The ground states of interacting one-dimensional metals are generically Luttinger liquids. Luttinger-liquid theory is usually considered for translation invariant systems. The Luttinger-liquid description remains valid for weak quasiperiodic modulations; however, as the quasiperiodic modulation gets increasingly strong, it is increasingly renormalized and eventually fails, as the system becomes localized. We explore how quasiperiodic modulation renormalizes the Luttinger parameter characterizing this emergent Luttinger liquid, using the renormalization of transmission coefficients across a barrier as a proxy that remains valid for general quasiperiodic modulation. We find, unexpectedly, that quasiperiodic modulation weakens the effects of short-range interactions, but enhances those of long-range interactions. We support the former finding with matrix-product numerics. We also discuss how interactions affect the localization phase boundary.
The ground states of interacting one-dimensional metals are generically Luttinger liquids. Luttinger-liquid theory is usually considered for translation invariant systems. The Luttinger-liquid description remains valid for weak quasiperiodic modulations; however, as the quasiperiodic modulation gets increasingly strong, it is increasingly renormalized and eventually fails, as the system becomes localized. We explore how quasiperiodic modulation renormalizes the Luttinger parameter characterizing this emergent Luttinger liquid, using the renormalization of transmission coefficients across a barrier as a proxy that remains valid for general quasiperiodic modulation. We find, unexpectedly, that quasiperiodic modulation weakens the effects of short-range interactions, but enhances those of long-range interactions. We support the former finding with matrix-product numerics. We also discuss how interactions affect the localization phase boundary.
We study the interplay between an on-site Hubbard repulsion and quasiperiodic potential in one-dimensional fermion chains using the density matrix renormalization group. We find that, at half filling, the quasiperiodic potential can destroy the Mott gap, leading to a metallic Luttinger liquid phase between the gapped Mott insulator at strong repulsion and the localized gapless Aubry-Andr\'e insulator at strong quasiperiodic potential. Away from half filling, the metallic phase of the interacting model persists to larger critical strengths of the potential than in the noninteracting case, suggesting interaction-stabilized delocalization at finite doping. We characterize the Luttinger liquid through its charge and spin correlations, structure factors, and entanglement entropy.
We study the interplay between electronic interactions and quasiperiodicity in a one-dimensional narrow-band system, focusing on ground-state and low-energy excitation properties. Using band projection as low-energy effective approach, we show that a projection restricted to first order in the interaction strength fails to reproduce the correlated phase diagram. This contrasts with the standard success of first-order band projection in translationally invariant flatband systems and highlights the essential role of virtual processes involving remote bands in quasiperiodic settings. By incorporating second-order interband contributions perturbatively, we obtain an effective Hamiltonian that quantitatively reproduces the exact phase iagram previously obtained using density matrix renormalization group calculations, including the transition between a Luttinger liquid and a charge-density-wave phase and the crossover to a quasifractal charge-density-wave regime at strong quasiperiodicity. We further use this controlled framework to investigate low-energy neutral excitations and the optical conductivity, identifying clear dynamical signatures distinguishing the different phases. Our results establish second-order band projection as a reliable tool for correlated quasiperiodic narrow-band systems and suggest a promising route for studying interacting quasiperiodic and moiré materials beyond one dimension.
The eect of electronic many-body interactions in one-dimensional (1D) systems is drastically enhanced compared to higher dimensions due to the restricted phasespace, often leading to strong electronic correlations. It can be shown that due to a breakdown of the Fermi liquid theory, the ground state at low temperatures becomes a Tomonaga-Luttinger liquid (TLL). In a TLL, the strength of electronic correlations is characterized by the Luttinger parameter K, ranging between K = 0 and K = 1, where K = 1 describes non-interacting electrons. Various systems, such as single-walled carbon nanotubes, 1D atomic metal chains, and MoS 2 mirror twin boundaries, have been reported to host TLLs, in which spin-degeneracy is intact.
We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
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.