Relaxation laws in classical and quantum long-range lattices

Transcription

1 Relaxaton laws n classcal and quantum long-range lattces R. Bachelard Grupo de Óptca Insttuto de Físca de São Carlos USP Quantum Non-Equlbrum Phenomena Natal RN 13/06/2016

2 Lattce systems wth long-range nteractons What s a lattce? ϕ r ϕ j r j j varables fxed postons Power-law nteracton: all-to-all couplng V v( ϕ, ϕ ) H = V + j, j α, j r, j r j p 2 2

3 Lattce systems wth long-range nteractons α > d for short-range nteractons The system s addtve and extensve H 2 p = V, j +, j 2 N N N: number of partcles d: dmenson of the space

4 Lattce systems wth long-range nteractons α > d for short-range nteractons The system s addtve and extensve H 2 p = V, j +, j 2 N N N: number of partcles d: dmenson of the space α < d for long-range nteractons The system s non-addtve and non-extensve H 2 p = V, j +, j 2 NxN α 1 / d N non-extensve t tn (1 α /d)/2 H H / N p p/ N (1 α /d) (1 α /d)/2 The system becomes extensve

5 Why studyng long-range systems? Rch nonequlbrum physcs due to ts slow relaxaton: Ensemble nequvalence Sze-dependent equlbraton tme N and t The lmts do not commute! Out-of-equlbrum phase transtons and crtcal ponts Phase reentrance

6 Long-range αxy chan A chan of rotators on a lattce, wth Hamltonan dynamcs H 2 p cos( ) J ϕ ϕ j = 2 2 j α, j ϕ : phase p : momentum

7 Long-range αxy chan A chan of rotators on a lattce, wth Hamltonan dynamcs H 2 p cos( ) J ϕ ϕ j = 2 2 j α, j ϕ : phase p : momentum { Mean-feld case: α=0 Nonequlbrum phase dagram Equlbrum phase transton [Antonazz et al., Phys Rev Lett. 99, (2007)]

8 Long-range αxy chan Just below the transton pont, the system slowly goes from an unmagnetzed phase to a magnetzed one

9 Long-range αxy chan Just below the transton pont, the system slowly goes from an unmagnetzed phase to a magnetzed one A sze-dependent relaxaton tme was observed [Y. Yamaguch..Phys Rev E 68, (2003)]

10 Long-range αxy chan: α>0

11 Long-range αxy chan: α>0

12 Long-range αxy chan τ relax N q αα Sze-dependent slow relaxaton tme (~power-law)

13 Long-range quantum Isng model H z z J σσ = h 2, j j α σ j z σ h x z = z-component of the Paul spn operator external magnetc feld partcle poston Integrable system, but that provdes a dynamcs for σσ xx (tt)

14 Long-range quantum Isng model H z z J σσ = h 2, j j α σ j z σ h x z = z-component of the Paul spn operator external magnetc feld partcle poston Integrable system, but that provdes a dynamcs for σσ xx (tt) σ x x 2Jt ( t) = σ (0)cos( 2ht) cos j j α

15 Thermodynamc lmt: N σ ( t) σ (0)cos 2 ( ht) 5+ 2 α d d/2 2 2 π α / d d exp JtN for 0 α ( d 2 α) ( d / 2) < Γ 2 2 π 4Jt d exp for α > (2 α d) Γ( d / 2) π 2 x x 1+ d 2 α d/2 d / α

16 Thermodynamc lmt: N σ ( t) σ (0)cos 2 ( ht) 5+ 2 α d d/2 2 2 π α / d d exp JtN for 0 α ( d 2 α) ( d / 2) < Γ 2 2 π 4Jt d exp for α > (2 α d) Γ( d / 2) π 2 x x 1+ d 2 α d/2 d / α

17 Relaxaton of the quantum Isng model For αα < dd/2 we have relaxaton tmes that scale as a power-law of the system sze N τ N q

18 Quantum Isng model vs. αxy chan τ relax N q

19 Quantum Isng model vs. αxy chan τ relax N q

20 Quantum Isng model vs. αxy chan τ relax N q

21 Normalzaton of the energy scale J σσ σ σ H = h = h 2 j 2 j z z j z z z J j z σ H σ α α, j, j H p J cos( ϕ ϕ ) p J cos( ϕ ϕ ) = = 2 2 j j H α α 2 2, j j 2 2, j j Ths normalzaton s necessary to have a well-defned thermodynamc lmt (very useful n statstcal physcs) provdes a sze-ndependent tme scale for short-tme dynamcs

22 Normalzaton of the energy scale J σσ σ σ H = h = h 2 j 2 j z z j z z z J j z σ H σ α α, j, j q α = q+ 1 d H p J cos( ϕ ϕ ) p J cos( ϕ ϕ ) = = 2 2 j j H α α 2 2, j j 2 2, j j q 1 α = q+ 1 2 d Usng the normalzed Hamltonan, the smlarty between the scalng laws of the two models s strkng R. Bachelard and M. Kastner, Phys. Rev. Lett. 110, (2013)

23 Concluson/perspectves Long-range classcal and quantum systems have smlar relaxatons, wth relaxaton tmes that exhbt dependence on the system sze (power-laws) Theory s stll mssng... Especally for the α=d/2 threshold Realzed n collaboraton wth Mchael Kastner NITheP, Stellenbosch, South Afrca R. Bachelard and M. Kastner, Phys. Rev. Lett. 110, (2013)

24 Example: Lght scatterng by cold atomc clouds Lght nduces a 1/r nteracton between the atoms n a d=3 space and a mean-feld nteracton n a d=1 space (optcal cavty) ν 0 ν 0 r j cst 3d experment@são Carlos SP Don t hestate to vst us n São Carlos! 1d experment@são Carlos SP (under constructon)