Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension

Aksoy, Ömer M.; Mudry, Christopher

doi:10.1103/physrevb.106.035117

Cited by 6 publications

(2 citation statements)

References 24 publications

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“…1 come with an additional minus sign because of the anti-periodic boundary conditions ( f = 1). 19 In fact, they are inverse of each other under the fermionic stacking operation [159][160][161]. 20 The upper corners (∆, J) = (0, ∞) and (∆, J) = (∞, ∞) are gapped and two-fold degenerate nder antiperiodic boundary conditions as a consequence of the dualization from Sec.…”

Section: Jordan-wigner Dual Interacting Majorana Chainmentioning

confidence: 97%

Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

Aksoy,

Mudry,

Furusaki

et al. 2024

SciPost Phys.

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Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2}ℤ2×ℤ2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2}ℤ2⊂ℤ2×ℤ2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZXYZ chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n}ℤn clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n}ℤn×ℤn global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n}ℤn charge and dipole symmetries.

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Section: Jordan-wigner Dual Interacting Majorana Chainmentioning

confidence: 97%

Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

Aksoy,

Mudry,

Furusaki

et al. 2024

SciPost Phys.

Self Cite

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“…The classification of (1+1)D invertible fermionic phases, including their complete stacking rules, has been presented in Refs. [63][64][65]. Let G f , G b and ω 2 be defined as usual.…”

Section: Examples Of Nontrivial Constraintsmentioning

confidence: 99%

Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D

Manjunath¹,

Calvera²,

Barkeshli³

2022

Preprint

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In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group G f of the ground state splits aswhere Z f 2 is generated by fermion parity, (−1) F . In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when G f is not of the form G b × Z f 2 . In this paper we study how G f together with the chiral central charge c− strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as U(1) f H, SU(2) f × H, U(2) f H, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr2RuO4, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models. CONTENTS 2 D. MZMs in reflection symmetric systems 1. TSCs with mirror symmetry

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Duality and stacking of bosonic and fermionic SPT phases

Turzillo,

You

2024

J. High Energ. Phys.

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We study the interplay of duality and stacking of bosonic and fermionic symmetry-protected topological phases in one spatial dimension. In general the classifications of bosonic and fermionic phases have different group structures under the operation of stacking, but we argue that they are often isomorphic and give an explicit isomorphism when it exists. This occurs for all unitary symmetry groups and many groups with antiunitary symmetries, which we characterize. We find that this isomorphism is typically not implemented by the Jordan-Wigner transformation, nor is it a consequence of any other duality transformation that falls within the framework of topological holography. Along the way to this conclusion, we recover the fermionic stacking rule in terms of $$ \mathcal{G} $$ G -pin partition functions, give a gauge-invariant characterization of the twisted group cohomology invariant, and state a procedure for stacking gapped phases in the formalism of symmetry topological field theory.

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Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension

Cited by 6 publications

References 24 publications

Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D

Duality and stacking of bosonic and fermionic SPT phases

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