Gauging defects in quantum spin systems: A case study

Bridgeman, Jacob C.; Hahn, Alexander; Osborne, Tobias J.; Wolf, Ramona

doi:10.1103/physrevb.101.134111

“…are both back-scattered within a wire or in-between wires. Similar to (38), a trivial topological phase of decoupled wires is obtained when all degrees of freedom are gapped by complete intra-wire interactions,…”

Section: Dihedral Twist Liquidsmentioning

confidence: 87%

“…In some cases, by tuning the relative strengths of intra-and inter-wire backscattering interactions, a gap-closing phase transition can be driven from one topological phase to another. For example, (32), (35), (38) and (39) describe exactly-solvable models for the U (1) l , SU (n) 1 , trivial and SO(2n) 1 topological order respectively. These models are special points in a moduli space of Hamiltonians and can go from one to another by tuning the density-density interaction strengths u intra ⊥ , u inter ⊥ , ũintra , ũinter and the sine-Gordon backscattering bare strength ∆ intra ⊥ , ∆ inter ⊥ , ∆ intra , ∆ inter .…”

Section: +1d Quantum Critical Phase Transitionsmentioning

confidence: 99%

Dihedral twist liquid models from emergent Majorana fermions

Teo¹,

Hu²

2021

Preprint

0

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We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N ) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral D k symmetry of SO(2n)1, and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G = Z2 for C = U (1) l , SU (n)1, and G = Z2, Z k , D k for C = SO(2n)1. We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories. CONTENTS I. Introduction1 A. Summary of results 2 II. Boson wires from interacting electrons 4 III. Topological phases in the orthogonal and unitary families 6 A. The SO(N ) 1 family 6 B. The U (1) l and SU (n) 1 family 7 C. Gauging the Z k or Z 2 symmetries in SO(2n) 1 9 1. The SO(2n) 1 /Z 2 twist liquid 9 2. The SO(2n) 1 /Z k twist liquid 11 IV. Dihedral twist liquids 12 A. The SU (n) 1 /Z 2 twist liquid 13 B. The U (1) l /Z 2 twist liquid 15 C. The SO(2n) 1 /D k twist liquid 17 V. Conclusion and discussion 21 Acknowledgments 23 A. The SO(N ) 1 current algebra and its primary fields 23 B. The anyon content of SO(n) 2 and U (1) l /Z 2 25 C. Discrete gauge theories with cyclic, dihedral or dicyclic gauge groups 27 1. The Z k gauge theory 31 2. The D k gauge theory 32 3. The Q 4k gauge theory 33 D. Useful group cohomology identities 33 References 41

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“…We derive a standalone effective theory for the edge of a quantum double model with group G. The only requirement we impose on the edge Hamiltonian is commutation with the bulk, encompassing special cases such as gapped edges [61][62][63][64][65], gapless points [69,70], and combinations thereof [72][73][74]. The theory is obtained by course-graining the 2D quantum-double bulk [24,[75][76][77][78] and projecting its edge into a 1D subspace defined by the bulk being in a ground state.…”

Section: Lattice Edge Theorymentioning

confidence: 99%

Spin chains, defects, and quantum wires for the quantum-double edge

Albert¹,

Aasen²,

Xu³

et al. 2021

Preprint

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Non-Abelian defects that bind Majorana or parafermion zero modes are prominent in several topological quantum computation schemes. Underpinning their established understanding is the quantum Ising spin chain, which can be recast as a fermionic model or viewed as a standalone effective theory for the surface-code edge -both of which harbor non-Abelian defects. We generalize these notions by deriving an effective Ising-like spin chain describing the edge of quantum-double topological order. Relating Majorana and parafermion modes to anyonic strings, we introduce quantum-double generalizations of non-Abelian defects. We develop a way to embed finite-group valued qunits into those valued in continuous groups. Using this embedding, we provide a continuum description of the spin chain and recast its non-interacting part as a quantum wire via addition of a Wess-Zumino-Novikov-Witten term and non-Abelian bosonization.

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“…When symmetry is present, these phases are further distinguished by their symmetryprotected or symmetry-enriched topological (SPT/SET) orders. [5,6,7,8,9,10,11,12,13,14,15] Symmetries in topological phases can be analyzed theoretically by studying the statistical properties of fluxes, referred to as twist defects [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. These topological point defects "rotate" the local winding low-energy degrees of freedom according to the symmetry actions.…”

Section: Introductionmentioning

confidence: 99%

Dihedral twist liquid models from emergent Majorana fermions

Teo

¹

,

Hu

²

2023

Quantum

1

0

View full text Add to dashboard Cite

We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral Dk symmetry of SO(2n)1, and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G=Z2 for C=U(1)l, SU(n)1, and G=Z2, Zk, Dk for C=SO(2n)1. We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.

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Gauging defects in quantum spin systems: A case study

Cited by 5 publications

References 82 publications

Dihedral twist liquid models from emergent Majorana fermions

Dihedral twist liquid models from emergent Majorana fermions

Spin chains, defects, and quantum wires for the quantum-double edge

Dihedral twist liquid models from emergent Majorana fermions

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