Auxiliary field Monte-Carlo studyof the QCD phase diagram at strong coupling
A. Ohnishi (YITP)
in collaboration withT.Z.Nakano (YITP/Kyoto U.), T. Ichihara (Kyoto U.)

Introduction
Auxiliary Field Monte-Carlo treatment of SC-LQCD
Monte-Carlo estimate of the phase boundary
SummaryWork in progress
Ohnishi @ NTFL (Feb. 21, 2012)

1

QCD Phase Diagram
RHIC/LHC/Early Universe

T
QGP
AGS/SPS/
FAIR/J-PARC
Quarkyonic

HadronMatterNucleonGas

Neutron Star

CSC

Nuclear
MatterOhnishi @ NTFL (Feb. 21, 2012)

μB

2

QCD phase diagram (Exp. & Theor. Studies)

QCD
QCDphasephasetransitiontransitionisisnot
notonly
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academicproblem,
problem,butbutalsoalsoaasubject
subjectwhich
whichwould
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ininHIC
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Ohnishi @ NTFL (Feb. 21, 2012)

3

QCD phase diagram (Exp. & Theor. Studies)
Lattice QCD at finite density has the sign problem.
→ Approx. methods and/or Effective model studies are necessary.
Approximate methods: Taylor exp. (LT04), Imag. μ , Canonical(LC04, 08), Reweighting (LR02, 04), Fugacity exp. (Nagata /Adams), Strong Coupling Lattice QCD
Effective models:
NJL, PNJL, PQM, ..

DirectDirectSampling
Samplingin
inLQCD
LQCD
atatfinite
finiteμμ→→Histogram
Histogrammethod
method(Ejiri)(Ejiri)ororSC-LQCD
SC-LQCD
Ohnishi @ NTFL (Feb. 21, 2012)4

Strong Coupling Lattice QCD
Pure YM

Wilson ('74), Creutz ('80),
Munster ('80, '81), Lottini,
Philipsen, Langelage's ('11)

YM+Quarks (MF)

Kawamoto ('80), Kawamoto, Smit ('81),
Damagaard, Hochberg, Kawamoto ('85),
Bilic, Karsch, Redlich ('92),
Fukushima ('03); Nishida ('03),
Kawamoto, Miura, AO, Ohnuma ('07).
Miura, Nakano, AO, Kawamoto ('09)
Nakano, Miura, AO ('10)

Ohnishi @ NTFL (Feb. 21, 2012)5

SC-LQCD with Polyakov Loop Effects at μ=0
T. Z. Nakano, K. Miura, AO, PRD 83 (2011), 016014 [arXiv:1009.1518 [hep-lat]]

P-SC-LQCD reproduces Tc(μ=0) in the strong coupling region( β= 2Nc/g2 ≤ 4)
MC data: SCL (Karsch et al. (MDP), de Forcrand, Fromm (MDP)), N τ =2 (de Forcrand,private), Nτ=4 (Gottlieb et al.('87), Fodor-Katz ('02)), N τ =8 (Gavai et al.('90))

Lattice Unit
Ohnishi @ NTFL (Feb. 21, 2012)6

Strong Coupling Lattice QCD
Pure YM

Wilson ('74), Creutz ('80),
Munster ('80, '81), Lottini,
Philipsen, Langelage's ('11)

YM+Quarks (MF)

YM+Q+Fluc. (MDP)(SCL(1/g2=0))

Kawamoto ('80), Kawamoto, Smit ('81),
Damagaard, Hochberg, Kawamoto ('85),
Bilic, Karsch, Redlich ('92),
Fukushima ('03); Nishida ('03),
Mutter, Karsch ('89),
Kawamoto, Miura, AO, Ohnuma ('07). de Forcrand, Fromm ('10),
Miura, Nakano, AO, Kawamoto ('09)
Nakano, Miura, AO ('10)

de Forcrand, Unger ('11)

Challenge:
Challenge:YM+Q+Fluc.+Finite
YM+Q+Fluc.+FiniteCoupling
CouplingEffects
Effectsde Forcrand, Fromm, Langelage, Miura, Philipsen, Unger ('11), AO, Nakano, Ichihara (in prep.)
Ohnishi @ NTFL (Feb. 21, 2012)7

Auxiliary
Auxiliary field
field effective
effective action
actioninin thethe Strong
Strong Coupling
Coupling Limit
Limit

