Bose-Fermi duality and entanglement entropies

TL;DR

This work analyzes whether entanglement and Renyi entropies are universal fingerprints of 1+1D CFTs and whether they can distinguish between theories like the massless Dirac fermion and the compact free boson. By examining replica-trick calculations, orbifold twists, and higher-genus partition functions, the authors show Renyi entropies respect Bose-Fermi duality when twists over all cycles are included, and reveal a novel real duality: a 45° rotation of momentum lattices maps many operator correlators and OPEs on the real axis between the ungauged Dirac fermion and the self-dual boson. They prove identical partition functions on Riemann surfaces with imaginary period matrices for certain configurations (notably N=2 and/or n=2), yet establish that for more general cases (N>2, n>2) the full entanglement spectrum can differ, thereby distinguishing between theories. The results illuminate when entanglement measures truly characterize a theory and when apparent coincidences arise from special symmetry constraints, offering a framework that connects universal real-space entanglement, higher-genus partition functions, and duality structure in CFTs.

Abstract

Entanglement (Renyi) entropies of spatial regions are a useful tool for characterizing the ground states of quantum field theories. In this paper we investigate the extent to which these are universal quantities for a given theory, and to which they distinguish different theories, by comparing the entanglement spectra of the massless Dirac fermion and the compact free boson in two dimensions. We show that the calculation of Renyi entropies via the replica trick for any orbifold theory includes a sum over orbifold twists on all cycles. In a modular-invariant theory of fermions, this amounts to a sum over spin structures. The result is that the Renyi entropies respect the standard Bose-Fermi duality. Next, we investigate the entanglement spectrum for the Dirac fermion without a sum over spin structures, and for the compact boson at the self-dual radius. These are not equivalent theories; nonetheless, we find that (1) their second Renyi entropies agree for any number of intervals, (2) their full entanglement spectra agree for two intervals, and (3) the spectrum generically disagrees otherwise. These results follow from the equality of the partition functions of the two theories on any Riemann surface with imaginary period matrix. We also exhibit a map between the operators of the theories that preserves scaling dimensions (but not spins), as well as OPEs and correlators of operators placed on the real line. All of these coincidences can be traced to the fact that the momentum lattice for the bosonized fermion is related to that of the self-dual boson by a 45 degree rotation that mixes left- and right-movers.

Figures (5)

• Figure 1: A line divided up into consecutive intervals. We will be considering the density matrix for the field theory on the intervals $A_{i = 1,2,3}$ upon tracing out the local degrees of freedom in the intervals $B_{i = 0,1,2,3}$.
• Figure 2: A: The reduced density matrix $\rho(\phi_1,\phi_2)$ computed via a path integral on the complex plane with cuts $A_i$ on the real line, boundary conditions $\phi = \phi_1$ at the top of the cut and $\phi = \phi_2$ on the bottom of the cut. B: The Riemann surface $\Sigma_{n,N}$ constructed as an $n$-fold branched cover of the complex plane, with branch cuts at $A_i$ glued together as shown.
• Figure 3: The cut plane used to calculate the density matrix for 2 intervals. The fields on each side of the slit are independent, as in (\ref{['gauged_density_matrix']}). For Rényi entropies of the reduced density matrix corresponding to the CFT vacuum, the fields are untwisted for a circle which encloses all cuts $A_i$. These can be deformed to the sum of the two loops shown which encircle the cuts. The bosonic fields can be twisted about these loops, so long as the ordered product is the identity.
• Figure 4: The momentum lattices for the ungauged Dirac fermion (left) and for the self-dual boson (right). We have bolded the canonical generating vectors for both lattices. Notice that they are related by a 45$^\circ$ rotation.
• Figure 5: The Riemann surface $\Sigma_{3,3}$ with a canonical basis of A- and B-cycles. The solid black lines are the branch cuts, oriented so that approaching from below takes one up a sheet and approaching from below takes one down a sheet. The blue dashed lines are A-cycles and the green solid lines are B-cycles, with notches corresponding to the index (eg $a_3,~a_4$ lie on the second sheet). We have pulled the B-cycles off of the branch points for clarity.