Universal Magnetic Oscillations of dc Conductivity in the Incoherent Regime of Correlated Systems

Using the dynamical mean field theory we investigate the magnetic field dependence of dc conductivity in the Hubbard model on the square lattice, fully taking into account the orbital effects of the field introduced via the Peierls substitution. In addition to the conventional Shubnikov–de Haas quantum oscillations, associated with the coherent cyclotron motion of quasiparticles and the presence of a well-defined Fermi surface, we find an additional oscillatory component with a higher frequency that corresponds to the total area of the Brillouin zone. These paradigm-breaking oscillations appear at elevated temperature. This finding is in excellent qualitative agreement with the recent experiments on graphene superlattices. We elucidate the key roles of the off-diagonal elements of the current vertex and the incoherence of electronic states, and explain the trends with respect to temperature and doping.

Figure 1

DMFT results for conductivity in the Hubbard model for $U=1D$. (a) Conductivity as a function of temperature and field at band filling $n_{\sigma }=0.4$. Color code is logarithmic: black means ${\log }_{10}{\sigma }_{\mathrm{dc}}^{xx}\approx -7.95$, white means ${\log }_{10}{\sigma }_{\mathrm{dc}}^{xx}\approx 2.12$. White line: onset points of the nonmonotonic behavior of ${\sigma }_{\mathrm{dc}}^{xx}(B){|}_T$. (b) Conductivity as a function of inverse magnetic field. Bottom to top: $T=0.0012,0.0049,0.0109,0.024D$; lines are plotted on the log scale, and offset for the sake of clarity. (c) Frequency spectrum of conductivity in the range $p/q\in [0.03,0.15]$ at different temperatures. Bottom to top: $T=0.001,0.009,0.016,0.029,0.064D$. Each spectrum is normalized to 1 and shifted for the sake of clarity. (b),(c) Vertical lines: peaks due to SdH oscillations (red), and BZ oscillations (blue). (d)–(f) Conductivity with respect to band filling and field at $T=0.005,0.03,0.1D$, respectively. Color code: white means $-8.22$, $-3.57$, $-3.12$, black means 2.14, 1.77, 1.03, respectively.

Figure 2

Phase diagrams showing the type of QOs observed in the range of field $p/q\in [0.03,0.15]$. (a) DMFT results in $(\delta ,T)$ plane, (b) DMFT results in $(U,T)$ plane, (c) FLA results in $(\mathrm{\Gamma },T)$ plane. Red: SdH only. Purple: both SdH and BZ, but SdH dominant. Blue: BZ dominant ($p/q=1$ peak stronger than $p/q\approx n_{\sigma }$ peak). Black shading and contours in (a),(b) denote the value of the field where nonmonotonic behavior starts in $1/{\sigma }_{\mathrm{dc}}^{xx}(B){|}_T$ (analogous to the white line in Fig. 1). Above the lime dashed line, no oscillations are detectable at any field strength. In (c), lines and symbols correspond to DMFT results, shading to FLA results. Lines are $\mathrm{\Gamma }(T)$ for various values of $U$. Purple squares indicate where the BZ oscillations start with increasing $T$, blue diamonds where the BZ becomes dominant, and lime circles where all QOs cease [corresponding to the top edge of blue and purple regions in (b)].

Figure 3

(a) Diagrammatic representation of the Kubo bubble. Left: in general; right: at the level of the DMFT. $(\tilde{\mathbf{k}},m)$ denotes eigenstates of the noninteracting Hamiltonian (see [26] for details). Red or lime triangles are the velocity vertices, in DMFT rewritten as a single factor depending on two kinetic energies, $v(ϵ,{ϵ}^{\prime })$. (b),(c) White panels: oscillation spectra of $v(ϵ,{ϵ}^{\prime })$ at a given $(ϵ,{ϵ}^{\prime })$. Gray panels: oscillation spectra of $v$ integrated over the relevant $(ϵ,{ϵ}^{\prime })$ domain, depending on model parameters ($T$ and $\mathrm{\Gamma }$), as indicated by the large curly bracket; (b) trend with respect to temperature. (c) Trend with respect to the scattering rate. (d) Field dependence of conductivity and the contributions of interband ($\epsilon \ne {\epsilon }^{\prime }$) and intraband ($\epsilon \approx {\epsilon }^{\prime }$) processes in FLA in four different parameter regimes.