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Bulletin of the American Physical Society

Fermion Hubbard model on non-bipartite lattices: flux problem, gauge fields and emergent chirality

Wayne Zheng

Published Oct 14, 2019• 0 citations
OpenAlex

Abstract

On some one dimensional (1D) and 2D non-bipartite lattices, from scratch we study both free and Hubbard interacting lattice fermions when there are some magnetic fluxes threaded or appropriate gauge fields coupled. On one hand, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-$1/2$ fermions at half-filling on a ring lattice with an odd number of sites, the optimal flux is determined to be $\pm{\pi}/2$. We prove this conclusion for Hubbard interacting fermions by means of a generalized reflection positivity technique. It can further lead to some applications towards 2D non-bipartite lattices such as triangle and Kagome. At half-filling the optimal flux pattern on the triangular lattice can be ascertained to be $[{\pi}/2, {\pi}/2]$. We find that chirality emerges in these optimal flux states. On the other hand, we verify these conclusions and further study other fillings away from half with the numerical exact diagonalization (ED) method and find that, when it deviates from half-filling, Hubbard interactions can even alter the optimal flux patterns on these lattices. Both in 1D and 2D, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states are addressed and interpreted as some non-Fermi liquid (NFL) features.

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Fermi–Hubbard model on nonbipartite lattices: flux problem and emergent chirality

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Abstract On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/2 fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be ± π /2. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be ±[ π /2, π /2], ±[ π /2, π /2, 0], respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states— both in 1D and 2D—are discussed and interpreted as some kind of non-Fermi liquid feature.

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Sign structure, electron fractionalization, and emergent gauge description of the Hubbard model

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The Fermi sign structure plays a crucial role for a Landau's Fermi liquid. In this work, we identify the exact sign structure for the bipartite Hubbard model at an arbitrary strength of the on-site Coulomb repulsion $U$. This general sign structure naturally reproduces the conventional fermion signs in the small $U$ limit, and the phase string signs of the $t\ensuremath{-}J$ model in the large $U$ limit. We focus on the half-filling case as an example to illustrate why such a generic sign structure is important to understand the transition from the weakly correlated Fermi-liquid regime to the strongly correlated Mott regime. In particular, we show that an electron fractionalization scheme with emergent partons and mutual Chern-Simons gauge fields provides a suitable framework to accurately handle the singular sign structure in two dimensions. A ground-state ansatz at half filling with incorporating the sign structure and the specific electron fractionalization is also proposed.

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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 $\mathbb{Z}_2$-gauge fields. We prove the stability of the $\pi$-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 $\pi$-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.

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Stability of the $$\pi $$-Flux Phase for $$\mathbb {Z}_{2}$$ Lattice Gauge Theory Coupled to Fermionic Matter

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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 $$\pi $$ π -flux phase, associated with magnetic flux equal to $$\pi $$ 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.

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Hamiltonian models of lattice fermions solvable by the meron-cluster algorithm

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We introduce a half-filled Hamiltonian of spin-half lattice fermions that can be studied with the efficient meron-cluster algorithm in any dimension. As with the usual bipartite half-filled Hubbard models, the na\"{\i}ve $U(2)$ symmetry is enhanced to $SO(4)$. On the other hand, our model has a novel spin-charge flip ${\mathbb{Z}}_{2}^{C}$ symmetry which is an important ingredient of free massless fermions. In this work we focus on one spatial dimension and show that our model can be viewed as a lattice-regularized two-flavor chiral-mass Gross-Neveu model. Our model remains solvable in the presence of the Hubbard coupling $U$, which maps to a combination of Gross-Neveu and Thirring couplings in one dimension. Using the meron-cluster algorithm we find that the ground state of our model is a valence bond solid when $U=0$. From our field theory analysis, we argue that the valence bond solid forms inevitably because of an interesting frustration between spin and charge sectors in the renormalization group flow enforced by the ${\mathbb{Z}}_{2}^{C}$ symmetry. This state spontaneously breaks translation symmetry by one lattice unit, which can be identified with a ${\mathbb{Z}}_{2}^{\ensuremath{\chi}}$ chiral symmetry in the continuum. We show that increasing $U$ induces a quantum phase transition to a critical phase described by the $SU(2{)}_{1}$ Wess-Zumino-Witten theory. The quantum critical point between these two phases is known to exhibit a novel symmetry enhancement between spin and dimer. Here we verify the scaling relations of these correlation functions near the critical point numerically. Our study opens up the exciting possibility of numerical access to similar novel phase transitions in higher dimensions in fermionic lattice models using the meron-cluster algorithm.

