Universal Scaling in Fast Quenches Near Lifshitz-Like Fixed Points Ali Mollabashi YITP Workshop on Quantum Information and String Theory 2019 5 June 2019 Ali Mollabashi Universality in Fast Quenches 5 June 2019 0 / 13
Quantum Quench ▸ How a closed system responds a time dependent parameter? Ali Mollabashi Universality in Fast Quenches 5 June 2019 1 / 13
Quantum Quench ▸ How a closed system responds a time dependent parameter? ▸ Understanding ▸ thermalization process ▸ cosmological fluctuations ▸ relaxation process (e.g. to GGE) ▸ critical dynamics Ali Mollabashi Universality in Fast Quenches 5 June 2019 1 / 13
Quantum Quench ▸ How a closed system responds a time dependent parameter? ▸ Understanding ▸ thermalization process ▸ cosmological fluctuations ▸ relaxation process (e.g. to GGE) ▸ critical dynamics ▸ If the system crosses / is driven to a critical point Ali Mollabashi Universality in Fast Quenches 5 June 2019 1 / 13
Categorizing Quantum Quenches ▸ How big is the the time scale of the quench duration compared to other scales in the theory? { Λ ,λ i ,λ f , ⋯ } Ali Mollabashi Universality in Fast Quenches 5 June 2019 2 / 13
Categorizing Quantum Quenches ▸ How big is the the time scale of the quench duration compared to other scales in the theory? { Λ ,λ i ,λ f , ⋯ } 1. Instantaneous Quenches ( δ t − 1 ≳ Λ) Evolution of certain far from equilibrium state with a fixed Hamiltonian [Calabrese-Cardy ’06 + many others] Ali Mollabashi Universality in Fast Quenches 5 June 2019 2 / 13
Categorizing Quantum Quenches ▸ How big is the the time scale of the quench duration compared to other scales in the theory? { Λ ,λ i ,λ f , ⋯ } 1. Instantaneous Quenches ( δ t − 1 ≳ Λ) Evolution of certain far from equilibrium state with a fixed Hamiltonian [Calabrese-Cardy ’06 + many others] 2. Smooth Quenches ( δ t − 1 ≪ Λ) ▸ Fast quenches 1 1 δ t − 1 ≪ λ d − ∆ ,λ d − ∆ , ⋯ i f ▸ Slow quenches 1 1 , ⋯ ≲ δ t − 1 ≪ Λ λ d − ∆ ,λ d − ∆ i f [Myers, Das, Galante, Nozaki, Das, Caputa, Heller, van Niekerk, ⋯ ] Ali Mollabashi Universality in Fast Quenches 5 June 2019 2 / 13
Reminder: Quantum Critical Point ▸ What happens near a critical point through a second order quantum phase transition? Ali Mollabashi Universality in Fast Quenches 5 June 2019 3 / 13
Reminder: Quantum Critical Point ▸ What happens near a critical point through a second order quantum phase transition? ▸ Correlation length ξ − 1 ∼ ∣ λ − λ c ∣ ν ▸ Energy scale of fluctuations ∆ ∼ ∣ λ − λ c ∣ z ν Ali Mollabashi Universality in Fast Quenches 5 June 2019 3 / 13
Reminder: Quantum Critical Point ▸ What happens near a critical point through a second order quantum phase transition? ▸ Correlation length ξ − 1 ∼ ∣ λ − λ c ∣ ν ▸ Energy scale of fluctuations ∆ ∼ ∣ λ − λ c ∣ z ν ▸ From the ratio of these two critical exponents, ∆ ∼ ξ − z z is defined as dynamical critical exponent Ali Mollabashi Universality in Fast Quenches 5 June 2019 3 / 13
