Rigorous derivation of the finite-bond-dimension scaling of the correlation length in tensor network renormalization

Tensor network renormalization (TNR) coarse-grains two-dimensional classical partition functions with a truncation bond dimension $D$, and at criticality the truncation induces a finite effective correlation length $\xi(D)$ on the coarse-grained transfer matrix. Numerical work supports the power law $\xi(D)\sim D^{\kappa}$ with $\kappa=6/[c(1+\sqrt{12/c})]$, where $c$ is the central charge of the infrared conformal field theory — the same exponent as matrix-product-state (MPS) finite-entanglement scaling — but no rigorous, model-independent derivation exists; even the matrix-product-state heuristic rests on still-unverified assumptions about the analytic structure of the critical entanglement spectrum. The problem is to establish this scaling law (or its correct form) from first principles within TNR: the exponent value and its universality across TNR variants and models of equal central charge, a controlled characterization of the truncation-induced effective perturbation of the fixed point, and the common origin of the scaling in MPS, corner-transfer-matrix renormalization group (CTMRG), and TNR truncations. Recent work has sharpened the mechanism question — the induced perturbation is not always the most relevant conformal operator — so the controlled theory is both open and urgent.

Background

Tensor network renormalization (TNR) is a real-space coarse-graining method for two-dimensional classical statistical models: the partition function is written as a contracted network of local tensors, and disentanglers (unitary tensors that cancel short-range entanglement between neighboring bonds) together with isometries (truncating maps that reduce the number of retained states) eliminate short-range entanglement at each step, so that at a critical point the coarse-grained network flows toward a scale-invariant fixed-point tensor encoding the universal physics (Evenbly and Vidal, Phys. Rev. Lett. 115, 180405 (2015)). Practical accuracy is limited by the bond dimension $D$, the number of states kept per internal index under truncation. Although TNR nominally handles arbitrarily large systems, Ueda and Oshikawa showed, through TNR finite-size scaling of transfer-matrix spectra for the critical Ising and three-state Potts models (Phys. Rev. B 108, 024413 (2023), summarized in Ueda's thesis arXiv:2401.18068), that a finite $D$ forces a finite effective correlation length $\xi(D)$ on the coarse-grained transfer matrix — a matrix-product-operator object whose leading eigenvector behaves like a matrix product state (MPS, a one-dimensional variational wavefunction whose expressive power is set by its bond dimension) of bond dimension $D$ — and that the finite-size spectra reveal this as an emergent relevant perturbation of the fixed-point Hamiltonian of the conformal field theory (CFT, the massless scale-invariant field theory describing the critical point, characterized by its central charge $c$; a "relevant" perturbation is one whose coupling grows rather than decays under coarse-graining, while an "irrelevant" perturbation is one whose coupling decays).

For one-dimensional quantum critical points an analogous finite-entanglement scaling is well established empirically: an MPS of bond dimension $\chi$ has a maximum correlation length $\xi(\chi)\sim\chi^{\kappa}$ with $\kappa=6/[c(1+\sqrt{12/c})]$. This heuristic formula is due to Tagliacozzo, de Oliveira, Iblisdir, and Latorre (Phys. Rev. B 78, 024410 (2008)), was developed by Pollmann et al. (Phys. Rev. Lett. 102, 255701 (2009)) and Pirvu et al. (Phys. Rev. B 86, 075117 (2012)), and combines the Calabrese–Lefevre form of the critical entanglement spectrum (the eigenvalues of the bipartite reduced density matrix, whose low-lying levels govern how much entanglement a truncated state can retain) with the CFT scaling of the bipartite entanglement entropy $S=(c/6)\ln\xi$; Vanhecke et al. (Phys. Rev. Lett. 123, 250604 (2019)) verified it numerically with high precision. The same exponent had earlier been observed in corner transfer matrix renormalization group (CTMRG, an iterative contraction method for two-dimensional classical models that truncates the corner and environment tensors to a finite bond dimension) finite-size scaling, and in higher-order tensor renormalization group calculations (H. Ueda, Okunishi, and Nishino, Phys. Rev. B 89, 075116 (2014)). Ueda's thesis presents numerical evidence that TNR obeys the same law — for the critical Ising model ($c=1/2$) the measured exponent is approximately $2$, close to the predicted $2.034\ldots$ — and argues heuristically, via the corner transfer matrix (the environment tensor obtained by contracting legs of the renormalized tensor, whose eigenvalues approximate the bipartite entanglement spectrum) and the four-renormalized-tensor representation of the half-system reduced density matrix, that MPS, CTMRG, and TNR truncations should share one scaling law. However, the thesis states explicitly that no mathematical proof is available at this point, and the underlying Calabrese–Lefevre assumptions on the analytic structure of the entanglement spectrum remain unverified (Schneider et al., SciPost Phys. 18, 142 (2025)). Moreover, Hodgkiss, Lootens, Ueda, Vanhecke, and Verstraete (arXiv:2607.15124 (2026)) recently computed the perturbations actually induced by finite bond dimension and found that, while a single critical Ising chain is dominated by the most relevant (magnetic-field) operator, two decoupled critical Ising chains acquire entangling subleading relevant perturbations that CFT scaling does not predict — so even the mechanism behind the scaling law is not yet under control.

Problem Statement

Provide a rigorous, model-independent derivation of the finite-bond-dimension scaling of the correlation length in tensor network renormalization. For a critical two-dimensional lattice model whose infrared limit is a conformal field theory of central charge $c$, coarse-grained by a TNR-type scheme truncated at bond dimension $D$: derive — or disprove and correct — the law $\xi(D)\sim D^{\kappa}$ with $\kappa=6/[c(1+\sqrt{12/c})]$, the same exponent as matrix-product-state finite-entanglement scaling. A complete resolution should include: (i) a controlled characterization of the effective perturbation of the fixed point induced by truncation, specifying which operators are induced and with what $D$-dependent couplings (the mechanism, recently shown to be subtler than perturbation by the most relevant CFT operator); (ii) the universality of the exponent across TNR variants (scheme families that share the coarse-graining philosophy but implement the truncation differently, for example Loop-TNR with loop-optimized disentanglers, Gilt-TNR with graph-independent local truncations of small tensor elements, HOTRG (higher-order tensor renormalization group) with direct directional coarse-graining, and plain TRG (tensor renormalization group, the precursor of TNR) with simple singular-value truncation) and across critical models with the same central charge; and (iii) the precise common origin of the scaling in MPS, CTMRG, and TNR truncations, including prefactors and crossover scales where accessible. An answer that establishes a corrected law (a different exponent, non-universal behavior, or universal logarithmic or irrelevant-operator corrections) is admissible provided it states exactly where the standard heuristic fails, agrees with the benchmark numerics, and additionally delivers components (i)–(iii) above; component (ii) is then to be read as determining the exponent's universality across TNR variants and models of equal central charge, or, if the exponent proves non-universal, the precise manner and scope of that non-universality.

References

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