Computing the renormalization group flow of two-dimensional ${\varphi }^4$ theory with tensor networks

We study the renormalization group flow of ${\varphi }^4$ theory in two dimensions. Regularizing space into a fine-grained lattice and discretizing the scalar field in a controlled way, we rewrite the partition function of the theory as a tensor network. Combining local truncations and a standard coarse-graining scheme, we obtain the renormalization group flow of the theory as a map in a space of tensors. Aside from qualitative insights, we verify the scaling dimensions at criticality and extrapolate the critical coupling constant $f_{\mathrm{c}}=\lambda /{\mu }^2$ to the continuum to find $f_{\mathrm{c}}^{\mathrm{cont}}=11.0861\left(90\right)$, which favorably compares with alternative methods.

Figure 1

RG flow in the space of tensors for $\lambda =0.1$ at five different values of ${\mu }_0^2$ using TRG (upper panel) and GILT-TNR (lower panel). For every linear system size (or equivalently number of RG steps), the 45 largest singular values of the coarse-grained tensors computed according to (24) are displayed. Each line corresponds to the flow of a given singular value. For both TRG and GILT-TNR, the maximal bond dimension is $\chi =110$, and the threshold parameter was chosen to be ${\epsilon }_{g}ilt=5\times {10}^{-8}$ for GILT-TNR. Whereas TRG yields nonuniversal fixed points, GILT-TNR provides the right fixed-point structure away from criticality and reveals a nontrivial fixed point at criticality that is characteristic of the underlying CFT.

Figure 2

RG flows in the space of tensors at criticality for four different values of $\lambda$ using the GILT-TNR algorithm at bond dimension $\chi =110$. After a number of iterations that grow as $\lambda$ gets close to zero, the same nontrivial fixed point, which is characteristic of the underlying CFT, is observed. On the right-hand panel is depicted a heuristic picture of the RG flow, which illustrates the fact that as we go to the continuum limit, the QFT UV fixed point, with a continuum of eigenvalues, starts being visible.

Figure 3

Critical coupling $f_c=\lambda /{\mu }_{\mathrm{c}}^2$ as a function of $\lambda$, computed for $\chi =110$ and ${\epsilon }_{g}ilt=5\times {10}^{-8}$. The error bars are estimated as described in Appendix pp2. The first fit up to linear order does not include the point $\lambda =0.1$ as it is expected to be valid only for smaller values of the coupling. The right-hand panel shows a narrower view of the same figure close to $\lambda =0$.

Figure 4

Example of convergence in bond dimension of the critical coupling for $\lambda =0.01$ and different values of ${\epsilon }_{gilt}$.