We study the Anderson orthogonality catastrophe, and the corresponding X-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the core-hole Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power-laws and dynamical exponents characterizing the X-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.
We study the Anderson orthogonality catastrophe, and the corresponding X-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the core-hole Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power-laws and dynamical exponents characterizing the X-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.
We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andr\'e model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wave functions and the density of states are fractal. We find, numerically, that the overlap between postquench and prequench wave functions, as well as the ``core-hole'' Green function, evolve in a complex, nonmonotonic way with system size and time, respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the postquench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wave functions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power laws and dynamical exponents characterizing the x-ray edge singularity. We discuss how these features can be probed in ultracold atomic gases using radio-frequency spectroscopy and Ramsey interference.
We present a systematic study of the role of Anderson orthogonality for the dynamics after a quantum quench in quantum impurity models, using the numerical renormalization group. As shown by Anderson in 1967, the scattering phase shifts of the single-particle wave functions constituting the Fermi sea have to adjust in response to the sudden change in the local parameters of the Hamiltonian, causing the initial and final ground states to be orthogonal. This so-called Anderson orthogonality catastrophe also influences dynamical properties, such as spectral functions. Their low-frequency behavior shows nontrivial power laws, with exponents that can be understood using a generalization of simple arguments introduced by Hopfield and others for the x-ray edge singularity problem. The goal of this work is to formulate these generalized rules as well as to numerically illustrate them for quantum quenches in impurity models involving local interactions. As a simple yet instructive example, we use the interacting resonant level model as testing ground for our generalized Hopfield rule. We then analyze a model exhibiting population switching between two dot levels as a function of gate voltage, probed by a local Coulomb interaction with an additional lead serving as charge sensor. We confirm a recent prediction that charge sensing can induce a quantum phase transition for this system, causing the population switch to become abrupt. We elucidate the role of Anderson orthogonality for this effect by explicitly calculating the relevant orthogonality exponents.
Advances in experimental tools allow to study quantum fidelity in unprecedented controlled settings. While the fidelity in metals, when adding a local impurity, is well known to show the Anderson orthogonality catastrophe (AOC), there remain outstanding questions in other quantum phases and settings. Here, we aim to tackle these by exploring systematically the ground-state fidelity of the Fermi liquid in the (extended) Aubry-Andr\'e (AA) model, which allows to explore the AOC in both localized extended and critical phases. We discover that the AOC is typically exponential in the critical regime of the AA model and at the mobility edge of the extended AA model for an extended impurity, while it decays in the AA model with a power law for a weak single-site impurity. We explain this in terms of critical correlations and multipoint correlations. The OC is found to be exponential in the insulating regime, due to a fundamentally different, statistical mechanism, which is explained in detail. Furthermore, we consider a parametric perturbation to the AA model, and find an exponential OC numerically, in agreement with an analytical derivation which we provide here.
We address the phenomenon of statistical orthogonality catastrophe in insulating disordered systems. In more detail, we analyse the response of a system of non-interacting fermions to a local perturbation induced by an impurity. By inspecting the overlap between the pre- and post-quench many-body ground states we fully characterise the emergent statistics of orthogonality events as a function of both the impurity position and the coupling strength. We consider two well-known one-dimensional models, namely the Anderson and Aubry–André insulators, highlighting the arising differences. Particularly, in the Aubry–André model the highly correlated nature of the quasi-periodic potential produces unexpected features in how the orthogonality catastrophe occurs. We provide a quantitative explanation of such features via a simple, effective model. We further discuss the incommensurate ratio approximation and suggest a viable experimental verification in terms of charge transfer statistics and interferometric experiments using quantum probes.
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
We consider the orthogonality catastrophe at the Anderson metal-insulator transition (AMIT). The typical overlap F between the ground state of a Fermi liquid and the one of the same system with an added potential impurity is found to decay at the AMIT exponentially with system size L as $F\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}c{L}^{\ensuremath{\eta}})$, where $\ensuremath{\eta}$ is the power of multifractal intensity correlations. Thus, strong disorder typically increases the sensitivity of a system to an added impurity exponentially. We recover, on the metallic side of the transition, Anderson's result that the fidelity F decays with a power law $F\ensuremath{\sim}{L}^{\ensuremath{-}q({E}_{F})}$ with system size L. Its power increases as the Fermi energy EF approaches the mobility edge EM as $q({E}_{F})\ensuremath{\sim}[({E}_{F}\ensuremath{-}{E}_{M})/{E}_{M}{]}^{\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$, where $\ensuremath{\nu}$ is the critical exponent of the correlation length ${\ensuremath{\xi}}_{c}$. On the insulating side of the transition, F is constant for system sizes exceeding the localization length $\ensuremath{\xi}$. While these results are obtained for the typical fidelity F, we find that logF is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of the fidelity F converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size L. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.