Abstract We consider the two-dimensional $$\mathbbm {Z}_{2}$$ Z 2 lattice gauge theory coupled to fermionic matter. In absence of electric fields, we prove that, at half-filling, the ground state of the gauge theory coincides with the
π π -flux phase, associated with magnetic flux equal to
π π in every elementary lattice plaquette, provided the fermionic hopping is large enough. This proves in particular the semimetallic behavior of the ground state of the model. Furthermore, we compute the magnetic susceptibility of the gauge theory, and we prove that it is given by the one of massless 2 d Dirac fermions, thus rigorously justifying recent numerical simulations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.
Kitaev's toric code is an exactly solvable model with
Z2-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for
Z2 lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a $\ensuremath{\pi}$-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.
We develop an exact map of all states and operators from two-dimensional lattices of spins-
1/2 into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of
Z2 lattice gauge theories because it does not rely on imposing local conservation laws and captures the motion of ``charges'' and ``fluxes'' on equal footing. This map allows to explicitly compute the Berry phases for the transport of fluxes in a large class of symmetry-enriched topologically ordered states with emergent
Z2 gauge fields that includes chiral, nonchiral, Abelian or non-Abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a noninteracting dispersion. The numerical complexity of computing such vison phases reduces, therefore, to computing overlaps of ground states of free-fermion Hamiltonians. Among other results, we establish numerically the conditions under which the Majorana-carrying flux excitation in Ising-topologically ordered states enriched by translations acquires the 0 or $\ensuremath{\pi}$ phase when moving around a single plaquette.
Emergent phenomena in interacting fermionic systems often arise as collective many-body effects whose formation can be understood by studying the interplay between geometry and symmetry. Understanding such phenomena from first principles remains one of the central challenges of Mathematical Physics, particularly in regimes where perturbative methods fail and non-trivial phases emerge through strongly correlated dynamics. This thesis investigates a class of lattice fermionic systems exhibiting \textit{Spontaneous Symmetry Breaking}, \textit{Flux ordering} and \textit{Topological Order}. Although the models considered belong to different physical frameworks, including Euclidean lattice field theories formulated in terms of Grassmann variables and Quantum Lattice Systems described by operators acting on Hilbert spaces, they share a common mathematical structure. In each case, geometric features encoded by lattice Dirac operators, magnetic fluxes, and discrete gauge fields interact with Reflection Positivity properties that allow for a rigorous non-perturbative analysis. A central role throughout the thesis is indeed played by \textit{Reflection Positivity} and one of its most striking consequences: the \textit{Chessboard Estimates}. These methods provide powerful tools for converting geometric information into quantitative bounds on energies, correlation functions, and phase stability. The first part of the thesis, based on \cite{fabbri2026chirallongrangeordereuclidean}, is devoted to lattice Gross–Neveu models in two dimensional Euclidean Spacetime. After a Hubbard–Stratonovich transformation, the fermionic theories are mapped to effective bosonic systems for which Reflection Positivity can be established. Combining Reflection Positivity methods with Peierls' Argument, we prove the existence of Chiral Long-Range Order for a class of lattice regularizations and obtain quantitative bounds relating the order parameter to the minimizers of the effective potential. The second part, based on \cite{GP}, focuses on fermions coupled to dynamical
Z2-gauge fields. We prove the stability of the
π-flux phase under gauging by showing that monopole excitations possess a strictly positive energy cost. These results imply the emergence of effective Dirac fermions at low energies and provide a rigorous characterization of the semimetallic phase of the model. Finally, we investigate the topological phase obtained by introducing a fermionic mass gap in the
π-flux phase, following \cite{bachmann2026anyonspifluxphasefermionic}. We prove the existence of an almost four-fold degenerate low-energy sector on the torus, separated from the rest of the spectrum by a uniform gap, and construct quasi-local loop and string operators whose algebra exhibits the characteristic toric-code braiding phases. This provides a rigorous realization of topological order emerging from interacting fermions coupled to a dynamical gauge field. Taken together, these results illustrate how Reflection Positivity methods and spectral properties of the Magnetic Laplacian can be combined to obtain rigorous information on the phase diagram of interacting fermionic systems. They reveal a common mechanism underlying symmetry breaking, flux stabilization, and topological order across a variety of lattice models.
We construct an exactly solvable extension of the spin-
1/2 Kitaev honeycomb model in which conserved
Z2 fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related
2/3-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.
We investigate how symmetry and topological order are coupled in the (
2+1)--dimensional
ZN rank-2 toric code for general
N, which is an exactly solvable point in the Higgs phase of a symmetric rank-2
U(1) gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles ``position-dependent excitations.'' As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by
N lattice sites in others, reminiscent of fracton topological order in
3+1 dimensions. We find that while there are
N2 flavors of
e charges and
2N flavors of
m fluxes, there are not
NN2+2N anyon types. Instead, there are
N6 anyon types, and we can use Chern-Simons theory with six
U(1) gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six
U(1) gauge fields. Thus, the realization of translation symmetry in the
U6(1) Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ torus is
N3gcd(Lx,N)gcd(Ly,N)gcd(Lx,Ly,N), where
gcd stands for ``greatest common divisor.'' We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.