Extending the iPEPS Disorder-Parameter Framework and Its Finite Correlation Length Scaling to Continuous, Higher-Form, and Subsystem Symmetries

Xu and Huang (Phys. Rev. Lett. 134, 146503 (2025)) established that disorder parameters — expectation values of symmetry transformations acting on a spatial subregion — can be evaluated in the thermodynamic limit within infinite projected entangled pair states (iPEPS), and proposed a finite correlation length (finite-entanglement) scaling theory for them at quantum criticality, for discrete 0-form symmetries (the Z2 Ising and q-state Potts models). The paper's outlook leaves open the extension of this framework to continuous global symmetries and its generalization to higher-form and subsystem symmetries, with the stated aim of detecting topological (fracton) phases and their transitions. The extension to continuous global symmetries, higher-form symmetries, and subsystem symmetries is open.

Background

A disorder operator for a global on-site symmetry group $G$ of a quantum many-body system is the product of the on-site symmetry transformations over all sites of a spatial subregion $R$, i.e. $\prod_{i\in R} U_i(g)$ with $g \in G$; its expectation value, the disorder parameter, is a non-local diagnostic of symmetric phases that is complementary to the usual local order parameter. In the symmetry-broken phase it decays exponentially with the volume $|R|$ of the subregion (area law), while in the symmetric phase it decays exponentially only with the boundary size $|\partial R|$ (perimeter law). Under a generalized duality (gauging) transformation, the disorder parameter of $G$ maps to the order parameter of a higher-form symmetry of a dual lattice gauge model — a $p$-form symmetry is a generalized global symmetry whose charged objects are $p$-dimensional, so that a 0-form symmetry is an ordinary global on-site symmetry while a 1-form symmetry has line-like charged objects and is diagnosed by Wilson loops, i.e. expectation values of closed strings of the gauge field — and the perimeter-versus-area distinction becomes the diagnosis of spontaneous higher-form symmetry breaking, the symmetry language of topological order.

Infinite projected entangled pair states (iPEPS) are variational tensor-network ansätze for the ground states of two-dimensional quantum systems, built from a local tensor with bond dimension $D$ repeated on an infinite lattice; contracting the resulting network requires an approximate environment, usually computed with the corner transfer matrix renormalization group (CTMRG) with environment bond dimension $\chi$. A finite-$D$ iPEPS has a single emergent length scale $\xi(D,\chi)$ — finite entanglement content induces a finite correlation length, which is the basis of finite correlation length scaling (called finite entanglement scaling in the literature), the analogue of finite-size scaling at criticality. Xu and Huang (Phys. Rev. Lett. 134, 146503 (2025)) showed that for a discrete 0-form symmetry ($\mathbb{Z}_2$ Ising, $q$-state Potts) the disorder parameter $\tilde{m}_g = \langle \prod_{i\in R} U_i(g) \rangle$ can be evaluated in the thermodynamic limit directly from dominant eigenvalues of iPEPS transfer matrices (the row-to-row contraction maps whose leading eigenvalues encode the correlations and normalization of the infinite network), giving $\tilde{m}_g \sim \exp(-\alpha_{\mathrm{peri}}|\partial R| - \alpha_{\mathrm{area}}|R|)$ with both coefficients extracted without any subsystem-size extrapolation, and proposed the scaling theory $\alpha_{\mathrm{peri}}(y,\xi) \sim \alpha_{\mathrm{peri}}(y) + \xi^{-1} f_0(y\,\xi^{d+z-\Delta_y})$ and $\alpha_{\mathrm{area}}(y,\xi) \sim \alpha_{\mathrm{area}}(y) + \xi^{-2} f_1(y\,\xi^{d+z-\Delta_y})$ near criticality, where $y$ is a relevant coupling, $z$ the dynamical critical exponent, and $\Delta_y$ the scaling dimension of the relevant field. At the $(2+1)$D Ising critical point they found $\alpha_{\mathrm{area}} \to 0$ as $\xi \to \infty$: the disorder parameter obeys a perimeter law at criticality, i.e. the dual 1-form symmetry is spontaneously broken there. In topological phases the same transfer-matrix eigenvectors give access to a constant correction $\gamma$, the topological disorder parameter, related to the quantum dimension of the symmetry defect (the asymptotic growth of the number of distinct fusion outcomes of the defect with itself).

