Mermin-Wagner theorems for quantum systems with multipole symmetries

We prove Mermin-Wagner-type theorems for quantum lattice systems in the presence of multipole symmetries. These theorems show that the presence of higher-order symmetries protects against the breaking of lower-order ones. In particular, we prove that the critical dimension in which the charge symmetry can be broken increases if the system admits higher multipole symmetries, e.g. $ d = 4 $ on the regular lattice $ \mathbb{Z}^d $ in the presence of dipole symmetry.

💡 Research Summary

The paper “Mermin‑Wagner theorems for quantum systems with multipole symmetries” extends the classic Mermin‑Wagner theorem to quantum lattice models that possess higher‑order conservation laws, i.e., multipole symmetries such as dipole, quadrupole, etc. The authors first set up a rigorous algebraic framework for quantum lattice systems on a countable metric space (L) embedded in (\mathbb{R}^d). The growth of the number of sites in a ball of radius (r) is bounded by (C r^\gamma); the exponent (\gamma) plays the role of an effective spatial dimension. Each site carries a finite‑dimensional Hilbert space and the quasi‑local algebra (\mathcal{U}) of observables is defined as the norm‑completion of the union of local algebras. Thermal equilibrium states at inverse temperature (\beta>0) are described by KMS states on (\mathcal{U}) with respect to a strongly continuous one‑parameter group of automorphisms (\alpha_t) generated by a translation‑invariant interaction (\Phi).

Multipole symmetries are introduced via a family of local charge operators (n_x) (self‑adjoint, uniformly bounded, finite support). For a multi‑index (a\in\mathbb{N}_0^d) the corresponding symmetry transformation is \