Exact Renormalization Group 2008

# Talks

#### Jürgen Berges

Non-thermal fixed points

Strongly correlated systems far from equilibrium can exhibit scaling solutions with a dynamically generated weak coupling. We show this by investigating isolated systems described by relativistic quantum field theories for initial conditions leading to nonequilibrium instabilities, such as parametric resonance or spinodal decomposition. The non-thermal fixed points prevent fast thermalization if classical-statistical fluctuations dominate over quantum fluctuations. We comment on the possible significance of these results for the heating of the early universe after inflation and the question of fast thermalization in heavy-ion collision experiments.

#### Claude Bervillier

Analytical approximation schemes for solving exact renormalization group equations

Abstract : We present and compare four new and efficient analytical approximation schemes to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. We consider, for the scalar field, the local potential approximations of the Wegner-Houghton equation in the dimension d=3 and of the Wilson-Polchinski equation for some values of $d\in \left[ 2,3\right]$. We also consider, for d=3, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method.

#### Jens Braun

Towards Bridging the Gap Between Quarks and Gluons and Baryonic Degrees of Freedom

I give a review of some of the current challenging problems of QCD. I discuss the chiral phase transition in QCD with its underlying mechanisms in terms of quarks and gluons and present results for the chiral phase boundary in the plane of temperature and number of (massless) quark flavors obtained from a functional renormalization group approach. Moreover, the dependence of the phase transition temperature on the quark chemical potential is discussed. The last part of the talk deals with the deconfinement phase transition in Yang-Mills theory. The order-parameter potential for SU(3) Yang-Mills theory, namely the Polyakov-loop potential, obtained from a functional renormalization group study is shown.

#### Leonie Canet

Strong-coupling regime of the Kardar-Parisi-Zhang equation

The celebrated Kardar-Parisi-Zhang equation, initially derived as a model to describe the kinetic roughening of a growing interface, has given rise to intense theoretical investigations for over two decades since it stands as a simple - yet unsolved - model for scaling phenomena and non-equilibrium phase transitions.

The problem is that the scaling rough phase of the KPZ equation corresponds to a strong-coupling fixed point and has remained out of reach of perturbative methods so far. We show that a non-perturbative renormalisation group approach provides a controlled analytical tool to describe this fixed-point and investigate the statistical properties of the rough interface profile.

#### Sebastian Diehl

Towards precision in the BCS-BEC crossover in ultracold fermion systems

The Functional Renormalization Group (FRG) is used for a study of the crossover problem. A unified picture for the whole phase diagram is obtained. Various effects beyond Mean field theory are included. The fluctuations of an effective, dynamically generated boson field are found to be crucial: Bosonic vacuum fluctuations contribute to the ratio of molecular to fermionic scattering length, while thermal fluctuations are necessary to establish the expected second order nature of the phase transition throughout the crossover. The FRG approach further enables us to reconstruct the effects of particle hole fluctuations, which impact e.g. on the critical temperature.

#### Nicholas Dupuis

Non-perturbative renormalization group approach to superfluidity

We use the non-perturbative renormalization group to solve the problem of infrared divergences occurring in the perturbation theory of interacting boson systems. Our approach reveals the instability of the Bogoliubov fixed point when $d\leq 3$ and yields the exact infrared behavior in all dimensions $d>1$. For $d\leq 3$, the fixed point is characterized by an SO(d+1) space-time symmetry. In one-dimension and for not too strong interactions, we obtain a good picture of the Luttinger-liquid behavior of the superfluid phase. We comment about the relevance of these results for the understanding of the Mott-superfluid transition in the Bose-Hubbard model.

#### Stefan Floerchinger

Functional renormalization for ultracold quantum gases

Using the functional renormalization group, we study the effect of thermal and quantum fluctuations in ultracold nonrelativistic quantum gases. For a single component Bose gas, we determine the phase diagram and calculate a branch of thermodynamic observables in both three and two spatial dimensions. We also investigate systems with Fermions using the methods of partial bosonization. For example, we investigate the phase diagram and especially the effect of particle-hole fluctuations in the BEC-BCS crossover of two Fermion species.

#### Thomas Gasenzer

Functional Renormalisation Group approach to Far-From-Equilibrium Quantum Field Dynamics

Dynamic equations for quantum fields far from equilibrium are derived by use of functional renormalisation group techniques. The obtained equations are non-perturbative and lead substantially beyond mean-field and quantum Boltzmann type approximations. The approach is based on a regularised version of the generating functional for correlation functions where times greater than a chosen cutoff time are suppressed. As a central result, a time evolution equation for the non-equilibrium effective action is derived, and the time-evolution of the Green functions is computed within a vertex expansion. It is shown that this agrees with the dynamics derived from the 1/N-expansion of the two-particle irreducible effective action.

