Bose-Einstein condensation and Silver Blaze in the 2-loop 2PI Gergely Mark´ o ´ Ecole Polytechnique CPHT October 2014, ACH triangle workshop • Motivation • Introduction to 2PI • Equations • Silver Blaze • Renormalization • Results • Conclusions ep (HAS-ELTE), U. Reinosa ( ´ Collaborators: Zs. Sz´ Ecole Polytechnique CPHT)
Motivation • Functional methods at finite density are of great interest, because of the phase diagram of strongly interacting matter. • In Andersen, PRD 75 065011 (2007) pion condensation is discussed in LO- 1 /N approximation of 2PI. Some general features are hidden, which are present in the 2-loop approximation. • Understanding the subtleties which lie in the renormalization of 2PI at finite µ . • Understanding the Silver Blaze phenomenon in a simple model. • Therefore as a first step we chose the charged scalar model, and included chemical potential in it.
Introduction to 2PI A bilocal source is introduced in the generating functional � Z [ J, K ] = e W [ J,K ] = � � D ϕ exp − S 0 − S int + ϕ · J + ϕ · K · ϕ The 2PI effective action defined through a double Legendre transform � � � d 4 x δW [ J, K ] δW [ J, K ] d 4 x d 4 y γ [ φ, G ] = W [ J, K ] − J ( x ) − K ( x, y ) δJ ( x ) δK ( x, y ) � �� � � �� � φ ( x ) [ φ ( x ) φ ( y )+ G ( x,y )] / 2 The physical ¯ φ ( x ) and ¯ G ( x, y ) are determined from stationarity conditions at vanishing sources ( J, K → 0 ) � � δγ [ φ, G ] δγ [ φ, G ] � � = 0 , = 0 � � � ¯ � ¯ δφ ( x ) δG ( x, y ) φ ( x ) G ( x,y )
γ [ φ, G ] can be written as shown in Cornwall et al., PRD 10 , 2428 (1974) γ [ φ, G ] = S 0 ( φ ) + 1 2Tr log G − 1 + 1 � � G − 1 2Tr 0 G − 1 + γ int [ φ, G ] S 0 is the free action, G 0 is the free propagator, γ int [ φ, G ] contains all the 2PI graphs constructed with vertices from S int ( φ + ϕ ) . The Tr is to be understood in all indices and as integration over coordinates. The 1PI effective action is recovered: Γ 1PI [ φ ] = γ [ φ, ¯ G ] . The chemical potential only enters through the free action S 0 and the free propagator G 0 .
Equations The symmetry of the theory is SO (2) in the presence of µ . We represent the field � � ϕ 1 as ϕ = , ϕ a ∈ R , and � ϕ a � = δ a, 1 φ . The free and full propagators are ϕ 2 � Z 0 Q 2 + m 2 � � � 0 − Z 0 µ 2 − 2 Z 0 µω G L G A G − 1 = and G = . Z 0 Q 2 + m 2 0 0 − Z 0 µ 2 2 Z 0 µω − G A G T The 2PI potential truncated at 2-loops can written as � T γ [ φ, G L , G T , G A ] = 1 + 1 � � log( G − 1 ( Q )) + G − 1 2( m 2 2 − µ 2 Z 2 ) φ 2 2 Tr 0 ( Q ) · G ( Q ) Q 48 + λ ( A +2 B ) + λ ( A ) + λ ( A +2 B ) + λ ( A ) + λ 4 φ 4 2 2 0 0 24 24 48 24 � + λ ( A +2 B ) − λ 2 ⋆ 0 3 + 48 144 � � � +2 3 − with G L = , G T = , G A = , φ = .