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$π$-Flux Tuning of Quantum Paramagnetism in the Triangular-Lattice Hubbard Model

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We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered $π$-flux model with zero and $π$ flux through oppositely oriented triangles. Although both reduce to the same Heisenberg model at leading order in the strong-coupling limit, their distinct charge fluctuation profiles yield different quantum paramagnetic regimes. Combining a strong-coupling expansion with infinite density matrix renormalization group (iDMRG) calculations, we find that the positive ring-exchange of the zero-flux model promotes spontaneous chirality, consistent with its notion of a chiral spin liquid at intermediate coupling. In the staggered-$π$-flux model, whose noninteracting limit is a Dirac semimetal, the ring exchange changes sign and disfavors chiral order and we find no resolvable scalar chirality. This renders flux-tuned ring-exchange an efficient discriminator between chiral and non-chiral quantum paramagnets at finite coupling.

arXiv:2610.30290•Oct 21, 2026•0 citations
On some one dimensional (1D) and 2D non-bipartite lattices, from scratch we study both free and Hubbard interacting lattice fermions when there are some magnetic fluxes threaded or appropriate gauge fields coupled. On one hand, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/21/21/2 fermions at half-filling on a ring lattice with an odd number of sites, the optimal flux is determined to be ±π/2\pm{\pi}/2±π/2. We prove this conclusion for Hubbard interacting fermions by means of a generalized reflection positivity technique. It can further lead to some applications towards 2D non-bipartite lattices such as triangle and Kagome. At half-filling the optimal flux pattern on the triangular lattice can be ascertained to be [π/2,π/2][{\pi}/2, {\pi}/2][π/2,π/2]. We find that chirality emerges in these optimal flux states. On the other hand, we verify these conclusions and further study other fillings away from half with the numerical exact diagonalization (ED) method and find that, when it deviates from half-filling, Hubbard interactions can even alter the optimal flux patterns on these lattices. Both in 1D and 2D, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states are addressed and interpreted as some non-Fermi liquid (NFL) features.
Abstract On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/2 fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be ± π /2. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be ±[ π /2, π /2], ±[ π /2, π /2, 0], respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states— both in 1D and 2D—are discussed and interpreted as some kind of non-Fermi liquid feature.
The Fermi sign structure plays a crucial role for a Landau's Fermi liquid. In this work, we identify the exact sign structure for the bipartite Hubbard model at an arbitrary strength of the on-site Coulomb repulsion UUU. This general sign structure naturally reproduces the conventional fermion signs in the small UUU limit, and the phase string signs of the $t\ensuremath{-}J$ model in the large UUU limit. We focus on the half-filling case as an example to illustrate why such a generic sign structure is important to understand the transition from the weakly correlated Fermi-liquid regime to the strongly correlated Mott regime. In particular, we show that an electron fractionalization scheme with emergent partons and mutual Chern-Simons gauge fields provides a suitable framework to accurately handle the singular sign structure in two dimensions. A ground-state ansatz at half filling with incorporating the sign structure and the specific electron fractionalization is also proposed.
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\mathbb{Z}_2Z2​-gauge fields. We prove the stability of the π\piπ-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\piπ-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.
Stability of the π\pi π-Flux Phase for Z2\mathbb {Z}_{2}Z2​ Lattice Gauge Theory Coupled to Fermionic Matter
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\pi π π -flux phase, associated with magnetic flux equal to π\pi 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.
We introduce a half-filled Hamiltonian of spin-half lattice fermions that can be studied with the efficient meron-cluster algorithm in any dimension. As with the usual bipartite half-filled Hubbard models, the na\"{\i}ve U(2)U(2)U(2) symmetry is enhanced to SO(4)SO(4)SO(4). On the other hand, our model has a novel spin-charge flip Z2C{\mathbb{Z}}_{2}^{C}Z2C​ symmetry which is an important ingredient of free massless fermions. In this work we focus on one spatial dimension and show that our model can be viewed as a lattice-regularized two-flavor chiral-mass Gross-Neveu model. Our model remains solvable in the presence of the Hubbard coupling UUU, which maps to a combination of Gross-Neveu and Thirring couplings in one dimension. Using the meron-cluster algorithm we find that the ground state of our model is a valence bond solid when U=0U=0U=0. From our field theory analysis, we argue that the valence bond solid forms inevitably because of an interesting frustration between spin and charge sectors in the renormalization group flow enforced by the Z2C{\mathbb{Z}}_{2}^{C}Z2C​ symmetry. This state spontaneously breaks translation symmetry by one lattice unit, which can be identified with a ${\mathbb{Z}}_{2}^{\ensuremath{\chi}}$ chiral symmetry in the continuum. We show that increasing UUU induces a quantum phase transition to a critical phase described by the SU(2)1SU(2{)}_{1}SU(2)1​ Wess-Zumino-Witten theory. The quantum critical point between these two phases is known to exhibit a novel symmetry enhancement between spin and dimer. Here we verify the scaling relations of these correlation functions near the critical point numerically. Our study opens up the exciting possibility of numerical access to similar novel phase transitions in higher dimensions in fermionic lattice models using the meron-cluster algorithm.
πππ-Flux Tuning of Quantum Paramagnetism in the Triangular-Lattice Hubbard Model
We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered πππ-flux model with zero and πππ flux through oppositely oriented triangles. Although both reduce to the same Heisenberg model at leading order in the strong-coupling limit, their distinct charge fluctuation profiles yield different quantum paramagnetic regimes. Combining a strong-coupling expansion with infinite density matrix renormalization group (iDMRG) calculations, we find that the positive ring-exchange of the zero-flux model promotes spontaneous chirality, consistent with its notion of a chiral spin liquid at intermediate coupling. In the staggered-πππ-flux model, whose noninteracting limit is a Dirac semimetal, the ring exchange changes sign and disfavors chiral order and we find no resolvable scalar chirality. This renders flux-tuned ring-exchange an efficient discriminator between chiral and non-chiral quantum paramagnets at finite coupling.