From Dia- to Paramagnetic Orbital Susceptibility of Massless Fermions

We study the orbital susceptibility of multiband systems with a pair of Dirac points interpolating between honeycomb and dice lattices. Despite having the same zero-field energy spectrum, these different systems exhibit spectacular differences in their orbital magnetic response, ranging from dia- to paramagnetism at Dirac points. We show that this striking behavior is related to a topological Berry phase varying continuously from $\pi$ (graphene) to 0 (dice). The latter strongly constrains interband effects, resulting in an unusual dependence of the magnetic response also at finite doping.

Figure 1

$\alpha \text{−}{\mathcal{T}}_3$ model. (a) ${\mathcal{T}}_3$ lattice. Thick links: nearest neighbors hoppings $t$ between $A$ and $B$ sites forming a honeycomb lattice. Thin links: additional hoppings $\alpha t$ connecting $C$ to $B$ sites. Varying $\alpha$ interpolates between the honeycomb lattice ($\alpha =0$) and the dice lattice ($\alpha =1$). (b) Zero field energy spectrum as a function of the wave vector $\mathit{k}$ for all $\alpha$.

Figure 2

(a) Numerically obtained orbital susceptibility $\chi$ [in units of the Landau band edge value $|{\chi }_L|=(1/16\pi )(e^{2}t{a}^2/{\hslash }^2)$] as a function of the chemical potential $\mu$ (in units of $t$) in the whole band for various $\alpha$ as indicated and for a temperature $T=0.02t$. Because of a sum rule discussed below, the orbital response at zero doping is systematically compensated by an opposite response at finite doping. (b) Susceptibility peak at zero chemical potential as a function of the inverse temperature for various $\alpha$ [same units as in (a)]. The slope is well fitted by $-3{\zeta }_{\gamma }(2)/\zeta (2)$, in agreement with Eq. (11).

Figure 3

(a) Landau level spectrum ${\epsilon }_n$ (each color corresponding to one valley) near zero energy as a function of the parameter $\alpha$ from graphene ($\alpha =0$) to dice ($\alpha =1$). (b) Function $w_{\alpha }(x)$ giving the dependence of the grand potential $\delta {\mathrm{\Omega }}_{\alpha }$ (at finite magnetic field and zero temperature) on the chemical potential $\mu$, see Eq. (8). Blue: graphene ($\alpha =0$). Red: dice lattice ($\alpha =1$). Black: critical case (${\alpha }_c=0.49065$).

Figure 4

(a) Field dependent part of the grand potential (per unit area) divided by the magnetic field $\delta {\mathrm{\Omega }}_{\alpha }/B$ (in units of $et/h$), at zero doping $\mu =0$ and temperature $T=0$, as a function of the magnetic field $B$ (in units of $(4\pi /\sqrt{3})(\hslash /e{a}^2)$) for various $\alpha$ as indicated. Dots are obtained numerically from the Hofstader spectrum on the corresponding lattice: $\delta {\mathrm{\Omega }}_{\alpha }/B$ is well fitted by a dependence $-(2/3)C_{\alpha }\sqrt{B}-(1/2)D_{\alpha }B$, corresponding to a magnetization ${\mathcal{M}}_{\alpha }=C_{\alpha }\sqrt{B}+D_{\alpha }B$. (b) Dimensionless parameters $C_{\alpha }/|C_0|$ and $D_{\alpha }/|D_0|$, obtained from the fit, as a function of $\alpha$, with $C_0=-(9\zeta (3/2)/8{\pi }^2\sqrt{2})(e^{3/2}ta/{\hslash }^{3/2})$ and $D_0\approx 0.094(e^{2}t{a}^2/{\hslash }^2)\approx 4.7|{\chi }_L|$. Dots are numerical results and the full line is the analytical prediction Eqs. (9) and (10).

Figure 5

Coalescence of the zero field spectrum (lower band near zero energy) into Landau levels ${\epsilon }_n$. (a) Graphene ($\alpha =0$): ${\epsilon }_n=-\sqrt{|n|}{\epsilon }_B$; $E_n=-\sqrt{n+1/2}{\epsilon }_B$ with $n\ge 0$. (b) Dice ($\alpha =1$): ${\epsilon }_n=-\sqrt{|n+1/2|}{\epsilon }_B$; $E_n=-\sqrt{n+1}{\epsilon }_B$ with $n\ge 0$. The area of each colored slice $[E_{n-1},E_n]$ counts the total number of states in this slice in zero field. In a field, these states condense into LL.