Topological quantum paramagnet in a quantum spin ladder

It has recently been found that bosonic excitations of ordered media, such as phonons or spinons, can exhibit topologically nontrivial band structures. Of particular interest are magnon and triplon excitations in quantum magnets, as they can easily be manipulated by an applied field. Here, we study triplon excitations in an $S=1/2$ quantum spin ladder and show that they exhibit nontrivial topology, even in the quantum-disordered paramagnetic phase. Our analysis reveals that the paramagnetic phase actually consists of two separate regions with topologically distinct triplon excitations. We demonstrate that the topological transition between these two regions can be tuned by an external magnetic field. The winding number that characterizes the topology of the triplons is derived and evaluated. By bulk-boundary correspondence, we find that the nonzero winding number implies the presence of localized triplon end states. Experimental signatures and possible physical realizations of the topological paramagnetic phase are discussed.

Figure 1

Schematic representation of the exchange interactions in the quantum spin ladder described by Eq. (1). The spins are shown as black circles, blue lines represent the intradimer exchange ($J$), and red lines interdimer interactions ($K$). The DM interaction ($D$), indicated in green, points in the $y$ direction into the plane of the ladder. In addition, the model exhibits an even-parity spin-anisotropic interaction ($\mathrm{\Gamma }$), which arises along with the odd-parity DM interaction due to spin-orbit coupling.

Figure 2

Triplon bands ($t_x$ and $t_z$) obtained from ${\mathcal{H}}_k$ [Eq. (2)] are plotted for different $h_y$. We see that the gap between the two modes vanishes to form a Dirac point at $h_y=\pm D$. Everywhere else in the dimer-quantum-paramagnetic phase, the two modes do not touch each other. For $|h_y|<D$ the phase is topologically nontrivial, else it is trivial. The parameters used are $D/J=\mathrm{\Gamma }/J=0.1$ and $K/J=0.01$.

Figure 3

Band structure of the quantum spin ladder [Eq. (2)] with open ends. Protected end states (green line) appear in the topological paramagnetic phase, $|h_y|<D$. Parameters used are the same as in Fig. 2.

Figure 4

Triplon end-state density profile $\rho -{\rho }_0$ plotted in the topological paramagnetic phase, $|h_y|<D$, near one of the ladder ends. In the topologically trivial phase, $|h_y|>D$, the end states are absent (black trace). Parameters used are the same as in Fig. 2.

Figure 5

Winding number $\mathcal{W}$ [Eq. (6)] as a function of applied field $h_y$. In the topological paramagnetic phase, $|h_y|<D, \mathcal{W}$ evaluates to $-1$, which, by the bulk-boundary correspondence, leads to the appearance of triplon end states (cf. Figs. 3 and 4). Parameters used are the same as in Fig. 2.