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arXiv:1206.3984 · Physical Review Letters

Quantum Breathing of an Impurity in a One-Dimensional Bath of Interacting Bosons

Sebastiano Peotta, Davide Rossini, Marco Polini, F. Minardi, Rosario Fazio

Published Jan 3, 2013• 55 citations
arXivHTMLPDF

Abstract

By means of the time-dependent density-matrix renormalization-group (TDMRG) method we are able to follow the real-time dynamics of a single impurity embedded in a one-dimensional bath of interacting bosons. We focus on the impurity breathing mode, which is found to be well described by a single oscillation frequency and a damping rate. If the impurity is very weakly coupled to the bath, a Luttinger-liquid description is valid and the impurity suffers an Abraham-Lorentz radiation-reaction friction. For a large portion of the explored parameter space, the TDMRG results fall well beyond the Luttinger-liquid paradigm.

DOI
10.1103/physrevlett.110.015302

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By means of the time-dependent density-matrix renormalization-group (TDMRG) method we are able to follow the real-time dynamics of a single impurity embedded in a one-dimensional bath of interacting bosons. We focus on the impurity breathing mode, which is found to be well described by a single oscillation frequency and a damping rate. If the impurity is very weakly coupled to the bath, a Luttinger-liquid description is valid and the impurity suffers an Abraham-Lorentz radiation-reaction friction. For a large portion of the explored parameter space, the TDMRG results fall well beyond the Luttinger-liquid paradigm.
We analyze the behavior of a mobile quantum impurity, restricted to a one-dimensional motion, in a bath formed by a two-leg bosonic ladder through a combination of field theory [Tomonaga-Luttinger liquid (TLL)] and numerical (density-matrix renormalization group) techniques. We compute the Green's function of the impurity as a function of time at different momenta. We find a power-law decay at zero momentum, which signals the breakdown of a quasiparticle description of the impurity motion. We compute the exponent both for the limits of weak and strong impurity-bath interactions. At small impurity-bath interaction, the impurity experiences the ladder as a single channel one-dimensional bath, but with an effective coupling reduced by a factor of 2\sqrt{2}2​. We compare the numerical results for the exponent at zero momentum with a semianalytical expression, initially established for the chain, and find excellent agreement without adjustable parameters. We find an increase of the exponent with increasing transverse tunneling in the bath. At small tunneling, the exponent is compatible with the TLL prediction, while larger tunneling shows strong deviations. Finally, we show that, as a function of the momentum of the impurity, two different regimes of decay of the Green's function exist, similar to the single chain case. The power-law regime occurs for small momentum, while at large momentum the Green's function shows a faster decay, corresponding to the one expected in a polaronic regime. In this last regime, we compute the lifetime of the polaron numerically. We compute the critical momentum marking the transition between these two regimes. We compare with analytical predictions based on the structure factor of the bath and find good agreement with the numerical results. Finally, we discuss the consequences of our results for cold atomic experiments.
Using a fermionic renormalization group method we present a simple real space picture of the strong influence an impurity has on the electronic properties of a Luttinger liquid. We compute the flow of the renormalized impurity potential for a single impurity over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. We confirm that low energy properties close to the impurity are as if the chain is cut in two pieces with open boundary conditions at the end points, but show that this universal behavior is only reached for extremely large systems. The accuracy of the renormalization group scheme is demonstrated by a direct comparison with data obtained from the density-matrix renormalization group method.
We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The impurity moves both along and across the legs and interacts with a bath of interacting bosonic particles present in the ladder. We use both analytical (Tomonaga-Luttinger liquid) and numerical [Density Matrix Renormalization Group (DMRG)] methods to compute the Green's function of the impurity. We find that for a small impurity-bath interaction, the symmetric mode of the impurity effectively couples only to the gapless mode of the bath while the antisymmetric mode of the impurity couples to both gapped and gapless modes of the bath. We compute the time dependence of the Green's function of the impurity, for impurity created in either the antisymmetric or symmetric mode with a given momentum. The latter case leads to a decay as a power law below a critical momentum and exponential above, while the former case exhibits both power-law and exponential decay depending on the transverse tunneling of the impurity. We compare the DMRG results with analytical results using the linked cluster expansion and find good agreement. In addition, we use DMRG to extract the lifetime of the quasiparticle, when the Green's function decays exponentially. We also treat the case of an infinite bath-impurity coupling for which both the symmetric and antisymmetric modes are systematically affected. For this case, the impurity Green's function in the symmetric mode decays as a power law at zero momentum. The corresponding exponent increases with increasing transverse tunneling of the impurity. We compare our results with other impurity problems for which the motion of either the impurity or the bath is limited to a single chain. Finally, we comment on the consequences of our findings for experiments with the ultracold gases.
Using the Density Matrix Renormalization Group (DMRG) and various analytical techniques (functional renormalization) we consider the effects of local impurities in Luttinger liquids. We find that the universal physics predicted by bosonization sets in only at extremely long, experimentally arguably irrelevant length scales or small energy scales respectively. The explicit construction of the RG flow allows to trace this behaviour to an extremely weak renormalization in real space of the impurities associated to the generation of a very long-ranged oscillating scattering potential.
We study the heating dynamics of a Luttinger liquid, upon suddenly coupling it to a dissipative environment. Within the Lindblad equation, the environment couples to local currents and heats the quantum liquid up to infinite temperatures. The fermionic single particle density matrix, and many other correlators, retain the initial Luttinger liquid correlations in space but decay exponentially in time with a rate that depends on the strength of the interaction. The spectrum of the time evolved density matrix is gapped, which collapses gradually as −ln⁡(t)-\ln(t)−ln(t). The von Neumann entropy crosses over from the early time −tln⁡(t)-t\ln(t)−tln(t) behaviour to a ln⁡(t)\ln(t)ln(t) growth for late times. The early time dynamics is captured numerically by performing simulations on spinless interacting fermions, using several numerically exact methods. Our results establish the validity of bosonization for the early time dynamics of 1D lattice models, and could be observed experimentally in bosonic Luttinger liquids.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.