Reminder: Quantum Critical Point ▸ What happens near a critical point through a second order quantum phase transition? ▸ Correlation length ξ − 1 ∼ ∣ λ − λ c ∣ ν ▸ Energy scale of fluctuations ∆ ∼ ∣ λ − λ c ∣ z ν ▸ From the ratio of these two critical exponents, ∆ ∼ ξ − z z is defined as dynamical critical exponent ▸ Critical points often have z ≠ 1 ! Ali Mollabashi Universality in Fast Quenches 5 June 2019 3 / 13
Reminder: Quantum Critical Point ▸ What happens near a critical point through a second order quantum phase transition? ▸ Correlation length ξ − 1 ∼ ∣ λ − λ c ∣ ν ▸ Energy scale of fluctuations ∆ ∼ ∣ λ − λ c ∣ z ν ▸ From the ratio of these two critical exponents, ∆ ∼ ξ − z z is defined as dynamical critical exponent ▸ Critical points often have z ≠ 1 ! ▸ How to model systems with Lifshitz-like ( z ≠ 1) fixed points? Ali Mollabashi Universality in Fast Quenches 5 June 2019 3 / 13
Lifshitz Symmetry ▸ Lifshitz scaling [Lifshitz ’41, Hertz ’76] x → λ ⃗ ⃗ t → λ z t , x Ali Mollabashi Universality in Fast Quenches 5 June 2019 4 / 13
Lifshitz Symmetry ▸ Lifshitz scaling [Lifshitz ’41, Hertz ’76] x → λ ⃗ ⃗ t → λ z t , x ▸ Algebra: standard Poincare algebra for H , P i and J ij & [ D , J ij ] = 0 [ D , P i ] = i P i [ D , H ] = i z H , , where J ij = − i ( x i ∂ j − x j p i ) H = − i ∂ t , D = − i ( z t ∂ t + x i ∂ i ) P i = − i ∂ i , Ali Mollabashi Universality in Fast Quenches 5 June 2019 4 / 13
Lifshitz Symmetry ▸ Lifshitz scaling [Lifshitz ’41, Hertz ’76] x → λ ⃗ ⃗ t → λ z t , x ▸ Algebra: standard Poincare algebra for H , P i and J ij & [ D , J ij ] = 0 [ D , P i ] = i P i [ D , H ] = i z H , , where J ij = − i ( x i ∂ j − x j p i ) H = − i ∂ t , D = − i ( z t ∂ t + x i ∂ i ) P i = − i ∂ i , ▸ No Boost symmetry: T 0 i ≠ T i 0 Ali Mollabashi Universality in Fast Quenches 5 June 2019 4 / 13
Lifshitz Symmetry ▸ Lifshitz scaling [Lifshitz ’41, Hertz ’76] x → λ ⃗ ⃗ t → λ z t , x ▸ Algebra: standard Poincare algebra for H , P i and J ij & [ D , J ij ] = 0 [ D , P i ] = i P i [ D , H ] = i z H , , where J ij = − i ( x i ∂ j − x j p i ) H = − i ∂ t , D = − i ( z t ∂ t + x i ∂ i ) P i = − i ∂ i , ▸ No Boost symmetry: T 0 i ≠ T i 0 ▸ Anisotropic scaling: z T 00 + T ii = 0 Ali Mollabashi Universality in Fast Quenches 5 June 2019 4 / 13
Outline + Statement of Results ▸ How does a theory with Lifshitz-like fixed point respond to time dependent parameters in the Hamiltonian? Ali Mollabashi Universality in Fast Quenches 5 June 2019 5 / 13
Outline + Statement of Results ▸ How does a theory with Lifshitz-like fixed point respond to time dependent parameters in the Hamiltonian? ▸ As a first step I report results we have found in fast quench regime Ali Mollabashi Universality in Fast Quenches 5 June 2019 5 / 13
