The Roberge- Weiss transition Michele Andreoli The Roberge-Weiss transition Lattice QCD Lattice basics Phase structure of QCD at imaginary chemical potential Montecarlo Fermion and generic number of flavors. determinant Imaginary µ A case study: N f = 8 The canonical approach RobergeWeiss Center symmetry Michele Andreoli RW symmetry Conseguences Phase transition Pisa Univ. & INFN A case study The aim of this work 28 September 2016 Simulation setup Numerical tools Polyakov loop Fermionic meas. Time series Scatter plots Phase histogram FSS scaling Binder cumulant Collapse plots Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 1 / 26
The Roberge- Lattice basics Weiss transition Michele A first principle non perturbative QCD formulation Andreoli 1 ˆ ψ , U ] ≡ � e − S � FG ψ , U ] = − β G ∑ 3 ReTr [ ∏ + ∑ ψ D ψ e − S [ ψ , ¯ S [ ψ , ¯ U ] ψ f M f ψ f ¯ and Z = D U D ¯ Lattice QCD Lattice basics P P f � �� � � �� � Montecarlo SG [ U ] SF [ ψ , ¯ ψ , U ] Fermion determinant Imaginary µ ◮ Fields The canonical lattice domain ◮ U x , µ ≈ e igaA µ ( x ) , ψ = ( ψ x 1 , ψ x 2 ,... ) , x ∈ Λ approach Λ = a Z 4 = { x | x µ a ∈ Z } RobergeWeiss ◮ Fermion matrix with naive µ Center symmetry ◮ ˆ H → ˆ H − µψ † ψ RW symmetry Conseguences ◮ Dirac M f = ( ∂ µ − igA µ ) γ µ + m f − µ f γ 0 Phase transition A case study ◮ Chemical potential as U(1) field: The aim of this work µ ∼ igA 0 , therefore U ± x , 0 → U ± x , 0 e ± µ a , Simulation setup ◮ Numerical tools ◮ “Staggered” (Kogut-Susskind) Polyakov loop � Fermionic meas. η x ; ν e + a µ f δν , 4 U x ; ν δ x , y − ˆ ◮ ( M f ) x , y = am f δ x , y + ∑ 4 Time series ν − ν = 1 2 � Scatter plots e − a µ f δν , 4 U † ν ; ν δ x , y + ˆ Phase histogram ν ´ N τ a x − ˆ d τ = aN τ = 1 β = FSS scaling ◮ skeleton M ( U , µ ) ∼ U 0 e µ a + U + 0 T 0 e − µ a Binder cumulant d 3 x = ( aN s ) 3 V = ´ Collapse plots β G = 2 Nc g 2 Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 2 / 26
The Roberge- Montecarlo Weiss transition Michele β G β G Andreoli ˆ ˆ 3 ∑ P ReTr [ ∏ P U ] det M [ U ] 3 ∑ P ReTr [ ∏ P U ] − ¯ ψ M ψ Z = D U D ¯ ψ D ψ e = ⇒ Z = D Ue ���� Lattice QCD ψ M ψ = det M ψ D ψ e − ¯ ´ D ¯ Lattice basics Montecarlo ◮ Probability Fermion determinant P [ U ] = e − S G [ U ] · det M [ U , µ ] Imaginary µ ◮ The canonical Z approach D Ue − S G [ U ] det M ( U , µ ) = � det M ( U , µ ) � G ◮ with Z = ´ RobergeWeiss Center ◮ Tools symmetry RW symmetry 1 ◮ det M = ∏ f det M Conseguences f ( U , µ f ) ( 1/4 root trick ) 4 Phase transition D φ D φ † e − φ † 1 ◮ Pseudo-fermions: det M = M φ ´ A case study In summing up: 1 The aim of this ◮ Multi-fermions: det M = ( det M n ) n Z = � det M ( U , µ ) � G work a i ◮ Remez algorithm: 1 M α = ∑ i Simulation setup M + b i Numerical tools ◮ RHMC (Rational Hybrid MonteCarlo) Polyakov loop Fermionic meas. ◮ generate U i with P ( U i ) ∼ e − S G [ U i ] det M [ U i ] Time series Scatter plots 1 Phase histogram ◮ calculate � O � = N conf ∑ O [ U i ] i , j = 1 ... N τ N 3 FSS scaling s Binder cumulant i ∈ conf Collapse plots Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 3 / 26
The Roberge- Why complex chemical potential? Weiss transition Michele Andreoli � 0 e − µ a � U 0 e µ a + U + Z ( µ ) = � det � G Lattice QCD Lattice basics Montecarlo 1) det M ( U , µ ) ⋆ = det M ( U , − µ ⋆ ) Fermion determinant 2) Z ( µ ) = Z ( − µ ) if �� G = �� G ⋆ Imaginary µ 3) Z ( µ ) ⋆ = Z ( − µ ⋆ ) The canonical approach RobergeWeiss ◮ Standard Montecarlo unfeasible if µ ∈ R (“sign problem”) Center symmetry The sign problem ◮ det M ( U , µ ) is real only if µ ⋆ = − µ ( see 1) RW symmetry Conseguences ◮ Way out: Imaginary chemical potential, Taylor Phase transition expansion, analytic cont., Reweighting at A case study µ = 0, etc The aim of this work ◮ Anyway, Nothing is wrong in the QCD formulation at Simulation setup Numerical tools imaginary µ : Polyakov loop ◮ after averanging on the background gauge fields, Fermionic meas. Time series ◮ Z ( µ ) = � det M ( U , µ ) � G is real ( see 2,3) Scatter plots Phase histogram FSS scaling Binder cumulant Collapse plots Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 4 / 26
