Predictive theory of finite-bond-dimension errors in the tensor renormalization group: convergence rates, effective correlation length, and spurious fixed points for universal critical data

The tensor renormalization group (TRG) of Levin and Nave coarse-grains the tensor-network representation of a classical partition function, and its only approximation is the truncation of intermediate indices to a fixed bond dimension $D$. Basu and Oganesyan (2022) computed critical exponents ($\gamma$, $\gamma/\nu$, $\delta$, $\eta$) and the susceptibility amplitude ratio of the isotropic and strongly anisotropic square-lattice Ising models and of the Yang-Lee edge critical point directly from TRG correlation functions, and documented three unexplained features: empirical convergence rates of order $1/D$ for some exponents and $1/D^5$ for others in the anisotropic model, no visible truncation-induced length scale in two-point correlators (in contrast to matrix-product-state pseudo-criticality), and spurious non-critical corner-double-line fixed points degrading convergence near the isotropic Onsager point. The open problem is a predictive account of how finite $D$ limits plain TRG: the mechanism and $D$-scaling of any truncation-induced effective correlation length, the origin of observable-dependent convergence rates, and the systematics of the spurious fixed points. Later work established the analogous result for disentangling schemes (Loop-TNR, HOTRG) — finite $D$ does impose a correlation length $\xi(D)\sim D^\kappa$ with the matrix-product-state finite-entanglement exponent, demonstrated numerically and mechanistically but not yet proven even for TNR — while the plain-TRG mechanism, the convergence-rate theory, and the spurious-fixed-point systematics remain open.

Background

A two-dimensional classical lattice model can be represented exactly as a network of local tensors whose contraction gives the partition function. The tensor renormalization group (TRG) of Levin and Nave (2007) coarse-grains such a network iteratively: each step decomposes local tensors by singular value decomposition (SVD — the factorization of a matrix into two orthogonal matrices and a diagonal matrix of non-negative singular values) and recombines them into tensors on a lattice of half the sites. The only approximation is keeping at most $D$ singular values per new index, where $D$ is the bond dimension, so all errors of the method are finite-$D$ errors. At criticality the quantities of interest are universal data — critical exponents and amplitude ratios that are independent of microscopic details and fixed, in two dimensions, by the conformal field theory (CFT) of the critical point.

Three features of the method frame the problem. First, in variational one-dimensional tensor methods, a matrix product state (MPS — the one-dimensional analogue of a truncated tensor network) with finite $D$ carries at most finite entanglement, so at criticality it mimics the system cut off at a finite correlation length $\xi(D)\sim D^\kappa$, with $\kappa=6/(c(1+\sqrt{12/c}))$ for central charge $c$ (Tagliacozzo et al. 2008). This finite-entanglement scaling converts truncation errors into finite-length effects and underlies systematic extrapolation. Whether an analogous mechanism exists in TRG is a priori unclear because TRG is not variational. Second, the iterated plain-TRG map is known to flow, in a range of temperatures near the critical temperature $T_c$ that presumably shrinks with $D$, to spurious non-critical fixed-point tensors of corner-double-line (CDL) type — tensors whose index structure carries only short-range correlations running around the corners — rather than to the critical fixed point (documented by Gu and Wen 2009 and Evenbly and Vidal 2015). Disentangling schemes were invented to remove these artifacts: tensor network renormalization (TNR, Evenbly and Vidal 2015), loop-TNR, Gilt-TNR based on graph-independent local truncations (Hauru et al. 2018), and the higher-order-SVD scheme HOTRG (Xie et al. 2012). Third, TRG tolerates non-positive statistical weights, so it can reach critical points that are inaccessible to Monte Carlo simulation because of the sign problem. The prime example is the Yang-Lee edge critical point — the endpoint of the branch cut of zeros of the ferromagnetic Ising partition function at complex magnetic field $h_c=\pm i|h_c|$ above $T_c$ — described by a complex $\phi^3$ field theory (Fisher 1978) and in two dimensions by the non-unitary minimal model $M(5,2)$ (Cardy 1985), with exact exponents $\sigma=-1/6$ and $\eta=-4/5$.

In this setting Basu and Oganesyan (arXiv:2204.06763, 2022) computed one- and two-point correlation functions directly within plain TRG at $D\le 28$ and extracted $\gamma$, $\gamma/\nu$, $\delta$, $\eta$ and the susceptibility amplitude ratio for three critical points: the isotropic Ising model at the Onsager point (the exactly solved critical point of the square-lattice Ising model), a strongly anisotropic (coupling anisotropy 10) Ising model, and the Yang-Lee critical point. They found quantitative agreement already at $D=16$, clear convergence trends in $D$ for the anisotropic and Yang-Lee cases but not for the isotropic model, empirical convergence of order $1/D$ for the pair $(\eta,\delta)$ and of order $1/D^5$ for the pair $(\gamma,\gamma/\nu)$ in the anisotropic model, no detectable finite-length signature in the two-point correlators, and they identified CDL-type spurious fixed points as a candidate culprit for the poor isotropic behavior. They explicitly stated that they had no theory for any of these observations.

Problem Statement

Construct a predictive theory of finite-bond-dimension truncation effects in the plain (Levin-Nave) tensor renormalization group as applied to critical points of two-dimensional classical models, so that the accuracy of universal data extracted from TRG correlation functions can be bounded and extrapolated in $D$ rather than merely observed. The single overarching objective — predictive control over TRG-extracted universal critical data — decomposes into the following jointly supporting questions, whose common object is the $D$-dependence of the iterated plain-TRG map near criticality:

1. Effective correlation length: determine whether finite-$D$ truncation in the iterated plain-TRG flow generates a well-defined effective correlation length in physical correlation functions, in analogy to matrix-product-state pseudo-criticality; establish the mechanism, including its relation to attraction toward spurious CDL fixed points near the critical point, and derive how the length scales with $D$ (or demonstrate rigorously that no such length is generated in plain TRG correlation functions at fixed $D$).

2. Convergence rates: derive the asymptotic $D$-dependence of errors in universal observables computed from TRG correlation functions — in particular explain why in the anisotropy-10 Ising model the exponents $\eta,\delta$ converge empirically as $1/D$ while $\gamma,\gamma/\nu$ converge as $1/D^5$ — and give predictions for the corresponding rates in other models and at other critical points, including the isotropic Onsager point where no clear convergence was observed.

3. Spurious fixed points: elucidate the role of non-critical corner-double-line fixed points in degrading convergence near the isotropic critical point, in particular how their presence and influence depend on the choice of tensor-network mapping, on lattice symmetry, and on weak explicit symmetry-breaking fields (the source observed long-distance errors of fixed sign that it attributed to the finite-dimensional approximation of the critical fixed-point tensor).

Beyond these core questions, the source's outlook mentions, as optional directions rather than required components of an answer: comparing the convergence trends of plain TRG with those of more sophisticated tensor-renormalization schemes and with exponent values obtained by other methods, and exploring whether such operator-level correlator computations remain viable and controllable for critical points inaccessible to Monte Carlo methods (Yang-Lee-type points with complex weights), for higher-dimensional models, and for models with gauge fields or topological terms. Answers to the core questions may be analytic (mechanism plus scaling laws), numerical-experimental (controlled $D$- and size-scans with scaling collapses), or a combination; an answer that refutes the premise — for example a demonstration that plain-TRG correlators at finite $D$ contain no well-defined induced length scale and that the observed convergence-rate patterns have a different origin — is equally admissible provided it covers the stated observables.

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