SLIDE 1 The critical endpoint in the 2D-gauge-Higgs model at topological angle θ = π
Daniel G¨
Work done in collaboration with Christof Gattringer and Tin Sulejmanpasic [arXiv: 1807.07793] Lattice 2018, East Lansing, 25.07.2018
SLIDE 2 Conventional lattice representation with Villain action
Matter fields φx ∈ C, Gauge angles Ax,µ ∈ [−π, π] Z =
- D[A] BG[A]
- D[φ] e−Sφ[φ,A] ,
BG[A] =
e− β
2 (Fx +2πnx )2−i θ 2π (Fx +2πnx ) ,
Fx = Ax,1 + Ax+ˆ
1,2 − Ax+ˆ 2,1 − Ax,2 ,
Sφ[φ, A] =
- x∈Λ
- (m2 + 4) |φx|2 + λ|φx|4 −
2
x eiAx,µφx+ˆ µ + c.c.
SLIDE 3 Conventional lattice representation with Villain action
Matter fields φx ∈ C, Gauge angles Ax,µ ∈ [−π, π] Z =
- D[A] BG[A]
- D[φ] e−Sφ[φ,A] ,
BG[A] =
e− β
2 (Fx +2πnx )2−i θ 2π (Fx +2πnx ) ,
Fx = Ax,1 + Ax+ˆ
1,2 − Ax+ˆ 2,1 − Ax,2 ,
Sφ[φ, A] =
- x∈Λ
- (m2 + 4) |φx|2 + λ|φx|4 −
2
x eiAx,µφx+ˆ µ + c.c.
Global charge conjugation symmetry C at θ = π: Ax,µ → −Ax,µ , φx → φx
∗
◮ Implemented exactly with Villain action
SLIDE 4 Worldline representation solves complex action problem
Z =
WH[j] WG[p]
δ( ∇ jx) δ(jx,1 + px − px−ˆ
2) δ(jx,2 − px + px−ˆ 1)
Dual variables: ◮ px ∈ Z ◮ jx,µ ∈ Z Constraints: ◮ Vanishing divergence for j-flux at each lattice point ◮ Combination of j- and p-flux has to cancel at each link Real and positive weights WH[j], WG[p] −1 −1 −1 −1 −1 +1 +1 +3 +1 +1 +1 +1 +1 +1 +1 +1
SLIDE 5
Dual MC-Updates
◮ Inserts loop around plaquette in either orientation: +1 −1 ◮ Fulfills constraints and ergodicity. Example configuration −1 −1 −1 −1 −1 +1 +1 +3 +1 +1 +1 +1 +1 +1 +1 +1
SLIDE 6
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 Z2 symmetry! ◮ Symmetry transformation: px
C
− − → p′
x ≡ −px − 1
, jx,µ
C
− − → j′
x,µ ≡ −jx,µ
, ∀x, µ
−2 +1 +1 +1 −1 −1 −1 −2 −2 −2 −1 −1 +1
C
SLIDE 7
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 Z2 symmetry! ◮ Symmetry transformation: px
C
− − → p′
x ≡ −px − 1
, jx,µ
C
− − → j′
x,µ ≡ −jx,µ
, ∀x, µ
−2 +1 +1 +1 −1 −1 −1 −2 −2 −2 −1 −1 +1
C
◮ Z2 nature: Applying transformation twice gives the identity transformation: px
C
− − → −px − 1
C
− − → −(−px − 1) − 1 = px jx,µ
C
− − → −jx,µ
C
− − → jx,µ
SLIDE 8
Observables
Topological charge, topological susceptibility, gauge action density: q = − 1 V ∂ ∂θ ln(Z) , χt = 1 V ∂2 ∂θ2 ln(Z) , sG = − 1 V ∂ ∂β ln(Z)
SLIDE 9 Observables
Topological charge, topological susceptibility, gauge action density: q = − 1 V ∂ ∂θ ln(Z) , χt = 1 V ∂2 ∂θ2 ln(Z) , sG = − 1 V ∂ ∂β ln(Z) In worldline representation: q = 1 V
2πβ
2π
χt = 1 V
2πβ
2π 2 −
2πβ
2π 2 , sG = 1 2βV
θ 2π )2
β
SLIDE 10 Observables
Topological charge, topological susceptibility, gauge action density: q = − 1 V ∂ ∂θ ln(Z) , χt = 1 V ∂2 ∂θ2 ln(Z) , sG = − 1 V ∂ ∂β ln(Z) In worldline representation: q = 1 V
2πβ
2π
χt = 1 V
2πβ
2π 2 −
2πβ
2π 2 , sG = 1 2βV
θ 2π )2
β
- Note: q is odd under C transformation at θ = π.
⇒ q is order parameter for breaking of C symmetry!
SLIDE 11
Breaking of C symmetry
◮ Conjectured: C symmetry is broken at large m2 and restored at sufficiently negative m2. [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 + m2 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.
SLIDE 12 Fist-Order transition as function of θ: (λ = 0.5, β = 3), M = 4 + m2
1 2 3 ∆θ 0.150 0.155 0.160 0.165
<sG>
8x8 16x16 32x32
1 2 3 ∆θ 0.124 0.125 0.126
<sG>
1 2 3 ∆θ
0.00 0.01 0.02 0.03 <q> M = 3.5
1 2 3 ∆θ
0.000 0.001 0.002
<q>
M = 2.0
SLIDE 13
Critical endpoint at ∆θ = 0: (λ = 0.5, β = 3), M = 4 + m2
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6
M
0.000 0.005 0.010 0.015 0.020 0.025 0.030
〈|q|〉
12x12 16x16 20x20 40x40 80x80 2.6 2.7 2.8 2.9 3.0 3.1
M
0.00 0.05 0.10 0.15 0.20
χt
SLIDE 14
Critical endpoint at ∆θ = 0: (λ = 0.5, β = 3), M = 4 + m2
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6
M
0.000 0.005 0.010 0.015 0.020 0.025 0.030
〈|q|〉
12x12 16x16 20x20 40x40 80x80 2.6 2.7 2.8 2.9 3.0 3.1
M
0.00 0.05 0.10 0.15 0.20
χt ◮ What is the scaling behavior? Critical exponents?
SLIDE 15 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|, q2 and Binder cumulant U:
dU dM
d dM ln |q|
d dM ln q2
∝ L
1 ν
- 3. Estimate MC from scaling of pseudo-critical mass defined as position of
maximum: Mpc(L) = MC + A L− 1
ν
- 4. Extract critical exponents β and γ from scaling of observables at critical
Mass MC: |q|(MC ,L) = L− β
ν Fq(x) ,
χt (MC ,L) = L
γ ν Fχ(x)
SLIDE 16 Critical exponents
16 32 64
L
4 8 16 32 64
max
d ln<q
2>/dM
d ln<|q|>/dM dU/dM 0.00 0.01 0.02 0.03
L
2.92 2.94 2.96 2.98
Mpc(L)
d ln<q
2>/dM
χt d ln<|q|>/dM d <|q|>/dM d U/dM 16 32 64 128
L
0.016 0.017 0.018 0.019 0.020
<|q|>(0,L)
16 32 64 128
L
0.01 0.1
χt (0,L)
◮ Final results for U(1) gauge-Higgs model: ν = 1.003(11) β = 0.126(7) γ = 1.73(7) ◮ 2-d Ising values: ν = 1 β = 0.125 γ = 1.75
SLIDE 17
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 Z2 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)
