Spectrum-Generating Algebra in Higher Dimensional Gauge Theories

Abstract: Non-equilibrium properties of strongly interacting gauge theories are often intractable with classical simulation methods. Due to recent developments of quantum simulations, studies of their properties in two spatial dimensions are becoming accessible. By demonstrating the existence of an approximate spectrum-generating algebra for a pure gauge plaquette ladder, we predict and verify the existence of Quantum Many-Body Scars in spin-1 Quantum Link Models. The analysis of the model is facilitated by a dualization process that maps the original gauge theory to a constrained spin chain. Was it not for the constraint, the system would have an exact spectrum-generating algebra. We propose a set of observables for diagnosing an approximate spectrum-generating algebra, which is expected to guide quantum simulators toward interesting physical regimes.

• The paper demonstrates that constrained gauge theories can host approximate spectrum-generating algebras that generate near-equally spaced eigenstate towers.
• The study employs dualization techniques to map gauge-invariant spin-1 models to constrained spin chains, enabling efficient numerical and analytical analysis.
• The research reveals that quantum many-body scars emerge as low-entanglement outlier states, offering promising avenues for quantum state engineering.

Spectrum-Generating Algebras and Quantum Many-Body Scars in Higher Dimensional Gauge Theories

Introduction and Model Construction

The paper analyzes the emergence of approximate spectrum-generating algebras (SGAs) and Quantum Many-Body Scars (QMBS) in higher-dimensional gauge theories, focusing on a pure-gauge spin-1 Quantum Link Model (QLM) formulated on a plaquette ladder geometry. The model features onsite and link-based spin-1 variables and can be efficiently mapped, via a dualization process, to a constrained spin-1 chain. This dualization is central to both analytical and numerical tractability; it allows for the application of methods familiar in the study of spin chains while retaining the essential gauge and topological constraints of the ladder model.

Figure 1: Original formulation with spin-1 variables $S^a_{n,i}$ defined on the links $(n,i)$ of a ladder.

Gauge symmetry is explicitly implemented by requiring the Hamiltonian $H$ to commute with the generators $G_n$ associated with the Gauss’ law at each vertex, as well as with global winding number operators. The Hilbert space is thus reduced to physical states satisfying $G_n \ket{\psi} = 0$ and zero global magnetization. Dualization reduces the Hamiltonian in the gauge-invariant sector to a constrained spin-1 chain structure, where local projectors enforce the absence of neighboring $+1$ and $-1$ spins, leading to significant Hilbert space fragmentation.

Spectrum-Generating Algebras and Broken Lie Algebra Structure

The unconstrained version of the dual spin-1 chain possesses an exact dynamical $\mathfrak{su}(2)$ symmetry, with raising and lowering operators generating full towers of equally-spaced energy eigenstates. Incorporating gauge constraints breaks this algebraic structure, but, crucially, the system retains an approximate SGA leading to "broken Lie algebra" towers for specific subsets of eigenstates.

This approximate SGA manifests as families of states in the spectrum characterized by nearly equal energy spacing and low entanglement entropy, distinct from the thermal bulk and forming the algebraic underpinning of QMBS. Operators $\tilde{H}^{\pm}$ generalize standard raising and lowering operators but act within the constrained Hilbert space. Their commutators, while no longer exactly generating $\mathfrak{su}(2)$, still approximate the algebra over subspaces with high overlap onto tower states.

Spectral Diagnostics: Entanglement and Broken Casimir Analysis

Spectral properties are probed via half-system entanglement entropy and a "broken Casimir" invariant $(n,i)$0, constructed from the constrained SGA operators. Numerical exact diagonalization for $(n,i)$1 reveals a set of outlier states at mid-spectrum, exhibiting entanglement significantly below the bulk, suggesting persistent non-thermalization for specific initial conditions.

Figure 2: Half-chain entanglement entropy for all eigenstates as a function of energy, with outlier low-entropy states (scars) identified.

Further analysis of $(n,i)$2 across eigenstates resolves the scar towers. The states with largest $(n,i)$3 correspond to the lowest-entropy outliers. State counting in these towers indicates $(n,i)$4 and $(n,i)$5 multiplets, with the $(n,i)$6 tower manifesting at zero momentum and $(n,i)$7 towers distributed across nonzero momentum sectors.

Figure 3: Expectation value of the broken Casimir for eigenstates resolved by momentum; prominent towers in zero and finite momentum sectors signal approximate SGA structures.

Non-Thermal Dynamics and Scar State Construction

The presence of approximate SGA towers enables the explicit construction of scarred product states. For the $(n,i)$8 tower, the state $(n,i)$9 (all spins $H$0 in the $H$1-basis) serves as the vacuum of $H$2. Iterated application of $H$3 generates the full tower. Initializing the system in $H$4 and evolving under the Hamiltonian produces robust, long-lived revivals in the magnetization and persistent elevation of the broken Casimir, in sharp contrast to generic product states.

Figure 4: Time-evolution of magnetization and broken Casimir starting from the scar state $H$5 demonstrates pronounced non-thermal revivals.

States in the $H$6 multiplet are constructed by substituting a single $H$7 spin among $H$8's and boosting with fixed momentum $H$9, i.e., $G_n$0. Their time evolution shows strong overlap with the $G_n$1 towers, exhibiting significant non-ergodic dynamics in their corresponding momentum sectors.

Figure 5: Time-evolution of magnetization and broken Casimir for momentum-boosted product states reveals sector-specific scarring dynamics.

Implications and Future Directions

This work places the study of QMBS in higher-dimensional gauge theories on rigorous algebraic footing, showing that even in strongly constrained Hilbert spaces, fragments of Lie algebraic structure can survive and drive robust non-ergodic dynamics. These findings have direct implications for quantum simulation platforms, particularly Rydberg arrays and digital quantum simulation with constrained local Hilbert spaces, where the identification of simple initial product states capable of evading thermalization is desirable for robust state engineering and quantum memory protocols.

The results also suggest that systematic algebraic and numerical diagnostics, such as the broken Casimir, can identify and characterize scarring phenomena in realistic lattice gauge theory models, even when the dimensionality and local constraints thwart analytic diagonalization. The methodology is extensible to larger volumes, higher spin QLMs, and plausibly to non-Abelian gauge theories.

On the theoretical front, the paper demonstrates that gauge symmetry, while not generically enforcing scarring, can enable the formation of highly atypical non-thermal eigenstates through Hilbert space fragmentation and emergent broken algebras. This adds to the growing body of evidence that constrained quantum dynamics in lattice gauge theories can widely support weak ergodicity breaking and non-thermal steady states, challenging the universality of the ETH in these models [see also (Garmroudi et al., 2023, Miao et al., 28 May 2025)].

Conclusion

The existence of approximate spectrum-generating algebra towers in the constrained Hilbert space of higher-dimensional gauge theories establishes a general algebraic mechanism for QMBS formation beyond one spatial dimension. The study provides explicit constructions of scarred states, demonstrates non-thermal time evolution, and introduces a robust diagnostic (the broken Casimir) for scarring. Further generalization to larger systems and other gauge groups, as well as experimental exploration in programmable quantum simulators, remains an open avenue for research in quantum dynamics and quantum information in strongly correlated gauge systems.