Information-theoretic principle of emergent 1-form symmetries

TL;DR

The paper introduces an information-theoretic principle for emergent 1-form symmetries, tying their existence to the decodability of symmetry charges under a fixed bare representation and to the preservation of information about state transformations. It develops a QEC-based protocol (e.g., MWPM) to detect emergent 1-form symmetries and to locate 2D topological quantum phase transitions, validating the approach on the toric code in a field and on deformed toric-code states that map to 2D RBIM along the Nishimori line. A key insight is that the loss of emergent 1-form symmetry is an information-theoretic transition reflected in post-measurement entanglement patterns, distinct from conventional order parameters, and can be probed via a disorder-parameter measured on QEC-recovered states. The framework further enables a planar-d geometry extension for detection, clarifies the connection to particle condensation and Higgs/confinement regimes in Z2 lattice gauge theory, and points to extensions to non-Abelian, higher-form, and SPT contexts as avenues for future work.

Abstract

Higher-form symmetries act on sub-dimensional spatial manifolds of a quantum system. They can emerge as an exact symmetry at low energies even when they are explicitly broken at the microscopic level, making them difficult to characterize. In this work, we propose that the emergence of 1-form symmetries is information-theoretic in nature, precisely characterized by the preservation of information about how quantum states transform under the symmetry. As a consequence, the loss of the emergent 1-form symmetry is an information-theoretic transition which we argue to be revealed from the long-range entanglement in the ensemble of post-measurement states. We analytically determine the regimes in which a 1-form symmetry emerges in product states on one- and two-dimensional lattices. In analytically intractable regimes, we demonstrate how to efficiently detect 1-form symmetries with a global quantum error correction (QEC) decoder and numerically examine the information-theoretic transition of the 1-form symmetry, including systems with $\mathbb{Z}_2$ topological order. As a practical application of our framework, we show that once the 1-form symmetry is detected to exist, a topological quantum phase transition characterized by the spontaneous breaking of the 1-form symmetry can be accurately determined by a disorder parameter. We further argue that our proposed theory for emergent 1-form symmetries offers new perspectives on particle condensation and suggests sharp information-theoretic phase boundaries between Higgs and confining regimes in the $\mathbb{Z}_2$ lattice gauge theory.

Figures (19)

• Figure 1: A geometric picture for emergent 1-form symmetries. We decompose the quantum state as a 1-form symmetric state acted on by charged string operators. (a) When the choice of the charged string operators (blue) for each charge configuration is unique, we can deform the 1-form symmetry (brown), such that it avoids the intersection with the string operator; the 1-form symmetry then effectively commutes with the charged string operators. (b) Assuming there exists no unique choice of the charged string operators for the same charge configuration (solid and dashed lines), the deformation will necessarily intersect with one set of the strings, preventing the deformed 1-form symmetry operators from effectively commuting with them. Therefore, when both configurations appear with comparable probability, one cannot commute the 1-form symmetry through the charged strings and the 1-form symmetry is absent.
• Figure 2: 1-form symmetries of product states. (a) In 1D, the $Z$ 1-form symmetry is generated by $Z_iZ_{i+1}$. Using the criterion we develop, the $Z$ 1-form symmetry is present for $\theta\in [0,\pi/2)$ and disappears at $\theta = \pi/2$. (b) In 2D, the $Z$ 1-form and $X$ 1-form symmetries are generated by $B_p$ and $A_v$, respectively. Via a mapping to the 2D RBIM along the Nishimori line, we find that the $Z$ and $X$ 1-form symmetries exist when $\theta\in[0,\theta_c)$ and $\theta\in(\pi/2-\theta_c,\pi/2]$, respectively, where $\theta_c = 0.22146(2)\pi$. The middle region $\theta \in [\theta_c, \pi/2-\theta_c]$ shown by the red line has neither the $Z$ 1-form symmetry nor the $X$ 1-form symmetry.
• Figure 3: Examples of 1-form symmetry operators and charged string operators in 2D. (a) The $Z$ and the $X$ 1-form symmetries are generated by the plaquette and the vertex operators. The 1-form symmetry charges for the $Z$ 1-form symmetry are created by $X$-string operators. (b) The coexistence of the non-contractible $X$ and $Z$ 1-form symmetry operators implies that a ground-state manifold has topological degeneracy. They cannot coexist at the same time in a trivial product state.
• Figure 4: Decoding of 1-form symmetry charge configurations. The Minimum-Weight Perfect Matching (MWPM) decoder takes the 1-form symmetry charge configuration (dots) and outputs the predicted sets of $X$-strings (green lines) required to remove all the $Z$ 1-form symmetry charges created by the actual $X$-strings (blue lines). If the $Z$ 1-form symmetry charges are dense, the algorithm fails by returning a set of recovery $X$-strings (green lines) that is inequivalent to the actual $X$-strings (blue lines).
• Figure 5: Numerical results for detecting $Z$ 1-form symmetry from the ground states of the toric code in a field. We consider the expectation values of the non-contractible $Z$ loop operator $\langle W_Z\rangle$ from the QEC-recovered states as the 1-form symmetry indicator. A value of +1 indicates the existence of the 1-form symmetry. The ground states are approximated by tensor-network states on the torus with bond dimension $D$ and system size $L_x\times L_y$. (a) The quantum phase diagram of the system, with the topological and the trivial phase. The dashed lines indicate cuts at which we have evaluated our protocol, illustrated in Fig. \ref{['Fig:scan_main']}b. (b) The emergent 1-form symmetries of the system. The black dots are the QEC thresholds from the numerical simulation. (c) Loop operator $\langle W_Z\rangle$ in the product-state limit $h_x^2+h_z^2=+\infty$. The black dashed lines indicate theoretical threshold from the MWPM decoder, the green dashed line is the theoretical threshold from the optimal decoder. (d) Loop operator $\langle W_Z\rangle$ for $h_z=2$. The black dashed lines indicate the crossing points of $\langle W_Z\rangle$ from different system sizes (symbols) and different tensor-network state bond dimensions $D$ (colors). (e) Loop operator $\langle W_Z\rangle$ at $h_z=0.2$. Peak of correlation length from the tensor network state at the bond dimension $D$ (dashed black). Location of the topological phase transition (red dashed) obtained from Ref. Youjin_TC_phase_diagram_2011.