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.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, 20190 citations

Transport through Periodically Driven Correlated Quantum Wires

Dante Marvin Kennes

We study correlated quantum wires subject to harmonic modulation of the onsite-potential concentrating on the limit of large times, where the response of the system has synchronized with the drive. We identify the ratio $Δε/Ω$ of the driving amplitude $Δε$ and the frequency of driving $Ω$ as the scale determining the crossover from a modified Luttinger liquid picture to a system that behaves effectively like a higher dimensional one. We exemplify this crossover by studying the frequency dependency of the boundary density of state $ρ_{\rm B}(ω)$ as well as the temperature dependency of the linear conductance $G(T)$ through the wire, if the latter is contacted to leads. Both observables are known to exhibit Luttinger liquid physics without driving given by characteristic power-law suppression as $ω\toε_{\rm F}$ (with $ε_{\rm F}$ the Fermi energy) or $T\to 0$ , respectively. With driving we find that this suppression is modified from a single power-law to a superposition of an infinite number of power laws. At small $Δε/Ω\ll 1$ only a few terms of this infinite sum are relevant as the prefactors of higher terms are suppressed exponentially. Thus a picture similar to the equilibrium Luttinger liquid one emerges. Increasing $Δε/Ω$ an increasing number of power laws contribute to the sum and approaching $Δε/Ω\gg 1$ the system behaves effectively two dimensional, for which the suppression is wiped out completely.We study correlated quantum wires subject to harmonic modulation of the onsite-potential concentrating on the limit of large times, where the response of the system has synchronized with the drive. We identify the ratio $Δε/Ω$ of the driving amplitude $Δε$ and the frequency of driving $Ω$ as the scale determining the crossover from a modified Luttinger liquid picture to a system that behaves effectively like a higher dimensional one. We exemplify this crossover by studying the frequency dependency of the boundary density of state $ρ_{\rm B}(ω)$ as well as the temperature dependency of the linear conductance $G(T)$ through the wire, if the latter is contacted to leads. Both observables are known to exhibit Luttinger liquid physics without driving given by characteristic power-law suppression as $ω\toε_{\rm F}$ (with $ε_{\rm F}$ the Fermi energy) or $T\to 0$, respectively. With driving we find that this suppression is modified from a single power-law to a superposition of an infinite number of power laws. At small $Δε/Ω\ll 1$ only a few terms of this infinite sum are relevant as the prefactors of higher terms are suppressed exponentially. Thus a picture similar to the equilibrium Luttinger liquid one emerges. Increasing $Δε/Ω$ an increasing number of power laws contribute to the sum and approaching $Δε/Ω\gg 1$ the system behaves effectively two dimensional, for which the suppression is wiped out completely.

arXiv:1801.02866Jan 9, 20183 citations

Transport in Luttinger Liquids

Thierry Giamarchi, T. Nattermann, Pierre Le Doussal

arXiv:cond-mat/0403487NATO science series. Series II, Mathematics, physics and chemistryJan 1, 20040 citations

Local spectral properties of Luttinger liquids: scaling versus nonuniversal energy scales

Dirk Schuricht, Sabine Andergassen, V. Meden

Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface, we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational-invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both set-ups have the same analytical form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In the second part we approximately compute the local spectral function of a microscopic lattice model-the extended Hubbard model-close to an open boundary using the functional renormalization group. We show that it follows the field theoretical prediction in the low-energy regime as a function of energy and temperature, and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction.Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface, we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational-invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both set-ups have the same analytical form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In the second part we approximately compute the local spectral function of a microscopic lattice model-the extended Hubbard model-close to an open boundary using the functional renormalization group. We show that it follows the field theoretical prediction in the low-energy regime as a function of energy and temperature, and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction.

arXiv:1111.7174Journal of Physics Condensed MatterDec 5, 20129 citations

Δε/Ω

of the driving amplitude

Δε

and the frequency of driving

Ω

as the scale determining the crossover from a modified Luttinger liquid picture to a system that behaves effectively like a higher dimensional one. We exemplify this crossover by studying the frequency dependency of the boundary density of state

ρ_{\rm B}(ω)

as well as the temperature dependency of the linear conductance

G(T)

through the wire, if the latter is contacted to leads. Both observables are known to exhibit Luttinger liquid physics without driving given by characteristic power-law suppression as

ω\toε_{\rm F}

(with

ε_{\rm F}

the Fermi energy) or

T\to 0

, respectively. With driving we find that this suppression is modified from a single power-law to a superposition of an infinite number of power laws. At small

Δε/Ω\ll 1

only a few terms of this infinite sum are relevant as the prefactors of higher terms are suppressed exponentially. Thus a picture similar to the equilibrium Luttinger liquid one emerges. Increasing

Δε/Ω

an increasing number of power laws contribute to the sum and approaching

Δε/Ω\gg 1

the system behaves effectively two dimensional, for which the suppression is wiped out completely.