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arXiv:2002.12289 · npj Quantum Materials

DC Hall coefficient of the strongly correlated Hubbard model

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

Published Jul 24, 2020• 24 citations
arXivHTMLPDF

Abstract

Abstract The Hall coefficient is related to the effective carrier density and Fermi surface topology in non-interacting and weakly interacting systems. In strongly correlated systems, the relation between the Hall coefficient and single-particle properties is less clear. Clarifying this relation would give insight into the nature of transport in strongly correlated materials that lack well-formed quasiparticles. In this work, we investigate the DC Hall coefficient of the Hubbard model using determinant quantum Monte Carlo in conjunction with a recently developed expansion of magneto-transport coefficients in terms of thermodynamic susceptibilities. At leading order in the expansion, we observe a change of sign in the Hall coefficient as a function of temperature and interaction strength, which we relate to a change in the topology of the apparent Fermi surface. We also combine our Hall coefficient results with optical conductivity values to evaluate the Hall angle, as well as effective mobility and effective mass based on Drude theory of metals.

DOI
10.1038/s41535-020-00254-w

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Abstract The Hall coefficient is related to the effective carrier density and Fermi surface topology in non-interacting and weakly interacting systems. In strongly correlated systems, the relation between the Hall coefficient and single-particle properties is less clear. Clarifying this relation would give insight into the nature of transport in strongly correlated materials that lack well-formed quasiparticles. In this work, we investigate the DC Hall coefficient of the Hubbard model using determinant quantum Monte Carlo in conjunction with a recently developed expansion of magneto-transport coefficients in terms of thermodynamic susceptibilities. At leading order in the expansion, we observe a change of sign in the Hall coefficient as a function of temperature and interaction strength, which we relate to a change in the topology of the apparent Fermi surface. We also combine our Hall coefficient results with optical conductivity values to evaluate the Hall angle, as well as effective mobility and effective mass based on Drude theory of metals.
Using determinant quantum Monte Carlo, we compare three methods of evaluating the dc Hall coefficient RH{R}_{H}RH​ 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 RH(0){R}_{H}^{(0)}RH(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 RH{R}_{H}RH​ constructed from the three-current correlator ${\ensuremath{\chi}}_{xy}^{\text{ZF}}$ can be determined under different simplifying assumptions and compared with RH(0){R}_{H}^{(0)}RH(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 RH{R}_{H}RH​, 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.
Exact formulas for the optical conductivity and the Hall conductivity of the two dimensional Hubbard model are derived in terms of an effective theory description of the local moment fluctuation in the system. In this framework, the quantum Monte Carlo simulation of the electromagnetic response of such a strongly correlated electron system becomes sign-problem-free in many physically relevant cases. In particular, it is sign-problem-free when we assume the widely used Millis-Monien-Pines form for the phenomenological susceptibility in the effective action of the fluctuating local moment, even though these local moments are now subjected to Landau damping as a result of their coupling to the itinerant quasiparticle on the fermi surface. This is true more generally when a φ4φ^{4}φ4 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 σxx(ω)σ^{xx}(ω)σxx(ω) and the Hall conductivity σxy(ω)σ^{xy}(ω)σxy(ω) of the cuprate superconductors. Both σxx(ω)σ^{xx}(ω)σxx(ω) and σxy(ω)σ^{xy}(ω)σxy(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 Imσxy(ω)\mathrm{Im}σ^{xy}(ω)Imσxy(ω) can be either positive or negative.(full-length abstract can be found in the main text.)
A path-integral field-theoretic derivation of electromagnetic linear response for the two-dimensional Hubbard model is given. We find, remarkably, that changes in the Fermi-surface topology associated with incommensurate planar spin-density-wave saddle points induce a change in sign of the Hall coefficient at dopings ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$=0.02--0.5 for U/t=2--10. The change in sign is not affected by short-range magnetic domains. We delinate from first principles an anomalous temperature dependence of the Hall carrier density at dopings close to ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$. An additional anisotropic component to the usual dc conductivity is nonvanishing for certain types of spirals. The paper extends the Bloch-Boltzmann theory to the case of untraditional Fermi liquids where the damping of the quasiparticles is \ensuremath{\Gamma}(\ensuremath{\epsilon})\ensuremath{\sim}max(kB{\mathit{k}}_{\mathit{B}}kB​T,\ensuremath{\epsilon}).
Abstract The Hall coefficient of the strongly interacting square lattice Hubbard model is calculated for temperatures between the antiferromagnetic interaction and the Mott gap scales. The leading order thermodynamic formula is evaluated for all doping concentrations. Second-order corrections of the thermodynamic formula are calculated and found to be negligible. The Hall coefficient diverges toward the Mott insulator. Below 45% doping the Hall sign is reversed relative to band structure-based Boltzmann’s equation. These results elucidate the effects of the Mott insulator on the charge carriers and their non-Fermi liquid transport.
The Hall coefficient exhibits anomalous behavior in lightly doped Mott insulators. For strongly interacting electrons its computation has been challenged by analytical and numerical obstacles. We calculate the leading contributions in the recently derived thermodynamic formula for the Hall coefficient. We obtain its doping and temperature dependence for the square lattice tJ-model at high temperatures. The second order corrections are evaluated to be negligible. Quantum Monte Carlo sampling extends our results to lower temperatures. We find a divergence of the Hall coefficient toward the Mott limit and a sign reversal relative to Boltzmann equation's weak scattering prediction. The Hall current near the Mott phase is carried by a low density of spin-entangled vacancies, which should constitute the Cooper pairs in any superconducting phase at lower temperatures.