Linked Cluster Expansions for the Functional Renormalization Group Rudrajit (Rudi) Banerjee (In collaboration with Max Niedermaier) PITT PACC Department of Physics and Astronomy University of Pittsburgh 36th Annual International Symposium on Lattice Field Theory 27th July 2018 Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Outline 1 Functional Renormalization Group 2 Linked Cluster Expansions and the Functional Renormalization Group 3 Critical Behavior of ϕ 4 Theory in Four Dimensions 4 Spatial Linked Cluster Expansions in Friedmann-Lemaˆ ıtre spacetimes 5 Conclusions and Outlook Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Functional Renormalization Group (FRG) The FRG is a reformulation of QFT; study non-linear response of functionals to scale dependent mode modulation – in functional integral replace action s [ ϕ ] → s [ ϕ ] + 1 2 ϕ · R k · ϕ . R k suppresses low energy modes. Modern formulations focus on the Legendre transform of the Polchinski equation, determining the Legendre Effective (aka Effective Average) Action. Legendre Effective Action Method Wetterich, Christof. “Exact evolution equation for the effective potential.” Physics Letters B 301.1 (1993): 90-94. � ∂ k R k � ∂ k Γ k [ φ ] = 1 2 Tr Γ ( 2 ) k [ φ ] + R k Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Successes of FRG • New approximation schemes: no expansion in conventional coupling constants. • Consistent with known results: ǫ expansion, large N expansion, . . . • Excellent effort to outcome ratio: relatively little effort yields fixed points, critical exponents, Wilson-type β -functions, some access to momentum-dependent correlation functions. • Computations feasible for any spacetime dimension D. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Weaknesses of FRG 1 Wetterich equation solved via truncation Ans¨ atze � Γ k [ φ ] = c n , k σ n [ φ ] n However, exact Γ[ φ ] is highly non-local, no structural characterization known. In particular, solution of Γ k [ φ ] flow eq. via (non-series) truncations is ad-hoc, no clear ordering principle. Non-local terms, e.g. φ∂ − 2 φ , ( ∂φ ) 2 φ 5 ∂ − 10 φ ? 2 To solve Wetterich equation, need initial condition(s) typically at k = Λ UV (it may be ill-posed at k = Λ UV ). With standard choice: Γ k =Λ UV [ φ ] = s bare [ φ ] , one makes implicit reference to perturbation theory. 3 No statement about asymptotic correctness or convergence of truncations is known. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Remedying the Weaknesses • Fix Weakness 2: use ultralocal + linking split of action in lattice formulation + 1 � s [ ϕ ] = s 0 ( ϕ x ) 2 ϕ · ℓ · ϕ , ultralocal linking x and specify ultralocal initial data at some k = k 0 via exact single site integrals depending on s 0 ( ϕ ) only (choose R k s.t. R k = k 0 = − ℓ ) [Dupuis-Machado, 2010]. • We address Weakness 1 via linked cluster expansion of Γ k [ φ ] via ℓ → ℓ + R k (potentially long ranged). • Perspective on Weakness 3: rigorous proofs for convergence of linked cluster expansion known in many other cases. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Linked Cluster Expansion (LCE) and the FRG + κ On lattice write action s [ ϕ ] = � x s 0 ( ϕ x ) 2 ϕ · ℓ · ϕ . ultralocal linking LCE is expansion of quantities in powers of κ , in particular ∞ κ l Γ l [ φ ] . � Γ κ [ φ ] = l = 0 FRGs entail closed recursion relations for Γ l s. Obtain solution to Wetterich eq. from solution to LCE recursion: � � Γ k [ φ ] = Γ κ [ φ ] � � ℓ → ℓ + R k However, direct iteration of recursion impractical beyond O ( κ 6 ) Solve recursions with GRAPHICAL METHODS instead. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Γ κ LCE Graph Rules Goal: Convert known LCE graph rules for Generating Functional W κ [ J ] [Wortis, 1974] into ones applicable to Γ κ [ φ ] LCE. Γ κ [ φ ] related to W κ [ J ] by modified Legendre transform: Γ κ [ φ ] := φ · J κ [ φ ] − W κ [ J κ [ φ ]] − κ δ W κ � � 2 φ · ℓ · φ , J κ [ φ ] = φ . δ J Insert κ -series expansions for Γ κ , W κ , and J κ , get mixed Γ m ( m < l ), W m ( m ≤ l ) recursion ( ∗ ) for Γ l . Our result : exact graph solution of the recursion. