thermodynamics and z(n) interfaces in large n matrix model
DESCRIPTION
Thermodynamics and Z(N) interfaces in large N matrix model. RIKEN BNL Shu Lin. YITP, Kyoto 11/18. SL, R. Pisarski and V. Skokov, arXiv:1301.7432. Outline. A brief review of matrix model Solution of matrix model in large N limit - PowerPoint PPT PresentationTRANSCRIPT
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Thermodynamics and Z(N) interfaces in large N matrix model
RIKEN BNLShu Lin
SL, R. Pisarski and V. Skokov, arXiv:1301.7432
1
YITP, Kyoto 11/18
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Outline
• A brief review of matrix model• Solution of matrix model in large N limit• Construct order-disorder and order-order
interfaces and calculate the corresponding interface tensions
• 1/N correction to solution of matrix model• Numerical results on the nonmonotonic behavior
as a function of N• Summary&Outlook
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SU(N) deconfinement phase transition
SU(2) second orderSU(N3) first order
Trace anomaly
M. Panero, Phys.Rev.Lett. (2009)
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More on trace anomaly
~ T2 ? M. Panero, Phys.Rev.Lett. (2009)
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SU(N) matrix modelDeconfinement phase transition order parameter:
Variables of matrix model: qi
Vpt ~ T4
Vnon ~ T2Tc2
A. Dumitru, Y. Guo, Y. Hidaka, C. Korthals, R. Pisarski, Phys.Rev. D (2012)
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SU(N) matrix model
For SU(N) group
Njiqqq jiij 1, -1/2 q 1/2
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SU(N) matrix modelT>>Td, Vpt dorminates, perturbative vacuum:
T<Td, confined vacuum driven by Vnon :
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SU(N) matrix model
Three parameters in nonperturbative potentialTwo conditions: Tc is the transition tempeprature; pressure vanishes at Tc
Free parameters determined by fitting to lattice data
Analytically solvable for two and three colorsNumerically solvable for higher color numbers
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Fitting to lattice data
Fitting for SU(3)
2
2
3333 ))()(()()(TTcTccTc c
c
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Fitting to lattice data
Polyakov loop l(T)Too sharp rise!!
‘t Hooft loop, order-order interface tension
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Matrix model in the large N limitwith
Eigenvalue density
Minimize Vtot(q) with respect to (q) subject to
Consider symmetric distribution due to Z(N) symmetry
In terms of (q), Vtot(q) becomes polynomial interaction of many body system
Sovable!
Pisarski, Skokov PRD (2012)
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First or second order phase transition?
0,)(~2/1,
1
5/21
lTTTTlTT
d
dd
)()(~
)(#)(~)(,5/3
5/7
TTTc
TTTTTpTT
dV
ddd
second?first?
second?first?discontinuous
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Distribution of eigenvaluesT<Td, confined
T>Td, deconfined
T=Td, transition
(q0)=0
Critical distribution identical to 2D U(N) LGT at phase transition—Gross-Witten transition!
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Construction of Z(N) vacuaapply Z(N) transform to the symmetric distribution
relabel
=k/N
Order-order interface is defined as a domain wall interpolating two Z(N) inequivalent vacua above Td
Order-disorder interface is defined as a domain wall interpolating confined and deconfined vacua at Td
Order-order interface tension can be related to VEV of spatial ‘t Hooft loop C. Korthals, A. Kovner and M. Stephanov, Phys Lett B (1999)
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Zero modes of the potential at T=Td
=1+b cos2q
“b mode”
b=0 b=1
“ mode” Emergent zero modes at naive large N limit
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Construction of order-disorder interfaceKinetic energy:
2
2
2 ),()2()(zzxqdzdx
NgTqK
Use b-mode for the order-disorder interface: potential energy V(q) vanishes
For an interface of length L, kinetic energy K(q) ~ 1/L vanishes in the limit
Interface tension (K(q)+V(q))/L vanishes for order-disorder interface
),(
)),'('()2()(2
2
2
zq
zqdqdzdq
NgTqK z
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Construction of order-order interface at T=Td
confineddeconfined
part I part II: Z(N) transform of part I
confined deconfined
Second possibility, use of mode: (z)
=0 0
Ruled out by divergent kinetic energy
q
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Construction of order-order interface at T>Td
Limiting cases:
T>Td, eigenvalue distribution gapped: eigenvalues q do not occupy the full range [-1/2,1/2]
near confineddeconfined
part I part III: Z(N) transform of part I
near confined deconfined
part II
q
dependence on obtained using ansatz of the interface
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Construction of order-order interface at T>Td
Another limiting case:
V ~ (1-2q0)4 , K ~ (1-2q0)2
Compare with the other limiting case
Noncommutativity of the two limitsRecall =k/NNoncommutativity of large N limit and Important to consider 1/N correction!
22
22
2
22/3
3
2 NTcTNg
Td
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Comparison with weak&strong coupling results
SYM strong coupling
Bhattaharya et al PRL (1992)Armoni, Kumar and Ridgway, JHEP (2009)Yee, JHEP (2009)
)(2732
)(sincossin34
)(334
43
3
2
32
2
32
kNkMT
Ng
TN
kNkNg
T
KK
SU(N) weak coupling
13
2 22
2
2
222/3
dTcTNg
TN
SS model
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1/N correction of thermodynamicsEuler-MacLaurin formula
variation
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1/N corrected eigenvalue distribution
to the order 1/N
solution
Imaginary part of Polyakov loop vanishes to order 1/N!
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Comparison with numerics
dTTd ,2 1/N expansion breaks down.
Numerical results suggest correction is organizied as N/1
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1/N correction to potentialAt T=Td, height of potential barrier can be obtained from numerical simulation
potential normalized by 1/(N2-1): barrier shows nonmonotoic behavior in N. maxiam at roughly N=5
tail
Correction to interface tension at T=Td
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Summary&Outlook
• Large N matrix model phase transition resembles that of Gross-Witten.
• In naive large N limit, interface tension vanishes at T=Td. Above Td, nontrivial dependence on T.
• Large N and phase transition limit do not commute. 1/N correction to the thermodynamics needed. It breaks down at Td.
• Numerical simulation shows nonmonotonic behavior in potential barrier as a function of N
• Magnetic degree of freedom?
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Thank you!
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mean field approximation?
Susceptbility of Polyakov loop for SU(3)
1307.5958 Lo et al
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2
2
3333 ))()(()()(TTcTccTc c
c