Probing the topology of the quantum analog of a classical skyrmion

In magnetism, skyrmions correspond to classical three-dimensional spin textures characterized by a topological invariant that keeps track of the winding of the magnetization in real space, a property that cannot be easily generalized to the quantum case since the orientation of a quantum spin is, in general, ill defined. Moreover, as we show, the quantum skyrmion state cannot be directly observed in modern experiments that probe the local magnetization of the system. However, we show that this novel quantum state can still be identified and fully characterized by a special local three-spin correlation function defined on neighboring lattice sites—the scalar chirality—which reduces to the classical topological invariant for large systems and which is shown to be nearly constant in the quantum skyrmion phase.

Figure 1

Schematic representation of the local magnetization measurements of the quantum skyrmion state realized on a 19-site cluster. Upon individual von Neumann measurements, this state collapses to different basis functions shown in the left panel. The average of the local magnetization over all basis functions results in a uniform magnetization pattern that could be observed in the SPSTM experiment (right panel).

Figure 2

Complete set of observables describing skyrmions in the classical and quantum cases. (a) and (b) Skyrmion number and average magnetization, (c) and (d) local magnetization pattern of the lattice, and (e) and (f) structure factors as a function of magnetic field for the classical (left panels) and quantum (right panels) problems for a 19-site triangular lattice with periodic boundary conditions. Roman numbers denote different phases. Since the classical ground state is many-fold degenerate we have chosen a representative pattern for the magnetization panel in (c). The inset in (a) shows three different types of classical skyrmions revealed in the intermediate phase by the scalar chirality.