arXiv:2103.04998 · Physical Review Research

Numerical approaches for calculating the low-field dc Hall coefficient of the doped Hubbard model

Wen O. Wang, Jixun K. Ding, Brian Moritz, Yoni Schattner, Edwin W. Huang, Thomas Devereaux

Published Jul 9, 2021• 10 citations

Abstract

Using determinant quantum Monte Carlo, we compare three methods of evaluating the dc Hall coefficient ${R}_{H}$ of the Hubbard model: the direct measurement of the off-diagonal current-current correlator ${\ensuremath{\chi}}_{xy}$ in a system coupled to a finite magnetic field (FF), ${\ensuremath{\chi}}_{xy}^{\text{FF}}$; the three-current linear response to an infinitesimal field as measured in the zero-field (ZF) Hubbard Hamiltonian, ${\ensuremath{\chi}}_{xy}^{\text{ZF}}$; and the leading order of the recurrent expansion ${R}_{H}^{(0)}$ in terms of thermodynamic susceptibilities. The two quantities ${\ensuremath{\chi}}_{xy}^{\text{FF}}$ and ${\ensuremath{\chi}}_{xy}^{\text{ZF}}$ can be compared directly in imaginary time. Proxies for ${R}_{H}$ constructed from the three-current correlator ${\ensuremath{\chi}}_{xy}^{\text{ZF}}$ can be determined under different simplifying assumptions and compared with ${R}_{H}^{(0)}$. We find these different quantities to be consistent with one another, validating previous conclusions about the close correspondence between Fermi surface topology and the sign of ${R}_{H}$, even for strongly correlated systems. These various quantities also provide a useful set of numerical tools for testing theoretical predictions about the full behavior of the Hall conductivity for strong correlations.

DOI: 10.1103/physrevresearch.3.033033

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arXiv:2103.04998 · Physical Review Research

Numerical approaches for calculating the low-field dc Hall coefficient of the doped Hubbard model

Wen O. Wang, Jixun K. Ding, Brian Moritz, Yoni Schattner, Edwin W. Huang, Thomas Devereaux

Published Jul 9, 2021• 10 citations

arXiv HTML PDF

Abstract

Using determinant quantum Monte Carlo, we compare three methods of evaluating the dc Hall coefficient

{R}_{H}

of the Hubbard model: the direct measurement of the off-diagonal current-current correlator

{\ensuremath{\chi}}_{xy}

in a system coupled to a finite magnetic field (FF),

{\ensuremath{\chi}}_{xy}^{\text{FF}}

; the three-current linear response to an infinitesimal field as measured in the zero-field (ZF) Hubbard Hamiltonian,

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

; and the leading order of the recurrent expansion

{R}_{H}^{(0)}

in terms of thermodynamic susceptibilities. The two quantities

{\ensuremath{\chi}}_{xy}^{\text{FF}}

and

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

can be compared directly in imaginary time. Proxies for

{R}_{H}

constructed from the three-current correlator

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

can be determined under different simplifying assumptions and compared with

{R}_{H}^{(0)}

. We find these different quantities to be consistent with one another, validating previous conclusions about the close correspondence between Fermi surface topology and the sign of

{R}_{H}

, even for strongly correlated systems. These various quantities also provide a useful set of numerical tools for testing theoretical predictions about the full behavior of the Hall conductivity for strong correlations.

DOI: 10.1103/physrevresearch.3.033033

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DC Hall coefficient of the strongly correlated Hubbard model

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Fedor Šimkovic, Riccardo Rossi, Michel Ferrero

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arXiv:2110.05863Physical Review ResearchDec 20, 202245 citations

Using determinant quantum Monte Carlo, we compare three methods of evaluating the dc Hall coefficient

{R}_{H}

of the Hubbard model: the direct measurement of the off-diagonal current-current correlator

{\ensuremath{\chi}}_{xy}

in a system coupled to a finite magnetic field (FF),

{\ensuremath{\chi}}_{xy}^{\text{FF}}

; the three-current linear response to an infinitesimal field as measured in the zero-field (ZF) Hubbard Hamiltonian,

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

; and the leading order of the recurrent expansion

{R}_{H}^{(0)}

in terms of thermodynamic susceptibilities. The two quantities

{\ensuremath{\chi}}_{xy}^{\text{FF}}

and

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

can be compared directly in imaginary time. Proxies for

{R}_{H}

constructed from the three-current correlator

{\ensuremath{\chi}}_{xy}^{\text{ZF}}

can be determined under different simplifying assumptions and compared with

{R}_{H}^{(0)}

. We find these different quantities to be consistent with one another, validating previous conclusions about the close correspondence between Fermi surface topology and the sign of

{R}_{H}

φ^{4}

term is included in the effective action and is thus not restricted to the Gaussian limit. Here we demonstrate the power of this framework by studying the effect of thermal fluctuation of the local moment on the optical conductivity

σ^{xx}(ω)

and the Hall conductivity

σ^{xy}(ω)

of the cuprate superconductors. Both

σ^{xx}(ω)

and

σ^{xy}(ω)

calculated are found to exhibit a two-component structure, with a Drude component at low energy and a mid-infrared component at higher energy. Depending on the relative importance of the hole pocket and the electron pocket on the reconstructed fermi surface and the coupling strength to the local moment, the Drude component in

\mathrm{Im}σ^{xy}(ω)

can be either positive or negative.(full-length abstract can be found in the main text.)

{\mathit{k}}_{\mathit{B}}

T,\ensuremath{\epsilon}).

K_ρ

. At sufficiently high temperature or frequency the Hall coefficient approaches the band value. We discuss the applications to Hall experiments made in quasi 1D organic compounds. In the second model, the complete temperature and doping dependencies of the high-frequency Hall coefficient $\RH$ are evaluated analytically and numerically for small, intermediate, and strong interactions using various approximation schemes. We find that $\RH$ follows the semiclassical

1/qn^*

law near T=0, but exhibits a striking

T

-linear behavior with an interaction- and doping-dependent slope at high temperature. We compare our results with Hall measurements performed in cobaltates.

T=0.067

and interactions up to

U=7

, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.