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This paper investigates the Casimir effect of a wedge and its holographic dual. We prove that the displacement operator universally determines the wedge Casimir effect in the smooth limit. Besides, we argue that the wedge Casimir energy increases with the opening angle and test it with several examples. Furthermore, we construct the holographic dual of wedges in AdS/BCFT in general dimensions. We verify that our proposal can produce the expected Casimir effect within smooth and singular limits. We observe that the Casimir energy density of a wedge increases with the brane tension. Next, we discuss the wedge contribution to holographic entanglement entropy and find it increases with the opening angle, similar to the wedge Casimir energy. Finally, we briefly discuss the holographic polygon in AdS$_3$/BCFT$_2$ and its generalization to higher dimensions., Comment: 38 pages, 7 figures, references added, minor revision published in JHEP

Cone holography is a codimension-$n$ doubly holographic model, which can be interpreted as the holographic dual of edge modes on defects. The initial model of cone holography is based on mixed boundary conditions. This paper formulates cone holography with Neumann boundary conditions, where the brane-localized gauge fields play an essential role. Firstly, we illustrate the main ideas in an AdS$_4$/CFT$_1$ toy model. We show that the $U(1)$ gauge field on the end-of-the-world brane can make the typical solution consistent with Neumann boundary conditions. Then, we generalize the discussions to general codimension-$n$ cone holography by employing brane-localized $p$-form gauge fields. We also investigate perturbative solutions and prove the mass spectrum of Kaluza-Klein gravitons is non-negative. Furthermore, we prove that cone holography obeys holographic $c$-theorem. Finally, inspired by the recently proposed chiral model in AdS/BCFT, we construct another type of cone holography with Neumann boundary conditions by applying massive vector (Proca) fields on the end-of-the-world brane., Comment: 31 pages, 2 figures, references added, revision published in JHEP

This paper investigates the brane-localized interactions, including DGP gravity and higher derivative (HD) gravity localized on the brane. We derive the effective action on the brane, which suggests the brane-localized HD gravity suffers the ghost problem generally. Besides, we obtain novel algebraic identities of the mass spectrum, which reveal the global nature and can characterize the phase transformation of the mass spectrum. We get a powerful ghost-free condition from the spectrum identities, which rules out one type of brane-localized HD gravity. We further prove the mass spectrum is real and non-negative $m^2\ge 0$ under the ghost-free condition. Furthermore, we discuss various constraints on parameters of brane-localized gravity in AdS/BCFT and wedge holography, respectively. They include the ghost-free condition of Kaluza-Klein and brane-bending modes, the positive definiteness of boundary central charges, and entanglement entropy. The ghost-free condition imposes strict constraint, which requires non-negative couplings for pure DGP gravity and Gauss-Bonnet gravity on the brane. It also rules out one class of brane-localized HD gravity. Thus, such HD gravity should be understood as a low-energy effective theory on the brane under the ghost energy scale. Finally, we briefly discuss the applications of our results., Comment: 57 pages, 7 figures, prove novel spectrum identities and analytical ghost-free conditions for brane-localized gravity, revision published in JHEP

In a recent paper [1], we proposed that the quantum states of black hole responsible for the Bekenstein-Hawking entropy are given by Bell states of Fermi quanta in the interior of black hole. In this paper, we include the effect of tunneling on these entangled states and show that partial tunneling of these Bell states of Fermi quanta give rises to the Page curve of Hawking radiation. We also show that the entirety of information initially stored in the black hole is returned to the outside via the Hawking radiation., Comment: 10 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:2209.03610

