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. Complex Mass Spectrum 2. Sinusoidal oscillation M ê T = 4 5 Mass Spectrum 4 0.008 3 0.006 < tr P † H r L trP H 0 L > C 2 0.004 1 0 1 2 3 4 5 6 0.2 0.002 0.1 0.000 Arg @ l j D 0.0 10 15 20 25 30 35 40 r - 0.1 - 0.2 0 1 2 3 4 5 6 m ê T

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

Introduction

A phase diagram of QCD Temperature T Quark-Gluon Plasma sQGP Critical Point I n h o m o g e n Quarkyonic e o Hadronic Phase u Matter s S ? uSC c B 2SC dSC Liquid-Gas CFL Color Superconductors 0 CFL- K , Crystalline CSC Baryon Chemical Potential m B Nuclear Superfluid Meson supercurrent Gluonic phase, Mixed phase <Fukushima and Hatsuda, 2010> • Lattice simulations at finite μ is hard: the sign problem. • Consider finite-density QCD in view of the Polyakov loop.

Polyakov loop • Wilson line in the temporal direction: Static quark, P ( ~ x ) R 1 /T x ) = P e i dx 4 A 4 ( x ) P ( ~ 0 1/T • Order parameter Low T: h tr P ( ~ x ) i = 0 ! F q = 1 Static anti-quark, P † ( ~ x ) Confined phase: unbroken Z(N) symmetry High T: h tr P ( ~ x ) i 6 = 0 ! F q = Finite 1/T Deconfined phase: broken Z(N) symmetry

Sign problem • Partition function of QCD Z DA e − S Y M det M ( µ ) Z = • Fermion determinant is complex det M ( − µ ) = [det M ( µ )] ∗ Sign problem No positive weight, no importance sampling. • There are many approaches.... A new approach: Lefschetz thimble <E. Witten, 2010> <AuroraScience Collaboration, 2012> <H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu, and T. Sano, 2013> and many more

Sign problem in the MFA • Effective models of the Polyakov loop - Unphysical results at conventional real saddle points. <Dumitru, Pisarski and Zschiesche, 2005> <Fukushima and Hidaka, 2007> - C omplex but CK -symmetric saddle point can fix the problems. <HN, M. Ogilvie, K. Pangeni, 2014 & 2015> • Lefschetz-thimble technique - Integration cycles respect CK symmetry. <Y. Tanizaki, HN, K. Kashiwa, 2015> CK symmetry seems to play a crucial role in the complex domain.

Charge conjugation ( C ) and complex conjugation ( K ) • CK symmetry in finite-density QCD Z DA e − S Y M det M ( µ ) Z = C : A µ → − A t µ det M ( µ ) = [det M ( − µ )] ∗ K : A µ → A ∗ CK µ • 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 a β = 1/T t x P † P i +1 L i H 0 = g 2 β T 0 = h P i +1 | e − aH 0 | P i i where 2 C <Marinov and Terentev 1979> <Menotti and Onofri 1981> etc

Lattice QCD in (1+1)-dim • The basis of group representation, r = r ¯ 1 3 3 8 1   1 0 0 0 T 0 = h r 0 | e � a g 2 β 2 C | r i 1 3 0 0 0   e 4 / 3 − →   1 ¯ 0 0 0 3   e 4 / 3 1 0 0 0 8 e 3 = r ’ ag 2 β - Set . = 1 2 - Show up to 6 highest eigenvalues: For pure SU(3), . r = 1 , 3 , ¯ 3 , 6 , ¯ 6 , 8 - 16 X 16 matrix is sufficient when there is a mixing.

Lattice QCD in (1+1)-dim • Inclusion of static quarks T = h r 0 | e � aH 0 / 2 det(1 + z 1 P ) det(1 + z 2 P † ) e � aH 0 / 2 | r i quark anti-quark • Symmetries - Particle-Antiparticle: (z 1 , z 2 ) → (z 2 , z 1 ) - Particle-Hole: (z 1 , z 2 ) → (1/z 1 , 1/z 2 ) because det(1 + zP ) = z N det(1 + P † /z ) • Alternative Hamiltonian, z 1 tr F P + z 2 tr F P † � � H = H 0 − β Real spectrum for low-lying eigenvalues. <P . Meisinger, M. Ogilvie and T. Wiser, 2010>

Lattice QCD in (1+1)-dim • Transfer matrix is not Hermitian. + z 2 + z 3 det(1 + z 1 P ) = 1 + z 1 1 1 Pure SU(3) With quarks (z 2 = 0) z 2     1 0 0 0 1 + z 3 z 1 0 1 1 e 2 / 3 e 2 / 3 1 z 2 1+ z 3 z 2 0 0 0   z 1 1 1 1   e 4 / 3   e 2 / 3 e 4 / 3 e 4 / 3 e 13 / 6  1   z 2 1+ z 3  0 0 0 z 1 z 1 1 1     e 4 / 3 e 2 / 3 e 4 / 3 e 4 / 3 e 13 / 6   1 z 2 1+ z 3 0 0 0 z 1 0 1 1 e 3 e 13 / 6 e 13 / 6 e 3 - Transfer matrix is real but not symmetric when z 1 and z 2 are not equal. - A manifestation of the sign problem.

