Particles, Holes, and Solitons: A Matrix Product State Approach

We introduce a variational method for calculating dispersion relations of translation invariant ($1+1$)-dimensional quantum field theories. The method is based on continuous matrix product states and can be implemented efficiently. We study the critical Lieb-Liniger model as a benchmark and excellent agreement with the exact solution is found. Additionally, we observe solitonic signatures of Lieb’s type II excitation. In addition, a nonintegrable model is introduced where a $U(1)$-symmetry breaking term is added to the Lieb-Liniger Hamiltonian. For this model we find evidence of a nontrivial bound-state excitation in the dispersion relation.

Figure 1

Elementary topological excitation branches for $\gamma =c/\rho \approx 60.16$ at $D=64$ with $\theta =\pi$. Blue diamonds are particle excitations and red stars hole excitations. (a) Dispersion relation of Lieb’s type I excitation obtained from the topological ansatz (blue diamonds) and from the Bethe-ansatz solution (black line). (b) Dispersion relation of Lieb’s type II excitation obtained from the topological ansatz (red stars), the trivial ansatz (green squares), and from the Bethe-ansatz solution (black line).

Figure 2

Dispersion relation for the Lieb-Liniger model with $D=64$ for $\gamma =c/\rho \approx 60.16$. Bulk excitations (dots) are trivial excitations. Blue diamonds (type I) and red stars (type II) are obtained by combining momenta and energy of two topological excitations (hole and particle) of Fig. 1.

Figure 3

The change in particle number expectation value for the hole state at momentum $p=0$ as a function of $\gamma$, as predicted by the Bethe solution [34] and with the cMPS excitation ansatz.

Figure 4

Top: Dispersion relation of ${\widehat{H}}^{\prime }$ with $D=22$ for $\gamma \approx 26.4$, $\mu =1$, and $u=1$. Blue dots are the pairwise sum of all topological excitations. Red circles are trivial excitations and the right-hand inset shows the topological spectrum. Bottom: Convergence of the two lowest trivial eigenvalues and the lowest two-kink excitation energy as a function of $D^{-1}$ at $p=0$.