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arXiv:1606.02243 · Physical Review Letters

Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

Stefan Kettemann

Published Sep 29, 2016• 12 citations
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Abstract

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 ${E}_{F}$ approaches the mobility edge ${E}_{M}$ 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 $\mathrm{log}F$ 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.

DOI
10.1103/physrevlett.117.146602

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We consider the orthogonality catastrophe at the Anderson metal-insulator transition (AMIT). The typical overlap FFF 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 LLL as F\ensuremath∼exp(\ensuremath−cL\ensuremathη)F\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}c{L}^{\ensuremath{\eta}})F\ensuremath∼exp(\ensuremath−cL\ensuremathη), where \ensuremathη\ensuremath{\eta}\ensuremathη 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 FFF decays with a power law F\ensuremath∼L\ensuremath−q(EF)F\ensuremath{\sim}{L}^{\ensuremath{-}q({E}_{F})}F\ensuremath∼L\ensuremath−q(EF​) with system size LLL. Its power increases as the Fermi energy EF{E}_{F}EF​ approaches the mobility edge EM{E}_{M}EM​ as q(EF)\ensuremath∼[(EF\ensuremath−EM)/EM]\ensuremath−\ensuremathν\ensuremathηq({E}_{F})\ensuremath{\sim}[({E}_{F}\ensuremath{-}{E}_{M})/{E}_{M}{]}^{\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}q(EF​)\ensuremath∼[(EF​\ensuremath−EM​)/EM​]\ensuremath−\ensuremathν\ensuremathη, where \ensuremathν\ensuremath{\nu}\ensuremathν is the critical exponent of the correlation length \ensuremathξc{\ensuremath{\xi}}_{c}\ensuremathξc​. On the insulating side of the transition, FFF is constant for system sizes exceeding the localization length \ensuremathξ\ensuremath{\xi}\ensuremathξ. While these results are obtained for the typical fidelity FFF, we find that logF\mathrm{log}FlogF is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of the fidelity FFF converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size LLL. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.
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.
Anderson orthogonality catastrophe in 2+12+12+1-D topological systems
In the thermodynamic limit, a many-body ground state has zero overlap with another state which is a slightly perturbed state of the original one, known as the Anderson orthogonality catastrophe (AOC). The amplitude of the overlap for two generic ground states typically exhibits exponential or power-law decay as the system size increases to infinity. In this paper, we show that for generic (2+1)(2+1)(2+1)-D topological systems at fixed points, there exists a universal topological response term in the scaling of the ground-state overlap. For Laughlin wave functions, in particular, we also find a leading term decaying faster than exponential, which is beyond AOC. Such finite-size scaling behaviors could be utilized to theoretically detect the gapless edge modes, distinguish the topology of quantum states or serve as a signature for topological phase transitions.
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Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
Abstract We review our recent results on Anderson localization in systems of two interacting particles coupled by contact interactions. Based on an exact mapping to an effective single-particle problem, we numerically investigate the occurrence of metal-insulator phase transitions for the pair in two-(2D) and three-dimensional (3D) disordered lattices. In two dimensions, we find that interactions cause an exponential enhancement of the pair localization length with respect to its single-particle counterpart, but do not induce a delocalization transition. In particular we show that previous claims of 2D interaction-induced Anderson transitions are the results of strong finite-size effects. In three dimensions we find that the pair undergoes a metal-insulator transition belonging to the same (orthogonal) universality class of the noninteracting model. We then explore the phase diagram in the space of energy E, disorder W and interaction strength U, which reveals a rich and counterintuitive structure, endowed with multiple metallic and insulating phases. We point out that this phenomenon originates from the molecular and scattering-like nature of the pair states available at given energy and disorder strength.
We study numerically the effects of classical magnetic impurities on the Anderson metal-insulator transition. We find that a small concentration of Heisenberg impurities enhances the critical disorder amplitude Wc{W}_{\mathrm{c}}Wc​ with increasing exchange coupling strength JJJ. The resulting scaling with JJJ is analyzed which supports an anomalous scaling prediction by Wegner due to the combined breaking of time-reversal and spin-rotational symmetry. Moreover, we find that the presence of magnetic impurities lowers the critical correlation length exponent $\ensuremath{\nu}$ and enhances the multifractality parameter ${\ensuremath{\alpha}}_{0}$. The new value of $\ensuremath{\nu}$ improves the agreement with the value measured in experiments on the metal-insulator transition (MIT) in doped semiconductors like phosphor-doped silicon, where a finite density of magnetic moments is known to exist in the vicinity of the MIT. The results are obtained by a finite-size scaling analysis of the geometric mean of the local density of states which is calculated by means of the kernel polynomial method. We establish this combination of numerical techniques as a method to obtain critical properties of disordered systems quantitatively.