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arXiv:1511.04077 · Physical review. B./Physical review. B

Bounds on corner entanglement in quantum critical states

Pablo Bueno, William Witczak‐Krempa

Published Jan 25, 2016• 44 citations
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Abstract

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 ${C}_{T}$. 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 ${C}_{T}$, 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 $\text{O}(N)$ 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.

DOI
10.1103/physrevb.93.045131

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Fortschritte der Physik•Dec 10, 2015•1 citation
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(\ensuremathθ)a(\ensuremath{\theta})a(\ensuremathθ) depending on the opening angle \ensuremathθ\ensuremath{\theta}\ensuremathθ, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit \ensuremathθ\ensuremath→\ensuremathπ\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremathθ\ensuremath→\ensuremathπ yields the stress tensor two-point function coefficient CT{C}_{T}CT​. Little is known about a(\ensuremathθ)a(\ensuremath{\theta})a(\ensuremathθ) 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(\ensuremathθ)a(\ensuremath{\theta})a(\ensuremathθ) is lower-bounded by a nontrivial function multiplied by the central charge CT{C}_{T}CT​, e.g., a(\ensuremathπ/2)\ensuremath≥(\ensuremathπ2ln2)CT/6a(\ensuremath{\pi}/2)\ensuremath{\ge}({\ensuremath{\pi}}^{2}ln2){C}_{T}/6a(\ensuremathπ/2)\ensuremath≥(\ensuremathπ2ln2)CT​/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.
We study the contribution to the entanglement entropy of (2+1)-dimensional conformal field theories (CFTs) coming from a sharp corner in the entangling surface. This contribution is encoded in a function a(θ) of the corner opening angle, and was recently proposed as a measure of the degrees of freedom in the underlying CFT. We show that the ratio a(θ)/C(T), where C(T) is the central charge in the stress tensor correlator, is an almost universal quantity for a broad class of theories including various higher-curvature holographic models, free scalars, and fermions, and Wilson-Fisher fixed points of the O(N) models with N=1,2,3. Strikingly, the agreement between these different theories becomes exact in the limit θ→π, where the entangling surface approaches a smooth curve. We thus conjecture that the corresponding ratio is universal for general CFTs in three dimensions.
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)O(N)O(N) models.
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.
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.
The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function a(θ) when the entangling surface contains a sharp corner with opening angle θ. In the limit of a smooth surface (θ → π), this corner contribution vanishes as a(θ) = σ(θ − π)2. In arXiv:1505.04804 , we provided evidence for the conjecture that for any d = 3 CFT, this corner coefficient σ is determined by C T , the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is an instance of a much more general relation connecting the analogous corner coefficient σ n appearing in the nth Rényi entropy and the scaling dimension h n of the corresponding twist operator. In particular, we find the simple relation h n /σ n = (n − 1)π. We show how it reduces to our previous result as n → 1, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as n → 0, σ n yields the coefficient of the thermal entropy, c S . We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict σ n for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval Rényi entropies of d = 2 CFTs.
In general three‐dimensional conformal field theories (CFTs), the entanglement entropy receives a logarithmic contribution whenever the entangling surface contains a sharp corner of opening angle θ. Such contribution is controlled by a regulator‐independent function which vanishes as in the smooth‐surface limit ( ). We review our recent conjecture that for general three‐dimensional CFTs, this corner coefficient σ is determined by CT, the coefficient appearing in the two‐point function of the stress tensor through the relation . We also comment on the extension of this conjecture to general Rényi entropies and higher‐dimensional CFTs.