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.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.

arXiv:1207.3548International Journal of Modern Physics BJul 26, 201223 citations

One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm

Adilet Imambekov, Thomas L. Schmidt, L. I. Glazman

For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.

arXiv:1110.1374Reviews of Modern PhysicsSep 14, 2012511 citations

Low temperature dynamics of nonlinear Luttinger liquids

Christoph Karrasch, R G Pereira, J Sirker

We develop a general nonlinear Luttinger liquid theory to describe the dynamics of one-dimensional quantum critical systems at low temperatures.To demonstrate the predictive power of our theory we compare results for the autocorrelation G(t) in the XXZ chain with numerical density-matrix renormalization group data and obtain excellent agreement.Our calculations provide, in particular, direct evidence that G(t) shows a diffusion-like decay, G t t 1 , ( ) in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory.We develop a general nonlinear Luttinger liquid theory to describe the dynamics of one-dimensional quantum critical systems at low temperatures.To demonstrate the predictive power of our theory we compare results for the autocorrelation G(t) in the XXZ chain with numerical density-matrix renormalization group data and obtain excellent agreement.Our calculations provide, in particular, direct evidence that G(t) shows a diffusion-like decay, G t t 1 , ( ) in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory.

arXiv:1410.2226New Journal of PhysicsSep 30, 201522 citations

Low temperature dynamics of nonlinear Luttinger liquids

Christoph Karrasch, Rodrigo G. Pereira, Jesko Sirker

We develop a general nonlinear Luttinger liquid theory to describe the dynamics of one-dimensional quantum critical systems at low temperatures. To demonstrate the predictive power of our theory we compare results for the autocorrelation G(t) in the XXZ chain with numerical density-matrix renormalization group data and obtain excellent agreement. Our calculations provide, in particular, direct evidence that G(t) shows a diffusion-like decay, $G(t)\sim 1/\sqrt{t},$ in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory.We develop a general nonlinear Luttinger liquid theory to describe the dynamics of one-dimensional quantum critical systems at low temperatures. To demonstrate the predictive power of our theory we compare results for the autocorrelation G(t) in the XXZ chain with numerical density-matrix renormalization group data and obtain excellent agreement. Our calculations provide, in particular, direct evidence that G(t) shows a diffusion-like decay, $G(t)\sim 1/\sqrt{t},$ in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory.

Jan 1, 201524 citations

Investigating the roots of the nonlinear Luttinger liquid phenomenology

L Markhof, Mikhail Pletyukhov, V. Meden

The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.

arXiv:1904.06220Apr 12, 20190 citations

Investigating the roots of the nonlinear Luttinger liquid phenomenology

L Markhof, Mikhail Pletyukov, V. Meden

SciPost PhysicsOct 9, 201914 citations

G(t)\sim 1/\sqrt{t},

in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory.