Open problems of the Newton-with-rotation method for critical tensor network renormalization group fixed points (Ebel–Kennedy–Rychkov)

In the tensor network approach to two-dimensional classical statistical mechanics, the critical point of a lattice model is encoded in a four-index tensor that is a fixed point of a real-space renormalization group (RG) map. Ebel, Kennedy, and Rychkov [Phys. Rev. X 15, 031047 (2025)] showed that, after composing the Gilt-TNR map with a $\pi/2$ lattice rotation to flip the marginal stress-tensor Jacobian eigenvalues from $1$ to $-1$, Newton's method solves the fixed-point equation directly, reaching $\sim 10^{-9}$ accuracy in the Hilbert–Schmidt distance at bond dimension $\chi=30$ for the critical 2D Ising and 3-state Potts models — orders of magnitude beyond the traditional shooting method. The paper closes with nine explicitly stated open problems, whose common objective is to turn this demonstrated computation into a robust, general, and mathematically rigorous fixed-point technology: enlarging the Newton convergence basin (ideally from the critical nearest-neighbor Ising tensor) via a globally differentiable map; applying the Newton search to other universality classes (tricritical Ising, Lee–Yang, coupled Potts); extending it to other disentangling algorithms such as TNR, whose differentiability is unproven; a constructive test separating total-derivative (redundant) Jacobian eigenvectors from genuine conformal quasiprimaries; robust gauge fixing for symmetry-breaking perturbations; the 'phantom' reflection-symmetric map puzzle behind the sufficiency of orthogonal gauge fixing; rotation-preserving maps that solve the corner-double-line (CDL) problem; the $\chi\to\infty$ Hilbert–Schmidt limit of the fixed-point tensors; and a rigorous computer-assisted existence proof of critical tensor RG fixed points.

Background

In the tensor network formulation of statistical mechanics, the partition function of a 2D lattice model such as the Ising model is rewritten as the full contraction of a periodic network of identical four-index tensors $A_{abcd}$, one per site. A tensor renormalization group (tensor RG) map replaces this network by a smaller equivalent one built from a new tensor, preserving the partition function exactly while truncating all indices to a fixed maximum size $\chi$, called the bond dimension. Iterating the map generates a discrete flow in the space of tensors; at a critical point the flow approaches a fixed-point tensor $A_*$ which encodes the universal properties of the transition. Linearizing the map around $A_*$ gives a Jacobian whose eigenvalues are related to the scaling dimensions $\Delta$ and spins $\ell$ of the operators of the conformal field theory (CFT) describing the transition: $\lambda=b^{2-\Delta}$ for a map of linear rescaling factor $b$ that does not rotate the lattice, and $\lambda=i^{\ell}b^{2-\Delta}$ when a $\pi/2$ rotation per step is included.

Two structural facts make finding the fixed point difficult. First, the earliest coarse-graining scheme (TRG, Levin and Nave, 2007) cannot eliminate corner-double-line (CDL) tensors — product-structured tensors whose short-range loop correlations accumulate and block the flow to the correct fixed point — so modern algorithms add disentangling steps: TNR (Tensor Network Renormalization, Evenbly and Vidal, 2015) inserts optimized disentangler unitaries, while Gilt-TNR (graph-independent local truncations, Hauru, Delcamp, and Mizera, 2018) truncates small singular values of a bond-space matrix. Second, the critical fixed point is not isolated: deformations that change the aspect ratio of the coordinate frame — the CFT stress tensors $T,\bar T$ with $\Delta=2$, $\ell=\pm 2$ — have Jacobian eigenvalue exactly $1$, so the fixed points form a two-parameter family and Newton root-finding on $f(A)=R(A)-A=0$, which requires $\nabla R-1$ to be invertible, breaks down. Ebel, Kennedy, and Rychkov (Phys. Rev. X 15, 031047 (2025)) overcame the second obstruction by composing the Gilt-TNR map with a $\pi/2$ lattice rotation, which turns the eigenvalues $1=e^{i\theta\ell}$ ($\ell=\pm 2$, $\theta=0$) into $-1$ ($\theta=\pi/2$), making the fixed point isolated and Newton-convergent. Starting from a tensor pre-processed by a few RG steps (obtained by the traditional 'shooting' method of fine-tuning the temperature and iterating RG, which only reaches accuracies $10^{-2}$–$10^{-5}$), the Newton method converges to the fixed point to $\sim 10^{-9}$ in the Hilbert–Schmidt (Frobenius) norm at $\chi=30$, for both the 2D Ising and 3-state Potts critical points.

