Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker

Berry curvature and quantum metric in copper-substituted lead phosphate apatite

“…which is a measure of the average distance between valence band states |n⟩ along the k µ direction in the E − < 0 and E + > 0 regions of the 3D BZ, a quantity that has been called the fidelity number [29]. Here Ṽ = 16π 3 /3 √ 3 is the dimensionless volume of the 3D BZ.…”

“…Likewisely, one can also introduce a quantum metric spectral function [28] g µν (k, ω) that frequencyintegrates to the quantum metric g µµ (k) = ´dω g µµ (k, ω) and is proportional the optical conductivity g µµ (k, ω) ∝ σ µµ (k, ω)/ω at momentum k, which can possibly be detected in pump-probe type of experiments [27]. Furthermore, one can integrate the quantum metric spectral function over momentum to obtain what we call the fidelity number spectral function G µµ (ω) = ´dD k (2π) D g µµ (k, ω) ∝ σ µµ (ω)/ω that corresponds to optical conductivity (and hence the optical absorption power) σ µµ (ω) measured in real space divided by frequency, which serves as a practical way to measure the quantum geometrical properties of insulators and semiconductors [29]. These spectral functions, however, do not apply to the present work simply because the material under question is metallic, and hence does not have optical absorption since infrared light cannot penetrate through.…”

“…On the other hand, the quantum metric is defined from the overlap of the valence band state at slightly different momenta [26] |⟨n(k)|n(k + δk)⟩| = 1 − g µν δk µ δk ν , which in the 2 × 2 Dirac Hamiltonian has the expression [27][28][29]…”

Investigating Quantum Geometry and Quantum Criticality by a Fidelity Marker

“…Moreover, a Ddimensional BZ can be viewed geometrically as a T D torus, implying that it can be parameterized by a set of periodic coordinates k = {k 1 , k 2 , ..., k D }. As a result, the momentum integration of the quantum metric can be interpreted as the average distance between neighboring Bloch states on the manifold, leading to a number that is the main focus of this thesis that we call the fidelity number [11]. This fidelity number can be regarded as a differential geometrical property of the manifold, and moreover, we will elaborate that it is equivalent to the so-called spread of Wannier functions [12,13,14], an important quantity that has been investigated quite intensively within the context of first-principle calculation of band structures.…”

“…Employing this identity, the derived expression used to obtain the fidelity number can be rewritten in a projector formalism. Consequently, this representation leads to a more concise expression that can also yield other insightful results, as the object called fidelity operator [11]. Building on a similar approach taken from the Chern number, the locality characteristic of the Wannier functions can be exploited in investigating the quantities of interest in this work.…”