Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1207.2502

The Heart of Fidelity

Gerd Bergmann

Published Jul 10, 2012• 0 citations
arXivHTMLPDF

Abstract

The multi-electron wave function of an interacting electron system depends on the size of the system, i.e. the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state |q> centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations.

DOI
10.48550/arxiv.1207.2502

Related papers

Factorization of the -Electron Wave Function in the Kondo Ground State

Gerd Bergmann

The multielectron wave function of an interacting electron system depends on the size of the system, that is, the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations.

ISRN Condensed Matter Physics•Oct 17, 2012•0 citations

Compact approximate solution to the Friedel-Anderson impurity problem

Gerd Bergmann

An approximate ground state of the Anderson-Friedel impurity problem is presented in a very compact form. It requires solely the optimization of two localized electron states and consists of four Slater states (Slater determinants). The resulting singlet ground-state energy lies far below the Anderson mean-field solution and agrees well with the numerical results by Gunnarsson and Sch\"onhammer, who used an extensive $1∕{N}_{f}$ expansion for a spin-$\frac{1}{2}$ impurity with double occupancy of the impurity level.

arXiv:cond-mat/0602196•Physical Review B•Oct 23, 2006•15 citations

Few-body nature of Kondo correlated ground states

Maxime Debertolis, Serge Florens, Izak Snyman

The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wave function. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite-size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.

arXiv:2011.12678•Physical review. B./Physical review. B•Jun 29, 2021•23 citations

Exact Overlaps in the Kondo Problem

Sergei L. Lukyanov, Hubert Saleur, Jesper Lykke Jacobsen +1 more

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.

arXiv:1411.1664•Physical Review Letters•Feb 24, 2015•13 citations

Scaling properties of the Anderson model in the Kondo regime studied by σGσW formalism

Catalin D. Spataru

The symmetric Anderson model for a single impurity coupled to two leads is studied at strong interaction using the GW approximation within the $\ensuremath{\sigma}G\ensuremath{\sigma}W$ formalism. We find that the low-energy properties show universal scaling behavior in the asymptotic regime. While the GW scaling functions are similar in form to the scaling functions known from the numerically exact solution, they are characterized by a different parameter value indicating that GW fails to describe correctly spin correlations between the impurity and lead electrons. We also compare the GW and exact Kondo scales for a broad range of the interaction strength. In contrast to the exponential behavior shown by the exact solution, the GW Kondo scale depends algebraically on the interaction strength.

arXiv:1011.1275•Physical Review B•Nov 10, 2010•7 citations

Mesoscopic Anderson box: Connecting weak to strong coupling

Dong E. Liu, Sébastien Burdin, Harold U. Baranger +1 more

We study the Anderson impurity problem in a mesoscopic setting, namely the ``Anderson box,'' in which the impurity is coupled to finite reservoir having a discrete spectrum and large sample-to-sample mesoscopic fluctuations. Note that both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. We study how the statistical fluctuations in these two problems are connected, using random matrix theory and the slave boson mean-field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise-type toy model. Third, we study the fluctuations of the mean-field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements. We find that the introduction of the strongly coupled impurity considerably changes the wave function but that a substantial correlation remains.

arXiv:1202.0626•Physical Review B•Apr 27, 2012•9 citations
The multi-electron wave function of an interacting electron system depends on the size of the system, i.e. the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state |q> centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations.
The multielectron wave function of an interacting electron system depends on the size of the system, that is, the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations.
An approximate ground state of the Anderson-Friedel impurity problem is presented in a very compact form. It requires solely the optimization of two localized electron states and consists of four Slater states (Slater determinants). The resulting singlet ground-state energy lies far below the Anderson mean-field solution and agrees well with the numerical results by Gunnarsson and Sch\"onhammer, who used an extensive 1∕Nf1∕{N}_{f}1∕Nf​ expansion for a spin-12\frac{1}{2}21​ impurity with double occupancy of the impurity level.
The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wave function. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite-size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.
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.
The symmetric Anderson model for a single impurity coupled to two leads is studied at strong interaction using the GW approximation within the $\ensuremath{\sigma}G\ensuremath{\sigma}W$ formalism. We find that the low-energy properties show universal scaling behavior in the asymptotic regime. While the GW scaling functions are similar in form to the scaling functions known from the numerically exact solution, they are characterized by a different parameter value indicating that GW fails to describe correctly spin correlations between the impurity and lead electrons. We also compare the GW and exact Kondo scales for a broad range of the interaction strength. In contrast to the exponential behavior shown by the exact solution, the GW Kondo scale depends algebraically on the interaction strength.
We study the Anderson impurity problem in a mesoscopic setting, namely the ``Anderson box,'' in which the impurity is coupled to finite reservoir having a discrete spectrum and large sample-to-sample mesoscopic fluctuations. Note that both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. We study how the statistical fluctuations in these two problems are connected, using random matrix theory and the slave boson mean-field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise-type toy model. Third, we study the fluctuations of the mean-field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements. We find that the introduction of the strongly coupled impurity considerably changes the wave function but that a substantial correlation remains.