Disorder operators for continuous global symmetries are well developed outside tensor networks. For a small $U(1)$ rotation angle on a rectangular region at a $(2+1)$D conformal critical point, the disorder operator exhibits universal power-law corner corrections whose exponent is fixed by the current central charge $C_J$ of the critical theory — a universal measure of its gapless current fluctuations — confirmed by large-scale quantum Monte Carlo (QMC) in the Bose-Hubbard model (Wang, Cheng, and Meng, Phys. Rev. B 104, L081109 (2021)), with parallel results for two-dimensional Fermi and non-Fermi liquids (Cai and Cheng) and for $(3+1)$D O(3) criticality (Liang et al.). QMC measurements of $U(1)$ disorder operators also underlie recent mixed-state topological-phase diagnostics (Shi, Sun, and Li) and symmetry-resolved entanglement computations (Chen et al.). In symmetry-broken phases with a continuous symmetry, Goldstone modes — the gapless excitations accompanying spontaneous breaking of a continuous symmetry — are expected to modify the naive area/perimeter structure, which is precisely the physics a tensor-network extension would have to capture.

For higher-form symmetries, adjacent tensor-network results exist but remain model-specific: the Fredenhagen-Marcu string order parameter (a non-local string operator diagnosing deconfinement and topological order in lattice gauge theories) has been evaluated in iPEPS for the toric code — the prototypical $\mathbb{Z}_2$ lattice gauge theory with $\mathbb{Z}_2$ topological order — in a magnetic field (Xu, Pollmann, and Knap); Wilson-loop area-law coefficients have been extracted from variationally optimized iPEPS for a continuous toric-code-to-double semion transition (the double semion being the other $\mathbb{Z}_2$ topological order, with semionic rather than bosonic anyonic excitations), where the loops act as disorder operators of the dual model (Zhang and Xu); and 't Hooft loops (disorder operators of the gauge field, dual to Wilson loops) have been imposed in infinite density-matrix renormalization group (iDMRG) computations to study the roughening transition of a $(2+1)$D $\mathbb{Z}_2$ gauge theory with matter (Xu, Knap, and Pollmann). For non-Abelian gauge groups the dual Wilson loops are matrix-product operators (MPOs) — one-dimensional tensor networks representing operators — implementing a non-invertible symmetry, one whose action is closed under fusion rather than inversion (as for the representation category of a non-Abelian group), so the corresponding disorder operators are genuinely non-invertible MPOs whose transfer-matrix structure differs from the invertible case. Subsystem symmetries — rigid symmetries acting on lower-dimensional subsets such as lines or fractals, whose breaking characterizes fracton phases, i.e. gapped phases with subdimensional restricted-mobility excitations (Vijay, Haah, and Fu) — have, however, never been paired with disorder operators, although fracton transitions themselves have been studied with exactly solvable tensor-network states (Zhu, Chen, Ye, and Trebst).

The outlook of Xu and Huang poses, as the natural continuation of their framework, its extension to continuous global symmetries and its generalization to higher-form and subsystem symmetries, with the explicit goal that such disorder parameters be useful to detect topological (fracton) phases and their transitions. The unresolved target is stated below.

Problem Statement

Extend the iPEPS disorder-parameter framework and its finite correlation length (finite-entanglement) scaling theory of Xu and Huang beyond discrete 0-form internal symmetries, as posed in the source outlook, with the common objective of a disorder-operator diagnostic of symmetric, topological, and fracton phases that is computable in tensor networks in the thermodynamic limit. Concretely, deliver for each of the following symmetry classes (i) a well-defined disorder-operator construction computable in iPEPS in the thermodynamic limit, (ii) the scaling law of the disorder parameter — all contributions that occur in the scaling form (area, perimeter, constant, corner, or algebraic, as applicable in each phase) and their dependence on the finite correlation length $\xi(D,\chi)$, including the phase-dependent limits — and (iii) validation on at least one benchmark model or transition, with comparison to independent results where they exist:

(1) Continuous global symmetries ($U(1)$, $SU(2)$, $O(N)$): construct the fixed-angle or group-averaged (twirled) disorder operator as a computable iPEPS object; determine how continuous-symmetry criticality and the Goldstone modes of symmetry-broken phases modify the area/perimeter structure of the disorder parameter; and establish the finite correlation length scaling theory of its coefficients at continuous quantum critical points such as the $(2+1)$D XY or O(3) universality classes, benchmarked against the field-theory and QMC corner corrections (for example those fixed by the current central charge).

(2) Higher-form symmetries: define and evaluate in tensor-network states the disorder parameter generalized to higher-form symmetries — Wilson/'t Hooft-loop-type extended operators, including the non-invertible MPO cases arising for non-Abelian groups — and develop their finite correlation length scaling at topological phase transitions, going beyond model-specific evaluations of particular string or loop order parameters.

(3) Subsystem symmetries: define the disorder operator for subsystem (fractal) symmetries, represent it in tensor-network states, determine its scaling form, and demonstrate its use in diagnosing fracton topological phases and their transitions.

All three symmetry classes are mandatory; a class counts as resolved only when its construction (i), its scaling law (ii), and its benchmark validation (iii) have all been delivered.

References

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