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arXiv:2406.08338 · Physical review. B./Physical review. B

Exact correlation functions for dual-unitary quantum circuits with exceptional points

Xi-Dan Hu, Dan-Bo Zhang

Published Jan 3, 2025• 7 citations
arXivHTMLPDF

Abstract

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 $XXZ$ 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 $Z$ transformation.

DOI
10.1103/physrevb.111.024301

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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 consider a unitary circuit where the underlying gates are chosen to be \check{R} R ̌ -matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.
We consider a class of quantum lattice models in 1+1 dimensions represented as local quantum circuits that enjoy a particular dual-unitarity property. In essence, this property ensures that both the evolution in time and that in space are given in terms of unitary transfer matrices. We show that for this class of circuits, generically nonintegrable, one can compute explicitly all dynamical correlations of local observables. Our result is exact, nonpertubative, and holds for any dimension d of the local Hilbert space. In the minimal case of qubits (d=2) we also present a classification of all dual-unitary circuits which allows us to single out a number of distinct classes for the behavior of the dynamical correlations. We find noninteracting classes, where all correlations are preserved, the ergodic and mixing one, where all correlations decay, and, interestingly, also classes that are both interacting and nonergodic.
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 investigate details of the chaotic dynamics of dual-unitary quantum circuits by evaluating all 2k2k2k-point out-of-time-ordered correlators. For the generic class of circuits, by writing the correlators as contractions of a class of quantum channels, we prove their exponential decay. This implies that local operators separated in time approach free independence. Along the way, we develop a replica trick for dual-unitary circuits, which may be useful and of interest in its own right. We classify the relevant eigenstates of the replica transfer matrix by paths in the lattice of noncrossing partitions, combinatorial objects central to free probability theory. Interestingly, the noncrossing lattice emerges even for systems without randomness and with small on-site Hilbert space dimension.
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 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.