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

Dissipation-induced Luttinger liquids in the Caldeira-Leggett formalism

S. Eigenhuis

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

It is well known that dissipation usually induces decoherence in quantum systems. However, recently it has been employed to engineer zero-energy Majorana modes on the boundary of a one-dimensional fermionic chain. In this thesis, by employing the Caldeira-Leggett theory of quantum dissipation, we study the equivalence between a particle on a one-dimensional lattice coupled to a thermal bath, and the massless Luttinger liquid with an impurity on the boundary in a closed system. In particular, the Luttinger liquid with a single impurity at the boundary of a semi-infinite chain is bosonized into the boundary sine-Gordon theory. Here, a zero-energy Majorana mode appears at the boundary when the Luttinger parameter K=1/2. The equivalence is shown to apply also in the case of helical Luttinger liquids, which play an important role in topological insulators. This equivalence is then generalized to the full sine-Gordon theory, such that massive Luttinger liquids can be mapped to a single-particle one-dimensional tunneling Hamiltonian with a dissipative Ohmic Caldeira-Leggett bath.

Related papers

Dissipative Luttinger liquids

Aaron J. Friedman

We investigate a one dimensional quantum fluid coupled to a dissipative bath. The quantum fluid is captured by the canonical Luttinger liquid; the bath is given by the model of Caldeira and Leggett, i.e. a tower of oscillators coupled linearly to the fluid density, $ρ$. The bath can be integrated out exactly, producing an effective interaction for the fluid that is nonlocal in time; we argue that the form corresponding to Ohmic dissipation is generic. Compared to previous works, we compute correlation functions for this minimal model without approximation, including at finite temperature $T>0$. From these and a Kubo calculation, we conclude that arbitrary dissipation destroys the perfect conductivity of the Luttinger liquid via Zeno localization, even in the absence of a spatial potential; from RG analysis of harmonic terms, we also find that the open Luttinger liquid is significantly more prone to localization by such potentials, in contrast to the usual intuition that baths make systems less localized.

arXiv:1910.06371•Oct 14, 2019•2 citations

Unification of electromagnetic noise and Luttinger liquid via a quantum dot

Karyn Le Hur, Mei-Rong Li

We investigate the effect of dissipation on a small quantum dot (resonant level) tunnel coupled to a chiral Luttinger liquid (LL) with the LL parameter $K$. The dissipation stems from the coupling of the dot to an electric environment, being characterized by the resistance $R$, via Coulomb interactions. We show that this problem can be mapped onto a Caldeira-Leggett model where the (ohmic) bath of harmonic oscillators is governed by the effective dissipation strength $\ensuremath{\alpha}={(2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K})}^{\ensuremath{-}1}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}}^{\ensuremath{-}1}={K}^{\ensuremath{-}1}+2R∕{R}_{K}$ and ${R}_{K}=h∕{e}^{2}$ the quantum of resistance. Experimental consequences are discussed and the limit $K=1∕{2}^{+}$ is thoroughly studied at small $R∕{R}_{K}$ through the spin-boson-fermion model.

arXiv:cond-mat/0410446•Physical Review B•Aug 9, 2005•61 citations

Hidden Caldeira-Leggett Dissipation in a Bose-Fermi Kondo Model

Mei-Rong Li, Karyn Le Hur, Walter Hofstetter

We show that the Bose-Fermi Kondo model (BFKM), which may find applicability both to certain dissipative mesoscopic qubit devices and to heavy-fermion systems described by the Kondo lattice model, can be mapped exactly onto the Caldeira-Leggett model. This mapping requires an ohmic bosonic bath and an Ising-type coupling between the latter and the impurity spin. This allows us to conclude unambiguously that there is an emergent Kosterlitz-Thouless quantum phase transition in the BFKM with an ohmic bosonic bath. By applying a bosonic numerical renormalization group approach, we thoroughly probe physical quantities close to the quantum phase transition.

