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 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.
In this paper, we explore quantum criticality in the disordered Aubry-Andr\'e (AA) model. For the pure AA model, it is well known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\mathrm{\ensuremath{\Delta}}$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\ensuremath{\xi}$ scales with $\mathrm{\ensuremath{\Delta}}$ as $\ensuremath{\xi}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}{\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}}$ with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}$ a new critical exponent, which is estimated to be ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}\ensuremath{\approx}0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfies a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization.
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.
We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-Andr\'e model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-Andr\'e model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-Andr\'e model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in the near future.
We explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andr\'e-Stark (AAS) model, where the Stark field of strength $\ensuremath{\varepsilon}$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, the inverse participation ratio (IPR), and the energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\ensuremath{\xi}$ scales with $\ensuremath{\varepsilon}$ as $\ensuremath{\xi}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.3$, a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andr\'e (AA) model and the pure Stark model. The IPR
I scales as $\mathcal{I}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{s}$ with the critical exponent $s\ensuremath{\approx}0.098$, which is also different from both of the pure models. The energy gap $\mathrm{\ensuremath{\Delta}}E$ scales as $\mathrm{\ensuremath{\Delta}}E\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{\nu}z}$ with the same critical exponent $z\ensuremath{\approx}2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and we study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and the quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.
In this work, we investigate the Stark localization near the Aubry-Andr\'e (AA) critical point. We perform careful studies for reporting system-dependent parameters, such as localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited state, for characterizing the localization-delocalization transition. We show that the scaling exponents possessed by these key descriptors of localization are quite different from that of a pure AA model or Stark model. Near the critical point of the AA model, in the presence of Stark field of strength
h, the localization length $\ensuremath{\zeta}$ scales as $\ensuremath{\zeta}\ensuremath{\propto}{h}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.29$, which is different from both the pure AA model $(\ensuremath{\nu}=1)$ and Stark model $(\ensuremath{\nu}\ensuremath{\approx}0.33)$. The IPR in this case scales as IPR $\ensuremath{\propto}{h}^{s}$ with $s\ensuremath{\approx}0.096$, which is again significantly different from both the pure AA model $(s\ensuremath{\approx}0.33)$ and Stark model $(s\ensuremath{\approx}0.33)$. The energy gap, $\mathrm{\ensuremath{\Delta}}$, scales as $E\ensuremath{\propto}{h}^{\ensuremath{\nu}z}$, where $z\ensuremath{\approx}2.37$, which is however the same as the pure AA model. Finally, we discuss how invoking a criticality-inducing additional control parameter may help in designing better many-body quantum sensors. Quantum critical sensors exploit the venerability of the wave function near the quantum critical point against small parameter shifts. By incorporating a control parameter in the form of the quasiperiodic field, i.e., the AA potential, we show a significant advantage can be drawn in estimating an unknown parameter, which is considered here to be the Stark weak field strength, with high precision. We compute and extract the scaling exponent associated with the quantum Fisher information,
FQ, which provides a bound on the uncertainty associated with measuring an unknown parameter. We demonstrate that by a control field in the form of the quasiperiodic potential, and setting it at AA criticality,
FQ scales with the system size as ${F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, where $\ensuremath{\beta}\ensuremath{\approx}6.7$, which is greater in comparison to the pure Stark case, for which $\ensuremath{\beta}$ is known to be 5.8. We, thus, pitch a generic idea of invoking quantum-criticality-inducing additional control parameters for obtaining enhanced quantum advantage in many-body critical sensors.