Bures metric over thermal state manifolds and quantum criticality

We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows one to complement the understanding of the phase diagram including crossover regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.

Figure 1

Vector field of the eigenvector associated to the highest eigenvalue of $g$, in the plane $(h,T)$ for the Ising model in transverse field.

Figure 2

Phase diagram of the Ising model in transverse field taking into account both critical points at $h=\pm 1$ and the purely classical Ising line $h=0$. The arrows indicate the direction of highest fidelity decrease (the direction of the arrows is conventional, but fixed once for all).

Figure 3

Contour plot of the highest eigenvalue of $g$, in the plane $(h,T)$ for the Ising model in transverse field.

Figure 4

Contour plot of the Gaussian curvature of $g$, in the plane $(h,T)$ for the Ising model in transverse field. The arrows indicate the zero curvature lines.