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Connected and One-Line-Irreducible Graphs W κ [ J ] LCE graph expansion → Connected graphs. Γ κ [ φ ] LCE graph expansion → One-Line-Irreducible (or 1PI) graphs. (a) (b) Analogous to perturbation theory. Considerable net computational gain: l |C l | |L l | 2 2 1 3 5 2 4 12 4 5 33 8 6 100 22 Table 1: Number of connected, one-line irreducible graphs with l edges. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Theorem For any l ≥ 2 the solution of the recursion ( ∗ ) is given by ( − ) l + 1 � � � µ Γ ( v | L ) Γ l [ φ ] = ℓ s ( e ) , t ( e ) Sym ( L ) L =( V , E ) ∈L e ∈ E v ∈ V | I ( v ) | ( − ) s ( T ) | Perm ( B ( v )) | µ Γ ( v | L ) = � � µ ( T ) . Sym ( T ) n = 1 T ∈T ( B ( v ) , n ) Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Theorem For any l ≥ 2 the solution of the recursion ( ∗ ) is given by ( − ) l + 1 � � � µ Γ ( v | L ) Γ l [ φ ] = ℓ s ( e ) , t ( e ) Sym ( L ) L =( V , E ) ∈L e ∈ E v ∈ V | I ( v ) | ( − ) s ( T ) | Perm ( B ( v )) | µ Γ ( v | L ) = � � µ ( T ) . Sym ( T ) n = 1 T ∈T ( B ( v ) , n ) • At order l draw all topologically distinct 1PI graphs with l edges. E.g. The following graphs contribute to Γ 4 : , , , . Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Theorem For any l ≥ 2 the solution of the recursion ( ∗ ) is given by ( − ) l + 1 � � � µ Γ ( v | L ) Γ l [ φ ] = ℓ s ( e ) , t ( e ) Sym ( L ) L =( V , E ) ∈L e ∈ E v ∈ V | I ( v ) | ( − ) s ( T ) | Perm ( B ( v )) | µ Γ ( v | L ) = � � µ ( T ) . Sym ( T ) n = 1 T ∈T ( B ( v ) , n ) • At order l draw all topologically distinct 1PI graphs with l edges. E.g. The following graphs contribute to Γ 4 : , , , . • Divide by the symmetry factor Sym ( L ) of the graph. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Theorem For any l ≥ 2 the solution of the recursion ( ∗ ) is given by ( − ) l + 1 � � � µ Γ ( v | L ) Γ l [ φ ] = ℓ s ( e ) , t ( e ) Sym ( L ) L =( V , E ) ∈L e ∈ E v ∈ V | I ( v ) | ( − ) s ( T ) | Perm ( B ( v )) | µ Γ ( v | L ) = � � µ ( T ) . Sym ( T ) n = 1 T ∈T ( B ( v ) , n ) • At order l draw all topologically distinct 1PI graphs with l edges. E.g. The following graphs contribute to Γ 4 : 1 , 1 , 1 , 1 . 48 4 8 8 • Divide by the symmetry factor Sym ( L ) of the graph. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
• In a graph L , for each edge connecting vertices v , v ′ write − ℓ v , v ′ , and for each vertex v a vertex weight µ Γ ( v | L ) . µ Γ ( v | L ) is a finite sum of products of exactly computable single site functions ̟ n ( φ ) , γ n ( φ ) , determined by the single site action action s 0 ( ϕ ) . Linked Cluster Expansions for the Functional Renormalization Group Banerjee
• In a graph L , for each edge connecting vertices v , v ′ write − ℓ v , v ′ , and for each vertex v a vertex weight µ Γ ( v | L ) . µ Γ ( v | L ) is a finite sum of products of exactly computable single site functions ̟ n ( φ ) , γ n ( φ ) , determined by the single site action action s 0 ( ϕ ) . µ Γ ( v 1 | L ) = ̟ 2 ( φ v 1 ) , v 1 v 2 v 3 µ Γ ( v 2 | L ) = ̟ 4 ( φ v 2 ) − γ 2 ( φ v 2 ) ̟ 3 ( φ v 2 ) 2 , µ Γ ( v 3 | L ) = ̟ 2 ( φ v 3 ) . µ Γ ( v | L ) can be obtained as a sum over labeled tree graphs. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
• In a graph L , for each edge connecting vertices v , v ′ write − ℓ v , v ′ , and for each vertex v a vertex weight µ Γ ( v | L ) . µ Γ ( v | L ) is a finite sum of products of exactly computable single site functions ̟ n ( φ ) , γ n ( φ ) , determined by the single site action action s 0 ( ϕ ) . µ Γ ( v 1 | L ) = ̟ 2 ( φ v 1 ) , v 1 v 2 v 3 µ Γ ( v 2 | L ) = ̟ 4 ( φ v 2 ) − γ 2 ( φ v 2 ) ̟ 3 ( φ v 2 ) 2 , µ Γ ( v 3 | L ) = ̟ 2 ( φ v 3 ) . µ Γ ( v | L ) can be obtained as a sum over labeled tree graphs. • The µ Γ ( v ) data can be stored in a look-up table. Proof ≈ 40 pages, R.B. and M.N. under review. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Critical Behavior of ϕ 4 Theory in Four Dimensions Reparametrize ϕ 4 action on lattice: − κ x − 1 ) 2 − λ � � � ϕ 2 x + λ ( ϕ 2 � s [ ϕ ] = ϕ x ℓ xy ϕ y 2 x x , y ultralocal hopping • Critical line κ c ( λ ) yields continuum limit: correlation length ξ → ∞ ⇐ ⇒ m R = 1 /ξ → 0. • κ c ( λ ) obtained by L¨ uscher-Weisz [L¨ uscher-Weisz, 1987] using LCE of generalized susceptibilities, e.g. x < ϕ x ϕ 0 > c = � l ≥ 0 κ l χ 2 , l . χ 2 := � Considerable effort required. FRG to LCE correspondence yields dramatic simplification. Linked Cluster Expansions for the Functional Renormalization Group Banerjee
Recommend
More recommend
