Spectrum of the Dirac Operator and Random Matrix Theory

Transcription

1 Spectrum of the Dirac Operator and Random Matrix Theory Marco Catillo Karl Franzens - Universität Graz 29 June 2016 Advisor: Dr. Leonid Glozman 1 / 23

2 Outline 1 Introduction 2 Overlap Dirac Operator 3 Random Matrix Theory 4 Numerical Analysis 5 Conclusions 2 / 23

3 What we already know The Dirac operator D plays an important rule on the calculation of the hadron spectroscopy. It is important to estimate hadron interpolators and correlators. The symmetry properties of D effect the values of the meson and baryon masses. The chiral condensate Σ = qq is an order parameter for chiral symmetry breaking. Therefore, from the Bank-Casher relation Σ = πρ(0), the presence of the symmetry breaking can be extrapolated from the distribution of Dirac eigenvalues ρ(λ). Different part of the spectrum of D give us useful informations about the symmetries in QCD, like the SU(4) symmetry. 3 / 23

4 What we want to do The importance to see the symmetries of the Dirac operator analyzing different part of the spectrum is crucial. The Random Matrix Theory (RMT ) gives us an useful tool to interpret symmetry properties of the ensembles of random matrix. Therefore we will apply the results of the RMT to the spectrum of the Dirac operator. 4 / 23

5 Lattice Setup Overlap Dirac operator D ov (m) N f = 2 lattice size: L 3 L t = configurations (JLQCD) a = (30)fm, L 1.9fm β = 2.30 m π = 289(2)MeV Topological charge Q = 0 S.Aoki et al (2008) 5 / 23

6 Overlap Dirac Operator ( D ov (m) = ρ + m ) ( + ρ m ) γ 5 sign (H w ) = 2 2 ( = 1 m ) D ov (0) + m 2ρ Neuberger(1998) where m is the lattice quark mass (m = 0.015), ρ is the simulation parameter (ρ = 1.6) and it sets the topological sector, H w = γ 5 D( ρ), D( ρ) is the Wilson-Dirac operator. 6 / 23

7 Chiral Symmetry The eigenvalues of D ov (m) comes in pairs: (λ ov (m), λ ov (m)). This is a consequence of the Chiral Symmetry on the lattice: {γ 5, D ov (0)} = 1 ρ Dov (0)γ 5D ov (0) Ginsparg-Wilson equation (1982) ( ) because λ ov (m) = 1 m λ 2ρ ov (0) + m. Im λ ov The eigenvalues ( are in) the circle with radius R = ρ 1 m 2ρ and centre in ( m + ρ, 0). 2 In this case the lagrangian is invariant under transformations: ( ) ψ(x) e iαa T a Dov (0) γ 5 1 2ρ ψ(x) m ( ) ρ 1 m 2ρ 2ρ Re λ ov ψ(x) ψ(x)e iαa T a γ 5 ( 1 ) Dov (0) 2ρ 7 / 23

8 Random Matrix Theory 8 / 23

9 RMT: General Remarks Hypothesis Suppose to have an N N matrix H in which each element is a random variable. The elements of H are statistically independent. How is made the distribution P(H)? We distinguish 3 different ensembles in which we associate a transformation law H H such that P(H)dH = P(H )dh : Matrices Properties Transformation law Ensemble β H real and symmetric H = O T HO, O O(N) GOE 1 H hermitian H = U HU, U U(N) GUE 2 H quaternion real H = U R HU, U Sp(N) GSE 4 β is called Dyson index. 9 / 23

10 RMT: General Remarks Distribution P(H) dh is the Haar measure P N,β (H)dH = N N,β exp ( β2 ( Tr H H) ) dh Joint probability for the eigenvalues of H ( P N,β (λ 1,..., λ N ) = C N,β exp 1 2 β i C N,β is the normalization constant. ) λ 2 i λ j λ k β j<k 10 / 23

11 Chiral Random Matrix Theory Chiral Ensembles P(W )dw = N(det D) N f e NΣ2 ( ) β Tr W W 4 dw where ( ) m iw D = iw is the Dirac operator, m W is an n m matrix, if Q = 0 then n = m = N/2, E. V. Shuryak and J. M. Verbaarschot, Nucl. Phys A560 (1993) Σ is a parameter and it is not necessarily connected to the chiral condensate, β is the Dyson index. These ensembles are invariant under transformation W UWV 1 : U, V orthogonal matrices, β = 1, chgoe, U, V unitary matrices, β = 2, chgue, U, V symplectic matrices, β = 4, chgse. 11 / 23