Equations The field expectation value φ and the components of the full propagator are determined from stationarity conditions: � � δγ [ φ, G L , G T , G A ] = δγ [ φ, G L , G T , G A ] � � 0 = � � � ¯ δφ δG L � φ, ¯ G L , ¯ G T , ¯ φ, ¯ G L , ¯ G T , ¯ G A G A � � δγ [ φ, G L , G T , G A ] = δγ [ φ, G L , G T , G A ] � � = , � � δG T δG A � � φ, ¯ G L , ¯ G T , ¯ φ, ¯ G L , ¯ G T , ¯ G A G A which yield equations for the gap masses defined from the inverse propagator: ¯ G T,L + Z 0 ( µ 2 − Q 2 ) , ¯ M 2 L,T ( Q ) = G L ¯ ¯ G T + ¯ G 2 A ¯ G A ¯ M 2 A ( Q ) = − + Z 0 2 µω, G L ¯ ¯ G T + ¯ G 2 A and the field equation with the structure 0 = ¯ φ ˜ f (¯ φ, ¯ G L ( φ = ¯ φ ) , ¯ G T ( φ = ¯ φ ) , ¯ G A ( φ = ¯ φ )) = ¯ φf (¯ φ ) .
Curvature masses To study the phase transition we monitor the curvature mass tensor. It is defined using the 1PI potential γ ( φ ) ≡ γ [ φ, ¯ G L , ¯ G T , ¯ G A ] as � � ab = ∂ 2 γ ( φ ) φ a φ b δ ab − φ a φ b + δ ab µ 2 = ˆ ˆ + ˆ M 2 M 2 M 2 . L T φ 2 φ 2 ∂φ a ∂φ b Evaluating the derivatives yield � φ 2 d f ( φ ) � ˆ L = 4¯ + 2 f (¯ ˆ T = 2 f (¯ M 2 φ ) + µ 2 , M 2 φ ) + µ 2 � � ¯ dφ φ T = µ 2 (Goldstone At ¯ ˆ L = ˆ T (symmetry restoration), at ¯ ˆ M 2 M 2 M 2 φ = 0 : φ � = 0 : theorem).
Numerics We solve the coupled field and gap equations iteratively. We discretize the propagators on a N τ × N s grid: ω n = 2 πnT, n ∈ [0 ..N τ − 1] , and k = ( s + 1) Λ , s ∈ [0 ..N s − 1] . N s • Numerical method was developed in Mark´ o et al., PRD 86 085031 (2012). • Rotation invariance ⇒ only 1D in momentum space. • Convolutions are done using FFT techniques. • Only adjustment needed: G A → ω n g A . While FFT is also applicable to odd functions, the stored frequencies would be shifted, which in the iterative process of solving the equations leads to loss of information.
Silver Blaze We can formulate our theory using complex fields as well: Φ = 1 2( ϕ 1 + iϕ 2 ) , and Φ ∗ = 1 √ √ 2( ϕ 1 − iϕ 2 ) . Then the Lagrangian is invariant under the gauge-transformation Φ → e iατ Φ , Φ ∗ → e − iατ Φ ∗ , µ → µ − iα . • Z µ = Z µ − iα provided that α = ω n , a Matsubara-frequency in order to maintain the periodicity of the fields. • T � = 0 : Periodicity in the imaginary µ direction (Roberge-Weiss periodicity). • T = 0 : ω n becomes continuous → analytic continuation: Z µ is µ -independent up to analyticity boundary µ c . This is the Silver Blaze property Cohen, Phys. Rev. Lett. 91 , 222001 (2003) • Generalization to n-point functions at T = φ = 0 : µ -dependence is just a shift of external frequencies.
Silver Blaze 0.0035 0.0002 0.003 0 -0.0002 0.0025 -0.0004 -0.0006 0.002 -0.0008 ω N τ = ∞ P 0 [A . u . ] -0.001 ω N τ =259 . 2 π T ⋆ 0.0015 -0.0012 ω N τ =172 . 8 π T ⋆ -0.0014 0.001 0 0.02 0.04 0.06 0.08 0.1 0.0005 µ c 0 -0.0005 0 0.05 0.1 0.15 0.2 0.25 0.3 µ/ T ⋆ • In any 2PI truncation the Silver Blaze • We use finite Matsubara-frequencies. We have to take the T → 0 limit such is realized. that 2 πN τ T → ∞ . • Provided UV regularization and discretization keeps the • On the lattice µ is introduced on links, gauge-transformation property. similarly to gauge fields.
