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

The ideas of the renormalization group (RG) and scale invariance have played central roles in physics over the last four decades. They are associated with the emergence of key concepts such as universality, self- similarity, scaling and asymptotic freedom. In addition, RG methods have allowed the numerical treatment of difficult few-body and many-body problems.

The RG provides a universal language spoken by scientists working in very different areas. Despite this universality, practical applications have often been developed independently in these areas, often without much communication among them. Generating closer interaction among workers in the many diverse fields where RG methods are applied is a desirable if long-overdue endeavor. Recent developments in the application of these methods to various areas of physics makes this particularly timely. The overall goal of the workshop is to bring together RG practitioners who are working in nuclear, particle, atomic and condensed matter physics. We intend the workshop to enable the communication of new exciting results across traditional boundaries, to stimulate exchange of theoretical techniques, and to generate new types of collaborative effort. Pedagogical introductions to technical aspects in specific areas will be provided. We plan to organize the workshop around five themes of common interest for all the disciplines represented.

1. Opening and closing conformal windows

Scale invariance and conformal symmetry appear at fixed points of the RG. Conformal windows open or close when certain parameters are varied, and in some cases only a discrete subgroup of the conformal symmetry survives. New realizations of conformal symmetry have been found in various disciplines which are worth discussing in a common place. Nonrelativistic fermions in the "unitary limit" are important both in nuclear physics and in atomic physics (Feshbach resonances). In few body problems, conformal symmetry can be anomalously broken and the RG flow tends to a limit cycle, where only a discrete remnant of conformal symmetry survives (Efimov effect). In lattice gauge theory, recent simulations have provided indications of new IR fixed points and conformal windows in QCD-like models with more flavors or higher order representations. The continuum physics seems quite different from QCD: conformal symmetry with absence of confinement and chiral symmetry breaking and the term "unparticle physics" has been used to describe the need for new observables. Graphs of RG flows or running of couplings look quite similar to those encountered in Hubbard models with competing next-to-nearest-neighbor interactions or Bose-Hubbard models, calculated with functional RG methods.

2. Effective interactions

Short-range interactions between nucleons can be described by an expansion around a nontrivial fixed point, describing the unitary limit. A consistent scheme has been demonstrated for renormalizing short-range interactions. Similar methods are also being applied to resonances in two-body scattering. An alternative to the Wilsonian RG is the similarity renormalization group, in which a unitary transformation is applied to a Hamiltonian to suppress its off-diagonal elements, so that it flows towards a diagonal form. There is interesting RG work to study Coulomb interactions in graphene, where the low-electronic excitations are well described as massless chiral Dirac fermions. Close to the Dirac point, the density of states vanishes linearly with energy, and finite-range interactions are irrelevant. The Coulomb interaction, however, is marginal, and logarithmically renormalizes the Fermi velocity until it becomes equal to the velocity of light at the fixed point. In recent years, the questions of chiral symmetry breaking and deconfinement, which are central in lattice gauge theory, have become increasingly important in condensed matter. Recent studies of chiral symmetry breaking in graphene have been done using lattice gauge theory methods. Other interesting developments have occured in condensed matter and cold atom physics. Starting with a given microscopic model RG approaches can provide effective models at low temperatures or energy scales. The energy cut-off of the problem is reduced via mode-elimination, and the RG expansion relies on a large-N expansion, with N being inversely proportional to the cut-off energy. This was shown to also rely on a 1/N expansion by making use of a double index notation, as introduced by t'Hooft in the context of gauge theory.

3. Global aspects of RG flows

The linearized behavior of RG flows near a fixed point has universal features that can be expressed in terms of critical exponents. However, calculating the RG flows between fixed points is usually a difficult nonlinear problem. We have mentioned before that similar global properties of RG flows are observed in condensed matter and lattice gauge models, while very different numerical methods are used to calculate the flows. A topic of common interest is the possibility of improving approximate numerical method such as the Migdal-Kadanoff approximation or local potential approximations. QCD confinement can be stated as the smoothness of RG flows between two fixed points. The already mentioned limit-cycles and their stability under perturbation provide other interesting examples of global behavior.

4. New types of boundaries in phase diagrams

Many interesting models have intricate phase diagrams. Popular examples are: temperature versus ratio of couplings in Hubbard models with competing interactions, temperature versus chemical potential in lattice QCD at finite temperature and density, and disorder parameter versus interaction strength in Bose-Hubbard models with on-site disorder. These phase diagrams have common features. The locations of the phase boundaries and the nature of the transitions are the subject of animated discussions in several communities and the time seems ripe to bring these communities together. The RG method is a common theoretical framework used to map phase diagrams, however, practical implementations can be very different in these communities. In ultra-cold gases of trapped atoms, Feshbach resonances can be used to tune the scattering lengths to very large values. Functional RG methods can be used to study the phase diagram of matter consisting of these particles and, for example, the crossover from BCS pairing to Bose-Einstein condensation of bound molecules. In quantum antiferromagnets, the concept of deconfined criticality has been introduced. The new type of critical theory is not expressed in terms of the order parameters of either state (as in Landau-Ginzburg-Wilson theory), but involves fractionalized degrees of freedom and an emergent topological conservation law.

5. RG and quantum information

There has been important progress in Density Matrix RG inspired numerical methods in condensed matter over the last years, as indicated by a number of recent workshops and conferences. A special focus has been put on the link between quantum many-body simulations and quantum information (mainly entanglement theory). A class of methods (Vidal's MERA - multiscale entanglement renormalization) can be directly related to conformal field theory. RG flows are usually understood in the space of cut-off hamiltonians, however, decimation-based methods such as White's density-matrix renormalization group (DMRG, today understood primarily as a variational method), can be constructed as an RG flow in the space of reduced density operators. More generally, the concepts introduced in quantum information provide new ways to look at the interplay between a subsystem and its environment that has been recently exploited in the RG context. It is desirable to disseminate these new developments broadly in the RG community.

There is a mandatory $90 registration fee for this workshop which will include expenses for catering and a workshop dinner.