12 0.5 rhzl Chiral Random Matrix Theory x 2 IJ2HxL2 - J1HxL J3HxLM 0.2 x 2 IJ0HxL2 + J1HxL 2 M Joint probability for the eigenvalues of D, with Q = ( P n,β (λ 1,..., λ n) = C n,β (λ 2 k + m2 )λ β 1 k exp 1 2 nσ2 β ) z λ 2 i λ 2 j λ 2 k β k i j<k 1 If we change variable z k = NΣλ k, and we consider the limit N, the distribution of the eigenvalues for the chgue (β = 2), N f = 2 and Q = 0 is given by 0.5 rhzl Massless limit ρ(z; 0) = z ( J (z) J 1 (z)j 3 (z) ) Quenched limit ρ(z; ) = z ( J (z) + J1 2 (z)) where J(z) is the Bessel function z rhz;0l rhz; L 12 / 23

13 ChRMT and QCD The chiral symmetry breaking SU L (N f ) SU R (N f ) SU V (N f ) leads to Nf 2 1 Goldstone bosons, U(x) SU(N f ). The effective lagrangian of these Goldstone boson can be written as L eff (U) = F 2 4 Tr ( µu (x) µu(x) ) { ΣRe e i θ ( ) } Nf Tr MU (x) where F 93MeV is the pion decay constant, Σ is the chiral condensate, M is the mass matrix. If the zero momentum modes p = 0 of the pion field U(x) are dominant then the partition function becomes Z eff = e d 4 xl eff (U) du P(W )dw The constraints on the form of the matrix elements of the Dirac operator depend by the color group and the choice of the representation of the quarks: Color group Quark representation Dirac operator β ensemble SU(2) fundamental real 1 chgoe SU(N c 3) fundamental complex 2 chgue SU(N c) adjoint quaternion real 4 chgse 13 / 23

14 An important consequence of the Bank-Casher formula (333) is thattheeigenvalu Introduction Overlap Dirac Operator immediately results in the Banks-Casher relation (333). Random Matrix Theory Numerical Analysis Conclusions The order of the limits in (333) is important. First we take the thermodynamiclim next the chiral limit and, finally, the field theory limit. near zero virtuality are spaced as QCD and ɛ regime λ = 1/ρ(0)= π/σv. (33 The p = 0 modes are dominant in this For case the average position of the smallest nonzero eigenvalue we obtain the estimate 1 L 1 λ min = π/σv. (34, V Σm 1 Λ QCD This should m π be contrasted with the eigenvalue spectrum of the non-interactingdi operator. Then the eigenvalues are those of a free Dirac particle in a box with eigenva spacing equal to λ 1/V We can expand the partition function in terms of ɛ 2 1/4 for mπ the eigenvalues 4πF p2 near λ = 0. Clearly, the presence gauge fields leads to a strong modification of (4πF the spectrum ) 2. near zerovirtuality.stro interactions result in the coupling of many degrees of freedom leading to extended sta At leading order of the ɛ expansion, and correlated the partition eigenvalues. function Because of isasymptotic perfectly freedom, described thespectraldensityof by ChRMT. Dirac operator for large λ behaves as Vλ 3.InFig.2weshowaplotofatypicalavera spectral density of the QCD Dirac operator for λ 0. The spectral density for negat λ is obtained by reflection with respect to the y-axis. More discussion of this figure w be given in section The spectrum of the Dirac operator can be schematically explained in this way: 1 λ min < λ < m c, we only have to include the p = 0 modes, 2 m c < λ < Λ QCD, the ɛ expansion can be applied and we have to consider p 0 modes, 3 λ > Λ QCD, the ɛ expansion is no longer applicable. ρ(λ) - λ min m c chrmt log λ Λ QCD 3 ~V λ FIGURE 2. Schematic picture of the average spectral density of QCD Dirac operator. (Taken from [83].) M. F. L. Golterman and K. C. L. Leung, Let us study the Leutwyler-Smilga sum rule for equally spaced eigenvalues,i.e. Phys. Rev. D57 (1998). λ n = πn. 14 / 23 (34

15 regime. The plot is normalized with 0:00212, which is with sea quark mass in the ran the value after the chiral extrapolation shown in Fig. 10. confined in a fixed topologica First of all, the physical volume at 2:30 is about 30% few cases with finite Q. larger than that at 2:35. Therefore, as explained We found a good agreeme above, the growth of O 3 is expected to be much milder lying eigenvalues in the re and the lattice data are consistent with this picture. The flat ChRMT, which implies strong region extends up to around In our case Σ = (251 ± 7 ± 11MeV ) 3 m V 30. Second, because breaking of chiral symmetr. the microscopic eigenvalue distribution approaches that of tracted the chiral condensate But V Σm 4.17 the quenched and m πl theory, 2.76, the lowest hence peak weis are shifted NOT towards in the ɛ regime! 11 MeV 3 from the lowest e Nevertheless for left. heavy Overall, quarks the number the low-lying of eigenvalues eigenvalues in the microscopic tion factor was calculated no should behave as if they region increases a lot. Unfortunately, the correspondence are in the quenched lattice. between ChPTand ChRMT is theoretically less clear, since And in this case ChRMT the sea quark can be masses applied are out in the of pthe regime. ɛ regime, In order to but the correspondence ChRMT and ChPT is less clear. Is our spectrum described by ChRMT? H. Fukaya at al. Phys. Rev. D ρ(λ)/v m =0.070 m =0.050 m =0.035 m =0.015 ChRMT (quenched) 3/(4π 2 )λ 3 +Σ/π contains a systematic error order effect in the expansio nation of will require larg press such finite size effects. Out of the regime (the masses) the Dirac eigenvalu reasonable agreement with C tracted in this region shows pendence, while its chiral l -regime result. Further information on the extracted in the regime by point functions or analyzing th with imaginary chemical pot work is a first step towards su ACKNOWLE We thank P. H. Damgaar λσv useful suggestions and comm edge the YITP workshop YIT FIG. 12 (color online). Eigenvalue histogram for the 2:30 Symmetries in Lattice Gauge 15 / 23

