Berry Geometry and the Quantum Metric

The quantum geometric tensor measures how rapidly an occupied state changes into the unoccupied subspace. Its real part is the quantum metric and its imaginary part is Berry curvature. The connection depends on a phase or occupied frame; the metric, curvature, Wilson loops, and integrated characteristic classes are gauge invariant when their domain and orientation are fixed.

Required background. Bloch and Wannier theory supplies the isolated occupied projector and its $U(N_{\rm occ})$ frame freedom.

Helpful background. Characteristic classes and Chern–Weil theory supplies the global interpretation of curvature integrals.

The quantum geometric tensor

For a normalized nondegenerate state $|u(\boldsymbol\lambda)\rangle$ and $P=|u\rangle\langle u|$, define

$$Q_{ij}=\langle\partial_i u|(1-P)|\partial_j u\rangle, \qquad g_{ij}=\operatorname{Re}Q_{ij}, \qquad \Omega_{ij}=-2\operatorname{Im}Q_{ij}.$$

With $\mathcal A_i=i\langle u|\partial_i u\rangle$, direct differentiation gives $\Omega_{ij}=\partial_i\mathcal A_j-\partial_j\mathcal A_i$. The projection $1-P$ removes the phase derivative parallel to $|u\rangle$, so $Q_{ij}$ is gauge invariant. Infinitesimally,

$$1-|\langle u(\boldsymbol\lambda)|u(\boldsymbol\lambda+d\boldsymbol\lambda)\rangle|^2 =g_{ij}\,d\lambda^i d\lambda^j+O(d\lambda^3),$$

which makes positivity of $g$ manifest Provost and Vallée 1980.

For an occupied multiplet, frame-independent projector formulas are

$$g_{ij}=\frac12\operatorname{Tr}(\partial_iP\,\partial_jP), \qquad \Omega_{ij}=i\,\operatorname{Tr}P[\partial_iP,\partial_jP].$$

The full non-Abelian curvature is matrix-valued; the trace above is the curvature relevant to the first Chern number. At an internal degeneracy, individual-band Abelian curvatures are not defined, but the multiplet projector remains valid if the multiplet stays separated from all other bands.

Two-band reconstruction

For $H(\mathbf k)=d_0(\mathbf k)\mathbf1+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$ with $|\mathbf d|>0$, the lower-band projector is $P_-=(1-\hat{\mathbf d}\cdot\boldsymbol\sigma)/2$. Pauli-matrix algebra yields

$$g_{ij}=\frac14\,\partial_i\hat{\mathbf d}\cdot\partial_j\hat{\mathbf d}, \qquad \Omega_{ij}=\frac12\hat{\mathbf d}\cdot (\partial_i\hat{\mathbf d}\times\partial_j\hat{\mathbf d}).$$

Thus the metric measures stretching of the map from parameter space to the Bloch sphere, while curvature measures its oriented area. Reversing $(k_x,k_y)$ reverses $\Omega_{xy}$ but not $g_{ij}$. The pointwise inequality $\det g\ge(\Omega_{12}/2)^2$ follows from positivity of $Q$ in two parameters; equality holds for special two-level geometries.

Numerical and physical interpretation

Finite differences of eigenvector phases are unreliable. On a momentum mesh, use overlaps around closed plaquettes for curvature and singular values of overlap matrices for a multiplet. Refine the mesh until the integrated Chern number and local peaks converge; a curvature hotspot alone is not a topological invariant.

The quantum metric enters localization bounds, superfluid weight in projected multiband settings, and optical sum rules, but each relation has additional dynamical hypotheses Peotta and Törmä 2015. Geometry of a projected band is not by itself evidence for a particular interacting phase.

Exercise

For $H=k_x\sigma_x+k_y\sigma_y+m\sigma_z$, evaluate $\Omega_{k_xk_y}$ in the lower band.

Solution

Here $\mathbf d=(k_x,k_y,m)$ and $D=\sqrt{k_x^2+k_y^2+m^2}$. Substitution into the two-band formula gives $\Omega_{k_xk_y}=m/(2D^3)$. It changes sign with $m$, is concentrated on the scale $|\mathbf k|\sim|m|$, and its sign would reverse if the momentum orientation or Berry-connection convention were reversed.

References

• Sebastiano Peotta and Päivi Törmä, “Superfluidity in Topologically Nontrivial Flat Bands,” Nature Communications 6 (2015) 8944, doi:10.1038/ncomms9944.
• J. P. Provost and G. Vallée, “Riemannian Structure on Manifolds of Quantum States,” Communications in Mathematical Physics 76 (1980) 289–301, doi:10.1007/BF02193559.