Complex spectrum of finite-density QCD Hiromichi Nishimura Johann - - PowerPoint PPT Presentation

Complex spectrum of finite-density QCD

Hiromichi Nishimura

Johann Wolfgang Goethe-Universität

Talk@Frankfurt

07 December 2015

<HN, M. Ogilvie, and K. Pangeni, in preparation>

Summary

1 2 3 4 5 6 1 2 3 4 5 Mass Spectrum MêT=4 1 2 3 4 5 6

• 0.2
• 0.1

0.0 0.1 0.2 mêT Arg@ljD

• 1. Complex Mass Spectrum
• 2. Sinusoidal oscillation

10 15 20 25 30 35 40 0.000 0.002 0.004 0.006 0.008 r <tr P†HrL trPH0L>C

• Introduction
• Polyakov loop, Sign problem, CK symmetry
• Formalism
• Strong-coupling lattice QCD, PT-symmetric system
• Results & Discussions
• Conclusions

Outline

Introduction

A phase diagram of QCD

sQGP

uSC dSC CFL 2SC

Critical Point

Quarkyonic Matter Quark-Gluon Plasma Hadronic Phase Color Superconductors

?

Temperature T Baryon Chemical Potential mB

I n h

• m
• g

e n e

• u

s S c B

Liquid-Gas

Nuclear Superfluid CFL-K , Crystalline CSC Meson supercurrent Gluonic phase, Mixed phase

• Lattice simulations at finite μ is hard: the sign problem.
• Consider finite-density QCD in view of the Polyakov loop.

<Fukushima and Hatsuda, 2010>

• Wilson line in the temporal direction:

Polyakov loop

P(~ x) = Pei

R 1/T dx4A4(x)

Confined phase: unbroken Z(N) symmetry Deconfined phase: broken Z(N) symmetry

• Order parameter

Low T: htrP(~

x)i = 0 ! Fq = 1 htrP(~ x)i 6= 0 ! Fq = Finite

High T:

1/T 1/T

Static quark, P(~

x)

Static anti-quark, P †(~

x)

• Fermion determinant is complex

det M (−µ) = [det M (µ)]∗ Z = Z DA e−SY M det M(µ)

• Partition function of QCD

No positive weight, no importance sampling. Sign problem

Sign problem

<H. Fujii, D. Honda, M. Kato,

• Y. Kikukawa, S. Komatsu, and T. Sano, 2013>

<AuroraScience Collaboration, 2012> <E. Witten, 2010> and many more

A new approach: Lefschetz thimble

• There are many approaches....

• Effective models of the Polyakov loop
• Lefschetz-thimble technique
• Unphysical results at conventional real saddle points.

<Y. Tanizaki, HN, K. Kashiwa, 2015>

• Integration cycles respect CK symmetry.

Sign problem in the MFA

<HN, M. Ogilvie, K. Pangeni, 2014 & 2015>

CK symmetry seems to play a crucial role in the complex domain.

• Complex but CK-symmetric saddle point can fix the problems.

<Dumitru, Pisarski and Zschiesche, 2005> <Fukushima and Hidaka, 2007>

CK

det M (µ) = [det M (−µ)]∗ C : Aµ → −At

µ

K : Aµ → A∗

µ

Z = Z DA e−SY M det M(µ)

Charge conjugation (C) and complex conjugation (K)

• CK symmetry in finite-density QCD
• CK symmetry is an antiunitary operation.

Setup

Lattice QCD in (1+1)-dim

• “Only quantum question”: What is the transfer matrix?
• Solvable for SU(N)

Yang-Mills on (1+1)-dim Lattice

Pi+1 P †

i

a

T0 = hPi+1| e−aH0 |Pii

H0 = g2β 2 C

where x t L β = 1/T

<Marinov and Terentev 1979> <Menotti and Onofri 1981> etc

• The basis of group representation, r

− →

    1

1 e4/3 1 e4/3 1 e3

   

