In this work we realize the 3 + 1 dimensional non-invertible ${\mathbb{Z}}_N$ chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional $U(1)$ rotor site Hilbert spaces. Specifically, our Hilbert space is that of a $U(1)$ lattice gauge theory coupled to a charge $1$ scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ${\mathbb{Z}}_N$ one-form symmetry $Z^{(1)}_m$ , at the lattice Hamiltonian level. We construct the generator of the ${\mathbb{Z}}_N$ chiral symmetry as as a unitary operator in the subspace of $Z^{(1)}_m$ -invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging $Z^{(1)}_m$ , we find a dual description of this subspace, as the subspace of a charge $1/N$ gauge theory invariant under an electric one-form symmetry $Z^{(1)}_e$ . We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the $Z^{(1)}_e$ symmetry.In this work we realize the 3 + 1 dimensional non-invertible ${\mathbb{Z}}_N$ chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional $U(1)$ rotor site Hilbert spaces. Specifically, our Hilbert space is that of a $U(1)$ lattice gauge theory coupled to a charge $1$ scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ${\mathbb{Z}}_N$ one-form symmetry $Z^{(1)}_m$ , at the lattice Hamiltonian level. We construct the generator of the ${\mathbb{Z}}_N$ chiral symmetry as as a unitary operator in the subspace of $Z^{(1)}_m$-invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging $Z^{(1)}_m$, we find a dual description of this subspace, as the subspace of a charge $1/N$ gauge theory invariant under an electric one-form symmetry $Z^{(1)}_e$. We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the $Z^{(1)}_e$ symmetry.

arXiv:2510.17969Oct 20, 20250 citations

Chiral Bosonic Mott Insulator on the Frustrated Triangular Lattice

S. A. Parameswaran, Michael P. Zaletel, Andreas R uegg +1 more

Bulletin of the American Physical SocietyMar 7, 20141 citation

Spin chains and chiral lattice fermions

H.B. Thacker

arXiv:hep-lat/9412019Nuclear Physics B - Proceedings SupplementsApr 1, 19954 citations

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

Nathan Seiberg B, Shu-Heng Shao

We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L} ( − 1 ) F L emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L} ( − 1 ) F L emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

arXiv:2307.02534SciPost PhysicsMar 1, 2024115 citations

Remarks on invertible phases with non-onsite symmetry

Ryohei Kobayashi, Kansei Inamura, Ken Shiozaki

We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic $\mathbb Z_4^F$ symmetry in (2+1) dimensions. For an onsite $\mathbb Z_4^F$ symmetry, symmetric invertible phases have integer chiral central charge $c_-\in\mathbb Z$ , whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local $\mathbb Z_4^F$ symmetry of a $p+ip$ superconductor with $c_-=1/2$ . We compute its lattice anomaly, and find that the symmetry indeed forbids $c_-\in \mathbb{Z}$ invertible states. The allowed invertible phases are consequently shifted to $c_-\in\mathbb Z+1/2$ . By gauging the fermion parity of the $p+ip$ superconductor, one obtains an Ising topological order enriched by a $\mathbb Z_4$ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the $p+ip$ state admits an enlarged non-onsite $U(1)^f$ symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite $U(1)^f$ symmetry again forbids Chern insulators carrying $c_-\in\mathbb{Z}$ , while being compatible with invertible states with $c_-\in\mathbb{Z}+1/2$ . This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic $\mathbb Z_4^F$ symmetry in (2+1) dimensions. For an onsite $\mathbb Z_4^F$ symmetry, symmetric invertible phases have integer chiral central charge $c_-\in\mathbb Z$, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local $\mathbb Z_4^F$ symmetry of a $p+ip$ superconductor with $c_-=1/2$. We compute its lattice anomaly, and find that the symmetry indeed forbids $c_-\in \mathbb{Z}$ invertible states. The allowed invertible phases are consequently shifted to $c_-\in\mathbb Z+1/2$. By gauging the fermion parity of the $p+ip$ superconductor, one obtains an Ising topological order enriched by a $\mathbb Z_4$ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the $p+ip$ state admits an enlarged non-onsite $U(1)^f$ symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite $U(1)^f$ symmetry again forbids Chern insulators carrying $c_-\in\mathbb{Z}$, while being compatible with invertible states with $c_-\in\mathbb{Z}+1/2$. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.

cond-mat.str-elCross-listedarXiv:2609.12042Sep 14, 20260 citations

Quantum Groups in Lattice Models

Vincent Pasquier

NATO ASI series. Series B : PhysicsJan 1, 19921 citation

In this work we realize the 3 + 1 dimensional non-invertible

{\mathbb{Z}}_N

chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional

U(1)

rotor site Hilbert spaces. Specifically, our Hilbert space is that of a

U(1)

lattice gauge theory coupled to a charge

1

scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic

{\mathbb{Z}}_N

one-form symmetry

Z^{(1)}_m

, at the lattice Hamiltonian level. We construct the generator of the

{\mathbb{Z}}_N

chiral symmetry as as a unitary operator in the subspace of

Z^{(1)}_m

-invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging

Z^{(1)}_m

, we find a dual description of this subspace, as the subspace of a charge

1/N

gauge theory invariant under an electric one-form symmetry

Z^{(1)}_e

. We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the

Z^{(1)}_e

symmetry.

\mathbb Z_4^F

symmetry in (2+1) dimensions. For an onsite

\mathbb Z_4^F

symmetry, symmetric invertible phases have integer chiral central charge

c_-\in\mathbb Z

, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local

\mathbb Z_4^F

symmetry of a

p+ip

superconductor with

c_-=1/2

. We compute its lattice anomaly, and find that the symmetry indeed forbids

c_-\in \mathbb{Z}

invertible states. The allowed invertible phases are consequently shifted to

c_-\in\mathbb Z+1/2

. By gauging the fermion parity of the

p+ip

superconductor, one obtains an Ising topological order enriched by a

\mathbb Z_4

symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the

p+ip

state admits an enlarged non-onsite

U(1)^f

symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite

U(1)^f

symmetry again forbids Chern insulators carrying

c_-\in\mathbb{Z}

, while being compatible with invertible states with

c_-\in\mathbb{Z}+1/2

. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.