Critical quench dynamics in confined quantum systems Critical quench dynamics in confined quantum systems Mario Collura and Dragi Karevski IJL, Groupe Physique Statistique - Universit´ e Henri Poincar´ e nov. 2009 Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Contents 1 Crossing a critical point Kibble-Zurek argument 2 Confining potential 3 Adiabatic approximation 4 Density of defects 5 Ising quantum chain Linear spatial modulation Transition amplitude Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Crossing a critical point Qualitative picture Time-dependent hamiltonian H ( t ) = H critical + g ( t ) V Power-law tuning parameter g ( t ) ∼ sgn ( t ) | t /τ | α = sgn ( t ) v | t | α driving the system through the critical point. The system remains in the instantaneous ground state | GS ( t ) � as long as it is protected by a finite gap ∆( t ) from the excited states. Breaking of the adiabaticity close to the critical point since the gap vanishes right at the QCP. Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Crossing a critical point Kibble-Zurek argument Adiabatic-Impulse approximation impulse Kibble-Zurek mechanism |2> |1> adiabatic Adiabatic: Sufficiently away adiabatic from the critical point no |2> transitions between |1> instantaneous eigenstates ^ t ^ 0 -t L 0 t R Impulse: Sufficiently close to the critical point critical | ϕ ( t ) � ≈ e − i α ( t ) | 0( t ) � t ∈ [ −∞ , − ˆ t L ] : slowing down ⇒ no change in the wave function except | ϕ ( t ) � ≈ e − i β ( t ) | 0( − ˆ t ∈ [ − ˆ t L , ˆ t R ] : t L ) � for an overall phase factor |� ϕ ( t ) | 0( t ) �| 2 = const . t ∈ [ˆ t R , + ∞ ] : Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Crossing a critical point Kibble-Zurek argument Kibble-Zurek time-scale τ KZ Kibble-Zurek timescale τ KZ τ 0 / ∆( τ KZ ) = ∆( τ KZ ) / | ˙ ∆( τ KZ ) | with ∆( t ) ∼ | g ( t ) | ν z ∼ v ν z | t | ν z α one has ℓ ∼ τ 1 / z τ KZ ∼ v − ν z / (1+ αν z ) ; KZ Scaling for defect density n ∼ ℓ − d ∼ v d ν/ (1+ ν z α ) A. Polkovnikov, PRB 72 , 161201(R) (2005) W. H. Zurek, U. Dorner and P. Zoller, Phys. Rev. Lett. 95 , 105701 (2005). B. Damski, Phys. Rev. Lett. 95 , 035701 (2005); B. Damski and W. H. Zurek, Phys. Rev. A 73 , 063405 (2006); ibid , Phys. Rev. Lett. 99 , 130402 (2007). Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Crossing a critical point Kibble-Zurek argument n ∼ τ − αν d / ( α z ν +1) ∼ v d ν/ (1+ ν z α ) A. Polkovnikov, PRB 72 , 161201(R) (2005) D. Sen, K. Sengupta, S. Mondal, PRL 101 , 016806 (2008) R. Barankov, A. Polkovnikov, PRL 101 , 076801 (2008) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Confining potential Power-law spatial inhomogeneity A power-law deviation in one direction of the quantum control parameter h form its critical value h c : x 2 h ( x , t ) − h c ≃ g ( t ) x ω , δ ( x , t ) ≡ x > 0 x v | t | α sgn ( t ) g ( t ) = g(t) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Confining potential Power-law spatial inhomogeneity A power-law deviation in one direction of the quantum control parameter h form its critical value h c : x 2 h ( x , t ) − h c ≃ g ( t ) x ω , δ ( x , t ) ≡ x > 0 x v | t | α sgn ( t ) g ( t ) = g(t) The perturbation introduces a crossover region in space-time ( x , t ) around the critical locus (0,0). Lenght-scale Time-scale τ ∼ ℓ ( τ ) z → τ ∼ v − z / y v ℓ ( t ) ∼ δ ( ℓ, t ) − ν → ℓ ( t ) ∼ | g ( t ) | − 1 / y g y v = y g + z α y g = (1 + ν ω ) /ν The exponent y v is the RG dimension of the perturbation field, such that under rescaling by a factor b the amplitude transforms as v ′ = b y v v . Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Confining potential Scaling arguments Under rescaling, the profile ϕ ( x , t , v ) associated to an operator ϕ with scaling dimension x ϕ transform as ϕ ( x , t , v ) = b − x ϕ ϕ ( xb − 1 , tb − z , vb y v ) Taking b = v − 1 / y v ∝ ℓ ∝ τ 1 / z one obtains ϕ ( x , t , v ) = v x ϕ / y v Φ( xv 1 / y v , tv z / y v ) Trap-size scaling ϕ ∼ ℓ − x ϕ associated to a finite size system with ℓ ∼ v − 1 / y v . T. Platini, D. Karevski and L. Turban, J. Phys. A 40 1467 (2007) B. Damski and W. H. Zurek, New J. Phys. 11 063014 (2009) M. Campostrini and E. Vicari, Phys. Rev. Lett. 102 , 240601 (2009) M. Collura, D. Karevski and L. Turban, J. Stat. Mech. P08007 (2009) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Adiabatic approximation Adiabatic approximation Time evolution of a quantum system described by a time-dependent Hamiltonian H ( t ) The system is initially in the instantaneous ground state of the Hamiltonian H ( t 0 ): | ϕ ( t 0 ) � = | 0( t 0 ) � At time t | ϕ ( t ) � = U ( t , t 0 ) | 0( t 0 ) � where the time evolution operator is � t U ( t , t 0 ) = ˆ T exp − i ds H ( s ) t 0 Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Adiabatic approximation Adiabatic expansion in the instantaneous eigenbasis Instantaneous eigenstates H ( t ) | k ( t ) � = E k ( t ) | k ( t ) � Adiabatic expansion up to first order Rate of change of the Hamiltonian: ∂ t H ( t ) ∼ ∂ t g ( t ) ∼ v → 0 R t R t | ϕ ( t ) � = e − i t 0 dsE 0 ( s ) | 0( t ) � + e − i t 0 dsE 0 ( s ) a k ( t 0 , t ) | k ( t ) � � k � =0 Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Adiabatic approximation Adiabatic expansion up to first order R t R t t 0 dsE 0 ( s ) | 0( t ) � + t 0 dsE 0 ( s ) a k ( t 0 , t ) | k ( t ) � | ϕ ( t ) � = e − i � e − i k � =0 � g ( t ) dg � k ( g ) | ∂ g H ( g ) | 0( g ) � e − i ϑ k ( g , g ( t )) a k ( t 0 , t ) = δ ω k 0 ( g ) g ( t 0 ) where � y v − 1 /α dg | g | 1 /α − 1 δω k 0 ( g ) ϑ k ( x , y ) = α x δω k 0 ( g ) = E k ( g ) − E 0 ( g ) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Adiabatic approximation Adiabatic expansion up to first order R t R t t 0 dsE 0 ( s ) | 0( t ) � + t 0 dsE 0 ( s ) a k ( t 0 , t ) | k ( t ) � | ϕ ( t ) � = e − i � e − i k � =0 � g ( t ) dg � k ( g ) | ∂ g H ( g ) | 0( g ) � e − i ϑ k ( g , g ( t )) a k ( t 0 , t ) = δ ω k 0 ( g ) g ( t 0 ) where For v ≪ 1, a k ≃ 0: � y instantaneous ground state v − 1 /α dg | g | 1 /α − 1 δω k 0 ( g ) ϑ k ( x , y ) = For v ≫ 1, exp ( − i ϑ k ) ∼ 1: α x sudden quench δω k 0 ( g ) = E k ( g ) − E 0 ( g ) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Density of defects Density of defects Density of defects � | a k | 2 n = k � =0 General scaling arguments ( ℓ ∼ g − 1 / y g ): � k ( g ) | ∂ g H ( g ) | 0( g ) � ∼ ℓ − z + y g G ( ℓ − z / k z ) δ ω k 0 ∼ ℓ − z Ω( ℓ − z / k z ); Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Density of defects Density of defects Density of defects � | a k | 2 n = k � =0 General scaling arguments ( ℓ ∼ g − 1 / y g ): � k ( g ) | ∂ g H ( g ) | 0( g ) � ∼ ℓ − z + y g G ( ℓ − z / k z ) δ ω k 0 ∼ ℓ − z Ω( ℓ − z / k z ); For a quench crossing the QCP, in order that the integral converges at g = 0 the scaling function G ( u ) / Ω( u ) = uf ( u ) at small u . n ∼ ℓ − d ∼ v d ν/ (1+ ν z α ) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Critical quench dynamics in confined quantum systems Density of defects Density of defects Density of defects � | a k | 2 n = k � =0 General scaling arguments ( ℓ ∼ g − 1 / y g ): � k ( g ) | ∂ g H ( g ) | 0( g ) � ∼ ℓ − z + y g G ( ℓ − z / k z ) δ ω k 0 ∼ ℓ − z Ω( ℓ − z / k z ); For a quench crossing the QCP, In the inhomogeneous case the in order that the integral convergence close to the critical converges at g = 0 the scaling point is not garanted. function G ( u ) / Ω( u ) = uf ( u ) at small u . n ∼ ℓ − d ∼ v d ν/ (1+ ν z α ) Mario Collura and Dragi Karevski Critical quench dynamics in confined quantum systems
Recommend
More recommend
