Exact emergent higher-form symmetries in bosonic lattice models

TL;DR

<3-5 sentence high-level summary>This work develops a framework in which higher-form (p-form) symmetries can emerge as exact symmetries at low energies in bosonic lattice models, and proves their robustness to local UV perturbations in the thermodynamic limit at zero temperature. By partitioning parameter space with energy scales and introducing a conjectured local unitary dressing, the authors define an exact emergent mid-IR symmetry described by an enlarged local algebra, applicable to both invertible and non-invertible cases. They instantiate the framework with three lattice models—the quantum clock model and emergent ${ m Z}_N^{(p)}$ and ${U(1)}^{(p)}$ gauge theories—revealing exact emergent and anomalous higher-form symmetries, including BF-theory descriptions in the continuum. A generalized Landau analysis of the Fradkin-Shenker model shows how exact emergent symmetries organize phase structure and universality classes, while finite-temperature results distinguish the fate of 1-form versus higher-p-form symmetries. The work highlights the physical consequences, including symmetry breaking, anomalies, and dynamical effects, and discusses boundary phenomena, anomaly inflow, and potential numerical approaches for identifying dressed symmetry operators.</p>

Abstract

Although condensed matter systems usually do not have higher-form symmetries, we show that, unlike 0-form symmetry, higher-form symmetries can emerge as exact symmetries at low energies and long distances. In particular, emergent higher-form symmetries at zero temperature are robust to arbitrary local UV perturbations in the thermodynamic limit. This result is true for both invertible and non-invertible higher-form symmetries. Therefore, emergent higher-form symmetries are $\textit{exact emergent symmetries}$: they are not UV symmetries but constrain low-energy dynamics as if they were. Since phases of matter are defined in the thermodynamic limit, this implies that a UV theory without higher-form symmetries can have phases characterized by exact emergent higher-form symmetries. We demonstrate this in three lattice models, the quantum clock model and emergent ${\mathbb{Z}_N}$ and ${U(1)}$ ${p}$-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a 2+1D lattice model that gives rise to $\mathbb{Z}_2$ gauge theory. Using exact emergent 1-form symmetries accompanied by their own energy/length scales, we show that the transition between the deconfined and Higgs/confined phases is continuous and equivalent to the spontaneous symmetry-breaking transition of a $\mathbb{Z}_2$ symmetry, even though the lattice model has no symmetry. Also, we show that this transition line must $\textit{always}$ contain two parts separated by multi-critical points or other phase transitions. We discuss the physical consequences of exact emergent higher-form symmetries and contrast them to emergent ${0}$-form symmetries. Lastly, we show that emergent 1-form symmetries are no longer exact at finite temperatures, but emergent $p$-form symmetries with ${p\geq 2}$ are.

Figures (16)

• Figure 1: The parameter space of a many-body Hamiltonian can be partitioned by its differing hierarchies of energy scales. A schematic depiction of this is shown here. The parameter space is partitioned into four regions, labeled I, II, III, and IV, with their differing energy scale hierarchies shown.
• Figure 2: The symmetries of a quantum many-body system generally depend on the energy scale of an observer. In particular, there can be emergent symmetries at low energies absent from the microscopic (UV) symmetries ${G_{\text{UV}}}$. These can be generalized symmetries and can be anomalous.
• Figure 3: (a) The low energy properties of $QFT_{ano}$ (${QFT_{af}}$ restricted to the ${ {\cal R} }$-symmetric sub-Hilbert space) can be exactly simulated by a boundary of a topological order $\EuScript{M}$ in one higher dimension. $QFT_{ano}$ uniquely determines $\EuScript{M}$. (b) The low energy properties of $QFT_{af}$ throughout all ${ {\cal R} }$ sectors can be simulated by including the boundary $\widetilde{ {\cal R} }$.
• Figure 4: (a) A lattice realization of Fig. \ref{['fig:symTO']}b, where qubits live on the links. The star and plaquette terms of the toric code model are shown both in the bulk and on the ${\widetilde{ {\cal R} }}$ boundary. (b) Since the bulk is topological, one can take the thin slab limit.
• Figure 5: (left) The parameter space of models C \ref{['2bodyU1Ham']} and D \ref{['ZNuvHam']} can be partitioned into three regions, I, II, and III, corresponding to its different energy scale hierarchies. These regions are not necessarily distinct phases of the models. Here, solid lines indicate a phase transition while dashed lines do not. (right) In these regions, we identify the exact emergent symmetries at each energy scale. Here ${p}$ is a positive integer and $d$ is the dimension of space such that ${d > p + 1}$ (${d > p}$) for model C (D). The emergent $U(1)^{(d-p-1)}$ symmetry in region III for model C is only nontrivial in the continuum limit. When ${J/U = 0}$ (${K/U = 0}$), the exact emergent ${U(1)^{(p)}}$ and ${\mathbb{Z}_N^{(p)}}$ (${U(1)^{(d-p-1)}}$ and ${\mathbb{Z}_N^{(d-p)}}$) symmetries of models C and D, respectively, are exact UV symmetries.