1 3 ¯ 3 8 1 3 ¯ 3 8

= r r’ =

T0 = hr0| ea g2β

2 C |ri

• Set .
• Show up to 6 highest eigenvalues: For pure SU(3), .
• 16 X 16 matrix is sufficient when there is a mixing.

ag2β 2 = 1

r = 1, 3, ¯ 3, 6, ¯ 6, 8

Lattice QCD in (1+1)-dim

• Inclusion of static quarks

<P . Meisinger, M. Ogilvie and T. Wiser, 2010>

quark anti-quark

T = hr0| eaH0/2 det(1 + z1P) det(1 + z2P †) eaH0/2 |ri

Lattice QCD in (1+1)-dim

H = H0 − β

• z1trF P + z2trF P †
• Alternative Hamiltonian,

Real spectrum for low-lying eigenvalues.

• Particle-Antiparticle: (z1, z2) → (z2, z1)
• Particle-Hole: (z1, z2) → (1/z1, 1/z2) because det(1 + zP) = zNdet(1 + P †/z)
• Symmetries

• Transfer matrix is not Hermitian.

    1

1 e4/3 1 e4/3 1 e3

   

      1 + z3

1 z1 e2/3 z2

1

e2/3 z2

1

e2/3 1+z3

1

e4/3 z1 e4/3 z2

1

e13/6 z1 e2/3 z2

1

e4/3 1+z3

1

e4/3 z1 e13/6 z1 e13/6 z2

1

e13/6 1+z3

1

e3

     

Pure SU(3) With quarks (z2 = 0)

• Transfer matrix is real but not symmetric when z1 and z2 are not equal.
• A manifestation of the sign problem.

Lattice QCD in (1+1)-dim

+ z2

1

+ z1 det(1 + z1P) = 1 + z3

1

• In (1+1)-dim with static quarks, the

results of mass spectrum are exact.

• They are also the results for higher

dimensions at leading order in strong coupling.

The leading diagrams for are the shortest possible paths.

h(~ x)(0)i

<Kogut and Sinclair, 1981>

Generalization to higher dimensions

φ(0) (~ x) φ(0) (~ x)

PT-symmetric (or CK-symmetric) Systems

Source: Jorge Cham (2015)

Non-Hermitian PT-symmetric QM

• “Classic” PT-symmetric Quantum Mechanics:
• Eigenvalues are either real or form a complex conjugate pair:

H = p − (ix)N

• N = 2: Harmonic oscillator
• N = 3: Non-hermitian Hamiltonian
• 1 < N < 2: Complex eigenvalues

<C. Bender and S. Boettcher, 1998>

If then Proof

H |ji = Ej |ji H (PT |ji) = PT H |ji = E∗

j (PT |ji)

Correlation functions in PT-symmetric system

<P . Meisinger and M. Ogilvie, 2014>

• PT-symmetric partition function

Z = tr T N = X

p

e−Lmp + X

q

⇣ e−Lmq + e−Lm∗

q

⌘

Manifestly real from CK symmetry.

T → diag(e−m0a, e−m1a, . . . )

• Three possible scenarios for a PT-symmetric system
• I. All mj are real.
• II. m0 is real but some mj form a complex conjugate pair.
• III. m0 is complex (two ground states).

← Our model

Correlation functions in PT-symmetric system

<P . Meisinger and M. Ogilvie, 2014>

• 2-point function

(L → ∞)

Complex complex pairs of mj give rise to a sinusoidal exponential decay.

• 1-point function (Polyakov loop)

htrF P(x)i =

1 Z "X

p

e−βmp hp| trF P |pi + X

q

⇣ e−βmq hq| trF P |qi + e−βm∗

q hq∗| trF P |q∗i

⌘#

Manifestly real from CK symmetry.

⌦ trF P †(x) trF P(0) ↵

C = ∞

X

j=1

e−xmj h0| trF P † |ji hj|trF P |0i

Results

Hermitian Case (z = z1= z2)

• Mass spectrum is real.
• The Polyakov loop and the conjugate

loop are the same.

• Invariant under z →1/z
• Peaks at z=1 (M=0).