Ohnishi @ NTFL (Feb. 21, 2012)8

Strong Coupling Expansion
Lattice QCD action (aniso. lattice, unrooted staggered Fermion)
χ1+1

UUS LQCD =S F S G S G =− 2 ∑ tr [U P U P ] f P
2gg plaq.
m0
χ
m011
−

S F = ∑ [ V  x −V  x  ]∑   x  [ x U x , j  x j − x  j U x , j  x ]∑  M x
2 x
2 x , j jx
V  x =e / 22

 x U x ,0  x0 , V −  x =e − /  x0 U x , 0  x , M x =x  x , a =a / , f P =1 or 1/ Strong coupling expansion(Strong coupling limit)

χIgnore plaquette action (1/g )2Integrate out spatial link variablesof min. quark number diagrams(1/d expansion)

U0
χ
V0+U0 +V0-m0Mx

Mx+jdUUU∫
abcd = ad  bc / N cm011
−S eff = ∑ [ V  x−V  x  ]−M x M x j  ∑ M x
2 ∑ x
2 x
4 Nc x, j
Ohnishi @ NTFL (Feb. 21, 2012)9

Introduction of Auxiliary Fields
Bosonization of MM term (Four Fermi (two-body) interaction)
1 x j y  x− j , y ]
∑2 jj,xx,yMeson matrix (V) has positive and negative eigen values

S F =− ∑ M x M x j =− ∑ M x V x , y M y [V x , y = s

f M k =∑ cos k j ,jf M  k =− f M  k  [ k =k  , ,  ]Negative mode = “High” momentum mode
→ Involves a factor exp(i π (x1+x2+x3)) = (-1)**(x1+x2+x3)
in coordinate representation
Bosonization of Negative mode: Extended HS transf.
→ Introducing “ i ” gives rise to the sign problem.
Miura, Nakano, AO (09), Miura, Nakano, AO, Kawamoto (09)exp  A B=∫ d  d  exp[−  − A B −i  A− B]≈exp [−   −A  −
 B]stationary
2Negative

2Ohnishi @ NTFL (Feb. 21, 2012)10

Phase cancellation mechanism in σMCBosonized effective action1∗∗S eff = ∑  x D x , y  y fk[∑Mkkk k ]22 x,y4 N c  k , f k 0
MD x , y = x 0 , y  x , y e  x=/2[]f M  k  e ikx  k ,   x =∑− / 

U x , 0 − x , y 0  x , y e∑k , f M k 02U


y ,0m0 x i  x  x
2  x x , y ,  x=2
2 Nc
k , f M  k 0

Fermion matrix is spatially separated→ Fermion det at each point
Imaginary part (π) involves
ε= (-1)x0+x1+x2+x3 = exp(i π(x0+x1+x2+x3))→ Phase cancellation of nearest neighborspatial site det for π field having low kf M  k  e ikx  kε=1ε=-1ε=-1ε=1

Ohnishi @ NTFL (Feb. 21, 2012)11

Auxiliary Field Monte-Carlo Integral
Effective action of Auxiliary Fields
∗∗
S eff =fk[∑Mkkk k ]24 N c  k , f M k 0−∑ log [ X N  x 3−2 X N  x 2 cosh 3 N    ]x

X N  x = X N [  x , ,  x ,]μ dependence appears only in the log.
σk, πk have to be generated in momentum space, while XN requires
σ(x) and π(x) → Fourier transf. in each step.
XN is complex, and this action has the sign problem.
But the sign problem is milder because of the phase cancellationand is less severe at larger μ.

Let'sLet'stry
tryatatfinite
finiteμμ!!
Ohnishi @ NTFL (Feb. 21, 2012)12

Auxiliary
Auxiliary Field
Field Monte-Carlo
Monte-Carlo (σMC)(σMC)
estimate
estimate ofof thethe phase
phase boundary
boundary

Ohnishi @ NTFL (Feb. 21, 2012)13

Numerical Calculation
44 asymmetric lattice + Metropolis sampling of σk and πk.
Metropolis sampling of full configuration (σk and πk) at a time.
(efficient for small lattice)
Initial cond. = const. σ
Chiral limit (m=0) simulation → Symmetry in σ ↔ - σ
Sign problem is not severe (<cos θ> ~ (0.9-1.0)) in a 44 lattice.
Computer: My PC (Core i7)

