Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement

TL;DR

The paper demonstrates that ’t Hooft anomalies of finite higher-form symmetries enforce intrinsic long-range entanglement in quantum many-body states, by introducing a generalized statistics invariant Θ that detects anomalies and forbids symmetric short-range entangled realizations when Θ ≠ 0 (mod $2\pi$). It proves a fidelity bound showing that the overlap with any SRE state decays exponentially with system size for states carrying anomalous higher-form symmetry, and extends this to mixed states prepared via local decoherence. As an explicit application, the authors decohere a (3+1)D ${\mathbb Z}_2$ toric code with fermionic loop excitations, revealing intrinsically mixed-state topological order (imTO) protected by a strong anomalous 1-form symmetry that violates remote detectability. They further connect these findings to a bulk-boundary perspective with a $(4+1)$D action and outline a program to classify imTO in higher dimensions using the algebraic data of strong higher-form symmetries. Overall, the work provides a robust entanglement-based diagnostic for imTO and broadens the role of higher-form anomalies in constraining quantum phases of matter.

Abstract

We show that generic gapped quantum many-body states which respect an anomalous finite higher-form symmetry have an exponentially small overlap with any short-range entangled (SRE) state. Hence, anomalies of higher-form symmetries enforce $intrinsic$ long-range entanglement, which is in contrast with anomalies of ordinary (0-form) symmetries which are compatible with symmetric SRE states (specifically, symmetric cat states). As an application, we show that the anomalies of strong higher-form symmetries provide a diagnostic for mixed-state topological order in $d \geq 2$ spatial dimensions. We also identify a new (3+1)D intrinsic mixed-state topological order that does not obey remote-detectability by local decoherence of the (3+1)D Toric Code with fermionic loop excitations. This breakdown of remote detectability, as encoded in anomalies of strong higher-form symmetries, provides a partial characterization of intrinsically mixed-state topological order.

Figures (6)

• Figure 1: The simplicial complex $X$ embedded in (a) 1d, (b) 2d, and (c) 3d space. This is a $d$-simplex with a center vertex $0$, dividing a single $d$-simplex into $d+1$ of them. This is regarded as a minimal triangulation of a $d$-sphere, once we identify infinity as a single point. Symmetry defects are supported at the $(d-p-1)$-simplices of the simplicial complex.
• Figure 2: (a): 1-form symmetry in (3+1)D. The symmetry operators are supported at 2-simplices, and the symmetry defects are supported at their boundaries. (b): The invariant for the ${\mathbb Z}_2$ 1-form symmetry in (3+1)D is defined by a sequence of 24 symmetry operators $U_{0jk}$ evaluated on a symmetric gapped state $\ket{\Psi}$. The thick blue or black lines denote the configurations of symmetry defects during the 24-step process. The arrow along the thick line indicates that the initial and final loops of defects are reversed. Therefore this 24-step process is sometimes called the loop-flipping process.
• Figure 3: The $d$-dimensional space embeds a disjoint set of disks (hypercubes) $\bigcup_j R_j$ with linear size $\gamma't$. The reduced density matrix $\rho_{\{R\}}$ on $\bigcup_j R_j$ has $p$-form symmetry $U_R(g,\Sigma)$ whose support is within a single region $R$. One can then define an invariant $U_\Theta(R)$ by the sequence of symmetry operators within a single region $R$.
• Figure 4: Definitions of the operators $S_e, \tilde{S}_e$.
• Figure 5: The configurations of surfaces $\Sigma_{0jk}$ embedded in the dual cubic lattice.