Emergent Gauge Symmetries in Quantum Condensed Matter Systems

Anthony J. Leggett, Fernando Sols

Victor L. Quito, Pedro L. S. Lopes, José A. Hoyos +1 more

Xiao-Gang Wen

Global symmetry (0-symmetry) acts on the whole space while higher $k$-symmetry acts on all the codimension-$k$ closed subspaces. The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries (unless we engineer fine-tuned toy models). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that the emergent spontaneously broken higher symmetries are nothing but a kind of topological order. Thus the study of emergent spontaneously broken higher symmetries is a study of topological order. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an emergent anomalous $U(1)$-1-symmetry below the energy gap of the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phase of an EM condensed matter system (induced by arbitrary fluctuations and condensations of electric charges and photons) must have a nontrivial bosonic topological order.

shifa liu

This paper presents a systematic extension of the Global Fractional Quantization (GFQ) framework from particle physics to condensed matter systems. We establish a rigorous theoretical foundation for the quantization of collective observables in many-body quantum systems, demonstrating that sums of specific physical quantities over entire bands, ensembles, or collective states exhibit discrete, rational relationships. The framework is built upon four fundamental postulates—Gauge Constraint, Gauge Invariance, Gauge Periodicity, and Global State Symmetry—applied to appropriately defined U(N) collective multiplets in condensed matter contexts. We provide detailed mathematical derivations with complete proofs and demonstrate the framework’s applicability to several key condensed matter systems: topological insulators, superconductors, magnetic systems,and correlated electron materials. The theory predicts novel quantization conditions for sums of Berry curvatures, total polarization, collective spin moments, and superconducting phase differences,which we validate through both analytical models and numerical calculations with comprehensive error analysis. Our results reveal a hierarchical structure of quantization laws that correlate with the system’s dimensionality, symmetry, and fundamental nature (topological vs. trivial, gapped vs. gapless). The GFQ framework offers a new unifying perspective on emergent quantization phenomena across diverse condensed matter platforms and provides testable predictions for future experiments.

Amogh Anakru, Zhen Bi

The emergence of fractonic topological phases and novel universality classes for quantum dynamics highlights the importance of dipolar symmetry in condensed matter systems. In this work, we study the properties of symmetry-breaking phases of the dipolar symmetries in fermionic models in various spatial dimensions. In such systems, fermions obtain energy dispersion through dipole condensation. Due to the nontrivial commutation between the translation symmetry and dipolar symmetry, the Goldstone modes of the dipolar condensate are strongly coupled to the dispersive fermions and naturally give rise to non-Fermi liquids at low energies. The infrared description of the dipolar symmetry-breaking phase is analogous to the well-known theory of a Fermi surface coupled to an emergent U(1) gauge field. We also discuss the crossover behavior when the dipolar symmetry is slightly broken and the cases with anisotropic dipolar conservation.

Arianna Montorsi, Mario Rasetti