Brick-wall Model Research Articles

Brickwall, normal modes, and emerging thermality

In this paper, we demonstrate how black hole quasinormal modes can emerge from a Dirichlet brickwall model normal modes. We consider a probe scalar field in a Baños-Teitelboim-Zanelli geometry with a Dirichlet brickwall and demonstrate that as the wall approaches the event horizon, the corresponding poles in the retarded correlator become dense and yield an effective branch cut. The associated discontinuity of the correlator carries the information of the black hole quasinormal modes. We further demonstrate that a nonvanishing angular momentum nonperturbatively enhances the pole condensing. We hypothesize that it is also related to quantum chaotic features of the corresponding spectral form factor, which has been observed earlier. Finally, we discuss the underlying algebraic justification of this approximate thermalization in terms of the trace of the algebra. Published by the American Physical Society 2024

Brick wall diagrams as a completely integrable system

We study the free energy of an integrable, planar, chiral and non-unitary four-dimensional Yukawa theory, the bi-fermion fishnet theory discovered by Pittelli and Preti. The typical Feynman-diagrams of this model are of regular “brick-wall”-type, replacing the regular square lattices of standard fishnet theory. We adapt A. B. Zamolodchikov’s powerful classic computation of the thermodynamic free energy of fishnet graphs to the brick-wall case in a transparent fashion, and find the result in closed form. Finally, we briefly discuss two further candidate integrable models in three and six dimensions related to the brick wall model.

New oxygen ion conducting composite solid electrolytes Sm2(WO4)3-WO3

Composite materials Sm2(WO4)3-WO3 were prepared by the solid state method and their transport properties have been examined by the electrochemical impedance technique and conductivity measurements versus oxygen partial pressure. It was shown that heterogeneous doping of the oxygen ion conductor Sm2(WO4)3 with a semiconductor WO3 even at low concentration of WO3 (volume fraction f < 0.13) led to an increase in ionic conductivity of the composites by more than an order of magnitude. This effect is caused by formation of the non-autonomous interface phase Sm2W6O21 covering grain boundaries of Sm2(WO4)3. At high concentrations of WO3 (f > 0.13) the composite materials comprise biphasic mixture with mixed ionic-electronic conductivity. Dependences of the conductivity on temperature, WO3 concentration and oxygen partial pressure were described in terms of the brick-wall model (at f < 0.13) and the generalized mixing equation (at f > 0.13) and conductivity parameters of the O2− ion conducting interface phase were determined.

Corrections to the thermodynamic quantities of Bose system by the generalized uncertainty principle

This paper investigated the Bose system in a spherical shell close to the black hole horizon. Several thermodynamic quantities of the Bose system are derived, which are different from those in the flat spacetime, by introducing the generalized uncertainty principle (GUP) into the grand partition function of statistical mechanics. The internal energy and the pressure of the Bose system appear to have a correction term of T 6, while the entropy has a T 5 correction term where both the coefficients are functions of the spacetime component g 00 and the brick wall model parameter ϵ. Taking the Schwarzschild black hole as an example, the physical quantities of the shell such as temperature, pressure and entropy are calculated for the final stage of black hole radiation.

Thermal, structural and transport properties of composite solid electrolytes (1-x)(C4H9)4NBF4–xAl2O3

Thermal, structural and electrical properties of composite solid electrolytes (1-x)(C4H9)4NBF4–xAl2O3 with nanocrystalline γ-alumina were investigated by DSC, X-ray diffraction, IR spectroscopy, impedance and electrochemical measurements. It was found that the melting enthalpy of (C4H9)4NBF4 in the composites strongly decreases and its value approaches to zero in the composites with x ≥ 0.9, where x is the molar fraction of alumina, indicating the transformation of (C4H9)4NBF4 to an interface-stabilized amorphous state. This effect was quantitatively interpreted in terms of the brick-wall model assuming that a layer of amorphous phase of the ionic salt is formed at salt/oxide interfaces. At the alumina concentration of x = 0.9, corresponding to a volume fraction of alumina f = 0.53, almost all the ionic salt gets into the interface layer the thickness of the amorphous layer is nearly 3 nm. These results agree with the results of X-ray diffraction studies and IR spectroscopy. Introduction of nanocrystalline γ-alumina into the (C4H9)4NBF4 matrix leads to a relative increase in conductivity by more than 2 orders of magnitude, conductivity goes through a maximum of 0.21 mS/cm at 130 °C for the composite with x = 0.9. This composite is characterized by a non-Arrhenius temperature dependence, typical for glassy electrolytes. It was shown that the electrochemical voltage for the composite 0.1(C4H9)4NBF4–0.9Al2O3 is nearly 4 V.

