Integrable boundary conditions for multi-species ASEP (Q2826699)

The asymmetric simple exclusion process (ASEP) describes particles hopping on a one-dimensional lattice with anisotropic rate and hard-core exclusion. Since 1998, it has become a well-known and much developed model in non-equilibrium statistical physics for driven stochastic processes [\textit{B. Derrida}, ``An exactly soluble non-equilibrium system: the asymmetric simple exclusion process'', Phys. Reports 301, No. 1--3, 65--83 (1998; \url{doi:10.1016/S0370-1573(98)00006-4})]. Its stationary state can be explicitly constructed by using a matrix-product formalism, it is integrable, and it shows a rich phenomenology (boundary-induced phase transitions, shock waves), applicable to many different fields. This paper contains two new results. The first is to provide integrable boundaries for ASEP with an arbitrary number of species (thus generalizing former results for two species [the first author et al., J. Phys. A, Math. Theor. 48, No. 17, Article ID 175002, 18 p. (2015; Zbl 1325.82005)] in a quantum inverse scattering framework that allows the construction of \(K\)-matrices solving the reflection equation through a Baxterization procedure [\textit{P. P. Kulish} and \textit{A. I. Mudrov}, Lett. Math. Phys. 75, No. 2, 151--170 (2006; Zbl 1144.16035)]). The integrable boundaries divide the set of all species into five different classes, according to their rate of progress. The second result is that the solutions obtained can be seen as a representation of a new algebra that contains the boundary Hecke algebra [\textit{D. Levy} and \textit{P. Martin}, J. Phys. A, Math. Gen. 27, No. 14, L521--L526 (1994; Zbl 0843.17008)]. The latter is shown to be insufficient to build the different solutions found in this paper, which are conjectured to be the only Markovian solutions to the reflection equation with at least two free parameters. This algebraic structure is expected to be useful to extend these results to permeable integrable boundaries.