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arXiv:2608.09695

Spread of Entanglement in Generalized Kicked Ising Chain

Tanay Pathak, Hiromi Ebisu, Tomaž Prosen

quant-phcond-mat.stat-mechhep-thnlin.CDSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

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 $q \ge 5$ 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.

Comments
23 pages, 6 figures

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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=2q=2q=2) to higher local dimensions (q>2q > 2q>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=3q=3q=3 and q=4q=4q=4, such points do not exist for q≥5q \ge 5q≥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=3q=3q=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.
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=2r= 2r=2, we derive exact expressions for all the n−n-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.
Understanding the dynamics of quantum correlations in many-body systems is a central problem in non-equilibrium quantum physics. We study the spread of mixed-state entanglement in a minimal model of quantum chaos, the kicked field Ising model. By combining the replica trick with the space-time duality of the model, we determine the exact spectrum of the partially transposed reduced density matrix. The resulting flat spectrum leads to exact relations between entanglement negativity, odd entropy and Rényi mutual information at early times. Numerical results further demonstrate that for equal tri-partitions and at late times, all entanglement measures saturate to the Haar-random values. In contrast, for unequal tri-partitions Rényi mutual information and negativity vanish at late times, implying that the corresponding reduced density matrix is factorizable. Extensive numerical simulations also show that the relation remains quantitatively valid for generic initial states, leading us to conjecture it for all initial states and all times.
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∣ |n| = r|t| ∣n∣=r∣t∣, where rrr 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 show that for dual-unitary kicked chains, built upon a pair of complex Hadamard matrices, correlators of strictly local, traceless operators vanish identically for sufficiently long chains. On the other hand, operators supported at pairs of adjacent chain sites, generically, exhibit nontrivial correlations along the light cone edges. In agreement with Bertini et. al. [Phys. Rev. Lett. 123, 210601 (2019)], they can be expressed through the expectation values of a transfer matrix TTT. Furthermore, we identify a remarkable family of dual-unitary models where an explicit information on the spectrum of TTT is available. For this class of models we provide a closed analytical formula for the corresponding two-point correlators. This result, in turn, allows an evaluation of local correlators in the vicinity of the dual-unitary regime which is exemplified on the kicked Ising spin chain.
We show that local correlators in a wide class of kicked chains can be calculated exactly at light-cone edges. Extending previous works on circuit lattice systems, the correlators between local operators are expressed through the expectation values of transfer matrices T\mathbf{T}T with small dimensions. For dual-unitary kicked chains, with spatial-temporal symmetry of the dynamics, this provides a full characterization of local correlators. Furthermore, we identify a remarkable family of dual-unitary models where an explicit information on the spectrum of T\mathbf{T}T is available. For this class of models we provide a closed analytical formula for the corresponding two-point correlators. The results are exemplified on the kicked Ising spin chain model.