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arXiv:1303.3633 · Journal of Statistical Mechanics Theory and Experiment

Logarithmic corrections to the free energy from sharp corners with angle 2π

Jean-Marie Stéphan, Jérôme Dubail

Published Sep 6, 2013• 30 citations
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Abstract

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.

DOI
10.1088/1742-5468/2013/09/p09002

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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.
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 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.
Abstract. We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entanglement entropy (EE) between the two parts A and B, using previous results by Calabrese and Cardy [1, 2], and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite-size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c/3 log t for the EE and c/4 log t for the LLE—, we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B). The decay of the EE is then non-universal, as noticed by Eisler and Peschel [3]. In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.
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.
We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entropy of entanglement (EE) between the two parts A and B , using previous results obtained by Calabrese and Cardy (2004 J. Stat. Mech. P06002; 2007 J. Stat. Mech. P10004), and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c /3log t for the EE and c /4log t for the LLE—we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B ). The decay of the EE is then non-universal, as noticed by Eisler and Peschel (2007 J. Stat. Mech. P06005). In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.
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.