Auxiliary field Monte-Carlo studyof the QCD phase diagram at strong coupling
Akira Ohnishi (YITP)
Takashi Z. Nakano (Kyoto U./ YITP)
Introduction
Strong coupling lattice QCD (mean field results)
Auxiliary field effective action in the Strong Coupling Limit
Monte-Carlo estimate of the phase boundary
Summary

AO, T. Z. Nakano, in prep.
Ohnishi, RG workshop (2011/09/07)

1

QCD Phase diagram
Phase transition at high T
Physics of early universe: Where do we come from ?
RHIC, LHC, Lattice MC, pQCD, ….

High μ transition
Physics of neutron stars:
Where do we go ?
RHIC-BES, FAIR, J-PARC,Astro-H, Grav. Wave, …
Sign problem in Lattice MC→ Model studies,Approximations,Functional RG, ...

Ohnishi, RG workshop (2011/09/07)

2

CP sweep during BH formation

AO, H. Ueda, T. Z. Nakano, M. Ruggieri, K. Sumiyoshi,
PLB, to appear [arXiv:1102.3753 [nucl-th]]Ohnishi, RG workshop (2011/09/07)

3

QCD based approaches to Cold Dense Matter
Effective Models(P)NJL, (P)QM, Random Matrix, …
E.g.: K.Fukushima, PLB 695('11)387 (PNJL+Stat.).

Functional (Exact, Wilsonian) RG
E.g.: T. K. Herbst, J. M. Pawlowski, B. J. Schaefer, PLB
696 ('11)58 (PQM-FRG).

Expansion / Extrapolation from μ=0

McLerran, Redlich, Sasaki ('09)

AC, Taylor expansion, … → μ/T<1Cumulant expansion of θ dist.
(S. Ejiri, ...)

Strong Coupling Lattice QCD
Mean field approaches

Herbst, Pawlowski, Schafer, ('11)

Monomer-Dimer-Polymer (MDP)
simulation
Ohnishi, RG workshop (2011/09/07)

4Strong Coupling Lattice QCD for finite μ
Mean Field approaches
Damagaard, Hochberg, Kawamoto ('85); Bilic, Karsch,Redlich ('92); Fukushima ('03); Nishida ('03);
Kawamoto, Miura, AO, Ohnuma ('07).

MDP simulation
Karsch, Mutter('89); de Forcrand, Fromm ('10); deForcrand, Unger ('11)

Partition function = sum of config.
weights of various loops.

Miura, Nakano, AO, Kawamoto,arXiv:1106.1219

Applicable only to Strong Coupling Limit(1/g2=0) at present

de Forcrand, Unger ('11)

Can
Canwe
weinclude
includebothbothfluctuation
fluctuationand
andfinite
finitecoupling
couplingeffects
effects??
→→One
Oneof
ofthe
thecandidatescandidates==Auxiliary
Auxiliaryfield
fieldMCMCOhnishi, RG workshop (2011/09/07)

5StrongStrong Coupling
Coupling Lattice
Lattice QCD
QCD

Ohnishi, RG workshop (2011/09/07)

6

SC-LQCD with Fermions & Polyakov loop: Outline
Effective Action & Effective Potential (free energy density)
χ- SLQCD
ημημ1+−+U +U Z =∫ D [ χ , χ2
̄ ,U 0, U j ]exp
22gm0χ
̄
Spatial link integral
NLO+∫ DU U ab U cd =δ ad δbc / NSCL=∫ D [ χ , χ̄ , U 0 ]exp

NNLOBosonization& MF Approx.

≈∫ D [χ , ̄χ , U 0 ]exp(−S eff [ χ , χ̄ ,U 0, Φstat. ])Polyakov loop

Fermion + U0 integral≈exp(−V F eff (Φ stat ;T ,μ)/T )Ohnishi, RG workshop (2011/09/07)

7

SC-LQCD with FermionsEffective Action with finite coupling corrections
Integral of exp(-SG) over spatial links with exp(-SF) weight → Seffn−1n
S eff =S SCL −log 〈 exp−S G 〉=S SCL −∑〈 S G 〉cn!
n=1
<SGn>c=Cumulant (connected diagram contr.) c.f. R.Kubo('62)

SCL (Kawamoto-Smit, '81)
NLO (Faldt-Petersson, '86)

NNLO (Nakano, Miura, AO, '09)
Ohnishi, RG workshop (2011/09/07)

8

SC-LQCD Eff. Pot. with Fermions & Polyakov loop
Effective potential [free energy density, NLO + LO(Pol. loop)]
aux. fields
w.f. ren.
zero point E.
thermalquad. coef.
Haar measure
Strong coupling lattice QCD with Polyakov loop (P-SC-LQCD)= Polyakov loop extended Nambu-Jona-Lasino (PNJL) model(Haar measure method, quadratic term fixed)+ higher order terms in aux. fields- quark momentum integral
Ohnishi, RG workshop (2011/09/07)

9P-SC-LQCD 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, RG workshop (2011/09/07)

10

Approximations in Pol. loop extended SC-LQCD
1/g2 expansion
with Pol. loops

Fluctuations
Fluctuations

1/dexpansiond=spatialdim.

