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Communications in Theoretical Physics

Time-Dependent Evolution of Two Coupled Luttinger Models of Spin-1/2 Fermions

Zhou Zong-Li, Ping Lou

Published Jun 1, 2009• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/51/6/33

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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.
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 $t\ensuremath{\geqslant}0$ 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.
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.
基于我们考虑由二断开的 Luttinger 液体组成的 spinful 系统的地理论的 bosonization，它在时间 t = 被联合 0 通过 chiral 密度密度相互作用。我们借助于 bosonization 和 Bogoliubov 转变在这个 1D 系统显示出著名纺纱费用分离。评估经分解一个粒子相等时间的关联功能， Cazalilla 的方法被扩大到纺纱费用部门。在关联的行为不与平衡的与一致的长时间限制，二的相等时间的关联联合了 spinful Luttinger 液体，这被发现。以便与试验性的结果作比较，管理非平衡系统的相等时间的关联的幂定律行为的改进代表在这个工作的结束被获得。
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger-liquid (LL) theory, which predicts phenomena such as spin-charge separation and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D ttt-JJJ model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2tJ=2tJ=2t) in terms of the excitations of the Bethe-ansatz solution. Furthermore we show that fractionalization with similar characteristics extends to the whole region corresponding to a repulsive LL.
Motivated by recent research achievement of quantum interacting systems in non-equilibrium, we consider a Luttinger model with a suddenly switched-on interaction proposed by Cazalilla [M.A. Cazalilla, Phys. Rev. Lett. 97 (2006) 156403 ]. In order to compare with real systems, we extend Cazalilla's scenario to the spinful system. To find the influence of initial states on the time evolution of some non-equilibrium systems, we mainly focus on the density-density propagator. By comparison and analysis, we discover the different behavior of this non-equilibrium system. Further, it is found that the propagator saves strong memory of initial state, and the effects of right-left interaction cancel out in total density-density propagator.
We study the time evolution of two fermionic one-dimensional models (spinless fermions with nearest-neighbor repulsion and the Hubbard model) exposed to an interaction quench for short and moderate times. The method used to calculate the time dependence is a semi-numerical approach based on the Heisenberg equation of motion. We compare the results of this approach with the results obtained by bosonization implying power law behavior. Indeed, we find that power laws describe our results well, but our results raise the issue of which exponents occur. For spinless fermions, it seems that the Tomonaga–Luttinger parameters work well, which also describe the equilibrium low-energy physics. But for the Hubbard model this is not the case. Instead, we find that exponents from the bosonization around the initial state work well. Finally, we discuss what can be expected for the long-time behavior.