Brick wall diagrams as a completely integrable system
Moritz Kade, Matthias Staudacher
备注：
v3: published in JHEP, 22 pages, fixed typo in subscript of eq. (3.8a) and (3.9a), improved labels of sub-equations in eq. (5.1) and (A.1), added further acknowledgements
我们研究一种可积、平面、手性且非幺正的四维Yukawa理论，即Pittelli和Preti发现的双费米子fishnet理论的自由能。该模型的典型Feynman图属于规则的“砖墙”型，取代了标准fishnet理论中的规则方格。我们以清晰的方式将A. B. Zamolodchikov对fishnet图热力学自由能的强大经典计算方法调整至砖墙情形，并得出了闭合形式的结果。最后，我们简要讨论了与砖墙模型相关的三维和六维中的另外两个候选可积模型。
英文摘要：
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.
We discuss the statistical-mechanical entropy of black hole calculated according to 't Hooft. It is argued that in presence of horizon the statistical mechanics of quantum fields depends on their UV behavior. The ``brick wall'' model was shown to provide a correct description when the ``brick wall'' parameter is less than any UV cut-off.
We calculate the statistical entropy of a scalar field on the background of three-dimensional De Sitter space in terms of the brick wall method and finally derive the perimeter law of the entropy.
Using the brick wall method we compute the statistical entropy of a scalar field in a nontrivial background, in two different cases. These background are generated by four and five dimensional black holes with four and three U(1) charges respectively. The Bekenstein entropy formula is generally obeyed, but corrections are discussed in the latter case.
We calculate the statistical entropy of a quantum field with an arbitrary spin propagating on the spherical symmetric black hole background by using the brick wall formalism at higher orders in the WKB approximation. For general spins, we find that the correction to the standard Bekenstein-Hawking entropy depends logarithmically on the area of the horizon. Furthermore, we apply this analysis to the Schwarzschild and Schwarzschild-AdS black holes and discuss our results.
We research the entropy of a black hole in curved space-times by 't Hooft`s approach, so-called the brick wall method. One of these space-time, a asymptotically dS space-time has two physical horizons; one is a black hole horizon and the other is a cosmological horizon. The others have only one horizon, a black hole horizon. Using this model, we calculate all thermodynamic quantities containing the background geometric effect and show that entropy is proportional to area of each boundary. Furthermore, we show that the Cardy-Verlinde formula can be rederived from the physical quantities of the …
The so-called ``brick-wall model'' is a semi-classical approach that has been used to explain black hole entropy in terms of thermal matter fields. Here, we apply the brick-wall formalism to thermal bulk fields in a Randall-Sundrum brane world scenario. In this case, the black hole entity is really a string-like object in the anti-de Sitter bulk, while appearing as a Schwarzchild black hole to observers living on the brane. In spite of these exotic circumstances, we establish that the Bekenstein-Hawking entropy law is preserved. Although a similar calculation was recently considered in the lit…
We calculate the entropy of the brane-world black hole in the Randall-Sundrum(RS) model by using the brick-wall method. The modes along the extra dimension are semi-classically quantized on the extra dimension. The number of modes in the extra dimension is given as a simple form with the help of the RS mass relation, and then the entropy for the scalar modes in the five-dimensional spacetime is described by the two-dimensional area of the black brane world.
We re-examine the brick-wall model in the context of spacetime foam. In particular we consider a foam composed by wormholes of different sizes filling the black hole horizon. The contribution of such wormholes is computed via a scale invariant distribution. We obtain that the brick wall divergence appears to be logarithmic when the cutoff is sent to zero.
Quantum field theory in the near-horizon region of a black hole predicts the existence of an infinite number of degenerate modes. Such a degeneracy is regulated in the brick wall model by the introduction of a short distance cutoff. In this Letter we show that states of the brick wall model with non zero energy admit a further degeneracy for any given finite value of the cutoff. The black hole entropy is calculated within the brick wall model taking this degeneracy into account. Modes with complex frequencies however do not exhibit such a degeneracy.
