Prof. Ioannis Papadimitriou
Holographic RG and applications
Holographic techniques provide a valuable
new tool for studying strongly coupled quantum systems, both in high energy and condensed matter
physics. The utility of such techniques stems from the fact that strongly coupled quantum
systems are related through the holographic duality to weakly coupled, classical, gravitational
theories in a higher dimensional space. Central to this connection is the Renormalization Group, which
becomes geometric in the dual gravity theory, with the extra dimension playing the role of
energy scale. In these lectures we will develop techniques for computing observables in strongly coupled
quantum field
theories from the dual holographic model. A number of concrete examples will be
discussed at the end of the lecture series.
Lecture
1:
Renormalization Group and global symmetries in QFT
1. The Local Renormalization Group as a
Hamiltonian flow
2. Global symmetries and Ward identities
3. UV divergences and renormalization of
composite operators
Lecture
2: AdS/CFT
correspondence and the holographic dictionary
1. AdS/CFT primer
2. A first
look at holographic renormalization
3. The holographic dictionary in
Hamiltonian language
Lecture
3: Radial Hamiltonian formulation of gravity theories
1.
Radial ADM formalism:
(a) Metric
(b) Scalars
(c) Vector fields
(d) p-forms
(e) Fermions
2.
Hamilton-Jacobi theory
Lecture
4:
Recursive solution of the Hamilton-Jacobi equation
1. The induced metric expansion
2. Dilatation operator expansion
3. Renormalized one-point functions and
Ward identities
4. Fefferman-Graham asymptotic expansions
Lecture
5: RG flows
and
holographic correlation functions
1. Poincare domain walls from a fake superpotential
2. Heating up the flow:
first order equations for planar black holes
3. Riccati equations and holographic two-point
functions
Lecture
6: Condensed matter applications
1. Two-point functions in a holographic
Kondo model
2. Lifshitz QFTs and hyperscaling violation
3. Holographic Hall conductivities
Prof. Ferenc Igloi
Strong
Disorder Renormalization Group Method
Disorder is an inevitable feature of real materials, even
extremely small amount of quenched disorder can modify the physical properties
of systems, in particular close to a critical point. There is a class of
models, in which the quenched disorder plays a dominant role over deterministic
(thermal, quantum or stochastic) fluctuations and the critical behavior is
governed by a so called infinite disorder fixed point (IDFP). The properties of an IDFP is conveniently
studied by the strong disorder renormalization group (SDRG) method. During the
SDRG procedure the degrees of freedom of the system with the smallest
characteristic time-scales are successively
eliminated and the scaling behavior of the renormalized model at the
fixed point is often asymptotically exact.
In our talk we shall concentrate on random quantum systems and
discuss their unusual critical properties, as well as their dynamical
singularities in the so called Griffiths region. We concentrate on one-dimensional
problems, for which many analytical results have been derived by the SDRG
approach. For higher dimensional problems we outline the numerical
implementation of the SDRG method. We explain the use of the SDRG method to
calculate the entanglement entropy in random quantum systems, speak about the
phenomena of many-body localization and its treatment trough a variant of the
SDRG method. We mention also the use of the SDRG for classical problems (random walk
in a random environment, reaction-diffusion problems with quenched disorder,
etc.).
Lecture 1.a
Quenched disorder and critical behavior
Organization of the course
Thermal and quantum phase transitions – scaling theory
The effect of quenched disorder – relevance-irrelevance
criteria
Infinite disorder critical behavior
Dynamical critical behavior and Griffiths singularities
Lecture 1.b Basic
ideas of strong disorder RG
Principles of the strong disorder RG
Classical example: 1d random walk in a random environment
Drift velocity,
persistence and order
RG rules
Quantum example: random transverse field Ising chain
Free-fermion representation
RG rules
Stochastic example: random contact process
Relations between the models
Lecture 2. Analytical solution of the SDRG equations for 1d random quantum
chains
Random transverse field Ising chain
Renormalization
of the distribution functions
Fixed
point solution
Scaling
of thermodynamic quantities
Random S=1/2 AF Heisenberg chains
RG
rules
Fixed-point solution
Random-singlet phase
Lecture 3.a Other 1d quantum models
Random quantum chains with discrete symmetry
Higher spin AF Heisenberg chains
Disorder induced cross-over effects
Heisenberg chain with random ferro- and antiferromagnetic
couplings
Disordered spin ladders
Random quantum chains with dissipation
Lecture 3.b Disordered boson systems, the superfluid-insulator
transition
Weak disorder limit
RG rules for strong disorder
Flow equations and phase diagram
The superfluid-insulator
transition
Numerical tests of the
strong disorder transition
Lecture 4. Random quantum systems in d›1 dimensions
Random transverse-field Ising model
Basic
elements of the numerical SDRG algorithm
Scaling
at the critical point
Disordered and ordered phases
Random transverse-field Ising model with long-range interactions
Random Heisenberg models
Lecture 5. Entanglement entropy of random quantum
systems
Basic notations of entanglement and the von Neumann entropy
Entanglement entropy through strong disorder RG
Analytical calculations for 1d random quantum spin chains
Numerical study of the random transverse-field Ising model in d›1
Dynamical entanglement entropy
Lecture 6. Many-body localization transition
Single-particle (Anderson) localization
Many-body localization
Phenomenology of many-body
localized systems
Strong-disorder RG study of the
1d problem
Prof. Chanyong Park
Holographic renormalization and entanglement entropy
Recently, the AdS/CFT
correspondence has been widely used in order to understand universal features of strongly interacting systems. In this
lecture, I will discuss how to extract various information of a strongly interacting system by using
the holographic technique. After briefly
reviewing the holographic renormalization and entanglement entropy,
I will discuss their relation and RG flow.
Lecture
1. Holographic renormalization
- Brief review on the AdS/CFT
correspondence
- Renormalization schemes of the dual
gravity
Lecture
2. Holographic entanglement entropy
- Central charge of the dual CFT
- a- and F-theorem
Prof. Sung-Sik Lee
Quantum
Renormalization
Group and Holography
The anti-de
Sitter/conformal field theory (AdS/CFT)
correspondence was originally conjectured
in the string theory. However, the holographic principle that relates field
theories to gravitational theories in one higher dimensional spaces is believed
to be valid beyond the particular approach to quantum gravity. The general
connection between two seemingly different theories is made through
renormalization group (RG), where the extra dimension in the dual gravitational
theory plays the role of scale in RG. However, a precise holographic mapping
can not be made through the conventional RG. This is because the scale
dependent couplings in the Wilsonian RG is fully deterministic, while the bulk
degrees of freedom in the dual gravitational theories are quantum mechanical.
In this lecture, an alternative formulation of RG
will be introduced, which fills the gap.
Quantum renormalization
group (RG) allows one to identify holographic duals for general quantum field
theories. In this approach, RG flow is projected onto a subspace of operators
while the coupling constants for those operators are promoted to quantum
mechanical variables. The partition function is given by the sum over all
possible RG paths, and the weight for each path is determined by a dynamical
action which includes gravity in one higher dimensional spacetime. An
application of quantum RG to a concrete lattice model will be presented, where
different phases exhibit geometries with different degrees of locality in the
bulk.