We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest-neighbor interaction and a local impurity in its Luttinger liquid phase one has to take into account (i) the impurity scaling, (ii) unusual finite-size $L$ corrections of the form $ln(L)/L$ , as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale ${L}_{u}$ , which becomes exceedingly large the closer the system is to its transition into the charge density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than ${L}_{u}$ . For small to intermediate two-particle interactions, for which the regime $L>{L}_{u}$ can be accessed, and taking into account the finite-size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by $1/16$ . While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of $1/16$ given the accessible system sizes. We employ the density matrix renormalization group.We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest-neighbor interaction and a local impurity in its Luttinger liquid phase one has to take into account (i) the impurity scaling, (ii) unusual finite-size $L$ corrections of the form $ln(L)/L$, as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale ${L}_{u}$, which becomes exceedingly large the closer the system is to its transition into the charge density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than ${L}_{u}$. For small to intermediate two-particle interactions, for which the regime $L>{L}_{u}$ can be accessed, and taking into account the finite-size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by $1/16$. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of $1/16$ given the accessible system sizes. We employ the density matrix renormalization group.

arXiv:1409.1671Physical Review BOct 27, 20143 citations

Renormalization group for one-dimensional fermions. A review on mathematical results

Guido Gentile, Vieri Mastropietro

Physics ReportsOct 1, 200169 citations

Renormalization Group for Non-Relativistic Fermions: II

R. Shankar

Cambridge University Press eBooksAug 31, 20170 citations

Orthogonality catastrophe in a one-dimensional system of correlated electrons

V. Meden, Peter Schmitteckert, Nic Shannon

We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization-group algorithm. Keeping up to $2000$ states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to $1/16$ based on predictions of boundary conformal field theory.We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization-group algorithm. Keeping up to $2000$ states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to $1/16$ based on predictions of boundary conformal field theory.

arXiv:cond-mat/9707082Physical review. B, Condensed matterApr 15, 199825 citations

Fermionic Functional Renormalization Group

Peter Kopietz, Lorenz Bartosch, Florian Schütz

Lecture notes in physicsDec 25, 20091 citation

Orthogonality catastrophe in a one-dimensional system of correlated electrons

No author provided

We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization group algorithm. Keeping up to 2000 states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Nevertheless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/16 based on predictions of boundary conformal field theory. [S0163-1829(98)01415-5].We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization group algorithm. Keeping up to 2000 states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Nevertheless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/16 based on predictions of boundary conformal field theory. [S0163-1829(98)01415-5].

Sep 10, 20260 citations

L

corrections of the form

ln(L)/L

, as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale

{L}_{u}

, which becomes exceedingly large the closer the system is to its transition into the charge density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than

{L}_{u}

. For small to intermediate two-particle interactions, for which the regime

L>{L}_{u}

can be accessed, and taking into account the finite-size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by

1/16

. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of

1/16

given the accessible system sizes. We employ the density matrix renormalization group.

2000

states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to

1/16

based on predictions of boundary conformal field theory.