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 T dependence of Dirac spectrum A modified model and Topological susceptibility Summary 28 Nov. @ U-Tokyo, Komaba

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

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

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  Conductance fluctuation in mesoscopic systems  D.R. Hofstadter, Phys. Rev. B14 (1976) 2239 # fluc. of e- levels for a mesoscopic system (with L s.t. coherence length > L > mean free path of e-)  Spectral density of chUE  Andreev reflection e- e- e- hole metal superconductor  R. Opperman, Physica A167 (1990) 301 Classification of ensembles  Symmetries as Time rev. spin rotation etc  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.

Random matrix model / / / /  A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996)  Random matrix model at T  0 Chiral symmetry: {DE , g5} = 0 Hermiticity: DE†= DE / ZQCD =  P det(iDE + mf ) YM / / / f with iDRM = 0 iW iW† 0 W  CN - × N + ZRM = S e-Q2/2Nt  DW e-N/2S2trW†W P det(iDRM + mf ) Q f | + g0 pT The lowest Matsubara freq. quasi 0 mode basis, i.e. topological charge: Q = N+ - N- Chiral restoration and Topological susceptibility

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

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 = 23 N = 25, 26, 27, 28 T / Tc

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

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

Leutwyler-Smilga model and Random Matrix Using singular value decomposition of S + m  V-1ULV, ZRM is rewritten with the part. func. ZL-S of c eff. theory for 0-momentum Goldstone modes ZL-S(Q,L) = DU e N S2trRe mLU-Q2/2Nt detUQ  H.Leutwyler, A.Smilga, PRD46(1992) ZRM(Q) = NQ  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/2 det(L2 + p2T2)|Q|/2 detL|Q| U : nonlinear representation of pions L : determines chiral condensate dL : fluctuations of sigma

meson masses in RMT ms2 S2 mp2 S2 m S m S T / Tc T / Tc Plausible chiral properties

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

Suppression of topological susceptibility Expansion by Q / N : - ln Z(Q)/Z(0) = b Q2 / N + O (Q3 / N2) a |Q| + b Q2 / N + O (Q3 / N2) × 1 1  0 (as N  ) 2 N sinh a/2   Q2 = 1 1 N 2b in RMM a  0 forT > 0 m S Q / N - ln Z(Q)/Z(0) T = 0 T > 0  T/Tc as N   Unphysical suppression of  at T  0 in RMM

Origin of the unphysical suppression ZRM = S e-Q2/2Nt  DS e- N /2S2trS†S det S + m ipT (N - |Q|)/2det(S + m)|Q| Q ipT S† + m SVD of S+m ZL-S(Q,L) = DU e N S2trRe mLU-Q2/2Nt detUQ  H.Leutwyler, A.Smilga, PRD46(1992) ZRM(Q) = NQ  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/2 det(L2 + p2T2)|Q|/2 detL|Q| This factor suppresses c We claim to tune NQ so as to cancel the factor at the saddle point.

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

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

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

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, c at T >0 is well-defined and decreases as T increases.  consistent with instanton 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 … _