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.
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.
We reinvestigate the momentum-resolved single-particle spectral function of the Tomonaga-Luttinger model. In particular, we focus on the role of the momentum dependence of the two-particle interaction V(q). Usually, V(q) is assumed to be a constant and integrals are regularized in the ultraviolet ``by hand'' employing an ad hoc procedure. As the momentum dependence of the interaction is irrelevant in the renormalization group sense, this does not affect the universal low-energy properties of the model, e.g., exponents of power laws, if all energy scales are sent to zero. If, however, the momentum k is fixed away from the Fermi momentum kF, with $|k\ensuremath{-}{k}_{\mathrm{F}}|$ setting a nonvanishing energy scale, the details of V(q) start to matter. We provide strong evidence that any curvature of the two-particle interaction at small transferred momentum q destroys power-law scaling of the momentum-resolved spectral function as a function of energy. Even for $|k\ensuremath{-}{k}_{\mathrm{F}}|$ much smaller than the momentum-space range of the interaction the spectral line shape depends on the details of V(q). The significance of our results for universality in the Luttinger liquid sense, for experiments on quasi-one-dimensional metals, and for recent results on the spectral function of one-dimensional correlated systems taking effects of the curvature of the single-particle dispersion into account (``nonlinear LL phenomenology'') is discussed.
In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin-1/2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.
We use bosonization methods to calculate the exact finite-temperature single-electron Green's function of a spinful Luttinger liquid confined by open boundaries. The corresponding local spectral density is constructed and analyzed in detail. The interplay between boundary, finite-size, and thermal effects are shown to dramatically influence the low-energy properties of the system. In particular, the well-known zero-temperature critical behavior in the bulk always crosses over to a boundary dominated regime in the vicinity of the Fermi level. Thermal fluctuations cause an enhanced depletion of spectral weight for small energies $\ensuremath{\omega}$, with the spectral density scaling as ${\ensuremath{\omega}}^{2}$ for $\ensuremath{\omega}$ much less than the temperature. Consequences for photoemission experiments are discussed.
We calculate one-particle spectra for a variety of models of Luttinger liquids with open boundary conditions. For the repulsive Hubbard model, the spectral weight close to the boundary is enhanced in a large energy range around the chemical potential. A power-law suppression, previously predicted by bosonization, only occurs after a crossover at energies very close to the chemical potential. Our comparison with exact spectra shows that the effects of boundaries can partly be understood within the Hartree-Fock approximation.
We calculate the single-particle spectral function A ( k , ω) of a one-dimensional Luttinger liquid by means of a functional renormalization group (RG) approach. Given an infrared energy cut-off Λ = Λ 0 e − l , our approach yields the spectral function in the scaling form, A Λ ( k F + p , ω) = τ Z l à l ( p ξ, ωτ), where k F is the Fermi momentum, Z l is the wavefunction renormalization factor, τ = 1/Λ is the timescale and ξ = v F /Λ is the length scale associated with Λ. At the Luttinger liquid fixed point ( l → ∞ ) our RG result for A ( k , ω) exhibits the correct anomalous scaling properties, and for k = ± k F agrees exactly with the well known bosonization result at weak coupling. Our calculation demonstrates that the field rescaling is essential for obtaining the crossover from Fermi liquid behaviour to Luttinger liquid behaviour from a truncation of the hierarchy of exact RG flow equations as the infrared cut-off Λ is reduced.