The Roberge- Complex µ : the canonical approach Weiss transition � 0 e − µ a � �� e ± βµ � � U 0 e µ a + U + Michele Z ( µ ) = � det � G and det M = e Tr ln M Tr ∏ τ U ± ∼ Andreoli ���� 0 ⊃ aN τ = β Lattice QCD Lattice basics ◮ Fugacity expansion ( Laurent expansion in ζ = e βµ ) Montecarlo βµ complex plan Fermion ∞ determinant ( e βµ ) N · z N ∑ Z GC ( µ ) = ◮ Imaginary µ N = − ∞ The canonical approach d ζ Z GC ( ζ ) RobergeWeiss ˛ ◮ z N = (Cauchy’s integral formula) Center ζ N + 1 2 π i symmetry RW symmetry ◮ Canonical Z C ( N ) ≡ z N Conseguences Phase transition ˛ d ( βµ ) A case study 2 π i Z GC ( µ ) · e − ( βµ ) · N (Laplace tras.) Z C ( N ) = ◮ The aim of this work Simulation setup ◮ Thermodynamic definitions: Numerical tools Polyakov loop H QCD − µ � ◮ Z GC ( µ ) = Tr [ e − β ( � N ) ] (“gran canonical”) Fermionic meas. ◮ Z C ( N ) = Tr [ e − β � Time series H QCD δ ( � N − N )] (“canonical”) Scatter plots Phase histogram FSS scaling Note: Binder cumulant Collapse plots ◮ Z GC ( µ ) and Z C ( N ) share the same information (Laplace transforms) Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 5 / 26
The Roberge- Z 3 center symmetry Weiss transition � � ˆ β G Michele 3 ∑ P ReTr [ ∏ P U ] U ± e ± βµ Tr ∏ Z = · D U e Andreoli � �� � 0 τ � �� � Lattice QCD e − SG = gauge Lattice basics det M = fermions Montecarlo Fermion determinant ◮ Center symmetry U 0 → ξ U 0 Imaginary µ ⇒ ξ N c = 1 , i.e The canonical det = 1 = ◮ approach ξ k = e k 2 π i 3 = { 1 , 2 π 3 i , 4 π RobergeWeiss 3 i } ∈ Z 3 Center symmetry ◮ The “gauge part” is invariant RW symmetry ◮ Tr [ ∏ P U ] → Tr [ ∏ P U ] , D U → D U Conseguences Phase transition ◮ The “fermion part” explicitly breaks A case study The aim of this ◮ P ∼ Tr [ ∏ τ U 0 ] (Polyakov loop) P work Simulation setup ◮ so P → ξ P Numerical tools Polyakov loop ◮ Order parameter Fermionic meas. Time series � P � � = 0 = ⇒ Z 3 broken ◮ Polyakov loop P ∼ Tr ∏ U 0 Scatter plots Phase histogram FSS scaling Note: Binder cumulant 1. In the SU(3) pure gauge, � P � � = 0 at high temperature, signalling the spontaneous Collapse plots Conclusions symmetry breakdown of the Z 3 symmetry Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 6 / 26
The Roberge- RW: the symmetry Weiss transition Michele Andreoli � � det M = e Tr ln M ∼ Lattice QCD U 0 e + βµ )+ h . c . Tr ( ∏ and Z = � det M ( U , µ ) � G Lattice basics τ Montecarlo ⊃ Fermion determinant ◮ The Roberge-Weiss symmetry Imaginary µ ◮ if U 0 → e i 2 π The canonical 3 k U 0 and βµ → βµ − i 2 π 3 k approach RobergeWeiss Z ( βµ ) = Z ( βµ − i 2 π Center 3 k ) ◮ symmetry RW symmetry ◮ Charge symmetry µ → − µ Conseguences Phase transition ◮ Z ( µ ) = Z ( − µ ) is even A case study ◮ θ ′ = − θ + k 2 π ⇒ θ ′ + θ = k π 3 = The aim of this 3 . 2 work ◮ Parity+Rotation=Reflection about θ = k π Simulation setup 3 Numerical tools ◮ If P i π ( U ) = P i π ( U ⋆ ) Polyakov loop Fermionic meas. ◮ RW ∼ charge symmetry Time series Scatter plots Phase histogram FSS scaling βµ complex plane Binder cumulant Collapse plots Conclusions Michele Andreoli (Pisa Univ. & INFN) The Roberge-Weiss transition 28 September 2016 7 / 26
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