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:1405.3300 · Physical Review B

Quantum impurity in a Tomonaga-Luttinger liquid: Continuous-time quantum Monte Carlo approach

Kazuo Hattori, Achim Rosch

Published Sep 2, 2014• 9 citations
arXivHTMLPDF

Abstract

We develop a continuous-time quantum Monte Carlo (CTQMC) method for quantum impurities coupled to interacting quantum wires described by a Tomonaga-Luttinger liquid. The method is negative-sign free for any values of the Tomonaga-Luttinger parameter, which is rigorously proved, and, thus, efficient low-temperature calculations are possible. Duality between electrons and bosons in one-dimensional systems allows us to construct a simple formula for the CTQMC algorithm in these systems. We show that the CTQMC for Tomonaga-Luttinger liquids can be implemented with only minor modifications of previous CTQMC codes developed for impurities coupled to noninteracting fermions. We apply this method to the Kane-Fisher model of a potential scatterer in a spin-less quantum wire and to a single spin coupled with the edge state of a two-dimensional topological insulator assuming an anisotropic XXZ coupling. Various dynamical response functions such as the electron Green's function and spin-spin correlation functions are calculated numerically and their scaling properties are discussed.

DOI
10.1103/physrevb.90.115103

Related papers

Critical behavior of Tomonaga-Luttinger liquids with a mobile impurity

Yasumasa Tsukamoto, Tatsuya Fujii, Norio Kawakami

The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as $c=1$ shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.

Physical review. B, Condensed matter•Aug 15, 1998•36 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

Continuous Time Quantum Monte Carlo in Combination with Machine Learning on the Hubbard Model

Hunpyo Lee

The acceleration of exact continuous time quantum Monte Carlo (CTQMC) approaches in multi-site or multi-orbital systems is extremely interesting work, because these approaches are very time-consuming in terms of numerical computation and might account for the nature of exotic behaviors such as high-temperature superconductivity and Mott insulator behavior observed in the strongly correlated materials. We extend the recently developed interaction-expansion CTQMC method in combination with a machine learning (CTQMC+ML) approach for the single-site and single-orbital systems to multi-site and multi-orbital ones. This method can be applied to explore the nonlocal correlation effects in lattice models and to study the electronic structure of real materials via an ab-initio density functional theory plus dynamical mean field theory approach. We find that our CTQMC+ML method for multi-site (and multi-orbital) systems accurately predicts the impurity Green’s function with less computational time than the CTQMC approaches, as in the case of the single-site and single-orbital version.

Journal of the Korean Physical Society•Nov 1, 2019•1 citation

Kondo Problems in Tomonaga–Luttinger Liquids

Akira Furusaki

Quantum impurity problems in Tomonaga-Luttinger liquids (TLLs) are reviewed with emphasis on their analogy to the Kondo problem in Fermi liquids. First, the problem of a static impurity in a spinless TLL is considered, which is related to the model studied in the context of the macroscopic quantum coherence. In the low-energy limit the TLL is essentially cut into two pieces when interaction is repulsive. The orthogonality catastrophe in a TLL is then discussed. Finally, the Kondo effect of a spin-1/2 impurity in a one-dimensional repulsively interacting electron liquids (a spinful TLL) is reviewed. Regardless of the sign of the exchange coupling, the impury spin is completely screened in the ground state. The leading low-temperature contributions to thermodynamic quantities come from boundary contributions of a bulk leading irrelevant operator.

arXiv:cond-mat/0409016•Journal of the Physical Society of Japan•Jan 1, 2005•25 citations