We propose a quantum model of the Schwarzschild black hole as a quantum mechanics of a system of fermionic degrees of freedom. The system has a constant density of states and a Fermi energy that is inversely proportional to the size of the system. Assuming equivalence principle, we show that the degeneracy pressure of the Fermi degrees of freedom is able to withstand the collapse of gravity if the radius of the system is given precisely by the horizon radius of the Schwarzschild black hole. In our model, the fermionic degrees of freedom at each energy level can be entangled in certain different ways, giving rise to a multitude of degenerate ground states of the system. The counting of these microstates reproduces precisely the Bekenstein-Hawking entropy. This simple Fermi model is universal and works also for the Reissner-Nordstr\"om charged black hole as well as black hole with a cosmological constant. From the properties of the Fermi variables, we propose that quantum gravity is characterized by a principle of {\it maximal capacity of states} where there can be no more than $V /l_P^3$ quantum states in any volume $V$. It implies a loss of spatial locality below the Planck length and suggests that any singularity predicted by general relativity is resolved and replaced by a quantum space in quantum gravity. In our model, a black hole spacetime is equipped with an uniform distribution of energy levels. This is another reason why black hole can be considered a simple harmonic oscillator of quantum gravity., Comment: v3: PRD version

It is controversial whether entanglement islands can exist in massless gravity theories. Recently, it is found that the massless entanglement island appears in wedge holography with DGP gravity on the branes. In this paper, we generalize the discussions to the codim-n holography named cone holography. For simplicity, we focus on the case with a codim-2 E brane and a codim-1 Q brane. We discuss the effective action, mass spectrum and holographic entanglement entropy for cone holography with DGP terms. We verify that there is massless gravity on the branes, and recover non-trivial entanglement islands and Page curves. Besides, we work out the parameter space which allows entanglement islands and Page curves. Compared with wedge holography, there are several new features. First, one can not add DGP gravity on the codim-2 E brane. That is because the energy density has to be a constant on codim-2 branes for Einstein gravity in bulk. Second, the Hartman-Maldacena surface ends only on the codim-1 Q brane. Third, the Hartman-Maldacena surface can be defined only in a finite time. We notice that this unusual situation also appears in AdS/dCFT and even in AdS/CFT. Fortunately, it does not affect the Page curve since it happens after Page time. Our results provide more support that the entanglement island is consistent with massless gravity theories., Comment: 32 pages, 16 figures, minor revision published in JHEP

Entanglement islands play an essential role in the recent breakthrough in resolving the black hole information paradox. However, whether entanglement islands can exist in massless gravity theories is controversial. It is found that entanglement islands disappear in the initial model of wedge holography with massless gravity on the brane. As a result, the entanglement entropy of Hawking radiation becomes a time-independent constant, and there is no Page curve. In this paper, we recover massless entanglement islands in wedge holography with suitable DGP gravity or higher derivative gravity on the branes. We study two typical cases. In the first case, we consider a black hole on the strong-gravity brane and a bath on the weak-gravity brane. It is similar to the usual double holography with non-gravitational baths. In the second case, we discuss two black holes on the two branes with the same gravitational strength. We recover massless entanglement islands and non-trivial Page curves in both cases. We also argue that the entanglement island is consistent with massless gravity. Our results strongly support that entanglement islands can exist in long-range theories of gravity., Comment: 42 pages, 20 figures, get a new lower bound for DGP parameter in sect. 2.4, add more discussions, revision published in JHEP

Entanglement islands play an essential role in the recent breakthrough in addressing the black hole information paradox. Inspired by double holography, it is conjectured that the entanglement islands can exist only in massive gravity. There are many pieces of evidence but also debates for this conjecture. This paper recovers the massless entanglement island in wedge holography with negative DGP gravity on the brane. However, the spectrum of negative DGP gravity includes a massive ghost, implying the model is unstable. Our work supports the view that there is no entanglement island in a well-defined braneworld model of massless gravity if one divides the radiation and black hole regions by minimizing entanglement entropy. However, such a partition results in a zero radiation region containing no information. Whether there are other physical non-trivial partitions of the radiation region is an open question and deserves further study., Comment: 6 pages, 3 figures, revision published in EPJC

