett. 29:663–667 (1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Honecker, A., Peschel, I. Matrix-product states for a one-dimensional lattice gas with parallel dynamics. J Stat Phys 88, 319–345 (1997). https://doi.org/10.1007/BF02508474
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02508474