We formulate a non-Abelian theory of structural consistency for systems in which dynamical transformations reside on the links of a simplicial network. Gauge covariance follows from freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of alternative paths. For an SU(2) model on the complete simplicial $2$-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven loss of global compatibility. In the high-disorder phase, the uniform compatibility $M_P$ decreases with system size, the integrated adjacent correlation weight $\mathcal I_{\mathrm{adj}}$ remains finite value. Moreover, the connected replica-overlap width approaches the numerical noise floor and its distribution narrows, providing no evidence for thermodynamic replica-symmetry breaking. Dense frustration therefore produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation weight survives.
We formulate a non-Abelian theory of structural consistency for systems in which dynamical transformations reside on the links of a simplicial network. Gauge covariance follows from freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of alternative paths. For an SU(2) model on the complete simplicial 2-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven loss of global compatibility. In the high-disorder phase, the uniform compatibility MP decreases with system size, the integrated adjacent correlation weight Iadj remains finite value. Moreover, the connected replica-overlap width approaches the numerical noise floor and its distribution narrows, providing no evidence for thermodynamic replica-symmetry breaking. Dense frustration therefore produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation weight survives.
Recent tensor-network samplings of modified Nishimori spin glasses have revealed robust finite-temperature critical transitions in two dimensions, defying the standard Edwards-Anderson lower critical dimension boundary (dl≈2.5). We present a theoretical framework demonstrating that the discrete Z2 gauge constraints utilized to bypass Monte Carlo kinetic traps fundamentally alter the system's universality class. By mapping the algorithmic disorder distribution to the 2D Ising Conformal Field Theory (CFT), we prove the emergent spatial variance generates a fractional momentum operator that drives the dynamic upper critical dimension to zero (du→0). This marginal topology dynamically suppresses the replica-coupling vertices, yielding an infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and a non-integrable replicon divergence that predicts a massive instability toward 1-step Replica Symmetry Breaking (1-RSB). Leveraging a spectral Corner Transfer Matrix Renormalization Group (CTMRG) architecture up to macroscopic scales (L=1024), we quantitatively validate the topological scaling argument G((T−Tc)ln(L/L0)). By isolating the continuum field theory from microscopic lattice artifacts, we recover the fundamental lattice metric L0≈0.94, unequivocally confirming the existence of a distinct, topologically driven spin-glass phase.
Non-reciprocity and geometric frustration enable many-body systems to avoid crystalline order and instead exhibit complex, liquid-like behavior. Here we show that their interplay is richer than the sum of its parts, leading to surprising structural and dynamical phenomena. In our minimal model, two copies of Ising gauge theory are non-reciprocally coupled in a way that crucially preserves a local Z2 symmetry. We discover that the combined Wilson loop observable of the two copies exhibits linear asymptotic scaling, with a quasiparticle-pair confinement length tuned by the strength of the non-reciprocal coupling. Key dynamical features are revealed in the behavior of individual deconfined excitations due to strong interactions induced by the non-reciprocity, leading to motion on a critical percolation cluster that follows a self-avoiding trail. Mapping from this quasiparticle dynamics onto the magnetic noise spectrum, we discover that non-reciprocity tunes topological logarithmic contributions and causes long-lived metastable states due to quasiparticle trapping. Our work opens the way for broader investigations of geometrically frustrated non-reciprocity.
Continuing the program in which the action constitutes the primary structure of dynamics and the phase ϕ=S/ℏ plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a U(1) connection. The present article unifies these two directions. We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix Aij to a complex Hermitian matrix Aij=Aijeiθij, in which θij represents a discrete U(1) connection. This results in a magnetic Laplacian K=D−A, whose global quadratic functional Sglobal[Ψ]=21∑(i,j)∈EAijψi−eiθijψj2 provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues λn to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism. The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.
