The critical endpoint in the 2D-gauge-Higgs model at topological - PowerPoint PPT Presentation

The critical endpoint in the 2D-gauge-Higgs model at topological angle θ = π Daniel G¨ oschl Work done in collaboration with Christof Gattringer and Tin Sulejmanpasic [arXiv: 1807.07793] Lattice 2018, East Lansing, 25.07.2018

Conventional lattice representation with Villain action Matter fields φ x ∈ C , Gauge angles A x ,µ ∈ [ − π, π ] � � D [ φ ] e − S φ [ φ, A ] , Z = D [ A ] B G [ A ] e − β 2 ( F x +2 π n x ) 2 − i θ 2 π ( F x +2 π n x ) , � � B G [ A ] = x ∈ Λ n x ∈ Z F x = A x , 1 + A x +ˆ 1 , 2 − A x +ˆ 2 , 1 − A x , 2 , � 2 �� ( m 2 + 4) | φ x | 2 + λ | φ x | 4 − � � � φ ∗ x e iA x ,µ φ x +ˆ S φ [ φ, A ] = µ + c . c . . x ∈ Λ µ =1

Conventional lattice representation with Villain action Matter fields φ x ∈ C , Gauge angles A x ,µ ∈ [ − π, π ] � � D [ φ ] e − S φ [ φ, A ] , Z = D [ A ] B G [ A ] e − β 2 ( F x +2 π n x ) 2 − i θ 2 π ( F x +2 π n x ) , � � B G [ A ] = x ∈ Λ n x ∈ Z F x = A x , 1 + A x +ˆ 1 , 2 − A x +ˆ 2 , 1 − A x , 2 , � 2 �� ( m 2 + 4) | φ x | 2 + λ | φ x | 4 − � � � φ ∗ x e iA x ,µ φ x +ˆ S φ [ φ, A ] = µ + c . c . . x ∈ Λ µ =1 Global charge conjugation symmetry C at θ = π : ∗ A x ,µ → − A x ,µ , φ x → φ x ◮ Implemented exactly with Villain action

Worldline representation solves complex action problem � � δ ( � ∇ � Z = W H [ j ] W G [ p ] j x ) δ ( j x , 1 + p x − p x − ˆ 2 ) δ ( j x , 2 − p x + p x − ˆ 1 ) { j , p } x Dual variables: ◮ p x ∈ Z +1 ◮ j x ,µ ∈ Z +1 +1 +1 +3 Constraints: ◮ Vanishing divergence for j -flux at +1 +1 +1 each lattice point ◮ Combination of j - and p -flux has to cancel at each link +1 +1 − 1 − 1 Real and positive weights +1 − 1 − 1 − 1 W H [ j ] , W G [ p ]

Dual MC-Updates Example configuration ◮ Inserts loop around plaquette in +1 either orientation: +1 +1 +1 +3 − 1 +1 +1 +1 +1 +1 +1 − 1 − 1 ◮ Fulfills constraints and ergodicity. +1 − 1 − 1 − 1

Charge conjugation symmetry at θ = π Also the dual form of the Villain action implements global charge conjugation symmetry at θ = π as an exact Z 2 symmetry! ◮ Symmetry transformation: C C → p ′ → j ′ p x − − x ≡ − p x − 1 , j x ,µ − − x ,µ ≡ − j x ,µ , ∀ x , µ +1 +1 − 1 − 2 − 2 C +1 − 1 − 1 − 2 +1 − 2 − 1 − 1

Charge conjugation symmetry at θ = π Also the dual form of the Villain action implements global charge conjugation symmetry at θ = π as an exact Z 2 symmetry! ◮ Symmetry transformation: C C → p ′ → j ′ p x − − x ≡ − p x − 1 , j x ,µ − − x ,µ ≡ − j x ,µ , ∀ x , µ +1 +1 − 1 − 2 − 2 C +1 − 1 − 1 − 2 +1 − 2 − 1 − 1 ◮ Z 2 nature: Applying transformation twice gives the identity transformation: C C p x − − → − p x − 1 − − → − ( − p x − 1) − 1 = p x C C − − → − j x ,µ − − → j x ,µ j x ,µ

Observables Topological charge, topological susceptibility, gauge action density: ∂ 2 � q � = − 1 χ t = 1 � s G � = − 1 ∂ ∂ ∂θ ln( Z ) , ∂θ 2 ln( Z ) , ∂β ln( Z ) V V V

Observables Topological charge, topological susceptibility, gauge action density: ∂ 2 � q � = − 1 χ t = 1 � s G � = − 1 ∂ ∂ ∂θ ln( Z ) , ∂θ 2 ln( Z ) , ∂β ln( Z ) V V V In worldline representation: � �� � � q � = 1 1 p x + θ � , V 2 πβ 2 π x �� 2 � �� 2 � ��� � χ t = 1 1 � 1 � p x + θ p x + θ � � − , V 2 πβ 2 π 2 πβ 2 π x x �� � �� 2 π ) 2 θ 1 1 − ( p x + � s G � = 2 β V β x

