Defect-preserving tensor-network coarse graining and the smearing-radius growth law at criticality

Guo and Wei (Phys. Rev. E 109, 034111 (2024)) showed that coarse-graining a critical two-dimensional classical lattice model with HOTRG (higher-order tensor renormalization group) combined with GILT (graph-independent local truncation) entanglement filtering, plus gauge fixing, accurately extracts conformal field theory (CFT) data of the vacuum — scaling dimensions, operator product expansion (OPE) coefficients, and correlation functions — but systematically corrupts localized information: point-like defects are smeared into clouds whose empirically observed smearing radius $r_s$ grows faster than the coarse-graining block size, and GILT artificially suppresses edge and corner modes, so defect correlation functions degrade without bound at large RG depth. Three of the six open directions posed in that work are addressed by later work: the finite-bond-dimension breaking of conformal invariance was quantified for Loop-TNR by Ueda and Oshikawa; the accuracy loss for operators with $\Delta\gtrsim 3$ was circumvented by Ebel, Kennedy and Rychkov's Newton-method fixed points together with finite-size-flow analyses; and the inverse problem of constructing a fixed-point tensor from conformal data was solved for rational CFTs (conformal field theories with finitely many primary operators) and the compactified boson by Cheng and collaborators. This task targets the surviving open core: defect-preserving coarse graining — modifying truncation and filtering decisions so that point-defect operators, their fusion, and edge modes survive with controlled fidelity while corner-double-line correlations are still removed — together with a quantitative theory of the smearing-radius growth law and its mitigation.

Background

Statistical systems at a continuous phase transition are described, in the scaling limit, by a conformal field theory (CFT) — a quantum field theory whose symmetry includes local scale and rotation transformations. In two dimensions a CFT is completely characterized by its data: the spectrum of local operators with their scaling dimensions $\Delta$ (which determine critical exponents), their conformal spins (the quantum number controlling how each operator transforms under spatial rotations), and the three-point couplings known as operator product expansion (OPE) coefficients $C_{ijk}$. Exact CFT data are known only in special cases, so extracting them numerically from lattice realizations is a basic tool of statistical mechanics.

A two-dimensional classical partition function can be rewritten exactly as the full contraction of a copy of a small tensor placed at every lattice site, with the local bond dimension given by the number of states per site. Real-space renormalization then becomes coarse-graining of this tensor network: the higher-order tensor renormalization group (HOTRG) of Xie et al. blocks tensors together and truncates each bond to a chosen bond dimension $\chi$ using higher-order singular value decompositions. At criticality the tensor is expected to flow toward a fixed-point tensor that encodes the CFT: scaling dimensions can be read off from the spectrum of the transfer matrix built from the fixed-point tensor (its eigenvalues on a cylinder scale as $\mu\sim b^{2-\Delta}$ for scale factor $b$), or from the eigenvalues of the linearized coarse-graining map (the lTRG method of Lyu, Xu and Kawashima), while OPE coefficients follow from the corresponding eigenvectors. Two obstacles had to be overcome for this picture to work. First, plain TRG-type schemes fail to remove corner-double-line (CDL) tensors — local correlation structures circulating around a single plaquette that keep the tensor away from a true fixed point; local entanglement filtering, introduced in the tensor-entanglement-filtering renormalization of Gu and Wen and implemented as GILT (graph-independent local truncations) by Hauru, Delcamp and Mizera, strips these short-range correlations from the tensor environment. Second, a tensor network has a large gauge freedom (redundant local reparametrizations), so the notion of a fixed point is meaningful only after gauge fixing; Guo and Wei adopted the minimal canonical form of Acuaviva et al. Their combined scheme (HOTRG + GILT + gauge fixing) reproduces the vacuum CFT data of the critical square-lattice Ising model — scaling dimensions up to $\Delta<3$, the OPE coefficient $C_{\sigma\sigma\varepsilon}$, and spin-spin correlation functions on the plane and on a torus — with accuracy comparable to the best contemporary methods.

The same work exposed what happens to localized information under such coarse graining. If a point-like defect — a modified bond or spin, or an inserted operator — is placed in the lattice and the whole network is coarse-grained, the defect does not remain a point: it is smeared into a cloud, described mathematically by replacing the defect operator by a Gaussian profile of radius $r_s$, so that defect two-point functions become convolutions of the exact ones with these profiles. The amount of smearing depends on where the defect sits inside its coarse-graining cell — defects near edges or corners of a block are smeared differently by HOTRG, and the GILT filtering step additionally suppresses edge modes artificially. When two smeared defects approach each other, they fuse at a coarser scale than in the exact theory. Most seriously, Guo and Wei observed empirically that the smearing radius grows faster than the linear size of the coarse-graining blocks, so that as the RG depth (number of coarse-graining steps) increases the smearing radius eventually overtakes physical distances and the information in defect correlation functions degrades without bound; their workaround is to sample defect positions randomly in the bulk, avoid edges and corners, and stop at moderate RG depth. None of these artifacts is explained by the extensive literature on the vacuum flow: the HOTRG truncation and the GILT filter are decided entirely from the translation-invariant environment and are blind to the defect subspaces they destroy. The authors therefore posed, as explicit future directions, the problems of making the coarse-graining and filtering decisions sensitive to defects while still removing CDL structures, and of understanding and mitigating the smearing-radius growth. They also posed sibling questions — quantitative control of the artificial length scale that truncation introduces into the vacuum flow, the accuracy loss for operators with $\Delta\gtrsim 3$, the inverse problem of constructing a fixed-point tensor from given CFT data, and extensions to quantum and three-dimensional models — which are outside the scope of this task.

Problem Statement

Consider a critical two-dimensional classical lattice model with known conformal field theory data (for definiteness the square-lattice Ising model at criticality, though the problem is posed for any critical model with known CFT data), represented as a tensor network that contains one or more point-like defects — localized modifications of bonds or spins, or inserted lattice operators — and coarse-grained by an algorithm of the HOTRG/GILT family at bond dimension $\chi$.

(1) Defect-preserving coarse graining. Construct modified coarse-graining and entanglement-filtering rules whose truncation and filtering decisions account for the presence of the defects: the tensor subspaces needed to represent point-like defect operators, the fusion behavior of two nearby defects, and the edge and corner modes of the coarse-graining cells must be preserved through many RG steps, while corner-double-line (CDL) short-range correlations are still removed and the defect-free part of the network still flows to the conformal fixed point with at least the quality of the unmodified vacuum algorithm.

(2) Theory and mitigation of smearing. Establish a quantitative characterization of the smearing of a point-like defect — the growth of the smearing radius $r_s$ with the coarse-graining block size, the RG depth, and the bond dimension $\chi$ — that explains the empirical observation that $r_s$ grows faster than the block size and therefore overtakes physical separations at large RG depth; and provide algorithmic or renormalization remedies for this growth, so that the degradation of defect observables with increasing RG depth is controlled.

The target setting is the two-dimensional classical case; the source's proposed extensions of the same defect-preserving objective to quantum lattice models (for example the transverse-field Ising chain) and to three-dimensional classical systems are natural continuations beyond this scope.

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