Topological and symmetry broken phases of Z_N parafermions in one dimension

We classify the gapped phases of Z_N parafermions in one dimension and construct a representative of each phase. Even in the absence of additional symmetries besides parafermionic parity, parafermions may be realized in a variety of phases, one for each divisor n of N. The phases can be characterized by spontaneous symmetry breaking, topology, or a mixture of the two. Purely topological phases arise if n is a unitary divisor, i.e. if n and N/n are co-prime. Our analysis is based on the explicit realization of all symmetry broken gapped phases in the dual Z_N-invariant quantum spin chains.Comment: 16 pages; v2: improved exposition and additional reference

Infinite Matrix Product States for long range SU(N) spin models

We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians associated with infinite matrix product states. The latter are constructed from correlators of primary fields in the SU(N) level 1 WZW model. Since the resulting groundstates are of Gutzwiller-Jastrow type, our models can be regarded as lattice discretizations of fractional quantum Hall systems. We then focus on two specific types of 1D spin chains with spins located on the unit circle, a uniform and an alternating arrangement. For an equidistant distribution of identical spins we establish an explicit connection to the SU(N) Haldane-Shastry model, thereby proving that the model is critical and described by a SU(N) level 1 WZW model. In contrast, while turning out to be critical as well, the alternating model can only be treated numerically. Our numerical results rely on a reformulation of the original problem in terms of loop models.Comment: 37 pages, 6 figure

On conformal field theories based on Takiff superalgebras

We revisit the construction of conformal field theories based on Takiff algebras and superalgebras that was introduced by Babichenko and Ridout. Takiff superalgebras can be thought of as truncated current superalgebras with Z-grading which arise from taking p copies of a Lie superalgebra g and placing them in the degrees s=0,...,p-1. Using suitably defined non-degenerate invariant forms we show that Takiff superalgebras give rise to families of conformal field theories with central charge c=p sdim(g). The resulting conformal field theories are defined in the standard way, i.e. they lend themselves to a Lagrangian description in terms of a WZW model and their chiral energy momentum tensor is the one obtained naturally from the usual Sugawara construction. In view of their intricate representation theory they provide interesting examples of conformal field theories.Comment: 13 pages, v2: Corrected several typos and minor mistakes, added a reference and streamlined some discussion

Reflection and Transmission for Conformal Defects

We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the reflectivity and transmissivity of the defect. Their properties are investigated and they are computed in a number of examples. We obtain a complete answer for all defects in the Ising model and between certain pairs of minimal models. In the case of two conformal field theories with an enhanced symmetry we restrict ourselves to non-trivial defects that can be obtained by a coset construction.Comment: 32 pages + 13 pages appendix, 12 figures; v2: added eqns (2.7), (2.8) and refs [6,7,39,40], version published in JHE

From symmetry-protected topological order to Landau order

Focusing on the particular case of the discrete symmetry group Z_N x Z_N, we establish a mapping between symmetry protected topological phases and symmetry broken phases for one-dimensional spin systems. It is realized in terms of a non-local unitary transformation which preserves the locality of the Hamiltonian. We derive the image of the mapping for various phases involved, including those with a mixture of symmetry breaking and topological protection. Our analysis also applies to topological phases in spin systems with arbitrary continuous symmetries of unitary, orthogonal and symplectic type. This is achieved by identifying suitable subgroups Z_N x Z_N in all these groups, together with a bijection between the individual classes of projective representations.Comment: 8 pages, 1 table. Version v2 corresponds to the published version. It includes minor revisions and additional reference