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Register a new userParticle-Time Duality in the Kicked Ising Chain I: The Dual Operator
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We demonstrate that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for $N$ spins at time $T$ is related to one of a non-unitary evolution operator for $T$ spins at time $N$. We investigate the spectrum of this dual operator with a focus on the different parameter regimes (chaotic, regular) of the spin chain. We present applications of this duality relation to spectral statistics in an accompanying paper.
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Previously, we demonstrated that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for $N$ spins at time $T$ is related to one of a non-unitary evolution operator for $T$ spins at time $N$. Using this duality relation we obtain the oscillating part of the density of states for a large number of spins. Furthermore, the duality relation explains the anomalous short-time behavior of the spectral form factor previously observed in the literature.
The field of quantum chaos originated in the study of spectral statistics for interacting many-body systems, but this heritage was almost forgotten when single-particle systems moved into the focus. In recent years new interest emerged in many-body aspects of quantum chaos. We study a chain of interacting, kicked spins and carry out a semiclassical analysis that is capable of identifying all kinds of genuin many-body periodic orbits. We show that the collective many-body periodic orbits can fully dominate the spectra in certain cases.
Taking one-dimensional random transverse Ising model (RTIM) with the double-Gaussian disorder for example, we investigated the spin autocorrelation function (SAF) and associated spectral density at high temperature by the recursion method. Based on the first twelve recurrants obtained analytically, we have found strong numerical evidence for the long-time tail in the SAF of a single spin. Numerical results indicate that when the standard deviation {sigma}_{JS} (or {sigma}_{BS}) of the exchange couplings J_{i} (or the random transverse fields B_{i}) is small, no long-time tail appears in the SAF. The spin system undergoes a crossover from a central-peak behavior to a collective-mode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when the standard deviation is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large {sigma}_{JS} or a disordered behavior for large {sigma}_{BS}. In this instance, the long-time tails in the SAFs appear, i.e., C(t)simt^{-2}. Similar properties are obtained when the random variables (J_{i} or B_{i}) satisfy other distributions such as the double-exponential distribution and the double-uniform distribution.
Despite considerable progress during the last decades in devising a semiclassical theory for classically chaotic quantum systems a quantitative semiclassical understanding of their dynamics at late times (beyond the so-called Heisenberg time $T_H$) is still missing. This challenge, corresponding to resolving spectral structures on energy scales below the mean level spacing, is intimately related to the quest for semiclassically restoring quantum unitarity, which is reflected in real-valued spectral determinants. Guided through insights for quantum graphs we devise a periodic-orbit resummation procedure for quantum chaotic systems invoking periodic-orbit self encounters as the structuring element of a hierarchical phase space dynamics. We propose a way to purely semiclassically construct real spectral determinants based on two major underlying mechanisms: (i) Complementary contributions to the spectral determinant from regrouped pseudo orbits of duration $T < T_H$ and $T_H-T$ are complex conjugate to each other. (ii) Contributions from long periodic orbits involving multiple traversals along shorter orbits cancel out. We furthermore discuss implications for interacting $N$-particle quantum systems with a chaotic classical large-$N$ limit that have recently attracted interest in the context of many-body quantum chaos.
We study the time evolution of the local magnetization in the critical Ising chain in a transverse field after a sudden change of the parameters at a defect. The relaxation of the defect magnetization is algebraic and the corresponding exponent, which is a continuous function of the defect parameters, is calculated exactly. In finite chains the relaxation is oscillating in time and its form is conjectured on the basis of precise numerical calculations.
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