Unusual corrections to scaling and convergence of universal Renyi properties at quantum critical points

Abstract

At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multi-sheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2 + 1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.

Cubic trihedral corner entanglement for a free scalar

“…For example, in d = 2 + 1, introducing a sharp corner in the entangling surface produces such a logarithmic contribution S univ α,corner = −a α (θ) log( /δ), where the universal coefficient a α (θ) is a function of the opening angle θ of the corner and the Rényi index α. [1][2][3][4][5]8,[25][26][27][28][29][30][31][32][33][34][35][36][37] In d = 3 + 1, geometric singularities in ∂A can also produce similar universal contributions in S α (A). 14,15,27,38,39 For the present discussion, let us focus on a threedimensional region A which is a polyhedron auch that the entangling surface ∂A consists of flat polygonal faces, straight edges and sharp corners or vertices.…”

“…Analogous isolation techniques have been used with success to study the corner coefficient for various (2 + 1)-dimensional critical systems. 3,8,32,35,36 At the most general level, the NLCE method can be used to study any property P that is well defined in the thermodynamic limit (such as an extensive or an intensive property). The NLCE calculates P for a lattice system L by summing contributions from individual clusters according to…”

Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice

“…An alternative strategy to the direct approach 21,22,24,33 is to perform finite-size scaling with the numerical linked-cluster expansion (NLCE). In addition to systematically accessing the limit L, → ∞, the NLCE eliminates the area-law piece and O(1/L) corrections to the Rényi entropy, providing direct access to the corner coefficient.…”

“…This procedure is repeated for every possible location r of the corner and the results are added; P (c) = r P r (c). Finally, we perform a fit of the total property P (L c ) against ln L cwhere L c is a measure of the cluster length scale 33 -in order to obtain a α (θ).…”

“…Previous works 9,[31][32][33] have computed a α (θ) in free theories at special angles like θ = π/2, π/4 that are "natural" for a square lattice (see Fig. 1) In this article we explore the universal content of the corner coefficient for a variety of opening angles θ beyond those commensurate with the lattice in two prototypical free CFTs, the Dirac fermion and complex relativistic boson, using two different methods.…”