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Physical review. B, Condensed matter

Optical mass in thet-Jmodel

Didier Poilblanc, Elbio R. Dagotto

Published Jul 1, 1991• 30 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.466

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

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A finite-size scaling analysis of the spectral function and of the optical conductivity of a single hole moving in an antiferromagnetic background is performed. It is shown that both the low-energy quasiparticle peak and the broad higher-energy structure are robust with increasing cluster size from 4\ifmmode\times\else\texttimes\fi{}4 to \ensuremath{\surd}26 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}26 sites. In the absence of spin fluctuations, for most static or dynamical quantities saturation occurs when the size exceeds a characteristic size ${\mathit{N}}_{\mathit{c}}$(${\mathit{J}}_{\mathit{z}}$). Typically, 16- and 26-site clusters give reliable results for ${\mathit{J}}_{\mathit{z}}$>0.75 and ${\mathit{J}}_{\mathit{z}}$>0.3, respectively. The hole optical mass is shown to be very large (>20) in agreement with the small bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a small gap \ensuremath{\propto}${\mathit{J}}_{\mathit{z}}$ separates the low-energy \ensuremath{\delta} function from the rest of the spectrum in the dynamical correlation functions. On the other hand, with ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ this gap seems to disappear with increasing system size as one would expect since the spin waves are gapless in the thermodynamic limit. The large momentum dependence of the quasiparticle weight in the isotropic case is inconsistent with a string picture but agrees well with the self-consistent Born approximation.An accurate estimation of the higher-energy part of the spectral functions of the t-J model can be made for momenta close to (0,0) or (\ensuremath{\pi},\ensuremath{\pi}); it consists of a broad featureless band. For momenta close to the Fermi surface, no evidence for excited string states is found. This may be either because the string scenario breaks down or because exact calculations at momenta near (\ensuremath{\pi}/2,\ensuremath{\pi}/2) are limited by the differing geometries of the finite clusters. In the Ising limit, the spectral function also depends crucially on the momentum, but a feature resembling a string excitation is seen for momenta close to (0,0). Our data on the finite-frequency optical conductivity of a single hole in the t-J model reveal that the total spectral weight at finite frequency has a strong size dependence and decreases with increasing size (for J not too small). This suggests that the optical mass and the optical absorption of the t-J model may have been overestimated in the past.

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Physical review. B, Condensed matter•Sep 1, 1990•100 citations
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.
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.
The behavior of the Drude weight is studied in the small-J/t regime of the t-J model by Lanczos diagonalizations of small clusters with arbitrary boundary conditions and a small number of holes. Although the total spin of the ground state in the J/t\ensuremath{\rightarrow}0 limit depends strongly on the boundary conditions, the Drude weight exhibits a rather universal behavior and indicates a sharp transition around J/t\ensuremath{\sim}0.1 for a hole concentration of \ensuremath{\sim}10%. This crossover is interpreted as a paramagnetic-ferromagnetic transition. For J/t larger than the crossover value, the Drude weight is somehow constant (and reduced by a factor \ensuremath{\sim}3 compared to the case of free holes), suggesting that coherent transport is almost insensitive to the variation of the antiferromagnetic correlation length. At small J/t (and finite doping), the exact nature of the ferromagnetic phase is still uncertain, since both fully polarized state or spin-polaron gas are found in small clusters depending on the boundaries.
The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​ and the incoherent hopping ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​\ensuremath{\sim}ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.
We study contributions to the spin-exchange scattering amplitude at two-loop (1/N2{\mathit{N}}^{2}N2) order for the metallic phase of the two-dimensional t-J model, for which a well-defined 1/N expansion is possible. We find for small hole dopings and small exchange couplings (t\ensuremath{\gg}J) a positive spin-exchange Landau parameter, which suggests that the spin-density fluctuations show a well-defined collective mode near the metal-insulator filling. This behavior results from the fact that the chirality fluctuations decouple from and are suppressed relative to the charge fluctuations for large mass enhancements. The result is interpreted as paralleling the collective spin and charge modes characteristic of the Luttinger liquid for low wave vectors.
A finite-size scaling analysis of the spectral function and of the optical conductivity of a single hole moving in an antiferromagnetic background is performed. It is shown that both the low-energy quasiparticle peak and the broad higher-energy structure are robust with increasing cluster size from 4\ifmmode\times\else\texttimes\fi{}4 to \ensuremath{\surd}26 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}26 sites. In the absence of spin fluctuations, for most static or dynamical quantities saturation occurs when the size exceeds a characteristic size Nc{\mathit{N}}_{\mathit{c}}Nc​(Jz{\mathit{J}}_{\mathit{z}}Jz​). Typically, 16- and 26-site clusters give reliable results for Jz{\mathit{J}}_{\mathit{z}}Jz​>0.75 and Jz{\mathit{J}}_{\mathit{z}}Jz​>0.3, respectively. The hole optical mass is shown to be very large (>20) in agreement with the small bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a small gap \ensuremath{\propto}Jz{\mathit{J}}_{\mathit{z}}Jz​ separates the low-energy \ensuremath{\delta} function from the rest of the spectrum in the dynamical correlation functions. On the other hand, with ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ this gap seems to disappear with increasing system size as one would expect since the spin waves are gapless in the thermodynamic limit. The large momentum dependence of the quasiparticle weight in the isotropic case is inconsistent with a string picture but agrees well with the self-consistent Born approximation.An accurate estimation of the higher-energy part of the spectral functions of the t-J model can be made for momenta close to (0,0) or (\ensuremath{\pi},\ensuremath{\pi}); it consists of a broad featureless band. For momenta close to the Fermi surface, no evidence for excited string states is found. This may be either because the string scenario breaks down or because exact calculations at momenta near (\ensuremath{\pi}/2,\ensuremath{\pi}/2) are limited by the differing geometries of the finite clusters. In the Ising limit, the spectral function also depends crucially on the momentum, but a feature resembling a string excitation is seen for momenta close to (0,0). Our data on the finite-frequency optical conductivity of a single hole in the t-J model reveal that the total spectral weight at finite frequency has a strong size dependence and decreases with increasing size (for J not too small). This suggests that the optical mass and the optical absorption of the t-J model may have been overestimated in the past.
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.