Entanglement at a two-dimensional quantum critical point: aT= 0 projector quantum Monte Carlo study

Abstract

Although the leading-order scaling of entanglement entropy is nonuniversal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in noninteracting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space-time dimensions-a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertexinduced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shapedependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

“…(2). Unlike other methods, such as direct calculation of square subregions in a toroidal lattice, 25 the NLCE offers an advantage that this sub-leading contribution can be isolated for each cluster individually. This procedure, detailed in Ref.…”

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

“…In two spatial dimensions, universal quantities appearing in the Renyi entropies have a rich dependence on subregion geometry. For example, various universal terms (sub-leading to the area law) can arise from geometries such as smooth circular bipartitions [11,55]; sharp corners [56][57][58][59]; bipartitioned infinite cylinders [10] or cylindrically-bifurcated tori [60][61][62][63].…”

Cornering Gapless Quantum States via Their Torus Entanglement

“…Furthermore, these EEs have proved to be useful diagnostics in the search for such exotic phases [9][10][11][12]. In contrast, for gapless states, analytical [3,[13][14][15][16][17][18][19][20][21][22][23] and numerical [14,[24][25][26][27][28][29] studies have revealed that the situation is more intricate and numerous open questions remain.…”