Mike Birse
U. Manchester

Yannick Meurice
U. Iowa

Shan-Wen Tsai
U.C. Riverside

Program Coordinator:
Inge Dolan
(206) 685-4286

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New applications of the renormalization group
method in nuclear, particle and condensed matter physics (INT-10-45W)

February 22 - 26, 2010


M.C. Banuls
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 renormalization procedures.

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 algorithms.

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.

Jan Pawlowski
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.

Daniel Litim
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.

Robert Perry
Title: The Similarity Renormalization Group with Applications to Two-Nucleon Observables

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 observables.

Michael Scherer
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.

T. DeGrand
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.

Hiromichi Nishimura
Title: Polyakov-Nambu-Jona Lasinio (PNJL) models in finite-temperature gauge theory

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.

Michael Ogilvie
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.

Anna Hasenfratz
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.

Jean-Paul Blaizot
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.

Richard Furnstahl
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 matter.

Christoph Husemann
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 electron filling.

Kai Schwenzer
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.

Hidenori Sonoda
Title: ERG differential equation for the Wilson-Fisher fixed point and its perturbative solution

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.

Sergej Moroz
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.

Nigel Goldenfeld
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.

R. Shankar
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.

Eddy Timmermans
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 internal 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.

Haiyuan Zou
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.

Shan-Wen Tsai
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.

Yannick Meurice
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.

Yuzhi 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.

David Kaplan
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.

Achim Schwenk
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.