Nakano, Miura, AO ('11)

1/g2 expansion
with quarks
Ohnishi, RG workshop (2011/09/07)

11

Approximations in Pol. loop extended SC-LQCD
Strong coupling expansion
Fermion terms: LO(1/g0, SCL), NLO(1/g2), NNLO (1/g4)
Plaquette action: LO (1/g2Nτ)Large dimensional approximation
1/d expansion (d=spatial dim.)→ Smaller quark # configs. are preferred.
Σj Mx Mx+j = O(1/d0) → M ∝ d-1/2 → χ ∝ d-1/4
Only LO (1/d0) terms are mainly evaluated.

(Unrooted) staggered Fermion
Nf=4 in the continuum limit.

Mean field approximation
Auxiliary fields are assumed
to be constant.

ThisThiswork:
work:
Auxiliary
AuxiliaryField
FieldMCMCininSCL
SCL

Ohnishi, RG workshop (2011/09/07)

12

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

Ohnishi, RG workshop (2011/09/07)

13Strong Coupling Expansion
Lattice QCD action (aniso. lattice, unrooted staggered Fermion)χS LQCD =S F +S G

1+S G =− 2 ∑ tr [ U P +U P ] f Pg plaq.

UχU+m012g

m011+−+S F = ∑ [ V ( x)−V ( x) ]+∑ η ( x) [ χ̄ x U x , j χ x+ ̂j− χ̄ x+ ̂j U x , j χ x ]+∑ γ M x2 x2γ 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 +U0
χ
V0+

V0-

m0

Mx

Mx+j

dUUU
∫
ab
cd = ad  bc / N cm01
1−S eff = ∑ [ V  x−V  x  ]−M x M x j  ∑ M x
2 ∑ x
2 x
4 Nc x, j
Ohnishi, RG workshop (2011/09/07)

14

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

S F =−α ∑ M x M x+ ̂j =−α ∑ M x V x , y M y [V x , y =(s)f M (k )=∑ cos k j ,f M ( ̄k )=− f M (k ) [ ̄k =k+(π , π , π) ]jNegative 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)22exp(α A B)=∫ d φ d ϕexp[−α(φ −( A+B)φ+ϕ −i ( A− B)ϕ)]≈exp[−α( ψ̄ ψ− A ψ− ψ̄ B)]stationary
Negative

Ohnishi, RG workshop (2011/09/07)

15

Introduction of Auxiliary Fields

3

Ω= L N τAppendix

16

Phase cancellation mechanism in σMCBosonized effective action1∗∗ΩS eff = ∑ χDχ+f(k)σσ+π̄x x , y y[∑Mk kk πk ]22 x,y4 N c γ k , f (k )>0Mμ/γ

D x , y =δ x+ 0̂ , y δ x , y e

σ (x)=2[]f M (k)e σ k , π(x)=∑U x ,0 −δ x , y+ 0̂ δ x , y e∑k , f M (k)>0−μ / γ

2U+y ,0m0
σ x +i ε x π x+2 Σx + γ δ x , y , Σx =22 Nc γikx

ikx

k , f M (k)>0Fermion 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 kOhnishi, RG workshop (2011/09/07)

f M (k)e π k

ε=1
ε=-1

ε=-1ε=117

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I2I1−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]

Nc

3= 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−

IN18

∣

Auxiliary Field Monte-Carlo Integral
Effective action of Auxiliary Fields∗∗
S eff = Ω 2f(k)σσ+π∑[Mk kk πk ]4 N c γ k , f M (k)>0−∑ log [ X N ( x ) −2 X N ( x)+2 cosh (3 N τ μ) ]3xX N (x)= X N [σ (x , τ) , π(x , τ)]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, RG workshop (2011/09/07)

19

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

Ohnishi, RG workshop (2011/09/07)

20

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, RG workshop (2011/09/07)

21

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, RG workshop (2011/09/07)

22

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  k exp −Re S eff Chiral susceptibility2=−1 ∂ log ZL3 ∂ m 20

Quark number density
1 γ 2 ∂ log Zρq =− 3
L N τ ∂μOhnishi, RG workshop (2011/09/07)

23

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=0Enlarged hadron phaseat medium T→ Consistent with MDP
de Forcrand, Fromm ('09); de Forcrand, Unger ('11)
Ohnishi, RG workshop (2011/09/07)

24

Clausius-Clapeyron Relation
First order phase boundary → two phases coexists q−s hdμ
P h = P q → dP h=dP q →=−dTρ q−ρ hdP h=ρ h d μ+s h dT , dP q =ρq d μ+s q dT
Continuum 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, RG workshop (2011/09/07)

25

SummaryWe have proposed an auxiliary field MC method (σMC),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.
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 ofTc at μ=0 is around half (LO in 1/d expansion in σMC).
Extension to NLO SC-LQCD is straightforward.
To do: Larger lattice, finite coupling, different Fermion, higher
1/d terms including baryonic action and chiral Polyakov coupling.
Ohnishi, RG workshop (2011/09/07)

26

Thank you

Ohnishi, RG workshop (2011/09/07)

27