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arXiv:1411.7773 · Physical Review B

Correlations and entanglement in quantum critical bilayer and necklace XY models

Johannes Helmes, Stefan Weßel

Published Sep 11, 2015• 20 citations
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Abstract

We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on a two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional ${\ensuremath{\phi}}^{4}$ theory. The second R\'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of $0.0674(7)$ $[0.0664(4)]$ for the bilayer (necklace) model, respectively. Furthermore, the presence of ${90}^{\ensuremath{\circ}}$ corners leads to an additive logarithmic term in both models. We estimate a contribution of $\ensuremath{-}0.010(2)$ $[\ensuremath{-}0.009(2)]$ due to each ${90}^{\ensuremath{\circ}}$ corner to the logarithmic correction for the bilayer (necklace) model, and compare our findings to recent numerical linked-cluster calculations and series expansion results on related models.

DOI
10.1103/physrevb.92.125120

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We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on a two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 theory. The second R'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of 0.0674(7)0.0674(7)0.0674(7) [0.0664(4)][0.0664(4)][0.0664(4)] for the bilayer (necklace) model, respectively. Furthermore, the presence of 90\ensuremath∘{90}^{\ensuremath{\circ}}90\ensuremath∘ corners leads to an additive logarithmic term in both models. We estimate a contribution of \ensuremath−0.010(2)\ensuremath{-}0.010(2)\ensuremath−0.010(2) [\ensuremath−0.009(2)][\ensuremath{-}0.009(2)][\ensuremath−0.009(2)] due to each 90\ensuremath∘{90}^{\ensuremath{\circ}}90\ensuremath∘ corner to the logarithmic correction for the bilayer (necklace) model, and compare our findings to recent numerical linked-cluster calculations and series expansion results on related models.
We examine the entanglement properties of the spin-half Heisenberg model on the two-dimensional square-lattice bilayer based on quantum Monte Carlo calculations of the second R\'enyi entanglement entropy. In particular, we extract the dominant area-law contribution to the bipartite entanglement entropy that shows a nonmonotonous behavior upon increasing the interlayer exchange interaction: a local maximum in the area-law coefficient is located at the quantum critical point separating the antiferromagnetically ordered region from the disordered dimer-singlet regime. Furthermore, we consider subleading logarithmic corrections to the R\'enyi entanglement entropy scaling. Employing different subregion shapes, we isolate the logarithmic corner term from the logarithmic contribution due to Goldstone modes that is found to be enhanced in the limit of decoupled layers. At the quantum critical point, we estimate a contribution of 0.016(1)0.016(1)0.016(1) due to each ${90}^{\ensuremath{\circ}}$ corner. This corner term at the SU(2) quantum critical point deviates from the Gaussian theory value, while it compares well with recent numerical linked cluster calculations on the bilayer model.
We consider a bilayer quantum spin model with anisotropic intralayer exchange couplings. By varying the anisotropy, the quantum critical phenomena changes from XY to Heisenberg to Ising universality class, with two modes, three modes, and one mode, respectively, becoming gapless simultaneously. We use series-expansion methods to calculate the second and third Renyi entanglement entropies when the system is bipartitioned into two parts. Leading area-law terms and subleading entropies associated with corners are separately calculated. We find clear evidence that the logarithmic singularity associated with the corners is universal in each class. Its coefficient along the Ising critical line is in excellent agreement with those obtained previously for the transverse-field Ising model. Our results provide strong evidence for the idea that the universal terms in the entanglement entropy provide an approximate measure of the low energy degrees of freedom in the system.
The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a subleading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2 + 1 dimensional O (3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1 + 1 dimensional conformal field theory.
We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence-bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high-temperature series expansions, and zero-temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field N\'eel state. Second, the corner logs do not agree with the known results for noninteracting Boson modes. And, third, even the finite-temperature mutual information shows an anomalous behavior as TTT goes to zero, suggesting that the $T\ensuremath{\rightarrow}0$ and $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in d>1d>1d>1, which deserves further attention.
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.
The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.