We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which are set theoretical solutions of the Yang-Baxter equations. We show that such systems are superintegrable: they possess an exponentially large set of conserved local charges, such that the charge densities propagate ballistically on the chain. For these quantities we observe a complete absence of "operator spreading". In addition, the models can also have other local charges which are conserved only additively. We discuss concrete models up to local dimensions $N\le 4$ , and show that they give rise to rich physical behaviour, including non-trivial scattering of particles and the coexistence of ballistic and diffusive transport. We find that the local update rules are classical versions of the "dual unitary gates" if the Yang-Baxter maps are non-degenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a non-integrable quantum mechanical deformation of the Yang-Baxter maps.We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which are set theoretical solutions of the Yang-Baxter equations. We show that such systems are superintegrable: they possess an exponentially large set of conserved local charges, such that the charge densities propagate ballistically on the chain. For these quantities we observe a complete absence of "operator spreading". In addition, the models can also have other local charges which are conserved only additively. We discuss concrete models up to local dimensions $N\le 4$, and show that they give rise to rich physical behaviour, including non-trivial scattering of particles and the coexistence of ballistic and diffusive transport. We find that the local update rules are classical versions of the "dual unitary gates" if the Yang-Baxter maps are non-degenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a non-integrable quantum mechanical deformation of the Yang-Baxter maps.

Mar 1, 202243 citations

On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum

Tomaž Prosen

We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy–momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang–Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators.We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy–momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang–Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators.

arXiv:2304.01130EntropyApr 30, 20234 citations

Integrable spin chains and cellular automata with medium-range interaction

Tamás Gombor, Balázs Pozsgay

We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three-site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three-site models, but we cannot exclude Yang-Baxter integrability with longer interaction ranges.We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three-site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three-site models, but we cannot exclude Yang-Baxter integrability with longer interaction ranges.

arXiv:2108.02053Physical review. ENov 22, 202182 citations

A Yang–Baxter integrable cellular automaton with a four site update rule

Balázs Pozsgay

Abstract We present a one dimensional reversible block cellular automaton, where the time evolution is dictated by a period 3 cycle of update rules. At each time step a subset of the cells is updated using a four site rule with two control bits and two action bits. The model displays rich dynamics. There are three types of stable particles, left movers, right movers and ‘frozen’ bound states that only move as an effect of scattering with the left and right movers. Multi-particle scattering in the system is factorized. We embed the model into the canonical framework of Yang–Baxter integrability by rigorously proving the existence of a commuting set of diagonal-to-diagonal transfer matrices. The construction involves a new type of Lax operator.Abstract We present a one dimensional reversible block cellular automaton, where the time evolution is dictated by a period 3 cycle of update rules. At each time step a subset of the cells is updated using a four site rule with two control bits and two action bits. The model displays rich dynamics. There are three types of stable particles, left movers, right movers and ‘frozen’ bound states that only move as an effect of scattering with the left and right movers. Multi-particle scattering in the system is factorized. We embed the model into the canonical framework of Yang–Baxter integrability by rigorously proving the existence of a commuting set of diagonal-to-diagonal transfer matrices. The construction involves a new type of Lax operator.

arXiv:2106.00696Journal of Physics A Mathematical and TheoreticalAug 16, 202138 citations

Many-body quantum chaos and dual-unitarity round-a-face

Tomaž Prosen

We propose a new type of locally interacting quantum circuits—quantum cellular automata—that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension d [DUIRF(d)], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF(2) for qubits (d=2) and provide, for d=3,4,…,7, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine dimDUBG(d) and show that they are of similar order though systematically larger than dimDUIRF(d) for d=2,3,…,7. It is remarkable that both sets have a rather complex topology for d≥3 in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions d≠d′ residing at even/odd lattice sites.We propose a new type of locally interacting quantum circuits—quantum cellular automata—that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension d [DUIRF(d)], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF(2) for qubits (d=2) and provide, for d=3,4,…,7, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine dimDUBG(d) and show that they are of similar order though systematically larger than dimDUIRF(d) for d=2,3,…,7. It is remarkable that both sets have a rather complex topology for d≥3 in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions d≠d′ residing at even/odd lattice sites.

arXiv:2105.08022Chaos An Interdisciplinary Journal of Nonlinear ScienceSep 1, 202139 citations

Many-body quantum chaos and dual-unitarity round-a-face

Tomaž Prosen

Sep 13, 202131 citations

N\le 4

, and show that they give rise to rich physical behaviour, including non-trivial scattering of particles and the coexistence of ballistic and diffusive transport. We find that the local update rules are classical versions of the "dual unitary gates" if the Yang-Baxter maps are non-degenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a non-integrable quantum mechanical deformation of the Yang-Baxter maps.