Due to the Weyl anomaly, an axial vector field produces novel anomalous chiral currents in spacetime with boundaries. Remarkably, the chiral current is not invariant under the gauge transformation of axial vector fields. As a result, more potential terms appear, and the chiral current becomes larger than the electric current near the boundary. This paper investigates the anomalous chiral current in AdS/BCFT. We find that the minimal holographic model of the axial vector field cannot produce the expected chiral current. To resolve this problem, we propose adding a suitable boundary action. Furthermore, we notice a similar situation for holographic condensation. Finally, we obtain the shape dependence of holographic chiral current and verify that it agrees with the field-theoretical result., Comment: 12 pages, 0 figure, revision accepted for publication in PLB

Torsion can be realized as dislocation in the crystal lattice of material. It is particularly interesting if the material has fermions in the spectrum, such as graphene, topological insulators, Dirac and Weyl semimetals, as it's transport properties can be affected by the torsion. In this paper, we find that, due to Weyl anomaly, torsion in Dirac and Weyl semimetals can induce novel chiral currents, either near a boundary or in a ``conformally flat space''. We briefly discuss how to measure this interesting effect in experiment. It is remarkable that these experiments can help to clarify the theoretical controversy of whether an imaginary Pontryagin density could appear in the Weyl anomaly., Comment: 15 pages, 1 figure, revision published in PRB

We propose that the quantum states of black hole responsible for the Bekenstein-Hawking entropy are given by a thin shell of Bell particles located at the region just underneath the horizon. We argue that the configuration can be stabilized by a new kind of degeneracy pressure which is suggested by a noncommutative geometry in the interior of the black hole. Black hole singularity is avoided. We utilize the work of Parikh and Wilczek \cite{Parikh:1999mf} to include the effect of tunneling on the Bell particles. We show that partially tunneled Bell particles give the Page curve of Hawking radiation, and the entirety of information initially stored in the black hole is returned to the outside via the Hawking radiation. In view of entropic force, the location of these Bell states is naturally related to the island and the quantum extremal surface., Comment: v2: 1+12 pages, 3 figures

The previous studies of the island and double holography mainly focus on codimension-one branes. This paper explores the island on the codimension-two brane in AdS/dCFT. The codimension-two brane is closely related to conical singularity, which is very different from the codimension-one brane. We analyze the mass spectrum of gravitons on the codimension-two brane and find that the larger the brane tension is, the smaller the gravitational mass is. The massless mode is forbidden by either the boundary or normalization conditions. We prove that the first massive gravitational mode is located on the codimension-two brane; the larger the tension, the better the localization. It is similar to the case of codimension-one brane and builds an excellent physical foundation for the study of black hole evolution on codimension-two branes. We find that the Page curve of eternal black holes can be recovered due to the island ending on the codimension-two brane. The new feature is that the extremal surface passing the horizon cannot be defined after some finite time in the no-island phase. Fortunately, this unusual situation does not affect the Page curve since it happens after Page time., Comment: 46 pages, 20 figures, reference added, minor revision published in JHEP

In this paper, we investigate AdS/BCFT for curvature-squared gravity. To warm up, we start with Gauss-Bonnet gravity. We derive the one point function of stress tensor and show that the central charge related to the norm of displacement operator is positive for the couplings obeying causality constraints. Furthermore, by imposing the null energy condition on the end-of-the-world brane, we prove the holographic g-theorem for Gauss-Bonnet gravity. This corrects a wrong point of view in the literature, which claims that the holographic g-theorem is violated for Gauss-Bonnet gravity. As a by-product, we obtain the boundary entropy and A-type boundary central charges in general dimensions. We also study AdS/BCFT for general curvature-squared gravity. We find that it is too restrictive for the shape of the brane and the dual BCFT is trivial if one imposes Neumann boundary conditions for all of the gravitational modes. Instead, we propose to impose Dirichlet boundary condition for the massive graviton, while imposing Neumann boundary condition for the massless graviton. In this way, we obtain non-trivial shape dependence of stress tensor and well-defined central charges. In particular, the holographic g-theorem is satisfied by general curvature-squared gravity. Finally, we discuss the island and show that the Page curve can be recovered for Gauss-Bonnet gravity. Interestingly, there are zeroth-order phase transitions for the Page curve within one range of couplings obeying causality constraints. Generalizing the discussions to holographic entanglement entropy and holographic complexity in AdS/CFT, we get new constraints for the Gauss-Bonnet coupling, which is stronger than the causality constraint., Comment: 49 pages, 29 figures, revision accepted for publication in JHEP, main improvements: prove that our g-function can recover the universal term of boundary entropy in general dimensions; add a toy model to explain the novel zeroth-order phase transition of the Page curve analytically

