Tensor network simulations for nonorientable surfaces

Abstract

In this study, we explore the geometric construction of the Klein bottle and the real projective plane ($\mathrm{RP}^2$) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on $\mathrm{RP}^2$ with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.

Figures (8)

• Figure 1: Illustrations of the partition functions on (a) the Klein bottle and (b) $\mathrm{RP}^2$. These partition functions can be cut in the middle as shown in the blue arrows(left panel), and then sewn again after flipping the latter half in $x$-direction. The resulting boundaries are crosscap states for the Klein bottle and rainbow and crosscap states for $\mathrm{RP}^2$. The schematic pictures of the crosscap and rainbow boundaries are shown in (c) and (d), respectively. (e) In the limit $L\gg\beta\gg1$, the bulk part of the partition function $Z^{\mathrm{RP}^2}(\beta,L)$ is dominated by the leading eigenvector $|i_0\rangle$ of the transfer matrix on a cylinder. The boundary contributions are then represented as the overlaps with $|i_0\rangle$ as denoted as $\langle\mathcal{R}|i_0\rangle$ and $\langle i_0|\mathcal{C}\rangle$.
• Figure 2: Tensor network representations of a crosscap boundary condition. The six triangles are the left-hand side isometries to renormalize the bulk tensor $T^{(0)}$ into $T^{(2)}$, with four smaller ones being $U^{(1)}$ and two larger ones $U^{(2)}$. The arrows indicate the order of two indices before amalgamation and truncation, which correspond to the order of $x_1$ and $x_2$ in $U^{(n)}_{xx_1x_2}$.
• Figure 3: Tensor network representations of a rainbow boundary condition. With the spatial reflection operator $O^{(n)}$ in between the isometries, the lower half of the indices are in the reverse order.
• Figure 4: Alternative way to construct a rainbow boundary condition in the tensor network. Instead of contracting a renormalized spatial reflection operator, we just reflect the vertically copied bulk tensor $T^{(n)}$ in the vertical direction, which means that the included isometries are also reflected.
• Figure 5: The rainbow free energy term $F_\mathcal{R}$ of the $q$-state Potts models ($q=2,3$ and $4$) obtained from HOTRG with the bond dimension $\chi = 30, 45$ and $80$. For each model, these values are obtained by both the renormalization of the spatial reflection operator as in Eq. \ref{['eq:fr-sro']} and by reflecting the left-hand side isometries as in Eq. \ref{['eq:fr-refl']}, which are consistent and indistinguishable in the figure. They are compared with the theoretical predictions denoted by the dashed lines.