Scalable computational approach to extract chemical bonding from real-space density functional theory calculations using finite-element basis: A projected orbital and Hamiltonian population analysis

We present an efficient and scalable computational approach for conducting projected population analysis from real-space finite-element (FE) based Kohn-Sham density functional theory calculations (DFT-FE). This work provides an important direction towards extracting chemical bonding information from large-scale DFT calculations on materials systems involving thousands of atoms while accommodating periodic, semi-periodic or fully non-periodic boundary conditions. Towards this, we derive the relevant mathematical expressions and develop efficient numerical implementation procedures that are scalable on multi-node CPU architectures to compute the projected overlap and Hamilton populations. This is accomplished by projecting either the self-consistently converged FE discretized Kohn-Sham eigenstates, or the FE discretized Hamiltonian onto a subspace spanned by localized atom-centered basis set. The proposed method is implemented in a unified framework within DFT-FE where the ground-state DFT calculations and the population analysis are performed on the same finite-element grid. We further benchmark the accuracy and performance of this approach on representative material systems involving periodic and non-periodic DFT calculations with LOBSTER, a widely used projected population analysis code. Finally, we discuss a case study demonstrating the advantages of our scalable approach to extract the chemical bonding information from increasingly large silicon nanoparticles up to a few thousand atoms.Comment: 9 Figures, 5 Tables, 53 pages with references and supplementary informatio

Algebraic approach to quantum black holes: logarithmic corrections to black hole entropy

The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As shown previously, for a neutral non-rotating black hole, such eigenvalues must be $2^{n}$-fold degenerate if one constructs the black hole stationary states by means of a pair of creation operators subject to a specific algebra. We show that the algebra of these two building blocks exhibits $U(2)\equiv U(1)\times SU(2)$ symmetry, where the area operator generates the U(1) symmetry. The three generators of the SU(2) symmetry represent a {\it global} quantum number (hyperspin) of the black hole, and we show that this hyperspin must be zero. As a result, the degeneracy of the $n$-th area eigenvalue is reduced to $2^{n}/n^{3/2}$ for large $n$, and therefore, the logarithmic correction term $-3/2\log A$ should be added to the Bekenstein-Hawking entropy. We also provide a heuristic approach explaining this result, and an evidence for the existence of {\it two} building blocks.Comment: 15 pages, Revtex, to appear in Phys. Rev.

Thermal Fluctuations and Black Hole Entropy

In this paper, we consider the effect of thermal fluctuations on the entropy of both neutral and charged black holes. We emphasize the distinction between fixed and fluctuating charge systems; using a canonical ensemble to describe the former and a grand canonical ensemble to study the latter. Our novel approach is based on the philosophy that the black hole quantum spectrum is an essential component in any such calculation. For definiteness, we employ a uniformly spaced area spectrum, which has been advocated by Bekenstein and others in the literature. The generic results are applied to some specific models; in particular, various limiting cases of an (arbitrary-dimensional) AdS-Reissner-Nordstrom black hole. We find that the leading-order quantum correction to the entropy can consistently be expressed as the logarithm of the classical quantity. For a small AdS curvature parameter and zero net charge, it is shown that, independent of the dimension, the logarithmic prefactor is +1/2 when the charge is fixed but +1 when the charge is fluctuating.We also demonstrate that, in the grand canonical framework, the fluctuations in the charge are large, $\Delta Q\sim\Delta A\sim S_{BH}^{1/2}$, even when $=0$. A further implication of this framework is that an asymptotically flat, non-extremal black hole can never achieve a state of thermal equilibrium.Comment: 25 pages, Revtex; references added and corrected, and some minor change

Black hole area quantization

It has been argued by several authors that the quantum mechanical spectrum of black hole horizon area must be discrete. This has been confirmed in different formalisms, using different approaches. Here we concentrate on two approaches, the one involving quantization on a reduced phase space of collective coordinates of a Black Hole and the algebraic approach of Bekenstein. We show that for non-rotating, neutral black holes in any spacetime dimension, the approaches are equivalent. We introduce a primary set of operators sufficient for expressing the dynamical variables of both, thus mapping the observables in the two formalisms onto each other. The mapping predicts a Planck size remnant for the black hole.Comment: 7 pages, uses MPLA style file (included). Revised version with changes in notation for clarity and consistency. To appear in MPL

