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arXiv:cond-mat/0607720 · Physical Review B

Time-dependent evolution of two coupled Luttinger liquids

Enrico Perfetto

Published Nov 30, 2006• 27 citations
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Abstract

We consider two disconnected Luttinger liquids which are coupled at $t=0$ through chiral density-density interactions. Both for $t<0$ and $t\ensuremath{\geqslant}0$ the system is exactly solvable by means of bosonization and this allows us to evaluate analytically the time dependence of correlation functions. We find that in the long-time limit the critical exponent governing the one-particle correlation function differs from the exponent dictated by the equilibrium ground state of the coupled system. We also discuss how this reflects on some physical quantities which are accessible to real experiments.

DOI
10.1103/physrevb.74.205123

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We consider two disconnected Luttinger liquids which are coupled at t=0t=0t=0 through chiral density-density interactions. Both for t<0t<0t<0 and t\ensuremath⩾0t\ensuremath{\geqslant}0t\ensuremath⩾0 the system is exactly solvable by means of bosonization and this allows us to evaluate analytically the time dependence of correlation functions. We find that in the long-time limit the critical exponent governing the one-particle correlation function differs from the exponent dictated by the equilibrium ground state of the coupled system. We also discuss how this reflects on some physical quantities which are accessible to real experiments.
Based on the field-theoretical bosonization we consider the spinful system consisting of two disconnected Luttinger liquids, which are coupled at time t = 0 through chiral density-density interactions. We show the well-known spin-charge separation in this 1D system by means of bosonization and Bogoliubov transformation. To evaluate analytically the one-particle equal-time correlation functions, Cazalilla's methods are extended to spin-charge sectors. It is found that in long time limit the behavior of correlations does not coincide with that of equilibrium equal-time correlations of two coupled spinful Luttinger liquids. In order to compare with experimental results, the improved exponent that governs the power-law behavior of equal-time correlations of the non-equilibrium system is obtained at the end of this work.
The evolution of correlations in the exactly solvable Luttinger model (a model of interacting fermions in one dimension) after a suddenly switched-on interaction is analytically studied. When the model is defined on a finite-size ring, zero-temperature correlations are periodic in time. However, in the thermodynamic limit, the system relaxes algebraically towards a stationary state which is well described, at least for some simple correlation functions, by the generalized Gibbs ensemble recently introduced by Rigol et al. (cond-mat/0604476). The critical exponent that characterizes the decay of the one-particle correlation function is different from the known equilibrium exponents. Experiments for which these results can be relevant are also discussed.
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.
For a one-dimensional model in which the two-body interactions are long-range and strong, the system almost crystallizes. The harmonic modes of such a lattice were used by Krivnov and Ovchinnikov to compute the ground-state wave function and the dynamical density-density correlations. We review this method, and apply it to the Calogero-Sutherland model, whose density-density correlation functions are exactly known for certain values of the coupling constant. We show numerically that the correlations obtained are quite accurate even if the coupling is not very large. Such comparisons have been made earlier by Forrester. The lattice method is considerably simpler than the ones used to derive the exact results, and yields expressions for the correlations- which are easily plotted. The equal-time correlations can be expanded in inverse powers of coupling; we show that the two leading order terms agree with the exact results which are known for integer values of the coupling. The strength-dependent power law fall-off is typical of a Luttinger liquid.In a general one-dimensional model where the two-body interaction decreases as a power of the relative distance, we argue, following Schulz, that at zero temperature the system behaves as a Luttinger liquid if the power exceeds 1, and as a Wigner crystal if it is less than 1.PACS Nos.: 63.20-e, 71.10Pm
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.
Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.