The chiral and the Anderson transition in QCD

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1 The chiral and the Anderson transition in QCD Tamás G. Kovács Institute for Nuclear Research, Debrecen March 11, 2015 Tamás G. Kovács The chiral and the Anderson transition in QCD 1/20

2 Collaboration: Falk Bruckmann (Regensburg U.) Gergely Endrődi (Regensburg U.) Matteo Giordano (Atomki, Debrecen) Ferenc Pittler (Eötvös U.) Sándor Katz (Eötvös U.) László Újfalusi (Budapest U. of Technology) Imre Varga (Budapest U. of Technology) Tamás G. Kovács The chiral and the Anderson transition in QCD 2/20

3 Localized modes in the QCD Dirac spectrum above T c Spectral density of the QCD Dirac operator (schematic) T < T c T T c T > T c localized mobility edge Tamás G. Kovács The chiral and the Anderson transition in QCD 3/20

4 Localized modes in the QCD Dirac spectrum above T c Spectral density of the QCD Dirac operator (schematic) T < T c T T c T > T c localized mobility edge ρ(0) ψψ order parameter of spontaneous χsb Mobility edge controlled by the temperature Goes to zero around T c localized modes disappear Does this have anything to do with the chiral transition? Tamás G. Kovács The chiral and the Anderson transition in QCD 3/20

5 Why would you care about the mobility edge? λ < λ c Modes localized on the length scale of 1/T Do not contribute to correlators on scales x 1/T λ > λ c Modes delocalized over the whole system Similar to quark modes below T c Contribute to low-energy physics λ c : effective bare quark mass Tamás G. Kovács The chiral and the Anderson transition in QCD 4/20

6 Why is there a mobility edge? Free theory at temperature T Antiperiodic temporal boundary condition Gap in the Dirac spectrum: πt lowest Matsubara mode Interactions switched on; T < T c 1/T > ξ (correlation length) Quarks do not feel the temporal boundary condition Matsubara picture does not apply Tamás G. Kovács The chiral and the Anderson transition in QCD 5/20

7 Temporal twist determines the lowest eigenvalues Interactions switched on; T > T c 1/T < ξ (correlation length) Lowest Matsubara mode phase of temporal twist twist: local Polyakov loop (-1 boundary condition) This is how the quark action breaks Z(3) symmetry Polyakov loop can only decrease the twist eigenvalues move down Matsubara bound becomes fuzzy gap disappears, gets sparsely populated effective bare quark mass: π N t λ c Tamás G. Kovács The chiral and the Anderson transition in QCD 6/20

8 Contribution of spatial and temporal derivatives to λ 0.04 D µ γ µ ψ = λψ 0.03 µ = spatial µ = temporal <ψ D µ γ µ ψ> λa Tamás G. Kovács The chiral and the Anderson transition in QCD 7/20

9 λ c /m ud sharply increases with the temperature N f = 2+1 flavor stout staggered, physical quark masses a=0.125 fm a=0.082 fm a=0.062 fm λ c /m ud T (MeV) Fit: almost linear with tiny quadratic term Tamás G. Kovács The chiral and the Anderson transition in QCD 8/20

10 Transition at λ c sharper for larger volume N f = 2+1 QCD with stout staggered quarks, physical point Spectral statistics at fixed T > T c, scan through the spectrum I L=3.0 fm L=4.5 fm L=6.0 fm λa Tamás G. Kovács The chiral and the Anderson transition in QCD 9/20

11 Finite size scaling works I(λ,µ,L)=f ( ) L 1/ν (λ λ c ) ν = 1.43(5) compatible with 3d unitary Anderson model M. Giordano,TGK and F. Pittler, PRL 112 (2014) I L=3.0 fm L=4.5 fm L=6.0 fm (λ-λ c ) L 1/ν Tamás G. Kovács The chiral and the Anderson transition in QCD 10/20

12 Anderson transition in the Dirac spectrum Singularity in spectral statistics at λ c 2 nd order phase transition Eigenmode correlation length diverges Above T c there is a critical point λ c in the spectrum Does not imply singularity in thermodynamic quantities like ψψ dλ mρ(λ) m 2 + λ 2 or ψψ(0) ψψ(x) How can this be connected to the chiral (phase) transition? Tamás G. Kovács The chiral and the Anderson transition in QCD 11/20

13 Localized modes disappear around T c N f = 2+1 flavor stout staggered, physical quark masses 80 mobility edge λ c /m ud quadratic fit T (MeV) Fit: λ c 0 at T = 163(2) MeV T c Tamás G. Kovács The chiral and the Anderson transition in QCD 12/20

14 Why look at critical eigenmodes at λ c? Strategy: understand behavior at λ c (fixed T) see what happens when T ց T c and λ c ց 0 Use knowledge about Anderson transitions Possible role of critical eigenmodes when λ c ց 0: ψ(0) 2 ψ(x) 2 correlator disconnected chiral susceptibility Tamás G. Kovács The chiral and the Anderson transition in QCD 13/20

15 What if there is a genuine phase transition? No QCD-like theory with finite T phase transition Hope: maybe QCD with N f = 3 light quarks? Toy model: light staggered quarks on coarse lattices Genuine first order phase transition Probably lattice artifact What happens to the Anderson picture at T c Look at T -dependence of spectral statistics at λ = 0 Tamás G. Kovács The chiral and the Anderson transition in QCD 14/20

16 The condensate and the mobility edge across the transition ψψ λ c Second order polynomial fit ψψ λc β Tamás G. Kovács The chiral and the Anderson transition in QCD 15/20

17 Unfolded level spacing distribution at T c is critical s= λ i+1 λ i λ i+1 λ i staggered at T c critical statistics p(s) s Tamás G. Kovács The chiral and the Anderson transition in QCD 16/20

18 Conclusions and outlook Above T c there is a localization-delocalization transition in the QCD Dirac spectrum Real Anderson transition, same universality class as 3d Anderson model Mobility edge: λ c effective quark mass above T c At T c mobility edge λ c 0 Final goal: understand connection between chiral and Anderson transition in QCD-like theories originally proposed by Garcia-Garcia and Osborn PR D75 (2007) Tamás G. Kovács The chiral and the Anderson transition in QCD 17/20

19 Mobility edge moves with the Matsubara frequency Qualitative understanding of: How does an imaginary chemical potential affect T c? Z(3) center symmetry breaking of the quark action. (Why do light quarks prefer the real P-loop sector?) T > T c hadron screening masses free quarks with Matsubara mass. Why? What are the quasiparticles? How does the system react to an external magnetic field around T c? Tamás G. Kovács The chiral and the Anderson transition in QCD 18/20

20 Dependence of the condensate on the magnetic field Bali et al. PRD (2012) Tamás G. Kovács The chiral and the Anderson transition in QCD 19/20

21 Reaction of the condensate to B, below and above T c Magnetic field shifts low Dirac eigenvalues down Localized and delocalized modes: different spectral statistics Localized Poisson statistics no repulsion Delocalized random matrix statistics strong repulsion At T > T c lowest modes localized react differently to B Tamás G. Kovács The chiral and the Anderson transition in QCD 20/20