The subject of this paper is the phase transition between symmetry protected topological states (SPTs). 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. Published by Elsevier B.V. This is an open access article under the CC BY license
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
In this paper we study the lattice model of Zn-1-symmetry protected topological states (1-SPT) in 3+1D for even n. We write down an exactly soluble lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with non-trivial self-statistics. For the n = 2 case, where the bulk classification is given by an integer m mod 4, we show that the boundary can be gapped with double semion topological order for m = 1 and toric code for m = 2. The bulk ground state wavefunction amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.
CONTENTSE. Evaluation of I 4 (b Zn ) 2 in a hypercube 15 F. Evaluation of Pij in the m=even case 15 G. Evaluation of Pij for general m 16 H. Calculation details for θq, θq 1 q 2 17 I. Evaluation of W i for (n, m) = (2, 1) 17 1. DS projector Hamiltonian 18 J. ω4, φ3 and φ2 18 K. Generalization of (38) and (40) to G-protected 1-SPT for finite unitary groups 19 References 20 arXiv:1908.02613v1 [cond-mat.str-el]
In an earlier work [1] we developed a holographic theory for the phase transition between bosonic symmetry-protected topological (SPT) states. This paper is a continuation of it. Here we present the holographic theory for fermionic SPT phase transitions. We show that in any dimension d, the critical states of fermionic SPT phase transitions has an emergent Z T 2 symmetry and can be realized on the boundary of a d + 1-dimensional bulk SPT with an extra Z T 2 symmetry.
The non-regularizability of free fermion field theories, which is the root of various quantum anomalies, plays a central role in particle physics and modern condensed matter physics. In this paper, we generalize the Nielsen-Ninomiya theorem to all minimal nodal free fermion field theories protected by the time reversal, charge conservation, and charge conjugation symmetries. We prove that these massless field theories cannot be regularized on a lattice.
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