Topological magnon bound states in periodically modulated Heisenberg XXZ chains

Strongly interacting topological states in multiparticle quantum systems pose great challenges to both theory and experiment. Recently, bound states of elementary spin waves (magnons) in quantum magnets have been experimentally observed in quantum Heisenberg chains comprising ultracold Bose atoms in optical lattices. Here, we explore a strongly interacting topological state called topological magnon bound state in the quantum Heisenberg chain under cotranslational symmetry. We find that the cotranslational symmetry is the key to the definition of a topological invariant for multiparticle quantum states, which enables us to characterize the topological features of multimagnon excitations. We calculate energy spectra, density distributions, correlations, and Chern numbers of the two-magnon bound states and show the existence of topological protected edge bound states. Our study not only opens a prospect to pursue topological magnon bound states, but also gives insights into the characterization and understanding of strongly interacting topological states.

Figure 1

Two-magnon excitation spectra. (a) Spectrum for periodic BC ($L_t=99$). (b) Magnon bound-state spectrum for periodic BC ($L_t=99$). (c) Spectrum for open BC ($L_t=100$). (d) Magnon bound-state spectrum for open BC ($L_t=100$). (e) Magnon density distribution $n_l=\left.〈\psi \right|{\widehat{n}}_l\left|\psi \right.〉$ corresponding to the eigenstates of the red pentagrams (A: edge state localized at right, B: bulk state, C: edge state localized at left) in the spectrum (d). (f) Two-magnon correlations ${\mathrm{\Gamma }}_{qr}$ corresponding to (e) (here, only the region with nonzero correlations is presented). (g) Two-magnon minor diagonal correlations ${\mathrm{\Gamma }}_{l,l+1}$ of (f). The other parameters are chosen as $\mathrm{\Delta }/J=100,\lambda /J=0.04,\beta =\frac{1}{3}$, and $\delta =\pi /6$.

Figure 2

The butterflylike spectrum for the two-magnon bound states. We diagonalize the Hamiltonian (1) with $\mathrm{\Delta }/J=100,L_t=100$, and different values of $\delta$: (a) $\delta =0$ and (b) $\delta =\pi /4$. The values of $\lambda$ are set as $\lambda /J=J{[\mathrm{\Delta }cos\left(\pi \beta \right)]}^{-1}$ for $\beta \ne \frac{1}{2}$, and $\lambda =0$ for $\beta =\frac{1}{2}$. The energies $E/J$ are added with a constant $\mathrm{\Delta }/J+2J/\mathrm{\Delta }$.

Figure 3

The two-magnon spectrum versus the periodic parameter $\delta$. (a), (b) Periodic BC ($L_t=99$); (c), (d) open BC ($L_t=100$). (b), (d) The enlarged magnon bound-state spectra in (a) and (c), respectively. The other parameters are chosen as $\mathrm{\Delta }/J=100,\lambda /J=0.04$, and $\beta =\frac{1}{3}$.

Figure 4

The multiparticle Chern number versus the total number of lattice sites $L_t$ for the three subbands of two-magnon bound states for the Hamiltonian (1) with $\mathrm{\Delta }/J=100,\lambda /J=0.04$, and $\beta =\frac{1}{3}$.

Figure 5

The bound-state spectrum for the systems under the periodic BC. The black circles denote the eigenenergies for the effective Hamiltonian (B16) and the red dots are the $L_t$ lowest eigenenergies for the original Hamiltonian (1). The parameters are chosen as $\lambda /J=0.04,\beta =\frac{1}{3},\delta =\pi /6,L_t=99$, and different values of $\mathrm{\Delta }/J$: (a) $\mathrm{\Delta }/J=1$, (b) $\mathrm{\Delta }/J=3$, (c) $\mathrm{\Delta }/J=5$, (d) $\mathrm{\Delta }/J=100$. The eigenenergies $E/J$ are added with a constant $\mathrm{\Delta }/J+2J/\mathrm{\Delta }$.

Figure 6

The bound-state spectrum for the systems under the open BC. The black circles denote the eigenenergies for the effective Hamiltonian (B21) and the red dots are the $L_t$ lowest eigenenergies for the original Hamiltonian (1). The parameters are given as $\lambda /J=0.04,\beta =\frac{1}{3},\delta =\pi /6,L_t=46$, and different values of $\mathrm{\Delta }/J$: (a) $\mathrm{\Delta }/J=1$, (b) $\mathrm{\Delta }/J=3$, (c) $\mathrm{\Delta }/J=5$, (d) $\mathrm{\Delta }/J=100$. The energies $E/J$ are added with a constant $\mathrm{\Delta }/J+2J/\mathrm{\Delta }$.

Figure 7

The energy gaps versus the deforming parameter $\eta$. We consider the system (C1) under the periodic BC. The parameters are chosen as $\mathrm{\Delta }/J=100,\lambda /J=0.04,\beta =\frac{1}{3}$, and $L_t=99$.

Figure 8

Three-magnon spectra versus the periodic parameter $\delta$. (a) Spectrum for periodic BC ($L_t=60$). (c) Spectrum for open BC ($L_t=58$). (b), (d) The enlarged magnon bound-state spectra in (a) and (c), respectively. (e) Magnon density distribution $n_l=\left.〈\psi \right|{\widehat{n}}_l\left|\psi \right.〉$ corresponding to the eigenstates of the pentagrams [(A, C, E, G): edge states, (B, D, F): bulk states] in (d). (f) Three-magnon correlations ${\mathrm{\Gamma }}_{l,l+1,l+2}=\left.〈\psi \right|{\widehat{S}}_l^{+}{\widehat{S}}_{l+1}^{+}{\widehat{S}}_{l+2}^{+}{\widehat{S}}_{l+2}^{-}{\widehat{S}}_{l+1}^{-}{\widehat{S}}_l^{-}\left|\psi \right.〉$, where ${\sum }_{l=1}^{L_t-2}{\mathrm{\Gamma }}_{l,l+1,l+2}\ge 0.9998$ for A–G. The other parameters are chosen as $\mathrm{\Delta }/J=100,\lambda /J=0.0001$, and $\beta =\frac{1}{5}$.