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Introduction Chiral condensate in RM model T dependence of Dirac spectrum

Topological susceptibility at finite temperature in a random matrix model. Munehisa Ohtani ( Univ. Regensburg ) with C. Lehner, T. Wettig ( Univ. Regensburg ) T. H atsuda ( Univ. of Tokyo ). Introduction Chiral condensate in RM model

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Introduction Chiral condensate in RM model T dependence of Dirac spectrum

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  1. Topological susceptibility at finite temperature in a random matrix model Munehisa Ohtani(Univ. Regensburg) with C. Lehner, T. Wettig (Univ. Regensburg) T. Hatsuda (Univ. of Tokyo) • Introduction • Chiral condensate in RM model • Tdependence ofDirac spectrum • A modified model and Topological susceptibility • Summary 28 Nov. @ U-Tokyo, Komaba

  2.  0 mode of +(-) chirality associated with an isolated (anti-) instanton ~ Index Theorem: -1  trFF = N+ - N- 32p2 _ # of I-I : Formation of instanton molecules V.Weinberg’s talk @ Lattice 07 ? quasi 0 modes begin to have a non-zero eigenvalue ? r(0) becomes sparse _ y y: chiral restoration T = 0.91Tc T = 1.01Tc T = 1.09Tc Introduction E.-M.Ilgenfritz & E.V.Shuryak PLB325(1994)263  Chiral symmetry breaking and instanton molecules _ Banks-Casher rel: y y= -pr (0) wherer (l) = 1/VS d(l- ln) = -1/p Im Tr( l-D+ie )-1

  3. (anti-)instanton molecule at high T q(x)2 The formation of instanton molecules suggests decreasing topological susceptibility as T  Instanton molecules &Topological susceptibility isolated (anti-)instantons at low T q(x)2 topological charge density q(x) d4x q(x)2 decreases as T  d4x q(x)2 =1/Vd4yd4x(q(x)2 +q(y)2 )/2  1/Vd4yd4xq(x)q(y) = Q2/V

  4.  Conductance fluctuation in mesoscopic systems D.R. Hofstadter, Phys. Rev. B14 (1976) 2239  Spectral density of chUE # fluc. of e- levels for a mesoscopic system (with L s.t. coherence length >L > mean free path of e-)  Andreev reflection superconductor R. Opperman, Physica A167 (1990) 301 Classification of ensembles Symmetries as Time rev. spin rotation etc hole e- e- e- metal  2D quantum gravity, zeros of Riemann z function,… P. DiFrancesco et.al., J. Phys. Rep. 254 (1995) 1;A.M. Odlyzko, Math. of Comp. 48 (1987) 273. Applications of Random Matrix  Energy levels of highly excited states in nucleus E.P. Wigner, Ann. Math.53(1951)36; M. Mehta, Random Matrices (1991) universality and symmetry

  5. ZRM = S e-Q2/2NtDWe-N/2S2trW†WP det(iDRM+mf )Qf 0 iW iW†0 with iDRM = + g0pT The lowest Matsubara freq. WCN-×N + | quasi 0 mode basis, i.e. topological charge: Q= N+- N- Random matrix model A.D.Jackson&J.J.M.Verbaarschot, PRD53(1996)  Random matrix model atT 0 / Chiral symmetry: {DE , g5} = 0Hermiticity: DE†= DE ZQCD = P det(iDE+mf ) YM / / / f • Chiral restoration and Topological susceptibility

  6. ZRM = S e-Q2/2NtDSe- N /2S2trS†Sdet S+ m ipT(N - |Q|)/2det(S+ m)|Q|QipTS†+ m In case of Nf = 1, integration by Scan be carried out exactly. Hubbard Stratonovitch transformation T.Wettig, A.Schäfer, H.A.Weidenmüller, PLB367(1996) 1) ZRM rewritten with fermions y2) integrate out random matrix WAction with 4-fermi int. 3) introduce auxiliary random matrix SCNf ×Nf4) integrate out y |

  7. Thermodynamic limit of chiral condensate _ y y = m lnZRM /VNf = det(iDRM+m) tr (iDRM+m)-1/det(iDRM+m)  _ _ y y/ y y0 Fixed m = 0.1/S slow convergence analytic calculation for N  N   N = 24 N = 25, 26,27, 28 N = 23 T /Tc

  8. dim.of matrix N N+ + N- ( V) plays a role of “1/ h ” Saddle pt. eq: (S- |Q|/Nm)((S+ m)2-Q2/N2m2 + p2T2) = (1-|Q|/N)(S+ m-|Q|/Nm) The saddle point eqs. for S, Q/N become exactin the thermodynamic limit. Saddle point equations ZRM = S e-Q2/2NtDSe- N /2S2trS†Sdet S+ m ipT(N - |Q|)/2det(S+ m)|Q|QipTS†+ m

