© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:0710.3474 · Physical Review B

Numerical study of transport through a single impurity in a spinful Tomonaga–Luttinger liquid

Yuji Hamamoto, Ken‐Ichiro Imura, Takeo Kato

Published Apr 2, 2008• 17 citations

Abstract

The single impurity problem in a spinful Tomonaga--Luttinger liquid is studied numerically using path-integral Monte Carlo methods. The advantage of our approach is that the system allows for extensive analyses of charge and spin conductance in the nonperturbative regime. By closely examining the behavior of conductances at low temperatures, in the presence of a finite backward scattering barrier due to the impurity, we identified four distinct phases characterized by either perfect transmission or reflection of charge and spin channels. Our phase diagram for an intermediate scattering strength is consistent with the standard perturbative renormalization group (RG) analysis in the limit of weak and strong backward scatterings, in the sense that all our phase boundaries interpolate the two limiting cases. Further investigations show, however, that precise location and form of our phase boundaries are not trivially explained by the standard RG analysis, e.g., some part of the phase diagram looks much similar to the weak backscattering limit, whereas some other part is clearly derived from the opposite limit. In order to give a more intuitive interpretation of such behaviors, we also reconsidered our impurity problem from the viewpoint of a quantum Brownian motion picture.

DOI: 10.1103/physrevb.77.165402

Related papers

Universal behavior of quantum impurity scattering in a Tomonaga-Luttinger liquid

Yu-Liang Liu

Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a $1/T$-type low-temperature behavior.

arXiv:cond-mat/9901032•Physical review. B, Condensed matter•Mar 15, 1999•8 citations

Circuit Quantum Simulation of a Tomonaga-Luttinger Liquid with an Impurity

A. Anthore, Zubair Iftikhar, E. Boulat +5 more

The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.

arXiv:1809.02017•Physical Review X•Sep 19, 2018•53 citations

A single impurity in Tomonaga—Luttinger liquids

Yuval Oreg, Alexander M. Finkel'stein

The problem of a single impurity in a one-dimensional Tomonaga—Luttinger liquid with a repulsive electron-electron interaction is discussed. We find that the renormalization group flow diagram for the parameters characterizing the impurity is rather complex. Apart from the fixed points corresponding to two weakly connected semi-infinite wires, the flow diagram contains additional fixed points which control the low-temperature physics when the bare potential of the impurity is not strong.

arXiv:cond-mat/9707162•Philosophical Magazine B•May 1, 1998•3 citations

Spin-gap phases in Tomonaga-Luttinger liquids

Masaaki Nakamura, Atsuhiro Kitazawa, K. Nomura

We give details of the analysis of critical properties of spin-gap phases in one-dimensional lattice electron models. In Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and triplet excitation spectra, using the c=1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the $t\ensuremath{-}J$ model, and the $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits.

arXiv:cond-mat/9904381•Physical review. B, Condensed matter•Sep 15, 1999•22 citations

A single impurity in Tomonaga-Luttinger liquids

Alexander M. Finkel'Stein Yuval Oreg

The problem of a single impurity in one dimensional Tomonaga -Luttinger liquids with a repulsive electron-electron interaction is discussed.We find that the renormalization group flow diagram for the parameters characterizing the impurity is rather complex.Apart from the fixed points corresponding to two weakly connected semi-infinite wires, the flow diagram contains additional fixed points which control the low temperature physics when the bare potential of the impurity is not strong.

Philosophical Magazine B•May 1, 1998•0 citations

7 A Single Impurity in Tomonaga–Luttinger Liquids.

Yuval Oreg, Er M. Finkel’stein

The problem of a single impurity in one dimensional Tomonaga-Luttinger liquids with a repulsive electron-electron interaction is discussed. We find that the renormalization group flow diagram for the parameters characterizing the impurity is rather complex. Apart from the fixed points corresponding to two weakly connected semi-infinite wires, the flow diagram contains additional fixed points which control the low temperature physics when the bare potential of the impurity is not strong.

•Aug 18, 2016•2 citations

The single impurity problem in a spinful Tomonaga--Luttinger liquid is studied numerically using path-integral Monte Carlo methods. The advantage of our approach is that the system allows for extensive analyses of charge and spin conductance in the nonperturbative regime. By closely examining the behavior of conductances at low temperatures, in the presence of a finite backward scattering barrier due to the impurity, we identified four distinct phases characterized by either perfect transmission or reflection of charge and spin channels. Our phase diagram for an intermediate scattering strength is consistent with the standard perturbative renormalization group (RG) analysis in the limit of weak and strong backward scatterings, in the sense that all our phase boundaries interpolate the two limiting cases. Further investigations show, however, that precise location and form of our phase boundaries are not trivially explained by the standard RG analysis, e.g., some part of the phase diagram looks much similar to the weak backscattering limit, whereas some other part is clearly derived from the opposite limit. In order to give a more intuitive interpretation of such behaviors, we also reconsidered our impurity problem from the viewpoint of a quantum Brownian motion picture.

Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a

1/T

-type low-temperature behavior.

The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.

We give details of the analysis of critical properties of spin-gap phases in one-dimensional lattice electron models. In Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and triplet excitation spectra, using the c=1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the $t\ensuremath{-}J$ model, and the $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits.