Positivity of center subsets for QCD Mkiang whteigs pitoisve Falk Bruckmann (Regensburg University) SIGN 2015, Atomki/Debrecen, Oct. 2015 with Jacques Bloch 1508.03522 Falk Bruckmann Positivity of center subsets for QCD 0 / 21
Introduction QCD at high density/with chemical potential µ has a sign problem: path integral weight det / D �≥ 0 ⇒ no importance sampling ‘subsets’: add up configurations until the weight is positive works in: random matrix model with µ Bloch 11 fugacity expansion Bloch, FB, Kieburg, Splittorff, Verbaarschot 12 lattice QCD in 0+1 dimensions Bloch, FB, Wettig 13 no gauge action, solvable (Bilic, Demeterfi 88, Ravagli, Verbaarschot 07) here: higher-dimensional lattices analytical & numerical evidence for positivity of subsets Falk Bruckmann Positivity of center subsets for QCD 1 / 21
Subset idea: an example ‘partition function’: (from the random matrix model) � ∞ f ( x ) = e − x 2 � 1 + x 2 − cosh ( µ ) x � Z = dx f ( x ) , ≶ 0 −∞ f � x � 2.0 1.5 1.0 0.5 2 x � 2 � 1 1 � 0.5 Falk Bruckmann Positivity of center subsets for QCD 2 / 21
Subset idea: an example ‘partition function’: (from the random matrix model) � ∞ f ( x ) = e − x 2 � 1 + x 2 − cosh ( µ ) x � Z = dx f ( x ) , ≶ 0 −∞ f � x � 2.0 1.5 1.0 0.5 2 x � 2 � 1 1 � 0.5 subsets: add values at x and − x = integrate over even part � ∞ dx 1 � � Z = f ( x ) + f ( − x ) 2 −∞ � �� � e − x 2 ( 1 + x 2 ) > 0 Falk Bruckmann Positivity of center subsets for QCD 2 / 21
Subsets in a random matrix model partition function for Φ 1 , 2 complex N × N matrices Osborn ’04 � � � e µ Φ 1 − e − µ Φ † m d Φ 1 , 2 e − N Tr (Φ 1 Φ † 1 +Φ 2 Φ † 2 ) det 2 Z ( µ ) = − e − µ Φ † 1 + e µ Φ 2 m subsets: measure and Gaussian invariant under Φ 1 , 2 → e i θ Φ 1 , 2 determinant changes (not a kind of gauge trafo!) add up the weight of those rotated matrices but before that: interpret the rotation as an imag. µ � � � e µ e i θ Φ 1 − e − µ e − i θ Φ † m 2 Z ( µ ) = d Φ 1 , 2 .. det − e − µ e − i θ Φ † 1 + e µ e i θ Φ 2 m = Z ( µ + i θ ) ∀ θ n − 1 = 1 � Z ( µ + ik / n ) n k = 0 Falk Bruckmann Positivity of center subsets for QCD 3 / 21
fugacity expansion: 2 N � e q µ Z q Z ( µ ) = q = − 2 N Z q : canonical partition functions range of q from det being a polynomial in e ± µ of order 2 N put together: 2 N n − 1 e q µ 1 � � e 2 π ik / n · q Z ( µ ) = Z q n q = − 2 N k = 0 � �� � δ q mod n , 0 sum of n th roots (and most of their powers) vanish choose n > 2 N : Z ( µ ) = e 0 Z 0 = Z ( µ = 0 ) “no mu, no cry” actually, the µ -dep. disappears on the level of the integrand ⇒ the integrand is that at µ = 0 and thus is positive Falk Bruckmann Positivity of center subsets for QCD 4 / 21
Subset definition in lattice QCD add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] � � � � � Z = DU w [ U ] = d U µ ( x ) w [ U ] µ, x SU ( 3 ) Falk Bruckmann Positivity of center subsets for QCD 5 / 21
Subset definition in lattice QCD add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] � � � � � � V µ ( x ) − 1 U µ ( x ) Z = DU w [ U ] = d w [ U ] = DU w [ VU ] µ, x SU ( 3 ) Falk Bruckmann Positivity of center subsets for QCD 5 / 21
Subset definition in lattice QCD add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] � � � � � � V µ ( x ) − 1 U µ ( x ) Z = DU w [ U ] = d w [ U ] = DU w [ VU ] µ, x SU ( 3 ) cf. group U(1): � 2 π � 2 π � � 1 d φ = 1 dU U = e i φ d ( V − 1 U ) = d ( − φ V + φ ) = 2 π 2 π 0 0 U ( 1 ) U ( 1 ) (translation invariance on the circle) Falk Bruckmann Positivity of center subsets for QCD 5 / 21
Center subsets in lattice QCD add up configurations ← multiplication of links with center elements center Z 3 : g = e 2 π ik / 3 1 3 with k = 0 , 1 , 2 commutes with all SU ( 3 ) group elements center subsets: � Z = DU w [ gU ] g ∈ Z 3 ⊗ Z 3 ⊗ . . . � 1 � = DU w [ gU ] # g g � �� � ? ≡ σ ( U ) > 0 if the subset weight σ is positive one can use importance sampling even if the original weight w was not positive Falk Bruckmann Positivity of center subsets for QCD 6 / 21
