4 - Large-N field theories for holography and condensed matter

Summary

The profound puzzles posed by quantum critical metals with Planckian dissipation and long-range entanglement, as observed in cuprates and heavy-fermion systems, cry out for a novel point of view. Holography can provide this new perspective. This book will propose that its concrete manifestation in terms of the AdS/CFT correspondence gives qualitatively new insights into these puzzles. The reason is that holography has to be understood above all as a “weak–strong” duality between two different descriptions of the same physics. In this regard it is qualitatively similar to the Kramers–Wannier or Abelian–Higgs duality we reviewed in chapter 2, but it takes the notion to a new level: it relates quantum field theories to a dual description that includes the gravitational force. For an extremely strongly coupled field theory, the weakly coupled theory is now Einstein's theory of general relativity. Vice versa, a strongly interacting gravitational theory has an equivalent description as a weakly coupled quantum field theory.

General relativity inherently contains the notion of a dynamically fluctuating space-time. The remarkable way in which this emerges in holography is by incorporating the renormalisation-group structure of the quantum field theory into the dualisation. As we previewed in the introduction, the renormalisation-group scale becomes part of the geometrical edifice as an additional space dimension in the gravitational theory.

It is still baffling that a quantitative duality relation can exist between two theories in different space-time dimensions. This paradox is resolved, however, by the holographic principle of quantum gravity. This lesson from black-hole physics insists that gravitational systems are less dense in information than conventional quantum field theories in a flat non-dynamical space-time, to the degree that the former can be encoded in a “holographic screen” with one dimension less. The dynamics of this “screen” can be thought of as the dynamics of the dual field theory.

In this chapter we will first provide a brief account of the conceptual and historical background of the holographic principle and in particular its manifestation within string theory (section 4.1). This is where the origin of the AdS/CFT correspondence lies. Fortunately one does not need all this material to understand holographic duality practically. In the remainder of this chapter we approach holography from a constructive angle instead, as was first put together in the excellent review [5].