Topological transition on the conformal manifold

Abstract

Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development. In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold. Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus. At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged. We call this continuous transition on the fermionic conformal manifold a topological transition. By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs. Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order. We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.

Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions

“…The critical point is Z 2 symmetric. More than that, it also has an additional Z 2 symmetry [11,[36][37][38][39].…”

“…For Z 2 ∨ Z 2 categorical symmetry, there is only one minimal gapless state Eq. (39). For categorical symmetry characterized by 2 + 1D S 3 = Z 3 Z 2 gauge theory there is also only one minimal gapless state (see Table I) [31]: , (46) where χ 6,5 h (τ ) are characters of the (6,5) minimal model (with central charge c = 4 5 ).…”

Lessons from the Ramond sector

“…11 The result is a square of relations for a quadruple of CFTs shown in Figure 1. We refer the reader to [29][30][31][32] for a pedagogical introduction to this subject. Given a Γ θ modular function, one could ask if its interpretation as the NS+ partition function of a fermionic CFT can be consistent with the square of maps shown in Figure 1.…”

“…At the level of the torus partition functions, this corresponds to summing over all four spin structures: periodic/anti-periodic boundary conditions on the space and time circles. This story has recently been modernized and enriched into more precise bosonization/fermionization maps [24][25][26][27][28][29][30][31][32][33][34][35]. Fermionic CFTs come in pairs related by tensoring with the fermionic symmetryprotected topological (SPT) phase (−1) Arf where Arf is the Arf invariant.…”

“…Table 1 summarizes the different contributions to ground state degeneracies. A minimal combination of partition functions that does not appear to suffer any negativity is the sum of (i) the Γ 0 (2) modular sum of the N = 1 vacuum character, (ii) two copies of the 32 , and (iii) eight copies of the free c = 3 2 theory. This resulting partition function has a Ramond ground state degeneracy of 32.…”