#### Carsten Honerkamp

Functional renormalization group for interacting fermions in 2D - recent applications and improvements

The functional renormalization group for fermions has become a powerful tool for the analysis of many-fermion systems, in particular for cases with competing interactions. Here review some recent applications of the method in the 'standard approximation' and describe efforts underway to improve the method beyond this level.

#### Christoph Husemann

Competing Orders in the Hubbard Model at van Hove Filling

One-loop renormalization group techniques have been successful in determining weak coupling instabilities of the two-dimensional Hubbard model using an N-patch discretization of momentum space. We propose a more efficient parametrization of the four-point vertex function, which is based on decomposing the effective two-fermion interaction in interacting fermion bilinears.

#### Yuji Igarashi

Quantum Master Equation for Yang-Mills Theory in ERG

We discuss a general functional method for construction of the Quantum Master Equation in the Batalin-Vilkovisky antifield formalism for Yang-Mills theory in the exact renormalization group.

#### Etsuko Itou

The BV Master Equation for the Gauge Wilson Action

The Wilson effective action for general Yang-Mills gauge theory is shown to satisfy the usual form of Batalin-Vilkovisky (BV) master equation, despite that a momentum cutoff apparently breaks the gauge invariance. In the case of Abelian gauge theory, in particular, it actually deduces the Ward-Takahashi identity for Wilson action recently derived by Sonoda.

#### Pawel Jakubczyk

Renormalization group for phases with broken discrete symmetry near quantum critical points

We extend the Hertz-Millis theory of quantum phase transitions in itinerant electron systems to phases with broken discrete symmetry. Using a set of coupled flow equations derived within the functional renormalization group framework, we compute the second order phase transition line T_c(delta), with delta a non-thermal control parameter, near a quantum critical point. We analyze the interplay and relative importance of quantum and classical fluctuations at different energy scales, and we compare the Ginzburg temperature T_G to the transition temperature T_c, the latter being associated with a non-Gaussian fixed-point.

#### Christoph Karrasch

A finite-frequency functional RG approach to the single impurity Anderson model

We use the Matsubara functional renormalization group to calculate finite-energy properties of the single impurity Anderson model. To this end, we account for the frequency-dependence of the two-particle vertex. The FRG approximation is shown to work well for arbitrary parameters (particularly finite temperatures) provided that the electron-electron interaction $U$ is not too large. In contrast, it turns out that aspects of (large $U$) Kondo physics which are described well by a simpler frequency-independent truncation scheme are no longer captured by the higher-order' frequency-dependent approximation. We suggest to parametrize the two-particle vertex not by three independent energy variables but by introducing three functions each of a single frequency, considerably reducing the numerical effort to integrate the FRG flow equations.

#### Andrey Katanin

The two-loop functional renormalization group approach to the one- and two-dimensional Hubbard model

I consider the application of the two-loop functional renormalization-group (fRG) approach to study the low-dimensional Hubbard models. This approach accounts for both, the universal and non-universal contributions to the RG flow. While the universal contributions were studied previously within the field-theoretical RG for the one-dimensional Hubbard model with linearized electronic dispersion and the two-dimensional Hubbard model with flat Fermi surface, the non-universal contributions to the flow of the vertices and susceptibilities appear to be important at large momenta scales. The two-loop fRG approach is also applied to the two-dimensional Hubbard model with a curved Fermi surface and the van Hove singularities near the Fermi level. I show that the vertices and susceptibilities in the end of the flow of the two-loop approch are suppressed in comparison with both the one-loop approach, the quasiparticle weight remains finite in two dimensions at not too low temperatures above the paramagnetic ground state.

#### Stefan Kehrein

Flow equations and nonequilibrium quantum many-body physics

Quantum many-body systems in nonequilibrium situations, either due to initial preparation in a non-thermal state or in driven systems, have recently become of considerable interest because of experimental realizations in ultracold atomic gases and in electronic nanostructures. Since the flow equation method retains the full Hilbert space, it is very well suited for studying such highly excited quantum systems. In this talk I will present two applications of this approach to models of paradigmatic importance in condensed matter physics: The Kondo model with external voltage bias and an interaction quench in a Fermi liquid.

