Quantum Critical Scaling of the Geometric Tensors

Berry phases and the quantum-information theoretic notion of fidelity have been recently used to analyze quantum phase transitions from a geometrical perspective. In this Letter we unify these two approaches showing that the underlying mechanism is the critical singular behavior of a complex tensor over the Hamiltonian parameter space. This is achieved by performing a scaling analysis of this quantum geometric tensor in the vicinity of the critical points. In this way most of the previous results are understood on general grounds and new ones are found. We show that criticality is not a sufficient condition to ensure superextensive divergence of the geometric tensor, and state the conditions under which this is possible. The validity of this analysis is further checked by exact diagonalization of the spin-$1/2$ $XXZ$ Heisenberg chain.

Figure 1

Scaling behavior of metric $g$. The points are the data obtained via Lanczos diagonalization of periodic chains of length $L$ up to 26, using equation $\mathcal{F}(\lambda ,\lambda +\delta \lambda )\approx 1-g\delta {\lambda }^2/2$ with $\delta \lambda =1.0\times {10}^{-3}$. The points at $|\lambda |<1$ are well fitted (solid lines) with $g/L=A_1+A_2{L}^{-1}+A_3{L}^{3-2{\Delta }_V^{(2)}}$. The contribution $A_3$ comes from the irrelevant operator with scaling dimension ${\Delta }_V^{(2)}=4K$. As this operator becomes rapidly irrelevant for $|\lambda |<1$, its contribution can be hardly observed. At $\lambda =1$ a better a fit is obtained with logarithmic corrections as expected at the isotropic point. In the massive regime we expect that the thermodynamic limit is approached exponentially fast. We obtain a good agreement by fitting our data with the phenomenological formula $g/L=A_1+A_2{e}^{-L/\xi }{L}^{-1/2}$, where $\xi$ is the correlation length as given by the Bethe ansatz [26]. As $\xi (\lambda =2)=8.35\dots$ we used only points with $L\ge 14$.