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arXiv:2312.13498 · SciPost Physics

Extracting subleading corrections in entanglement entropy at quantum phase transitions

Menghan Song, Jiarui Zhao, Zi Yang Meng, Cenke Xu, Meng Cheng

Published Jul 11, 2024• 27 citations
arXivHTMLPDF

Abstract

We systematically investigate the finite size scaling behavior of the Rényi entanglement entropy (EE) of several representative 2d quantum many-body systems between a subregion and its complement, with smooth boundaries as well as boundaries with corners. In order to reveal the subleading correction, we investigate the quantity “subtracted EE” S^s(l) = S(2l) - 2S(l) S s ( l ) = S ( 2 l ) − 2 S ( l ) for each model, which is designed to cancel out the leading perimeter law. We find that (1) for a spin-1/2 model on a 2d square lattice whose ground state is the Neel order, the coefficient of the logarithmic correction to the perimeter law is consistent with the prediction based on the Goldstone modes; (2) for the (2+1)d ( 2 + 1 ) d O(3) Wilson-Fisher quantum critical point (QCP), realized with the bilayer antiferromagnetic Heisenberg model, a logarithmic subleading correction exists when there is sharp corner of the subregion, but for subregion with a smooth boundary our data suggests the absence of the logarithmic correction to the best of our efforts; (3) for the (2+1)d ( 2 + 1 ) d SU(2) J-Q _2 2 and J-Q _3 3 model for the deconfined quantum critical point (DQCP), we find a logarithmic correction for the EE even with smooth boundary.

DOI
10.21468/scipostphys.17.1.010

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We systematically investigate the finite size scaling behavior of the Rényi entanglement entropy (EE) of several representative 2d quantum many-body systems between a subregion and its complement, with smooth boundaries as well as boundaries with corners. In order to reveal the subleading correction, we investigate the quantity “subtracted EE” S^s(l) = S(2l) - 2S(l) S s ( l ) = S ( 2 l ) − 2 S ( l ) for each model, which is designed to cancel out the leading perimeter law. We find that (1) for a spin-1/2 model on a 2d square lattice whose ground state is the Neel order, the coefficient of the logarithmic correction to the perimeter law is consistent with the prediction based on the Goldstone modes; (2) for the (2+1)d ( 2 + 1 ) d O(3) Wilson-Fisher quantum critical point (QCP), realized with the bilayer antiferromagnetic Heisenberg model, a logarithmic subleading correction exists when there is sharp corner of the subregion, but for subregion with a smooth boundary our data suggests the absence of the logarithmic correction to the best of our efforts; (3) for the (2+1)d ( 2 + 1 ) d SU(2) J-Q _2 2 and J-Q _3 3 model for the deconfined quantum critical point (DQCP), we find a logarithmic correction for the EE even with smooth boundary.
The subleading corner logarithmic corrections in entanglement entropy (EE) are crucial for revealing the universal characteristics of quantum critical points (QCPs), but they are challenging to detect. Motivated by recent developments in the stable computation of EE in (2+12+12+1)-dimensional [(2+12+12+1)D] quantum many-body systems, we have developed a method for directly measuring the corner contribution in EE with lower computational cost. The cornerstone of our approach is to measure the subtracted corner entanglement entropy (SCEE), defined as the difference between the EEs of subregions with the same boundary length for smooth and cornered boundaries during a sign-problem-free quantum Monte Carlo simulation. Our improved method inherently eliminates not only the area-law term of EE but also the subleading log corrections arising from Goldstone modes, leaving the universal corner contribution as the leading term of the SCEE with greatly improved data quality. Utilizing this advanced approach, we calculate the SCEE of the bilayer Heisenberg model on both square and honeycomb lattices across their (2+12+12+1)D OOO(3) QCPs with different opening angles on the entanglement boundary and obtain accurate values of the corresponding universal corner log coefficients. These findings will encourage further theoretical investigations to access controlled universal information for interacting conformal field theories at (2+12+12+1) dimensions.
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.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.
The deconfined quantum critical point (DQCP) in square lattice S=1/2 quantum antiferromagnets has been extensively studied with a large body of evidence pointing to a weakly first-order transition scenario. Recent studies, which focused on entanglement at this nearly continuous DQCP in square lattice J-Q models, have observed conflicting bipartite entanglement entropy (EE) scaling behavior. One bipartition choice gave scaling coefficients in remarkable agreement with predictions from the unitary CFT corresponding to the putative DQCP. While another equally natural choice gave scaling coefficients in complete violation of unitary CFT that may be attributed to lack of scale invariance at the known weakly first-order behavior of the model. This motivates the exploration of DQCP behavior via entanglement measures in lattice models with distinct crystalline symmetries. Here we study a S=1/2 honeycomb model that hosts a nearly continuous transition between Néel and valence-bond-solid ground states relevant to probing DQCP. Using large-scale quantum Monte Carlo simulations, we compute the Rényi EE for a variety of bipartitions and test the CFT based description of the DQCP on the honeycomb lattice. For smooth bipartitions, we find no evidence of logarithmic corrections, in accordance with CFT, thereby essentially ruling out contributions from Goldstone modes. For subsystems with corners, CFT predicts universal logarithmic contributions, which we extract for corners with 60 and 120 degree angles and find close agreement with an emergent SO(5) CFT. While we observe scaling consistent with a critical system in the majority of cases, we also demonstrate an intriguing counterexample of the hexagon subsystem that exhibits a subtle period three oscillation. This results in three separate finite-size series, where the sign of the logarithmic term apparently changes depending on the series.
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.
We study the Rényi entanglement entropy (EE) of the two-dimensional J-Q model, the emblematic quantum spin model of deconfined criticality at the phase transition between antiferromagnetic and valence-bond-solid ground states. State-of-the-art quantum Monte Carlo calculations of the EE reveal critical corner contributions that scale logarithmically with the system size, with a coefficient in remarkable agreement with the form expected from a large-N conformal field theory with SO(N=5) symmetry. However, details of the bipartition of the lattice are crucial in order to observe this behavior. If the subsystem for the reduced density matrix does not properly accommodate valence-bond fluctuations, logarithmic contributions appear even for cornerless bipartitions. We here use a 45° tilted cut on the square lattice. Beyond supporting an SO(5) deconfined quantum critical point, our results for both the regular and tilted cuts demonstrate important microscopic aspects of the EE that are not captured by conformal field theory.