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

We study the Riemannian metric and the Euler characteristic number of the Bloch band in a one-dimensional spin model with multisite 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 phase transition in zero temperature can be distinguished by the Euler number of the Bloch band.

Figure 1

The trace of the Riemannian metric $\text{Tr}\mathcal{G}$ as a function of the external field $h$ and the crystal momentum $k$ with the fixed Hamiltonian parameters: (a) the three-site spins exchange interactions $\delta =0$ and the anisotropy parameter $\gamma =1$; (b) $\delta =0.3$ and $\gamma =0.9$.

Figure 2

The cyclic quantum distance $l$ of the Bloch band as a function of $h$, with the fixed anisotropy parameter $\gamma =1/3$ and different three-site spins coupled coefficients $\delta$, where the integral path is along the diagonal line in the extended Brillouin zone (inset).

Figure 3

The cyclic quantum distance $l$ of the Bloch band as a function of $h$, with the fixed three-site spins coupled coefficient $\delta =0.3$ and different anisotropy parameters $\gamma$.

Figure 4

(a) The derivative of the cyclic ground-state distance $d{l}_e/dh$ with different lattice sizes $N$, where the Hamiltonian parameters $\gamma =0.7$ and $\delta =0.3$; (b) with the increasing of the lattice sizes, the positions of the maximum points of $d{l}_e/dh$ tend as $N^{-1.0074}$ and $N^{-0.7122}$ to the critical points $h_c=0.7$ and $h_c=1.3$, respectively.

Figure 5

The Euler number $\chi$ of the Bloch band as a function of the external field $h$ and three-site spins coupled coefficients $\delta$. The ferromagnetic phase in this model can be marked by a nontrivial Euler number $\chi =4$, and the Euler number $\chi \to 0$ quickly with the increasing of the external field $h$ in the paramagnetic phase.

Figure 6

The Euler number $\chi$ with several groups of the anisotropy parameter $\gamma$ and three-site spins coupled coefficients $\delta$.

Figure 7

The trajectories of the two-dimensional vector $\mathbf{d}\left(k\right)$ with the Hamiltonian parameters $\delta =0.7$, $\gamma =1$ and (a) $h=-0.3$; (b) $h=1$; (c) $h=1.7$; (d) $h=2$.

Figure 8

The trajectories of the three-dimensional vector $\mathbf{d}(k,\phi )$ with the Hamiltonian parameters $\delta =0.7$, $\gamma =1$ and (a) $h=-0.3$; (b) $h=1$; (c) $h=1.7$; (d) $h=2$.

Figure 9

The trajectories of the three-dimensional vector $\mathbf{d}(k,\phi )$ with the Hamiltonian parameters $\delta =0.7$, $\gamma =1$ and $h=1$ can be split into two equal disjoint topological spheres (a) $\mathbf{d}(k,\phi )$ with $k\in [-\pi ,0]$; and (b) $\mathbf{d}(k,\phi )$ with $k\in [0,\pi ]$.

Figure 10

The nearest neighbors spin-spin correlations $C_{i,i+1}^x$, $C_{i,i+1}^y$, and $C_{i,i+1}^z$ as functions of $h$, with the fixed parameters (a) $\gamma =1,\delta =0$; (b) $\gamma =0,\delta =0$; (c) $\gamma =-1,\delta =0$; (d) $\gamma =0.3,\delta =0.4$; (e) $\gamma =0,\delta =0.4$; (f) $\gamma =-0.3,\delta =0.4$.

Figure 11

The spin-spin correlations $C_{i,i+1}^x$, $C_{i,i+1}^y$, and $C_{i,i+1}^z$ as functions of $h$ with the fixed parameters $\gamma =0.3$, and (a) $\delta =0$; (b) $\delta =0.5$; (c) $\delta =1$. The spin-spin correlations $C_{i,i+1}^x$, $C_{i,i+1}^y$, and $C_{i,i+1}^z$ as functions of $\delta$ with the fixed parameters $\gamma =0.3$, and (d) $h=0$; (e) $h=0.5$; (f) $h=1$.