Exact results using dual-unitarity largely rely on nearest-neighbor structures, while finite-range interactions typically lead to complications. Going beyond the usual nearest-neighbor setting, we introduce an analytically tractable family of finite-range kicked Ising models that admit exact closed-form entanglement dynamics. The construction is based on a staggered structure in which dual-unitarity is present on sublattices that are then coupled to each other. The central observation is that these inter-sublattice couplings do not obstruct the dual-unitarity of the resulting model. For the minimal interaction range of $r= 2$, we derive exact expressions for all the $n-$Rényi entanglement entropies at all times and show that the result is the sum of the two coupled sublattice contributions. Our framework extends naturally to larger finite interaction ranges and to systems with heterogeneous local Hilbert spaces, without additional assumptions. It thus provides a controlled setting for studying exact entanglement growth beyond strictly nearest-neighbor dual-unitary models.
Exact results using dual-unitarity largely rely on nearest-neighbor structures, while finite-range interactions typically lead to complications. Going beyond the usual nearest-neighbor setting, we introduce an analytically tractable family of finite-range kicked Ising models that admit exact closed-form entanglement dynamics. The construction is based on a staggered structure in which dual-unitarity is present on sublattices that are then coupled to each other. The central observation is that these inter-sublattice couplings do not obstruct the dual-unitarity of the resulting model. For the minimal interaction range of r=2, we derive exact expressions for all the n−Rényi entanglement entropies at all times and show that the result is the sum of the two coupled sublattice contributions. Our framework extends naturally to larger finite interaction ranges and to systems with heterogeneous local Hilbert spaces, without additional assumptions. It thus provides a controlled setting for studying exact entanglement growth beyond strictly nearest-neighbor dual-unitary models.
p-Body ≃ Range p−1: Exact Order-Range Mapping and Dual-Unitarity
We introduce and study a periodically kicked version of a one-dimensional spin-1/2 Ising model with contiguous p-body interactions. At specific interaction strengths of the model, we establish an exact Floquet interaction order-range mapping: the Floquet evolution operator of the p-body Ising model is exactly mapped to a two body Ising model with interactions having maximum range p−1 and exactly determined couplings. Upon further tuning the kicking strength, the mapped model becomes dual-unitary, thus providing an explicit family of p-body dual unitary models. We then determine exact entanglement dynamics of the p-body model for a class of solvable initial states at all times. We further illustrate the nontrivial structures that can emerge from the order-range mapping using a minimal example of p=3 and generalize it for all odd p. These findings provide a systematic route to constructing and studying p-body dual-unitary Floquet models.
Many-body Floquet models with exact space--time symmetry, such as the kicked Ising spin chain (KIC), provide natural examples of systems with dual-unitary dynamics. The requirement of exact space--time symmetry is, however, highly restrictive, as it permits only nearest-neighbor interactions. Based on a pair of Hadamard matrices, we construct a wide family of dual-unitary kicked spin chains with long-range interactions. We show that local two-point correlations in such models propagate along the light-cone edges ∣n∣=r∣t∣, where r is the interaction range, and can be derived analytically for operators with local support. This approach is illustrated using the example of a kicked Ising spin chain with next-to-next-neighbor interactions.
We investigate the dynamics of entanglement in a generalized version of the kicked Ising chain, extending the model from the standard qubit case (local dimension q=2) to higher local dimensions (q>2). We identify the existence of ''dual-unitary'' points where the model's space-time duality allows for exact analytical solutions. Our analysis reveals that while a few unique dual-unitary points exist analytically for systems with local dimensions q=3 and q=4, such points do not exist for q≥5 due to the lack of a unique kicking strength that satisfies the required matrix element conditions. Utilizing the transfer matrix method and a replica trick specifically adapted for higher dimensions, we derive exact expressions for the growth of entanglement entropy in the q=3 (kicked Potts-type) model starting from a class of solvable initial states. Our results demonstrate that at the dual-unitary point, both Rényi and von Neumann entanglement entropies grow linearly with time until reaching a maximum value determined by the subsystem size.
We study the nonequilibrium dynamics of kicked Ising models in 1 + 1 dimensions which have interactions alternating between odd and even bonds in time. These models can be understood as quantum circuits tiling space-time with the generalized space-time dual properties of triunitarity (three “arrows of time”) at the global level, and also second-level dual unitarity at the local level, which constrains the behavior of pairs of local gates underlying the circuit under a space-time rotation. We identify a broad class of initial product states wherein the effect of the environment on a small subsystem can be exactly represented by influence matrices with simple Markovian structures, resulting in the subsystem's full dynamics being efficiently computable. We further find additional conditions under which the dynamics of entanglement can be solved for all times, yielding rich phenomenology ranging from linear growth at half the maximal speed allowed by locality, followed by saturation to maximum entropy (i.e., thermalization to infinite temperature), to entanglement growth with saturation to extensive but submaximal entropy. Intriguingly, for certain parameter regimes, we find a nonchaotic class of dynamics which is neither integrable nor Clifford, exemplified by nonzero operator entanglement growth but with a spectral form factor which exhibits large, apparently time-quasiperiodic revivals.
Exactly solvable models in quantum many-body dynamics provide valuable insights into many interesting physical phenomena and serve as platforms for rigorously investigating fundamental theoretical questions. Nevertheless, they are extremely rare and existing solvable models and solution techniques have serious limitations. In this paper, we present a family of exactly solvable unitary circuits that model quantum many-body dynamics in discrete space and time. Unlike many previous solvable models, one can exactly compute the full quantum dynamics initialized from any matrix product state in this family of models. The time evolution of local observables and correlations, the linear growth of Rényi entanglement entropy, spatiotemporal correlations, and out-of-time-order correlations are all exactly computable. A key property of these models enabling the exact solution is that any time-evolved local operator is an exact matrix product operator with a finite bond dimension, even at arbitrarily long times, which we prove using the underlying C * -(weak) Hopf algebra structure along with tensor network techniques. We lay down the general framework for construction and solution of this family of models, and give several explicit examples. In particular, we study in detail a model constructed out of a C * -weak Hopf algebra that is very close to a Floquet version of the PXP model, and the exact results we obtain may shed light on the phenomenon of quantum many-body scars, and more generally, Floquet quantum dynamics in constrained systems.
Dual-unitary circuits are a class of locally interacting quantum many-body systems displaying unitary dynamics also when the roles of space and time are exchanged. These systems have recently emerged as a remarkable framework where certain features of many-body quantum chaos can be studied exactly. In particular, they admit a class of ``solvable'' initial states for which, in the thermodynamic limit, one can access the full nonequilibrium dynamics. This reveals a surprising property: when a dual-unitary circuit is prepared in a solvable state the quantum entanglement between two complementary spatial regions grows at the maximal speed allowed by the local structure of the evolution. Here we investigate the fate of this property when the system is prepared in a generic pair-product state. We show that in this case, the entanglement increment during a time step is submaximal for finite times, however, it approaches the maximal value in the infinite-time limit. This statement is proven rigorously for dual-unitary circuits generating high enough entanglement, while it is argued to hold for the entire class.