Title: Tensor Network States: ground states and time evolution
The term Tensor Network States is used to describe a number of
families, representing different ansatzes for the efficient
description of the state of a quantum many-body system. The first of
these families, Matrix Product States (MPS), which lie at the basis of
Density Matrix Renormalization Group methods, have become the most
precise method for the study of one dimensional quantum many-body
systems. Their natural generalization to two or higher dimensions, the
Projected Entanglement Pair States (PEPS) are good candidates to
describe the physics of higher dimensional lattices. Other families,
like Tree Tensor States can also be understood in terms of
Quantum information gives us some tools to understand why these
families are expected to be good ansatzes for the physically relevant
states, and some of the limitations connected to the simulation
In this talk I will introduce some of these families, describe the
existing algorithms and their limitations and finally focus on recent
developments on their capability to simulate time evolution.
Philippe de Forcrand
Title: Nuclear physics from lattice QCD at strong coupling
The ab initio determination of nuclear matter properties, starting from the QCD
Lagrangian via lattice simulations, remains a distant goal. I show that this goal can be reached in a limiting case,
where the bare gauge coupling is infinite. The full phase diagram as
a function of temperature and chemical potential can be determined.
The nuclear potential can be obtained, and its features understood.
The masses of nuclei of various atomic numbers and shapes can be
measured and described in simple terms.
Title: The QCD phase diagram with RG methods
I will review the progress made in our understanding of the QCD phase
diagram within an RG approach to QCD and effective QCD models.
In particular this includes a discussion of the confinement-deconfinement
phase transition/cross-over, the chiral phase transition/cross-over,
as well as their interrelation.
Title: Quantum gravity and the renormalization group
It has been suggested long ago that gravity may well exist as a local
quantum field theory in the metric field, provided that the relevant gravitational
couplings achieve a non-trivial ultraviolet fixed point under the renormalization group.
This scenario known as 'asymptotic safety' has by now received strong support based on
numerous RG studies in the continuum and Monte Carlo simulations on the lattice. I review
the basic renormalisation group set-up for gravity, and highlight some of the recent results.
Title: The Similarity Renormalization Group with Applications to Two-Nucleon
Unitary transformations can be used to decouple low-energy and high-energy
degrees of freedom in the Hamiltonian, leading to a resolution-dependent
"renormalization" of all observables. This decoupling allows us to use a
small number of hadronic degrees of freedom (e.g., nucleons only or
nucleons plus pions) to perform microscopic, precision nuclear many-body
calculations for low-energy structure and reactions. I will focus on the
nucleon-nucleon interaction, which is drastically softened by SRG
transformations, and on the evolution of other one- and two-body
Title: Towards a quantitative FRG approach for the BCS-BEC crossover
I present our investigations on the phase transition and the BCS-BEC crossover for an ultracold gas of fermionic atoms within a functional renormalization group approach. The phase diagram is discussed as a function of the scattering length and the temperature and I compute the gap and the critical temperature for the phase transition to superfluidity. The approach allows for a description of the few-body physics and the many-body physics within the same formalism. Aiming at a quantitative description the truncation scheme is subsequently extended to include the effects of particle-hole fluctuations, the fermionic self-energy and higher-order interaction vertices. Our results are in agreement with BCS theory including the Gorkov correction for small negative scattering lengths and with an interacting Bose gas for small positive scattering lengths. At the unitarity point our result for the gap agrees with QMC simulations, while the critical temperature differs.
Title: Introduction to recent RG studies for QCD and beyond
This talk will (1) introduce lattice QCD and its connection with
RG to researchers who do not do elementary particle physics (2) describe
the most heavily used RG method in recent lattice simulations -- the Schrodinger functional and
(3) give a brief overview of recent lattice work on theories of beyond-Standard Model physics.
