QCD AT FINITE SIGN PROBLEM fermion determinant is complex [det M ( - PowerPoint PPT Presentation

N ONPERTURBATIVE SIMULATIONS AT FINITE CHEMICAL POTENTIAL Gert Aarts Swansea University Budapest, April 2009 – p.1

O UTLINE sign problem at finite chemical potential a revived approach: stochastic quantization relativistic Bose gas: phase structure sign and Silver Blaze problems analytical understanding: complex Langevin dynamics in mean field approximation Budapest, April 2009 – p.2

QCD AT FINITE µ SIGN PROBLEM fermion determinant is complex [det M ( µ )] ∗ = det M ( − µ ) fluctuating sign det M ( µ ) = | det M ( µ ) | e iϕ severe sign problem in thermodynamic limit: average phase factor in phase quenched theory � e iϕ � pq = e − Ω∆ f → 0 as Ω → ∞ Ω = four-volume Budapest, April 2009 – p.3

P HASE TRANSITIONS AT FINITE DENSITY QCD AND QCD LIKE THEORIES T µ lattice QCD most effective around µ � T , T ∼ T c sign problem severe in hadronic and exotic phases Budapest, April 2009 – p.4

P HASE TRANSITIONS AT FINITE DENSITY QCD AND QCD LIKE THEORIES 2 0.8 m � 0.07 1.75 0805.1939 [hep-lat] 1.5 0.6 Han & Stephanov 1.25 0.4 T 1 0.75 CP 0.2 0.5 0 1st order 0.25 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Μ model study of sign problem in random matrix theory severe at small T and µ � = 0 Budapest, April 2009 – p.4

P HASE TRANSITIONS AT FINITE DENSITY QCD AND QCD LIKE THEORIES complex action: S ∗ ( µ ) = S ( − µ ) intruiging questions: how severe is the sign problem in practice? thermodynamic limit? phase transitions? how relevant is the sign problem? Silver Blaze problem? Cohen ’03 . . . Budapest, April 2009 – p.4

QCD AT FINITE µ SIGN PROBLEM important configurations differ in an essential way from those obtained at µ = 0 or with | det M | cancelation between configurations with ‘positive’ and ‘negative’ weight how to pick the dominant configurations in the path integral? Budapest, April 2009 – p.5

QCD AT FINITE µ SIGN PROBLEM important configurations differ in an essential way from those obtained at µ = 0 or with | det M | cancelation between configurations with ‘positive’ and ‘negative’ weight how to pick the dominant configurations in the path integral? radically different approach: complexify all degrees of freedom stochastic quantization and complex Langevin dynamics Budapest, April 2009 – p.5

R EADING MATERIAL original suggestion Parisi & Wu ’81, Parisi, Klauder ’83 lots of activity in 80’s Damgaard and Hüffel, Physics Reports ’87 application to finite µ : three-dimensional spin models Karsch & Wyld ’85, . . . stopped because of numerical problems (runaways, instabilities) renewed interest: Minkowski dynamics Berges, Borsanyi, Sexty, Stamatescu ’05-.. Budapest, April 2009 – p.6

R EADING MATERIAL this talk: can stochastic quantization evade the sign problem? – the relativistic Bose gas at finite chemical potential 0810.2089 [hep-lat], PRL complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas, 0902.4686 [hep-lat] QCD with static quarks + related models: with I.O. Stamatescu: stochastic quantization at finite chemical potential, 0807.1597 [hep-lat], JHEP with I.O.S.: Lattice proceedings, 0809.5527 [hep-lat] SEWM proceedings: 0811.1850 [hep-ph] Budapest, April 2009 – p.7

S TOCHASTIC QUANTIZATION L ANGEVIN DYNAMICS field theory Parisi & Wu ’81 Dφ e − S � path integral Z = Langevin dynamics in “fifth” time direction ∂φ ( x, θ ) = − δS [ φ ] δφ ( x, θ ) + η ( x, θ ) ∂θ Gaussian noise � η ( x, θ ) η ( x ′ , θ ′ ) � = 2 δ ( x − x ′ ) δ ( θ − θ ′ ) � η ( x, θ ) � = 0 reach equilibrium as θ → ∞ motivated by Brownian motion Budapest, April 2009 – p.8

S TOCHASTIC QUANTIZATION L ANGEVIN DYNAMICS force ∂S/∂φ complex: Parisi, Klauder ’83 complexify Langevin dynamics φ → φ R + iφ I example: real scalar field coupled Langevin eqs ∂φ R ∂θ = − Re δS � φ → φ R + iφ I + η � δφ � ∂φ I ∂θ = − Im δS � � δφ � φ → φ R + iφ I observables: analytic extension � O ( φ ) � → � O ( φ R + iφ I ) � Budapest, April 2009 – p.8

S TOCHASTIC QUANTIZATION L ANGEVIN DYNAMICS associated Fokker-Planck equation ∂P [ φ, θ ] δ � δ δS [ φ ] � � d d x = δφ ( x, θ ) + P [ φ, θ ] ∂θ δφ ( x, θ ) δφ ( x, θ ) P [ φ ] ∼ e − S stationary solution: real action: formal proofs of convergence P [ φ, θ ] = e − S [ φ ] � e − λθ P λ [ φ ] + Z λ> 0 complex action: theoretical status less clear cut but all other methods fail! Budapest, April 2009 – p.8

P HASE TRANSITIONS AT FINITE DENSITY QCD AND QCD LIKE THEORIES intruiging questions: how severe is the sign problem? thermodynamic limit? phase transitions? Silver Blaze problem? Cohen ’03 . . . study in a model with a phase diagram with similar features as QCD at low temperature ⇒ relativistic Bose gas at nonzero µ Budapest, April 2009 – p.9