We compute the black hole horizon entanglement entropy for a massless scalar field in the brick wall model by incorporating the minimal length. Taking the minimal length effects on the occupation number $n(ω,l)$ and the Hawking temperature into consideration, we obtain the leading UV divergent term and the subleading logarithmic term in the entropy. The leading divergent term scales with the horizon area. The subleading logarithmic term is the same as that in the usual brick wall model without the minimal length.
The brick-wall model seeks to explain the Bekenstein-Hawking entropy as a wall-contribution to the thermal energy of ambient quantum fields raised to the Hawking temperature. Reservations have been expressed concerning the self-consistency of this model. For example, it predicts large thermal energy densities near the wall, producing a substantial mass-correction and, presumably, a large gravitational back-reaction. We re-examine this model and conclude that these reservations are unfounded once the ground state---the Boulware state---is correctly identified. We argue that the brick-wall model…
Using the brick wall method, we study statistical entropy for spherically symmetric black holes in Hořava-Lifshitz gravity. In particular, a Lifshitz scalar field is considered in order to incorporate foliation preserving diffeomorphism which eventually gives a modified dispersion relation. Finally, we obtain the area law without UV cutoff for $z > 3$, and discuss some of consequences in connection with the generalized uncertainty principle.
The statistical entropy of a scalar field on the warped AdS$_3$ black hole in the cosmological topologically massive gravity is calculated based on the brick-wall method, which is different from the Wald's entropy formula giving the modified area law due to the higher-derivative corrections in that the entropy still satisfies the area law. It means that the entropy for scalar excitations on this background is independent of higher-order derivative terms or the conventional brick wall method has some limitations to take into account the higher-derivative terms.
In this work we review, in the framework of the so-called brick wall model, the divergence problem arising in the one loop calculations of various thermodynamical quantities, like entropy, internal energy and heat capacity. Particularly we find that, if one imposes that entanglement entropy is equal to the Bekenstein-Hawking one, the model gives problematic results. Then a proposal of solution to the divergence problem is made following the zeroth law of black hole mechanics.
The brick wall method in calculations of the entropy of black holes can be applied to the FRW cosmology in order to study the statistical entropy. An appropriate cutoff satisfying the covariant entropy bound can be chosen so that the entropy has a definite bound. Among the entropy for each of cosmological eras, the vacuum energy-dominated era turns out to give the maximal entropy which is in fact compatible with assumptions from the brick wall method.
The properties of the thermal radiation are discussed by using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity. There is no burst at the last stage of the emission of a Schwarzshild black hole. When the new equation of state density is utilized to investigate the entropy of a scalar field outside the horizon of a static black hole, the divergence appearing in the brick wall model is removed, without any cutoff. The entropy proportional to the horizon area is derived from the contribution of the vicinity of the horizon.
A semi-classical reasoning leads to the non-commutativity of the space and time coordinates near the horizon of Schwarzschild black hole. This non-commutativity in turn provides a mechanism to interpret the brick wall thickness hypothesis in 't Hooft's brick wall model as well as the boundary condition imposed for the field considered. For concreteness, we consider a noncommutative scalar field model near the horizon and derive the effective metric via the equation of motion of noncommutative scalar field. This metric displays a new horizon in addition to the original one associated with the S…
[Abridged] We compute the canonical entropy of a quantum scalar field around static and spherically symmetric black holes through the brick wall approach at the higher orders (in fact, up to the sixth order in \hbar) in the WKB approximation. We explicitly show that the brick wall model generally predicts corrections to the Bekenstein-Hawking entropy in all spacetime dimensions. In four dimensions, we find that the corrections to the Bekenstein-Hawking entropy are of the form (A^n \log A), while, in six dimensions, the corrections behave as (A^m + A^n \log A), where A denotes the area of the b…