  9. _ _ y y/ y y0 mS T /Tc Chiral condensate _ y y = m lnZRM /VNf =1 N tr S0+ m ipT-1 whereS0: saddle pt. valueVNfipTS0†+ m (Q = 0 at the saddle pt.) The 2nd order transitionin the chiral limit

  10. H.Leutwyler, A.Smilga, PRD46(1992) ZL-S(Q,L) = DUeN S2trRemLU-Q2/2NtdetUQ ZRM(Q) = NQDLZL-S(Q,L)e-N/2S2trL2det(L2+ p2T2)N/2 detL|Q| det(L2+ p2T2)|Q|/2 U : nonlinear representation of pionsL: determines chiral condensate dL : fluctuations of sigma Leutwyler-Smilga model and Random Matrix Using singular value decomposition of S+ m  V-1ULV, ZRM is rewritten with the part. func. ZL-Sof c eff. theory for 0-momentum Goldstone modes

  11. meson masses in RMT ms2S2 mp2S2 mS mS T /Tc T /Tc Plausible chiral properties

  12. _ y y = m lnZ/VNf =Tr( iD+m)-1 ( pr (l) ? instanton molecule lS T /Tc Eigenvalue distribution of Dirac operator _r (l) = 1/VS d(l- ln) = -1/p Im Tr( l-D+ie )-1 = 1/p Rey y|m -il r (0) becomes sparse as T 

  13.   Q2= 1 1N 2b × 1 1  0 (as N  )2 N sinha/2 - ln Z(Q)/Z(0) T = 0 T > 0 in RMM a  0 forT > 0  mS mS Q/N Q/N as N   T/Tc Suppression of topological susceptibility Expansion byQ/N: a |Q| + bQ2/N+ O (Q3/N2) - ln Z(Q)/Z(0) = bQ2/N+ O (Q3/N2) Unphysical suppressionof  at T  0 in RMM

  14. H.Leutwyler, A.Smilga, PRD46(1992) SVD of S+m ZL-S(Q,L) = DUeN S2trRemLU-Q2/2NtdetUQ ZRM(Q) = NQDL ZL-S(Q,L)e-N/2S2trL2det(L2+ p2T2)N/2 detL|Q| det(L2+ p2T2)|Q|/2 This factor suppresses cWe claim totuneNQ so as to cancel the factor at the saddle point. Origin of the unphysical suppression ZRM = S e-Q2/2NtDSe- N /2S2trS†Sdet S+ m ipT(N - |Q|)/2det(S+ m)|Q|QipTS†+ m

  15. where _  y y in the conventional model is reproduced. ( cancelled factor =1 at Q= 0 i.e. saddle pt. eq. does not change  cat T = 0 in the conventional model is reproduced. ( cancelled factor =1 also at T= 0 i.e. quantities at T= 0 do not change Modified Random Matrix model We propose a modified model: ZmRM= S DL ZL-S(Q,L)e-N/2S2trL2det(L2+ p2T2)N/2Q  cat T > 0 is not suppressed in the thermodynamic limit.

  16. mS = 0.1 mS = 0.01 mS T /Tc B.Alles, M.D’Elia, A.Di Giacomo, PLB483(2000) topological susceptibility in the modified model whereL0 : saddle pt. value c = 1+ Nf 1-1t m(m+L0)S 2 · Decreasing c as T · Comparable withlattice results

  17. With eigenfunctions Dovn = lnn , we can show that ln + ln*=  n*(Dov+ g5Dovg5) n = a n* g5Dovg5Dov n= a ln* lnDov (g5n )= g5Dov†n = ln* (g5n )  Hermitian operator g5iDW : diagonalized by unitary matrix sign(g5iDW)= U†diag(-1,-1,-1,1) U U = exp(iatcf c) = Dov=  iDRM(at T=0 ) as a  0 Overlap operator and RMT Ginsparg-Wilson rel: {Dov , g5} = a Dovg5 Dovg5-Hermiticity: Dov†= g5Dov g5 Dov= 1/a (1+g5 sign(g5iDW))

  18. Summary and outlook • Chiral restoration and topological susceptibility c are studied in a random matrix model formation of instanton molecules connects them via Banks-Casher relation and the index theorem. • Conventional random matrix model : 2nd order chiral transition & unphysical suppression of c for T >0 in the thermodynamic limit. • We propose a modified model in which y y &c|T=0 are same as in the original model, cat T >0 is well-defined anddecreases as T increases.consistent withinstanton molecule formation, lattice results • Outlook: To find out the random matrix with quasi 0 mode basis from which the modified model are derived, Extension to finite chemical potential, Nf dependence … _

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