#### Bertram Klein

Critical scaling behavior in the O(N) model in infinite and finite volume

We use the functional RG to obtain universal scaling functions for O (N) models in three dimensions in the presence of an explicitly symmetry-breaking field. Our results are in good agreement with those of O(N) spin model simulations on the lattice. Applying the same RG technique to a finite-volume system, we are also able to determine the finite-size scaling functions for this universality class. These scaling functions are relevant for the analysis of the chiral phase transition behavior in lattice simulations of Quantum Chromodynamics, where both symmetry-breaking and finite-volume cutoff effects are important.

#### Peter Kopietz

Functional renormalization group approach to interacting fermions with partial bosonization: from weak to strong coupling

We show that for certain model systems the functional renormalization group approach to interacting Fermi systems with partial bosonization via suitable Hubbard Stratonovich fields developed by Schtz, Bartosch and Kopietz [Phys. Rev. B 72, 035107 (2005)] can be used to explore the strong coupling regime. In particular, use this method to show that strong interactions can give rise to a confinement transition in quasi one-dimensional metals, where in the strong coupling regime the curvature of the Fermi surface is completely smoothed out [Ledowski and Kopietz, Phys. Rev. B 76, 121403(R), 2007]. We also present preliminary results for the Anderson impurity model, where a non-perturbative treatment of the spin-flip scattering is crucial to remove the unphysical ferromagnetic instability encountered at the mean-field level and to describe the strong coupling Fermi liquid fixed point.

#### Hans Christian Krahl

A Functional Renormalization Group approach to the Hubbard model

Within the two-dimensional repulsive t-t'-Hubbard model, an attractive coupling in the d-wave pairing channel is induced by fluctuations of antiferromagnetic spin waves. We investigate this coupling using functional renormalization group equations. The momentum dependent d-wave coupling can be bosonized by the use of scale dependent field transformations. We propose an effective coarse grained model for the Hubbard model which is based on the exchange of antiferromagnetic and $d$-wave collective bosons.

#### Oliver Lauscher

Projected flow equations and competing ordering tendencies in the 2D Hubbard model

The competition of different ordering tendencies in the two-dimensional $t$-$t'$-Hubbard model is analyzed on the basis of the fermionic weak-coupling RG approach. We study the RG flow of the 4-point vertex in a truncated interaction space spanned by the on-site term, two spin-spin interaction terms, a d-wave SC term and a $d$-density wave term. By constructing appropriate projectors and applying them onto the RG equation for the 4-point vertex we derive the set of RG equations for the couplings parametrizing the truncated interaction subspace. These equations lend themselves to profound numerical studies. First results at zero temperature are already available.

#### Daniel Litim

Optimisation and the functional RG

I review optimisation ideas for flow equations from a conceptual and a practical point of view, and illustrate their relevance and applicability through a number of examples.

#### Volker Meden

Functional RG for transport through quantum dots

I discuss the application of approximation schemes which are based on the functional RG to study correlation effects in quantum dots. As a first example I describe transport through multi-level dots. Depending on the ratio between the typical single-paricle level spacing and the typical level broadening the transmission amplitude and the transmission phase show characteristic behavior. Secondly, I discuss the Josephson current through a single-level dot coupled to superconducting leads. The physics is goverend by a phase transition. Our approach allows for a detailed study of the dependence on all parameters. I emphasize that both studies are relevant in connection with recent and ongoing experiments. I give a brief outlook on the possible extensions of the approximate method (see also the talks by C. Karrasch and M. Pletyukhov).

#### Yannick Meurice

Linear and Nonlinear Aspects of Finite Size Scaling

Starting with the general theory of finite size scaling, we compare the size of linear and nonlinear effects for the estimation of Binder cumulants of various spin and gauge models. We propose a new strategy to resolve the nonuniversal corrections using improved actions in the limit of small lattice size. We discuss the applicability of the method for the hierarchical model, the 3D Ising model and 4D lattice gauge theory.

#### Dominique Mouhanna

Nonperturbative renormalization group approach to frustrated magnets

Frustrated magnets, i.e. magnetic systems with competing interactions, display a puzzling phenomenology: they exhibit scaling behaviors without universality. Moreover the perturbative approaches to these systems 1) fail to describe this specific critical behavior and 2) are in conflict. I show, in this talk, how a nonperturbative renormalization group approach allows both to correctly described the phenomenology of frustrated magnets and to get a coherent picture of the different theoretical approaches used. As a surprising consequence of our approach, it appears that a high-order - 6 loops - perturbative approach performed at fixed dimension leads to spurious predictions for these systems.

