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AIP conference proceedings

Collapse and Revival Starting from a Luttinger Liquid

Salvatore R. Manmana

Published Jan 1, 2006• 3 citations
Publisher

Abstract

We investigate the collapse and revival of the initial state of a system of spinless fermions on a one‐dimensional lattice with nearest‐neighbor Coulomb repulsion V and nearest‐neighbor hopping t at half filling which we push out of equilibrium. Starting from a Luttinger liquid phase, we suddenly increase the interaction to a value associated with an insulating ground state. Using a variant of the time‐dependent density matrix renormalization group method (DMRG) that approximates the time‐evolution operator within a Krylov subspace, we calculate the full time evolution of quantities like the momentum distribution 〈nk〉 and the local density 〈ni〉. In the atomic limit, t = 0, the density and the density‐density correlation function are time‐independent, while 〈nk〉 changes with time, showing that in out‐of‐equilibrium situations density‐density correlations and one‐particle propagators are no longer connected by a single anomalous dimension as in the equilibrium case. Since in this limit the initial state is reobtained after Trevival = 2π/(V/t), the collapse and revival of the initial Luttinger liquid equilibrium state is discussed. Preliminary results providing insight into the behavior in the thermodynamic limit for various parameter values with finite hopping amplitude t ≠ 0 are presented.

DOI
10.1063/1.2178043

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We investigate the collapse and revival of the initial state of a system of spinless fermions on a one‐dimensional lattice with nearest‐neighbor Coulomb repulsion V and nearest‐neighbor hopping t at half filling which we push out of equilibrium. Starting from a Luttinger liquid phase, we suddenly increase the interaction to a value associated with an insulating ground state. Using a variant of the time‐dependent density matrix renormalization group method (DMRG) that approximates the time‐evolution operator within a Krylov subspace, we calculate the full time evolution of quantities like the momentum distribution 〈nk〉 and the local density 〈ni〉. In the atomic limit, t = 0, the density and the density‐density correlation function are time‐independent, while 〈nk〉 changes with time, showing that in out‐of‐equilibrium situations density‐density correlations and one‐particle propagators are no longer connected by a single anomalous dimension as in the equilibrium case. Since in this limit the initial state is reobtained after Trevival = 2π/(V/t), the collapse and revival of the initial Luttinger liquid equilibrium state is discussed. Preliminary results providing insight into the behavior in the thermodynamic limit for various parameter values with finite hopping amplitude t ≠ 0 are presented.
We study one-dimensional spinless fermions at zero and finite temperature TTT using the density-matrix renormalization group. We consider nearest- as well as next-nearest-neighbor interactions; the latter render the system inaccessible by a Bethe ansatz treatment. Using an infinite-system algorithm we demonstrate the emergence of Luttinger liquid physics at low energies for a variety of static correlation functions as well as for thermodynamic properties. The characteristic power-law suppression of the momentum distribution n(k)n(k)n(k) function at T=0T=0T=0 can be directly observed over several orders of magnitude. At finite temperature, we show that n(k)n(k)n(k) obeys a scaling relation. The Luttinger liquid parameter and the renormalized Fermi velocity can be extracted from the density response function, the specific heat, or the susceptibility without the need to carry out any finite-size analysis. We illustrate that the energy scale below which Luttinger liquid power laws manifest vanishes as the half-filled system is driven into a gapped phase by large interactions.
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 time evolution of a one-dimensional system of strongly correlated electrons (a ``sample'') that is suddenly coupled to a smaller, initially empty system (a ``nanoprobe''), which can subsequently move along the system. Our purpose here is to study the role of interactions in this model system when it is far from equilibrium. We therefore take both the sample and the nanoprobe to be described by a Hubbard model with on-site repulsive interactions and nearest-neighbor hopping. We compute the behavior of the local particle density and the local density of states (LDOS) as a function of time using time-dependent matrix product states at quarter and at half filling, fillings at which the chain realizes a Luttinger liquid or a Mott insulator, respectively. This allows us to study in detail the oscillation of the particles between the sample and the nanoprobe. While, for noninteracting systems, the LDOS is time independent, in the presence of interactions, the backflow of electrons to the sample will lead to nontrivial dynamics in the LDOS. In particular, studying the time-dependent LDOS allows us to study how the Mott gap closes locally and how this melting of the Mott insulator propagates through the system in time after such a local perturbation---a behavior that we envisage can be investigated in future experiments on ultrashort time scales or on optical lattices using microscopy setups.
We investigate the local density of states of the one-dimensional half-filled spinless fermion model with nearest-neighbour hopping t > 0 and interaction V in its Luttinger liquid phase -2t < V ≤ 2t. The bulk density of states and the local density of states in open chains are calculated over the full band width ∼4t with an energy resolution ≤0.08t using the dynamical density-matrix renormalization group (DDMRG) method. We also perform DDMRG simulations with a resolution of 0.01t around the Fermi energy to reveal the power-law behaviour D(ϵ) ∼ |ϵ - ϵ(F)|(α) predicted by the Luttinger liquid theory for bulk and boundary density of states. The exponents α are determined using a finite-size scaling analysis of DDMRG data for lattices with up to 3200 sites. The results agree with the exact exponents given by the Luttinger liquid theory combined with the Bethe Ansatz solution. The crossover from boundary to bulk density of states is analysed. We have found that boundary effects can be seen in the local density of states at all energies, even far away from the chain edges.
One parametric one-dimensional spinless fermion models is formulated and solved exactly by the Bethe ansatz method. The Hamiltonian contains the kinetic terms describing nearest-neighbor hopping of electrons with different hopping integrals for many particle configurations. The Luttinger liquid behavior is studied for the case repulsion effective interactions. The ground state energy, the critical exponent, describing the asymptotic behavior of the density-density correlation function at long distances, the Fermi velocity have been calculated numerically for an arbitrary density of electrons. The model displays the condensation of particles in the case of the attractive interaction.
Using the density matrix renormalization group method, we systematically investigate the evolution of the Luttinger integral in the one-dimensional generalized ttt-VVV model as a function of filling and interaction strength, and identify three representative phases. In the weak-coupling regime, the zero-frequency Green's function exhibits a branch-cut structure at the Fermi momentum, and the Luttinger integral accurately reflects the particle density, indicating that the Luttinger theorem holds. As the interaction increases, the spectral weight near the Fermi momentum is gradually suppressed. Interestingly, in the strong coupling regime near half-filling, this singularity is progressively destroyed, accompanied by the emergence of momentum-space zeros in the real part of the Green's function, leading to a novel non-Fermi liquid metallic phase beyond the classic Luttinger liquid paradigm, where the Luttinger surface is no longer defined by a single singularity. While finite spectral weight remains at the original Fermi momentum, the singularity gradually diminishes. Meanwhile, zeros with negligible spectral weight appear away from this momentum, significantly affecting the integral. At exact half-filling, a single-particle gap opens, and the Green's function becomes nearly vanishing across the entire momentum space, indicating the complete suppression of low-energy electronic states consistent with the nature of an insulating charge-density-wave phase. These results suggest that the breakdown of the Luttinger theorem is not triggered by a single mechanism, but rather results from the interplay between interaction-driven evolution of excitation modes and the breaking of particle-hole symmetry, ultimately leading to a continuous reconstruction of the generalized Fermi surface from topologically protected to correlation-driven.