The entanglement gap and a new principle of adiabatic continuity

TL;DR

This paper defines the entanglement gap in FQH states, introduces the conformal limit to analyze the entanglement spectrum, and proposes a new principle of adiabatic continuity to identify topological connections between states.

Contribution

It provides a complete definition of the entanglement gap, introduces the conformal limit procedure, and demonstrates a new adiabatic continuity principle for topological state analysis.

Findings

The entanglement spectrum reorganizes into a low-energy part separated by a full gap.

Finite-size effects in level counting serve as fingerprints of FQH states.

Level spacing decreases with system size, indicating gapless edge states.

Abstract

We give a complete definition of the entanglement gap separating low-energy, topological levels, from high-energy, generic ones, in the "entanglement spectrum" of Fractional Quantum Hall (FQH) states. By removing the magnetic length inherent in the FQH problem - a procedure which we call taking the "conformal limit", we find that the entanglement spectrum of an incompressible ground-state of a generic (i.e. Coulomb) lowest Landau Level Hamiltonian re-arranges into a low-(entanglement) energy part separated by a full gap from the high energy entanglement levels. As previously observed, the counting of these levels starts off as the counting of modes of the edge theory of the FQH state, but quickly develops finite-size effects which we show can also serve as a fingerprint of the FQH state. As the sphere manifold where the FQH resides grows, the level spacing of the states at the same angular momentum goes to zero, suggestive of the presence of relativistic gapless edge-states. By using the adiabatic continuity of the low entanglement energy levels, we investigate whether two states are topologically connected.