Tomonaga Luttinger Research Articles

One-dimensional fermionic systems after interaction quenches and their description by bosonic field theories

We study the time evolution of two fermionic one-dimensional models (spinless fermions with nearest-neighbor repulsion and the Hubbard model) exposed to an interaction quench for short and moderate times. The method used to calculate the time dependence is a semi-numerical approach based on the Heisenberg equation of motion. We compare the results of this approach with the results obtained by bosonization implying power law behavior. Indeed, we find that power laws describe our results well, but our results raise the issue of which exponents occur. For spinless fermions, it seems that the Tomonaga–Luttinger parameters work well, which also describe the equilibrium low-energy physics. But for the Hubbard model this is not the case. Instead, we find that exponents from the bosonization around the initial state work well. Finally, we discuss what can be expected for the long-time behavior.

Tunability of Electronic States in Ultrathin Gold Nanowires

The electronic state in ultrathin gold nanowires is tuned by careful engineering of the device architecture via a chemical methodology. The electrons are localized to an insulating state (showing variable range hopping transport) by simply bringing them close to the substrate, while the insertion of an interlayer leads to a Tomonaga Luttinger liquid state.

Single-particle excitation spectrum in 1D ultracold fermionic optical lattices

We investigate the properties of fermions trapped in a one-dimensional (1D) optical lattice by using the density-matrix renormalization (DMRG) group method. Owing to a harmonic confinement potential inherent in real experiments, system becomes inhomogeneous structure. Under certain conditions for weak-repulsive interaction, we find that the single-particle excitation spectrum structure changes to band branching and discrete bound-state states as increasing the trap strength. Additionally, we consider the case of strong-repulsive interacting regimes which local density profile show coexist with central Mott-plateau phase with surrounded by metallic regions. As increasing the trap strength, we find the breakdown of 1D Tomonaga-Luttinger (TL) liquid which is alternative to an effective doping into the Mott phase. We will present the various kinds of striking spectra, show the comparison of repulsive interaction case with attractive interaction one.

Spin ladders and quantum simulators for Tomonaga–Luttinger liquids

Magnetic insulators have proven to be usable as quantum simulators for itinerant interacting quantum systems. In particular the compound (C5H12N)2CuBr4 (for short: (Hpip)2CuBr4) was shown to be a remarkable realization of a Tomonaga–Luttinger liquid (TLL) and allowed us to quantitatively test the TLL theory. Substitution weakly disorders this class of compounds and thus allows us to use them to tackle questions pertaining to the effect of disorder in TLL as well, such as that of the formation of the Bose glass. In this paper we present, as a first step in this direction, a study of the properties of the related (Hpip)2CuCl4 compound. We determine the exchange couplings and compute the temperature and magnetic field dependence of the specific heat, using a finite temperature density matrix renormalization group procedure. Comparison with the measured specific heat at zero magnetic field confirms the exchange parameters and Hamiltonian for the (Hpip)2CuCl4 compound, giving the basis needed to begin studying the disorder effects.

Au-induced quantum chains on Ge(001)—symmetries, long-range order and the conduction path

Atomic nanowires on the Au/Ge(001) surface are investigated for their structural and electronic properties using scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES). STM reveals two distinct symmetries: a c(8 × 2) describing the basic repeating distances, while the fine structure on top of the wires causes an additional superstructure of p(4 × 1). Both symmetries are long-range ordered as judged from low-energy electron diffraction. The Fermi surface is composed of almost perfectly straight sheets. Thus, the electronic states are one-dimensionally confined. Spatial dI/dV maps, where both topography and density of states (DOS) are probed simultaneously, reveal that the DOS at low energies, i.e. the conduction path, is oriented along the chain direction. This is fully consistent with the recently reported Tomonaga–Luttinger liquid phase of Au/Ge(001), with the density of states being suppressed by a power-law towards the Fermi energy.

Photoemission spectroscopy and the unusually robust one-dimensional physics of lithium purple bronze

Temperature-dependent photoemission spectroscopy in Li0.9Mo6O17 contributes to evidence for one-dimensional (1D) physics that is unusually robust. Three generic characteristics of the Luttinger liquid are observed: power law behavior of the k-integrated spectral function down to temperatures just above the superconducting transition, k-resolved lineshapes that show holon and spinon features, and quantum critical (QC) scaling in the lineshapes. Departures of the lineshapes and the scaling from expectations in the Tomonaga–Luttinger model can be partially described by a phenomenological momentum broadening that is presented and discussed. The possibility that some form of 1D physics obtains even down to the superconducting transition temperature is assessed.

