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Physical review. B, Condensed matter

Twisted boundary conditions in cluster calculations of the optical conductivity in two-dimensional lattice models

Didier Poilblanc

Published Nov 1, 1991• 137 citations
Publisher

Abstract

We investigate the extended t-J model (which may include a small next-nearest-neighbor antiferromagnetic coupling or a small hole-hole nearest-neighbor repulsion) by exact-diagonalization techniques. The calculation is done for one and two holes on a 4\ifmmode\times\else\texttimes\fi{}4 torus with arbitrary twisted boundary conditions. The role of the boundary conditions on the ground-state energy, the band width, the charge stiffness, and the optical conductivity is extensively studied. In the two-dimensional space of the twist angles (to be viewed as fluxes through the holes of the torus) the ground-state energy surface exhibits multiple level crossings and may present, locally, a negative curvature (or stiffness) characteristic of a paramagnetic behavior. For two holes and sufficiently large J/t (J/t>0.7) the minima of the energy surface in the parameter space of the fluxes are separated by half a flux quantum. The optical conductivity is calculated by averaging over the boundary conditions in order to minimize finite-size effects and to converge more rapidly toward the thermodynamic limit. We find three different contributions to the optical conductivity; (i) a Drude peak at \ensuremath{\omega}=0, (ii) a broad absorption band in the range J2t, and (iii) a tail extending to larger frequencies (2t7t) associated to the diffusive character of the hole states at these energies.Typically, at low doping density (6%) for a wide range of J/t, and at intermediate doping (12%) for small J/t, these three components account for roughly 30%, 50%, and 20% of the total spectral weight, respectively. At small ratio J/t, both Drude weight and finite-frequency absorption scale almost proportionally with the doping density. However, with increasing J/t, the effective hole-hole attraction (for 12% doping) leads to a transfer of weight from finite frequency to zero frequency. In contrast to the quasiparticle mass, in the J/t\ensuremath{\rightarrow}0 limit, the optical mass defined through the weight of the Drude \ensuremath{\delta}-function peak does not increase like \ensuremath{\sim}t/J but, rather, approaches a limit around \ensuremath{\sim}5. We discuss our results in connection with the optical experiments in the high-${\mathit{T}}_{\mathit{c}}$ cuprates and suggest that the observed mid-infrared band is due to the strong coupling between the mobile holes and the spin excitations of the Mott insulator, as expected from a description in terms of the t-J Hamiltonian.

