Luttinger Model Research Articles

LUTTINGER MODEL AND LUTTINGER LIQUIDS

The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.

AN EXPANDED LUTTINGER MODEL

This paper generalizes Luttinger's model by introducing curvature (d2ε(k)/dk2≠0) into the kinetic energy. An exact solution for arbitrary interactions is still possible in principle, but it now requires disentangling the eigenvalue spectrum of an harmonic string of interacting boson fields at each value of q. The additional boson fields, extracted from the excitation spectrum of the Fermi sea, are self-selected according to the nature and strength of the dispersion.

QUANTUM HALL EDGE PHYSICS AND ITS ONE-DIMENSIONAL LUTTINGER LIQUID DESCRIPTION

We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.

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.

Incoherent Scatterer in a Luttinger Liquid: A New Paradigmatic Limit

We address the problem of a Luttinger liquid with a scatterer that allows for both coherent and incoherent scattering channels. The asymptotic behavior at zero temperature is governed by a new stable fixed point: A Goldstone mode dominates the low energy dynamics, leading to universal behavior. This limit is marked by equal probabilities for forward and backward scattering. Notwithstanding this nontrivial scattering pattern, we find that the shot noise as well as cross-current correlations vanish. We thus present a paradigmatic picture of an impurity in the Luttinger model, alternative to the Kane-Fisher picture.

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.

Thermalization and quantum correlations in exactly solvable models

The generalized Gibbs ensemble introduced for describing few-body correlations in exactly solvable systems following a quantum quench is related to the nonergodic way in which operators sample, in the limit of infinite time after the quench, the quantum correlations present in the initial state. The nonergodicity of the correlations is thus shown analyticallyto imply the equivalence with the generalized Gibbs ensemble for quantum Ising and XX spin chains as well as for the Luttinger model the thermodynamic limit, and for a broad class of initial states and correlation functions of both local and nonlocal operators.

Anomalous Hall Effect in Geometrically Frustrated Magnets

Geometrically frustrated conducting magnets display extraordinarily large anomalous Hall effects (AHEs) that could be used to realise materials required for the emerging field of spintronics. While the intrinsic Berry phase developed in collinear ferromagnets is well explained through the effects of spin-orbit interactions within the Karplus and Luttinger model, its origins in frustrated magnets are not. The direct space mechanism based on spin chirality that was originally applied to the pyrochlore Nd2Mo2O7appears unsatisfactory. Recently, an orbital description based on the Aharonov-Bohm effect has been proposed and applied to both the ferromagnetic pyrochlores Nd2Mo2O7and Pr2Ir2O7; the first of which features long-ranged magnetic order while the latter is a chiral spin liquid. Two further examples of geometrically frustrated conducting magnets are presented in this paper—the kagome-like Fe3Sn2and the triangular PdCrO2. These possess very different electronic structures to the 3-dimensional heavy-metal pyrochlores and provide new opportunities to explore the different origins of the AHE. This paper summarises the experimental findings in these materials in an attempt to unite the conflicting theoretical arguments.

Four-terminal resistance of an interacting quantum wire with weakly invasive contacts

We analyze the behavior of the four-terminal resistance, relative to the two-terminal resistance of an interacting quantum wire with an impurity, taking into account the invasiveness of the voltage probes. We consider a one-dimensional Luttinger model of spinless fermions for the wire. We treat the coupling to the voltage probes perturbatively, within the framework of non-equilibrium Green function techniques. Our investigation unveils the combined effect of impurities, electron–electron interactions and invasiveness of the probes on the possible occurrence of negative resistance.

On the universal relations for the prefactors in correlation functions of 1D quantum liquids

For various one-dimensional quantum liquids in the framework of the Luttinger model (bosonization) we establish the relations between the coefficients before the power-law asymptotics of the correlators (prefactors) and the formfactors of the corresponding local operators. The derivation of these relations in the framework of the bosonization procedure allows to substantiate the prediction for the formfactors corresponding to the low-lying particle–hole excitations. We also obtain the formulas for the summation over the particle–hole states corresponding to the power-law asymptotics of the correlators.

Spin splitting of holes in symmetric GaAs/GaxAl1−xAs quantum wells

The in-plane Zeeman splitting of holes in symmetric GaAs/Ga${}_{x}$Al${}_{1\ensuremath{-}x}$As quantum wells with various orientations is studied using the Luttinger model. Our calculations extend previous theories by taking fully into account the wave-vector and magnetic-field dependences of the spin $g$ factor as well as an effect of bulk inversion asymmetry. The spin-orbit coupling in two-dimensional systems is known to lock the hole spin parallel to the growth direction, resulting in a weak spin response to the in-plane magnetic field $\mathbf{B}$. We demonstrate that, for higher values of the wave vector $\mathbf{k}$, the hole spin rotates toward the well plane and becomes pinned to the momentum. As a consequence, the in-plane Zeeman splitting strongly increases and is anisotropic with respect to relative orientations of $\mathbf{B}$ and $\mathbf{k}$. We also show that, for low-symmetry growth directions, the $g$ factor at moderate fields is highly asymmetric under the inversion of $\mathbf{B}$. This asymmetry results entirely from the properties of the Luttinger Hamiltonian and has no analog in nonmagnetic systems. The bulk inversion asymmetry is shown to influence the spin splitting mainly at low fields and to produce an additional field-inversion asymmetry of $g$, sensitive to the value of the Dresselhaus parameter ${C}_{k}$. Our calculations of the heavy-hole $g$ factor are in satisfactory agreement with the few available experimental data. We propose additional experiments to verify our theoretical findings.