Microscopic study of edge excitations of spin-polarized and spin-unpolarizedν=2/3fractional quantum Hall effect

The edge of spin unpolarized or spin polarized $\nu=2/3$ fractional quantum Hall states is predicted by the effective theory to support a backward moving neutral mode in addition to a forward moving charge mode. We study this issue from a microscopic perspective where these states are identified with effective filling factor of 2 of composite fermions, but with an effective magnetic field that is antiparallel to the external field. A simple counting from the composite fermion description suggests that there might be two backward moving edge modes, but explicit calculations show that one of these is projected out of the low energy sector, while the remaining mode provides a good microscopic account of the actual counter propagating edge mode. The forward moving modes are identified as "Schur modes," obtained by multiplying the ground state wave function by the symmetric Schur polynomials. The edge of the 2/3 spin unpolarized state provides a particularly striking realization of "spin charge separation" in a one dimensional Tomonaga Luttinger liquids, with the spin and charge modes moving in opposite directions.

Luttinger-liquid theory of purple bronze Li0.9Mo6O17in the charge regime

Molybdenum purple bronze Li$_{0.9}$Mo$_{6}$O$_{17}$ is an exceptional material known to exhibit one dimensional (1D) properties for energies down to a few meV. This fact seems to be well established both in experiments and in band structure theory. We use the unusual, very 1-dimensional band dispersion obtained in \emph{ab-initio} DFT-LMTO band calculations as our starting point to study the physics emerging below 300meV. A dispersion perpendicular to the main dispersive direction is obtained and investigated in detail. Based on this, we derive an effective low energy theory within the Tomonaga Luttinger liquid (TLL) framework. We estimate the strength of the possible interactions and from this deduce the values of the TLL parameters for charge modes. Finally we investigate possible instabilities of TLL by deriving renormalization group (RG) equations which allow us to predict the size of potential gaps in the spectrum. While $2k_F$ instabilities strongly suppress each other, the $4k_F$ instabilities cooperate, which paves the way for a possible CDW at the lowest energies. The aim of this work is to understand the experimental findings, in particular the ones which are certainly lying within the 1D regime. We discuss the validity of our 1D approach and further perspectives for the lower energy phases.

SOME EXPERIMENTAL TESTS OF TOMONAGA–LUTTINGER LIQUIDS

The Tomonaga–Luttinger–Liquid (TLL) has been the cornerstone of our understanding of the properties of one dimensional systems. This universal set of properties plays in one dimension, the same role than Fermi liquid plays for the higher dimensional metals. I will give in these notes an overview of some of the experimental tests that were made to probe such TLL physics. In particular I will detail some of the recent experiments that were made in spin systems and which provided remarkable quantitative tests of the TLL physics.

LONG TIME CORRELATIONS OF NONLINEAR LUTTINGER LIQUIDS

An overview is given of the limitations of Luttinger liquid theory in describing the real time equilibrium dynamics of critical one-dimensional systems with nonlinear dispersion relation. After exposing the singularities of perturbation theory in band curvature effects that break the Lorentz invariance of the Tomonaga–Luttinger model, the origin of high frequency oscillations in the long time behaviour of correlation functions is discussed. The notion that correlations decay exponentially at finite temperature is challenged by the effects of diffusion in the density–density correlation due to umklapp scattering in lattice models.

Quench dynamics of the Tomonaga–Luttinger model with momentum-dependent interaction

We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.

Phonon-Induced Backscattering in Helical Edge States

A single pair of helical edge states as realized at the boundary of a quantum spin Hall insulator is known to be robust against elastic single particle backscattering as long as time reversal symmetry is preserved. However, there is no symmetry preventing inelastic backscattering as brought about by phonons in the presence of Rashba spin orbit coupling. In this Letter, we show that the quantized conductivity of a single channel of helical Dirac electrons is protected even against this inelastic mechanism to leading order. We further demonstrate that this result remains valid when Coulomb interaction is included in the framework of a helical Tomonaga Luttinger liquid.

On the solutions of Landau transport equations in one-dimension and their equivalence with the Tomonaga–Luttinger model predictions

By using linear response theory, we show that Landau transport equations in one-dimension (1D) and Tomonaga–Luttinger model provide exactly the same description for density and spin-density modes and conductance in 1D systems. Landau transport equations in 1D can be derived rigorously from the time-dependent Hartree and adiabatic time-dependent local spin density (TDLSDA) theories without making use of the concept of Landau's quasi-particles. In 1D, in the long wavelength limit, there is a perfect equivalence between Landau and the random-phase-approximation (RPA) and TDLSDA responses.

Spectral properties of trapped one-dimensional ultracold fermions loaded on optical lattices

We examine spectral properties on one-dimensional (1D) trapped ultracold Fermi atoms loaded on optical lattices by using the dynamical density-matrix renormalization group (DDMRG) method. We find that spectra are very rich due to the interplay of free or repulsive interaction and the harmonic-trap potential. One of the rich examples is that a 1D Tomonaga-Luttinger (TL) liquid spectrum emerges beneath multiple bound-state flat levels in the Mott-plateau phase and the TL spectrum gradually disappears with a partial breakdown of the central Mott plateau by increasing the trap strength. A more striking example is the growth of a distinct low-energy broadened band from the bound-state flat levels together with the breakdown. It demonstrates a carrier doping effect into the Mott phase in optical lattice systems.

