Predictive theory of bond-dimension convergence and truncation-induced length scales in plain tensor renormalization group

The tensor renormalization group (TRG) of Levin and Nave extracts universal critical data from two-dimensional classical lattice models, with accuracy controlled only by the truncation bond dimension $D$ of the singular-value-decomposition coarse-graining step. Basu and Oganesyan computed critical exponents and an amplitude ratio from one- and two-point correlation functions at the isotropic and anisotropic (coupling anisotropy 10) Ising critical points and at the Yang-Lee edge singularity, and documented an unexplained convergence structure: empirical $1/D$ error decay for the exponents $\eta$ and $\delta$ but $1/D^5$ for $\gamma$ and $\gamma/\nu$ at anisotropy 10, no clear convergence trend at the isotropic Onsager point, and no theory connecting $D$ to errors in universal observables. The analogous finite-bond-dimension effect is now understood for disentangling schemes such as tensor network renormalization (TNR), the higher-order TRG (HOTRG), and the corner-transfer-matrix renormalization group (CTMRG): finite $D$ induces a correlation length $\xi(D) \propto D^\kappa$ matching matrix-product-state finite-entanglement scaling, generated by an emergent symmetry-respecting relevant perturbation; fixed-point computations have also reached very high accuracy via Newton and rotation techniques. However, for plain TRG the observable-resolved rate puzzle, the possible truncation-induced effective correlation length in critical correlators, the role of corner-double-line spurious fixed points near the isotropic critical point, and the quantitative comparison with disentangling schemes all remain open. The problem asks for a predictive truncation-error theory for plain TRG, validated against these explicit empirical anchors.

Background

The partition function of a two-dimensional classical lattice model such as the Ising model on a square lattice can be rewritten exactly as the contraction of a periodic network of identical local tensors, each carrying one index per leg. The tensor renormalization group (TRG) introduced by Levin and Nave coarse-grains such a network by splitting tensors along their legs with singular value decompositions (SVDs) and re-contracting them, halving the linear lattice size at each step; the only uncontrolled approximation is the truncation of the SVD spectra to a fixed bond dimension $D$ on each leg. Iterating the map contracts a macroscopic lattice onto a few tensors from which thermodynamic and long-distance quantities are read off. Unlike variational tensor methods, plain TRG has no error functional, so the magnitude of finite-$D$ errors and their scaling with $D$ are not built into the method and must be understood empirically or by separate theory.

Basu and Oganesyan applied plain TRG with impurity tensors (operator insertions whose renormalized versions track local operators along the coarse-graining flow) to compute universal critical data directly from one- and two-point correlation functions: the correlation-function decay exponent $\eta$, the critical-isotherm exponent $\delta$, the susceptibility exponent $\gamma$, the ratio $\gamma/\nu$ (with $\nu$ the correlation-length exponent), the Yang-Lee exponent $\sigma$ (the power-law exponent of the magnetization as a function of the distance from the edge singularity in the complex magnetic-field plane), and the universal susceptibility amplitude ratio $A = A_>/A_<$ between the disordered and ordered phases. They studied three exactly known two-dimensional critical points: the isotropic Ising (Onsager) critical point; an anisotropic Ising point with coupling anisotropy 10, which is the multicritical corner point of a critical ordered phase with modulated quasi-periodic order; and the Yang-Lee edge singularity, the endpoint of the line of Fisher zeros (loci in the complex magnetic-field plane where the Ising partition function vanishes) at the purely imaginary magnetic field $h = ih_c$ above $T_c$, a non-unitary conformal critical point whose complex Boltzmann weights defeat Monte Carlo importance sampling but not TRG. Their empirical findings define the problem: at the anisotropic and Yang-Lee points the exponents converge with $D$ (exact values reproduced to better than one percent at $D = 24$-$28$), with fitted error decay of order $1/D$ for the pair $\eta, \delta$ but of order $1/D^5$ for the pair $\gamma, \gamma/\nu$ at anisotropy 10, while at the isotropic point no comparably clear convergence trend appears; the authors state that they have 'no clear understanding of this result nor theory for what one might expect for the asymptotic convergence in bond dimension'.

The natural conceptual reference is the matrix product state (MPS), the one-dimensional variational tensor ansatz, for which finite-$D$ effects at criticality are well understood: bounding the entanglement content induces an effective correlation length $\xi_D \propto D^\kappa$ with $\kappa = 6/[c(\sqrt{12/c}+1)]$, where $c$ is the central charge of the underlying conformal field theory, rounding the transition into pseudo-criticality (finite-entanglement scaling; see also Burgelman et al. for pseudo-criticality traps in two-dimensional tensor-network simulations). Whether the non-variational, multiplicative truncation of TRG generates any analogous physical length scale was left open by the source: empirically, in their two-point functions, such a scale, if present, was 'either astronomically large or does not appear in simple two-point correlations'.

