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 $\chi_{xy}$ in a system coupled to a finite magnetic field (FF), $\chi_{xy}^{\text{FF}}$; the three-current linear response to an infinitesimal field as measured in the zero-field (ZF) Hubbard Hamiltonian, $\chi_{xy}^{\text{ZF}}$; and the leading order of the recurrent expansion $R_H^{(0)}$ in terms of thermodynamic susceptibilities. The two quantities $\chi_{xy}^{\text{FF}}$ and $\chi_{xy}^{\text{ZF}}$ can be compared directly in imaginary time. Proxies for $R_H$ constructed from the three-current correlator $\chi_{xy}^{\text{ZF}}$ 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.
Highlights
Transport measurements are among the most common and accessible experimental probes, and are often among the first to be performed following the discovery of new materials
In previous work [15], we investigated the dc Hall coefficient RH of the Hubbard model using determinant quantum Monte Carlo (DQMC) to evaluate the leading order of the recurrent expansion RH(0), showing a strong temperature dependence—increasing with decreasing temperature—mimicking the behavior seen in cuprates [16]
We verified that the small discrepancies between ZF and finite magnetic field (FF) results are reduced for larger lattice sizes, as shown in Fig. 6 of Appendix C
Summary
Transport measurements are among the most common and accessible experimental probes, and are often among the first to be performed following the discovery of new materials. Current-current correlators χαβ (τ ) (α, β = x or y direction) measured in imaginary time are analytically continued to real frequency to obtain all components of the conductivity tensor σαβ (ω) In this approach, explicitly adding a magnetic field B raises the computational complexity by requiring complex (as opposed to real) calculations. One could consider the zerofield limit by expanding the off-diagonal part of χαβ up to linear terms in B This method still requires analytic continuation, but avoids measurements in a finite field. In previous work [15], we investigated the dc Hall coefficient RH of the Hubbard model using DQMC to evaluate the leading order of the recurrent expansion RH(0), showing a strong temperature dependence—increasing with decreasing temperature—mimicking the behavior seen in cuprates [16]. We close with a discussion of our results and the challenges that remain for an evaluation of the full frequency dependence of the conductivities in the Hubbard model in a magnetic field
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