arXiv:cond-mat/0501755•Physical Review Letters•Aug 19, 2005•39 citations

Tunneling Processes of a Heavy Particle Coupled to Metallic Electron Systems

Kaoru Awaka, Gen Tatara, Hidetoshi Fukuyama

A new formalism to treat the tunneling processes of a heavy particle coupled to degenerate electron systems has been presented. This simple and transparent method follows that of Caldeira and Leggett in a general framework, but the term representing the dissipation is given in terms of the imaginary part of the retarded density correlation function as far as the coupling is weak. Hence our method can incorporate the microscopic details of the electronic systems such as mutual interactions and randomness. It is found that for the Luttinger liquid, the dissipation can be either sub- or super-Ohmic depending on the type of interaction, while near the Anderson transition there exists a crossover from Ohmic to sub-Ohmic dissipation. In the case of sub-Ohmic dissipation, the tunneling rate is found to vanish discontinuously above some critical value as a function of the strength of dissipation.

Journal of the Physical Society of Japan•Jun 15, 1993•9 citations

On the spectral properties of inhomogeneous Luttinger liquids

Manuel Simonato

The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.

•Apr 1, 2019•0 citations

Dissipation-induced long-range order in the one-dimensional Bose-Hubbard model

Ana Beatriz Landim Ribeiro, Paul McClarty, Pedro Ribeiro +1 more

Understanding the stability of strongly correlated phases of matter when coupled to environmental degrees of freedom is crucial for identifying the conditions under which these states may be observed. Here, we focus on the paradigmatic one-dimensional Bose-Hubbard model, and study the stability of the Luttinger-liquid and Mott insulating phases in the presence of local particle exchange with site-independent baths of noninteracting bosons. We perform a numerically exact analysis of this model by adapting the recently developed wormhole quantum Monte Carlo method for retarded interactions to a continuous-time formulation with worm updates; we show how the wormhole updates can be easily implemented in this scheme. For an Ohmic bath, our numerical findings confirm the scaling prediction that the Luttinger-liquid phase becomes unstable at infinitesimal bath coupling. We show that the ensuing phase is a long-range-ordered superfluid with spontaneously broken U(1) symmetry. While the Mott insulator remains a distinct phase for small bath coupling, it exhibits diverging compressibility and noninteger local boson occupation in the thermodynamic limit. Upon increasing the bath coupling, this phase undergoes a transition to a long-range-ordered superfluid. Finally, we discuss the effects of super-Ohmic dissipation on the Luttinger-liquid phase. Our results are compatible with a stable dissipative Luttinger-liquid phase that transitions to a long-range-ordered superfluid at a finite system-bath coupling.

