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arXiv:1107.0230 · Physical Review B

Finite-size effects in transport data from quantum Monte Carlo simulations

Rubem Mondaini, Karim Bouadim, Thereza Paiva, Raimundo R. dos Santos

Published Mar 26, 2012• 27 citations
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Abstract

We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ sites. While the compressibility, the dc conductivity, and the gap are known to suffer from ``closed-shell'' effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant ($\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ($\ensuremath{\Delta}\ensuremath{\tau}$) on $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$, on the average density ($\ensuremath{\rho}$), and on the double occupancy ($d$): we have found that $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$ and $\ensuremath{\rho}$ are more strongly dependent on $\ensuremath{\Delta}\ensuremath{\tau}$ away from closed-shell configurations, but $d$ follows the $\ensuremath{\Delta}{\ensuremath{\tau}}^{2}$ dependence in both closed- and open-shell cases.

DOI
10.1103/physrevb.85.125127

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We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to 16\ifmmode×\else\texttimes\fi1616\ifmmode\times\else\texttimes\fi{}1616\ifmmode×\else\texttimes\fi16 sites. While the compressibility, the dc conductivity, and the gap are known to suffer from ``closed-shell'' effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant (\ensuremath⟨sign\ensuremath⟩\ensuremath{\langle}\text{sign}\ensuremath{\rangle}\ensuremath⟨sign\ensuremath⟩), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval (\ensuremathΔ\ensuremathτ\ensuremath{\Delta}\ensuremath{\tau}\ensuremathΔ\ensuremathτ) on \ensuremath⟨sign\ensuremath⟩\ensuremath{\langle}\text{sign}\ensuremath{\rangle}\ensuremath⟨sign\ensuremath⟩, on the average density (\ensuremathρ\ensuremath{\rho}\ensuremathρ), and on the double occupancy (ddd): we have found that \ensuremath⟨sign\ensuremath⟩\ensuremath{\langle}\text{sign}\ensuremath{\rangle}\ensuremath⟨sign\ensuremath⟩ and \ensuremathρ\ensuremath{\rho}\ensuremathρ are more strongly dependent on \ensuremathΔ\ensuremathτ\ensuremath{\Delta}\ensuremath{\tau}\ensuremathΔ\ensuremathτ away from closed-shell configurations, but ddd follows the \ensuremathΔ\ensuremathτ2\ensuremath{\Delta}{\ensuremath{\tau}}^{2}\ensuremathΔ\ensuremathτ2 dependence in both closed- and open-shell cases.
We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to 16 × 16 sites. While the compressibility, the dc conductivity, and the gap are known to suffer from “closed-shell” effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant (〈sign〉), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ( τ ) on 〈sign〉, on the average density (ρ), and on the double occupancy (d): we have found that 〈sign〉 and ρ are more strongly dependent on τ away from closed-shell configurations, but d follows the τ 2 dependence in both closedand open-shell cases.
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 study transport properties of the half-filled two-dimensional (2D) Hubbard model with spatially varying interactions, where a pattern of interacting and non-interacting sites is formed. We use Determinantal Quantum Monte Carlo method to calculate the double occupation, effective hopping and Drude weight. These data point to two phase transitions, driven by fermionic correlations. The first is the expected metal to a Mott insulating state. The second one, is an exotic transition from a Mott insulating state to a highly anisotropic metal, that takes place at large values of the fermion-fermion interaction. This second transition occurs when the layers formed by the spatially varying interactions decouple due to the suppression of the hopping between interacting and non-interacting sites, leading to fermionic transport along the non-interacting ones.
We use determinant quantum Monte Carlo simulations and exact diagonalization to explore insulating behavior in the Hubbard model with a bimodal distribution of randomly positioned local site energies. From the temperature dependence of the compressibility and conductivity, we show that gapped, incompressible Mott insulating phases exist away from half-filling when the variance of the local site energies is sufficiently large. The compressible regions around this Mott phase are metallic only if the density of sites with the corresponding energy exceeds the percolation threshold, but are Anderson insulators otherwise.
The Drude weight D and the dc conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) is finite for all T>0, displaying Fermi-liquid behavior, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$\ensuremath{\propto}1/T2{\mathit{T}}^{2}T2, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.
Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.