Ohnishi @ NTFL (Feb. 21, 2012)14

Results (1): σ distribution
Fixed μ/T simulation: μ/T= 0 ~ 2.4
Low μ region: Second order(Single peak: finite σ → zero)High μ region: First order(Dist. func. has two peaks)

Ohnishi @ NTFL (Feb. 21, 2012)15

Results (2): Susceptibility and Quark density
Weight factor <cos θ>〈cos 〉=Z / Z abs
Z =∫ D  k D  k exp−S eff =∫ D  k D  k exp −Re S eff e i Z abs =∫ D  k D  k exp −Re S eff Chiral susceptibility2=−

1 ∂ log Z
L3 ∂ m 20

Quark number density
1  2 ∂ log Zq =− 3
L N  ∂
Ohnishi @ NTFL (Feb. 21, 2012)16

Results (3): Phase diagram
By taking T = γ2/Nτ ,γ dep. of the phase boundary
becomes small. Bilic et al. ('92)
Definitions of phase boundary
2= σ2 + π2 dist. peak:finite or zero (red curve)
Chiral susceptibility peak(blue)

Fluctuation effectReduction of Tc at μ=0
Enlarged hadron phaseat medium T→ Consistent with MDP
de Forcrand, Fromm ('09); de Forcrand, Unger ('11)
Ohnishi @ NTFL (Feb. 21, 2012)17

Results (4): Larger Lattice
Can we go to a larger lattice ?
dμ/dT > 0 at low T.
→ How about in a larger lattice ?
Suggestion by de Forcrand→ low T behavior is sensitive to Nτ→ 43 x 8

Ohnishi @ NTFL (Feb. 21, 2012)18

SummaryWe have proposed an auxiliary field MC method (σMC)
in SC-LQCD.
To simulate the SCL quark-U0 action (LO in strong coupling (1/g0)
and 1/d (1/d0) expansion) without further approximation.
c.f. Determinantal MC by Abe, Seki
Sign problem is mild in small lattice (<cos θ> ~ (0.9-1) for 44),
because of the phase cancellation coming from nearest neighborinteraction.
Extension to NLO SC-LQCD is straightforward.

Phase boundary is obtained and found to be compatible withrecent MDP results.
Phase boundary is moderately modified from MF results byfluctuations, if T= γ2/Nτ and μ = γ2μ0 scaling is adopted.
σMC results are compatible with MDP results, while the shift of Tc
at μ=0 is around half (LO in 1/d expansion in σMC).
Ohnishi @ NTFL (Feb. 21, 2012)19

Future work
To do:
Larger lattice (84, 163x8, ...)
Finite coupling effects (NLO, NNLO, Polyakov loop, ...)
Higher 1/d terms including baryonic action
Polyakov coupling (back reaction)
Unrooted staggered fermion corresponds to 4 flavors (tates)in continuum→ Different Fermion (e.g. staggered-Wilson fermion).

Ohnishi @ NTFL (Feb. 21, 2012)20

Thank you

Ohnishi @ NTFL (Feb. 21, 2012)21

Clausius-Clapeyron Relation
First order phase boundary → two phases coexists q −s hd
P h=P q  dP h=dP q =−dT
 q− hdP h =h d s h dT , dP q= q d s q dTContinuum theory→ Quark matter has larger entropy and density (dμ /dT < 0)Strong coupling lattice
SCL: Quark density is larger
than half-filling, and “Quark hole”
carries entropy → dμ/dT > 0
NLO, NNLO → dμ/dT < 0AO, Miura, Nakano, Kawamoto ('09)
Ohnishi @ NTFL (Feb. 21, 2012)22

Introduction of Auxiliary Fields

3

Ω= L N τAppendix

23

Fermion Determinant
Faldt, Petersson, 1986

Fermion action is separated to each spatial point and bi-linear→ Determinant of Nτ x Nc matrixexp−V eff /T =∫ dU 0∣eI2

I1−e−

0−e⋮
−e U−I3∣e− U 0ee⋱−−e=∫ dU 0 det [ X N [ ]⊗1c e− /TIN
Nc x NτNU −1 e /TU]

Nc3= X N −2 X N 2 cosh 3 N   2∣I1e0I  /2=[  xi  x x]/2 N c  m0 /−−eIe
2X N =B N B N −2 2 ; N −1
B N = 0 −e− I 3
B N =I N B N −1 B N −2
⋮Appendixe⋱−e−IN24∣