Topological spin excitations in Harper-Heisenberg spin chains

Many-body spin systems represent a paradigmatic platform for the realization of emergent states of matter in a strongly interacting regime. Spin models are commonly studied in one-dimensional periodic chains, whose lattice constant is on the order of the interatomic distance. However, in cold atomic setups or functionalized twisted van der Waals heterostructures, long-range modulations of the spin physics can be engineered. Here we show that such superlattice modulations in a many-body spin Hamiltonian can give rise to observable topological boundary modes in the excitation spectrum of the spin chain. In the case of an XY spin-$1/2$ chain, these boundary modes stem from a mathematical correspondence with the chiral edge modes of a two-dimensional quantum Hall state. Our results show that the addition of many-body interactions does not close some of the topological gaps in the excitation spectrum, and the topological boundary modes visibly persist in the isotropic Heisenberg limit. These observations carry through when the spin moment is increased and a large-spin limit of the phenomenon is established. Our results show that such spin superlattices provide a promising route to observe many-body topological boundary effects in cold atomic setups and functionalized twisted van der Waals materials.

Figure 1

(a) Sketch of a spin-chain superlattice with Heisenberg couplings (magenta arrows) modulated in space [cf. Eq. (1)] that can be engineered in a cold atom setup. Moreover, the model of panel (a) naturally arises in atomically engineered lattices on top of twisted van der Waals materials as shown in panel (b). (c) The calculated dynamical structure factor [cf. Eq. (2)] for a uniform $S=1/2$ chain, showing the emergence of different confined modes and a gapless excitation spectrum. (d) The calculated dynamical structure factor of a modulated chain with $\alpha =\pi /\sqrt{2},\lambda =0.5$ and $\varphi =0.6\pi$, showing the emergence of a spectral gap and in-gap edge excitations (cyan circles). As the pumping parameter $\varphi$ is varied, we expect topological in-gap modes to traverse the bulk gap as sketched in panel (e).

Figure 2

Single-particle superlattices. (a) The energy spectrum of the off-diagonal Harper model [Eq. (4)] as a function of the pumping parameter $\varphi$. Topological in-gap bound states appear on both sides of the chain (red and blue) and traverse the gap. The appearance of these states is in correspondence with the topological bulk response of the topological pump and can be mapped to the chiral edge modes of the 2D quantum Hall effect [21]. (b) Sketch of the 2D quantum Hall model [Eq. (5)] that is mappable to the off-diagonal Harper pump [Eq. (4)]. The Hofstadter spectra in a bulk site averaged over $\varphi$ (c) and the corresponding dynamical charge response in a bulk site averaged over $\varphi$ (d) [cf. Eq. (6)]. Panels (c), (d) highlight their characteristic topological spectral gaps for arbitrary $\alpha$. Note the different energy axis in panels (c), (d) due to the fact that the dynamical correlator includes contributions from transitions that can have absolute energy $0\le \omega \le W$, where $W={\omega }_{\max }-{\omega }_{\min }$ is the full bandwidth of the bulk spectrum (c). (e) There are bulk gaps that remain open in the dynamical charge response as a function of $\varphi$. (f) Same as panel (e) but evaluated at the boundary, showing in-gap boundary excitations. We used $\lambda =0.8$ and $\alpha =\pi /\sqrt{2}$ in panels (a), (c), (d), (e), (f), panels (c), (d) are averaged over $\varphi$.

