Origin and universality of the finite correlation length $\xi(D)$ in variationally optimized Lorentz-invariant gapless iPEPS

When an infinite projected entangled pair state (iPEPS) with bond dimension $D$ is variationally optimized for a Lorentz-invariant gapless two-dimensional quantum system, the optimal state acquires a finite spatial correlation length $\xi(D)$, which acts as an emergent infrared cutoff and supports a finite-correlation-length scaling framework for extrapolating energies and order parameters to the thermodynamic limit (Rader and Läuchli, Phys. Rev. X 8, 031030 (2018)). The mechanism, however, has no first-principles derivation: the conjecture that finite $D$ truncates the Euclidean path integral sampled by the wave function to a finite temporal extent $\xi_\tau(D)$, which Euclidean invariance converts into a spatial $\xi$, is unproven; the putative power law $\xi\sim D^\kappa$ ($\kappa\approx 5$ for the 3d Ising conformal field theory, $\kappa\approx 2$ for continuous-symmetry breaking) is unexplained; and it is unknown whether $\kappa$ is fixed by universal data of the infrared field theory or by the imposed tensor symmetries. The question remains open: the one-dimensional analogue has only recently been understood (effective-field-theory formulations in 2023–2025 and the 2026 identification of the perturbations actually induced by finite bond dimension), and the Fermi-surface extension has been solved for Gaussian fermionic tensor networks, but for variational $(2+1)$d iPEPS no derivation of $\xi(D)$, no settlement of the universality of $\kappa$, and no explanation of the boundary-MPS entanglement behavior exist. The problem asks for this theory, its universality content, and its validation across universality classes.

Background

An infinite projected entangled pair state (iPEPS) is a variational wave function for two-dimensional quantum lattice systems obtained by placing a local tensor on every site of an infinite lattice and contracting virtual (bond) indices between neighboring sites; the bond dimension $D$ of these indices is the single variational resource parameter. Ground states of gapped local Hamiltonians satisfy an area law for the entanglement entropy, which projected entangled pair states (PEPS) reproduce for any $D$, but gapless states generically require ever-growing entanglement, so how well a finite-$D$ iPEPS can approximate a gapless state is a basic open issue. Evaluating an iPEPS requires contracting an infinite two-dimensional tensor network, which is done approximately by a boundary matrix product state (boundary MPS) of bond dimension $\chi$, or equivalently by corner-transfer-matrix renormalization group (CTMRG); the spatial correlation length $\xi$ of the state is extracted from the spectrum of the resulting double-layer row-to-row transfer matrix.

In one dimension the analogous situation is theoretically under control. For a critical chain approximated by an infinite matrix product state (iMPS, the one-dimensional analogue of an iPEPS, with local tensors contracted along a chain) of bond dimension $\chi$, the truncation induces a finite correlation length growing as $\xi\sim\chi^{\kappa}$ with $\kappa$ fixed by the central charge $c$ (the coefficient counting degrees of freedom of the 1+1d conformal field theory, CFT), and observables obey finite-entanglement scaling in $\xi$ in complete analogy with finite-size scaling in the system length. Rader and Läuchli established the two-dimensional counterpart empirically: for the square-lattice transverse-field Ising model at criticality (a $(2+1)$d Lorentz-invariant quantum critical point in the 3d Ising universality class) and for the $S=1/2$ Heisenberg and XY models (spontaneously broken continuous symmetry with gapless Goldstone modes, i.e. gapless excitations that unavoidably accompany such symmetry breaking), the energetically optimal iPEPS always displays a finite $\xi(D)$, and $\xi(D)$ plays the role of a finite-size cutoff on which energies and order parameters scale with their field-theory critical exponents. A companion paper by Corboz, Czarnik, Kapteijns and Tagliacozzo reported the same phenomenon for interacting spinless fermions on the honeycomb lattice, together with a geometric 'towers and valleys' picture of why the optimal state cannot be scale invariant.

