Anderson átmenet a kvark-gluon plazmában

Transcription

1 Anderson átmenet a kvark-gluon plazmában Kovács Tamás György Atomki, Debrecen 25. április 22. Kovács Tamás György Anderson átmenet a kvark-gluon plazmában /9

2 Együttműködők: 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) Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

3 Theory of strong interactions: QCD Lagrangian: L = 4 F µνf µν + ψ[d(a)+m]ψ F µν : field strength corresponding to A gluons ψ: quarks D(A)=γ µ ( µ ia µ ) Dirac operator Quantization path integral O physical quantity O Dψ D ψ DA O exp ( d 4 x L ) Integration in -dimensional function space Mathematical definition??? How to calculate it??? Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 3/9

4 QCD on the lattice Continuous space-time 4-dimensional cubic lattice Integration over function spaces finite dim. integrals Mathematically well-defined Continuum limit (how to get rid of the lattice?) lattice spacing a ξ lattice a= M phys ξ lattice tune system to critical point tune a few parameters (gauge coupling, quark masses) to fix physics in a limit Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 4/9

5 Lattice QCD ψ i C 3 (on lattice sites) U 3 U 4 U ψ Different basis at each lattice site U i SU(3) (complex rotation) U 2 U ψ vector potential U e ia ψ 2 2 basis transformation (ψ) dynamical variables on links Discretization Derivative: µ ψ a (ψ 2 ψ ) Covariant derivative: D µ ψ a (ψ 2 U ψ ) Gluon action: 4 F µνf µν g 2 tr(u U 2 U 3 U 4 ) Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 5/9

6 Wick rotation Analytic continuation in time t it Minkowski space-time 4d Euclidean space e ilt e Ht Physical quantities = statistical averages Monte Carlo numerical simulation temporally finite box of size L t finite temperature T = /L t Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 6/9

7 Lattice Dirac operator Partition function (integrating out quarks): Z = Dψ D ψ DU e S g[u] ψ{d[u]+m}ψ = DU det{d[u]+m} e S g [U] Statistical physics system (4-dimensional) Dynamical variables: U i SU(N c ) on lattice links Dirac operator: D[U] Discretized differential operator depending on U-s V V sparse matrix (V volume) (D[U]+M) appears in physical quantities Small eigenvalues (eigenvectors) physically important Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 7/9

8 The structure of the Dirac operator Symmetries of D Imλ Spectrum imaginary, symmetric around Reλ probability distribution of [U] random D[U] with given distribution distribution of D[U] physical quantities eigenvalue ststistics of D[U] bulk termodynamics What do we know about the spectral statistics of D[U]? Is the detailed dynamics important or it is already given by the symmetries? Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 8/9

9 Spectrum of sparse random matrices Anderson model one-electron Hamiltonian in disorderer crystal random on-site energies constant hopping terms to nearest neighbor sites all other matrix elements zero localized states at the band edge delocalized states at the band center mobility edge controlled by disorder localized mobility edge Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 9/9

10 Localization and spectral statistics Level spacing distribution Level spacing: λ n+ λ n Unfolding: s = λ n+ λ n λ n+ λ n Two extremes: λ n statistically independent (Poisson) p(s)=exp( s) Random matrix statistics.8 (a) Poisson Random matrix.6 p(s) s Kovács Tamás György Anderson átmenet a kvark-gluon plazmában /9

11 How about the Dirac operator? λ = special point (symmetry). Transition at T c 2MeV : ρ (spectral density) ρ() T < T c hadronic phase ρ()= T > T c quark-gluon plasma λ statistics of low eigenvalues of D[U] described ρ() by random matrix theory analytically (σ-model) + numerically (lattice QCD) Kovács Tamás György Anderson átmenet a kvark-gluon plazmában /9

12 Eigenvalue statistics T > T c (quark-gluon plasma) in different regions of the spectrum unfolded level spacing distribution s= λ n+ λ n λ n+ λ n Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

13 Eigenvalue statistics T > T c (quark-gluon plasma) in different regions of the spectrum unfolded level spacing distribution s= λ n+ λ n.8.6 (a) exponential Wigner QCD λ n+ λ n p(s) s Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

14 Eigenvalue statistics T > T c (quark-gluon plasma) in different regions of the spectrum unfolded level spacing distribution s= λ n+ λ n.8 (a) exponential Wigner QCD.8 λ n+ λ n (b) exponential Wigner QCD p(s).6.4 p(s) s s Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

15 Eigenvalue statistics T > T c (quark-gluon plasma) in different regions of the spectrum unfolded level spacing distribution s= λ n+ λ n.8 (a) exponential Wigner QCD.8 λ n+ λ n (b) exponential Wigner QCD p(s).6.4 p(s) s.8 (c) exponential Wigner QCD s.6 p(s) s Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

16 Eigenvalue statistics T > T c (quark-gluon plasma) in different regions of the spectrum unfolded level spacing distribution s= λ n+ λ n.8 (a) exponential Wigner QCD.8 λ n+ λ n (b) exponential Wigner QCD p(s).6.4 p(s).6.4 p(s) s (c) exponential Wigner QCD p(s) s (d) exponential Wigner QCD s s Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 2/9

17 Integrated level spacing distribution.5 I= p(s)ds I 5.5 L=3. fm L=4.5 fm L=6. fm λa Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 3/9

18 Finite size scaling verify critical behavior and compute ν Parameters: λ (location in the spectrum) relevant µ leading irrelevant I(λ, µ, L) dimensionless, RG invariant I(λ,µ,L) = I(b /ν λ, b y µ µ, b L) Only one relevant variable Close to the critical point: b yµ µ Block all systems to standard size b L Scaling function: I(λ,µ,L)=f ( ) L /ν (λ λ c ) For finite L I(λ) analytic f analytic Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 4/9

19 Finite size scaling I(λ,µ,L)=f ( ) L /ν (λ λ c ) Is it possible to choose ν and λ c to have data collapse? I 5.5 L=3. fm L=4.5 fm L=6. fm λa Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 5/9

20 Finite size scaling I(λ,µ,L)=f ( ) L /ν (λ λ c ) Yes! fit polynomial to f(x)= a +a x+a 2 x and λ c,ν I.5 L=3. fm L=4.5 fm L=6. fm (λ-λ c ) L /ν Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 6/9

21 Corrections to scaling Use only systems larger than L min for the fit system sizes: L 3 = 24 3,28 3,32 3,36 3,4 3,48 3, dimensional K. Slevin & T. Ohtsuki, PRL 82 (999).6 O K. Slevin & T. Ohtsuki, PRL 78 (997).5 ν U.4 S Y. Asada, K. Slevin & T. Ohtsuki, J.Phys.Soc.Jpn. 74 (25) L min ν compatible with unitary Anderson model Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 7/9

22 The temperature controls the mobility edge a=.25 fm a=.82 fm a=.62 fm λ c /m ud T (MeV) for fixed lattice size the system gets more ordered! Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 8/9

23 Conclusions Phase transition or not? The transition to QGP is only a cross-over Have we found a genuine phase transition? No! In QCD no thermodynamic quantity is singular Genuine transition in QCD-like models? Relevance of localized Dirac modes in QGP Mobility edge effective gap (like large m q ) Screening masses External magnetic field and QCD thermodynamics TGK, PRL 4 (2); TGK, F. Pittler, PRL 5 (2); References: F. Bruckmann, TGK, S. Schierenberg, PRD 84 (2); TGK, F. Pittler, PRD 86 (22); M. Giordano, TGK, F. Pittler, PRL 2 (24) Kovács Tamás György Anderson átmenet a kvark-gluon plazmában 9/9