R ELATIVISTIC B OSE GAS AT NONZERO µ PHASE TRANSITIONS AND THE S ILVER B LAZE continuum action � | ∂ ν φ | 2 + ( m 2 − µ 2 ) | φ | 2 � d 4 x S = + µ ( φ ∗ ∂ 4 φ − ∂ 4 φ ∗ φ ) + λ | φ | 4 � complex scalar field, d = 4 , m 2 > 0 S ∗ ( µ ) = S ( − µ ) as in QCD Budapest, April 2009 – p.10

R ELATIVISTIC B OSE GAS AT NONZERO µ PHASE TRANSITIONS AND THE S ILVER B LAZE lattice action � � x φ x ) 2 � 2 d + m 2 � φ ∗ x φ x + λ ( φ ∗ S = x 4 � � � � x e − µδ ν, 4 φ x +ˆ ν e µδ ν, 4 φ x φ ∗ ν + φ ∗ − x +ˆ ν =1 complex scalar field, d = 4 , m 2 > 0 S ∗ ( µ ) = S ( − µ ) as in QCD Budapest, April 2009 – p.10

R ELATIVISTIC B OSE GAS AT NONZERO µ PHASE TRANSITIONS AND THE S ILVER B LAZE tree level potential in the continuum V ( φ ) = ( m 2 − µ 2 ) | φ | 2 + λ | φ | 4 condensation when µ 2 > m 2 , SSB T <φ> = 0 Silver Blaze <φ> = 0 problem µ Budapest, April 2009 – p.10

R ELATIVISTIC B OSE GAS AT NONZERO µ COMPLEX L ANGEVIN √ write φ = ( φ 1 + iφ 2 ) / 2 ⇒ φ a ( a = 1 , 2) φ R a + iφ I complexification φ a → a complex Langevin equations ∂φ R ∂θ = − Re δS � a + η a � δφ a � φ a → φ R a + iφ I a ∂φ I ∂θ = − Im δS � a � δφ a � φ a → φ R a + iφ I straightforward to solve numerically, m = λ = 1 lattices of size N 4 , with N = 4 , 6 , 8 , 10 no instabilities etc Budapest, April 2009 – p.11

R ELATIVISTIC B OSE GAS COMPLEX L ANGEVIN 2 − φ I 2 � | φ | 2 → 1 � φ R + iφ R a φ I field modulus squared a a a 2 1.2 4 4 4 6 4 8 4 10 0.8 2 > Re <| φ| 0.4 0 0 1 0.5 1.5 µ Silver Blaze! Budapest, April 2009 – p.12

R ELATIVISTIC B OSE GAS COMPLEX L ANGEVIN 2 − φ I 2 � | φ | 2 → 1 � φ R + iφ R a φ I field modulus squared a a a 2 0.3 4 4 4 6 4 8 4 10 2 > Re <| φ| 0.2 0.1 0 0.5 1 µ second order phase transition in thermodynamic limit Budapest, April 2009 – p.12

R ELATIVISTIC B OSE GAS COMPLEX L ANGEVIN density � n � = (1 / Ω) ∂ ln Z/∂µ 6 4 4 4 6 4 8 4 10 4 Re < n> 2 0 0 0.5 1 1.5 µ Silver Blaze Budapest, April 2009 – p.12

R ELATIVISTIC B OSE GAS COMPLEX L ANGEVIN density � n � = (1 / Ω) ∂ ln Z/∂µ 0.3 4 4 4 6 4 8 4 10 0.2 Re < n> 0.1 0 0 0.25 0.5 0.75 1 1.25 µ second order phase transition in thermodynamic limit Budapest, April 2009 – p.12

S ILVER B LAZE AND THE SIGN PROBLEM RELATIVISTIC B OSE GAS Silver Blaze and sign problems are intimately related complex action e − S = | e − S | e iϕ phase quenched theory � Dφ | e − S | Z pq = different physics QCD: phase quenched = finite isospin chemical potential different onset: m N / 3 versus m π / 2 Budapest, April 2009 – p.13

S ILVER B LAZE AND THE SIGN PROBLEM PHASE QUENCHED phase quenched theory in this case: real action chemical potential appears only in the mass parameter (in continuum notation) V = ( m 2 − µ 2 ) | φ | 2 + λ | φ | 4 dynamics of symmetry breaking, no Silver Blaze Budapest, April 2009 – p.14

S ILVER B LAZE AND THE SIGN PROBLEM COMPLEX VS PHASE QUENCHED density 0.3 0.3 4 4 4 4 4 4 6 6 4 4 8 8 4 4 10 10 0.2 0.2 Re < n> < n> pq 0.1 0.1 0 0 0 0.25 0.5 0.75 1 1.25 0 0.25 0.5 0.75 1 1.25 µ µ complex phase quenched phase e iϕ = e − S / | e − S | does precisely what is expected Budapest, April 2009 – p.15

H OW SEVERE IS THE SIGN PROBLEM ? AVERAGE PHASE FACTOR complex action e − S = | e − S | e iϕ full and phase quenched partition functions � � Dφ e − S Dφ | e − S | Z full = Z pq = average phase factor in phase quenched theory � e iϕ � pq = Z full = e − Ω∆ f → 0 Ω → ∞ as Z pq exponentially hard in thermodynamic limit Budapest, April 2009 – p.16

H OW SEVERE IS THE SIGN PROBLEM ? AVERAGE PHASE FACTOR 1 4 4 4 6 0.8 4 8 4 10 i ϕ > pq 0.6 Re <e 0.4 0.2 0 0 1 0.5 1.5 µ average phase factor � e iϕ � pq Budapest, April 2009 – p.16