Nonlinear Zeeman Splitting of Holes in Doped GaAs Heterostructures

spin orientation. We observed a nonlinear behavior of valence-band g factor in strong magnetic fields. To explain obtained results, a detailed theoretical calculation was carried out based on the Luttinger model for valence-band states. We examined the spin splitting of hole levels under the influence of both external magnetic field and built-in electric field existing in doped heterostructures. Changes of hole g factor with the width of the structure and the density of two-dimensional carriers are discussed.

The Luttinger liquid in superlattice structures: atomic gases, quantum dots and the classical Ising chain

We study the physical properties of a Luttinger liquid in a superlattice that is characterized by alternating two tunneling parameters. Using the bosonization approach, we describe the corresponding Hubbard model by the equivalent Tomonaga–Luttinger model. We analyze the spin–charge separation and transport properties of the superlattice system. We suggest that cold Fermi gases trapped in a bichromatic optical lattice and coupled quantum dots offer the opportunity to measure these effects in a convenient manner. We also study the classical Ising chain with two tunneling parameters. We find that the classical two-point correlator decreases as the difference between the two tunneling parameters increases.

Finite-Temperature Fidelity Susceptibility for One-Dimensional Quantum Systems

We calculate the fidelity susceptibility χf for the Luttinger model and show that there is a universal contribution linear in temperature T (or inverse length 1/L). Furthermore, we develop an algorithm--based on a lattice path integral approach--to calculate the fidelity F(T) in the thermodynamic limit for one-dimensional quantum systems. We check the Luttinger model predictions by calculating χf(T) analytically for free spinless fermions and numerically for the XXZ chain. Finally, we study χf at the two phase transitions in this model.

Spin-charge Separation of the Luttinger Model after an Interaction Quench

A Luttinger model of spin-½ fermions is considered after the interaction is suddenly switched on at time t = 0. By means of the bosonization technique, we evaluate analytically the one-particle correlation functions in detail, mainly involving equal-time correlations and propagators. The critical exponent which governs the power-law behavior of equal-time correlations for this spinful non-equilibrium system is obtained. In comparison with the published results, the difference between critical exponents of correlations in spinful and spinless non-equilibrium systems is found and explained. Furthermore, it is found that the propagator exhibits different power-law behavior from other equal-time correlations in this non-equilibrium system.

Partially Gapped Fermions in 2D

We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t'-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t'-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.

A Two-Dimensional Analogue of the Luttinger Model

We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.

Quantum quench dynamics of the Luttinger model

The dynamics of the Luttinger model and the sine-Gordon model (at the Luther-Emery point and in the semiclassical approximation) after a quantum quench is studied. We compute in detail one and two-point correlation functions for different types of quenches: from a non-interacting to an interacting Luttinger model and vice-versa, and from the gapped to the gapless phase of the sine-Gordon model and vice-versa. A progressive destruction of the Fermi gas features in the momentum distribution is found in the case of a quench into an interacting state in the Luttinger model. The critical exponents for spatial correlations are also found to be different from their equilibrium values. Correlations following a quench of the sine-Gordon model from the gapped to the gapless phase are found in agreement with the predictions of Calabrese and Cardy [Phys. Rev. Lett. {\bf 96} 136801 (2006)]. However, correlations following a quench from the gapped to the gapless phase at the Luther-Emery and the semi-classical limit exhibit a somewhat different behavior, which may indicate a break-down of the semiclassical approximation or a qualitative change in the behavior of correlations as one moves away from the Luther-Emergy point. In all cases, we find that the correlations at infinite times after the quench are well described by a generalized Gibbs ensemble [M. Rigol \emph{et al.} Phys. Rev. Lett. {\bf 98}, 050405 (2007)], which assigns a momentum dependent temperature to each eigenmode.

Is the Luttinger Liquid a New State of Matter?

We are demonstrating that the Luttinger model with short range interaction can be treated as a type of Fermi liquid. In line with the main dogma of Landau’s theory one can define a fermion excitation renormalized by interaction and show that in terms of these fermions any excited state of the system is described by free particles. The fermions are a mixture of renormalized right and left electrons. The electric charge and chirality of the Landau quasi-particle is discussed.