Tuning the Chern number and Berry curvature with spin-orbit coupling and magnetic textures

Timothy M. McCormick and Nandini Trivedi

Phys. Rev. A 91, 063609 – Published 8 June 2015

Abstract

We obtain the band structure of a particle moving in a magnetic spin texture, classified by its chirality and structure factor, in the presence of spin-orbit coupling. This rich interplay leads to a variety of novel topological phases characterized by the Berry curvature and their associated Chern numbers. We suggest methods of experimentally exploring these topological phases by Hall drift measurements of the Chern number and Berry phase interferometry to map the Berry curvature.

Received 11 March 2015

DOI:https://doi.org/10.1103/PhysRevA.91.063609

Authors & Affiliations

Timothy M. McCormick and Nandini Trivedi

Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 91, Iss. 6 — June 2015

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic illustration of the hopping processes which lead to the matrix elements $t^{'}$ (top) and $t^{''}$ (bottom). The black arrows represent the fixed texture at a site $i$ and its neighbor $i + δ$ . The faded red arrows show how a particle hops and then projects along the local texture and the dashed blue arrow represents the hopping. Due to the nature of the spin-orbit coupling, the hopping amplitudes now depend on the direction of the hopping as well as the orientation of the spin texture at each site.
Reuse & Permissions
Figure 2
(a)–(d) Real-space spin configurations for several spin textures: (a) a ferromagnetic texture, (b) an antiferromagnetic texture, (c) $4 \times 1$ spirals on a $12 \times 12$ lattice, and (d) $3 \times 3$ merons on a $12 \times 12$ lattice. (e)–(h) The spin structure factors for several textures on a $12 \times 12$ site lattice: (e) a ferromagnetic texture, (f) an antiferromagnetic texture, (g) spirals with a $4 \times 1$ unit cell in the $y$ direction, and (h) sixteen $3 \times 3$ merons.
Reuse & Permissions
Figure 3
(Left) Energy $E$ in units of the hopping matrix element $t$ for a particle in a ferromagnetic texture is identical to that of a free particle with a $cos (α)$ modulated bandwidth. (Right) The bands (in units of the hopping matrix element $t$ ) for an antiferromagnetic texture are identical to that of the Rashba portion of the Hamiltonian in Eq. (1) with spin-preserving hopping with a $sin (α)$ . There are Dirac cones at the time-reversal invariant momenta for the antiferromagnetic bands.
Reuse & Permissions
Figure 4
Energy $E$ in units of the hopping matrix element $t$ in the reduced Brillouin zone for a $4 \times 1$ spiral texture with $α = 0$ (left), $α = π / 4$ (middle), and $α = π / 2$ (right). We note the presence of zero-energy bands for $α = π / 4$ .
Reuse & Permissions
Figure 5
(a) Sum of Chern numbers $n_{2}$ as a function of $α$ for the lowest two bands of a system with unit cells of $3 \times 3$ merons. These lowest two bands are well separated from the others except for $0.5 < α < 0.67$ and $1.2 < α < 1.4$ and so, excluding these regions, the Chern number can be interpreted as the Hall conductivity. Spin-orbit coupling has a nontrivial effect on the topological nature of the bands for a fixed texture. (b),(c),(d) Energy in units of the hopping amplitude $t$ for $3 \times 3$ merons, along the principal axes of symmetry $Γ \to X \to M \to Γ$ in the Brillouin zone. (b) $α = 0$ , (c) $α = 0.16 π$ , and (d) $α = 0.5 π$ . (e) A diagram of the Brillouin zone. The red arrows show the path shown in the horizontal axis of (b)–(d) above. (f),(g),(h) Berry curvature $F_{2}$ for the second band of a $3 \times 3$ meron system for (f) $α = 0$ , (g) $α = 0.16 π$ , and (h) $α = π / 2$ . Peaks in the Berry curvature identify values of $k$ which contribute to the Chern number.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information