Discrete symmetries of unitary minimal conformal theories

Ruelle, Philippe; Verhoeven, O.

doi:10.1016/s0550-3213(98)00639-7

“…In this appendix we briefly review the results of Ref. 11,23 which answers this question. The key idea of Ref.11 is the following.…”

Section: H the On-site Global Symmetry Of Conformal Field Theoriesmentioning

confidence: 96%

“…In appendix H we summarize the argument in Ref. 11 where it is shown that the maximum on-site internal symmetry ("on-site" symmetries are the ones consisting of product over local transformations that act on the local, e.g. site or group of sites, Hilbert space.)…”

Section: The Constraint On the Central Chargementioning

confidence: 98%

The phase transitions between Z×Z bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases

Tsui

¹

,

Huang

²

,

Jiang

³

et al. 2017

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The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here we study a specific class of such phase transitions in 1+1 dimensions -the phase transition between bosonic topological phases protected by Z n × Z n . We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D. arXiv:1701.00834v2 [cond-mat.str-el] 6 Apr 2017 2. Exactly solvable "fixed point" Hamiltonians for the SPTs Each SPT phase is characterized by an exactly solvable "fixed point" Hamiltonian. In appendix A we briefly review the construction of these Hamiltonians using the "cocycles" associated with the cohomology group [5,6] . For the case relevant to our discussion the following lattice Hamiltonians can be derived [7] so that its ground state belong to the "0" and "1" topological classes of

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“…The unitary minimal Virasoro models are classified by pairs A 2m ; G with G a simply-laced Lie algebra with Coxeter number q 2m or 2m 2, and m 1 an integer [11]. All the required data are encoded in the Dynkin diagrams of A 2m and G [8][9][10].…”

mentioning

confidence: 99%

“…The D 4 model has an S 3 internal symmetry, for which all twisted torus and cylinder partition functions have been computed in [10]. It has four physical boundary conditions, labeled by the nodes of the D 4 Dynkin diagram, with the state j4i being fully invariant under S 3 .…”

mentioning

confidence: 99%

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Kramers-Wannier Dualities via Symmetries

Ruelle

¹

2005

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Topological Phase Transitions

Ma

¹

,

Wang

²

2019

Phase Transition Dynamics

0

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The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. The subject of this dissertation is the phase transition between these symmetry protected topological states (SPTs). There are two main parts in this dissertation. In the first part we consider spatial dimension d and symmetry group G so that the cohomology group, H d+1 (G, U (1)), contains at least one Z 2n or Z factor. We show that the phase transition between the trivial SPT and the root states that generate the Z 2n or Z groups can be induced on the boundary of a d+1 dimensional G × Z T 2-symmetric SPT by a Z T 2 symmetry breaking field. Moreover we show these boundary phase transitions can be "transplanted" to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase. In the second part, we study the phase transition between bosonic topological phases protected by Z n × Z n in 1+1 dimensions. We find a direct transition occurs when n = 2, 3, 4 and in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint(c ≥ 1) on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.

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Discrete symmetries of unitary minimal conformal theories

Cited by 30 publications

References 26 publications

The phase transitions between Z×Z bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases

The phase transitions between Z×Z bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases

Kramers-Wannier Dualities via Symmetries

Topological Phase Transitions

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