We study the low-temperature thermodynamics of a spin- $S$ magnetic impurity coupled to $m>~2$ degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting electrons or it acquires a new leading term governed by a critical exponent ${\ensuremath{\alpha}}_{m}{=(g}_{m}^{\ensuremath{-}1}\ensuremath{-}1)/m,$ where ${g}_{m}<~1$ is a generalized ( $m$ channel) Luttinger-liquid charge parameter measuring the strength of the repulsive electron-electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened $(m=2S).$ We study the low-temperature thermodynamics of a spin-$S$ magnetic impurity coupled to $m>~2$ degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting electrons or it acquires a new leading term governed by a critical exponent ${\ensuremath{\alpha}}_{m}{=(g}_{m}^{\ensuremath{-}1}\ensuremath{-}1)/m,$ where ${g}_{m}<~1$ is a generalized ($m$ channel) Luttinger-liquid charge parameter measuring the strength of the repulsive electron-electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened $(m=2S).$

arXiv:cond-mat/9710158Physical review. B, Condensed matterJan 1, 199813 citations

Boundary Interaction Changing Operators and Dynamical Correlations in Quantum Impurity Problems

Frédéric Lesage, Hubert Saleur

Recent developments have made possible the computation of equilibrium dynamical correlators in quantum impurity problems. In many situations, however, one is interested in correlators subject to a nonequilibrium initial preparation. This is the case, for instance, for the occupation probability $P(t)$ in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for $P(t)$ : for instance, we find that at large times (or small $g$ ), the leading behavior for $g<\frac{1}{2}$ is $P(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}t$, with the universal ratio $\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}$.Recent developments have made possible the computation of equilibrium dynamical correlators in quantum impurity problems. In many situations, however, one is interested in correlators subject to a nonequilibrium initial preparation. This is the case, for instance, for the occupation probability $P(t)$ in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for $P(t)$: for instance, we find that at large times (or small $g$), the leading behavior for $g<\frac{1}{2}$ is $P(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}t$, with the universal ratio $\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}$.

arXiv:cond-mat/9712019Physical Review LettersMay 18, 199889 citations

Complexity of Quantum Impurity Problems

Sergey Bravyi, David Gosset

arXiv:1609.00735Communications in Mathematical PhysicsAug 31, 201789 citations

Quantum Impurity Problems

Gergely Zaránd

Jan 1, 20050 citations

A Numerically Exact Approach to Quantum Impurity Problems in Realistic Lattice Geometries

Andrew Allerdt, Adrian E. Feiguin

We review a completely unbiased and controlled numerical approach to quantum impurity problems consisting of exactly mapping the complete lattice Hamiltonian onto an equivalent one dimensional system through a unitary transformation. The resulting dimensional and entanglement reduction allows one to study single particle problems on arbitrary $d$ -dimensional lattices using the density matrix renormalization group (DMRG) method. The real-space resolution allows one to position the impurities at the boundary or bulk of the sample and to study screening effects due to edge or surface modes. We describe how to generalize this approach to multi-impurity problems, discuss applications and possible extensions.We review a completely unbiased and controlled numerical approach to quantum impurity problems consisting of exactly mapping the complete lattice Hamiltonian onto an equivalent one dimensional system through a unitary transformation. The resulting dimensional and entanglement reduction allows one to study single particle problems on arbitrary $d$-dimensional lattices using the density matrix renormalization group (DMRG) method. The real-space resolution allows one to position the impurities at the boundary or bulk of the sample and to study screening effects due to edge or surface modes. We describe how to generalize this approach to multi-impurity problems, discuss applications and possible extensions.

Frontiers in PhysicsJun 12, 201916 citations

We study the low-temperature thermodynamics of a spin-

S

magnetic impurity coupled to

m>~2

degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting electrons or it acquires a new leading term governed by a critical exponent ${\ensuremath{\alpha}}_{m}{=(g}_{m}^{\ensuremath{-}1}\ensuremath{-}1)/m,$ where

{g}_{m}<~1

is a generalized (

m

channel) Luttinger-liquid charge parameter measuring the strength of the repulsive electron-electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened

(m=2S).

P(t)

in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for

P(t)

: for instance, we find that at large times (or small

g

), the leading behavior for

g<\frac{1}{2}

is $P(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}t$, with the universal ratio $\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}$.

d

-dimensional lattices using the density matrix renormalization group (DMRG) method. The real-space resolution allows one to position the impurities at the boundary or bulk of the sample and to study screening effects due to edge or surface modes. We describe how to generalize this approach to multi-impurity problems, discuss applications and possible extensions.