Statistical entropy of a class of regular black holes by brick Wall Model Journal of Physics: Conference Series

In the paper we study the entropy of the scalar field in the background of regular black holes (Heyward and nonsingular)by using the brick-wall method suggested by the’t Hooft. We show the leading term of entropy follows the Bekenstein-Hawking area law and the brickwall method predicts the corrections to the entropy of the regular black hole. The general structure of the coefficient of logarithmic sub-leading corrections for the both black holes are same. The coefficients of the logarithmic corrections are-1/180 for both regular black holes which differ from the Schwarzschild black hole.

T−J model on the effective brick-wall lattice for the recently discovered high-temperature superconductor Ba2CuO3+δ

Layered copper oxides have highest superconducting transition temperatures at ambient pressure. Its mechanism remains a grand challenge in condensed matter physics. The essential physics lying in 2-dimensional copper-oxygen layers is well described by a single band Hubbard model or its strong coupling limit t-J model in 2-dimensional square lattice. Recently discovered high temperature superconductor Ba$_2$CuO$_{3+\delta}$ with $\delta \sim 0.2$ has different crystal structure with large portion of in-plane oxygen vacancies. We observe that an oxygen vacancy breaks the bond of its two neighboring copper atoms, and propose ordered vacancies in Ba$_2$CuO$_{3+\delta}$ lead to extended t-J model on an effective brick-wall lattice. For the nearest neighbor hopping, the brick-wall model can be mapped onto t-J model on honeycomb lattice. Our theory explains the superconductivity of Ba$_2$CuO$_{3+\delta}$ at high charge carrier density, and predict a time reversal symmetry broken pairing state.

The grain size effect in dielectric diffusion and electrical conduction of PZnTe-PZT ceramics

This study investigates systematically the microstructure, dielectric diffusion and electrical conduction behavior in xPb(Zn0.5Te0.5)O3-(1-x)Pb(Zr0.5Ti0.5)O3 (PZnTe-PZT) ceramics. The grain size has an obvious drop in PZT and PZnTe-PZT system. EDS reveals that the Te element segregates at the grain boundary, which can inhibit grain growth during the sintering process and result in the difference of element contents between grain and grain boundary. The permittivity peak become broaden when the PZnTe add in and this phenomenon is explained by the brick-wall model, which is due to the difference in grain size. The diffusion behavior is described by the empirical expression and the W2/3M-H parameter and it shows that the co-action of the grain size effect and the doping effect influence the dielectric diffusion of PZnTe-PZT ceramics. The electrical conduction characteristics in different temperature and frequency show the difference of activation energy for ac conductivity between the PZT and the PZnTe-PZT, which are related to the grain size and the element segregation in grain boundary.

Entropy of Vaidya Black Hole on Event Horizon with Generalized Uncertainty Principle Revisited**Supported by the National Natural Science Foundation of China under Grant Nos. 11675139, 11605137, 11435006, 11405130, and the Double First-Class University Construction Project of Northwest University, the China Postdoctoral Science Foundation under Grant No. 2017M623219, and Shaanxi Postdoctoral Science Foundation

In this note, we recalculate the entropy of the Vaidya black hole on the event horizon by considering the generalized uncertainty principle based on the brick-wall model. The result shows that we need not impose a cut-off by hand anymore and the result satisfies the Bekenstein-Hawking law as well.