DOI
10.1103/physrevb.44.9562

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Physical review. B, Condensed matter•Oct 1, 1991•164 citations
We investigate the extended t-J model (which may include a small next-nearest-neighbor antiferromagnetic coupling or a small hole-hole nearest-neighbor repulsion) by exact-diagonalization techniques. The calculation is done for one and two holes on a 4\ifmmode\times\else\texttimes\fi{}4 torus with arbitrary twisted boundary conditions. The role of the boundary conditions on the ground-state energy, the band width, the charge stiffness, and the optical conductivity is extensively studied. In the two-dimensional space of the twist angles (to be viewed as fluxes through the holes of the torus) the ground-state energy surface exhibits multiple level crossings and may present, locally, a negative curvature (or stiffness) characteristic of a paramagnetic behavior. For two holes and sufficiently large J/t (J/t>0.7) the minima of the energy surface in the parameter space of the fluxes are separated by half a flux quantum. The optical conductivity is calculated by averaging over the boundary conditions in order to minimize finite-size effects and to converge more rapidly toward the thermodynamic limit. We find three different contributions to the optical conductivity; (i) a Drude peak at \ensuremath{\omega}=0, (ii) a broad absorption band in the range J2t, and (iii) a tail extending to larger frequencies (2t7t) associated to the diffusive character of the hole states at these energies.Typically, at low doping density (6%) for a wide range of J/t, and at intermediate doping (12%) for small J/t, these three components account for roughly 30%, 50%, and 20% of the total spectral weight, respectively. At small ratio J/t, both Drude weight and finite-frequency absorption scale almost proportionally with the doping density. However, with increasing J/t, the effective hole-hole attraction (for 12% doping) leads to a transfer of weight from finite frequency to zero frequency. In contrast to the quasiparticle mass, in the J/t\ensuremath{\rightarrow}0 limit, the optical mass defined through the weight of the Drude \ensuremath{\delta}-function peak does not increase like \ensuremath{\sim}t/J but, rather, approaches a limit around \ensuremath{\sim}5. We discuss our results in connection with the optical experiments in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates and suggest that the observed mid-infrared band is due to the strong coupling between the mobile holes and the spin excitations of the Mott insulator, as expected from a description in terms of the t-J Hamiltonian.
Twisted boundary conditions are used to study the behavior of the optical mass in the two-dimensional t-J model. A 4\ifmmode\times\else\texttimes\fi{}4 torus with two holes is shown to exhibit a paramagnetic behavior under certain conditions for J/t0.7. However, it is shown that by averaging the charge stiffness over the test fluxes (i.e., the boundary conditions) one can reduce finite-size effects and extract the optical mass relevant to a macroscopic system. For small J/t, the optical mass appears to deviate from the (larger) quasiparticle effective mass defined by the quasiparticle dispersion relation. Our numerical work also reveals large weight in the optical conductivity at frequencies characteristic of the spin fluctuations. Finally, we note that no sign of flux quantization characteristic of pairing is observed in this torus geometry for J/t0.65.
The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.
We calculate the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of the Hubbard and t-J models on 10- and 16-site lattices, respectively, using a Lanczos method. Results are presented for various values of the couplings U/t, J/t, and band fillings, using free- and periodic-boundary conditions. We discuss the behavior of the kinetic energy in the ground state, the distribution of spectral weight of \ensuremath{\sigma}(\ensuremath{\omega}) between low and high energies and the sum rule. Indications of a Drude-like peak at low energy in the one-hole subspace are observed in both models. Assuming that holes behave like independent (mass-renormalized) particles in the normal state of the superconductors, our results also suggest that the optical midinfrared broadband observed experimentally can be explained by hole excitations of strongly correlated systems.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in the two-dimensional Hubbard model is examined by applying the exact diagonalization technique to small square clusters with periodic boundary conditions up to $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ sites. Spectral-weight distributions at half-filling and their doping dependence in the 20-site cluster are found to be similar to those in a $\sqrt{18}\ifmmode\times\else\texttimes\fi{}\sqrt{18}$ cluster, but different from $4\ifmmode\times\else\texttimes\fi{}4$ results. The results for the 20-site cluster enable us to perform a systematic study of the doping dependence of the spectral-weight transfer from the region of the Mott-gap excitation to lower-energy regions. We discuss the dependence of the Drude weight and the effective carrier number on the electron density at a large on-site Coulomb interaction.
The hole dynamics in a quantum antiferromagnetic background for a small doping concentration is studied within the t-J model description. The noncrossing approximation for the hole self-energy is used. The correct quasiparticle pole structure (without artificial broadening) in the spectral density function is employed. Dyson's equation for the hole Green's function is solved numerically by introducing a lattice in momentum space. The evolution of the density of states and the momentum-distribution function for holes is investigated as a function of hole-magnon coupling constant t at fixed J. The momentum distribution function shows a sharp drop at the Fermi surface for several different hole-magnon coupling constants. This indicates the validity of the Fermi liquid description for the t-J model in the low doping concentration. The volume (surface) enclosed by the Fermi surface (curve) is found to be invariant for all the coupling strengths we have studied. The incoherent spectrum below the chemical potential in the density of states becomes more significant for stronger interaction constants. These features numerically verify Luttinger's theorem for a small density of doped holes in the t-J model. The optical conductivity is computed using the lowest-order diagrams. As the hole-magnon coupling constant t increases, absorption becomes stronger at \ensuremath{\omega}\ensuremath{\simeq}2J, the characteristic frequency for spin waves. This feature may be associated with the experimentally observed midinfrared absorption band in underdoped cuprates.
We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}t2{\mathit{t}}^{2}t2/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.