Entropy Corrections for Schwarzschild and Reissner-Nordstr\"om Black Holes

Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first order fluctuation corrections is given by: {\cal S} = S_{BH} - \ln[\f{3}{R} (S_{BH}/4\p)^{1/2} -2]^{-1} + (1/2) \ln(4\p), where $S_{BH}=A/4$ is its Bekenstein-Hawking entropy and $R$ is the radius of the cavity. We extend our results to four dimensional Reissner-Nordstr\"om black holes, for which the corresponding expression is: {\cal S} = S_{BH} - \f{1}{2} \ln [ {(S_{BH}/\p R^2) ({3S_{BH}}/{\p R^2} - 2\sqrt{{S_{BH}}/{\p R^2 -\a^2}}) \le(\sqrt{{S_{BH}}/{\p R^2}} - \a^2 \ri)}/ {\le({S_{BH}}/{\p R^2} -\a^2 \ri)^2} ]^{-1} +(1/2)\ln(4\p). Finally, we generalise the stability analysis to Reissner-Nordstr\"om black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character.Comment: 6 pages, Revtex. References added, minor changes. Version to appear in Class. Quant. Gra

A comment on black hole entropy or does Nature abhor a logarithm?

There has been substantial interest, as of late, in the quantum-corrected form of the Bekenstein-Hawking black hole entropy. The consensus viewpoint is that the leading-order correction should be a logarithm of the horizon area; however, the value of the logarithmic prefactor remains a point of notable controversy. Very recently, Hod has employed statistical arguments that constrain this prefactor to be a non-negative integer. In the current paper, we invoke some independent considerations to argue that the "best guess" for the prefactor might simply be zero. Significantly, this value complies with the prior prediction and, moreover, seems suggestive of some fundamental symmetry.Comment: 10 pages and Revtex; (v2) imperative title change and added one reference; (v3) minor content and style changes throughout; 7 new citations; (v4) 8 new citations, an addendum and other minor changes; (v5) yet more references, some points clarified, and a recent criticism is addressed (addendum 2

Anti-de Sitter black holes, perfect fluids, and holography

We consider asymptotically anti-de Sitter black holes in $d$-spacetime dimensions in the thermodynamically stable regime. We show that the Bekenstein-Hawking entropy and its leading order corrections due to thermal fluctuations can be reproduced by a weakly interacting fluid of bosons and fermions (`dual gas') in $\Delta=\alpha(d-2)+1$ spacetime dimensions, where the energy-momentum dispersion relation for the constituents of the fluid is assumed to be $\epsilon = \kappa p^\alpha$. We examine implications of this result for entropy bounds and the holographic hypothesis.Comment: Minor changes to match published version. 9 Pages, Revte

Spectrum of rotating black holes and its implications for Hawking radiation

The reduced phase space formalism for quantising black holes has recently been extended to find the area and angular momentum spectra of four dimensional Kerr black holes. We extend this further to rotating black holes in all spacetime dimensions and show that although as in four dimensions the spectrum is discrete, it is not equispaced in general. As a result, Hawking radiation spectra from these black holes are continuous, as opposed to the discrete spectrum predicted for four dimensional black holes.Comment: 11 pages, Revtex4. Minor changes to match version to appear in Class. Quant. Gra

Black Hole Entropy from Spin One Punctures

Recent suggestion, that the emission of a quantum of energy corresponding to the asymptotic value of quasinormal modes of a Schwarzschild black hole should be associated with the loss of spin one punctures from the black hole horizon, fixes the Immirzi parameter to a definite value. We show that saturating the horizon with spin one punctures reproduces the earlier formula for the black hole entropy, including the $ln (area)$ correction with definite coefficient (- 3/2) for large area.Comment: 4 pages. RevTe

Varying Fine Structure Constant and Black Hole Physics

Recent astrophysical observations suggest that the value of fine structure constant $\alpha=e^2/\hbar c$ may be slowly increasing with time. This may be due to an increase of $e$ or a decrease of $c$, or both. In this article, we argue from model independent considerations that this variation should be considered adiabatic. Then, we examine in detail the consequences of such an adiabatic variation in the context of a specific model of quantized charged black holes. We find that the second law of black hole thermodynamics is obeyed, regardless of the origin of the variation, and that interesting constraints arise on the charge and mass of black holes. Finally, we estimate the work done on a black hole of mass $M$ due to the proposed $\alpha$ variation.Comment: 7 Pages, Revtex. Reference added, minor changes. Version to appear in Class. Quant. Gra