• Introduction • Chiral condensate in RM model• T dependence of Dirac spectrum • A modified model and Topological susceptibility • Summary

Topological susceptibility at finite Topological susceptibility at finite temperaturetemperature

in a random matrix modelin a random matrix modelMunehisa Ohtani (Univ. Regensburg) with C. Lehner, T. Wettig (Univ.

Regensburg)

T. Hatsuda (Univ. of Tokyo)

28 Nov. @ U-Tokyo, Komaba

_

_

: chiral restoration

# of I-I : Formation of instanton molecules

?

?

IntroductionIntroduction

_ 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

Index Theorem: 1 tr FF = N+ N

~

32 2

0 mode of +() chirality associatedwith an isolated (anti-) instanton

quasi 0 modes begin to have a non-zero eigenvalue

(0) becomes sparse

T = 0.91Tc T = 1.09TcT = 1.01Tc

V.Weinberg’s talk @ Lattice 07

Instanton molecules &Topological susceptibilityInstanton molecules &Topological susceptibility

topological charge density q(x)

q(x)2

isolated (anti-)instantonsat low T

d4x q(x)2 decreases 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

The formation of instanton molecules suggests

decreasing topological susceptibility as T

(anti-)instanton moleculeat high T

q(x)2

Applications of Random MatrixApplications 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 elevels for a mesoscopic system

(with L s.t. coherence length L mean free path of e) Spectral density of chUE

Andreev reflection

eee

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 function,… P. DiFrancesco et.al., J. Phys. Rep. 254 (1995) 1; A.M. Odlyzko, Math. of Comp. 48 (1987) 273.

Random matrix model at T 0

Chiral restoration and Topological susceptibility

A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996)

Chiral symmetry: {DE , 5} = 0 Hermiticity: DE

†= DE

Random matrix modelRandom matrix model

ZQCD = det(iDE + mf ) YM / ///f

0 T The lowest Matsubara freq.

quasi 0 mode basis, i.e.

topological charge: Q = N+ N

with iDRM = 0 iW iW† 0

W CN × N +

ZRM = eQ2/2N DW eN/22trW†W det(iDRM + mf )

Q f

|

Hubbard Stratonovitch transformationHubbard Stratonovitch transformation T.Wettig, A.Schäfer, H.A.Weidenmüller, PLB367(1996)

1) ZRM rewritten with fermions integrate out random matrix W Action with 4-fermi int.

3) introduce auxiliary random matrix S CNf × Nf

integrate out

ZRM = eQ2/2N DS eN /22trS†S det S + m iT

(N|Q|)/2

det(S + m)|Q|

Q iT S† + m

In case of Nf = 1, integration by S can be carried out exactly.

|

Thermodynamic limit of chiral condensateThermodynamic limit of chiral condensate

_ _

/ 0

T / Tc

N = 2 3

N = 2 4

N = 25, 26, 27, 28

Fixed m = 0.1/

N

_

= m lnZRM /VNf = det(iDRM + m) tr (iDRM + m)1/det(iDRM + m)

slow convergence

analytic calculation for N

Saddle point equationsSaddle 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 = eQ2/2N DS eN /22trS†S det S + m iT

(N|Q|)/2

det(S + m)|Q|

Q iT S† + m

Saddle pt. eq:

(S Q|/Nm)((S + m)2Q2/N2m2 + 2T2) = (1|Q|/N)(S + mQ|/Nm)

Chiral condensateChiral condensate

_

= m lnZRM /VNf =1 N tr S0 + m iT

1

whereS0 : saddle pt. value

VNf

iT

S0†

+ m

0.5

1

1.5

2

0.5

1

1.5

2

00.250.5

0.75

1

0.5

1

1.5

2

_ _

/ 0

T / Tc

m

The 2nd order transitionin the chiral limit

(Q = 0 at the saddle pt.)

Leutwyler-Smilga model and Random Matrix Leutwyler-Smilga model and Random Matrix

Using singular value decomposition of S + m V1UV, ZRM is rewritten

with the part. func. ZL-S of eff. theory for 0-momentum Goldstone modes

ZRM(Q) = NQ DZL-S(Q,) eN/2 2tr2det(2 + 2T2)N/2 det(2 + 2T2)|Q|/2

det|Q|

ZL-S(Q,) = DU eN 2trRe mUQ2/2N detUQ

H.Leutwyler, A.Smilga, PRD46(1992)

U : nonlinear representation of pions : determines chiral condensate: fluctuations of sigma

0.51

1.5

2

0.5

1

1.5

00.51

1.52

0.51

1.5

2T / Tc

m

meson masses in RMTmeson masses in RMT

0.5

1

1.5

2

0.5

1

1.5

00.20.40.60.8

0.5

1

1.5

2

m

T / Tc

m

m

Plausible chiral properties

0

0.5

1

1.5

2

2

0

2

0

0.25

0.5

0.75

1

0

0.5

1

1.5

2

()

T / Tc

Eigenvalue distribution of Dirac operatorEigenvalue distribution of Dirac operator _

() = 1/V (n) = 1/ Im Tr( D+i)1 = 1/Re|m i _

= m lnZ /VNf = Tr( iD+m)1

(

(0) becomes

sparse as T

instanton molecule

?

T/Tc

as N

Q2

= 1

1

N 2

Suppression of topological susceptibility Suppression of topological susceptibility

ln Z(Q)/Z(0) = Q2 / NQ3 / N2|Q| Q2 / NQ3 / N2Expansion by Q / N :

× 1

1 0 (as N )

2 N sinh /2

in RMM for

m

Q / N

ln Z(Q)/Z(0)

Q / N m

Unphysical suppressionof at T in RMM

Origin of the unphysical suppressionOrigin of the unphysical suppression

ZRM(Q) = NQ DZL-S(Q,) eN/2 2tr2det(2 + 2T2)N/2 det(2 + 2T2)|Q|/2

det|Q|

ZL-S(Q,) = DU eN 2trRe mUQ2/2N detUQ

H.Leutwyler, A.Smilga, PRD46(1992)

This factor suppresses We claim to

tune NQ so as to cancel the factor at the saddle point.

ZRM = eQ2/2N DS eN /22trS†S det S + m iT

(N|Q|)/2

det(S + m)|Q|

Q iT S† + m

SVD of S+m

Modified Random Matrix modelModified Random Matrix model

ZmRM = DZL-S(Q,) eN/2 2tr2det(2 + 2T2)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 / Tc

m

topological susceptibility in the modified model topological susceptibility in the modified model

m

m

· Decreasing as T · Comparable with lattice results

B.Alles, M.D’Elia, A.Di Giacomo, PLB483(2000)

1

+ Nf

11

m(m+0)

where0 : saddle pt. value

Overlap operator and RMTOverlap operator and RMT

Dov= 1/a (1+5 sign(5iDW)) Ginsparg-Wilson rel: {Dov , 5} = a Dov 5 Dov

5-Hermiticity: Dov†= 5 Dov 5

With eigenfunctions Dovn = nn , we can show that

n + n* = n

*(Dov+ 5Dov5) n = a n* 5Dov5Dov n= a n

* n

Dov (5 n ) = 5 Dov†n = n

* (5 n )

Hermitian operator 5iDW : diagonalized by unitary matrix sign(5iDW) = U† diag(1,1,1,1) U

U = exp(i acc) =

Dov = iDRM ( at T=0 ) as a 0

Summary and outlookSummary 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 : 2nd 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 with quasi 0 mode basis from which the modified model are derived, Extension to finite chemical potential, Nf dependence …

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