Corner Contributions Research Articles

Exact results for corner contributions to the entanglement entropy and Rényi entropies of free bosons and fermions in 3d

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.

Corner contributions to holographic entanglement entropy

The entanglement entropy of three-dimensional conformal field theories contains a universal contribution coming from corners in the entangling surface. We study these contributions in a holographic framework and, in particular, we consider the effects of higher curvature interactions in the bulk gravity theory. We find that for all of our holographic models, the corner contribution is only modified by an overall factor but the functional dependence on the opening angle is not modified by the new gravitational interactions. We also compare the dependence of the corner term on the new gravitational couplings to that for a number of other physical quantities, and we show that the ratio of the corner contribution over the central charge appearing in the two-point function of the stress tensor is a universal function for all of the holographic theories studied here. Comparing this holographic result to the analogous functions for free CFT's, we find fairly good agreement across the full range of the opening angle. However, there is a precise match in the limit where the entangling surface becomes smooth, i.e., the angle approaches $\pi$, and we conjecture the corresponding ratio is a universal constant for all three-dimensional conformal field theories. In this paper, we expand on the holographic calculations in our previous letter arXiv:1505.04804, where this conjecture was first introduced.

Hydrodynamic Modeling of Moored Ship Motion in an Irregular Domain

In this study, a hydrodynamic model for the analysis of moored ship motion is presented in a realistic harbor with highly irregular geometry. An accurate description of the harbor geometry, bathymetry and the associated wave's characteristics such as diffraction, refraction and partial reflection is required for the analysis of moored ship motion. Further, Fluid domain is divided into bounded, unbounded or open sea and ship region. In each region, wave field is determined by using Boundary Element Method (BEM) with the corner contribution and Chebyshev point discretization. Then, the hydrodynamic coefficient such as added mass and damping coefficients were determined based on the equation of motion with six degree of freedom, which represent the six different component of moored ship motion as surge, sway, heave, roll, pitch and yaw. The current numerical model is validated through previous well defined model based on moored ship motion. Moreover, current numerical scheme is implemented on realistic Pohang New Harbor (PNH), which is situated in Pohang city, South Korea to analyze the wave field in ship region under the various resonance frequencies for monochromatic incident waves.

Corner contribution to the entanglement entropy of strongly interacting O(2) quantum critical systems in 2+1 dimensions

In a D=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 N in the associated O(N) continuum $\phi^4$ 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) 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.

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

We consider the random-bond Potts model in the large-Q limit and calculate the excess entropy, SΓ, of a contour, Γ, which is given by the mean number of Fortuin–Kasteleyn clusters which are crossed by Γ. In two dimensions, SΓ is proportional to the length of Γ, to which—at the critical point—there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: limQ → ∞c(Q)/lnQ = 0.74(2), close to previous estimates calculated at finite values of Q.

Entanglement across a cubic interface in3+1dimensions

We calculate the area, edge and corner Renyi entanglement entropies in the ground state of the transverse-field Ising model, on a simple-cubic lattice, by high-field and low-field series expansions. We find that while the area term is positive and the line term is negative as required by strong subadditivity, the corner contributions are positive in 3-dimensions. Analysis of the series suggests that the expansions converge up to the physical critical point from both sides. The leading area-law Renyi entropies match nicely from the high and low field expansions at the critical point, forming a sharp cusp there. We calculate the coefficients of the logarithmic divergence associated with the corner entropy and compare them with conformal field theory results with smooth interfaces and find a striking correspondence.

Corner contribution to percolation cluster numbers in three dimensions

In three-dimensional critical percolation we study numerically the number of clusters, $N_{\Gamma}$, which intersect a given subset of bonds, $\Gamma$. If $\Gamma$ represents the interface between a subsystem and the environment, then $N_{\Gamma}$ is related to the entanglement entropy of the critical diluted quantum Ising model. Due to corners in $\Gamma$ there are singular corrections to $N_{\Gamma}$, which scale as $b_{\Gamma} \ln L_{\Gamma}$, $L_{\Gamma}$ being the linear size of $\Gamma$ and the prefactor, $b_{\Gamma}$, is found to be universal. This result indicates that logarithmic finite-size corrections exist in the free-energy of three-dimensional critical systems.

Corner contribution to cluster numbers in the Potts model

For the two-dimensional Q-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number N_Gamma of clusters that intersect a given contour Gamma. To leading order, N_Gamma is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of Gamma. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed.