During the last few years, exact solutions that describe black holes that are bound to a two-brane in a four dimensional anti-de Sitter bulk have been constructed. In situations wherein there is a negative cosmological constant on the brane, for large masses, these solutions are exactly the rotating BTZ black holes on the brane and, in fact, describe rotating BTZ black strings in the bulk. We evaluate the canonical entropy of a free and massless scalar field (at the Hawking temperature) around the rotating BTZ black string using the brick wall model. We explicitly show that the Bekenstein-Hawk…
We consider the entropy of a quantum scalar field on a background black hole geometry in asymptotically anti-de Sitter space-time, using the ``brick wall'' approach. In anti-de Sitter space, the theory has no infra-red divergences, and all ultra-violet divergences can be absorbed into a renormalization of the coupling constants in the one-loop effective gravitational Lagrangian. We then calculate the finite renormalized entropy for the Schwarzschild-anti-de Sitter and extremal Reissner-Nordstrom-anti-de Sitter black holes, and show that, at least for large black holes, the entropy is entirely …
We present evidence which confirms a suggestion by Susskind and Uglum regarding black hole entropy. Using a Pauli-Villars regulator, we find that 't Hooft's approach to evaluating black hole entropy through a statistical-mechanical counting of states for a scalar field propagating outside the event horizon yields precisely the one-loop renormalization of the standard Bekenstein-Hawking formula, $S=\A/(4G)$. Our calculation also yields a constant contribution to the black hole entropy, a contribution associated with the one-loop renormalization of higher curvature terms in the gravitational act…
Motivated by the works on Equivalence Principle in the context of linear Generalized Uncertainty Principle and, independently, in the context of quadratic Generalized Uncertainty Principle, we expand these endeavors in the context of Generalized Uncertainty Principle when both linear and quadratic terms in momentum are include. We demonstrate how the definitions of equations of motion change upon that expansion. We also show how to obtain an analogue of Liouville theorem in the presence of linear and quadratic Generalized Uncertainty Principle. We employ the corresponding modified invariant un…
Within the framework of the "brick wall model", a novel method is developed to compute the contributions of a scalar field to the thermodynamic quantities of black holes. The relations between (transverse) momenta and frequencies in Rindler space are determined numerically with high accuracy and analytically with an accuracy of better than 10 % and are compared with the corresponding quantities in Minkowski space. In conflict with earlier results, the thermodynamic properties of black holes turn out to be those of a low temperature system. The resulting discrepancy for partition function and e…
We discuss the relationship between the bulk-boundary correspondence in Rehren's algebraic holography (and in other 'fixed-background' approaches to holography) and in mainstream 'Maldacena AdS/CFT'. Especially, we contrast the understanding of black-hole entropy from the viewpoint of QFT in curved spacetime -- in the framework of 't Hooft's 'brick wall' model -- with the understanding based on Maldacena AdS/CFT. We show that the brick-wall modification of a Klein Gordon field in the Hartle-Hawking-Israel state on 1+2-Schwarzschild AdS (BTZ) has a well-defined boundary limit with the same temp…
We study the 't Hooft's brick wall model for black holes in a holographic context. The brick wall model suggests that without an appropriate near horizon IR cut-off, the free energy of the probe fields show the divergence due to the large degenerate states near the horizons. After studying the universal nature of the divergence in various holographic setting in various dimensions, we interpret the nature of the divergence in a holographic context. The free energy divergence is due to the large degeneracy and continuity of the low energy spectrum in the boundary theory at the deconfinement phas…
By using the 't Hooft's "brick wall" model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropies arising from the quantum scalar field in different coordinate settings, such as the Painlevé and Lemaitre coordinates. At first glance, it seems that the entropies would be different from that in the standard Schwarzschild coordinate since the metrics in both the Painlevé and Lemaitre coordinates do not possess the singularity at the event horizon as that in the Schwarzschild-like coordinate. However, after an exact calculation we find that, up to the sublead…
Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons
Universidad de Oviedo（奥维耶多大学）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）
One-point functions of local operators are studied, at weak and strong coupling, for the ABJM theory in the presence of a 1/2 BPS domain wall. In the underlying quantum spin chain the domain wall is represented by a boundary state which we show is integrable yielding a compact determinant formula for one-point functions of generic operators.
