Deviations from exponential decay dynamics have been proposed for a wide variety of systems in high-energy, atomic, and molecular physics. This work examines the quantum dynamics of a simple hierarchical local random matrix model. The hierarchical structure is imposed by two physically motivated constraints: an exponential size scaling of matrix elements, and a quantum number ``triangle rule,'' which introduces correlations in the quantum-state space by mimicking the nodal structure of wave functions in a coordinate Hamiltonian. These correlations lead to a systematic slowing of dephasing dynamics compared to exponential decays. A generalized Lorentzian line shape is introduced as the Fourier transform of a polynomial survival amplitude to describe the average behavior of these decays. The model is brought into a representation that can be compared directly with the golden rule. In this representation, the deviations from exponentiality arise from energy-dependent correlations among the coupling matrix elements that persist even for large systems. Finally, the effects of relaxing the size scaling and ``triangle rule'' constraints are studied. Sparsity of the random matrix alone is not sufficient to produce slow asymptotic dynamics; both types of constraints are required.
Deviations from exponential decay dynamics have been proposed for a wide variety of systems in high-energy, atomic, and molecular physics. This work examines the quantum dynamics of a simple hierarchical local random matrix model. The hierarchical structure is imposed by two physically motivated constraints: an exponential size scaling of matrix elements, and a quantum number triangle rule,'' which introduces correlations in the quantum-state space by mimicking the nodal structure of wave functions in a coordinate Hamiltonian. These correlations lead to a systematic slowing of dephasing dynamics compared to exponential decays. A generalized Lorentzian line shape is introduced as the Fourier transform of a polynomial survival amplitude to describe the average behavior of these decays. The model is brought into a representation that can be compared directly with the golden rule. In this representation, the deviations from exponentiality arise from energy-dependent correlations among the coupling matrix elements that persist even for large systems. Finally, the effects of relaxing the size scaling and triangle rule'' constraints are studied. Sparsity of the random matrix alone is not sufficient to produce slow asymptotic dynamics; both types of constraints are required.
The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground-state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPSs). The stochastic model displays slow dynamics, including power-law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations-which we compute accurately via numerical MPSs-including an active-inactive phase transition and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.
We study a quantum-mechanical system, prepared, at t=0, in a model state, that subsequently decays into a sea of other states whose energy levels form a discrete spectrum with given statistical properties. An important quantity is the survival probability P(t), defined as the probability, at time t, to find the system in the original model state. Our main purpose is to analyze the influence of the discreteness and statistical properties of the spectrum on the behavior of P(t). Since P(t) itself is a statistical quantity, we restrict our attention to its ensemble average 〈P(t)〉, which is calculated analytically using random-matrix techniques, within certain approximations discussed in the text. We find, for 〈P(t)〉, an exponential decay, followed by a revival, governed by the two-point structure function of the statistical spectrum, thus giving a nonzero asymptotic value for large t's. The analytic result compares well with a number of computer simulations, over a time range discussed in the text.
We study the dynamics of a simple model for quantum decay, where a single state is coupled to a set of discrete states, the pseudocontinuum, each coupled to a real continuum of states. We find that for constant matrix elements between the single state and the pseudocontinuum the decay occurs via one state in a certain region of the parameters, involving the Dicke and quantum Zeno effects. When the matrix elements are random, several cases are identified. For a pseudocontinuum with small bandwidth there are weakly damped oscillations in the probability to be in the initial single state. For intermediate bandwidth one finds mesoscopic fluctuations in the probability with amplitude inversely proportional to the square root of the volume of the pseudocontinuum space. They last for a long time compared to the nonrandom case.
Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems-the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction-the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic," in the sense that their eigenfunction statistics deviate from random matrix theory.
We present a method to estimate the quantum Fisher information (QFI) of many-body quantum states in the presence of decoherence, where its direct evaluation requires the full spectral resolution of the density matrix. We show that, for many-body wave functions known analytically in the occupation-number basis, systematic lower bounds to the QFI can be mapped onto expectation values over a classical probability distribution defined by the wave function amplitudes. This mapping enables efficient estimation via Markov-chain Monte Carlo sampling, with a computational cost that scales as a `slow' exponential (ebL with b≲0.6) and remains manageable for system sizes well beyond exact diagonalization. We specify this framework to Jastrow-Gutzwiller wave functions. We characterize their metrological content by identifying the observables that maximize the QFI and the corresponding scaling with L. Then, we analyze the QFI under three physically motivated noise channels: local dephasing, local amplitude damping, and global depolarizing. We compare polynomial and Krylov-based lower bounds across these channels, relating their behavior to the effective rank of the noisy density matrix and to the structure of the operator generating the parameter encoding. The framework extends naturally to other analytically known wave functions and to a broader class of information-theoretic quantities beyond the QFI.
We explore the effects of spatial locality on the dynamics of random quantum systems subject to a Markovian noise. To this end, we study a model in which the system Hamiltonian and its couplings to the noise are random matrices whose entries decay as power laws of distance, with distinct exponents α_{H}, α_{L}. The steady state is always featureless, but the rate at which it is approached exhibits three phases depending on α_{H} and α_{L}: a phase where the approach is asymptotically exponential as a result of a gap in the spectrum of the Lindblad superoperator that generates the dynamics, and two gapless phases with subexponential relaxation, distinguished by the manner in which the gap decreases with system size. Within perturbation theory, the phase boundaries in the (α_{H},α_{L}) plane differ for weak and strong decoherence, suggesting phase transitions as a function of noise strength. We identify nonperturbative effects that prevent such phase transitions in the thermodynamic limit.