In this paper, we study the effective action, the mass spectrum and the first law of entanglement entropy for a novel doubly holographic model called wedge holography. We work out the effective action of quantum gravity on the branes. In the perturbative formulation, it is given by an infinite sum of Pauli-Fierz actions. In the non-perturbative formulation, the effective action is composed of a higher derivative gravity and a matter action. Usually, a higher derivative gravity can be renormalizable but suffers the ghost problem. For our case, since the effective theory on the brane is equivalent to Einstein gravity in the bulk, it must be ghost-free. We notice that the matter action plays an important role in eliminating the ghost. We also provide evidences that the higher derivative gravity on the brane is equivalent to a ghost-free multi-gravity. Besides, we prove that the effective action yields the correct Weyl anomaly. Interestingly, although the effective action on the brane is an infinite tower of higher derivative gravity, the holographic Weyl anomaly is exactly the same as that of Einstein gravity. We also analyze the mass spectrum of wedge holography. Remarkably, there is always a massless mode of gravitons on the end-of-the-world branes in wedge holography. This happens because one imposes Neumann boundary condition on both branes. On the other hand, the massless mode disappears if one imposes Dirichlet boundary condition on one of the branes as in brane world theory and AdS/BCFT. Finally, we verify the first law of entanglement entropy for wedge holography. Interestingly, the massive fluctuations are irrelevant to the first order perturbation of the holographic entanglement entropy. Thus, in many aspects, the effective theory on the brane behaves like massless Einstein gravity., Comment: 41 pages, 14 figures, references added, minor revision accepted for publication in JHEP

Parallel worlds are imaginative ideas in quantum mechanics and cosmology. The superpositions of parallel worlds are novel states of quantum gravity and have no classical correspondences generally. In this letter, we investigate the superposition or the Schr\"odinger's cat state of optical parallel worlds, which could be realized in laboratory and may shed some light on the detection of parallel universes in a real world. We propose two realizable experimental schemes, which enable to explore the mysterious `parallel universes' by a Mach-Zehnder interferometer. The first one is based on an atomic ensemble in a superposition state, which is a fat Schr\"odinger's cat state. The second one is to prepare a photon in a superposition of different paths, where each path lies in an optical parallel universe., Comment: 5 gages, 5 figure, typos corrected, references added, comments welcome!

According to Witten [1], the conformal boundary condition of gravity, which specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and leads to well-defined perturbation theory of gravity about any classical solution. The conformal boundary condition was previously considered in [2, 3] in the context of AdS/BCFT, wherein the equation of motion of the end-of-the-world was derived and emphasized. In this paper, we investigate further other consequences of the conformal boundary condition in AdS/BCFT. We derive the boundary central charges of the holographic Weyl anomaly and show that they are exactly the same for conformal boundary condition and Dirichlet boundary condition. We analysis the metric perturbation with conformal boundary condition (CBC), Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) imposed on the end-of-the-world brane and show that they admit an interpretation as the fluctuation of the extrinsic curvature (case of CBC and DBC) and the induced metric (case of NBC) of Q respectively. In all cases, the fluctuation modes are massive, which are closely relevant to the massive island formation in the literature. Our results reveal that there are non-trivial gravitational dynamics from extrinsic curvatures on the conformal and Dirichlet branes, which may have interesting applications to the island. We also discuss, in passing, the localization of gravitons in brane world theory. We find that, contrary to NBC, the graviton for CBC/DBC is located on the brane with non-positive tension instead of non-negative tension., Comment: v3: 41 pages, 6 figures, appendix added on localization of gravity, references added, JHEP version

