Lorentz lectures of prof. Subir Sachdev (Harvard): May/June 2012

1. The superfluid-insulator quantum phase transition: field theory
vs. holography

Experiments on ultracold atoms in optical lattices provide a remarkable
realization of the the superfluid-insulator quantum phase transition in
two spatial dimensions.  I will show how the low energy physics of this
transition is described by a conformal field theory (CFT). However,
many experimentally interesting questions on the finite temperature
dynamics of this CFT cannot be accurately addressed by conventional
field-theoretic methods.  I will argue that the gauge-gravity duality
of string theory provides a natural and powerful formalism to answer
such questions. Sample computations of experimental observables by this
method will be presented.

2. Non-Fermi liquid metals: field theory vs. holography

The Fermi liquid is the canonical metallic state of quantum matter:
it is adiabatically connected to the free electron model of occupied
states inside a Fermi surface. However numerous experiments in many modern
materials demand the existence of other types of metallic states. I will
review the current field-theoretic understanding of such non-Fermi liquid
metals. In two spatial dimensions, all present theories of non-Fermi
liquid metals lead to strong-coupling problems for which no reliable
solutions are available. In recent years, ideas from gauge-gravity
duality have given a new perspective on these exotic metals, and offer
hope of reliable solutions in certain limits: these developments will
be reviewed using some simple models.

3 & 4. The onset of antiferromagnetism in metals: from the cuprate
superconductors to the heavy fermion compounds.

I will discuss the theory of the onset of antiferromagnetism in metals,
with concomitant Fermi surface reconstruction.  The non-Fermi liquid
metal obtained near the critical point is described by a field theory
which is strongly coupled in two spatial dimensions. I will describe
the onset of unconventional superconductivity near this critical point:
it involves a subtle interplay between the breakdown of fermionic
quasiparticle excitations on the Fermi surface, and the strong pairing
glue provided by the antiferromagnetic fluctuations. The net result is
a logarithm-squared enhancement of the pairing vertex for generic Fermi
surfaces, with a universal dimensionless co-efficient independent of the
strength of interactions: this is expected to lead to high temperature
superconductivity at the scale of the Fermi energy. I will also discuss
the possibility that the antiferromagnetic critical point can be replaced
by an intermediate `fractionalized Fermi liquid' phase, in which there is
Fermi surface reconstruction but no long-range antiferromagnetic order.
Connections will often be made to experiments on the underdoped cuprates
and the heavy-fermion materials.