Renormalization Renormalization is based on Mark´ o et al., PRD 87 105001 (2013). • Prescriptions on 2- and 4-point functions. • At T = T ⋆ , µ = ¯ φ = 0 . • No new counterterms are needed compared to µ = 0 case. • Except for field renormalization, which is special in the homogeneous 2-loop approximation. • At 2-loop order: no diagram in the gap equation has momentum dependent divergence. But the field equation has the setting-sun at zero external momentum (homogeneity). • Shift of external frequencies by µ in n-point functions: need for Z 2 .
Renormalization In line with the other prescriptions, we require: � � d dµ 2 ˆ M 2 � = 1 − α , φ =0 � � T ⋆ ,µ =0 � � � � d d dµ 2 ˆ dµ 2 ¯ M 2 M 2 � � = . φ =0 � φ =0 � � � T ⋆ ,µ =0 T ⋆ ,µ =0 Which lead to the following expressions for the field normalizations: � � � � � ∂ T [ ¯ � ∂ S [ ¯ D , ¯ D ∗ , ¯ Z 0 + λ 2 − λ 2 � � D ] D ] ⋆ ⋆ � � Z 2 = 6 B ⋆ [ G ⋆ ](0) , � � ∂µ 2 ∂µ 2 18 � � T ⋆ ,µ =0 T ⋆ ,µ =0 � ∂ T [ ¯ � α + λ ⋆ D ] D − 1 ( Q ) = ( ω n + iµ ) 2 + q 2 + ¯ , with ¯ M 2 � Z 0 = φ =0 . � ∂µ 2 3 � T ⋆ ,µ =0 • Z 0 is finite, as the tadpole has no µ • α dependence only through Z 0 . We dependent divergence. choose Z 0 , no new parameter.
Transition line The transition temperature at chemical potential µ , is determined by φ =0; T = T c ( µ ) ,µ = µ 2 . ˆ M 2 The µ = 0 existence of T c splits the m 2 ⋆ − λ ⋆ parameter plane in two 70 ¯ T c ( µ = 0) = 0 T c ( µ = 0) = 0 60 Points of investigations Λ p =50T ⋆ 50 Λ p =100T ⋆ 40 λ ⋆ 30 BEC 20 SSB 10 (A) (B) 0 0 0.2 0.4 0.6 0.8 1 1.2 m 2 ⋆ / T 2 ⋆
Comparison to Hartree-Fock The density is defined as � � ρ = 1 ∂ ln Z φ 2 + µ = µ ¯ ( ¯ G L ( Q ) + ¯ ω n ¯ G T ( Q )) − 2 G A ( Q ) . βV ∂µ Q Q We compare the iso-density lines at given parameters in the H-F and the 2-loop approximations: 1 0.8 0.9 0.7 0.8 0.6 T 2L c T HF 0.7 0.5 c T / T ⋆ T / T ⋆ ¯ T c 0.6 0.4 0.5 ¯ M HF φ = µ 0.3 T , ¯ 0.4 0.2 T 2L c T HF 0.3 0.1 c ¯ M HF φ = µ ¯ T c T , ¯ 0.2 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 µ/ T ⋆ µ/ T ⋆ (A) (B)
Comparison to Lattice Lattice results are from Gattringer et al., Nucl. Phys. B 869 , 56 (2013). Difficulties: • Cut-off effects are not small, as the inverse lattice spacing there is not small ( aµ c ≈ 1 . 15 ). • We did not use the lattice action → even the bare theories differ. Choice of parameters based on the reproduction of the T c ( µ ) curve. 0.5 0.45 0.4 0.35 0.3 T /µ c 0.25 0.2 0.15 Lattice : λ b = 1 , η = 9 2PI : λ ⋆ = 12 . 5 , m 2 ⋆ / T 2 ⋆ = 0 . 6 0.1 2PI : λ ⋆ = 11 . 5 , m 2 ⋆ / T 2 ⋆ = 0 . 8 0.05 2PI : λ ⋆ = 10 . 5 , m 2 ⋆ / T 2 ⋆ = 1 . 0 0 1 1.01 1.02 1.03 1.04 1.05 1.06 µ/µ c
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