arXiv:2311.07683•Physical review. B./Physical review. B•Sep 24, 2024•6 citations
It is well known that dissipation usually induces decoherence in quantum systems. However, recently it has been employed to engineer zero-energy Majorana modes on the boundary of a one-dimensional fermionic chain. In this thesis, by employing the Caldeira-Leggett theory of quantum dissipation, we study the equivalence between a particle on a one-dimensional lattice coupled to a thermal bath, and the massless Luttinger liquid with an impurity on the boundary in a closed system. In particular, the Luttinger liquid with a single impurity at the boundary of a semi-infinite chain is bosonized into the boundary sine-Gordon theory. Here, a zero-energy Majorana mode appears at the boundary when the Luttinger parameter K=1/2. The equivalence is shown to apply also in the case of helical Luttinger liquids, which play an important role in topological insulators. This equivalence is then generalized to the full sine-Gordon theory, such that massive Luttinger liquids can be mapped to a single-particle one-dimensional tunneling Hamiltonian with a dissipative Ohmic Caldeira-Leggett bath.
We investigate a one dimensional quantum fluid coupled to a dissipative bath. The quantum fluid is captured by the canonical Luttinger liquid; the bath is given by the model of Caldeira and Leggett, i.e. a tower of oscillators coupled linearly to the fluid density, ρρρ. The bath can be integrated out exactly, producing an effective interaction for the fluid that is nonlocal in time; we argue that the form corresponding to Ohmic dissipation is generic. Compared to previous works, we compute correlation functions for this minimal model without approximation, including at finite temperature T>0T>0T>0. From these and a Kubo calculation, we conclude that arbitrary dissipation destroys the perfect conductivity of the Luttinger liquid via Zeno localization, even in the absence of a spatial potential; from RG analysis of harmonic terms, we also find that the open Luttinger liquid is significantly more prone to localization by such potentials, in contrast to the usual intuition that baths make systems less localized.
We investigate the effect of dissipation on a small quantum dot (resonant level) tunnel coupled to a chiral Luttinger liquid (LL) with the LL parameter KKK. The dissipation stems from the coupling of the dot to an electric environment, being characterized by the resistance RRR, via Coulomb interactions. We show that this problem can be mapped onto a Caldeira-Leggett model where the (ohmic) bath of harmonic oscillators is governed by the effective dissipation strength $\ensuremath{\alpha}={(2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K})}^{\ensuremath{-}1}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}}^{\ensuremath{-}1}={K}^{\ensuremath{-}1}+2R∕{R}_{K}$ and RK=h∕e2{R}_{K}=h∕{e}^{2}RK​=h∕e2 the quantum of resistance. Experimental consequences are discussed and the limit K=1∕2+K=1∕{2}^{+}K=1∕2+ is thoroughly studied at small R∕RKR∕{R}_{K}R∕RK​ through the spin-boson-fermion model.
We show that the Bose-Fermi Kondo model (BFKM), which may find applicability both to certain dissipative mesoscopic qubit devices and to heavy-fermion systems described by the Kondo lattice model, can be mapped exactly onto the Caldeira-Leggett model. This mapping requires an ohmic bosonic bath and an Ising-type coupling between the latter and the impurity spin. This allows us to conclude unambiguously that there is an emergent Kosterlitz-Thouless quantum phase transition in the BFKM with an ohmic bosonic bath. By applying a bosonic numerical renormalization group approach, we thoroughly probe physical quantities close to the quantum phase transition.
A new formalism to treat the tunneling processes of a heavy particle coupled to degenerate electron systems has been presented. This simple and transparent method follows that of Caldeira and Leggett in a general framework, but the term representing the dissipation is given in terms of the imaginary part of the retarded density correlation function as far as the coupling is weak. Hence our method can incorporate the microscopic details of the electronic systems such as mutual interactions and randomness. It is found that for the Luttinger liquid, the dissipation can be either sub- or super-Ohmic depending on the type of interaction, while near the Anderson transition there exists a crossover from Ohmic to sub-Ohmic dissipation. In the case of sub-Ohmic dissipation, the tunneling rate is found to vanish discontinuously above some critical value as a function of the strength of dissipation.
The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.
Understanding the stability of strongly correlated phases of matter when coupled to environmental degrees of freedom is crucial for identifying the conditions under which these states may be observed. Here, we focus on the paradigmatic one-dimensional Bose-Hubbard model, and study the stability of the Luttinger-liquid and Mott insulating phases in the presence of local particle exchange with site-independent baths of noninteracting bosons. We perform a numerically exact analysis of this model by adapting the recently developed wormhole quantum Monte Carlo method for retarded interactions to a continuous-time formulation with worm updates; we show how the wormhole updates can be easily implemented in this scheme. For an Ohmic bath, our numerical findings confirm the scaling prediction that the Luttinger-liquid phase becomes unstable at infinitesimal bath coupling. We show that the ensuing phase is a long-range-ordered superfluid with spontaneously broken U(1) symmetry. While the Mott insulator remains a distinct phase for small bath coupling, it exhibits diverging compressibility and noninteger local boson occupation in the thermodynamic limit. Upon increasing the bath coupling, this phase undergoes a transition to a long-range-ordered superfluid. Finally, we discuss the effects of super-Ohmic dissipation on the Luttinger-liquid phase. Our results are compatible with a stable dissipative Luttinger-liquid phase that transitions to a long-range-ordered superfluid at a finite system-bath coupling.