Title: Polyakov-Nambu-Jona Lasinio (PNJL) models in finite-temperature
Two important phase transitions in finite-temperature quantum
chromodynamics (QCD) are the deconfinement transition and the chiral
transition. They are in the universality class of Z(3) and O(4),
respectively, in two different limits of the quark mass. The extension of
the Nambu-Jona Lasinio model with the Polyakov loop (PNJL) is an effective
theory for finite-temperature gauge theory, which successfully
incorporates the order parameters of the two phase transitions, and
reproduces much of the known physics of deconfinement and chiral symmetry
restoration. With various representations and boundary conditions of the
fundamental fields in PNJL models, an interplay between the order
parameters gives rise a rich phase structure.
Title: Quark Confinement, Lost and Regained
One of the important problems of modern strong-interaction physics is determining the phase diagram of SU(N) gauge theories. Recent progress has provided us with an unexpectedly rich phase structure that gives us new insight into the nature of quark confinement. Our understanding of the confinement-deconfinement transition at finite temperature is based on the Polyakov loop as the order parameter for confinement, global Z(N) symmetry, and Svetitisky-Yaffe universality. Recent results from both theory and lattice simulations have demonstrated the existence of a class of models where confinement is unexpectedly regained when one or more spatial dimensions is small. Both perturbative and non-perturbative contributions to the Polyakov loop effective potential can be calculated. The deconfined and confined phases are separated by novel new phases including partially confined phases. These results also lead to new perspectives on many interesting topics, such as the construction of the large-N limit in lattice gauge theories and the nature of the conformal window.
Conformal or Walking? Monte Carlo Renormalization Group studies of technicolor-inspired models
Models with many fermions and/or fermions in higher representations can be candidates for extended technicolor or unparticle theories. The phenomenologically most interesting models show "walking" or develop an infrared fixed point in strong gauge coupling. Lattice methods can be used to explore the phase diagram of these models, and Monte Carlo renormalization group (MCRG) methods are especially effective in identifying an infrared fixed point and measuring anomalous dimensions (critical indexes). In this talk I discuss the MCRG method as applied to SU(3) gauge models with many fermion flavors and discuss some new results in the 8, 12 and 16 flavor models.
Title: Solutions of renormalization group flow equations with full momentum dependence
We demonstrate the power of a recently-proposed approximation scheme for
the non-perturbative renormalization group that gives access to correlation
functions over their full momentum range.
We solve numerically the leading-order flow equations obtained within
this scheme, and compute the two-point functions of the $O(N)$ theories at
criticality, in two and three dimensions.
Excellent results are obtained for both universal and non-universal
quantities at modest numerical cost.
Title: Similarity RG and Many-Body Forces
I will focus on applications to nuclear physics. The talk will include
both explicit treatments of three-body (and higher-body) interactions
and in-medium SRG (based on normal-ordering games that allow
truncation at density-dependent two-body interactions).
Applications are to few-body nuclei as well as infinite nuclear
Title: Channel-decomposed one-loop RG for the 2D Hubbard model
By using the functional renormalization group we compute detailed momentum
dependencies of the scale-dependent interaction vertex of the 2D
(t,t')-Hubbard model. Compared to previous studies we improve accuracy and
clearness by separating dominant parts from a remainder term. The former
explicitly describe the interaction of fermion bilinears, such as Cooper
pairs or spin operators. Applying the method to the repulsive Hubbard
model we find antiferromagnetism, d-wave superconductivity, and
ferromagnetism in dependence on next-to-nearest neighbor hopping t' and
Title: The infrared fixed points of QCD and their physical implications
The infrared fixed points of local Green functions encode interesting long-range aspects of a theory.
In particular in the case of QCD they should reflect strong interactions and confinement. I discuss
the qualitative aspects of the possible fixed points without explicitly solving the renormalization
group flow. It turns out that the infrared scaling laws of different fixed points of Landau gauge
QCD provide a mechanism for both the long-range interaction between static colors sources in
quenched QCD and its screening in the dynamical theory.
Title: ERG differential equation for the Wilson-Fisher fixed point and its
I formulate a flow equation for the Wilson action of a real scalar field
theory in 3 dimensions. I then use the equation to construct the flow
between the gaussian fixed point and the Wilson-Fisher fixed point. Up
to this point the discussion is formal but exact. Finally, I introduce
perturbative expansions to solve the flow equation and obtain the
critical exponents. Better approximations to the flow equations are wanted.
