Topological Transition on the Conformal Manifold

TL;DR

This paper investigates topological transitions within the conformal manifold of (1+1)d fermionic CFTs, revealing how stacking invertible fermionic topological orders induces continuous topological transitions characterized by dualities and crossover of critical exponents.

Contribution

It introduces the concept of topological transitions on the fermionic conformal manifold, linking stacking of invertible fermionic topological orders to dualities and critical phenomena.

Findings

Stacking IFTOs induces topological transitions without changing the fermionic CFT.

Critical exponents of order and disorder operators crossover at transition points.

Examples include the Luttinger model and new $c=2$ gapless phases.

Abstract

Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under developments. In this paper, we study a general phenomenon in the space of (1+1)$d$ critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFT), a.k.a. 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)$d$ $\mathbb{Z}_2$ 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 new class of $c=2$ examples describing $U(1)\times SU(2)$-protected gapless phases.