Continuous Matrix Product States for Quantum Fields

We define matrix product states in the continuum limit, without any reference to an underlying lattice parameter. This allows us to extend the density matrix renormalization group and variational matrix product state formalism to quantum field theories and continuum models in 1 spatial dimension. We illustrate our procedure with the Lieb-Liniger model.

Figure 1

Energy density as a function of the interaction parameter $c$ for different values of $D=2$, 4, 6, 8 (from top to bottom). The result for $D=8$ is indistinguishable from the one given by the Bethe Ansatz. The inset shows the relative error $\Delta E=(e-e_{\mathrm{Bethe}})/e_{\mathrm{Bethe}})$ (where $E_{\mathrm{Bethe}}$ is the energy given by the Bethe Ansatz solution), as a function of $D$ for $c=0.2$, 2, 20, and 200 ($x$, *, $+$, and $o$, respectively). We show the results for up to $D=10$; the saturation of the accuracy with $D=10$ is due to insufficient convergence of the results.

Figure 2

(a) Off-diagonal elements of the one-particle reduced density operator as a function of the distance in a logarithmic scale for (from top to bottom): $c=0.2$, 2, 20, 200, and $c=2000$. For reference we have also drawn a straight line with $\mathrm{\text{slope}}=1/2$, which is the slope corresponding to the Tonk-Girardeau limit ($c\to \infty$). As it can be seen, the slope of the curves approaches $1/2$ as $c$ increases. (b) Two-body density-density correlation function for the same values of $c$ (in the left, from bottom to top). For large $c$ one can observe the Friedel oscillations corresponding to the Tonks-Girardeau limit. All the results have been calculated with $D=14$.