Tensor renormalization group in bosonic field theory

We compute the partition function of a massive free boson in a square lattice using a tensor network algorithm. We introduce a singular value decomposition of continuous matrices that leads to very accurate numerical results. It shows the emergence of a corner double line fixed-point structure. In the massless limit, we reproduce the results of conformal field theory including a precise value of the central charge.

Figure 1

Relative error in the free energy per site $\delta f$ as a function of the mass $m$ for a lattice with $L_1=L_2=2^{30}$ and maximal bond dimensions ${\chi }_{\max }$.

Figure 2

Left: $\delta f$ for $m=1.2\times {10}^{-6}$ as a function of the bond dimension for $L_1=L_2=2^{30}$. Right: Singular values of $B_4^{\varphi }$ with no truncation. For large masses, they always join in equal value pairs. The two top curves correspond to doubly degenerate singular values.

Figure 3

Singular values of $B_n^{\pi }$. Left: RG flow for $m={10}^{-5}$ and ${\chi }_{\max }=24$. Right: As a function of the mass for $n=8$ and ${\chi }_{\max }=8$.

Figure 4

Left: Indicator $P_{\mathrm{CDL}}$ for $m={10}^{-5}$ and ${\chi }_{\max }=24$. The step at which it drops coincides with Fig. 3 (left). Right: RG cycles needed to reach a CDL structure. The theoretical argument (17) is shown in black.

Figure 5

Left: Singular values of $B_4^{\pi }$ without truncation. For small masses, the largest singular value is approximately $m^{-2}$. For large masses, they all converge to $m^{-2}$. Right: Singular values of $B_5^{\pi }$ for $m={10}^{-6}$.

Figure 6

Plot of the function $b(n,L_1)$ for $n=1,...,L_1-1$ and $L_1=100$. The red curve is $e^{-\pi n_1/L_1}$, and the blue curve is $e^{-\pi (L_1-n_1)/L_1}$.