Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

Abstract

In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in [1], [2] and [3] in the context of KPZ universality class. We show that they may be mapped onto an integrable sl(2) Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature.Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a "dual model" of probability theory is possible and useful.The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of N = 4 super Yang-Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the sl(2|1) superstring that has been derived directly from N = 4 SYM.

The non-compact XXZ spin chain as stochastic particle process

“…The Hamiltonian of the non-compact XXZ spin chain depends on the parameters q and s. In the rational limit q → 1 we recover the non-compact XXX chain, cf. [12]. The similarity transformation relating the spin chain and the stochastic process becomes trivial in this limit.…”

“…Furthermore, as the mapping depends on the choice of the basis, the stochastic process related to a Hamiltonian may be difficult to identify. Recently, a relation between rational non-compact spin chains and stochastic processes was pointed out in [12]. The types of spin chains considered appeared previously in relation to highenergy QCD and N = 4 super Yang-Mills theory in the context of the AdS/CFT correspondence, see [13] for further references as well as [14] and references thereof.…”

“…The types of spin chains considered appeared previously in relation to highenergy QCD and N = 4 super Yang-Mills theory in the context of the AdS/CFT correspondence, see [13] for further references as well as [14] and references thereof. It was found in [12] that the Hamiltonian density of such spin chains yields the rational limits of a class of stochastic transport models that were known to be integrable. More precisely, the multiparticle asymmetric diffusion model (MADM) and a more general q-Hahn process that were defined in [15] and [16] respectively.…”

“…Here we focus on connecting q-deformed non-compact (XXZ) spin chains with stochastic particle processes. From the many aspects that were studied in [12], which presumably do have a q-analog, we restrict ourselves to setting the stage and relate the Hamiltonian of the non-compact XXZ chain to the Markov matrix of a stochastic q-Hahn process.…”

Generic triangular solutions of the reflection equation: ${U}_{q}\left(\hat{s{l}_{2}}\right)$ case

“…Although it is beyond the scope of the present paper, we expect that our results will be useful to construct Baxter Q-operators for integrable systems with triangular boundaries (by taking limits of K-operators as discussed in [6]). As for the rational case (q = 1 case; the XXX-model associated with the Yangian Y (sl 2 )), diagonal K-operators for Baxter Q-operators appeared first in [8], and a generic non-diagonal K-operator was proposed recently 2 in [9].…”