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arXiv:0902.0836 · Physical Review B

Spectral function of spinless fermions on a one-dimensional lattice

Rodrigo G. Pereira, Steven R. White, Ian K. Affleck

Published Apr 23, 2009• 119 citations
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Abstract

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.

DOI
10.1103/physrevb.79.165113

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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=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ is given by the two-spinon contribution.
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.
We calculate the exact spectral function of a single impurity repulsively interacting with a bath of fermions in one-dimensional lattices, by deriving the explicit expression of the form factor for both regular Bethe states and the irregular spin-flip state and η-pairing state, based on the exactly solvable one-dimensional Hubbard model. While at low impurity momentum Q∼0 the spectral function is dominated by two power-law Fermi singularities, at large momentum we observe that the two singularities develop into two-sided distributions and eventually become anomalous Fermi singularities at the boundary of the Brillouin zone (i.e., Q=±π), with the power-law tails extending toward low energy. Near the quarter filling of the Fermi bath, we also find two broad polaron peaks at large impurity momentum, collectively contributed by many excited many-body states with non-negligible form factors. Our exact results of those distinct features in one-dimensional Fermi polarons, which have no correspondences in two and three dimensions, could be readily probed in cold-atom laboratories by trapping highly imbalanced two-component fermionic atoms into one-dimensional optical lattices.
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 consider the one-dimensional gas of fermions with spin S interacting via an attractive δ -function potential using the Bethe Ansatz solution. In zero magnetic field the atoms form bound states of N = 2 S + 1 fermions, i.e. generalized Cooper states with each atom having a different spin component. For low energy excitations the system is a Luttinger liquid and is properly described by a conformal field theory with conformal charge c = 1 . The linear dispersion of a Luttinger liquid is asymptotically exact in the low-energy limit where the band curvature terms in the dispersion are irrelevant. For higher energy excitations, however, the spectral function displays deviations in the neighborhood of the single-particle (hole) energy, which can be described by an effective X-ray edge type model. Using the Bethe Ansatz solution we obtain expressions for the critical exponents for the single-particle (hole) Green's function. This model can be relevant in the context of ultracold atoms with effective total spin S confined to an elongated optical trap.
We study the charge dynamic structure factor of the one-dimensional Hubbard model with finite on-site repulsion UUU at half-filling. Numerical results from the time-dependent density matrix renormalization group are analyzed by comparison with the exact spectrum of the model. The evolution of the line shape as a function of UUU is explained in terms of a relative transfer of spectral weight between the two-holon continuum that dominates in the limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$ and a subset of the two-holon-two-spinon continuum that reconstructs the electron-hole continuum in the limit $U\ensuremath{\rightarrow}0$. Power-law singularities along boundary lines of the spectrum are described by effective impurity models that are explicitly invariant under spin and $\ensuremath{\eta}$-spin SU(2)\text{SU}(2)SU(2) rotations. The Mott-Hubbard metal-insulator transition is reflected in a discontinuous change of the exponents of edge singularities at U=0U=0U=0. The sharp feature observed in the spectrum for momenta near the zone boundary is attributed to a van Hove singularity that persists as a consequence of integrability.
The momentum, fermionic density, spin density, and interaction dependencies of the exponents that control the (k,ω)-plane singular features of the one-fermion spectral functions of a one-dimensional gas of spin-1/2 fermions with repulsive delta-function interaction both at zero and finite magnetic field are studied in detail. Our results refer to energy scales beyond the reach of the low-energy Tomonaga-Luttinger liquid and rely on the pseudofermion dynamical theory for integrable models. The one-fermion spectral weight distributions associated with the spectral functions studied in this paper may be observed in systems of spin-1/2 ultra-cold fermionic atoms in optical lattices.