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arXiv:1207.0011 · Physical Review B

Luttinger liquid physics from the infinite-system density matrix renormalization group

Christoph Karrasch, Joel E. Moore

Published Oct 31, 2012• 33 citations
arXivHTMLPDF

Abstract

We study one-dimensional spinless fermions at zero and finite temperature $T$ 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 $n(k)$ function at $T=0$ can be directly observed over several orders of magnitude. At finite temperature, we show that $n(k)$ 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.

DOI
10.1103/physrevb.86.155156

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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.
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We present time-dependent density-matrix renormalization-group (DMRG) results for strongly interacting one-dimensional fermionic systems at finite temperature. When interactions are strong the characteristic spin energy can be greatly suppressed relative to the characteristic charge energy, allowing for the possibility of spin-incoherent Luttinger liquid physics when the temperature is high compared to the spin energy, but small compared to the charge energy. Using DMRG we compute the spectral properties of the $t\text{\ensuremath{-}}J$ model at arbitrary temperatures with respect to both spin and charge energies. We study the full crossover from the Luttinger liquid regime to the spin-incoherent regime, focusing on small J/tJ/tJ/t, where the signatures of spin-incoherent behavior are more manifest. Our method allows us to access the analytically intractable regime where temperature is on the order of the spin energy, $T\ensuremath{\sim}J$. Our results should be helpful in the interpretation of experiments that may be in the crossover regime, $T\ensuremath{\sim}J$, and apply to one-dimensional cold atomic gases where finite-temperature effects are appreciable. The technique may also be used to guide the development of analytical approximations for the crossover regime.
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 investigate the phase diagrams of a one-dimensional lattice model of fermions and of a spin chain with interactions extending up to next-nearest-neighbor range.In particular, we investigate the appearance of regions with dominant pairing physics in the presence of nearest-neighbor and next-nearest-neighbor interactions.Our analysis is based on analytical calculations in the classical limit, bosonization techniques and large-scale density-matrix renormalization group numerical simulations.The phase diagram, which is investigated in all relevant filling regimes, displays a remarkably rich collection of phases, including Luttinger liquids, phase separation, charge-density waves, bond-order phases, and exotic cluster Luttinger liquids with paired particles.In relation with recent studies, we show several emergent transition lines with a central charge c = 3/2 between the Luttinger liquid and the cluster Luttinger liquid phases.These results could be experimentally investigated using highly tunable quantum simulators.