Elementary Excitations in Gapped Quantum Spin Systems

For quantum lattice systems with local interactions, the Lieb-Robinson bound serves as an alternative for the strict causality of relativistic systems and allows the proof of many interesting results, in particular, when the energy spectrum exhibits an energy gap. In this Letter, we show that for translation invariant systems, simultaneous eigenstates of energy and momentum with an eigenvalue that is separated from the rest of the spectrum in that momentum sector can be arbitrarily well approximated by building a momentum superposition of a local operator acting on the ground state. The error satisfies an exponential bound in the size of the support of the local operator, with a rate determined by the gap below and above the targeted eigenvalue. We show this explicitly for the Affleck-Kennedy-Lieb-Tasaki model and discuss generalizations and applications of our result.

Figure 1

Sketch of the energy-momentum diagram. The points illustrate the eigenvalues of the Hamiltonian for a finite periodic system, whereas the lines and band illustrate the distribution of eigenvalues in the thermodynamic limit.

Figure 2

Lowest variational excitation energies $E_{\min }^{(\ell )}$ obtained with the Ansatz from Eq. (10). (a) Shows the $E_{\min }^{(\ell )}$ as function of the momentum $p$ for $\ell =1,\dots ,5$, as well as the approximate position of the two-magnon and three-magnon continuum based on the numerical results for the one-magnon dispersion relation with $\ell =5$. (b) Illustrates the exponential convergence of $E_{\min }^{(\ell )}$ by plotting $E_{\min }^{(\ell )}-E_{\min }^{(\ell +1)}$ for different values of $p$ [as indicated by vertical lines in (a)]. Inset (c) shows, as a function of $p$, the energy gap above the magnon dispersion and the two- or three-magnon continuum, as well as the exponential rate of convergence of the variational energy, corresponding to the slope of the lines in (b) (but also for additional values of the momentum).