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arXiv:2004.08386 · Physical review. B./Physical review. B

Exact local correlations in kicked chains

Boris Gutkin, Petr A. Braun, Maram Akila, Daniel Waltner, Thomas Guhr

Published Nov 11, 2020• 40 citations
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Abstract

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 $\mathbf{T}$ 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 $\mathbf{T}$ 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.

DOI
10.1103/physrevb.102.174307

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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.
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.
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 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.
Dual-unitary quantum circuits can provide analytic spatiotemporal correlation functions of local operators from transfer matrices, enriching our understanding of quantum dynamics with exact solutions. Nevertheless, a full understanding is still lacking as the case of a nondiagonalizable transfer matrix with exceptional points has barely been explored. In this paper, we give an inverse approach for constructing dual-unitary quantum circuits with exceptional points in the transfer matrices by establishing relations between transfer matrices and local unitaries. As a consequence of the coalesce of eigenvectors, the correlation functions exhibit a polynomial modified exponential decay, which is significantly different from pure exponential decay, especially at early stages. Moreover, we point out that the Hamiltonian evolution of a kicked XXZXXZXXZ spin chain can be approximately mapped to a dual-unitary circuit with exceptional points by Trotter decomposition. Finally, we investigate the dynamics approaching and at exceptional points, showing that behaviors of correlation functions are distinct by ZZZ transformation.
We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for a choice of observables.
We consider the problem of local correlations in the kicked, dual-unitary coupled maps on D-dimensional lattices. We demonstrate that for D>=2, fully dual-unitary systems exhibit ultra-local correlations: the correlations between any pair of operators with local support vanish in a finite number of time steps. In addition, for D=2D=2D=2, we consider the partially dual-unitary regime of the model, where the dual-unitarity applies to only one of the two spatial directions. For this case, we show that correlations generically decay exponentially and provide an explicit formula for the correlation function between the operators supported on two and four neighbouring sites.