Volume extrapolation via eigenvector continuation

We develop an extension of eigenvector continuation (EC) that makes it possible to extrapolate simulations of quantum systems in finite periodic boxes across large ranges of box sizes. The formal justification for this approach, which we call finite-volume eigenvector continuation (FVEC), is provided by matching periodic functions at different box sizes. As concrete FVEC implementation we use a discrete variable representation based on plane-wave states and present several applications calculated within this framework.

Figure 1

Positive-parity energy spectrum of two particles in finite volume as a function of the box size $L$ for a Gaussian potential (11) with $R=2$ and $V_0=-4.0$ in natural units (see text). Solid lines show the three lowest energy levels calculated in DVR basis with $N=32$. Dashed lines indicate FVEC results obtained based on training data from four different box sizes (solid circles).

Figure 2

Positive-parity finite-volume energy spectrum of three bosons interacting via the potential (12). Solid lines show the exact states calculated in DVR bases with $N\le 28$, whereas dashed lines indicate FVEC results obtained based on training data at five different box sizes (solid circles). The FVEC calculation was performed using $8\times 5=40$ training states, which includes the $A_1^{+}$ ground state not shown in the plot. See text for details.

Figure 3

Negative-parity $S_z=1/2$ finite-volume energy spectrum of three neutrons interacting via a separable contact potential fit to reproduce the neutron-neutron scattering length $a_{nn}=-18.9\mathrm{fm}$. Solid lines show the exact states calculated in DVR bases with $N\le 22$, whereas dashed lines indicate FVEC results obtained based on $N=22$ training data at three different box sizes (solid circles). The first and third levels shown in the plots are $T_1^{-}$ states with total spin $S=1/2$. The second level is a (noninteracting) $S=3/2 T_1^{-}$ state, whereas the fourth level is a $T_2^{-}$ state with $S=1/2$. A total number of $3\times 8=24$ training data were used to generate this plot, covering a subset of states from the four three-dimensional multiplets (see text for details).

Figure 4

FVEC calculation with uncertainty estimates for two particles interacting via a Gaussian potential with range $R=2$ and depth $V_0=-4.0$ (in natural units). A pool of 16 training data sets with $6\le L\le 9$ (indicated as dark shaded bands) was used to estimate the FVEC uncertainty by considering all combinations of $N_{\text{EC}}=4$ (left panel) and $N_{\text{EC}}=5$ (right panel) out of the overall pool. The range of all these individual calculations is shown as shaded bands.