Generalization to higher dimensions φ (0) � ( ~ x ) • 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. φ (0) The leading diagrams for h � ( ~ x ) � (0) i are the shortest possible paths. <Kogut and Sinclair, 1981> � ( ~ x )

PT -symmetric (or CK -symmetric) Systems

Source: Jorge Cham (2015)

Non-Hermitian PT -symmetric QM H = p − ( ix ) N • “Classic” PT -symmetric Quantum Mechanics: <C. Bender and S. Boettcher, 1998> - N = 2: Harmonic oscillator - N = 3: Non-hermitian Hamiltonian - 1 < N < 2: Complex eigenvalues • Eigenvalues are either real or form a complex conjugate pair: H | j i = E j | j i Proof If H ( PT | j i ) = PT H | j i = E ∗ j ( PT | j i ) then

Correlation functions in PT -symmetric system <P . Meisinger and M. Ogilvie, 2014> • PT -symmetric partition function T → diag( e − m 0 a , e − m 1 a , . . . ) ⇣ ⌘ Z = tr T N = X X e − Lm q + e − Lm ∗ e − Lm p + q p q Manifestly real from CK symmetry. • Three possible scenarios for a PT -symmetric system I. All m j are real. II. m 0 is real but some m j form a complex conjugate pair. ← Our model III. m 0 is complex (two ground states).

Correlation functions in PT -symmetric system <P . Meisinger and M. Ogilvie, 2014> • 1-point function (Polyakov loop) h tr F P ( x ) i = "X ⌘# 1 ⇣ e − β m p h p | tr F P | p i + e − β m q h q | tr F P | q i + e − β m ∗ X q h q ∗ | tr F P | q ∗ i Z p q Manifestly real from CK symmetry. • 2-point function ∞ e − xm j h 0 | tr F P † | j i h j | tr F P | 0 i X tr F P † ( x ) tr F P (0) ⌦ ↵ C = ( L → ∞ ) j =1 Complex complex pairs of m j give rise to a sinusoidal exponential decay.

Results

Hermitian Case (z = z 1 = z 2 ) 6 • Mass spectrum is real. 5 Mass Spectrum 4 • The Polyakov loop and the conjugate 3 loop are the same. 2 1 • Invariant under z → 1/z 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.8 Polyakov Loops • Peaks at z=1 (M=0). 0.6 0.4 0.2 - Mass Spectrum = Re[m j - m 0 ] 0.0 - Arg[ λ j ] = Im[m j - m 0 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 z - <TrP> - <TrP † >

5 Finite density (z 2 =0) Mass Spectrum 4 • Mass spectrum becomes complex: 3 complex conjugate pairs form. 2 1 • Invariant under z 1 → 1/z 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.1 Arg @ l j D • The peak of mass spectrum at z 1 =1 0.0 (M= μ ). - 0.1 - 0.2 • The Polyakov loop and the conjugate 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 loop are different. 0.8 Polyakov Loops - Mass Spectrum = Re[m j - m 0 ] 0.6 0.4 - Arg[ λ j ] = Im[m j - m 0 ] 0.2 - <TrP> 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 - <TrP † > z 1

M ê T = 4 Fermion Mass / T = 4 5 Mass Spectrum 4 3 • Mass spectrum becomes complex: 2 complex conjugate pairs form. 1 0 1 2 3 4 5 6 0.2 • The peak of the spectrum at the 0.1 Hermitian point, μ = M. Arg @ l j D 0.0 - 0.1 • Similar structure as the case of z 2 =0 - 0.2 because M/T is large. 0 1 2 3 4 5 6 1.0 0.8 Polyakov Loops - Mass Spectrum = Re[m j - m 0 ] 0.6 0.4 - Arg[ λ j ] = Im[m j - m 0 ] 0.2 - <TrP> 0.0 0 1 2 3 4 5 6 - <TrP † > m ê T

M ê T = 2 Fermion Mass / T = 2 5 Mass Spectrum 4 3 • For lower mass, the symmetric 2 structure disappears. 1 0 1 2 3 4 5 6 • Lower eigenvalues become real at 0.2 lower μ . 0.1 Arg @ l j D 0.0 - 0.1 - 0.2 1.0 0 1 2 3 4 5 6 Polyakov Loops 0.8 - Mass Spectrum = Re[m j - m 0 ] 0.6 0.4 - Arg[ λ j ] = Im[m j - m 0 ] 0.2 - <TrP> 0.0 0 1 2 3 4 5 6 - <TrP † > m ê T

M ê T = 1 Fermion Mass / T = 1 5 Mass Spectrum 4 3 • The larger eigenvalues become real at 2 lower μ . 1 0 1 2 3 4 5 6 0.2 0.1 Arg @ l j D 0.0 - 0.1 - 0.2 1.0 0 1 2 3 4 5 6 Polyakov Loops 0.8 - Mass Spectrum = Re[m j - m 0 ] 0.6 0.4 - Arg[ λ j ] = Im[m j - m 0 ] 0.2 - <TrP> 0.0 0 1 2 3 4 5 6 - <TrP † > m ê T

M ê T = 0 Fermion Mass / T = 0 5 Mass Spectrum 4 3 • Spectrum remains complex for 2 any M/T< ∞ . 1 0 1 2 3 4 5 6 0.2 0.1 Arg @ l j D 0.0 - 0.1 - 0.2 1.0 0 1 2 3 4 5 6 Polyakov Loops 0.8 - Mass Spectrum = Re[m j - m 0 ] 0.6 0.4 - Arg[ λ j ] = Im[m j - m 0 ] 0.2 - <TrP> 0.0 0 1 2 3 4 5 6 - <TrP † > m ê T

2-point function • Sinusoidal modulation. • Loss of spectral positivity. 0.008 ag 2 β /2 = 0.1, z1 = 0.8, z2 = 0 0.006 < tr P † H r L trP H 0 L > C 0.004 0.002 0.000 10 15 20 25 30 35 40 r

Discussions