Yang-Baxter Integrable Models Research Articles

Dynamics of fluctuations in quantum simple exclusion processes

We consider the dynamics of fluctuations in the quantum asymmetric simple exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP describes a chain of spinless fermions with random hoppings that are induced by a Markovian environment. We show that fluctuations of the fermionic degrees of freedom obey evolution equations of Lindblad type, and derive the corresponding Lindbladians. We identify the underlying algebraic structure by mapping them to non-Hermitian spin chains and demonstrate that the operator space fragments into exponentially many (in system size) sectors that are invariant under time evolution. At the level of quadratic fluctuations we consider the Lindbladian on the sectors that determine the late time dynamics for the particular case of the quantum symmetric simple exclusion process (Q-SSEP). We show that the corresponding blocks in some cases correspond to known Yang-Baxter integrable models and investigate the level-spacing statistics in others. We carry out a detailed analysis of the steady states and slow modes that govern the late time behaviour and show that the dynamics of fluctuations of observables is described in terms of closed sets of coupled linear differential-difference equations. The behaviour of the solutions to these equations is essentially diffusive but with relevant deviations, that at sufficiently late times and large distances can be described in terms of a continuum scaling limit which we construct. We numerically check the validity of this scaling limit over a significant range of time and space scales. These results are then applied to the study of operator spreading at large scales, focusing on out-of-time ordered correlators and operator entanglement.

Yang-Baxter integrable Lindblad equations

We consider Lindblad equations for one dimensional fermionic models and quantum spin chains. By employing a (graded) super-operator formalism we identify a number of Lindblad equations than can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. Employing Bethe Ansatz techniques we show that the late-time dynamics of some of these models is diffusive.

One-dimensional sums and finitized characters of 2 × 2 fused RSOS models

Tartaglia and Pearce have argued that the nonunitary fused Forrester–Baxter models are described, in the continuum scaling limit, by the minimal models constructed as the higher-level conformal cosets at integer fusion level and fractional level with . These results rely on Yang–Baxter integrability and are valid in Regime III for models determined by the crossing parameter in the interval . Combinatorially, Baxter’s one-dimensional sums generate the finitized branching functions as weighted walks on the Dynkin diagram. The ground state walks terminate within shaded n-bands, consisting of n contiguous shaded 1-bands. The shaded 1-bands occur between heights where , . These results do not extend to the interval since, for these models, there are no shaded n bands to support the ground states. Here we consider the models in the interval and investigate the associated one-dimensional sums. In this interval, we verify that the one-dimensional sums produce new finitized Virasoro characters of the minimal models with . We further conjecture finitized bosonic forms and check that these agree with the ground state one-dimensional sums out to system sizes N = 12. The models thus realize new Yang–Baxter integrable models in the universality classes of the minimal models . For the series with , the spin-1 one-dimensional sums were previously analysed by Jacob and Mathieu without the underlying Yang–Baxter structure. Finitized Kac characters for the logarithmic minimal models are also obtained for by taking the logarithmic limit with .

Some Recent Results on Pair Correlation Functions and Susceptibilities in Exactly Solvable Models

Using detailed exact results on pair-correlation functions of Z-invariant Ising models, we can write and run algorithms of polynomial complexity to obtain wavevector-dependent susceptibilities for a variety of Ising systems. Reviewing recent work we compare various periodic and quasiperiodic models, where the couplings and/or the lattice may be aperiodic, and where the Ising couplings may be either ferromagnetic, or antiferromagnetic, or of mixed sign. We present some of our results on the square-lattice fully-frustrated Ising model. Finally, we make a few remarks on our recent works on the pentagrid Ising model and on overlapping unit cells in three dimensions and how these works can be utilized once more detailed results for pair correlations in, e.g., the eight-vertex model or the chiral Potts model or even threedimensional Yang-Baxter integrable models become available.