Quantum distance and the Euler number index of the Bloch band in a one-dimensional spin model

Abstract

We study the Riemannian metric and the Euler characteristic number of the Bloch band in a onedimensional spin model with multi-site spins exchange interactions. The Euler number of the Bloch band originates from the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. We study this approach analytically in a transverse field XY spin chain with three-site spin coupled interactions. We define a class of cyclic quantum distance on the Bloch band and on the ground state, respectively, as a local characterization for quantum phase transitions. Specifically, we give a general formula for the Euler number by means of the Berry curvature in the case of two-band models, which reveals its essential relation to the first Chern number of the band insulators. Finally, we show that the ferromagnetic-paramagnetic phases transition in zero-temperature can be distinguished by the Euler number of the Bloch band.

Crossdimensional universality classes in static and periodically driven Kitaev models

“…which is found to be equal to the square of the stroboscopic Berry curvature F (k) for any parametrization of d(k) [58,93,94]. We now detail the first type of TPT occurred in the periodically driven Kitaev model.…”

Cross-dimensional universality classes in static and periodically driven Kitaev models

“…which is found to be equal to the square of the stroboscopic Berry curvature F (k) for any parametrization of d(k) 55,89,90 . We now detail the first type of TPT occurred in the periodically driven Kitaev model.…”

Measurement of interaction-dressed Berry curvature and quantum metric in solids by optical absorption

“…The model contains only one filled band and one empty band, and the modulus of momentum is restricted to 0 ≤ k ≤ π/a such that the integration in the self-energy is finite, where a = 1 represents a lattice constant. In the noninteracting and zero temperature limit, the square root of the determinant of the quantum metric is equal to half of the module of the Berry curvature 11,[13][14][15]47 det…”

“…10 Generically, how the quantum state |ψ(k) rotates in the Hilbert space as the parameter changes from k to k+δk defines the quantum metric according to | ψ(k)|ψ(k + δk) | = 1 − 1 2 g ψ µν δk µ δk ν . [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] This aspect is particularly important to describe quantum phase transitions, since the quantum metric generally diverges near the critical point k c regardless of any detail of the system, giving rise to the notion of fidelity susceptibility. [25][26][27][28][29][30][31] Despite the ubiquity of Berry curvature and quantum metric behind numerous quantum phenomena, their very definition becomes rather ambiguous in realistic materials subject to many-body interactions and at finite temperature.…”