16 P(s) distribution An important prediction of the RMT is given by the nearest neighbor spacing distribution p(s): where p β (s) = a β s β e b β s ensemble a β b β chgoe (β = 1) π/2 π/4 chgue (β = 2) 32/π 2 4/π chgse (β = 4) /729π 3 64/9π Figure: Nearest neighbor spacing p(s) for β = 1 (solid line), β = 2 (dotted line), β = 4 (dashed line). s n = ξ n+1 ξ n and ξ n is given by ξ n ξ(λ n) = λn 0 ρ(λ)dλ. 16 / 23

17 Numerical Analysis 17 / 23

18 Stereographic Projection The choice of the stereographic projection is not unique. 1 From the center (ρ + m 2, 0): λ = 1 Im λov (m) Re λov (m) ρ+ m 2 ( = ρ + m ) tg(θ) where θ ( π, π]. 2 First 200 eigenvalues From the point (2ρ, 0): λ ov (m) λ ov (0) Im λ ov λ = 1 Im λov (m) Re λov (m) 2ρ ( 3 Dm = D ov (m) 1 D m = K + m, {K, γ 5 } = 0, K = K, = ρ sin(θ) cos 2 (θ/2). ) Dov (0) 1 2ρ, m (ρ + m 2, 0) 2ρ Re λ ov 0.1 Im(λ) λ = Im 1 λov (m) λov (0) 2ρ = 2ρtg(θ/2) Re(λ) 18 / 23

19 Unfolding Procedure S(λ) = 1 M M δ(λ λ k ) k=1 λ η(λ) = S(λ)dλ = 1 M 0 M θ(λ λ k ) k=1 η(λ) = ξ(λ) + η fl (λ) η(λ) Graph 1 Histogram η(λ) Quintic Polynomial, χ = where the smooth part is ξ(λ) = λ 0 ρ(λ)dλ, η fl (λ) is the fluctating part. Consequences: The density distribution ρ (ξ) = 1, λ(mev ) in a small region of the spectrum such that in the limit M it contains a lot of levels, then ρ(λ) = 1 D, with D = λ n+1 λ n, therefore ξ(λ) = λ D. 19 / 23

20 s Lowest Eigenmodes Histogram of the s variable defined in Eq. (26). The bin size e number of eigenvalues is 50 and the number of configurations is nsider the It is set well-known { 1,..., 50}. that This the is lowest the distribution eigenvalues p(s). are The described solid by the RMT. p(s) in For Gaussian different Unitary definitions Ensembles of the (chgue), stereographic the dashed projection line is and different choice of the fit n Gaussian in the Orthogonal unfolding Ensembles procedure (chgoe), p(s) doesn t the dotted change. line is the aussian Symplectic Ensembles (see the Eq. (34)). Range eigenvalues: Stereographic projection 2 k/j h k i/h j i RMT 2/ / / / / / p(s) chgoe chgue chgse 0.4 atio h k i/h j i for all 1 apple j apple k apple 4 and the same values computed Table: Ratio λ MT. We donote with k / λ the j for all 1 j k 4 and the same values computed error. 0.2 with the RMT. σ is the error. 10 The shape of p(s) is in agreement with the predictions of ChGUE s 20 / 23

21 Higher Eigenmodes For higher eigenvalues the agreement with the RMT is not obvious. Range eigenvalues: Stereographic projection 2 Range eigenvalues: Stereographic projection chgoe 1.2 chgoe 1 chgue 1 chgue chgse chgse p(s) p(s) s s But the shape of p(s) for higher eigenvalues and in general for all eigenvalues is still in agreement with the predictions of GUE. 21 / 23

22 Conclusions The distribution of first 200 eigenvalues doesn t depend by the stereographic projection and by the definition of the unfolding procedure. p(s) doesn t change in different regions of the spectrum and in general if we consider all Dirac spectrum. The applicability of RMT is not restricted to lowest eigenvalues but also to higher eigenvalues, but it is not clear the relation with the QCD. We guess that it can be a consequence of the SU(4) symmetry confinement breaking. 22 / 23

23 Thank you! 23 / 23