Fate of Bosonic Topological Edge Modes in the Presence of Many-Body Interactions

TL;DR

Bosonic topological edge modes in quantum magnets can persist even when full many-body interactions are present, challenging the view that interactions universally suppress such edge states. The authors combine DMRG and time-evolution to compute the dynamical structure factor and resolve boundary modes in a spin-ladder model across three regimes: strong field (topologically trivial), strong rail coupling (no magnetic order in 1D), and strong spin-orbit coupling (edge modes robust despite non-conserved triplon number). Key findings include robust, localized in-gap edge modes with fractionalized boundary occupancy n_t ≈ 0.43, a topological phase diagram inferred from dynamical responses, and the breakdown of harmonic theories near strong SOC while edge modes remain identifiable. The work supports the relevance of bosonic edge states for 2D extensions and material candidates, and suggests that boundary physics and many-body coherence can govern the visibility of topological signatures in experiments.

Abstract

Many magnetic materials are predicted to exhibit bosonic topological edge modes in their excitation spectra, because of the nontrivial topology of their magnon, triplon, or other quasi-particle band structures. However, there is a discrepancy between theory prediction and experimental observation, which suggests some underlying mechanism that intrinsically suppresses the expected experimental signatures, like the thermal Hall current. Many-body interactions that are not accounted for in the non-interacting quasi-particle picture are most often identified as the reason for the absence of the topological edge modes. Here we report persistent bosonic edge modes at the boundaries of a ladder quantum paramagnet with gapped triplon excitations in the presence of the full many-body interaction. We use tensor network methods to resolve topological edge modes in the time-dependent spin-spin correlations and the dynamical structure factor, which is directly accessible experimentally. We further show that signatures of these edge modes survive even when the non-interacting quasi-particle theory breaks down, discuss the topological phase diagram of the model, demonstrate the fractionalization of its low-lying excitations, and propose potential material candidates.

Figures (11)

• Figure 1: (a) Schematic of the spin ladder. On each rung, two spin-1/2 sites are (strongly) coupled through antiferromagnetic Heisenberg interaction. Along the side rails, the spins interact (weakly) antiferromagnetically. In this limit the spins form singlet pairs along the rungs. (b) The same model but with anti-symmetric (DM) and symmetric (pseudo-dipolar) exchange in the $y$-direction given by $D_y$ and $\Gamma_y$. These terms introduce a winding and open a topological gap in the excitation spectrum.
• Figure 1: (a) Schematic of the model of the quantum paramagnet from Eq. \ref{['eq:hamiltonian']}. (b) The same model in the basis of triplon operators $t^\gamma$ up to bilinear terms. $D_y$, $\Gamma_y$ (green) and $h_y$ (orange) couple the $t^x$ and $t^z$ chains along the correspondingly colored bonds. $J$ is a chemical potential. The full Hamiltonian is given in Eq. \ref{['aeq:triplon_hamiltonian']}.
• Figure 1: (a) DSF for $K/J=0.01$, $h_y/J=0.05$ and $D_y=\Gamma_y=0.1$ on a 64-rung ladder (128 spin-half sites). (b) Cut through the DSF shown in (a) at $k=\pi/2$. At $\omega/J=1$ there is a quasi-particle peak corresponding to the localized topological boundary mode.
• Figure 2: (a) DSF of the topological quantum paramagnet with $L=32$ rungs and $K/J = 0.01$, $D_y/J = \Gamma_y/J = 0.1$ and $h_y = 0$ (corresponding to the red plot in (b)) The solid lines show the spectrum of the effective low-energy models from Ref. joshi2017topological for the same parameters. There is a localized boundary mode at $\omega/J=1$. (b) DSF for different values of $h_y / D_y$ at $k=\pi/2$ (dashed line in (a)). The first two curves lie in the topological phase region (see (e)) with the in-gap modes marked by arrows. At $h_y / D_y = 1$ the gap is closed. The last curve lies in the trivial region and is gapped, with no mode in-between. All peaks are centered at $\omega\approx J$ (and shifted for clarity). (c) Many-body spectrum of a 4-rung system with the same parameters as in (a). The spectrum is separated into sectors of different triplon particle numbers. The color indicates $N_t$. The inset shows the 1-triplon sector with the edge modes marked by arrows. (d) $\rho_i^{\text{top}}$ for different values of $h_y / D_y$, but with $L=8$. The in-gap modes are localized at the two boundaries of the system with fractional particle numbers $n_t$ at each termination. For finite magnetic fields the in-gap mode spreads into the bulk and vanishes at the topological phase transition at $h_y/D_y=1$. (e) The topological phase diagram of the model.
• Figure 2: Real part of the spin-spin correlation function at time $t=50/J$ in the (a) trivial and (b) the topological case with $K=0.01J$ and field strengths $h_y/D_y = 0.5$ and $h_y/D_y = 1.5$, respectively, on a ladder with $L=12$ rungs. The topological auto-correlation at this time step has peaks at the boundaries, which are absent in the trivial case. (c) Time dependence of the boundary auto-correlation $\mathrm{Re}\ C^{zz}_{11}$ for $K=0.01J$, $D_y=\Gamma_y = 0.1J$ and no magnetic field on a ladder with $L=8$ rungs for different boundary conditions. For PBC (no in-gap mode), it decays rapidly whereas for OBC (in-gap mode) it does not decay. (d) The decay envelopes of (c) for different values of magnetic field. For finite field strengths the correlator decays also for OBC, but more slowly than for PBC.