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arXiv:1002.3032 · Physical Review B

Density-matrix renormalization-group study of coupled Luttinger liquids

S. Moukouri, Eitan Eidelstein

Published Oct 29, 2010• 2 citations
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Abstract

We report a two-step density-matrix renormalization-group computation of the equal-time single-particle Green's function, the density-density correlations, and the low-frequency spectral weight function of a spinless fermion model in an anisotropic two-dimensional lattice at half filling. We find that at weak couplings the density-density correlations have the universal decay of a Fermi liquid; the spectral weight function displays a sharp quasiparticle peak. But in the vicinity of a quantum critical point, these correlations strongly deviate from a Fermi-liquid prediction and a pseudogap opens in the spectral weight function.

DOI
10.1103/physrevb.82.165132

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We report a two-step density-matrix renormalization-group computation of the equal-time single-particle Green's function, the density-density correlations, and the low-frequency spectral weight function of a spinless fermion model in an anisotropic two-dimensional lattice at half filling. We find that at weak couplings the density-density correlations have the universal decay of a Fermi liquid; the spectral weight function displays a sharp quasiparticle peak. But in the vicinity of a quantum critical point, these correlations strongly deviate from a Fermi-liquid prediction and a pseudogap opens in the spectral weight function.
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.
We study the spectral function of interacting one-dimensional fermions for an integrable lattice model away from half-filling. The divergent power-law singularity of the spectral function near the single-particle or single-hole energy is described by an effective x-ray edge type model. At low densities and for momentum near the zone boundary, we find a second divergent singularity at higher energies which is associated with a two-particle bound state. We use the Bethe ansatz solution of the model to calculate the exact singularity exponents for any momentum and for arbitrary values of chemical potential and interaction strength in the critical regime. We relate the singularities of the spectral function to the long-time decay of the fermion Green's function and compare our predictions with numerical results from the time-dependent density-matrix renormalization group (tDMRG). Our results show that the tDMRG method is able to provide accurate time decay exponents in the cases of power-law decay of the Green's function. Some implications for the line shape of the dynamical structure factor away from half-filling are also discussed. In addition, we show that 85% of the total spectral weight in the zero field Heisenberg model near wave-vector $q=\ensuremath{\pi}$ is given by the two-spinon contribution.
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 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.
We present a class of one-dimensional, generic, spinless fermion lattice Hamiltonians that express quasi-Fermi liquid physics, manifesting both Luttinger and Fermi liquid features solely due to irrelevant interactions. Using infinite matrix product state techniques, we unveil its universal structure by calculating static and dynamic responses. Key features include a finite discontinuity in the momentum distribution at the Fermi level, despite power-law singularities in the spectral function protected by particle-hole symmetry. Away from half filling, Landau quasiparticles emerge. Charge dynamics show either high-energy bound states or a concentration of spectral weight within the continuum for attractive or repulsive interactions, respectively. These universal features are realized across multiple models and energy scales, thus reifying the quasi-Fermi liquid as a unique paradigm for one-dimensional fermions.