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.
We study the relaxation dynamics of a nonequilibrium Tomonaga-Luttinger model after a sudden interaction switch-on (``quench''), focusing on a double-step initial momentum distribution function. In the framework of the nonequilibrium bosonization, the results are obtained in terms of singular Fredholm determinants that are evaluated numerically and whose asymptotics are found analytically. While the quasiparticle weights decay exponentially with time after the quench, this is not a relaxation into a thermal state, in view of the integrability of the model. The steady-state distribution emerging at infinite times retains two edges which support Luttinger-liquid-like power-law singularities smeared by dephasing. The obtained critical exponents and the dephasing length are found to depend on the initial nonequilibrium state.
We provide evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: the leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density-matrix renormalization group.
We study the dynamics resulting from an abrupt change of the two-particle interaction in two models of closed one-dimensional Fermi systems: (a) the field theoretical Tomonaga-Luttinger model and (b) a microscopic lattice model. Using a nonperturbative approach which is controlled for small two-particle interactions, we are able to reach large times allowing us to access the properties of the steady state of the lattice model. Comparing those to the exact solution of the full dynamics in the Tomonaga-Luttinger model, we provide evidence for universal Luttinger liquid behavior.
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.
The dynamics of the Luttinger model after a quantum quench is studied. We compute in detail one- and two-point correlation functions for two types of quenches: from a noninteracting to an interacting Luttinger model and vice versa. In the former case, the noninteracting Fermi gas features in the momentum distribution and other correlation functions are destroyed as time evolves. In the infinite-time limit, equal-time correlations are power laws but the critical exponents are found to differ from their equilibrium values. In all cases, we find that these correlations are well described by a generalized Gibbs ensemble [M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007)], which assigns a momentum-dependent temperature to each eigenmode.