Validity of the Tomonaga Luttinger liquid relations for the one-dimensional Holstein model

For the one-dimensional Holstein model, we show that the relations among the scaling exponents of various correlation functions of the Tomonaga Luttinger liquid (LL), while valid in the thermodynamic limit, are significantly modified by finite-size corrections. We obtain analytical expressions for these corrections and find that they decrease very slowly with increasing system size. The interpretation of numerical data on finite-size lattices in terms of LL theory must therefore take these corrections into account. As an important example, we re-examine the proposed metallic phase of the zero-temperature, half-filled one-dimensional Holstein model without employing the LL relations. In particular, using quantum Monte Carlo calculations, we study the competition between the singlet pairing and charge ordering. Our results do not support the existence of a dominant singlet pairing state.

Thermodynamics and spin-charge separation of one-dimensional strongly repulsive three-component fermions

The low-temperature thermodynamics of one-dimensional strongly repulsive SU(3) fermions in the presence of a magnetic field is investigated via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the model at low temperatures in a weak magnetic field are derived via the Wiener–Hopf method. It is shown that the low-energy physics can be described by spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU(3) Heisenberg spin chain. Beyond the Tomonaga–Luttinger liquid regime, the equation of state is given in terms of the polylog function for a weak external field. The results obtained are essential for further study of quantum criticality in strongly repulsive three-component fermions.

Performance analysis of metallic carbon nanotubes as nanotechnology circuit interconnects

Carbon nanotubes (CNTs) have emerged as a potential nanoelectronic material that can replace traditional copper as the material for very large scale integrated circuits (VLSI) interconnects.In this article, we have quantified the performance expected out of CNTs-based interconnect systems based on realistic characterization of their Tomonaga Luttinger theory-based transmission line model. We have presented performance estimates for bus systems consisted of single-walled nanotubes and crystalline bundles of CNTs. Results indicate that the bundles offer at leasta 10 times performance advantage over isolated nanotubes. However, this is at the expense of larger signal damping associated with them. Published 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:2505–2508, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26314

Breaking the Law

Physics Conduction of heat in solids is often achieved through the same means as electrical conduction—via charge carriers—and materials that are good electrical conductors (such as metals) also tend to conduct heat efficiently. For metals, the ratio between the thermal and electrical conductivities is determined by the Wiedemann-Franz (WF) law. However, in one-dimensional systems, a distinct state known as the Tomonaga-Luttinger (TL) liquid is predicted to occur, wherein the spin and charge degrees of freedom of an electron separate into spinons and holons; because heat is conducted by both but electricity only by holons, a violation of the WF law may occur. Wakeham et al. report such a violation in the purple bronze Li0.9Mo6O17, which is a conductor consisting of weakly interacting one-dimensional chains. The violation they find is as large as a factor of 100,000 at the lowest temperatures accessed and consistent with the scenario of spin-charge separation in a TL liquid, even though the material is only approximately one-dimensional. If, as the authors speculate, its TL liquid nature is a result of strong correlations within the chains, it may be possible to tune the dimensionality of the system, and thus effectively recombine spinons and holons, by appropriate chemical substitution. Nat. Commun. 2 , 10.1038/ncomms1406 (2011).

Atomically controlled quantum chains hosting a Tomonaga–Luttinger liquid

Atom assemblies on surfaces represent the ultimate lower size limit for electronic circuits, and their conduction properties are governed by quantum phenomena. A fundamental prediction for a line of atoms confining the electrons to one dimension is the Tomonaga–Luttinger liquid1. Yet, astonishingly, this has not been observed in surface systems so far. Here we scrutinize self-organized chains of single-atom width by scanning tunnelling spectroscopy and photoemission. The low-energy spectra univocally show power-law behaviour. Even more, the density of states obeys universal scaling with energy and temperature. This demonstrates paradigmatic Tomonaga–Luttinger liquid properties2,3 encountered at the atomic scale, with bearing for the conductivity of wires and junctions. Local control enables us to study modified interactions due to defects or bridging atoms not previously possible.

Enhanced coherence of a quantum doublet coupled to Tomonaga–Luttinger liquid leads

We use boundary field theory to describe the phases accessible to a tetrahedral qubit coupled to Josephson junction chains acting as Tomonaga–Luttinger liquid leads. We prove that, in a pertinent range of the fabrication and control parameters, an attractive finite coupling fixed point emerges due to the geometry of the composite Josephson junction network. We show that this new stable phase is characterized by the emergence of a quantum doublet which is robust not only against the noise in the external control parameters (magnetic flux, gate voltage) but also against the decoherence induced by the coupling of the tetrahedral qubit with the superconducting leads. We provide protocols allowing to read and to manipulate the state of the emerging quantum doublet and argue that a tetrahedral Josephson junction network operating near the new finite coupling fixed point may be fabricated with todayʼs technologies.