Several further concepts are needed for the open problems. (i) The Jacobian spectrum contains two kinds of eigenvalues: those associated with CFT quasiprimaries (operators invariant under the global conformal group), which are universal and predicted by $\lambda=b^{2-\Delta}$ up to the spin phase, and those associated with total-derivative (redundant) operators, which are non-universal and unpredictable from the CFT. Several eigenvalues observed at the Ising fixed point match no quasiprimary and were conjecturally assigned to derivative operators. (ii) Tensor networks have a gauge freedom: inserting an invertible matrix on a bond and its inverse on the conjugate bond leaves the partition function invariant, so RG iterates must be gauge-fixed to converge to a definite tensor; the paper's combined continuous/discrete gauge-fixing step is only partly robust, and for $\mathbb{Z}_2$-odd (symmetry-breaking) perturbations it occasionally fails. A puzzle noted in the paper is that fixing the gauge with orthogonal transformations only — rather than general invertible ones — empirically suffices for many maps that do not even preserve reflection symmetry; a satisfactory explanation exists only for maps that do preserve reflection symmetry, while for the others the paper offers a heuristic argument via a 'phantom' reflection-symmetric RG map that was not constructed. (iii) Kennedy and Rychkov initiated a rigorous analytic theory of tensor RG maps acting on infinite-dimensional tensor spaces in Banach norms [J. Stat. Phys. 187 (2022) 33; Ann. Henri Poincaré 25 (2024) 773], with rigorous results for the high- and low-temperature fixed points, extended to three dimensions by Ebel [arXiv:2401.04229]; no analogous rigorous construction exists for the critical fixed point. The published paper concludes with a list of nine open problems, which together ask whether the Newton-with-rotation computation can be made robust (large convergence basin, stable gauge fixing), general (other algorithms, other universality classes, the $\chi\to\infty$ limit), trustworthy (operator identification, conceptual underpinnings), and rigorous (mathematical existence proofs). Companion and related work has addressed adjacent questions: the same authors' transfer-matrix and lattice-dilatation-operator study accounts for the absence of non-universal eigenvalues in 'frozen' linearized tensor RG maps, a computer-assisted construction rigorously establishes the high-temperature fixed point, and lattice-symmetry tensor RG schemes with rotation-symmetric CDL filtering have been constructed; these results are adjacent to, and leave open, the items below.

Problem Statement

The common objective is to develop the Newton-with-rotation solution of critical tensor RG fixed-point equations, demonstrated by Ebel–Kennedy–Rychkov for Gilt-TNR at $\chi=30$ on the 2D Ising and 3-state Potts models, into a robust, general, and mathematically rigorous fixed-point technology, by resolving the open problems listed in Section 5 of the source paper (numbering as in the published version). Each of the nine items is an independently posed open problem that can be settled on its own: fully resolving any one of the nine items resolves this task, and the remaining items are optional; partial progress that leaves an item's question open does not resolve it.

1. (Basin and differentiability) Enlarge the domain of Newton convergence — ideally so that the iteration converges from the bond-dimension-2 nearest-neighbor critical Ising tensor itself, without first applying RG steps — by resolving the discrete gauge-fixing obstruction of the source's Section 4.2.1 and obtaining a globally differentiable tensor RG map.
2. (Generality across universality classes) Apply the Newton search to locate and study critical fixed points of other universality classes: the tricritical Ising point (initializing from the tricritical point of the Blume–Capel model, whose location is known from Monte Carlo), the Lee–Yang fixed point (Ising model in a critical imaginary magnetic field, which is inaccessible to Monte Carlo due to the sign problem), and fixed points with no exactly solved CFT, such as the coupled 3-state Potts model. Resolving this item requires the Newton search to be carried out for all three named targets, not for only one of them.
3. (Generality across algorithms) Set up the Newton method for other disentangling tensor RG algorithms — notably TNR — combined with the $\pi/2$ rotation; in particular, determine whether such maps are differentiable, or even continuous, given their internal iterative optimization of disentanglers which may become trapped in local minima.
4. (Operator identification) Provide a constructive test, via explicit computations involving the Jacobian eigenvectors and the fixed-point tensor, that unambiguously confirms which eigenvectors are associated with total-derivative (redundant, non-universal) operators as opposed to genuine CFT quasiprimaries.
5. (Gauge robustness) Design a fully robust gauge-fixing procedure for symmetry-breaking (in particular $\mathbb{Z}_2$-odd) perturbations of the fixed point, replacing the present discrete gauge-fixing step that fails for some parameter choices and step sizes.
6. (Phantom map) Explain why gauge fixing with orthogonal gauge transformations suffices for tensor RG maps that possess fixed points although they do not preserve reflection symmetry: either exhibit the nearby reflection-symmetric 'phantom' RG map postulated in the source's heuristic argument (Section D.3.2), or find a rigorous argument that does not postulate its existence.
7. (Rotation-invariant maps) Construct practical tensor RG maps that manifestly preserve lattice rotations while solving the CDL problem, so that the Newton method can be implemented directly within the subspace of rotation-invariant tensors, bypassing the eigenvalue-1 problem entirely, and assess whether such maps can be implemented efficiently.
8. (Bond-dimension limit) Gather evidence for or against the hypothesis that the finite-$\chi$ fixed-point tensors of Gilt-TNR converge, as $\chi\to\infty$, to a limiting Hilbert–Schmidt tensor — for instance via higher-$\chi$ computations (up to $\chi=120$) with Jacobians evaluated by automatic differentiation rather than finite differences — or establish that some tensor RG map exists for which this convergence holds.
9. (Rigor) Prove the existence of critical tensor RG fixed points at the mathematical level of rigor — the program of Kennedy–Rychkov — by reformulating the fixed-point equation as a contraction in a neighborhood in an appropriate Banach space of infinite-dimensional tensors, for example via a computer-assisted construction extending the rigorous results already established for the high-temperature fixed point.

The items share a common theme: items 1–3 and 7 concern robustness and generality of the method, items 4–6 its trustworthiness and conceptual foundations, item 8 its controlled continuum limit, and item 9 its rigorous status.

References

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