Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group

TL;DR

The paper addresses how to extract full conformal-field-theory data from high-precision fixed-point tensors produced by tensor-network renormalization, focusing on the 2D Ising CFT. It compares two locality-based approaches—transfer matrix (TM) and lattice dilatation operator (LDO)—applied to fixed points obtained via a rotating Newton method, showing excellent agreement with the Ising spectrum up to $Δ=4\tfrac{1}{8}$ and reasonable accuracy beyond. The main contributions include a detailed construction of TM (direct and crossed) and LDO for Gilt-TNR, extended spin information (modulo integers), and the demonstration that non-universal Jacobian eigenvalues can be explained by their equivalence to LDO operators. The results reinforce the connection between tensor RG fixed points and CFT data, and they illuminate the roles of rotation, locality, and fixed-point accuracy in accurately recovering scaling dimensions and spins, with potential extension to higher dimensions and OPE data in the future.

Abstract

Tensor network renormalization group maps study critical points of 2d lattice models like the Ising model by finding the fixed point of the RG map. In a prior work arXiv:2408.10312 we showed that by adding a rotation to the RG map, the Newton method could be implemented to find an extremely accurate fixed point. For a particular RG map (Gilt-TNR) we studied the spectrum of the Jacobian of the RG map at the fixed point and found good agreement between the eigenvalues corresponding to relevant and marginal operators and their known exact values. In this companion work we use two further methods to extract many more scaling dimensions from this Newton method fixed point, and compare the numerical results with the predictions of conformal field theory (CFT). The first method is the well-known transfer matrix (TM), while the second method we refer to as the lattice dilatation operator (LDO). We introduce some extensions of these method that provide also spins of the CFT operators, modulo an integer. With comparable computing resources, the TM and LDO methods perform equally well. The agreement for the scaling dimensions and spins is excellent up to the scaling dimesnion 4.125, and reasonably good up to 2 units higher. Some of the eigenvalues of the Jacobian of the RG map can come from perturbations associated with total derivative interactions and so are not universal. In some past studies (arXiv:2102.08136, arXiv:2305.09899) such non-universal eigenvalues did not appear in the Jacobian. We explain this surprising result by showing that their RG map has the unusual property that the Jacobian is equivalent to the LDO operator.

Figures (6)

• Figure 1: $N\times M$ tensor network built out of a four-legged tensor $A$. We are assuming periodic boundary conditions, so that the outgoing bonds on network sides are contracted pairwise.
• Figure 2: A non-rotating RG map (left) vs a rotating RG map (right)
• Figure 3: The distance between two consecutive RG iterates as a function of the RG step. The Gilt-TNR parameters are $\chi=30$, $\epsilon_{\rm gilt}=6\times 10^{-6}$. Figure from Ebel:2024nof.
• Figure 4: Convergence of the Newton method with a fixed approximate Jacobian for different ranks $s$ of the $P_s$ projector in \ref{['eq:approxJ']}. Figure from Ebel:2024nof.
• Figure 5: Exact spectrum of the critical 2D Ising CFT.

Theorems & Definitions (2)

• Definition 3.1
• Remark 4.1