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Abstract
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting 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 vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence 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.
Highlights
At a quantum critical point (QCP), the Renyi [1] entanglement entropies contain the muchcelebrated ability to access the central charge of the associated conformal field theory (CFT) in 1 + 1 space-time dimensions [2, 3]
For the transverse-field Ising model (TFIM) simulation discussed in this paper, the procedure for efficiently calculating the Renyi entropy closely follows that used in another T = 0 projector quantum Monte Carlo (QMC) – the valencebond basis QMC for the spin-1/2 Heisenberg model [39, 53], which has a detailed description in Ref. [37]
Using a novel “projector” quantum Monte Carlo (QMC) method that operates at zero temperature [51], we have performed a detailed numerical study of the Renyi entanglement entropy, S2, at the quantum critical point of the transverse-field Ising model in 2+1 spacetime dimensions
Summary
At a quantum critical point (QCP), the Renyi [1] entanglement entropies contain the muchcelebrated ability to access the central charge of the associated conformal field theory (CFT) in 1 + 1 space-time dimensions [2, 3]. If we look at the well-studied 2+1 dimensional quantum Lifshitz model [14, 23, 24, 25] (the field-theory associated with the Rokhsar-Kivelson (RK) Hamiltonian [26]) there is a proposed functional form [22] for the universal subleading term to the area law that is in good agreement with our data with a size-independent coefficient. Additional universal subleading terms to this area law are possible, they may have a complicated dependence on the geometry of the bipartition Typically believed, it is not generally known if particular geometric features, for example the number of vertices or the Euler characteristic [14, 31, 17, 18], give rise to certain universal numbers that can be compared reliably between field theories and quantum lattice models. The geometries amenable to study on finite-sizes lattice are sometimes different than those that can be studied with continuum field theories, as we discuss
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