Wave field analysis in a harbor with irregular geometry through boundary integral of Helmholtz equation with corner contributions

A mathematical model has been designed to analyze the wave characteristics in a harbor, averting any possible situation of occasional extreme wave hazards due to the seasonal weather conditions. Helmholtz equation has been considered with appropriate boundary conditions on the basis of small amplitude hypothesis. Further, a solution of the Helmholtz equation is given by Green’s identity formula and then it is utilized to predict the ocean surface wave fields due to refraction and diffraction of various directional incident waves in the context of the shallow water waves. The corner angle coefficient is firstly determined at each corner segment on the boundary of the harbor and then corner contribution on the boundary is imposed by multiplying the corner angle coefficient to the boundary wave field. Further, the Chebyshev point discretization is implemented on the boundary of harbor to improve the numerical accuracy at the sharp corners. This numerical model is validated through the quantitative comparison with existing studies in literature. Further the logarithmic convergence rate is achieved for the proposed numerical scheme as well. We have implemented this numerical model practically on the Pohang New Harbor (PNH), Pohang, South Korea to compute the resonance modes for various directional incident waves. Furthermore, the wave elevation is obtained in the whole interior domain of the harbor at various modes of resonance with specific incident waves. In conclusion, this numerical scheme is a competent tool to foster the prediction of the ocean surface wave field on the boundary and interior of the harbors with complex geometries.

Entanglement entropy at generalized Rokhsar-Kivelson points of quantum dimer models

We study the $n=2$ R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.

Corner contribution to percolation cluster numbers

We study the number of clusters in two-dimensional ($2d$) critical percolation, ${N}_{\ensuremath{\Gamma}}$, which intersect a given subset of bonds, $\ensuremath{\Gamma}$. In the simplest case, when $\ensuremath{\Gamma}$ is a simple closed curve, ${N}_{\ensuremath{\Gamma}}$ is related to the entanglement entropy of the critical diluted quantum Ising model, in which $\ensuremath{\Gamma}$ represents the boundary between the subsystem and the environment. Due to corners in $\ensuremath{\Gamma}$ there are universal logarithmic corrections to ${N}_{\ensuremath{\Gamma}}$, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large-scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.

Introduction [Measurements Corner

Antenna-measurement practitioners who have the luxury of a perfect measurement facility will fi nd little of interest in this issue's Measurements Corner contribution. For the rest of us, the paper by Álvarez, et al. offers a collections of methods based on maximum-likelihood estimation for characterizing and correcting errors due to stray multipath in antenna-meas urement facilities. Both detailed derivations and experimental examples are presented.

Corner diffraction coefficients for the quarter plane

The current near a right-angled corner on a perfectly conducting flat scatterer illuminated by a plane wave is expressed as a sum of three currents. The first is the physical optics current, which describes the surface effect. The second is the fringe wave current, which is found from the half-plane solution and accounts for the distortion of the current caused by the edges. The third is the corner current, which is found from the numerical solution to the electric-field integral equation applied to the square plate, and accounts for the distortion of the current caused by the corner. It is found that the corner current for the right-angled corner, illuminated from a forward direction, consists mainly of two edge waves propagating along the edges forming the corner. Analytical expressions for these edge wave currents are constructed from the numerical results. A corner diffracted field is calculated by evaluating the asymptotic corner contributions to the radiation integral over the sum of the three currents. It is found that the corner contribution from the edge wave currents in some cases is of the same size as the corner contributions from the physical optics current and the fringe wave current.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Interfacial contribution to cluster free energy

The Monte Carlo technique of “overlapping distributions” is used for computing directly the free energy difference F N + 1 − F N between two clusters containing respectively N + 1 and N solute atoms, in the square and the simple cubic Ising model with nearest neighbour interactions. High accuracy results are obtained within reasonable computer times. The capillarity approximation gives a good fit to the data, provided the following is taken into account: (i) the specific bulk and surface energies are given their macroscopic equilibrium values; (ii) a curvature correction to the surface specific energy has to be introduced: it is positive for two dimensions and negative for three dimensions; (iii) a size independent term is to be added in three dimensions; it may be viewed as a corner contribution; (iv) the coefficient of the logarithmic term is given its more recent value. When introduced in the master equation which describes the kinetics of the cluster population, within the simplifying assumptions of the classical nucleation theory, good agreement is found, for the shapes of the cluster size distributions, with the numerical experiments on the kinetic Ising model in two dimensions. However, the time scales of both computations do not match linearly. Possible reasons for this discrepancy are discussed.

Edge and corner contributions in Weyl's problem

The smoothed asymptotic eigenvalue distributions for scalar and electromagnetic waves are calculated for a finite cylindrical domain with arbitrary cross section shape. Four terms of the expansion are given including the edge and corner contributions unknown hitherto.