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 Gambassi SISSA and INFN, Trieste (Italy) Trieste, 19th September 2016

Quench dynamics

Quench dynamics

Quench dynamics Bath T initial equilibrium driven P 0 ( { σ } ) P eq ( { σ } , T ) distribution by reservoir Example: Metropolis algorithm E M MC steps MC steps

Quench dynamics Bath T initial equilibrium driven P 0 ( { σ } ) P eq ( { σ } , T ) distribution by reservoir Example: Metropolis algorithm E M t relax MC steps MC steps

Quench dynamics Bath T ' T c initial equilibrium driven P 0 ( { σ } ) P eq ( { σ } , T ) distribution by reservoir T ' T c equilibrium: correlation length and time diverges quench: too! Equilibrium is attained in infinite time t relax Is there some universal AGING behaviour? Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05

Aging dynamics New critical exponents? at criticality: Janssen, Schaub, Schmittman ’89 ( ξ = ∞ ) rev: Calabrese & Gambassi ’05 1) two-time correlation functions G R ( q, t, t 0 ) = q � 2+ η + z G R ( q z t, q z t 0 ) η , z (response function) equilibrium G C ( q, t, t 0 ) = q � 2+ η G C ( q z t, q z t 0 ) exponents (correlation function)

Aging dynamics New critical exponents? at criticality: Janssen, Schaub, Schmittman ’89 ( ξ = ∞ ) rev: Calabrese & Gambassi ’05 1) two-time correlation functions G R ( q, t, t 0 ) = q � 2+ η + z G R ( q z t, q z t 0 ) η , z (response function) equilibrium G C ( q, t, t 0 ) = q � 2+ η G C ( q z t, q z t 0 ) exponents (correlation function) if t 0 ⌧ t G C ( q, t, t 0 ) = q � 2+ η ( t/t 0 ) θ � 1 e G C ( q z t, q z t 0 ) G R ( q, t, t 0 ) = q � 2+ η + z ( t/t 0 ) θ e G R ( q z t, q z t 0 ) new non-equilibrium exponent ! θ

Aging dynamics New critical exponents? at criticality: ( ξ = ∞ ) Janssen, Schaub, Schmittman ’89 rev: Calabrese & Gambassi ’05 2) magnetization ext. magnetic no ext. magnetic Modify quench field for t<0 field for t>0 protocol: ∼ t θ 0 h φ ( t ) i M ( t ) = M 0 t θ 0 M ( M 0 t θ 0 + β /z ν ) θ 0 = θ + (2 − z − η ) /z 0 cf. equilibrium Take home message : breaking of time-translational systems with spatial invariance (or fluctuation-dissipation theorem) boundaries generates new exponents Sieberer et al. ’13, ‘14 Marino & Diehl ‘16

Theoretical description coarse graining: Hohenberg- criticality Hohenberg & Halperin models Halperin ‘77 e.g., Ising model: lattice continuum φ ( x ) spin variables field σ i microscopic Hamiltonian effective Hamiltonian Glauber dynamics of Ising model: Model A Gaussian noise h 6 φ 2 i r 2 � r � g ˙ φ = φ + ζ h ζ ( r , t ) ζ ( r 0 , t 0 ) i = 2 T δ ( d ) ( r � r 0 ) δ ( t � t 0 ) τ 0 φ 2 R Initial condition is stochastic: 0 / 2 φ 0 P [ φ 0 ] ∝ e − Initial correlation length: τ − 1 / 2 → 0 0

Response functional Martin, Siggia, Rose ‘73 Janssen ‘76 De Dominicis ‘76 Z Z O ( φ )e − S [ φ , e φ ] O ( φ ) h δ [ φ � φ ( t )] i ζ , φ 0 = h O ( t ) i = New field: response field e φ φ ] contain info also on initial conditions S [ φ , e This action is viable for renormalization

Renormalization and critical exponents Equilibrium: some divergencies, renormalization required renormalization critical exps. r, φ , e φ ν , η , z Quench: new divergences, additional renormalization required initial slip exp. renormalization e θ φ 0 - second order in expansion ✏ value of : rev: Calabrese & - large-N limit of O(N) θ Gambassi ’05 - Monte Carlo