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

Accelerated continuous time quantum Monte Carlo method with machine learning

Taegeun Song, Hunpyo Lee

An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a dynamical mean field theory (DMFT) approach for the description of strongly correlated systems. The inversion of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix with ${k}^{2}$ operations given by the diagram expansion order $k$ in the CTQMC fast update and the multiplication of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix, and the noninteracting properties with $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{m\ensuremath{-}1}$ operations to measure the $m$-point correlators, are computationally time consuming. Here we propose the CTQMC method in combination with a machine learning technique, which eliminates the $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}$ and $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{3}$ operations for the two-point impurity Green's functions ${G}_{\ensuremath{\sigma}}(i{\ensuremath{\omega}}_{n})$ and four-point vertices ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$, respectively. This method not only predicts the accurate physical properties at low temperature, but also dramatically decreases the computational times of ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$ for the nonlocal extension of DMFT approximation.

arXiv:1901.01501•Physical review. B./Physical review. B•Jul 31, 2019•12 citations
We develop a continuous-time quantum Monte Carlo (CTQMC) method for quantum impurities coupled to interacting quantum wires described by a Tomonaga-Luttinger liquid. The method is negative-sign free for any values of the Tomonaga-Luttinger parameter, which is rigorously proved, and, thus, efficient low-temperature calculations are possible. Duality between electrons and bosons in one-dimensional systems allows us to construct a simple formula for the CTQMC algorithm in these systems. We show that the CTQMC for Tomonaga-Luttinger liquids can be implemented with only minor modifications of previous CTQMC codes developed for impurities coupled to noninteracting fermions. We apply this method to the Kane-Fisher model of a potential scatterer in a spin-less quantum wire and to a single spin coupled with the edge state of a two-dimensional topological insulator assuming an anisotropic XXZ coupling. Various dynamical response functions such as the electron Green's function and spin-spin correlation functions are calculated numerically and their scaling properties are discussed.
The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as c=1c=1c=1 shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.
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.
The acceleration of exact continuous time quantum Monte Carlo (CTQMC) approaches in multi-site or multi-orbital systems is extremely interesting work, because these approaches are very time-consuming in terms of numerical computation and might account for the nature of exotic behaviors such as high-temperature superconductivity and Mott insulator behavior observed in the strongly correlated materials. We extend the recently developed interaction-expansion CTQMC method in combination with a machine learning (CTQMC+ML) approach for the single-site and single-orbital systems to multi-site and multi-orbital ones. This method can be applied to explore the nonlocal correlation effects in lattice models and to study the electronic structure of real materials via an ab-initio density functional theory plus dynamical mean field theory approach. We find that our CTQMC+ML method for multi-site (and multi-orbital) systems accurately predicts the impurity Green’s function with less computational time than the CTQMC approaches, as in the case of the single-site and single-orbital version.
Quantum impurity problems in Tomonaga-Luttinger liquids (TLLs) are reviewed with emphasis on their analogy to the Kondo problem in Fermi liquids. First, the problem of a static impurity in a spinless TLL is considered, which is related to the model studied in the context of the macroscopic quantum coherence. In the low-energy limit the TLL is essentially cut into two pieces when interaction is repulsive. The orthogonality catastrophe in a TLL is then discussed. Finally, the Kondo effect of a spin-1/2 impurity in a one-dimensional repulsively interacting electron liquids (a spinful TLL) is reviewed. Regardless of the sign of the exchange coupling, the impury spin is completely screened in the ground state. The leading low-temperature contributions to thermodynamic quantities come from boundary contributions of a bulk leading irrelevant operator.
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/T1/T1/T-type low-temperature behavior.
An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a dynamical mean field theory (DMFT) approach for the description of strongly correlated systems. The inversion of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix with k2{k}^{2}k2 operations given by the diagram expansion order kkk in the CTQMC fast update and the multiplication of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix, and the noninteracting properties with $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{m\ensuremath{-}1}$ operations to measure the mmm-point correlators, are computationally time consuming. Here we propose the CTQMC method in combination with a machine learning technique, which eliminates the $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}$ and $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{3}$ operations for the two-point impurity Green's functions ${G}_{\ensuremath{\sigma}}(i{\ensuremath{\omega}}_{n})$ and four-point vertices ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$, respectively. This method not only predicts the accurate physical properties at low temperature, but also dramatically decreases the computational times of ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$ for the nonlocal extension of DMFT approximation.