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 [Sasamoto-Wadati], [Barraquand-Corwin] and [Povolotsky] in the context of KPZ universality class. We show that they may be mapped onto an integrable $\mathfrak{sl}(2)$ 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 $\mathcal{N}=4$ 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 $\mathfrak{sl}(2|1)$ superstring that has been derived directly from $\mathcal{N}=4$ SYM.

35 pages, 2 figures, v2: typos fixed and references added, v3: typo fixed, v4,5: minor correction

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• Duality and hidden equilibrium in transport models
• Exact solution of an integrable non-equilibrium particle system
• Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space
• Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steady state of the open SSEP
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Integrable heat conduction model
• KPZ fluctuations in finite volume
• Integrable boundaries for the q-Hahn process
• Supersymmetries in non-equilibrium Langevin dynamics
• Generic triangular solutions of the reflection equation: $U_q(\widehat{s{l}_2})$ case
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• The steady state of the boundary-driven multiparticle asymmetric diffusion model
• BC-type open $SL(2,\mathbb{C})$ spin chain
• Steady state representations for the harmonic process