Observables Topological charge, topological susceptibility, gauge action density: ∂ 2 � q � = − 1 χ t = 1 � s G � = − 1 ∂ ∂ ∂θ ln( Z ) , ∂θ 2 ln( Z ) , ∂β ln( Z ) V V V In worldline representation: � �� � � q � = 1 1 p x + θ � , V 2 πβ 2 π x �� 2 � �� 2 � ��� � χ t = 1 1 � 1 � p x + θ p x + θ � � − , V 2 πβ 2 π 2 πβ 2 π x x �� � �� 2 π ) 2 θ 1 1 − ( p x + � s G � = 2 β V β x Note: � q � is odd under C transformation at θ = π . ⇒ � q � is order parameter for breaking of C symmetry!

Breaking of C symmetry ◮ Conjectured: C symmetry is broken at large m 2 and restored at sufficiently negative m 2 . [Komargodski et.al., ArXiv: 1705.04786] ◮ 2-d Ising transition between the two regimes? ◮ � q � corresponds to the Ising magnetization. ◮ We cannot observe symmetry breaking on a finite lattice = ⇒ study �| q |� . ◮ M = 4 + m 2 drives the system through the phase transition. Corresponds to temperature in Ising model. ◮ ∆ θ = θ − π plays the role of the external magnetic field in Ising model. ∆ θ = 0 corresponds to the symmetrical point.

Fist-Order transition as function of θ : ( λ = 0 . 5, β = 3), M = 4 + m 2 M = 2.0 M = 3.5 0.03 <q> 0.002 <q> 0.02 0.001 0.01 0.000 0.00 -0.01 -0.001 -0.02 -0.002 -0.03 3 ∆θ 3 ∆θ -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 <s G > <s G > 0.165 0.126 0.160 8x8 0.125 0.155 16x16 32x32 0.150 0.124 3 ∆θ 3 ∆θ -3 -2 -1 0 1 2 -3 -2 -1 0 1 2

Critical endpoint at ∆ θ = 0: ( λ = 0 . 5, β = 3), M = 4 + m 2 0.20 0.030 χ t 〈 |q| 〉 0.025 0.15 0.020 0.015 0.10 0.010 12x12 16x16 0.05 20x20 0.005 40x40 80x80 0.000 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 2.6 2.7 2.8 2.9 3.0 3.1 M M

Critical endpoint at ∆ θ = 0: ( λ = 0 . 5, β = 3), M = 4 + m 2 0.20 0.030 χ t 〈 |q| 〉 0.025 0.15 0.020 0.015 0.10 0.010 12x12 16x16 0.05 20x20 0.005 40x40 80x80 0.000 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 2.6 2.7 2.8 2.9 3.0 3.1 M M ◮ What is the scaling behavior? Critical exponents?

Finite Size Scaling for determination of critical exponents We follow the following procedure: 1. Study emerging divergences in observables as we increase the volume. 2. Determine exponent ν from scaling of �| q |� , � q 2 � and Binder cumulant U : dU d d � � � 1 dM ln � q 2 � dM ln �| q |� ∝ L max , max , � � � ν dM � � � max 3. Estimate M C from scaling of pseudo-critical mass defined as position of maximum: M pc ( L ) = M C + A L − 1 ν 4. Extract critical exponents β and γ from scaling of observables at critical Mass M C : �| q |� ( M C , L ) = L − β γ ν F q ( x ) , ν F χ ( x ) χ t ( M C , L ) = L

Critical exponents 64 M pc (L) 2 >/dM d ln<q max d ln<|q|>/dM 2.98 dU/dM 32 ◮ Final results for U(1) gauge-Higgs model: 16 2.96 ν = 1 . 003(11) 8 2 >/dM d ln<q 2.94 β = 0 . 126(7) χ t 4 d ln<|q|>/dM γ = 1 . 73(7) d <|q|>/dM d U/dM 2.92 16 32 64 0.00 0.01 0.02 0.03 - ν L L χ t (0,L) <|q|> (0,L) 0.020 ◮ 2-d Ising values: 0.1 0.019 ν = 1 0.018 β = 0 . 125 0.017 γ = 1 . 75 0.016 0.01 16 32 64 128 16 32 64 128 L L

Summary ◮ We study the critical endpoint of the U(1) gauge-Higgs model at topological angle θ = π . ◮ The Villain action implements the charge conjugation symmetry at θ = π as an exact Z 2 symmetry. ◮ Complex action problem is solved by simulating in the world line representation. ◮ We identify the critical endpoint and determine the critical exponents from a finite size scaling analysis. ◮ We show that the critical endpoint is in the 2d Ising universality class: ν = 1 . 003(11) , β = 0 . 126(7) , γ = 1 . 73(7)