Weyl anomaly leads to novel anomalous currents in a spacetime with boundaries. Recently it is found that the anomalous current can be significantly enhanced by the high temperature for free theories, which could make the experimental measurement easier. In this paper, we investigate holographic anomalous currents at a finite temperature. It is found that the holographic current is still enhanced by the high temperature in dimensions higher than three. However, the temperature dependence is quite different from that of free theories. This may be due to the fact that the holographic CFT is strongly coupled and there is non-zero resistance in the holographic model. Remarkably, the temperature dependence of holographic anomalous currents is universal in the high temperature limit, which is independent of the choices of background magnetic fields., Comment: 11 pages, 4 figures, published in PLB

Recently it is found that Weyl anomaly leads to novel anomalous currents in the spacetime with a boundary. However, the anomalous current is suppressed by the mass of charge carriers and the distance to the boundary, which makes it difficult to be measured. In this paper, we explore the possible mechanisms for the enhancement of anomalous currents. Interestingly, we find that the anomalous current can be significantly enhanced by the high temperature, which makes easier the experimental detection. For free theories, the anomalous current is proportional to the temperature in the high temperature limit. Note that the currents can be enhanced by thermal effects only at high temperatures. In general, this is not the case at low temperatures. For general temperatures, the absolute value of the current of Neumann boundary condition first decreases and then increases with the temperature, while the current of Dirichlet boundary condition always increases with the temperature. It should be mentioned that the enhancement does not have an anomalous nature. In fact, the so-called anomalous current in this paper is not always related to Weyl anomaly. Instead, it is an anomalous effect due to the boundary., Comment: 16 pages, 6 figures,references added, minor revision accepted for publication in EPJC

We propose a novel codimension-n holography, called cone holography, between a gravitational theory in $(d+1)$-dimensional conical spacetime and a CFT on the $(d+1-n)$-dimensional defects. Similar to wedge holography, the cone holography can be obtained by taking the zero-volume limit of holographic defect CFT. Remarkably, it can be regarded as a holographic dual of the edge modes on the defects. For one class of solutions, we prove that the cone holography is equivalent to AdS/CFT, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. In general, cone holography and AdS/CFT are different due to the infinite towers of massive Kaluza-Klein modes on the branes. We test cone holography by studying Weyl anomaly, Entanglement/R\'enyi entropy and correlation functions, and find good agreements between the holographic and the CFT results. In particular, the c-theorem is obeyed by cone holography. These are strong supports for our proposal. We discuss two kinds of boundary conditions, the mixed boundary condition and Neumann boundary condition, and find that they both define a consistent theory of cone holography. We also analyze the mass spectrum on the brane and find that the larger the tension is, the more continuous the mass spectrum is. The cone holography can be regarded as a generalization of the wedge holography, and it is closely related to the defect CFT, entanglement/R\'enyi entropy and AdS/BCFT(dCFT). Thus it is expected to have a wide range of applications., Comment: 43 pages, 10 figures, minor revision published in PRD

Recently, a codimension two holography called wedge holography is proposed as a generalization of AdS/CFT. It is conjectured that a gravitational theory in $d+1$ dimensional wedge spacetime is dual to a $d-1$ dimensional CFT on the corner of the wedge. In this paper, we give an exact construction of the gravitational solutions for wedge holography from the ones in AdS/CFT. By applying this construction, we prove the equivalence between wedge holography and AdS/CFT for vacuum Einstein gravity, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. The equivalence to AdS/CFT can be regarded as a "proof" of wedge holography in a certain sense. As an application of this powerful equivalence, we derive easily the holographic Weyl anomaly, holographic Entanglement/R\'enyi entropy and correlation functions for wedge holography. Besides, we discuss the general solutions of wedge holography and argue that they correspond to the AdS/CFT with suitable matter fields. Interestingly, we notice that the intrinsic Ricci scalar on the brane is always a constant, which depends on the tension. Finally, we generalize the discussions to dS/CFT and flat space holography. Remarkably, we find that AdS/CFT, dS/CFT and flat space holography can be unified in the framework of codimension two holography in asymptotically AdS. Different dualities are distinguished by different types of spacetimes on the brane., Comment: 29 pages, 4 figures, references added, revision accepted for publication in JHEP