#### Sandor Nagy

Renormalization of the Sine-Gordon model

The critical behaviour of the correlation length of two-dimensional sine-Gordon model has been determined by the functional renormalization group method performed in the internal space including wave-function renormalization. The results provides the treatment of the Kosterlitz-Thouless-Berezinski type phase transition directly in the sine-Gordon model. It is also obtained that the role of the higher Fourier modes is not negligible.

#### Jan Martin Pawlowski

Strong correlations in gauge theories

In recent years much progress has been made in the understanding of the strongly correlated sector of QCD. Most notably we understand, to a large extend, the confinement mechanism and the related confinement scenario. Moreover we have access to quantitative computations of physical observables such as the order parameters of the confinement-deconfinement and chiral phase transition.

I review the present status of the functional RG approach to gauge theories. Results on the confinement mechanism and the confinement-deconfinement phase transition in QCD are summarised, including a comparison to other approaches, such as lattice gauge theories and Dyson-Schwinger equations. I close with a discussion of prospects for a first principle study of the QCD phase diagram and the remaining problems.

#### Roberto Percacci

Wilsonian investigations into the UV properties of gravity

I will review recent progress suggesting that gravity may be renormalizable at a nontrivial fixed point, along the lines of the "asymptotic safety" programme. Whenever possible, connection with other approaches will be made.

#### Massimo Pietroni

Dealing with non-linearities in Cosmology

To extend the program of precision cosmology to the study of the Large Scale Structure of the Universe it is mandatory to deal with non-linear fluctuations of the density and velocity fields. We will present two, RG-inspired, approaches to resum perturbative contributions to all orders. Results for the matter power spectrum will be presented and compared to the state of the art of other approaches.

#### Frank Reininghaus

Dephasing rates within nonequilibrium RG: A generic approach

We consider a generic model for a local quantum system coupled to reservoirs and present a general solution to the problem how relaxation and dephasing rates can be implemented within nonequilibrium renormalization group. Generalizing previous RG-methods to a specific frequency representation and using a cutoff on the imaginary frequency axis, we show that decay rates always cut off the RG flow and find the physical meaning of these rates. We illustrate the method for the nonequilibrium Kondo model in a finite magnetic field h and present results for the magnetic susceptibility and the differential conductance in the weak coupling regime, where the RG equations can be solved analytically in a controlled way by expanding in the exchange coupling J. We find that the conductance is enhanced at specific values V=h/n (n=1,2, ...) of the voltage.

#### Urko Reinosa

Renormalization and gauge symmetry of 2PI effective actions

In recent years, functional methods based on two-particle-irreducible effective actions have known a renewed interest in problems involving quantum fields in- and out-of-equilibrium. Still, the application of these techniques to quantum gauge fields suffers from difficulties related to symmetry and renormalization. I will discuss these problems and report on recent progress in tackling them.

#### Martin Reuter

Background independence and asymptotic safety in Quantum Einstein Gravity

We review some basic concepts of the asymptotic safety approach to quantum gravity and its implementation in terms of the effective average action, with an emphasis on its background independence. As any consistent theory of quantum gravity is supposed to explain rather than postulate spacetime, the requirement of background independence is the crucial difference between matter quantum field theories and gravity. By means of a simple example (conformally reduced gravity) we demonstrate that the background independent quantization of Einstein gravity leads to a RG flow which differs significantly from the one obtained on a rigid background. In particular background independence seems to be important for asymptotic safety.

#### Oliver Rosten

Invariants of the ERG

I will construct a functional, the dual action', which, formally at any rate, is an invariant (ie has vanishing flow) of Polchinski-like ERG equations. The relationship between the dual action and the Wilsonian effective action can be easily inverted. However, if the integrated ERG kernel is massless, this construction generically suffers from infra-red problems. I will discuss what this means, and under what circumstances the dual action might be a useful.

#### Bernd-Jochen Schaefer

Exploring the QCD phase diagram with Functional RG

The phase structure of various effective quark-meson models for two and three quark flavors are presented. The impact of the Polyakov loop dynamics on the phase diagrams is also briefly addressed. All models exhibit a critical endpoint where fluctuations become important. The influence of fluctuations around criticality is demonstrated by an comparison of the mean-field approximation with a renormalization group analysis.

#### Krishnendu Sengupta

Superfluid-Insuator transitions of bosons on a Kagome lattice at non-integer fillings

In this talk, I shall describe the quantum phase transition of bosons in a Kagome lattice at non-integer fillings and describe the possible Mott states of the systems. Our results are based on a dual description of the bosons in terms of vortices and are supplemented by Quantum Monte Carlo studies. These results are also relevant for XXZ models on Kagome lattice. We shall describe the critical theory for this transition and point out the possible role of NPRG in extracting relevant information from this critical theory.

