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arXiv:2404.13876 · Physical review. B./Physical review. B

Extracting universal corner entanglement entropy during the quantum Monte Carlo simulation

Yuan Da Liao, Menghan Song, Jiarui Zhao, Zi Yang Meng

Published Dec 3, 2024• 6 citations
arXivHTMLPDF

Abstract

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 ($2+1$)-dimensional [($2+1$)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 ($2+1$)D $O$(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 ($2+1$) dimensions.

DOI
10.1103/physrevb.110.235111

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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.
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.
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 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.
The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.
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.
In a D=2+1D=2+1D=2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components NNN in the associated O(N)\text{O}(N)O(N) continuum ${\ensuremath{\phi}}^{4}$ field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2)\text{O}(2)O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the R\'enyi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories.