Electrical conductivity in the Hubbard model: Orbital effects of magnetic field

Calculation of conductivity in the Hubbard model is a challenging task. Recent years have seen much progress in this respect and numerically exact solutions are now possible in certain regimes. In this paper we discuss the calculation of conductivity for the square-lattice Hubbard model in the presence of a perpendicular magnetic field, focusing on orbital effects. We present the relevant formalism in all detail and in full generality, and then discuss the simplifications that arise at the level of the dynamical mean field theory (DMFT). We prove that the Kubo bubble preserves gauge and translational invariance, and that in the DMFT the vertex corrections cancel regardless of the magnetic field. We present the DMFT results for the spectral function and both the longitudinal and Hall conductivities in several regimes of parameters. We analyze thoroughly the quantum oscillations of the longitudinal conductivity and identify a high-frequency oscillation component, arising as a combined effect of scattering and temperature, in line with recent experimental observations in moiré systems.

Figure 1

Real-space diagrammatic representation of a contribution to the connected part of the current-current correlation function. $F$ is the full vertex, red segments indicate a term in the current operator connecting two nearest-neighbor sites in the $x$ direction; swapping the direction of the red arrow changes its sign. Top panels: for generic case [left, Eq. (74)] and DMFT [right, Eq. (76)]. Bottom panels: after the transformation in Eqs. (75) (left) and (77) (right).

Figure 2

Noninteracting density of states, as a function of frequency and magnetic field (the Hofstadter butterfly [55]).

Figure 3

Spectrum of the translation-invariant Green's function $\mathrm{Im}{\overline{G}}_{\mathbf{k}}\left(\omega \right)$. The examples are given for $U=0$ with a broadening $\eta =0.02D$, and at three different values of $p/q$, as indicated in panel titles. The spectrum is not negative definite, and is therefore not indicative of the physical spectral function.

Figure 4

Local density of states as a function of magnetic field. Left: full frequency range. Right: quasiparticle part of the spectrum. The result is obtained with the DMFT(NRG). The parameters are $U=2.5D$, $\langle n_{\sigma }\rangle =0.425$, $T=0.025D/k_{\mathrm{B}}$. The calculation was performed with $L=q=997$.

Figure 5

Upper panel: conductivity obtained with the full DMFT(NRG) calculation (black line), and the simplified calculation where the self-energy is taken from the zero-field DMFT(NRG) calculation, and the chemical potential is either corrected to fix the overall density (lime line) or not (red line). Vertical lines indicate $p/q=1/i$ with $i$ integer, which coincides with the dips in the inverse conductivity. Lower panel: dependence on the magnetic field of the density of states at the Fermi level (lime line) and the scattering rate (black line). Vertical lines indicate $p/q=\langle n_{\sigma }\rangle /i$, which roughly coincides with the dips in both the scattering rate and the density of states at the Fermi level.

Figure 6

Cross-check between the zero-field formalism and the finite-field formalism at weak fields. Solid lines are the $B_z=0$ DMFT(NRG) result, obtained within the zero-field formalism, at different values of density. We keep $U=2.5D$ fixed. Dashed lines with symbols are obtained within finite $B_z$ formalism, at three lowest values of the field, at $L=q=1999$.

Figure 7

Left panels: the $T$ dependence of the inverse conductivity at different values of the field and fixed coupling and density. Right panel: the same results with rescaled temperature $T\to T/T^{*}\left(U\right)$. $T^{*}\left(U\right)$ is given in the inset.

Figure 8

Inverse conductivity on the linear scale. Columns are different dopings, rows are different values of interaction. Different curves are different values of the field.

Figure 9

Magnetic-field dependence of the inverse conductivity at two values of coupling constant, and various temperatures. Shaded circles correspond to the first extremum ${(p/q)}^{*}$ (see text).

Figure 10

Different components of the conductivity and resistivity tensors, showing the effect of the Hall component on the relation between the longitudinal resistivity ${\rho }^{xx}$ and the inverse longitudinal conductivity $1/{\sigma }^{xx}$.

Figure 11

Hall conductivity at $U=0$. Top: temperature dependence of the Hall conductivity for a selection of magnetic fields. Gray lines indicate the value at $T=0$, which is the Chern number for the corresponding topological insulator. Center: field dependence of the Hall conductivity at zero temperature. Bottom: chemical potential dependence of the Hall conductivity for a selection of magnetic fields.

Figure 12

Zoom-in on the oscillatory part of the magnetic-field dependence of inverse conductivity, in a range of weak fields. Different curves are different temperatures. Lime and black points denote the apparent antinodes of the wave, i.e., the amplitude of oscillation, up to a sign. Inset: dependence of the amplitude of oscillation vs temperature, on the logarithm scale, revealing exponential decay with $T$.

Figure 13

The $(B_z,T)$ phase diagram for the quantum oscillations. The diagram indicates the minimal magnetic field for observing significant quantum oscillations (nonmonotonic behavior) at a given temperature. Left: results for a set of model parameters. Right: rescaled results showing reasonable overlap.

Figure 14

Fourier spectra of the oscillatory component of the inverse dc conductivity. All spectra are normalized to 1. Left: temperature dependence at fixed electron density. Right: density dependence at fixed temperature. Both panels: vertical red dashed lines correspond to SdH frequency $p/q=\langle n_{\sigma }\rangle$ and its higher harmonics; vertical blue dashed lines correspond to $p/q=1$ oscillation frequency and its higher harmonics.

Figure 15

Oscillation spectra of the current vertex (as quantified by $Y$) at various $\mathrm{\Gamma },T$. Spectra are colored gray where no pronounced peaks are observed, red where the standard SdH $p/q=\langle n_{\sigma }\rangle$ peaks and its higher harmonics are observed, purple where we also observe the $p/q=1$ peaks, but the SdH peaks are dominant, and blue where $p/q=1$ peak is stronger than $p/q=\langle n_{\sigma }\rangle$ peak.

Figure 16

Evolution of the oscillation spectrum of the current vertex (as quantified by $Y$) with chemical potential, for two values of $\mathrm{\Gamma }$.

Figure 17

Longitudinal dc conductivity within FLA: the total result and the contributions from interband and intraband processes. Left: different plots correspond to different temperatures at a fixed scattering rate. Right: different plots correspond to different ${\mathrm{\Gamma }}^{\prime }\mathrm{s}$ at a fixed temperature.

Figure 18

(a) Oscillation spectra of longitudinal dc conductivity obtained within the FLA, at different $\mathrm{\Gamma }$ and $T$. Coloring is analogous to Fig. 15. (b) The phase diagram of the FLA toy model. Grayscale color coding in the background and the black contours correspond to the onset field for the nonmonotonous behavior ${(p/q)}^{*}$.

Figure 19

The doping-field dependence of longitudinal dc conductivity within FLA and DMFT. Color code is logarithmic: white is $-2.40$ and $-3.12$, respectively; black is 0.98 and 1.03, respectively.

Figure 20

Dependence of total free energy and its components on the magnetic field.

Figure 21

Benchmark with the data from Markov et al. [38]. Our data: DMFT(NRG solver). Reference data: DMFT(exact diagonalization solver with five bath sites)$+$Padé analytical continuation used to obtain continuous spectra.