observables can be measured, too alternative view: part of the integral = center sum performed deterministically, while remaining part = integral over the coset subject to sampling why center? gauge theories w/o center ( SU ( N ) adj , G 2 ) have no sign problem confinement criterion on Polyakov loop: tr P → g tr P (approx.) preserved/broken at low/high T (and µ ?) technically easy seemingly sufficient below: connected to imag. µ and diagrammatics similarity to clusters in Pott model Alford, Chandrasekharan, Cox, Wiese 01 Falk Bruckmann Positivity of center subsets for QCD 7 / 21
Center subsets in 0+1 dimensions no plaquette ⇒ w = det / D temporal links can be gauged to a single link ← Polyakov loop P original weight, m = 0, one flavor: 3 × 3 ( 1 + e µ/ T P )( 1 + e − µ/ T P † ) w ( P ) = det = e − 3 µ/ T + e − 2 µ/ T 2 tr P + e − µ/ T � 2 tr P † + ( tr P ) 2 � + e 3 µ/ T + e 2 µ/ T 2 tr P † + e µ/ T � 2 tr P + ( tr P † ) 2 � + 2 + 2 | tr P | 2 � > 0 ( Re w matters, still � > 0 ) center subsets: 1 � � = e − 3 µ/ T + e 3 µ/ T + 2 + 2 | tr P | 2 > 0 w ( P )+ w ( e 2 π i / 3 P )+ w ( e − 2 π i / 3 P ) 3 positive, used for importance sampling � only baryon chemical potential µ B = 3 µ enters (see below) Falk Bruckmann Positivity of center subsets for QCD 8 / 21
Collective subsets and imag. µ lattice implementation of µ on one (say last) time slice: e µ/ T U 0 , e − µ/ T U † 0 ‘collective subsets’: same center element, only on that time slice: 2 2 e µ/ T U 0 → e µ/ T · 1 e 2 π ik / 3 U 0 = 1 � � e µ/ T + 2 π ik / 3 · U 0 3 3 k = 0 k = 0 & analogously for U † 0 2 D µ/ T → 1 � det / det / D µ/ T + 2 π ik / 3 3 k = 0 & plaquette unchanged = adding/averaging 3 complex µ ’s = Roberge-Weiss symmetry (on integrals) utilized on integrands Falk Bruckmann Positivity of center subsets for QCD 9 / 21
fugacity expansion: 2 e q µ/ T 1 � � � e q µ/ T / e 2 π ik / 3 · q det / / D µ/ T = D q → D q 3 q q k = 0 � �� � δ q mod 3 , 0 nonzero triality terms removed from the path integrand; the corresponding integrals = canonical partition functions vanish anyhow expansion in 3 µ = µ B these collective subsets attenuate the sign problem, but in general do not solve it ⇒ independent center multiplications on all temporal links Falk Bruckmann Positivity of center subsets for QCD 10 / 21
Positivity of center subsets: numerical evidence 1+1 dimensional QCD one (unrooted) staggered quark µ = 0 . 3, m = 0 (sign problem worst) no plaquette = strong coupling approx. reweighting factors on 100,000 configurations (* means 1,000): N t × N x 2 × 2 4 × 2 6 × 2 8 × 2 10 × 2 phase-quenched 0.8134(3) 0.4361(4) 0.233(2) 0.130(2) 0.071(1) sign-quenched 0.9271(2) 0.6150(5) 0.355(3) 0.203(2) 0.109(2) collective subsets 0.9778(9) 0.777(4) 0.500(6) 0.303(8) 0.178(3) full subsets 1.0 1.0 1.0 ∗ 1.0 ∗ 1.0 ∗ N t × N x 2 × 4 4 × 4 6 × 4 2 × 6 2 × 8 pq 0.7934(5) 0.295(1) 0.0961(9) 0.7364(6) 0.6725(7) sq 0.9197(3) 0.442(2) 0.149(1) 0.8917(4) 0.8523(5) collective 0.959(1) 0.557(6) 0.214(8) 0.912(3) 0.867(2) full 1.0 ∗ 1.0 ∗ 1.0 ∗ 1.0 ∗ 1.0 ∗ Falk Bruckmann Positivity of center subsets for QCD 11 / 21
higher dimensions on 200 random configurations: full subsets give reweighting factors 1 . 0 for 2 3 , 2 2 × 4 and 2 4 with gauge action β = 1 , 2 , 3 , 4 , 5 on 2 × 6 reweighting factors 1 . 0, 1 . 0, 0 . 984 ( 7 ) , 0 . 964 ( 13 ) , 0 . 972 ( 17 ) = mild sign problem subsets can always be used to attenuate the sign problem reweighting factor has to increase (Cauchy-Schwarz) or combined with other methods Falk Bruckmann Positivity of center subsets for QCD 12 / 21
A first physics result quark number density as a function of µ and T (in lattice units) 3 3 N x =2 N x =6 3 2.5 2 2 n n 2 1 1 1.5 1 0 0 0.5 0 0.5 0.5 0.4 0.4 0.3 0.3 T T 0.2 0.2 0.6 0.6 0.5 0.5 0.1 0.4 0.1 0.4 0.3 0.3 µ µ 0.2 0.2 0 0 0.1 0.1 0 0 ( N t = 2 , . . . , 12 and N t = 2 , . . . , 8) Silver-Blaze property visible Falk Bruckmann Positivity of center subsets for QCD 13 / 21
Positivity of center subsets: analytical proof for N t = 2 massless staggered determinant at µ = 0 the usual argument: D is antihermitian and chiral ∗ / ⇒ eigenvalues in pairs ± i · real ⇒ det / D µ = 0 ≥ 0 chiral ∗ : anticommutes with η 5 , η 5 ( x ev , od ) = ± 1 argument using quarks as Grassmannians: � D ¯ ψ D ψ exp ( ¯ det / ψ / D µ = 0 = D µ = 0 ψ ) and even-odd decomposition Falk Bruckmann Positivity of center subsets for QCD 14 / 21
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