We present an analysis of quantum energy flow and localization in molecules from the point of view of a scaling approach. This scaling approach is based on earlier scaling theories of Anderson localization in disordered metals. The picture provides a simple general framework for describing both energy flow and the localization of eigenstates. This framework can be applied to molecules whose Hamiltonians are described by local random matrices, in which transitions between states nearby in quantum space occur more easily than those between distant states. In this theory the central quantity of interest is the survival probability. It is the squared overlap of a wave function at a later time with its initial condition. Its long time behavior varies depending on whether the eigenstates are localized or not. In the scaling description the survival probability approaches its long time limit varying inversely with time raised to some power. Different power laws are valid in different time regimes, depending on the degree of localization and the size of the molecule’s phase space. Near the localization transition we find that the survival probability decays as t−1. This decay is much slower than when energy flows easily, for a large number of dimensions participating in the energy flow.
We present an analysis of quantum energy flow and localization in molecules from the point of view of a scaling approach. This scaling approach is based on earlier scaling theories of Anderson localization in disordered metals. The picture provides a simple general framework for describing both energy flow and the localization of eigenstates. This framework can be applied to molecules whose Hamiltonians are described by local random matrices, in which transitions between states nearby in quantum space occur more easily than those between distant states. In this theory the central quantity of interest is the survival probability. It is the squared overlap of a wave function at a later time with its initial condition. Its long time behavior varies depending on whether the eigenstates are localized or not. In the scaling description the survival probability approaches its long time limit varying inversely with time raised to some power. Different power laws are valid in different time regimes, depending on the degree of localization and the size of the molecule’s phase space. Near the localization transition we find that the survival probability decays as t−1. This decay is much slower than when energy flows easily, for a large number of dimensions participating in the energy flow.
Many-dimensional quantum calculations of energy flow are presented, revealing dynamical scaling in agreement with theoretical predictions. A model system of coupled anharmonic oscillators which undergoes a localization transition as a function of coupling strength was studied. For strong coupling, action diffusion is clearly seen. For near critical coupling, the average survival probability varies inversely with time but shows no simple relationship to the average extent of the spread of the wave function in action space.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.
We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as $1/{r}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ satisfying $d\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}2d$, where d is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder ${W}^{*}=W/{L}^{2d\ensuremath{-}\ensuremath{\alpha}}lnL$, where W is the disorder strength and L the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder W with the system size L and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.
The notion of scale-invariant dynamics is well established at late times in quantum chaotic systems, as illustrated by the emergence of a ramp in the spectral form factor (SFF). Building on the results of the preceding Letter [Phys. Rev. Lett. 131, 060404 (2023)], we explore features of scale-invariant dynamics of survival probability and SFF at criticality, i.e., at eigenstate transitions from quantum chaos to localization. We show that, in contrast to the quantum chaotic regime, the quantum dynamics at criticality do not only exhibit scale invariance at late times, but also at much shorter times that we refer to as mid-time dynamics. Our results apply to both quadratic and interacting models. Specifically, we study Anderson models in dimensions three to five and power-law random banded matrices for the former, and the quantum sun model and the ultrametric model for the latter, as well as the Rosenzweig-Porter model.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
Building on recent progress in the study of Anderson and many-body localization via the renormalization group (RG), we examine the scaling theory of localization in the quantum Random Energy Model (QREM). The QREM is known to undergo a localization-delocalization transition at finite energy density, while remaining fully ergodic at the center of the spectrum. At zero energy density, we show that RG trajectories consistently flow toward the ergodic phase, and are characterized by an unconventional scaling of the fractal dimension near the ergodic fixed point. When the disorder amplitude is rescaled, as suggested by the forward scattering approximation approach, a localization transition emerges also at the center of the spectrum, with properties analogous to the Anderson transition on expander graphs. At finite energy density, a localization transition takes place without disorder rescaling, and yet it exhibits a scaling behavior analogous to the one observed on expander graphs. The universality class of the model remains unchanged under the rescaling of the disorder, reflecting the independence of the RG from microscopic details. Our findings demonstrate the robustness of the scaling behavior of random graphs and offer new insights into the many-body localization transition.