A known structural obstruction is the corner-double-line (CDL) tensor: a tensor that factors into products of small corner matrices around each plaquette, carries only short-range correlations, and is a non-critical fixed point of any local coarse-graining step, including plain TRG. A range of temperatures around the isotropic critical point flows into such spurious CDL fixed points, degrading the critical data; the source found that tiny symmetry-breaking fields appear to improve convergence near that point, while how the artifacts depend on the choice of network-to-tensor mapping, SVD ordering, normalization conventions, and lattice symmetries remained, in their words, 'unclear, based on published previous work'. Disentangling algorithms — tensor network renormalization (TNR), the tensor-entanglement-filtering renormalization (TEFR), loop-TNR, Gilt-TNR (graph-independent local truncations), and the higher-order TRG (HOTRG) — were built specifically to strip CDL structure, at additional computational cost.

Later work has clarified the neighboring picture without solving the plain-TRG problem. Ueda and Oshikawa showed that for disentangling schemes (Loop-TNR, HOTRG) finite $D$ at criticality induces a finite correlation length $\xi(D) \propto D^\kappa$ with the MPS exponent, generated by an emergent relevant perturbation that respects the model symmetries (the thermal operator for Ising; a symmetry-breaking magnetic one appears only through machine-precision roundoff). Huang, Chan, Kao and Chen found the same $D^\kappa$ crossover length for HOTRG strip transfer-matrix computations, and Chan and Chen turned the crossover-scale separation into an operational framework for extracting conformal data across three tensor-network schemes. Ebel, Kennedy and Rychkov obtained very high-quality fixed-point tensors for Gilt-TNR via Newton and rotation techniques (accuracy about $10^{-9}$ at bond dimension 30) and, with Ebel's high-temperature three-dimensional analysis, started a computer-assisted rigorous program. Operator-level TRG computations of conformal data (Bal et al.; Lyu, Xu and Kawashima) likewise exist without error-scaling predictions. None of this addresses plain TRG: Ueda and Oshikawa explicitly note that the finite-$D$ effect is 'even stronger and non-trivial for TRG', and the observable-resolved rate pattern, the isotropic failure, and the CDL implementation-dependence in plain TRG remain without a theory.

Problem Statement

Construct a predictive theory of finite-bond-dimension truncation effects in the plain (Levin-Nave) tensor renormalization group when it is used to extract universal critical data, replacing the current empirical monitoring of $D$-convergence with controlled error scaling. Concretely, for the three benchmark critical points of the source study — the isotropic Ising (Onsager) critical point, the anisotropy-10 Ising critical point, and the Yang-Lee edge singularity at imaginary magnetic field — the theory should deliver, as one linked objective:

(a) a prediction of how the error of each universal observable (the exponents $\eta$, $\delta$, $\gamma$, $\gamma/\nu$, the Yang-Lee exponent $\sigma$, and the amplitude ratio $A$; 'error' meaning the deviation of the TRG estimate from the exactly known value) scales with the bond dimension $D$ of plain TRG, explaining in particular the empirically observed approximately $1/D$ decay for $\eta$ and $\delta$ versus the approximately $1/D^5$ decay for $\gamma$ and $\gamma/\nu$ at anisotropy 10, and the absence of a comparably clear convergence trend at the isotropic point;

(b) a determination of whether finite-$D$ truncation in the multiplicative TRG flow induces a well-defined effective correlation length in critical correlation functions — the analog of matrix-product-state finite-entanglement pseudo-criticality, now established for disentangling schemes — and, if so, its scaling with $D$ and its observable signatures (the source found any such length to be 'astronomically large or [not appearing] in simple two-point correlations');

(c) an identification of the mechanism behind the degradation near the isotropic critical point, in particular the role of spurious non-critical corner-double-line (CDL) fixed points and how their onset and influence depend on implementation choices (network-to-tensor mapping, SVD ordering, normalization conventions) and on tiny symmetry-breaking fields, which empirically appear to improve convergence.

Deliverables (a)-(c) form the scientific core — a truncation-error mechanism with observable-resolved rate predictions for the benchmark critical points; together they constitute predictive control over TRG-based universal-data extraction. The source's outlook additionally flags, as optional directions rather than as part of this problem's completion boundary, extension to higher-dimensional models, critical points with several relevant perturbations, gauge fields and topological terms, and exploration of the convergence trends of more sophisticated tensor-renormalization schemes in comparison with plain TRG; these are recorded as adjacent future work.

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