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 6 Mass Spectrum 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 z Polyakov Loops

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

Finite density (z2=0)

• Mass spectrum becomes complex:

complex conjugate pairs form.

• Invariant under z1 →1/z1
• The peak of mass spectrum at z1=1

(M=μ).

• The Polyakov loop and the conjugate

loop are different.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 Mass Spectrum 0.0 0.5 1.0 1.5 2.0 2.5 3.0

• 0.2
• 0.1

0.0 0.1 0.2 Arg@ljD 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 z1 Polyakov Loops

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

Fermion Mass / T = 4

• Mass spectrum becomes complex:

complex conjugate pairs form.

• The peak of the spectrum at the

Hermitian point, μ = M.

• Similar structure as the case of z2=0

because M/T is large.

1 2 3 4 5 6 1 2 3 4 5 Mass Spectrum MêT=4 1 2 3 4 5 6

• 0.2
• 0.1

0.0 0.1 0.2 Arg@ljD 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 mêT Polyakov Loops

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

Fermion Mass / T = 2

• For lower mass, the symmetric

structure disappears.

• Lower eigenvalues become real at

lower μ.

1 2 3 4 5 6 1 2 3 4 5 Mass Spectrum MêT=2 1 2 3 4 5 6

• 0.2
• 0.1

0.0 0.1 0.2 Arg@ljD 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 mêT Polyakov Loops

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

Fermion Mass / T = 1

1 2 3 4 5 6 1 2 3 4 5 Mass Spectrum MêT=1 1 2 3 4 5 6

• 0.2
• 0.1

0.0 0.1 0.2 Arg@ljD 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 mêT Polyakov Loops

• The larger eigenvalues become real at

lower μ.

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

Fermion Mass / T = 0

• Spectrum remains complex for

any M/T<∞.

1 2 3 4 5 6 1 2 3 4 5 Mass Spectrum MêT=0 1 2 3 4 5 6

• 0.2
• 0.1

0.0 0.1 0.2 Arg@ljD 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 mêT Polyakov Loops

• Mass Spectrum = Re[mj - m0]
• Arg[λj] = Im[mj - m0]
• <TrP>
• <TrP†>

2-point function

10 15 20 25 30 35 40 0.000 0.002 0.004 0.006 0.008 r <tr P†HrL trPH0L>C

ag2β/2 = 0.1, z1 = 0.8, z2 = 0

• Sinusoidal modulation.
• Loss of spectral positivity.

Discussions

QCD & Liquid-Gas

• Radial distribution g(r) for a liquid of size σ.
• Often relevant near liquid-gas transition.

<Reichman and Charbonneau, 2005>

• Oscillation in density correlation function

T μ

“Hadron gas” “QGP-liquid” QCD phase diagram

<Steinheimer et al, 2014>

• Oscillation in Polyakov-loop correlators
• Due to screening between two quarks.
• Color-charge density oscillation.
• Maybe relevant for heavy-ion collisions??

Phenomenology

10 20 30 39

Ê

100 200 300 400 50 100 150 200 250 300 Μ HMeVL T HMeVL

αs = 1

PNJL “Type B”

• Complex CK-symmetric saddle point of the effective potential

αs = 1

25 35 45 55

Ê

100 200 300 400 50 100 150 200 250 300 Μ HMeVL T HMeVL

PNJL “Type A”

hA4(r)A4(0)i ⇠ Exp [r κR] r (κR cos [r κI] + κI sin [r κI])

Mab = ∂2Veff ∂Aa

4∂Ab 4

mev = κR ± iκI − →

<HN, M. Ogilvie, K. Pangeni, 2015>

Heavy-quark

• Complex CK-symmetric saddle point of the effective potential
• B
• μ ()

()

• Lattice simulations could differentiate the models of confinement.

• We constructed the transfer matrix of Lattice QCD in strong-coupling and

heavy-quark limits.

• The mass matrix is non-hermitian, but CK-symmetric.
• Mass spectrum becomes complex when chemical potential is non-zero.
• Oscillation of correlation functions of the Polyakov loops. Should be
• bservable in lattice simulations.

Conclusions