Title: Nonrelativistic inverse square potential, scale anomaly, and complex extension
I will review the properties of an inverse square potential problem in quantum mechanics and
provide different physical applications (Efimov effect, BKT transition, black holes physics,
AdS/CFT). I will present our functional renormalization group study of this system and will
argue that an extension to the complex values of the emergent contact coupling is natural.
It provides a deeper mathematical understanding and can be physically motivated. In addition,
a geometric description in terms of RG flows on the Riemann sphere will be introduced.
Finally, a close connection between RG analysis of this simple quantum mechanical system and large-flavor QCD will be discussed.
Title: The Renormalization Group Far From Equilibrium: Singular Perturbations, Pattern Formation and Hydrodynamics
Renormalization and the renormalization group (RG) were originally developed by physicists
attempting to understand the divergent terms in perturbation theory and the short distance
behaviour of quantum electrodynamics. During the last twenty years, these methods have been
used to unify the construction of global approximations to ordinary and partial differential
equations. Early examples included similarity solutions and travelling waves, which exhibit
the same anomalous scaling properties found in quantum field theories, but here manifested
in such problems as flow in porous media, the propagation of turbulence and the spread of
advantageous genes. Fifteen years ago, these methods were extended to asymptotic problems
with no special power-law scaling structure, enabling a vast generalization that includes
and unifies all known singular perturbation theory methods, but with greater accuracy and
calculational efficiency. Applications range from cosmology to viscous hydrodynamics.
This talk will provide an introduction and overview to these developments, trying where possible to
point out connections to recent developments that are the focus of the workshop.
Maria A. H. Vozmediano
Title: Coulomb interactions and disorder in undoped graphene
The recent synthesis of a single layer of graphite (graphene) and the experimental observation of some
unusual electronic and structural properties has prompted a real revolution in the theory of condensed
matter systems. Under a theoretical point of view the fact, confirmed by the experiments, that the low
energy elementary excitations are well described by massless Dirac fermions implies a revision of the Landau
fermi liquid paradigm. In this talk I will give a general overview of the graphene physics at the neutrality
point and address the problem of the Coulomb interactions and disorder. A renormalization group analysis allows
to classify graphene as a strange Fermi liquid described by a Lorentz covariant infrared fixed point whose
effective coupling constant is the fine structure constant.
Title: Renormalization group for nonrelativistic fermions
The general idea of Kadanoff and Wilson of eliminating high energy modes takes a new twist
in this case because the low energy modes are not near the origin in momentum space, but near the Fermi surface.
Consequently the fixed point is defined by coupling functions and their functional flows. Landau's Fermi liquid
theory is a byproduct as are all the usual instabilities of the Fermi liquid. A leisurely introduction to these
topics will be given in this blackboard talk.
Title: Pseudospins and effective spin-spin interactions in cold atom physics
By Raman coupling two selected hyperfine states of alkali atoms
in an external magnetic field cold atom experiments simulate
magnetic-like spin 1/2 dynamics in a controlled quantum many-body
environment of unusual accessibility. We describe the effective spin
(or pseudospin) degrees of freedom in accordance with choosing
one hyperfine state as 'spin-up', the other as 'spin-down' state
for systems of indistinguishable fermionic atoms, systems of
bosonic atoms and/or mixtures of distinguishable neutral atoms
with spinor Bose-Einstein condensates (BEC's) and or fermions.
We derive the spin-spin interactions which for distinguishable
atoms are anisotropic. However, the combination of short-range
neutral atom interactions and indistinguishability leads to
short-range ising-like spin-spin interactions. In the degenerate
state approximation we show how a magnetically controlled Feshbach
resonance can tune the ratio and relative sign of spin-independent
to spin-dependent interactions. We discuss implications for
cold atom simulations of spin domain formation in spin 1/2
Bose-Einstein condensates, coupling neutral atom optical lattice
spins via mediated spin-spin interactions, hybrid spin-charge vortices
and macroscopic quantum tunneling.