What is missing is the theory behind these observations. Rader and Läuchli conjectured that truncating the bond dimension limits the temporal extent $\xi_\tau(D)$ of the Euclidean path integral represented by the wave function (the contraction of an iPEPS with its complex conjugate can be read as a two-dimensional Euclidean path integral in space and imaginary time), and that the Euclidean (Lorentz) invariance of the targeted fixed point then forces a matching finite spatial $\xi$; known finite-$D$ iPEPS with algebraic correlations, such as Rokhsar–Kivelson-type $(2+0)$d wave functions (superpositions of classical lattice configurations at a fine-tuned weighting) with built-in space-time asymmetry, evade this by having essentially zero $\xi_\tau$. Numerically they observed a rough power law $\xi\sim D^{\kappa}$ with $\kappa\approx 5$ for the 3d Ising CFT and $\kappa\approx 2$ shared by the two continuous-symmetry-breaking cases, whose prefactors differ by a factor consistent with the inverse ratio of their numbers of Goldstone modes (the XY model has one, the Heisenberg model two). By analogy with the one-dimensional central-charge formula, they speculated that $\kappa$ might be fixed by universal data of the infrared theory, for instance the $F$-coefficient of the $F$-theorem (a monotone counting the degrees of freedom of a 2+1d CFT, the analogue of the central charge in one dimension), the central charge, or the number of Goldstone modes, with the imposed tensor symmetries (such as $U(1)$ or $SU(2)$) possibly entering only through prefactors. They also compared the entanglement entropy $S_{\mathrm{bMPS}}$ of the boundary MPS obtained from contracting the iPEPS as a function of $\xi$: genuine $(2+1)$d states showed faster growth than the quantitatively understood logarithmic growth of $(2+0)$d wave functions, with an unexplained irregularity in the XY data. This diagnostic matters practically because $S_{\mathrm{bMPS}}$ dictates how large the environment bond dimension $\chi$ must be for a faithful contraction, and conceptually because it should reveal whether the correlation volume of the optimal state extends into the imaginary-time direction. The source work further flagged the extension of the framework to other Lorentz-invariant universality classes (Wilson–Fisher $O(N)$ fixed points for $N>1$, i.e. the universal critical theories of $N$-component order parameters; Gross–Neveu–Yukawa theories governing ordering transitions of interacting Dirac fermions; quantum electrodynamics in three spacetime dimensions, QED3, invoked for deconfined criticality (continuous transitions between phases with unrelated order parameters, mediated by fractionalized excitations) and for algebraic spin liquids (gapless spin liquids with power-law correlations); non-collinear magnetic order (spiral or incommensurate order in which the magnetization direction varies across the lattice) or $SU(N)$ magnetic order) and, as a frontier flagged as open, gapless phases with a Fermi surface, whose low-energy description is not Lorentz invariant.

Problem Statement

Develop the theory of finite-bond-dimension effects for variationally optimized iPEPS approximating Lorentz-invariant gapless $(2+1)$-dimensional quantum systems, as posed by Rader and Läuchli (Phys. Rev. X 8, 031030 (2018)). The overarching objective is a predictive first-principles account of the induced finite correlation length $\xi(D)$, its universality content, and its boundary-entanglement signature, validated against the existing and foreseeable iPEPS data. Concretely:

1. Mechanism. Derive from first principles why the energetically optimal finite-$D$ iPEPS for a Lorentz-invariant gapless $(2+1)$d system has a finite spatial correlation length $\xi(D)$; in particular prove, refine, or refute the conjecture that finite $D$ imposes a finite temporal extent $\xi_\tau(D)$ on the Euclidean path integral sampled by the wave function, which Euclidean invariance then converts into a finite spatial $\xi$. The derivation must disentangle the intrinsic variational finite-$D$ effect from the separate finite-$\chi$ error of the contraction environment, since both produce finite correlation lengths.

2. Functional form and universality. Establish the asymptotic functional form of $\xi(D)$: whether $\xi\sim D^{\kappa}$ holds, and whether $\kappa$ is a universal number fixed by infrared data of the field theory (for example an $F$-theorem-type coefficient, the central charge, or the number of Goldstone modes) and whether the imposed tensor symmetries enter only through prefactors, in parallel with the one-dimensional formula fixing $\kappa$ through the central charge $c$.

3. Boundary diagnostic. Explain the scaling of the boundary-MPS entanglement entropy $S_{\mathrm{bMPS}}$ with $\xi$, in particular why genuine $(2+1)$d Lorentz-invariant states show faster growth than $(2+0)$d anisotropic wave functions with their quantitatively understood logarithmic scaling, and the origin of the irregular XY-model behavior observed in the source work.

4. Range of validity. Delimit the framework by validating it in the settings where data exist (3d Ising / Wilson–Fisher criticality, $O(2)$ and $O(3)$ continuous symmetry breaking) and clarifying the frontier cases flagged by the source: Wilson–Fisher $O(N)$ fixed points beyond $N=1$, Gross–Neveu–Yukawa criticality of interacting Dirac fermions, QED3-type descriptions of gapless spin liquids and deconfined criticality, non-collinear or $SU(N)$ magnetic order, and gapless phases with a Fermi surface, where the Gaussian (free-fermion) fermionic tensor network case has been solved since the source appeared, leaving interacting metals as the open instance.

A complete answer settles items 1–3 and carries out the delimitation of item 4: it is a mechanism plus functional form whose predictions (exponents, prefactors, symmetry dependence, boundary-entanglement scaling) are checkable against existing and new iPEPS data, together with a statement of which of the settings and frontier classes listed in item 4 the framework covers and which remain open. Answers that settle items 1–3 while delimiting item 4 are acceptable scientific completions of the core; full resolution of every frontier class in item 4 is not required.

References

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