$S$ Matrix from Matrix Product States

We use the matrix product state formalism to construct stationary scattering states of elementary excitations in generic one-dimensional quantum lattice systems. Our method is applied to the spin-1 Heisenberg antiferromagnet, for which we calculate the full magnon-magnon $S$ matrix for arbitrary momenta and spin, the two-particle contribution to the spectral function, and higher order corrections to the magnetization curve. As our method provides an accurate microscopic representation of the interaction between elementary excitations, we envisage the description of low-energy dynamics of one-dimensional spin chains in terms of these particlelike excitations.

Figure 1

The angle of the $S$-matrix elements for the sectors of total spin $S=2$ (green, middle curve), $S=1$ (blue, upper curve), and $S=0$ (red, lower curve) at total momentum $\kappa =0$ and different relative momenta ${\kappa }_{\mathrm{rel}}={\kappa }_1-{\kappa }_2$. The linear regime for small relative momenta is clearly visible, as well as the deviations from that regime at higher relative momenta. Calculations were done with bond dimension $D=64$.

Figure 2

The two-particle contribution for the spectral function $S(\kappa ,\omega )$ at momentum $\kappa =(\pi /10)$. Calculations were done with bond dimension $D=48$.

Figure 3

The magnetization $m$ versus applied magnetic field $h$ for the spin-1 Heisenberg chain. Our results (red, upper curve) with bond dimension $D=64$ are compared to the hard-core boson square-root dependence (blue, lower curve) [11] and first order corrections by the scattering length $a_2$ (green, middle curve) [15]. The direct MPS calculations (black dots) were done at the same bond dimension of $D=64$. The bottom-right inset provides a close-up of the phase transition. The top-left inset provides our result for the LL parameter $K$ in function of the magnetization $m$ (red), compared to the linear relation based on the scattering length (green) [31].