Title: Fisher zeros, singularities of the gap equation and zeros of the beta
function for nonlinear O(N) sigma models at finite volume
Motivated by attempts to solve related problems in 4-dimensional gauge
theories, we study the map between the mass gap M2 and the 't Hooft
coupling for nonlinear O(N) sigma models on finite
lattices of linear size L and in the large-N limit. We show that the
map requires a Riemann surface with q+1 sheets and 2q cuts in the
't Hooft coupling plane, where q is an integer of order LD. We provide
a close form
expression for the partition function in the approximation where the
non-zero modes of the Lagrange multiplier are neglected and use this
form to calculate the Fisher's zero at finite L and N.
We show empirically that these zeros appear on "strings" coming from
the origin in the 't Hooft coupling plane and ending approximately at
the singular points of the map discussed above. The density of zeros on
these strings scales like 1/L2 and 1/N. The singular points correspond
to the complex zeros of the beta function. We discuss recent progress
related to the connection between discrete and continuous scaling.
Title: Effects of retardation in the renormalization group for fermions
When fermion-fermion interactions are frequency-dependent, there are
important retardation effects in the RG flows of vertices and
self-energies. I will discuss these effects and show how
Eliashberg-like theories are obtained from the flow equations. I will
also discuss the role of retardation effects when there is competition
between two or more instability channels, such as density-wave phases
and BCS pairing.
Title: Complex zeros of the beta function, confinement and discrete scaling
We discuss the relation between the zeros of the partition function (Fisher's zeros) and the zeros of the beta function in the complex coupling plane, for lattice gauge theories and O(N) sigma models.
Confinement corresponds to situations where the Fisher's zeros stay away
from the real axis in the infinite volume limit.
We discuss the relationship between continuous and discrete scaling and in particular, limit cycles (log-periodic corrections to scaling) in models with discrete RG transformations. We present evidence for numerical instabilities in recent attempts to
interpolate between discrete RG steps. We discuss the possible (ir)relevance of
these phenomena for MCRG.
Some questions will be discussed
in more detail by Haiyuan Zou and Yuzhi "Louis" Liu.
Title: Numerical instabilities associated with block spinning non-integer numbers of sites
We propose an extension of the recursion formula of Dyson's hierarchical model where the number of sites blocked becomes an arbitrary number bD instead of 2 in the original formula. We show that when bD is an integer, the polynomial approximations developed for bD=2 remain valid. The value of the critical exponent γ depends slightly on b and changes by 0.0012 between bD=2 and bD=8. When bD is not integer, the polynomial approximation breaks down at a degree lmax which decreases with bD. We explain this instability by considering bD=2 + ζ and expanding at first order in ζ.
Ryan M. Kalas
Title: Narrow Feshbach resonances and finite-range interactions in cold atom gases
One of the flexible aspects of cold atom gases is that the strength of the effective inter-atomic interactions can be tuned, from weak to strong, attractive to repulsive, by utilizing a feature known as a Feshbach resonance. After reviewing how a magnetic Feshbach resonance works, I will go on to describe narrow Feshbach resonances. Narrow resonances are harder to deal with experimentally because one has to control an external magnetic field more precisely, but they present interesting new opportunities. In contrast to standard broad resonances, where the effective inter-particle interactions can be taken to be zero-range, the strong energy dependence of a narrow resonance leads to finite-range interactions. I will describe a separable potential model of a resonance that allows us to extract the salient features of a narrow resonance. I will conclude with a simple example that shows how the two-body narrow resonance physics can lead to new many-body effects.
Title: Conformality Lost
Fixed point annihilation is shown to be a generic mechanism for the transition between conformal and nonconformal phases of a theory. A general feature of this mechanism is an "infinite order" phase transition, as seen in the Berezinski-Kosterlitz-Thouless transition. We suggest that the transition in large N QCD as a function of the number of quark flavors is of this class, with implications that the theory should have a nontrivial unstable fixed point.
Title: Renormalization group for nuclear forces and nuclear structure
I will discuss RG approaches to nuclear forces, open problems and applications of renormalization-group evolved interactions to neutron-rich nuclei and neutron stars. The talk will also present first results for light and medium-mass nuclei based on normal-ordered flow equations, suggesting that the accuracy is comparable to coupled-cluster calculations.