Critical quench dynamics in confined quantum systems

Transcription

1 Critical quench dynamics in confined quantum systems IJL, Groupe Physique Statistique - Université Henri Poincaré 27 nov. 2009

2 Contents 1 Crossing a critical point Kibble-Zurek argument 2 Confining potential 3 Adiabatic approximation 4 5 Ising quantum chain Transition amplitude

3 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.

4 Crossing a critical point Kibble-Zurek argument P AI Kibble-Zurek mechanism Adiabatic: Sufficiently away from the critical point no transitions between instantaneous eigenstates Impulse: Sufficiently close to the critical! Q point critical slowing down no change in the wave function except for an overall phase factor IG. 2: Transition probability when the system starts time volution from a ground state at t i and evolves to +. Dots: exact expression (11). Solid line: AI predicion (12) with α = π/2 determined from diabatic solution in ppendix A. Dashed thick line: lowest order diabatic result, πτ Q/2, coming from (A5) and (A6) with η =1/2. ading to Eqs. (A5) and (A6) is not only much easier hen determination of exact LZ solution [18], but also eally elementary. Therefore, we expect that it can be one comparably easily for any model of interest. Adiabatic-Impulse (a) approximation (b) 2> 1> 1> 2> 0 t adiabatic impulse -t ^ 0 L ^tr 4 adiabatic FIG. 3: The same as in Fig. 1 but for nonsymmetric Landau- Zener problem with δ > 1 see (13). t [, ˆt L ] : ϕ(t) e iα(t) 0(t) t [ ˆt L, ˆt R ] : ϕ(t) e iβ(t) 0( ˆt L ) agreement we have seen above is accidental and has something t to[ˆt do R, with + ] the symmetry : ϕ(t) 0(t) of the Landau-Zener 2 = const. problem. The second problem was preliminarily considered in [9], but without comparing the AI prediction to exact analytic one being interesting on its own. Finally, the third problem is an example where AI approximation correctly suggests at a first sight unexpected symmetry between this problem and the one considered in Sec. III B.

5 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 one has (t) g(t) νz v νz t νzα τ KZ v νz/(1+ανz) ; l τ 1/z KZ Scaling for defect density n l d v dν/(1+νzα) A. Polkovnikov, PRB 72, (R) (2005) W. H. Zurek, U. Dorner and P. Zoller, Phys. Rev. Lett. 95, (2005). B. Damski, Phys. Rev. Lett. 95, (2005); B. Damski and W. H. Zurek, Phys. Rev. A 73, (2006); ibid, Phys. Rev. Lett. 99, (2007).

6 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 δ(x, t) h(x, t) h c g(t)x ω, x > 0 x g(t) = v t α sgn(t) g(t)

7 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 δ(x, t) h(x, t) h c g(t)x ω, x > 0 x g(t) = v t α sgn(t) g(t) The perturbation introduces a crossover region in space-time (x, t) around the critical locus (0,0). Lenght-scale Time-scale l(t) δ(l, t) ν l(t) g(t) 1/yg y g = (1 + νω)/ν τ l(τ) z τ v z/yv y v = y g + zα 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 yv v.

8 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 xϕ ϕ(xb 1, tb z, vb yv ) Taking b = v 1/yv l τ 1/z one obtains ϕ(x, t, v) = v xϕ/yv Φ(xv 1/yv, tv z/yv ) Trap-size scaling ϕ l xϕ associated to a finite size system with l v 1/yv. T. Platini, D. Karevski and L. Turban, J. Phys. A (2007) B. Damski and W. H. Zurek, New J. Phys (2009) M. Campostrini and E. Vicari, Phys. Rev. Lett. 102, (2009) M. Collura, D. Karevski and L. Turban, J. Stat. Mech. P08007 (2009)

9 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 U(t, t 0 ) = ˆT exp i t t 0 dsh(s)

10 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 ϕ(t) = e i R t dse 0(s) t 0 0(t) + e i R t dse 0(s) t 0 a k (t 0, t) k(t) k 0

11 Adiabatic approximation Adiabatic expansion up to first order ϕ(t) = e i R t dse 0(s) t 0 0(t) + e i R t dse 0(s) t 0 a k (t 0, t) k(t) k 0 a k (t 0, t) = g(t) g(t 0) dg k(g) g H(g) 0(g) e iϑ k (g,g(t)) δω k0 (g) where ϑ k (x, y) = v 1/α α y δω k0 (g) = E k (g) E 0 (g) x dg g 1/α 1 δω k0 (g)

