We introduce a bare-bone random matrix quantum impurity model, by hybridizing a localized spinless electronic level with a bath of random fermions in the Gaussian Orthogonal Ensemble (GOE). While stripped out of correlations effects, this model reproduces some salient features of the impurity charge distribution obtained in previous works on interacting disordered impurity models. Computing by numerical sampling the impurity charge distribution in our model, we find a crossover from a Gaussian distribution (centered on half a charge unit) at large hybridization, to a bimodal distribution (centered both on zero and full occupations of the charge) at small hybridization. In the bimodal regime, a universal $(-3/2)$ power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number $N$ of fermionic orbitals in the bath. In the Gaussian regime and in the limit $N\to\infty$ , we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.We introduce a bare-bone random matrix quantum impurity model, by hybridizing a localized spinless electronic level with a bath of random fermions in the Gaussian Orthogonal Ensemble (GOE). While stripped out of correlations effects, this model reproduces some salient features of the impurity charge distribution obtained in previous works on interacting disordered impurity models. Computing by numerical sampling the impurity charge distribution in our model, we find a crossover from a Gaussian distribution (centered on half a charge unit) at large hybridization, to a bimodal distribution (centered both on zero and full occupations of the charge) at small hybridization. In the bimodal regime, a universal $(-3/2)$ power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number $N$ of fermionic orbitals in the bath. In the Gaussian regime and in the limit $N\to\infty$, we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.

arXiv:2507.22586Jul 30, 20250 citations

An exactly solvable impurity model

Igor N. Karnaukhov

The European Physical Journal BSep 1, 19931 citation

Effects of weak random disorder in the one-dimensional Hubbard model

Anders W. Sandvik, Douglas J. Scalapino

We introduce the notion of an impurity susceptibility as a measure of the effect of disorder on the properties of interacting many-particle systems. If finite, the impurity susceptibility gives the lowest-order correction to a given physical observable in the presence of disorder. If divergent, it signals that arbitrarily weak disorder leads to a qualitatively different physical behavior. Using a quantum Monte Carlo technique, impurity susceptibilities for various properties of the one-dimensional Hubbard model are calculated. Two types of disorder are considered; random hopping matrix elements and random potentials. For the local magnetic moment and the kinetic energy the impurity susceptibilities are finite. The impurity susceptibility of the q=2 ${\mathit{k}}_{\mathit{F}}$ spin response is divergent for both types of disorder, indicating a quenching of the T\ensuremath{\rightarrow}0 divergence of this quantity. The susceptibility of the q=0 spin response to random hopping diverges at half filling, as a consequence of a divergent magnetic susceptibility in the presence of this type of disorder. We also find large, possibly divergent, impurity susceptibilities for the q=2 ${\mathit{k}}_{\mathit{F}}$ and 4 ${\mathit{k}}_{\mathit{F}}$ charge-density responses.We introduce the notion of an impurity susceptibility as a measure of the effect of disorder on the properties of interacting many-particle systems. If finite, the impurity susceptibility gives the lowest-order correction to a given physical observable in the presence of disorder. If divergent, it signals that arbitrarily weak disorder leads to a qualitatively different physical behavior. Using a quantum Monte Carlo technique, impurity susceptibilities for various properties of the one-dimensional Hubbard model are calculated. Two types of disorder are considered; random hopping matrix elements and random potentials. For the local magnetic moment and the kinetic energy the impurity susceptibilities are finite. The impurity susceptibility of the q=2${\mathit{k}}_{\mathit{F}}$ spin response is divergent for both types of disorder, indicating a quenching of the T\ensuremath{\rightarrow}0 divergence of this quantity. The susceptibility of the q=0 spin response to random hopping diverges at half filling, as a consequence of a divergent magnetic susceptibility in the presence of this type of disorder. We also find large, possibly divergent, impurity susceptibilities for the q=2${\mathit{k}}_{\mathit{F}}$ and 4${\mathit{k}}_{\mathit{F}}$ charge-density responses.

Physical review. B, Condensed matterApr 15, 199311 citations

Sparse modeling of large-scale quantum impurity models with low symmetries

Hiroshi Shinaoka, Yuki Nagai

Quantum embedding theories can be used for obtaining quantitative descriptions of correlated materials. However, a critical challenge is solving an effective impurity model of correlated orbitals embedded in an electron bath. Many advanced impurity solvers require the approximation of a bath continuum using a finite number of bath levels, producing a highly nonconvex, ill-conditioned inverse problem. To address this drawback, this study proposes an efficient fitting algorithm for matrix-valued hybridization functions based on a data-science approach, sparse modeling, and a compact representation of Matsubara Green's functions. The efficiency of the proposed method is demonstrated by fitting random hybridization functions with large off-diagonal elements and those of a 20-orbital impurity model for a high- ${T}_{\mathrm{c}}$ compound, LaAsFeO, at low temperatures $(T)$ . The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.Quantum embedding theories can be used for obtaining quantitative descriptions of correlated materials. However, a critical challenge is solving an effective impurity model of correlated orbitals embedded in an electron bath. Many advanced impurity solvers require the approximation of a bath continuum using a finite number of bath levels, producing a highly nonconvex, ill-conditioned inverse problem. To address this drawback, this study proposes an efficient fitting algorithm for matrix-valued hybridization functions based on a data-science approach, sparse modeling, and a compact representation of Matsubara Green's functions. The efficiency of the proposed method is demonstrated by fitting random hybridization functions with large off-diagonal elements and those of a 20-orbital impurity model for a high-${T}_{\mathrm{c}}$ compound, LaAsFeO, at low temperatures $(T)$. The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.

arXiv:2007.03955Physical review. B./Physical review. BJan 19, 202112 citations

Density Matrix Method

World Scientific lecture and course notes in chemistryJun 1, 19911 citation

(-3/2)

power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number

N

of fermionic orbitals in the bath. In the Gaussian regime and in the limit

N\to\infty

, we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.

{\mathit{k}}_{\mathit{F}}

spin response is divergent for both types of disorder, indicating a quenching of the T\ensuremath{\rightarrow}0 divergence of this quantity. The susceptibility of the q=0 spin response to random hopping diverges at half filling, as a consequence of a divergent magnetic susceptibility in the presence of this type of disorder. We also find large, possibly divergent, impurity susceptibilities for the q=2

{\mathit{k}}_{\mathit{F}}

and 4

{\mathit{k}}_{\mathit{F}}

charge-density responses.

{T}_{\mathrm{c}}

compound, LaAsFeO, at low temperatures

(T)

. The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.