Local probes for charge-neutral edge states in two-dimensional quantum magnets

The bulk-boundary correspondence is a defining feature of topological states of matter. However, for quantum magnets in two dimensions such as spin liquids or topological magnon insulators, a direct observation of topological surface states has proven challenging because of the charge-neutral character of the excitations. Here we propose spin-polarized scanning tunneling microscopy as a spin-sensitive local probe to provide direct information about charge-neutral topological edge states. We show how their signatures, imprinted in the local structure factor, can be extracted by specifically employing the strengths of existing technologies. As our main example, we determine the dynamical spin correlations of the Kitaev honeycomb model with open boundaries. We show that by contrasting conductance measurements of bulk and edge locations, one can extract direct signatures of the existence of fractionalized excitations and nontrivial topology. The broad applicability of this approach is corroborated by a second example of a kagome topological magnon insulator.

Figure 1

Spin-polarized scanning tunneling microscopy (SP-STM). We propose tunneling from a metallic and magnetic substrate to an STM tip via inelastic spin flips of an insulating magnetic layer ($S$) in between. A tunneling electron can excite a mode with energy $\omega$ in $S$ provided the applied bias voltage exceeds this energy. The resulting conductance is proportional to the spin-dependent densities of states in tip, substrate, and the sample $S$; cf. Eq. (1). Tuning the spin polarization in tip and substrate allows for selectively probing different types of spin excitations in the sample.

Figure 2

Topological magnon insulator. (a) The STM tip is placed at the edge (pos A) or over the 2D bulk (pos B) of the kagome layer, with color gradients indicating the range of the response. For numerical evaluations, a unit cell of $L_y=181$ sites along the $y$ direction is used. Inset: lattice vectors (green) and directions of the DM interaction (orange). (b) Energy spectrum for the magnon Hamiltonian with $J=1.0$ and DM term $D=0.2$, containing edge states within the gaps of the three bulk bands. (c) Conductance $\partial I/\partial V$ using Eq. (1). While the response at tip position B exhibits a flat conductance throughout all band gaps, at the edge (pos A), a finite response within the first gap is acquired, yielding a clear signature for the existence of topological edge magnons.

Figure 3

Kitaev spin liquid. (a) Geometry of the STM setup and sketch of the Kitaev model given by Eq. (4). OBCs are introduced by setting the strength of the dashed bonds to $J_b\to 0$. Probing the local spin noise in the bulk requires the creation of a gapped flux pair (blue). Moving to the boundary, these fluxes are gapless (orange), allowing for the detection of gapless Majorana edge modes. (b) Integrated DOS for the fermionic spectrum in the presence of a boundary flux pair, both for PBCs ($J_b=1$, gray) and OBCs ($J_b=0$, blue). The gray-shaded area marks the gapped continuum band in the bulk; the dashed gray line marks the energy of the bound state. The blue circles with attached dashed lines show the energies of the in-gap edge modes at the given system size of $56\times 56$ unit cells. For OBCs, the continuum band is shifted to lower energies due to the vanishing flux gap. (c) The wave function $\psi (x,y)$ of the fermionic bound state on $40\times 40$ unit cells delocalizes upon reducing the coupling $J_b$ across the boundary (located at $y=0$). (d) Components of the dynamical structure factor for $56\times 56$ unit cells. A finite frequency broadening was introduced for the sharp delta response from the bound state. (e) Conductance for the tip at positions A and B, and different polarizations of tip and substrate. In the bulk, the fermion bound state creates a sharp step within the gap. On the boundary, the step is replaced by a continuum due to the dispersive edge modes, starting from zero bias; see the inset.

Figure 4

Kitaev model. The three types of bonds are labeled according to the anisotropic exchange interaction of the Hamiltonian given by Eq. (4) of the main text. The interactions along a line of $z$ bonds through the system are weakened by a factor $J_b<1$, yielding open boundary conditions for $J_b=0$. Inserting bond fermions (red bonds) flips the flux $W_p$ of the two adjacent plaquettes. In the bulk, these flux excitations are gapped (blue plaquettes), while boundary plaquettes cost no energy and lead to degeneracies in the spectrum (orange plaquettes).

Figure 5

Majorana pairings: The Majorana fermions ${\widehat{b}}_i^{\alpha }$ (colored according to $\alpha$) are paired up to form the gauge fermions ${\widehat{\chi }}_{ij}^{}$ living on bonds $(i,j)$, whose occupation numbers commute with the Hamiltonian. If we introduce open boundaries by setting the exchange $J_b=0$ on the line of vertical bonds shown here (dotted bonds), there arises an ambiguity in how to pair up the resulting “dangling” Majoranas (shown in blue). (a) The original pairing along the former bond is still valid and produces the usual ultralocal expression for the spin structure factor. (b) The Majoranas can now also be paired up in longer-range bonds $(i,j)$, and the resulting fermion occupation numbers still commute with the Hamiltonian, allowing for longer-range contributions to the structure factor. The different pairings are related by a basis change within the degenerate ground-state manifold.

Figure 6

Geometry of the TMI setup [compare Fig. 2 of the main text]. We consider open boundaries in the $y$ direction, implying translational invariance only along the ${\mathit{a}}_2$ lattice vector. The number of sites within a unit cell is $6W+1$; the position of a site $n$ is specified by $(x_n,l_n)$, with $x_n$ labeling the unit cell and $l_n\in \{0,\cdots ,6W\}$ labeling the site within a unit cell as depicted here.