12 Adiabatic approximation Adiabatic expansion up to first order ϕ(t) = e i R t dse 0(s) t 0 0(t) + e i R t dse 0(s) t 0 a k (t 0, t) k(t) k 0 a k (t 0, t) = g(t) g(t 0) dg k(g) g H(g) 0(g) e iϑ k (g,g(t)) δω k0 (g) where ϑ k (x, y) = v 1/α α y δω k0 (g) = E k (g) E 0 (g) x dg g 1/α 1 δω k0 (g) For v 1, a k 0: instantaneous ground state For v 1, exp( iϑ k ) 1: sudden quench

13 n = k 0 a k 2 General scaling arguments (l g 1/yg ): δω k0 l z Ω(l z /k z ); k(g) g H(g) 0(g) l z+yg G(l z /k z )

14 n = k 0 a k 2 General scaling arguments (l g 1/yg ): δω k0 l z Ω(l z /k z ); k(g) g H(g) 0(g) l z+yg G(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 dν/(1+νzα)

15 n = k 0 a k 2 General scaling arguments (l g 1/yg ): δω k0 l z Ω(l z /k z ); k(g) g H(g) 0(g) l z+yg G(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. In the inhomogeneous case the convergence close to the critical point is not garanted. n l d v dν/(1+νzα)

16 n = k 0 a k 2 General scaling arguments (l g 1/yg ): δω k0 l z Ω(l z /k z ); k(g) g H(g) 0(g) l z+yg G(l z /k z ) Inhomogeneous QCP τ KZ ( τ 0 Ω 0 zα y g ) yg /y v v z/y v n [ (τ KZ )] d/z ( τ 0 Ω 0 zα y g ) dα/yv v d/y v

17 Ising quantum chain Ising quantum chain in time-dependent inhomogeneous transverse field H(t) = 1 L 1 σ x 2 nσn+1 x 1 2 n=1 L h n (g)σn z n=1 h n (g) = 1 + g(t)n ω, g(t) = v t α sgn(t)

18 Ising quantum chain Ising quantum chain in time-dependent inhomogeneous transverse field H(t) = 1 L 1 σ x 2 nσn+1 x 1 2 n=1 L h n (g)σn z n=1 h n (g) = 1 + g(t)n ω, g(t) = v t α sgn(t) In linear case ω = 1 Exact solution using fermionic mapping For general ω Numerical diagonalization + Finite-size analyses

19 Ising quantum chain Transition amplitude Up to the first order correction we can write the evolution of the Ising chain ground state 0(g 0 ) as ϕ(t) 0(g) + pq a pq (t 0, t)η q(g)η p(g) 0(g) Defects density Fidelity Excess energy n pq a pq (t 0, t) 2 e pq δω pq a pq (t 0, t) 2

20 Ising quantum chain Transition amplitude Linear case ω = 1 Transition amplitude a pq (t 0, t) for a quench starting at a value g 0 = g(t 0 ) and ending at a new value g t = g(t). Quenches that do not cross the critical point a pq (t 0, t) = F pq A φpq ( g 0, g(t) ) e iθpq(t) where Θ pq(t) = πh( g 0) + φ pq g(t) 2+α 2α φ pq = 2Ω pq v 1/α A φ (x, y) = 2α 2 + α α + 2 sgn(g0) he 1 iφx 2+α 2α E 1 iφy 2+α 2α i The spatial inhomogeneity modifies the dependence on g of the scaling function F pq(g) = G pq(g)/(2ω pq(g)) close to g = 0 such that it leads to a complete breakdown of the approximation

21 n e n e n e Ising quantum chain Transition amplitude Quench to the critical point: g i = 1, g f = 0 ω = 1, α = 2 ω = 2, α = 2 ω = 2, α = v v v ω = 1, α = 2 ω = 2, α = 2 ω = 2, α = v v v For v 1, n v 1/4,1/5,1/6 and e n 2. The dashed lines correspond to the asymptotic sudden-quench values n sq and e sq 0.136

22 Ising quantum chain Transition amplitude Conclusion Scaling theory for the non-linear quench of a power-law perturbation, such as a confining potential, close to a critical point Power law behavior of the density of defects with the ramping rate with an exponent depending on the space-time properties of the potential. First order adiabatic calculation and exact results on an inhomogeneous transverse field Ising chain