Tables of integrals

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Matter at high density and low temperature is expected to be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects. At the highest densities, where the QCD coupling is weak, rigorous calculations are possible, and the ground state is a particularly symmetric state, the color-flavor locked (CFL) phase. The CFL phase is a superfluid, an electromagnetic insulator, and breaks chiral symmetry. The effective theory of the low-energy excitations in the CFL phase is known and can be used, even at more moderate densities, to describe its physical properties. At lower densities the CFL phase may be disfavored by stresses that seek to separate the Fermi surfaces of the different flavors, and comparison with the competing alternative phases, which may break translation and/or rotation invariance, is done using phenomenological models. We review the calculations that underlie these results and then discuss transport properties of several color-superconducting phases and their consequences for signatures of color superconductivity in neutron stars.

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Color superconductivity in dense quark matter

It has been shown that a gravitational dual to a superconductor can be obtained by coupling anti-de Sitter gravity to a Maxwell field and charged scalar We review our earlier analysis of this theory and extend it in two directions First, we consider all values for the charge of the scalar field Away from the large charge limit, backreaction on the spacetime metric is important While the qualitative behaviour of the dual superconductor is found to be similar for all charges, in the limit of arbitrarily small charge a new type of black hole instability is found We go on to add a perpendicular magnetic field B and obtain the London equation and magnetic penetration depth We show that these holographic superconductors are Type II, ie, starting in a normal phase at large B and low temperatures, they develop superconducting droplets as B is reduced

Holographic Superconductors

Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems 1 . The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated with a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture.

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Lectures on Holographic Superfluidity and Superconductivity

We have studied a holographically dual description of superconductor in $(2+1)$ dimensions in the presence of an applied magnetic field and observed that there exists a critical value of magnetic field, below which a charged condensate can form via a second-order phase transition.

Critical magnetic field in a holographic superconductor

A density wave in quark matter is discussed at finite temperature, which occurs along with the chiral condensation, and is described by a dual standing wave in scalar and pseudoscalar condensates on the chiral circle. The mechanism is quite similar to that for the spin density wave suggested by Overhauser and entirely reflects many-body effects. It is found within a mean-field approximation for the Nambu--Jona-Lasinio model that the chiral-condensed phase with the density wave develops at a high-density region just outside the usual chiral-transition line in phase diagram. A magnetic property of the density wave is also elucidated.

Chiral symmetry and density waves in quark matter

The most fundamental strings in high-density color superconductivity are the non-Abelian semisuperfluid strings which have color-gauge flux tubes but behave as superfluid vortices in the energetic point of view. We show that in addition to the usual translational zero modes, these vortices have normalizable orientational zero modes in the internal space, associated with the color-flavor locking symmetry broken in the presence of the strings. The interaction among two parallel non-Abelian semisuperfluid strings is derived for general relative orientational zero modes to show the universal repulsion. This implies that the previously known superfluid vortices, formed by spontaneously broken $U(1{)}_{\mathrm{B}}$, are unstable to decay. Moreover, our result proves the stability of color superconductors in the presence of external color-gauge fields.

Non-Abelian strings in high-density QCD: Zero modes and interactions

We consider the thermodynamics of chiral models in the mean-field approximation and discuss the relevance of the (frequently omitted) fermion vacuum loop. Within the chiral quark-meson model and its Polyakov loop extended version, we show that the fermion vacuum fluctuations can change the order of the phase transition in the chiral limit and strongly influence physical observables. We compute the temperature-dependent effective potential and baryon-number susceptibilities in these models, with and without the vacuum term, and explore the cutoff and the pion mass dependence of the susceptibilities. Finally, in the renormalized model the divergent vacuum contribution is removed using dimensional regularization.

Vacuum fluctuations and the thermodynamics of chiral models

In this paper, two important transport observables in the relativistic heavy ion collider (RHIC) experiment\char22{}damping rate and jet quenching parameter\char22{}are calculated from an AdS/QCD model. A quark moving in the viscous medium such as the quark-gluon-plasma is modeled by an open string whose end point travels on the boundary of a deformed ${\mathrm{AdS}}_{5}$ black hole. The correction introduced via the deformed ${\mathrm{AdS}}_{5}$ is believed to help us better understand the data which is expected to be measured in the RHIC.

Eiji Nakano

Papers

Critical magnetic field in a holographic superconductor

Chiral symmetry and density waves in quark matter

Non-Abelian strings in high-density QCD: Zero modes and interactions

Vacuum fluctuations and the thermodynamics of chiral models

Drag force, jet quenching, and an AdS/QCD model