Variational matrix product ansatz for dispersion relations

A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically nontrivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-½ $XXZ$ antiferromagnet and the spin-1 Heisenberg antiferromagnet, and we obtain surprisingly accurate results.

Figure 1

Spectrum of the lowest-lying excitations of the spin-½ $XXZ$ antiferromagnet with anisotropy parameter $\Delta =4$ at $D=33$. Black circles indicate topologically nontrivial excitations, gray squares indicate topologically trivial excitations.

Figure 2

(a) Simulation results for ${\Delta }_{XXZ}^{\left(D\right)}$ as a function of the anisotropy $\Delta$ for various values of $D$ ranging from 10 to 400 (gray circles), as well as the exact result with the Bethe ansatz (black line). (b) Absolute error on the energy gap vs absolute error on the energy density for various values of $\Delta$ and $D$.

Figure 3

Spectrum of the lowest-lying excitations of the spin-1 Heisenberg antiferromagnet at $D=30$, labeled by their spin $S$ degeneracy.