We quantize a scalar field at finite temperature T in the background of a classical black hole, adopting 't Hooft's ``brick wall'' model with generic mixed boundary conditions at the brick wall boundary. We first focus on the exactly solvable case of two dimensional space-time. As expected, the energy density is integrable in the limit of vanishing brick wall thickness only for T=T_H - the Hawking temperature. Consistently with the most general stress energy tensor allowed in this background, the energy density shows a surface contribution localized on the horizon. We point out that the usual …
Entropy of the Kerr-Newman black hole is calculated via the brick wall method with maintaining careful attention to the contribution of superradiant scalar modes. It turns out that the nonsuperradinat and superradiant modes simultaneously contribute to the entropy with the same order in terms of the brick wall cutoff $ε$. In particular, the contribution of the superradiant modes to the entropy is negative. To avoid divergency in this method when the angular velocity tends to zero, we propose to intr oduce a lower bound of angular velocity and to treat the case of the angular momentum per unit …
We compute the black hole entropy in the context of the Modified Dispersion Relations using the brick wall model. An explicit dependence of the radial coordinate approaching the horizon is shown 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. By assuming a specific form for the radial coordinate $r(E/E_{P}) $, we examine a possible candidate for $r(E/E_{P}) $. A comparison with the 't Hooft approach is presented.
ABJM theory in the presence of a half-BPS domain wall is dual to the D2-D4 probe brane system with nonzero worldvolume flux. The ABJM domain wall was recently shown to be integrable to lowest order in perturbation theory and bond dimension. In the present paper we show that the string theory dual of this system is integrable, namely that the string boundary conditions on the probe D4-brane preserve the integrability of the Green-Schwarz sigma model. Our result suggests that the ABJM domain wall is integrable to all loop orders and for any value of the bond dimension.
By using the brick wall method we calculate the thermodynamic potential of the complex scalar field in a charged Kerr black hole. Using it we show that in the Hartle-Hawking state the leading term of the entropy is proportional to $\frac{ A _H}{ε^2}$, which becomes divergent as the system approaches the black hole horizon. The origin of the divergence is that the density of states diverges at the horizon.
We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry. For a big class of non-compact Calabi-Yau 3-folds we construct complex integrable systems of Hitchin type with the base given by the moduli space of deformations of those 3-folds. Then Donaldson-Thomas invariants of the Fukaya category of such a Calabi-Yau 3-fold can be (conjecturally) described in two more ways: in terms of the attractor flow on the base of the corresponding complex integrable system and in terms of the sk…
Using brick wall method the entropy of charged dilaton-axion black hole is determined for both asymptotically flat and non-flat cases. The entropy turns out to be proportional to the horizon area of the black hole confirming the Beckenstien, Hawking area-entropy formula for black holes. The leading order logarithmic corrections to the entropy are also derived for such black holes.
The six-vertex model on an $N\times N$ square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ``reconstruction'' formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.
We discuss the integrability and wall-crossing properties of Kondo problems, where an 1d impurity is coupled to a 2d chiral CFT and triggers a defect RG flow. We review several new and old examples inspired by constructions in four-dimensional Chern-Simons theory and by affine Gaudin models.
In this thesis, we construct a class of exact ASDYM 1-solitons and multi-solitons on 4-dimensional real spaces with the Euclidean signature $(+, +, +, +)$, the Minkowski signature $(+, - , -, -)$, and the split signature ($+$, $+$, $-$, $-$) (the Ultrahyperbolic space). They are new results and successful applications of the Darboux transformation introduced by Nimmo, Gilson, Ohta. In particular, the principal peak of the Lagrangian density Tr$F_{μν}F^{μν}$ is localized on a 3-dimensional hyperplane in 4 dimensional space. Therefore, we use the term "soliton walls" to distinguish them from the…
It is shown how a integrable mechanical system provides all the localized static solutions of a deformation of the linear O(N)-sigma model in two space-time dimensions. The proof is based on the Hamilton-Jacobi separability of the mechanical analogue system that follows when time-independent field configurations are being considered. In particular, we describe the properties of the different kinds of kinks in such a way that a hierarchical structure of solitary wave manifolds emerges for distinct N.
