Challenges forChallenges forFunctional Functional

Renormalization Renormalization

Exact renormalization Exact renormalization group equationgroup equation

important success in important success in many areasmany areas

first rate tool if perturbative expansions first rate tool if perturbative expansions or numerical simulations fail or are or numerical simulations fail or are difficultdifficult

models with fermionsmodels with fermions gravitygravity models with largely different length models with largely different length

scalesscales non-perturbative renormalizabilitynon-perturbative renormalizability

different laws at different different laws at different scalesscales

fluctuations wash out many details fluctuations wash out many details of microscopic lawsof microscopic laws

new structures as bound states or new structures as bound states or collective phenomena emergecollective phenomena emerge

key problem in Physics !key problem in Physics !

scale dependent lawsscale dependent laws

scale dependent ( running or flowing scale dependent ( running or flowing ) couplings) couplings

flowing functionsflowing functions

flowing functionalsflowing functionals

flowing actionflowing action

Wikipedia

flowing actionflowing action

microscopic law

macroscopic law

infinitely many couplings

functional functional renormalizationrenormalization

transition from microscopic to transition from microscopic to effective theory is made continuouseffective theory is made continuous

effective laws depend on scale keffective laws depend on scale k flow in space of theoriesflow in space of theories flow from simplicity to complexity if flow from simplicity to complexity if

theory is simple for large ktheory is simple for large k or opposite , if theory gets simple for or opposite , if theory gets simple for

small ksmall k

challenges for functional challenges for functional renormalizationrenormalization

reliability and error reliability and error estimatesestimates

accessibilityaccessibility exploration of new terrainexploration of new terrain precision and benchmarkingprecision and benchmarking

truncation errortruncation error

structure of structure of truncated flow equationtruncated flow equation

g : flowing data

ς: flow generators

flowing dataflowing data

typically, g can be viewed as functions typically, g can be viewed as functions

effective potential U(effective potential U(ρρ ) )

inverse relativistic propagator P(pinverse relativistic propagator P(p22))

inverse non-relativistic propagator P( inverse non-relativistic propagator P( ωω , , pp22))

momentum dependent four-fermion vertexmomentum dependent four-fermion vertex

truncationtruncation

finite number of functionsfinite number of functions

functions parameterized by finite set of datafunctions parameterized by finite set of data

e.g.e.g.

polynomial expansionpolynomial expansion

function values at given argumentsfunction values at given arguments

( can be single coupling)( can be single coupling)

truncation : limitation to restricted set truncation : limitation to restricted set of dataof data

(finite set for numerical purposes)(finite set for numerical purposes)

exact flow equationsexact flow equations

for given set of data :for given set of data :

flow generators flow generators ςς can be can be computed computed exactlyexactly as formal as formal expressionsexpressions

O(N) – scalar modelO(N) – scalar model

first order derivative expansion

flowing data g : U(ρ ), Z(ρ ), Y(ρ )

exact generator for U:

momentum integrationmomentum integration

one loop form of exact flow equation :

σ(q) : input functions

one d-dimensional momentum integration necessary,( sometimes analytical integration possible )

specificationspecification

For finite set of data the system of flow equations is not closed !

σ(q) cannot be computed uniquely from gone needs prescription how σ(q) is determined in terms of g

specification parametersspecification parameters

specification typically involves specification parameters s

Lowest order derivative Lowest order derivative expansion for scalar expansion for scalar

O(N) modelO(N) model

Z(Z(ρρ )= Z )= Z Y(Y(ρρ )=0 )=0 flowing data : U(flowing data : U(ρρ ), Z ), Z

Flow equation for average Flow equation for average potentialpotential

Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact

Scalar field theoryScalar field theory

specification (1)specification (1)

one has to specify the exact definition of Z one has to specify the exact definition of Z fromfrom

inverse propagator P(q) inverse propagator P(q)

P: second functional derivative at minimum P: second functional derivative at minimum of effective potential - Goldstone mode or of effective potential - Goldstone mode or radial moderadial mode

Z=Z=∂∂P/P/∂∂qq22 at q at q22=0 =0 or or Z=(P(qZ=(P(q22=ck=ck22)-P(0))/ck)-P(0))/ck22

c is one of the specification parametersc is one of the specification parameters

Which forms of effective Which forms of effective action are compatible with action are compatible with

given data g ?given data g ?

specification (2)specification (2)

different forms of inverse propagator are compatible with a given definition of Z

choice of inverse choice of inverse propagatorpropagator

physics knowledge can physics knowledge can be put into choice of be put into choice of general form of input general form of input

functions !functions !

flow parameters wflow parameters w

specification parametersspecification parameters cutoff parameterscutoff parameters bosonization parametersbosonization parameters

error estimateerror estimate

vary w within certain vary w within certain priorspriors

accessibilityaccessibility

public program with public program with structurestructure

for first step : individual routines from users / library

numerical momentum numerical momentum integrationintegration

update of flowupdate of flow

precision and precision and benchmarkingbenchmarking

benchmarks (1)benchmarks (1)

universal critical physics:universal critical physics:

O(N)-models : exponents, amplitude O(N)-models : exponents, amplitude ratios, equation of state ( including ratios, equation of state ( including non-perturbative physics as Kosterlitz -non-perturbative physics as Kosterlitz -Thouless transition )Thouless transition )

non-abelian non-linear sigma-models in non-abelian non-linear sigma-models in d=2 : d=2 : mass gap, correlation functionsmass gap, correlation functions

Ising model on lattice and other Ising model on lattice and other exactly solvable modelsexactly solvable models

benchmarks (2)benchmarks (2)

quantum mechanicsquantum mechanics

a) as d=1 functional integrala) as d=1 functional integral

b) limit n b) limit n →→ 0 , T 0 , T →→ 0 of many body 0 of many body system, system,

d=3+1d=3+1

scattering length for atoms, dimers, scattering length for atoms, dimers, Efimov effect with Efimov parameter Efimov effect with Efimov parameter

BCS-BEC crossover for ultracold atoms :BCS-BEC crossover for ultracold atoms :

all T, n, aall T, n, a

Floerchinger, Scherer , Diehl,…see also Diehl, Gies, Pawlowski,…

BCS BEC

free bosons

interacting bosons

BCS

Gorkov

BCS – BEC crossoverBCS – BEC crossover

explore new terrain explore new terrain

up to you !up to you !

Phase transitions in Phase transitions in Hubbard ModelHubbard Model

Anti-ferromagnetic Anti-ferromagnetic andand

superconducting superconducting order in the order in the

Hubbard modelHubbard modelA functional renormalization A functional renormalization

group studygroup study

T.Baier, E.Bick, …C.Krahl, J.Mueller, S.Friederich

Phase diagramPhase diagram

AFSC

Mermin-Wagner theorem Mermin-Wagner theorem ??

NoNo spontaneous symmetry spontaneous symmetry breaking breaking

of continuous symmetry in of continuous symmetry in d=2 d=2 !!

not valid in practice !not valid in practice !

Phase diagramPhase diagram

Pseudo-criticaltemperature

challenges for functional challenges for functional renormalizationrenormalization

reliability and error reliability and error estimatesestimates

accessibilityaccessibility exploration of new terrainexploration of new terrain precision and benchmarkingprecision and benchmarking

conclusionsconclusions

functional renormalization :functional renormalization :

bright futurebright future

substantial work substantial work aheadahead

end