Lattice gauge theories provide a first-principles, non-perturbative formulation of the fundamental interactions of the Standard Model, and their Hamiltonian reformulation on a quantum simulator offers a promising route to regimes---real-time dynamics, finite baryon density---that lie beyond the reach of Euclidean Monte Carlo methods. With the ultimate goal of overcoming these shortcomings, over the last decade, quantum simulations of Abelian lattice gauge theories have expanded from the first proofs of principle to the experimental study of confinement and string breaking by pair production. One of the next frontiers lies in the simulation of non-Abelian theories, which are significantly more demanding both in local Hilbert space dimension and in symmetry-protection requirements, and ultimately essential for addressing open problems in high-energy physics. In this thesis, we investigate dihedral non-Abelian lattice gauge theories DN---whose small order and finite local dimension make D3 and D4 natural targets for near-term quantum platforms---as a minimal yet physically rich arena for the study of non-Abelian gauge dynamics. After introducing dihedral groups and the associated lattice gauge theories, we propose a gauge-fixed ladder model that reduces the effective dimensionality while preserving the essential physics. We use it to show that the D3 theory on a ladder displays a smooth crossover---rather than a sharp phase transition---between electric- and magnetic-dominated regimes, with the nonstabilizerness peaking in the crossover region. Furthermore, we introduce a formulation of the model compatible with the analysis of the Eigenstate Thermalization Hypothesis. We then study string breaking in pure-gauge DN ladders in the presence of static charges through tensor-network DMRG simulations in the rishon formalism. We show that for odd N static charges are screened by gluelumps---charge-glueball bound states enabled by the trivial centre of the group---while for even N a fundamental flux string persists. Turning to the quantum-simulation side, we introduce a dynamical post-selection (DPS) protocol in which the local gauge charges are tested by mid-circuit measurements on auxiliary registers, and we show that gauge protection emerges through a sharp quantum Zeno transition in the spectrum of an associated Liouvillian. We verify the generality of this picture on Z2, Z3, and truncated U(1) gauge groups, identifying an optimal window of measurement frequencies that survives also in the presence of noisy measurements. Finally, we benchmark DPS against an alternative post-processed symmetry verification scheme on a two-plaquette D3 model designed for state-of-the-art qudit hardware. We find that both protocols significantly improve the reconstructed dynamics, with DPS tracking the exact evolution over longer times. Taken together, the results of this thesis position dihedral gauge theories as a concrete stepping stone from Abelian proofs of principle toward genuinely non-Abelian quantum simulation on near-term devices.
We study the effects of uncorrelated quenched disorder to the phase diagram and continuous transitions of three-dimensional lattice Z_{2} gauge Higgs models. For this purpose, we consider two types of quenched disorder, associated with the sites and plaquettes of the cubic lattice. In both cases, for sufficiently weak disorder, the phase diagram remains similar to that of the pure system, showing two different phases (one of them being a topologically ordered phase), separated by two different continuous transition lines. However, the quenched disorder changes the universality classes of the critical behaviors along some of the transition lines. The random-plaquette disorder turns out to be relevant along the topological Z_{2} gauge transition line, so the critical behaviors belong to the different random-plaquette Z_{2} gauge (RPZ_{2}G) universality class with length-scale exponent ν=ν_{rp}≈0.82; on the other hand, it turns out to be irrelevant along the other Ising^{×} transition line (a variant of the Ising transitions with a gauge-dependent order parameter), leaving unchanged its asymptotic critical behaviors with ν=ν_{I}≈0.63. The random-site disorder leads to a substantially different scenario: it destabilizes the Ising^{×} critical behaviors of the pure model, changing them into those of the randomly dilute Ising^{×} (RDI^{×}) universality class with ν=ν_{RDI}≈0.68, while the critical behaviors along the other Z_{2} gauge topological transition line remains stable, with ν=ν_{I}≈0.63.
We report an intrinsic "Bubble Phase" in the two-dimensional Kitaev-Bose-Hubbard model, driven purely by phase frustration between complex hopping and anisotropic pairing. By combining Inhomogeneous Gutzwiller Mean-Field Theory with a Bogoliubov-de Gennes stability analysis augmented by a novel Energy Penalty Method, we demonstrate that this phase spontaneously fragments into coherent islands, exhibiting the hallmark Bose glass signature of finite compressibility without global superfluidity. Notably, we propose a unified framework linking disorder-driven localization to deterministic phase frustration, identifying the Bubble Phase as a pristine, disorder-free archetype of the Bose glass. Our results provide a theoretical blueprint for realizing glassy dynamics in clean quantum simulators.