Figure 3

(a) Dynamical structure factor $\chi \left(\omega \right)$ [Eq. (2)] in the different sites of the chain averaged over $\varphi$ for a Heisenberg spin chain $S=1/2$ of Eq. (1). It is observed that the bulk of the chain shows a spectral gap, whereas at the edge the average spectral function ${\langle \chi \left(\omega \right)\rangle }_{\varphi }$ signals the appearance of edge excitations. This can be explicitly seen by looking at the dynamical structure factor $\chi \left(\omega \right)$ in the bulk (b), left edge (c), and right edge (d) as a function of $\varphi$, showing states that thread through the gap at the edge while the bulk shows a robust spectral gap at the high energy part of the spectrum. The edge modes that pump with $\varphi$ can be adiabatically connected to the ones found in the noninteracting limit Fig. 2. Panel (e) shows the bulk dynamical correlator in the anisotropic case $\mathrm{\Delta }\ne 1$, showing that the main excitation gap remains open as one goes from the noninteracting $\left(\mathrm{\Delta }=0\right)$ to the fully interacting $\left(\mathrm{\Delta }=1\right)$ limit. We took $\lambda =0.5$ for panels (a)–(e), $\alpha =0.66\pi$ for panels (a), (e) and $\alpha =0.7\pi$ for (b)–(d).

Figure 4

(a)–(d) Dynamical structure factor $\chi \left(\omega \right)$ Eq. (2) averaged over $\varphi$, for a Heisenberg spin chain $S=1/2$ of Eq. (1), for different modulation wave vectors $\alpha$. Panel (a) [zoom in (c)] shows the bulk and panel (b) [zoom in (d)] shows the edge $\chi \left(\omega \right)$. It is observed that whereas the bulk (a), (c) shows a spectral gap, the edge (b), (d) shows a nonzero spectral weight, reflecting the emergence of the edge modes at arbitrary modulation frequencies $\alpha$. In the absence of $ZZ$ interaction in the Heisenberg model Eq. (1), panels (a), (c) would be equivalent to panel Fig. 2. We took $\lambda =0.8$ for panels (a)–(d).

Figure 5

(a) Dynamical structure factor in the different sites of the chain for a uniform $S=1$ Heisenberg model, showing the emergence of topological modes at the edge (red circles). Panel (b) shows the dynamical structure factor for a modulated $S=1$ chain, showing the coexistence of the preexisting zero modes (red circles) with the topological pumping modes (cyan circles). Panel (c) shows the dynamical structure factor for the different sites of a $S=1$ chain averaged over $\varphi$. Bulk (d), left edge (e), and right edge (f) dynamical structure factors as a function of $\varphi$, showing states that thread through the gap in the edge while the bulk shows an excitation gap in the high energy part of the spectrum. We took $\alpha =\pi /\sqrt{2}$ for panels (b)–(f), $\lambda =0.4$ for panel (b), and $\lambda =0.7$ in panels (c)–(f).

Figure 6

(a)–(d) Dynamical structure factor $\chi \left(\omega \right)$ Eq. (2) averaged over $\varphi$, for a Heisenberg spin chain $S=1$ of Eq. (1), for different modulation wave vectors $\alpha$. Panel (a) [zoom in (c)] shows the bulk and panel (b) [zoom in (d)] shows the edge $\chi \left(\omega \right)$. It is observed that whereas the bulk (a), (c) shows a spectral gap, the boundary (b), (d) shows a nonzero spectral weight, reflecting the emergence of the boundary modes at arbitrary modulation frequencies $\alpha$. In contract with the $S=1/2$ chain of Fig. 4, the present case cannot be adiabatically connected to the free fermion Hamiltonian Eq. (4). We took $\lambda =0.8$ for panels (a)–(d).

Figure 7

Dynamical structure factor $\chi \left(\omega \right)$ at the edge (a), (c) and bulk (b), (d) for a Harper-Heisenberg chain of Eq. (1), for spins $S=3/2$ (a), (b) and $S=2$ (c), (d). In particular, panels (b), (d) show the emergence of a spectral gap in the bulk, that hosts pumping modes at the edge (a), (c). This situation is analogous to the pumping shown for $S=1/2$ in Fig. 3, $S=1$ in Fig. 5, and ultimately the free fermion case of Fig. 2. In comparison with the $S=1/2$ case of Fig. 3, a simple mapping with the free fermionic case of Fig. 2 cannot be performed. We took $\alpha =\pi /\sqrt{2}$ and $\lambda =0.5$ for panels (a)–(d).