Universal short-time dynamics: FRG for a temperature quench - - PowerPoint PPT Presentation
Universal short-time dynamics: FRG for a temperature quench arXiv:1606.06272 Alessio Chiocchetta SISSA and INFN, Trieste (Italy) In collaboration with: Jamir Marino ThP , Cologne (Germany) Sebastian Diehl ThP , Cologne (Germany) Andrea
SISSA and INFN, Trieste (Italy)
In collaboration with:
Jamir Marino ThP
, Cologne (Germany)
Sebastian Diehl ThP
, Cologne (Germany)
Andrea Gambassi SISSA and INFN, Trieste (Italy)
Trieste, 19th September 2016
arXiv:1606.06272
Quench dynamics
Quench dynamics
Quench dynamics
Example: Metropolis algorithm
M E
MC steps MC steps
Bath
initial distribution
Peq({σ}, T) P0({σ})
equilibrium driven by reservoir
T
Quench dynamics
Example: Metropolis algorithm
M E
MC steps MC steps
Bath
initial distribution
Peq({σ}, T) P0({σ})
equilibrium driven by reservoir
trelax T
Bath
T ' Tc equilibrium: correlation length and time diverges
quench: too! Equilibrium is attained in infinite time
T ' Tc
Quench dynamics
initial distribution
Peq({σ}, T) P0({σ})
equilibrium driven by reservoir
trelax
Is there some universal behaviour?
Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05
New critical exponents? at criticality: 1) two-time correlation functions
(ξ = ∞) GR(q, t, t0) = q2+η+zGR(qzt, qzt0) GC(q, t, t0) = q2+ηGC(qzt, qzt0)
Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05
Aging dynamics
η, z
equilibrium exponents
(response function) (correlation function)
New critical exponents? at criticality: 1) two-time correlation functions
t0 ⌧ t
if
(ξ = ∞) GR(q, t, t0) = q2+η+zGR(qzt, qzt0) GC(q, t, t0) = q2+ηGC(qzt, qzt0) GC(q, t, t0) = q2+η(t/t0)θ1 e GC(qzt, qzt0) GR(q, t, t0) = q2+η+z(t/t0)θ e GR(qzt, qzt0)
Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05
Aging dynamics
η, z
equilibrium exponents
(response function) (correlation function)
new non-equilibrium exponent !
M(t) = M0 tθ0 M(M0tθ0+β/zν)
∼ tθ0
hφ(t)i
Take home message: breaking of time-translational invariance (or fluctuation-dissipation theorem) generates new exponents
field for t<0 no ext. magnetic field for t>0
θ0 = θ + (2 − z − η)/z
Sieberer et al. ’13, ‘14
systems with spatial boundaries
Aging dynamics
New critical exponents? at criticality: (ξ = ∞)
Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05
2) magnetization
Modify quench protocol:
Marino & Diehl ‘16
coarse graining: Hohenberg- Halperin models criticality e.g., Ising model:
σi φ(x)
lattice continuum spin variables field microscopic Hamiltonian effective Hamiltonian
Glauber dynamics of Ising model: Model A
˙ φ = h r2 r g 6φ2i φ + ζ
hζ(r, t)ζ(r0, t0)i = 2Tδ(d)(r r0)δ(t t0)
P[φ0] ∝ e−
R τ0φ2
0/2
Gaussian noise
Initial condition is stochastic:
Initial correlation length:
Theoretical description
Hohenberg & Halperin ‘77
φ0 τ −1/2 → 0
hO(t)i = Z O(φ) hδ[φ φ(t)]iζ,φ0 = Z O(φ)e−S[φ,e
φ]
New field: response field e
φ
This action is viable for renormalization
S[φ, e φ] contain info also on initial conditions
Response functional
Martin, Siggia, Rose ‘73 Janssen ‘76 De Dominicis ‘76
Equilibrium: Quench: new divergences, additional renormalization required
value of :
renormalization some divergencies, renormalization required
e φ0 θ
✏ θ
Renormalization and critical exponents
ν, η, z
critical exps. renormalization initial slip exp.
rev: Calabrese & Gambassi ’05
FRG equation
Ψ = (ϕ, e ϕ) Sk[Ψ] = S[Ψ] + ∆Sk[Ψ]
Modified action:
∆Sk[Ψ]imposes a IR cutoff parametrized by Γk[Φ] Sk[Ψ]
effective action
dΓk dk = 1 2 Z tr "✓ δ2Γk δΦtδΦ + Rk ◆−1 dRk dk #
Functional renormalization group
k
Wetterich ‘93
(coarse-grained over volume )
∼ k−d
How to include the quench? An ansatz for has to be used.
Γk[Φ]
Γ[φ, e φ] = Γ0[φ0, e φ0] + Z
x
ϑ(t t0)e φ ✓ Z ˙ φ + Kr2φ + ∂U ∂φ De φ ◆
initial conditions quench potential
t
Formally similar to a boundary problem:
bulk effective action boundary effective action
Functional renormalization group
How to include the quench? An ansatz for has to be used.
Γk[Φ]
Γ[φ, e φ] = Γ0[φ0, e φ0] + Z
x
ϑ(t t0)e φ ✓ Z ˙ φ + Kr2φ + ∂U ∂φ De φ ◆
initial conditions quench potential
Γ0 = Z ✓ − Z2 2τ0 e φ2
0 + Z0 e
φ0φ0 ◆
determined from and causality arguments
P[φ0]
Functional renormalization group
Breaking of TTI
Step 1: write an ansatz for the potential
Fourier transform not useful Our approach: 3 steps Simplest ansatz:
U(φ) U(φ) = τ 2φ2 + g 4!φ4
U(φ) U(φ)
φ φ τ > 0 τ < 0
How to perform FRG with a boundary
Breaking of TTI
Step 2:
Fourier transform not useful Our approach: 3 steps
dΓ dk = 1 2 Z
x
tr ϑ(t − t0)G(x, x)dR dk σ
⇣ Γ(2) + R ⌘−1 = (G−1 − V )−1
field independent field dependent
rewrite FRG eq. as follows:
How to perform FRG with a boundary
Breaking of TTI
Step 2:
Fourier transform not useful Our approach: 3 steps
dΓ dk = 1 2 Z
x
tr ϑ(t − t0)G(x, x)dR dk σ
Z
y
G0(x, y) V (y)G(y, x0) = G0(x, x0) +
+1
X
n=1
Gn(x, x0)
Dyson-like equation Each term can be evaluated!
∼ φ2n
How to perform FRG with a boundary
Breaking of TTI
Step 3:
Fourier transform not useful Our approach: 3 steps
dΓk dk ∝ Z
r
Z ∞ dt e φφ [1 + f(t)]
replace these terms into FRG eq.
vanishing function constant value: renormalizes bulk
Z ∞ dt e φ(t)φ(t)f(t) =
∞
X
n=0
cn dn dtn h e φ(t)φ(t) i
Bulk renormalizes the boundary!
How to perform FRG with a boundary
Functional renormalization group
U = g 4!(φ2 − φ2
m)2 + λ
6!(φ2 − φ2
m)3
more refined ansatz:
Monte Carlo FRG RG ✏2
✏
RG Functional renormalization group
U = g 4!(φ2 − φ2
m)2 + λ
6!(φ2 − φ2
m)3
more refined ansatz:
Conclusion & outlook
Summary
Near future...
see Jamir Marino’s talk!
... far future