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

2. 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:  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 Lattice 07
quasi 0 modes begin to have a non-zero eigenvalue
r (0) becomes sparse

3. 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

4. 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.

5. Random matrix model / / / /
 A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996)
 Random matrix model at T  0
Chiral symmetry: {DE , g5} = Hermiticity: DE†= DE /
ZQCD =  P det(iDE + mf ) YM / / / f
with iDRM = 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

6. 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.

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

8. 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)

9. The 2nd order transition in the chiral limit
Chiral condensate _
 y y  = m lnZRM /VNf =1 N tr S0 + m ipT 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

10. 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

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

12. 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 ?

13. 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) ×
 0 (as N  ) 2 N sinh a/2
  Q2 = N b
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

14. 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.

15. Modified Random Matrix model
We propose a modified model:
ZmRM = S  DL ZL-S(Q,L) e-N/2 S2trL2det(L2 + p2T2)N/ 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.

16. topological susceptibility in the modified model
c = Nf 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

17. 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

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 : nd 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 … _