Luttinger-liquid universality in the time evolution after an interaction quench

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.

Figure 1

Time evolution of the $Z$ factor out of the noninteracting ground state of a $1d$ metallic Fermi system after switching on two-particle terms at time $t=0$. Dashed lines show the universal asymptotic power law $t^{-{\gamma }_{\mathrm{st}}(K)}$ with an exponent determined by the equilibrium LL parameter $K$. (a) TL model. The plots displays boxlike two-particle interactions $g(k)$ of strength $g=g(0)$; the asymptotics are universal for any $g(k)$. Time is given in units of $(v_{\mathrm{F}}{k}_c{)}^{-1}$. (b, c, Inset) Spinless lattice fermions of Eq. (6) at filling $\nu$ featuring nearest ($\Delta$) and next-nearest (${\Delta }_2$) neighbor interactions.

Figure 2

Time evolution of the kinetic energy per length $d{e}_{\mathrm{kin}}/dt$. (a) TL model for different (Gaussian and quartic) two-particle potentials $g(k)$. For any continous $g(k)$, $d{e}_{\mathrm{kin}}/dt$ asymptotically falls off as $ϵ(K,v)/t^3$ with a universal prefactor. $e_{\mathrm{kin}}$ and $t$ are given in units of $v_{\mathrm{F}}{k}_c^2$ and $(v_{\mathrm{F}}{k}_c{)}^{-1}$, respectively. (b) Spinless lattice fermions. Solid lines show DMRG data; dashed lines display $t^{-3}$ power laws, where the ratio of prefactors is chosen according to the TL prediction. Inset: DMRG data before taking the $t$ derivative.

Figure 3

Imaginary time evolution $\exp (-\tau H)|E_{\mathrm{gs}}^0⟩\stackrel{\tau \to \infty }{\to }|E_{\mathrm{gs}}⟩$ towards the interacting ground state $|E_{\mathrm{gs}}⟩$. The main part shows DMRG data for the $\tau$ derivative of the total energy in the lattice model. The ${\tau }^{-3}$ decay predicted by bosonization manifests over several orders of magnitude. The prefactor agrees with the TL formula ${ϵ}_{\mathrm{it}}(K,v)$ for all parameters (this is illustrated in the insets). The imaginary-time energy dynamics are, thus, universal in the LL sense.