Geometric orbital susceptibility: Quantum metric without Berry curvature

The orbital magnetic susceptibility of an electron gas in a periodic potential depends not only on the zero field energy spectrum but also on the geometric structure of cell-periodic Bloch states which encodes interband effects. In addition to the Berry curvature, we explicitly relate the orbital susceptibility of two-band models to a quantum metric tensor defining a distance in Hilbert space. Within a simple tight-binding model allowing for a tunable Bloch geometry, we show that interband effects are essential even in the absence of Berry curvature. We also show that for a flat band model, the quantum metric gives rise to a very strong orbital paramagnetism.

Figure 1

(a) Tunable brick-wall lattice with a staggered on-site potential $\mathrm{\Delta }$: It interpolates between a square ($\lambda =1$) and a deformed honeycomb (brick wall) ($\lambda =0$). (b),(d) Berry curvature term $-\mathcal{M}{\left(\mathit{k}\right)}^2$, respectively, for $\lambda =1$ and 0. (c),(e) Metric term $Z_g\left(\mathit{k}\right)$, respectively, for $\lambda =1$ and 0. For clarity, the area of the plot covers twice the first Brillouin zone. Blue and red colors denote, respectively, positive and negative values. (f) Evolution of the magnetic susceptibility versus chemical potential $\mu$ as a function of $\lambda$. The first column is the Pauli spin susceptibility which is proportional to the zero field density of states. It is normalized such that ${\chi }_{\text{spin}}=-3{\chi }_{\text{orb}}$ at the band edges, as in the absence of a lattice. The four next columns concern the orbital response, respectively, the susceptibility ${\chi }_{\text{orb}}$ (red), ${\chi }_{\text{LP}}$ (blue), ${\chi }_{\mathrm{\Omega }}$ (green), and ${\chi }_g$ (magenta). They are normalized to the Landau susceptibility ${\chi }_{\text{L}}$ at the band edges, which itself depends on $\lambda :{\chi }_{\text{L}}\left(\lambda \right)={\chi }_{\text{L}}\left(1\right)\sqrt{\frac{1+3\lambda }{3+\lambda }}$. The half bandwidth, equal to $\sqrt{{\mathrm{\Delta }}^2+3+\lambda }$, is also normalized to unity. We fix $\mathrm{\Delta }/t=0.4$ and $T=0.001t$.

Figure 2

(a) The susceptibility contributions ${\chi }_{\text{orb}}$, ${\chi }_{\text{LP}}$, ${\chi }_{\mathrm{\Omega }}$, and ${\chi }_g$ as a function of the chemical potential $\mu$ for the lattice model [this is identical to Fig. 1 with $\lambda =0$]. (b) Similar quantities but for the low energy (linearized) model.

Figure 3

Top: (a) The checkerboard lattice (here there is no on-site potential) and (b) the corresponding energy spectrum. Bottom: (c) Pauli spin susceptibility and orbital susceptibility ${\chi }_{\text{orb}}$ with its different contributions as a function of the chemical potential. Units for $\chi$ and $\mu$ are similar to Fig. 1.