#### Hidenori Sonoda

Simple recipe for realization of symmetry using ERG

The talk will consist of three parts. In part 1 I briefly describe how to modify the ERG differential equation to facilitate perturbative calculations of the critical exponents of the Wilson-Fisher fixed point. In part 2 I explain how to construct perturbatively renormalizable theories using ERG. I emphasize that the correlation functions calculated with a cutoff action is independent of the cutoff. In part 3 I describe a general framework to incorporate symmetry (global and local) using ERG.

#### Philipp Strack

Fermion-boson RG for the ground-state of fermionic superfluids

We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter, and fluctuations thereof, are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. The flow equations derived capture the influence of fluctuations on non-universal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. In the infrared regime, transverse order parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations in agreement with the exact behavior of an interacting Bose gas.

#### Haruhiko Terao

Conformal extension of the Higgs sector and the little hierarchy problem

After a brief review of the little hierarchy problem of the SM, we consider a new possibility in the Higgs sector above the TeV scale. There it is assumed that the Higgs scalar acquires a large anomalous dimension induced by a conformally invariant Yukawa coupling. We examine the infrared fixed point of gauge-Yukawa theories by means of the ERG equations in a rough aaproximation scheme and show that sufficiently large anomalous dimensions can be realized there. We also present a concrete phenomenological model, which may pass the EW precision tests.

Effective field theory with a variable ultraviolet cutoff

The properties of strongly gravitating systems suggest that field theory overcounts the states of a system. Reducing the number of degrees of freedom, without abandoning the notion of effective field theory, may be achieved through a connection between the ultraviolet and infrared cutoffs. I present an implementation of this idea within the Wilsonian approach to the renormalization group. An exact flow equation describes the evolution of the effective action. I discuss the implications for the existence of infrared fixed points and the running of couplings.

#### Shan-Wen Tsai

Effects of retardation in the functional renormalization group approach to interacting fermions

When fermion-fermion interactions are frequency dependent, there are important retardation effects in the functional RG flows of vertices and correlation functions. I will discuss these effects and show that they may change the phase diagram and the critical energy scales. I will also present results for the case of electron-phonon systems, and for mixtures of fermionic cold atoms in the presence of a Bose-Einstein condensate.

#### Michael Weyrauch

Beyond the static approximation in the fermionic renormalization group

I briefly review a variant of the fermionic fRG using an additive cutoff instead of the standard multiplicative cutoff for the free propagator. I apply this approach to Hubbard models of 1D rings and quantum dot systems. Comparisons are made with DMRG calculations.

#### Peter Wölfle

Transport of interacting electrons through a potential barrier: nonperturbative RG approach

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2, and subject to a potential barrier of arbitrary strength, as a function of temperature. We first map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the diagrams contributing to the renormalization group (RG) Beta-function are identified. A universal part of the Beta-function is given by a ladder series and summed to all orders in g2. First non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature dependent conductance is solved analytically. Our result agrees with know limiting cases. [D.N. Aristov, P. Wölfle, Europhys. Lett. 82, 27001 (2008)]

#### Nicolas Wschebor

Precise NPRG calculation of critical exponents of the O(N) model

A manageable approximation, formulated within the NPRG, that goes beyond derivative expansion and field truncations schemes, will be presented. It will be shown that it makes possible to calculate in a standard personal computer the critical exponents for the O(N) with at least the same accuracy that best field theoretical methods.

# Posters

#### Michael Scherer

Particle-hole fluctuations in the BEC-BCS crossover

The effect of particle-hole fluctuations for the BEC-BCS crossover is investigated by use of functional renormalization. We compute the critical temperature for the whole range in the scattering length a. On the BCS side for small negative a we recover the Gorkov approximation, while on the BEC side of small positive a the particle-hole fluctuations play no important role. In the unitarity limit of infinite scattering length our result agrees with numerical simulations. A key ingredient for our treatment is the computation of the momentum dependent four-fermion vertex and its bosonization in terms of an effective bound state exchange.

#### Richard Schmidt

Trion formation in ultracold fermion gases

Using the functional renormalization group, we investigate the formation of bound states in a gas of three different nonrelativistic Fermion species. We employ a scale dependent bosonization and fermionization technique to investigate a model with global U(3) symmetry. Close to a Feshbach resonance, the fermions indeed form a fermionic bound state or trion.

 University Faculty Institute