Recently it is found that, due to Weyl anomaly, a background scalar field induces a non-trivial Fermi condensation for theories with Yukawa couplings. For simplicity, the paper consider only scalar type Yukawa coupling and, in the BCFT case, only for a specific boundary condition. In these cases, the Weyl anomaly takes on a simple special form. In this paper, we generalize the results to more general situations. First, we obtain general expressions of Weyl anomaly due to a background scalar and pseudo scalar field in general 4d BCFTs. Then, we derive the general form of Fermi condensation from the Weyl anomaly. It is remarkable that, in general, Fermi condensation is non-zero even if there was not a non-vanishing scalar field background. Finally, we verify our results with free BCFT with Yukawa coupling to scalar and pseudo-scalar background potential with general chiral bag boundary condition and with holographic BCFT. In particular, we obtain the shape and curvature dependence of the Fermi condensate from the holographic one point function., Comment: 32 pages, 1 figure, revision published in JHEP

Recently it is found that, due to Weyl anomaly, an external magnetic field can induce anomalous currents near a boundary. In this note, we study anomalous currents for complex scalars and Dirac fields in general dimensions. We develop a perturbation method to calculate Green's function in the spacetime with boundaries. By applying this method, we obtain anomalous currents up to the linear order of magnetic fields in a half space and in a strip. To the best of our knowledge, the results for Dirac fermions and for strips are new. It is remarkable that, unlike the scalars and holographic BCFT, the anomalous currents of Dirac fields are independent of boundary conditions in general dimensions. Besides, the currents of Dirac fields are always larger than those of complex scalars. Finally, we find an exact formal expression of the anomalous current in a half space. The result is expressed in momentum integrals, which can be evaluated numerically. We find that the mass suppresses the anomalous currents as expected., Comment: 23 pages, 5 figures, accepted for publication in PRD

Fermi condensation is usually a phenomena of strongly correlated system. In this letter, we point out a novel mechanism for condensation of Dirac fermions due to Weyl anomaly. The condensation has its physical origin in the nontrivial response of the fermion vacuum to changes in the background spacetime (either boundary location or the background metric), and can be felt when a background scalar field is turned on. The scalar field can be, for example, the Higgs field in a fundamental theory or the phonon in condensed matter system. For a spacetime with boundaries, the induced Fermi condensate is inversely proportional to the proper distance from the boundary. For a conformally flat spacetime without boundaries, Fermi condensation depends on the conformal factor and its derivatives. We also generalize the Banks-Casher relation which relates the Fermi condensate to the zero mode density of the Dirac operator to a local form. Due to its universal nature, this anomaly induced Fermi condensate can be expected to have a wide range of applications in physics., Comment: 5 pages, 1 figure, revision published in PRD

Recently it is found that Weyl anomaly leads to new anomalous currents in an external electromagnetic field in the curved spacetime. For simplicity, the initial works mainly focus on weak gravitational fields and the anomalous current is obtained for conformally flat spaces with small scale factors. In this paper, we generalize the results to the case with arbitrary scale factors. Firstly, we derive the transformation law of current under Weyl transformation, from which one can read off the anomalous current in general conformally flat spaces. Secondly, by using the Weyl transformation of currents we provide a new derivation of the Weyl-anomaly-induced current near the boundary. Thus we have obtained the two kinds of anomalous currents in the literature from a unified formula. Finally, we extend the discussions to n-form fields and find similar anomalous currents., Comment: 18 pages, 0 figure, match the published version in PLB