Outline + Statement of Results ▸ How does a theory with Lifshitz-like fixed point respond to time dependent parameters in the Hamiltonian? ▸ As a first step I report results we have found in fast quench regime ▸ We study theories under relevant deformations in two distinct regimes ▸ In strongly coupled regime (via holographic models) ▸ In free field theories Ali Mollabashi Universality in Fast Quenches 5 June 2019 5 / 13
Outline + Statement of Results ▸ How does a theory with Lifshitz-like fixed point respond to time dependent parameters in the Hamiltonian? ▸ As a first step I report results we have found in fast quench regime ▸ We study theories under relevant deformations in two distinct regimes ▸ In strongly coupled regime (via holographic models) ▸ In free field theories ▸ The respond of the system is universal : ▸ only depends on ∆ ▸ independent of (i) quench details (ii) state ▸ free theory matches with holography Ali Mollabashi Universality in Fast Quenches 5 June 2019 5 / 13
Holographic Setup ▸ EMD theory 16 π G N ∫ d d + 1 x √ − g ( R + Λ − 1 2 ( ∂χ ) 2 − 1 S = 2 ( ∂ϕ ) 2 − 1 2 m 2 ϕ 2 − V ( ϕ )) − 1 4 e λχ F 2 − 1 where m 2 = ∆ ( ∆ − d z ) and d z ∶ = d + z − 1. Ali Mollabashi Universality in Fast Quenches 5 June 2019 6 / 13
Holographic Setup ▸ EMD theory 16 π G N ∫ d d + 1 x √ − g ( R + Λ − 1 2 ( ∂χ ) 2 − 1 S = 2 ( ∂ϕ ) 2 − 1 2 m 2 ϕ 2 − V ( ϕ )) − 1 4 e λχ F 2 − 1 where m 2 = ∆ ( ∆ − d z ) and d z ∶ = d + z − 1. ▸ Take th following solution ( z ≥ 1 ) [Taylor ’08] ds 2 = − f ( t , r ) dr 2 r 4 f ( t , r ) + g ( t , r ) 2 d ⃗ r 2 z − 2 dt 2 + x 2 in pure Lifshitz background f ( t , r ) = r − 2 , g ( t , r ) = r − 1 Ali Mollabashi Universality in Fast Quenches 5 June 2019 6 / 13
Holographic Scenario t = δ t z ˆ ▸ define: r = δ t ˆ r , t , ⋯ and take δ t → 0 t scalar respond t = r z z δt Pure Lifshitz . . . . . . . r Ali Mollabashi Universality in Fast Quenches 5 June 2019 7 / 13
Holographic Scenario r , t = δ t z ˆ ▸ In terms of dimensionless parameters r = δ t ˆ t r d z − ∆ [ p s ( ˆ r ∆ [ p r ( ˆ ϕ ( ˆ r ) = δ t d z − ∆ ˆ t ) + ⋯ ] + δ t ∆ ˆ t ) + ⋯ ] t , ˆ ▸ Source profile ⎧ ⎪ ⎪ ˆ p s ( t ) = δ p ⎨ t κ 0 < t < δ t ⎪ ⎪ ⎩ 1 δ t ≤ t Ali Mollabashi Universality in Fast Quenches 5 June 2019 8 / 13
Holographic Scenario r , t = δ t z ˆ ▸ In terms of dimensionless parameters r = δ t ˆ t r d z − ∆ [ p s ( ˆ r ∆ [ p r ( ˆ ϕ ( ˆ r ) = δ t d z − ∆ ˆ t ) + ⋯ ] + δ t ∆ ˆ t ) + ⋯ ] t , ˆ ▸ Source profile ⎧ ⎪ ⎪ ˆ p s ( t ) = δ p ⎨ t κ 0 < t < δ t ⎪ ⎪ ⎩ 1 δ t ≤ t ▸ Response profile p r ( ˆ t ) = a κ ⋅ δ p ⋅ δ t d z − 2∆ ⋅ ˆ dz − 2∆ + κ t z ▸ From holographic renormalization for 2∆ = d z + 2 nz there is logarithmic enhancement [Andrade-Ross ’12] Ali Mollabashi Universality in Fast Quenches 5 June 2019 8 / 13
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