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arXiv:2503.01831 · Physical review. B./Physical review. B

Z 2 gauging and self-dualities of the X X model and its cousins

Lei Su

Published Jun 23, 2025• 4 citations
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Abstract

In this work, we investigate the one-dimensional $XX$ lattice model and its cousins through the lens of momentum and winding $U(1)$ symmetries. We distinguish two closely related $\mathbb{Z}_2$ symmetries based on their relation to the $U(1)$ symmetries, and establish a web of $\mathbb{Z}_2$-gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective $\mathbb{Z}_2$-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the $XX$ model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the $c =1$ compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.

DOI
10.1103/c16r-fdfb

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In this work, we investigate the one-dimensional XXXXXX lattice model and its cousins through the lens of momentum and winding U(1)U(1)U(1) symmetries. We distinguish two closely related Z2\mathbb{Z}_2Z2​ symmetries based on their relation to the U(1)U(1)U(1) symmetries, and establish a web of Z2\mathbb{Z}_2Z2​-gauging relations among these models, rooted in two fundamental seeds: the XY±YX XY \pm YXXY±YX models. These two seeds, each self-dual under gauging of the respective Z2\mathbb{Z}_2Z2​-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the XXXXXX model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the c=1c =1c=1 compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.
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We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2 ℤ 2 symmetry. We introduce a subsystem \mathbb Z_2 ℤ 2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.
Self-dual S3S_3S3​ gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2\mathbb{Z}_2Z2​ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3S_3S3​ quantum double D(S3)\mathcal{D}(S_3)D(S3​) on a tensor product Hilbert space in which the Z2em\mathbb{Z}^{\mathrm{em}}_2Z2em​ anyon-permutation symmetry, exchanging the non-Abelian chargeon CCC and fluxon FFF, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em\mathbb{Z}_2^{\mathrm{em}}Z2em​-preserving bosonic perturbations, driving D(S3)\mathcal{D}(S_3)D(S3​) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n)\mathcal{D}(D_{2n})D(D2n​). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
Given a two-dimensional quantum lattice model with an Abelian gauge theory interpretation, we investigate a duality operation that amounts to gauging its invertible 1-form symmetry, followed by gauging the resulting 0-form symmetry in a twisted way via a choice of discrete torsion. Using tensor networks, we introduce explicit lattice realisations of the so-called condensation defects, which are obtained by gauging the 1-form symmetry along submanifolds of spacetime, and employ the same calculus to realise the duality operators. By leveraging these tensor network operators, we compute the non-trivial interplay between symmetry-twisted boundary conditions and charge sectors under the duality operation, enabling us to construct isometries relating the dual Hamiltonians. Whenever a lattice gauge theory is left invariant under the duality operation, we explore the possibility of promoting the self-duality to an internal symmetry. We argue that this results in a symmetry structure that encodes the 2-representations of a 2-group.
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