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arXiv:1606.03096 · Physical review. B./Physical review. B

Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice

Johannes Helmes, Lauren E. Hayward Sierens, Anushya Chandran, William Witczak‐Krempa, Roger G. Melko

Published Sep 23, 2016• 41 citations
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Abstract

A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions, when the partition involves a corner of angle $\ensuremath{\theta}$, the subleading term is logarithmic with coefficient ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ for the $\ensuremath{\alpha}$-R\'enyi entropy. In the smooth limit $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi},{a}_{1}(\ensuremath{\theta})$ yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general R\'enyi indices and angles, ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ is richer and few general results exist. We study ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in $\ensuremath{\theta},\ensuremath{\alpha}$ regimes hitherto unexplored. We derive field theory estimates for ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$, including exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ for various theories, including interacting $O(N)$ models.

DOI
10.1103/physrevb.94.125142

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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 ($2+1$)-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.

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A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions, when the partition involves a corner of angle \ensuremathθ\ensuremath{\theta}\ensuremathθ, the subleading term is logarithmic with coefficient a\ensuremathα(\ensuremathθ){a}_{\ensuremath{\alpha}}(\ensuremath{\theta})a\ensuremathα​(\ensuremathθ) for the \ensuremathα\ensuremath{\alpha}\ensuremathα-R'enyi entropy. In the smooth limit \ensuremathθ\ensuremath→\ensuremathπ,a1(\ensuremathθ)\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi},{a}_{1}(\ensuremath{\theta})\ensuremathθ\ensuremath→\ensuremathπ,a1​(\ensuremathθ) yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general R'enyi indices and angles, a\ensuremathα(\ensuremathθ){a}_{\ensuremath{\alpha}}(\ensuremath{\theta})a\ensuremathα​(\ensuremathθ) is richer and few general results exist. We study a\ensuremathα(\ensuremathθ){a}_{\ensuremath{\alpha}}(\ensuremath{\theta})a\ensuremathα​(\ensuremathθ) focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in \ensuremathθ,\ensuremathα\ensuremath{\theta},\ensuremath{\alpha}\ensuremathθ,\ensuremathα regimes hitherto unexplored. We derive field theory estimates for a\ensuremathα(\ensuremathθ){a}_{\ensuremath{\alpha}}(\ensuremath{\theta})a\ensuremathα​(\ensuremathθ), including exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of a\ensuremathα(\ensuremathθ){a}_{\ensuremath{\alpha}}(\ensuremath{\theta})a\ensuremathα​(\ensuremathθ) for various theories, including interacting O(N)O(N)O(N) models.
The entanglement entropy in many gapless quantum systems receives a contribution from the corners in the entangling surface in 2+1d, which is characterized by a universal function $a(\ensuremath{\theta})$ depending on the opening angle $\ensuremath{\theta}$, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi}$ yields the stress tensor two-point function coefficient CT{C}_{T}CT​. Little is known about $a(\ensuremath{\theta})$ beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the two- and three-point correlators of the stress tensor. We conjecture that it encodes four-point data, making it much richer. Further, we establish strong constraints on this and higher-order smooth-limit coefficients. We also show that $a(\ensuremath{\theta})$ is lower-bounded by a nontrivial function multiplied by the central charge CT{C}_{T}CT​, e.g., $a(\ensuremath{\pi}/2)\ensuremath{\ge}({\ensuremath{\pi}}^{2}ln2){C}_{T}/6$. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the R\'enyi entropies. We illustrate our findings using O(N)\text{O}(N)O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d.
In quantum field theories defined on a space-time with boundaries, the entanglement entropy exhibits subleading, boundary-induced corrections to the ubiquitous area law. At critical points described by conformal field theories (CFTs), and when the entangling surface intersects the physical boundary of the space, new universal terms appear in the entropy and encode valuable information about the boundary CFT. In 2+12+12+1 dimensions, the universal subleading boundary term is logarithmic with coefficient $b(\ensuremath{\theta})$ depending on the angle $\ensuremath{\theta}$ at which the entangling surface intersects the boundary, as well as on the boundary conditions (BCs). In this paper, we conduct a numerical study of $b(\ensuremath{\theta})$ for free bosons on finite-size square lattices. We find a surprisingly accurate fit between our lattice results and the corresponding holographic function available in the literature. We also comment on the ratio ${b}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\pi}/2)/{A}_{T}$, where AT{A}_{T}AT​ is the central charge in the near boundary expansion of the stress tensor, for which a holographic analysis suggests that it may be a universal quantity. Though we show evidence that this ratio is violated for the free boson with Dirichlet BCs, we conjecture its validity for free bosons evenly split between Dirichlet and Neumann BCs.
We study subleading corrections to the corner free energy in classical two-dimensional critical systems, focusing on a generic boundary perturbation by the stress-tensor of the underlying conformal field theory (CFT). In the particular case of an angle 2π, we find that there is an unusual correction of the form L −1 log L , where L is a typical length scale in the system. This correction also affects the one-point function of an operator near the corner. The prefactor can be seen as semi-universal, in the sense that it depends on a single non-universal quantity, the extrapolation length. Once this ultraviolet cutoff is known, the term is entirely fixed by the geometry of the system, and the central charge of the CFT. Such a corner appears for example in the bipartite fidelity of a one-dimensional quantum system at criticality, which allows for several numerical checks in free fermion systems. We also present an exact result in the XX and Ising chains that confirms this analysis. Finally, we consider applications to the time evolution of the (logarithmic) Loschmidt echo and the entanglement entropy following a local quantum quench. Due to subtle issues in analytic continuation, we find that the logarithmic term in imaginary time transforms into a time-dependent L −2 correction for the entanglement entropy, and a L −2 log L term for the Loschmidt echo.
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.
In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient σ for the EE and σn for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for σ and σn for all n=2,3,… . The results for σ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that σ/CT=π2/24 in all 3d CFTs, where CT is the central charge. For the Rényi entropies, the ratios σn/CT do not indicate similar universality. However, in the limit n→∞, the asymptotic values satisfy a simple relationship and equal 1/(4π2) times the asymptotic values of the free energy of free scalars/fermions on the n-covered 3-sphere.
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.