Self-consistency of a Modification Method Based on Membrane Model with Minimal Length Considered**Supported by the National Natural Science Foundation of China under Grant Nos. 11675139, 11605137, 11435006, 11405130, the Double First-Class University Construction Project of Northwest University, the China Postdoctoral Science Foundation under Grant No. 2017M623219, and Shaanxi Postdoctoral Science Foundation

By adopting a result from generalized uncertainty principles (GUP), we modify the inner bound of the membrane model to a physical fixed value and get the cut-offs naturally rather than by hand, which are both in brick-wall model and membrane model, and the semi-classical quantization condition could always be valid as well. We also calculate the entropies of Schwarzschild-de Sitter black hole and find the GUP we choose qualitatively shows that the requirement of mass in this method is the same as the natural requirement of the Schwarzschild-de Sitter black hole, which means the method might be self-consistent.

Maximal distant entanglement in Kitaev tube

We study the Kitaev model on a finite-size square lattice with periodic boundary conditions in one direction and open boundary conditions in the other. Based on the fact that the Majorana representation of Kitaev model is equivalent to a brick wall model under the condition t = Δ = μ, this system is shown to support perfect Majorana bound states which is in strong localization limit. By introducing edge-mode fermionic operator and pseudo-spin representation, we find that such edge modes are always associated with maximal entanglement between two edges of the tube, which is independent of the size of the system.

Why the entropy of spacetime is independent of species of particles: the species problem

The Hawking radiation emits all species of particles, but the Bekenstein–Hawking entropy is independent of the number of the species of particles. This is the so-called species problem—a puzzling problem for a long time. In this paper, we suggest a solution to this problem. A result of the scheme is that the black hole atmosphere has a mass equaling 3/8 mass of a classical Schwarzschild black hole, which agrees with ’t Hooft’s brick wall model.

Evidence of Kittel type behaviour of the permittivity of a nanostructured high sensitivity piezoelectric

The broadband dielectric properties of high sensitivity piezoelectric 0.36BiScO3-0.64PbTiO3 ceramics with average grain sizes from 1.6 μm down to 26 nm were investigated in the 100–500 K temperature range. The grain size dependence of the dielectric permittivity was analysed within the effective medium approximation. It was found that the generalised core-shell (or brick wall) model correctly explains the size dependence down to the nanoscale. For the first time, the grain bulk and boundary properties were obtained without making any assumptions of values of the parameters or simplifications. Two contributions to dielectric permittivity of the grain bulk are described. The first is the size-independent one, which follows the Curie-Weiss law. The second one is shown to plausibly follow the Kittel's law. This seems to suggest the unexpected persistence of mobile ferroelectric domains at the nanoscale (26 nm grains). Alternative explanations are discussed.

Gravity’s Rainbow and Black Hole Entropy

We consider the effects of Gravity’s Rainbow on the computation of black hole entropy using a dynamical brick wall model. An explicit dependence of the radial coordinate approaching the horizon is proposed to analyze the behavior of the divergence. We find that, due to the modification of the density of states, the brick wall can be eliminated. The calculation is extended to include rotations and in particular to a Kerr black hole in a comoving frame.

Zero, normal and super-radiant modes for scalar and spinor fields in Kerr–anti de Sitter spacetime

We study zero and normal modes for scalar and spinor fields in Kerr–anti de Sitter spacetime as a bound state problem with Dirichlet and Neumann boundary conditions. Zero mode is defined as its momentum near the horizon to be zero: , and is shown not to exist as a physical state for both scalar and spinor fields. Physical normal modes must satisfy the spectrum condition as results of (i) non-existence of zero mode and (ii) the analyticity with respect to the rotation parameter of Kerr–anti de Sitter spacetime. Under the spectrum condition, the super-radiant modes are shown to be type 2 super-radiance, which is characterized by negative frequency and positive momentum on the horizon . The type 2 super-radiant modes are necessary for the completeness relation of normal modes and are stable. Applying the spectrum condition to black hole thermodynamics, the brick wall model is shown to be well-defined.

Black hole radiation with modified dispersion relation in tunneling paradigm: Static frame

To study possible deviations from the Hawking's prediction, we assume that the dispersion relations of matter fields are modified at high energies and use the Hamilton–Jacobi method to investigate the corresponding effects on the Hawking radiation in this paper. The preferred frame is the static frame of the black hole. The dispersion relation adopted agrees with the relativistic one at low energies but is modified near the Planck mass mp. We calculate the corrections to the Hawking temperature for massive and charged particles to O(mp−2) and massless and neutral particles to all orders. Our results suggest that the thermal spectrum of radiations near horizon is robust, e.g. corrections to the Hawking temperature are suppressed by mp. After the spectrum of radiations near the horizon is obtained, we use the brick wall model to compute the thermal entropy of a massless scalar field near the horizon of a 4D spherically symmetric black hole. We find that the subleading logarithmic term of the entropy does not depend on how the dispersion relations of matter fields are modified. Finally, the luminosities of black holes are computed by using the geometric optics approximation.