It was recently discovered that for a boundary system in the presence of a background magnetic field, the quantum fluctuation of the vacuum would create a non-uniform magnetization density for the vacuum and a magnetization current is induced in the vacuum. It was also shown that this "magnetic Casimir effect" of the vacuum is closely related to another quantum effect of the vacuum, the Weyl anomaly. Furthermore, the phenomena can be understood in terms of the holography of the boundary system. In this paper, we generalize this four dimensional effect to six dimensions. We use the AdS/BCFT holography to show that in the presence of a 3-form magnetic field strength $H$,a string current is induced in a six dimensional boundary conformal field theory. This allows us to determine the gauge field contribution to the Weyl anomaly in six dimensional conformal field theory in a $H$-flux background. For the (2,0) superconformal field theory of $N$ M5-branes, the current has a magnitude proportional to $N^3$ for large $N$. This suggests that the degree of freedoms scales as $N^3$ in the (2,0) superconformal theory of $N$ multiple M5-branes. The prediction we have for the Weyl anomaly is a new criteria that the (2,0) theory should satisfy., Comment: 19 pages, 2 figures, match the published version in JHEP

In this paper, we investigate Casimir effect, Weyl anomaly and displacement operator for boundary conformal field theory in general dimensions. We find universal relations between them. In particular, they are all determined by the central charge of boundary conformal field theory. We verify these relations by studying free BCFTs and holographic BCFTs. As a byproduct, we obtain the holographic two point function of stress tensor when the bulk boundary is perpendicular to the AdS boundary., Comment: 22pages, 1 figure, match the published version in JHEP

Neumann boundary condition plays an important role in the initial proposal of holographic dual of boundary conformal field theory, which has yield many interesting results and passed several non-trivial tests. In this paper, we show that Dirichlet boundary condition works as well as Neumann boundary condition. For instance, it includes AdS solution and obeys the g-theorem. Furthermore, it can produce the correct expression of one point function, the boundary Weyl anomaly and the universal relations between them. We also study the relative boundary condition for gauge fields, which is the counterpart of Dirichlet boundary condition for gravitational fields. Interestingly, the four-dimensional Reissner-Nordstrom black hole with magnetic charge is an exact solution to relative boundary condition under some conditions. This holographic model predicts that a constant magnetic field in the bulk can induce a constant current on the boundary in three dimensions. We suggest to measure this interesting boundary current in materials such as the graphene., Comment: 28 pages, 3 figures, match the published revision of JHEP

Recently, it is found that when an external magnetic field parallel to the boundary is applied, Weyl anomaly gives rises to a new anomalous current transport in the vicinity of the boundary. At the leading order of closeness from the boundary, the current is determined universally by the central charge of the theory. In this paper, we give a holographic proof for the existence and universality for this transport phenomena. We show that the current is independent of boundary conditions in four dimensions while it depends on boundary conditions in other dimensions. We also study the backreaction of the bulk Maxwell fields on the AdS spacetime and obtain the holographic Weyl anomaly for 5d BCFTs in presence of the background field strength., Comment: 16 pages, 1 figure, accepted for publication in JHEP

We show that when an external magnetic field parallel to the boundary is applied, the Weyl anomaly gives rises to a new anomalous current in the vicinity of the boundary. The induced current is a magnetization current in origin: the movement of the virtual charges near the boundary give rise to a nonuniform magnetization of the vacuum and hence a magnetization current. Unlike other previously studied anomalous current phenomena such as the chiral magnetic effect or the chiral vortical effect, this induced current does not rely on the presence of a material system and can occur in vacuum. Similar to the Casimir effect, our discovered phenomenon arises from the effect of the boundary on the quantum fluctuations of the vacuum. However this induced current is purely quantum mechanical and has no classical limit. We briefly comment on how this induced current may be observed experimentally., Comment: 6 pages, 1 figures, title changed, revision published in PRL

A maximally entangled state is a quantum state which has maximum von Neumann entropy for each bipartition. Through proposing a new method to classify quantum states by using concurrences of pure states of a region, one can apply Bell's inequality to study intensity of quantum entanglement of maximally entangled states. We use a class of seven-qubit quantum states to demonstrate the method, where we express all coefficients of the quantum states in terms of concurrences of pure states of a region. When a critical point of an upper bound of Bell's inequality occurs in our quantum states, one of the quantum state is a ground state of the toric code model on a disk manifold. Our result also implies that the maximally entangled states does not suggest local maximum quantum entanglement in our quantum states., Comment: 10 pages, 3 figures