A pioneering treatise presenting how the new mathematical techniques of holographic duality unify seemingly unrelated fields of physics. This innovative development morphs quantum field theory, general relativity and the renormalisation group into a single computational framework and this book is the first to bring together a wide range of research in this rapidly developing field. Set within the context of condensed matter physics and using boxes highlighting the specific techniques required, it examines the holographic description of thermal properties of matter, Fermi liquids and superconductors, and hitherto unknown forms of macroscopically entangled quantum matter in terms of general relativity, stars and black holes. Showing that holographic duality can succeed where classic mathematical approaches fail, this text provides a thorough overview of this major breakthrough at the heart of modern physics. The inclusion of extensive introductory material using non-technical language and online Mathematica notebooks ensures the appeal to students and researchers alike.

We present a holographic model of a topological Weyl semimetal. A key ingredient is a timereversal breaking parameter and a mass deformation. Upon varying the ratio of mass to timereversal breaking parameter the model undergoes a quantum phase transition from a topologically nontrivial semimetal to a trivial one. The topological nontrivial semimetal is characterised by the presence of an anomalous Hall effect. The results can be interpreted in terms of the holographic renormalization group (RG) flow leading to restoration of time-reversal at the end point of the RG flow in the trivial phase.Weyl semimetals are an exciting new class of 3D materials with exotic transport properties [1,2]. They are characterised by pointlike singularities in the Brillouin zone at which conduction and valence bands touch. Around these points the electronic quasiparticle excitations can be described by either left-or right-handed Weyl spinors. The Nielsen-Ninomiya theorem guarantees that left-and right-handed Weyl spinors always appear in pairs [3]. When time-reversal symmetry is broken the left-and right-handed quasiparticles can sit at different points in the Brillouin zone. Effectively the Weyl fermions are separated by an (axial) vector in the momentum space. The wave function of a Weyl spinor can be understood as a monopole of the Berry curvature in momentum space. Left-handed Weyl fermions have monopole charge +1 and the right-handed ones have monopole charge −1 [4][5][6]. Since the monopole charge in momentum space is a topological invariant it is still present in fermionic two-point correlation functions when interactions are taken into account [7]. However at strong coupling such semiclassical reasoning based on fermionic wave functions or correlators might not be available and the dynamical variables are physical operators with the quantum numbers of fermion bilinears. The question arises then if it is possible to construct a model at strong coupling that has the essential physical properties of a Weyl semimetal, in particular, if there exists any strongly coupled model in which a quantum phase transition between a topological and a topologically trivial state persists even in the absence of the notion of singularities in the dispersion relations of fermionic two-point correlations functions? A tool to answer these questions is the AdS/CFT correspondence ("holography"). It has already proved to be extremely useful for the understanding of strongly correlated relativistic systems, including superconductors [8], strange metals [9,10], lattice systems [11], etc. In particular, the modern understanding of anomaly related transport phenomena such as the chiral magnetic and chiral vortical effects, is based to a considerable part on research using holographic models [12-14] 1 . We first review a quantum field theoretical model model with same local properties around the band touching points as a Weyl semimetal. It takes the form of a "Lorentz breaking" Dirac system [17] with LagrangianHere / X = γ µ X µ , A µ is the electromagne...

We calculate the shear viscosity of field theories with gravity duals using Kubo-formula by calculating the Green function of dual transverse gravitons and confirm that the value of the shear viscosity is fully determined by the effective coupling of transverse gravitons on the horizon. We calculate the effective coupling of transverse gravitons for Einstein and GaussBonnet gravities coupled with matter fields, respectively. Then we apply the resulting formula to the case of AdS Gauss-Bonnet gravity with F 4 term corrections of Maxwell field and discuss the effect of F 4 terms on the ratio of the shear viscosity to entropy density.