Introduction instanton molecules and topological susceptibility Random matrix model Chiral condensate and Dirac spectrum A modified model and Topological susceptibility Summary Topological susceptibility at finite temperature in a random matrix model Chiral 07, 14 RCNP Munehisa Ohtani (Univ. Regensburg) with C. Lehner, T. Wettig (Univ. Regensburg) T. Hatsuda (Univ. of Tokyo)

Introduction _ Banks-Casher rel:     =  (0) where  ( ) = 1 / V  (  n ) =  1 /  Im Tr(  D+i  )  1  E.-M.Ilgenfritz & E.V.Shuryak PLB325(1994)263  Chiral symmetry breaking and instanton molecules _ _      : chiral restoration # of I-I  : Formation of instanton molecules ? ? Index Theorem:  1  tr FF = N +  N  ~ 32  2  0 mode of +(  ) chirality associated with an isolated (anti-) instanton quasi 0 modes begin to have a non-zero eigenvalue  (0) becomes sparse

Instanton molecules &Topological susceptibility topological charge density q(x) q(x)2q(x)2 isolated (anti-)instantons at low T  d 4 x q(x) 2 decreases as T   d 4 x q(x) 2  1 / V  d 4 yd 4 x(q(x) 2  q(y) 2 ) / 2  1 / V  d 4 yd 4 x q(x)q(y) = Q 2 /V The formation of instanton molecules suggests decreasing topological susceptibility as T  (anti-)instanton molecule at high T q(x)2q(x)2

 Random matrix model at T  0 Chiral restoration and Topological susceptibility  A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996) Chiral symmetry: {D E,  5 } = 0 Hermiticity: D E † = D E Random matrix model Z QCD =   det(iD E + m f )  YM / / // f  0  T The lowest Matsubara freq. quasi 0 mode basis, i.e. topological charge: Q = N +  N  with iD RM = 0 iW iW † 0 W  C N  × N + Z RM =  e  Q 2 /2N   D W e  N/2  2 trW † W  det(iD RM + m f ) Q f |

Hubbard Stratonovitch transformation  T.Wettig, A.Schäfer, H.A.Weidenmüller, PLB367(1996) 1) Z RM rewritten with fermions   integrate out random matrix W Action with 4-fermi int. 3) introduce auxiliary random matrix S  C N f × N f  integrate out  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. Z RM =  e  Q 2 /2N   D S e  N /2  2 trS † S det S + m i  T (N  |Q|)/2 det(S + m) |Q|  Q i  T S † + m |

Chiral condensate _      =  m lnZ RM /VN f = 1 N tr S 0 + m i  T  1 where S 0 : saddle pt. value VN f i  T S 0 † + m _ _    /     0 T / T c m m  The 2 nd order transition in the chiral limit ( Q = 0 at the saddle pt. )

 ( ) T / T c  Eigenvalue distribution of Dirac operator _  ( ) = 1 / V  ( n ) =  1 /  Im Tr( D+i  )  1 = 1 /  Re      | m   i _    =  m lnZ /VN f = Tr( iD+m  )  1 ((  (0) becomes sparse as T  instanton molecule ?

 T/TcT/Tc as N      Q 2  = 1 1 N 2  Suppression of topological susceptibility  ln Z(Q)/Z(0) =  Q 2 / N  Q 4 / N 3   |Q|  Q 2 / N  Q 3 / N 2  Expansion by Q / N : × 1 1  0 (as N   ) 2 N sinh  /2 in RMM  for  m m  Q / NQ / N  ln Z(Q)/Z(0) Q / NQ / N m m   Unphysical suppression of  at T  in RMM

Leutwyler-Smilga model and Random Matrix Using singular value decomposition of S + m  V  1 U  V, Z RM is rewritten with the part. func. Z L-S of chiral eff. theory for 0-momentum Goldstone modes Z RM (Q) = N Q  D  Z L-S (Q,  ) e  N/2  2 tr  2 det(  2 +  2 T 2 ) N/2 det(  2 +  2 T 2 ) |Q|/2 det  |Q| Z L-S (Q,  ) =  D U e  N  2 trRe m  U  Q 2 /2N  detU Q  H.Leutwyler, A.Smilga, PRD46(1992) This factor suppresses   We claim to tune N Q so as to cancel the factor at the saddle point.

Modified Random Matrix model Z mRM =   D  Z L-S (Q,  ) e  N/2  2 tr  2 det(  2 +  2 T 2 ) N/2 Q We propose a modified model: where _      in the conventional model is reproduced. cancelled factor =1 at Q = 0 i.e. saddle pt. eq. does not change ((   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   at T > 0 is not suppressed in the thermodynamic limit.

T / T c m m  topological susceptibility in the modified model  1 + N f 1  1  m(m+  0 )  where  0 : saddle pt. value m  m  · Decreasing  as T  · Comparable with lattice results  B.Alles, M.D’Elia, A.Di Giacomo, PLB483(2000) 

Summary and outlook  Chiral restoration and topological susceptibility  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 : 2 nd order chiral transition & unphysical suppression of  for T >0 in the thermodynamic limit.  We propose a modified model in which     &   are same as in the original model,  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 before H-S transformation from which the modified model are derived, Extension to finite chemical potential, N f dependence … _