Functional renormalization group Modified action: S k [ Ψ ] = S [ Ψ ] + ∆ S k [ Ψ ] Ψ = ( ϕ , e ϕ ) ∆ S k [ Ψ ] imposes a IR cutoff parametrized by k (coarse-grained over effective action Γ k [ Φ ] S k [ Ψ ] volume ) ∼ k − d "✓ δ 2 Γ k ◆ − 1 d R k # Z d Γ k d k = 1 FRG equation tr δ Φ t δ Φ + R k 2 d k Wetterich ‘93

Functional renormalization group How to include the quench? An ansatz for has to be used. Γ k [ Φ ] ✓ ◆ Z φ + K r 2 φ + ∂ U Z ˙ Γ [ φ , e φ ] = Γ 0 [ φ 0 , e ϑ ( t � t 0 ) e ∂φ � D e φ 0 ] + φ φ x quench initial potential conditions Formally similar to a boundary problem: bulk effective action boundary effective action 0 t

Functional renormalization group How to include the quench? An ansatz for has to be used. Γ k [ Φ ] ✓ ◆ Z φ + K r 2 φ + ∂ U Z ˙ Γ [ φ , e φ ] = Γ 0 [ φ 0 , e ϑ ( t � t 0 ) e ∂φ � D e φ 0 ] + φ φ x quench initial potential conditions determined Z ✓ ◆ − Z 2 from and P [ φ 0 ] e 0 + Z 0 e 0 φ 2 Γ 0 = φ 0 φ 0 causality 2 τ 0 arguments

How to perform FRG with a boundary Breaking of TTI Fourier transform not useful Our approach: 3 steps Step 1: write an ansatz for the potential U ( φ ) U ( φ ) τ > 0 Simplest ansatz: φ U ( φ ) = τ 2 φ 2 + g 4! φ 4 U ( φ ) τ < 0 φ

How to perform FRG with a boundary Breaking of TTI Fourier transform not useful Our approach: 3 steps Step 2: rewrite FRG eq. as follows:  � d Γ d k = 1 ϑ ( t − t 0 ) G ( x, x )d R Z tr d k σ 2 x ⌘ − 1 Γ (2) + R ⇣ = ( G − 1 − V ) − 1 G = 0 field field independent dependent

How to perform FRG with a boundary Breaking of TTI Fourier transform not useful Our approach: 3 steps  � d Γ d k = 1 ϑ ( t − t 0 ) G ( x, x )d R Z Step 2: tr d k σ 2 x Z Dyson-like G ( x, x 0 ) = G 0 ( x, x 0 ) + G 0 ( x, y ) V ( y ) G ( y, x 0 ) equation y + 1 Each term can X = G 0 ( x, x 0 ) + G n ( x, x 0 ) be evaluated! n =1 ∼ φ 2 n

How to perform FRG with a boundary Breaking of TTI Fourier transform not useful Our approach: 3 steps Step 3: replace these terms into FRG eq. constant value: renormalizes bulk Z Z ∞ d Γ k d t e φφ [1 + f ( t )] d k ∝ vanishing function 0 r Bulk i� Z ∞ h X � ∞ d n renormalizes d t e e � φ ( t ) φ ( t ) f ( t ) = φ ( t ) φ ( t ) c n � the d t n 0 t =0 n =0 boundary!

Functional renormalization group more refined m ) 2 + λ U = g 4!( φ 2 − φ 2 6!( φ 2 − φ 2 m ) 3 ansatz:

Functional renormalization group more refined m ) 2 + λ U = g 4!( φ 2 − φ 2 6!( φ 2 − φ 2 m ) 3 ansatz: ��� ���� RG ✏ 2 ���� θ ● ��� ���� FRG θ ��� ���� �� ���� � RG ��� ✏ ��� ��� ��� ��� ��� ��� � Monte Carlo

Conclusion & outlook Summary • Critical quench generates new critical exponents • FRG can be used to compute non-equilibrium exponents • Good agreement with numerics and perturbative RG Near future... • Full potential see Jamir • Application to quenches in isolated quantum systems Marino’s • O(N) and Potts models talk! • Other HH models (conserved dynamics) ... far future • non-thermal fixed points? • coarsening dynamics • towards new exp. platforms (cold atoms, exciton-polaritons…)

Thank you for your attention!