Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.30363

Disentangling the Toric Code

Lei Gioia, Salvatore D. Pace, Ruben Verresen, Shu-Heng Shao, Ryan Thorngren

cond-mat.str-elhep-thmath-phquant-phSubmitted Sep 24, 2026• Announced Sep 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Commuting-projector Hamiltonians with integer spectrum such as the 1+1d Ising ferromagnet and the 2+1d toric code may be regarded as the generators of non-on-site $U(1)$ symmetries. In this paper we study whether these symmetries can be made on-site in the presence of finite and infinite-dimensional ancillae. In the finite-dimensional case, we prove that the fractionalized excitations, such as domain walls and anyons, make it impossible to on-site these Hamiltonians. On the other hand, we show that when infinite dimensional ancillae (e.g., rotors) are allowed, both the 1+1d Ising ferromagnet and the 2+1d toric code can be disentangled, yielding on-site symmetries. This is not in contradiction with the non-trivial ground state order of these Hamiltonians, since the ancilla Hamiltonians are no longer bounded from below. Our results indicate agreement between lattice and quantum field theory anomalies in the presence of infinite-dimensional ancillae, while with finite-dimensional ancillae the obstruction to on-siteability may even be non-invertible.

Comments
14+9 pages, 4 figures

Related papers

Disentangling Anomaly-Free Symmetries of Quantum Spin Chains

Sahand Seifnashri, Wilbur E. Shirley

We clarify the lore that anomaly-free symmetries are either on-site or can be transformed into on-site symmetries. We prove that any finite, internal, anomaly-free symmetry in a 1+1D lattice Hamiltonian system can be disentangled into an on-site symmetry by introducing ancillae and applying conjugation via a finite-depth quantum circuit. We provide an explicit construction of the disentangling circuit using Gauss's law operators and emphasize the necessity of adding ancillae. Our result establishes the converse to a generalized Lieb-Schultz-Mattis theorem by demonstrating that any anomaly-free symmetry admits a trivially gapped Hamiltonian.

arXiv:2503.09717•Physical Review Letters•Jan 15, 2026•2 citations

Phase diagram of a symmetry-enriched toric code

José Garre-Rubio, Mohsin Iqbal, David T. Stephen

We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site $\mathbb{Z}_2\times\mathbb{Z}_2$, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.

•Nov 5, 2020•0 citations

A classification of invertible phases of bosonic quantum lattice systems in one dimension

Anton Kapustin, Nikita Sopenko, Bowen Yang

We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G, the only obstruction is an index valued in degree-2 cohomology of G. We show that two invertible G-invariant states are in the same phase if and only if their indices coincide.

arXiv:2012.15491•Journal of Mathematical Physics•Aug 1, 2021•21 citations

Realizing the fusion rules of Ising anyons without lattice defects

Miguel Jorge Bernabé Ferreira, Pramod Padmanabhan, Paulo Teotonio-Sobrinho

The global $e$-$m$ exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.

•Aug 6, 2015•0 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.

cond-mat.str-elCross-listedarXiv:2609.12042•Sep 14, 2026•0 citations

Anomaly-Free Symmetries with Obstructions to Gauging and Onsiteability

Wilbur E. Shirley, Carolyn Zhang, Wenjie Ji +1 more

We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.

arXiv:2507.21267•Physical Review Letters•Jan 20, 2026•2 citations
Commuting-projector Hamiltonians with integer spectrum such as the 1+1d Ising ferromagnet and the 2+1d toric code may be regarded as the generators of non-on-site U(1)U(1)U(1) symmetries. In this paper we study whether these symmetries can be made on-site in the presence of finite and infinite-dimensional ancillae. In the finite-dimensional case, we prove that the fractionalized excitations, such as domain walls and anyons, make it impossible to on-site these Hamiltonians. On the other hand, we show that when infinite dimensional ancillae (e.g., rotors) are allowed, both the 1+1d Ising ferromagnet and the 2+1d toric code can be disentangled, yielding on-site symmetries. This is not in contradiction with the non-trivial ground state order of these Hamiltonians, since the ancilla Hamiltonians are no longer bounded from below. Our results indicate agreement between lattice and quantum field theory anomalies in the presence of infinite-dimensional ancillae, while with finite-dimensional ancillae the obstruction to on-siteability may even be non-invertible.
We clarify the lore that anomaly-free symmetries are either on-site or can be transformed into on-site symmetries. We prove that any finite, internal, anomaly-free symmetry in a 1+1D lattice Hamiltonian system can be disentangled into an on-site symmetry by introducing ancillae and applying conjugation via a finite-depth quantum circuit. We provide an explicit construction of the disentangling circuit using Gauss's law operators and emphasize the necessity of adding ancillae. Our result establishes the converse to a generalized Lieb-Schultz-Mattis theorem by demonstrating that any anomaly-free symmetry admits a trivially gapped Hamiltonian.
We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.
We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G, the only obstruction is an index valued in degree-2 cohomology of G. We show that two invertible G-invariant states are in the same phase if and only if their indices coincide.
The global eee-mmm exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.
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 Z4F\mathbb Z_4^FZ4F​ symmetry in (2+1) dimensions. For an onsite Z4F\mathbb Z_4^FZ4F​ symmetry, symmetric invertible phases have integer chiral central charge c−∈Zc_-\in\mathbb Zc−​∈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 Z4F\mathbb Z_4^FZ4F​ symmetry of a p+ipp+ipp+ip superconductor with c−=1/2c_-=1/2c−​=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c−∈Zc_-\in \mathbb{Z}c−​∈Z invertible states. The allowed invertible phases are consequently shifted to c−∈Z+1/2c_-\in\mathbb Z+1/2c−​∈Z+1/2. By gauging the fermion parity of the p+ipp+ipp+ip superconductor, one obtains an Ising topological order enriched by a Z4\mathbb Z_4Z4​ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ipp+ipp+ip state admits an enlarged non-onsite U(1)fU(1)^fU(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)fU(1)^fU(1)f symmetry again forbids Chern insulators carrying c−∈Zc_-\in\mathbb{Z}c−​∈Z, while being compatible with invertible states with c−∈Z+1/2c_-\in\mathbb{Z}+1/2c−​∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.