Effect of resistivity ratio on energy storage and dielectric relaxation properties of 0–3 dielectric composites

The brick wall model for ceramics was employed to simulate the dielectric response of 0–3 composites. By applying equivalent circuit method, the effective field, energy storage density and effective dielectric permittivity of the composites were studied. The results indicate that the resistivity ratio between the matrix and the fillers plays a crucial role for all these properties. A composite with slightly less resistive matrix with respect to the fillers favors energy storage and maintains large effective dielectric permittivity providing a useful guideline for improved performances. The low effective dielectric permittivity usually measured on ceramic–polymer 0–3 dielectric composites can also be understood based on our simulation results.

The generalized uncertainty relation and the quantum entropy of static spherical black hole surrounded by quintessence due to spin fields

By using the brick-wall model, the quantum entropies of static spherical black hole surrounded by quintessence due to the Weyl neutrino, electromagnetic, massless Rarita–Schwinger, and gravitational fields for the source-free case are investigated from a generalized uncertainty relation. It is shown that in addition to the usual quadratically and logarithmically divergent terms, there exist additional quadratic, biquadratic, and logarithmic divergences at ultraviolet σ → 0, which not only depend on the black hole characteristics but also on the spins of the fields and the gravity correction factor. These additional terms describe the contribution of the quantum fields to the entropy and the effect of gravitational interactions on it. After the smallest length scale is taken into account, we find that the contribution of the gravitational interactions to the entropy is larger than the usual dominant term and becomes a part of the whole dominant term, so it is very important and cannot be neglected.

Free-fall frame black hole in gravity’s rainbow

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use "Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. In this scenario, one could rewrite the rainbow metric $g_{\mu\nu}\left( E\right) $ in terms of some orthonormal frame fields and use the modified equivalence principle to determine the energy dependence of $g_{\mu\nu}\left( E\right) $. Obviously, the form of $g_{\mu\nu}\left( E\right) $ depends on the choice of the orthonormal frame. For a static black hole, there are two natural orthonormal frames, the static one hovering above it and freely falling one along geodesics. The cases with the static orthonormal frame have been extensively studied by many authors. The aim of this paper is to investigate properties of rainbow black holes in the scenario with the free-fall orthonormal frame. We first derive the metric of rainbow black holes and their Hawking temperatures in this free-fall scenario. Then, the thermodynamics of a rainbow Schwarzschild black hole is studied. Finally, we use the brick wall model to compute the thermal entropy of a massless scalar field near the horizon of a Schwarzschild rainbow black hole in this free-fall scenario.

Near-horizon geometry and the entropy of a minimally coupled scalar field in the Kerr black hole

In this article we will discuss a Lorentzian sector calculation of the entropy of a minimally coupled scalar field in the Kerr black hole background. We will use the brick wall model of t' Hooft. In the Kerr black hole, complications arise due to the absence of a global timelike Killing field and the presence of the ergosphere. Nevertheless, it is possible to calculate the entropy of a thin shell of matter field in the near-horizon region using the brick wall model. The corresponding leading order entropy of the nonsuperradiant modes is found to be proportional to the area of the horizon and is logarithmically divergent. Thus, the entropy of a three dimensional system in the near-horizon region is proportional to the boundary surface. This is similar to that of the black hole entropy itself. The corresponding internal energy remains finite if the entropy is chosen to be of the order of the black hole entropy itself. For a fixed value of the brick wall cut-off, the leading order entropy in the Kerr black hole is found to be half of the corresponding term in the Schwarzschild black hole. This is consistent with the preferential emission of particles in the Kerr black hole with azimuthal angular momentum in the same direction as that of the black hole itself. However, we can obtain the Schwarzschild case expression by including a subleading term and taking the appropriate limit. The results obtained in this article may be relevant to entropy bound and holographic principle.