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arXiv:1104.3058 · Physical Review B

Anderson orthogonality and the numerical renormalization group

Andreas Weichselbaum, Wolfgang Münder, Jan von Delft

Published Aug 10, 2011• 17 citations
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Abstract

Anderson orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say $|{G}_{\mathrm{I}}\ensuremath{\rangle}$ and $|{G}_{\mathrm{F}}\ensuremath{\rangle}$, become orthogonal in the thermodynamic limit of large particle number $N$, in that $|\ensuremath{\langle}{G}_{\mathrm{I}}|{G}_{\mathrm{F}}\ensuremath{\rangle}|\ensuremath{\sim}{N}^{\ensuremath{-}\frac{1}{2}{\ensuremath{\Delta}}_{\mathrm{AO}}^{2}}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We show that the numerical renormalization group offers a simple and precise way to calculate the exponent ${\ensuremath{\Delta}}_{\mathrm{AO}}$: the overlap, calculated as a function of Wilson chain length $k$, decays exponentially $\ensuremath{\sim}{e}^{\ensuremath{-}k\ensuremath{\alpha}}$, and ${\ensuremath{\Delta}}_{\mathrm{AO}}$ can be extracted directly from the exponent $\ensuremath{\alpha}$. The results for ${\ensuremath{\Delta}}_{\mathrm{AO}}$ so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground-state properties change upon switching from $|{\mathrm{G}}_{\mathrm{I}}\ensuremath{\rangle}$ to $|{\mathrm{G}}_{\mathrm{F}}\ensuremath{\rangle}$: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.

DOI
10.1103/physrevb.84.075137

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Anderson orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say ∣GI\ensuremath⟩|{G}_{\mathrm{I}}\ensuremath{\rangle}∣GI​\ensuremath⟩ and ∣GF\ensuremath⟩|{G}_{\mathrm{F}}\ensuremath{\rangle}∣GF​\ensuremath⟩, become orthogonal in the thermodynamic limit of large particle number NNN, in that ∣\ensuremath⟨GI∣GF\ensuremath⟩∣\ensuremath∼N\ensuremath−12\ensuremathΔAO2|\ensuremath{\langle}{G}_{\mathrm{I}}|{G}_{\mathrm{F}}\ensuremath{\rangle}|\ensuremath{\sim}{N}^{\ensuremath{-}\frac{1}{2}{\ensuremath{\Delta}}_{\mathrm{AO}}^{2}}∣\ensuremath⟨GI​∣GF​\ensuremath⟩∣\ensuremath∼N\ensuremath−21​\ensuremathΔAO2​ for N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. We show that the numerical renormalization group offers a simple and precise way to calculate the exponent \ensuremathΔAO{\ensuremath{\Delta}}_{\mathrm{AO}}\ensuremathΔAO​: the overlap, calculated as a function of Wilson chain length kkk, decays exponentially \ensuremath∼e\ensuremath−k\ensuremathα\ensuremath{\sim}{e}^{\ensuremath{-}k\ensuremath{\alpha}}\ensuremath∼e\ensuremath−k\ensuremathα, and \ensuremathΔAO{\ensuremath{\Delta}}_{\mathrm{AO}}\ensuremathΔAO​ can be extracted directly from the exponent \ensuremathα\ensuremath{\alpha}\ensuremathα. The results for \ensuremathΔAO{\ensuremath{\Delta}}_{\mathrm{AO}}\ensuremathΔAO​ so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground-state properties change upon switching from ∣GI\ensuremath⟩|{\mathrm{G}}_{\mathrm{I}}\ensuremath{\rangle}∣GI​\ensuremath⟩ to ∣GF\ensuremath⟩|{\mathrm{G}}_{\mathrm{F}}\ensuremath{\rangle}∣GF​\ensuremath⟩: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.
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 $F\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}c{L}^{\ensuremath{\eta}})$, 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 FFF decays with a power law $F\ensuremath{\sim}{L}^{\ensuremath{-}q({E}_{F})}$ with system size LLL. Its power increases as the Fermi energy EF{E}_{F}EF​ approaches the mobility edge EM{E}_{M}EM​ as $q({E}_{F})\ensuremath{\sim}[({E}_{F}\ensuremath{-}{E}_{M})/{E}_{M}{]}^{\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$, where $\ensuremath{\nu}$ is the critical exponent of the correlation length ${\ensuremath{\xi}}_{c}$. On the insulating side of the transition, FFF is constant for system sizes exceeding the localization length $\ensuremath{\xi}$. 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.
It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz. Using a strategy based on the integrable structure of the corresponding quantum field theory, we propose an exact formula for this overlap, which we check against extensive density matrix renormalization group calculations.
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.
This thesis contributes to the field of strongly correlated electron systems with studies in two distinct fields thereof: the specific nature of correlations between electrons in one dimension and quantum quenches in quantum impurity problems. In general, strongly correlated systems are characterized in that their physical behaviour needs to be described in terms of a many-body description, i.e. interactions correlate all particles in a complex way. The challenge is that the Hilbert space in a many-body theory is exponentially large in the number of particles. Thus, when no analytic solution is available - which is typically the case - it is necessary to find a way to somehow circumvent the problem of such huge Hilbert spaces. Therefore, the connection between the two studies comes from our numerical treatment: they are tackled by the density matrix renormalization group (DMRG) [1] and the numerical renormalization group (NRG) [2], respectively, both based on matrix product states. The first project presented in this thesis addresses the problem of numerically finding the dominant correlations in quantum lattice models in an unbiased way, i.e. without using prior knowledge of the model at hand. A useful concept for this task is the correlation density matrix (CDM) [3] which contains all correlations between two clusters of lattice sites. We show how to extract from the CDM, a survey of the relative strengths of the system’s correlations in different symmetry sectors as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. We demonstrate this by a DMRG study of a one-dimensional spinless extended Hubbard model [4], while emphasizing that the proposed analysis of the CDM is not restricted to one dimension. The second project presented in this thesis is motivated by two phenomena under ongoing experimental and theoretical investigation in the context of quantum impurity models: optical absorption involving a Kondo exciton [5, 6, 7] and population switching in quantum dots [8, 9, 10, 11, 12, 13, 14, 15]. It turns out that both phenomena rely on the various manifestations of Anderson orthogonality (AO) [16], which describes the fact that the response of the Fermi sea to a quantum quench (i.e. an abrupt change of some property of the impurity or quantum dot) is a change of the scattering phase shifts of all the single-particle wave functions, therefore drastically changing the system. In this context, we demonstrate that NRG, a highly accurate method for quantum impurity models, allows for the calculation of all static and dynamic quantities related to AO and present an extensive NRG study for population switching in quantum dots.