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WORLD SCIENTIFIC eBooks

Topological Phase Transitions

Published Mar 1, 1998• 0 citations
Publisher

Abstract

B.4 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 2 × Z 2 clock model . . . . . . . . . . . . . . . . . . .B.5 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 3 × Z 3 clock model . . . . . . . . . . . . . . . . . . .B.6 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 4 × Z 4 clock model . . . . . . . . . . . . .

DOI
10.1142/9789812386298_0018

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B.4 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 2 × Z 2 clock model . . . . . . . . . . . . . . . . . . .B.5 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 3 × Z 3 clock model . . . . . . . . . . . . . . . . . . .B.6 The modular invariant partition function and the primary scaling operators of the orbifold critical Z 4 × Z 4 clock model . . . . . . . . . . . . .
We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of Z_3 Z 3 parafermion or u(1)_6 u ( 1 ) 6 conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of k k -state clock type models.
We consider nonchiral symmetry-protected topological phases of matter in two spatial dimensions protected by a discrete symmetry such as ZK{\mathbb{Z}}_{K}ZK​ or ${\mathbb{Z}}_{K}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{K}$ symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons KKK-matrix theories and fermionic phases relevant to topological superconductors as an example, we demonstrate explicitly that when the modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge state.