Topological Transition on the Conformal Manifold

TL;DR

The paper develops a unified framework for topological transitions on the fermionic conformal manifold, showing that along a 1D locus the adjacent CFTs differ by stacking an invertible fermionic topological order (IFTO). At the transition point the IFTO acts as a symmetry, while away from it the two sides are related by IFTO stacking; in bosonic language this corresponds to a Z2^B orbifold across the transition. The authors connect these gapless topological transitions to bulk (2+1)d Z2 topological order via electromagnetic duality, and provide concrete examples including the c=1 Luttinger/Dirac-Thirring system and a c=2 theory with U(1)×SU(2) symmetry. Central tools include bosonization/fermionization, torus partition functions across sectors, duality defects (Kramers-Wannier), and lattice UV realizations, establishing a correspondence between IFTO-stackings, Z2 orbifolds, and KW dualities. The work broadens the understanding of gapless topological phenomena and provides a practical framework to analyze and engineer topological transitions in 1+1d fermionic systems and their bosonic avatars.

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.

Figures (11)

• Figure 1: We can couple the (2+1)$d$ bosonic $\mathbb{Z}_2$ gauge theory to a (1+1)$d$ bosonic theory with a non-anomalous $\mathbb{Z}_2^{\cal B}$ symmetry, or to a (1+1)$d$ fermionic theory. The (2+1)$d$ electromagnetic duality implements either the Kramer-Wannier duality $(\mathbb{Z}_2^{\cal B}$ orbifold) when the (1+1)$d$ boundary is bosonic, or the IFTO stacking ($\mathbb{Z}_2^{\rm IFTO})$ when the boundary is fermionic.
• Figure 2: Top: The Landau phase transition between the symmetry breaking phase (the order phase) and the symmetric phase (the disorder phase). Bottom: The topological transition between two families of gapless phases with power-law decaying correlation functions for both the order and the disorder operators. In one family, the scaling dimension of the order operator is smaller than that of the disorder operator, i.e.$\Delta_{\rm ord}<\Delta_{\rm dis}$, while in the other family we have $\Delta_{\rm ord}>\Delta_{\rm dis}$.
• Figure 3: The commutative diagram of fermionic CFTs $\cal F$, $\mathcal{F}'$ and their bosonizations $\cal B$ and $\mathcal{B}'$.
• Figure 4: The action of topological defect lines on local and non-local operators in the Ising CFT Frohlich:2004ef. The $\mathbb{Z}_2^{\cal B}$ defect line $\eta$ acts on operators with $\pm1$ sign. The duality defect $\cal N$ exchanges the local, order operator $\sigma(z,\bar{z})$ with the non-local, disorder operator $\mu(z,\bar{z})$, which lives at the end of the $\mathbb{Z}_2^{\cal B}$ defect line $\eta$.
• Figure 5: The duality defect $\cal N$ in the self-dual bosonic CFT ${\cal B}_0$ becomes a duality interface between ${\cal B}$ and ${\cal B}'$ after renormalization group flow. The duality defect $\cal N$ in $\cal B$ turns into the $\mathbb{Z}_2^{\rm IFTO}$ symmetry defect of $\cal F$ under fermionization.