Spatio-temporal correlations across the melting of 2 D Wigner molecules Amit Ghosal IISER KOLKATA • Coulomb interacting particles in 2D confinements. • Static & Dynamic responses across ‘melting’. • Effect of ‘Disorder/irregularity’ on melting. Computational Tools • Molecular dynamics and Classical (Metropolis) Monte Melzer Group Carlo with Simulated Annealing at finite T . • Path integral Quantum Monte Carlo (QMC) at low T ; variational and diffusion QMC at T = 0. B. Meer, et.al. , PNAS’14 Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Crystal of Coulomb particles and its melting: Wigner Crystal Melting Competetion between (1934) PE & KE Coulomb repulsion forces particles to stay as far as possible from each other, localizing them in a crystal. Kinetic Energy delocalizes them. KE ∼ k B T (equipartition) ⇒ Thermal / Classical melting [Gann, Chakravarty & Chester, 1979] KE ∼ Quantum (zero-point) fluctuations ⇒ Quantum melting. [Tanatar & Ceperley, 1989] • In confinements, Wigner Crystal ⇒ “Wigner Molecule” N N q 2 1 � � � x 2 + y 2 H = r j | + V conf ( r i ); r = | � r | = 4 πǫ | � r i − � i < j i (a) Irregular: V Ir conf ( r ) = a { x 4 / b + by 4 − 2 λ x 2 y 2 + γ ( x − y ) xyr } (b) Circular: V Cr conf ( r ) = α r 2 , with α = m ω 2 / 2 Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Thermal melting of Wigner Molecules (WM) T=0.002 T=0.015 T=0.065 (b)T=0.0150 (b)T=0.0650 (a)T=0 Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Static Correlations: EPJB 86 , 499, (2013), arXiv:1701.02338 � N b BOO: ψ 6 ( i ) = � N 1 l =1 e i 6 θ ki i =1 a − 1 � Lindemann: 1 � N r 0 i | 2 � L = �| � r i − � k =1 N b N i 12 N=141 T=0.002 (a) V Ir (b) V Cr 5 10 Conf T=0.006 Conf T=0.010 N=101 8 T=0.020 4 P(| ψ 6 |) N=500 N=500 T=0.030 N=75 6 T=0.050 3 N=50 LR 4 2 2 1 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 | ψ 6 | | ψ 6 | 0 5 0.02 0.04 0.06 0.08 0.1 T T=0.002 (c) V Ir (d) V Cr T=0.006 Conf Conf Specific Heat: c V = d ˆ dT = T − 2 � � E 2 � − � E � 2 � E 4 T=0.010 P( φ 6 ) T=0.020 N=500 N=500 T=0.030 3 0.05 T=0.050 0.04 c = 137 Γ 2 N=145 T c 0.03 1 0.02 50 100 150 N N=121 0 c V 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 φ 6 φ 6 N=76 m 6 ( i ): projection of ψ 6 ( i ) onto mean local orientation field. � � Nb � 1 � ψ ∗ N=56 m 6 ( i ) = 6 ( i ) k =1 ψ 6 ( k ) � Larsen & Grier, PRL ’96 � � Nb • Also studied g ( r ) , g 6 ( r ), Generalized susceptibilities: χ ψ , χ φ 0.02 0.04 T 0.06 0.08 0.1 Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Take-home messages from static correlations & questions: 1. Crossover from ‘solid’-like to ‘liquid’-like behavior discerned from independent observables (unique T x within tolerance). 2. No apparent distinction between T X (within errorbars) in circular and irregular confinements. 3. Qualitative responses are more-or-less independent of N (for 100 ≥ N ≤ 100) though there are differences in details. What can dynamics tell us about the ‘solid’ and ‘liquid’ in traps? Can motional signatures distinguish the crossover based on the nature of the confinement? (e.g., circular vs. irregular) Can we access generic signatures of disordered dynamics in traps? EPL, 114, 46001 (2016); arXiv:1701.02338; and unpublished Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Displacements [∆ � r ( t ) = { � r ( t ) − � r (0) } ] in ‘solid’ • Spatially correlated inhomogeneous motion at large t even at low T in irregular traps. (a)V Ir (b) V Ir (C)V Ir Conf Conf Conf T = 0.006 T = 0.006 T = 0.006 t = 100 t = 1000 t = 9500 y r/r 0 ≤ 0.15 r/r 0 >1.0 0.50<r/r 0 ≤ 1.0 0.15<r/r 0 ≤ 0.50 (d) V Cr (e) V Cr (f) V Cr T=0.006 T=0.006 T=0.006 Conf Conf Conf t=100 t=1000 t=9500 y x x x Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
spatio-temporal density correlations: Dynamical (spatio-temporal) information best extracted from Van-Hove correlation function: G ( r , t ) = � � N � r − | � r i ( t ) − � � i , j =1 δ r j (0) | � • Self-part G s ( r , t ) (when i = j ): probability to move on an average a distance r in time t . 10 0 10 0 t = 200 t = 200 10 0 Irregular 10 0 Circular G s (r,t) 400 G s (r,t) 400 700 (T=0.006) (T=0.006) 10 -2 10 -2 1000 700 G s (r,t) G s (r,t) 10 -4 1000 10 -4 0 1 2 r/r 0 0 0.5 1 2 r/r 0 10 -2 10 -2 t= 1 10 100 1000 9500 10 -4 10 -4 0.5 1 r/r 0 2 2.5 0 0.5 1 r/r 0 2 2.5 10 0 10 0 1 t = 1 Irregular Circular (T=0.030) 10 (T=0.030) 10 100 100 200 200 G s (r,t) G s (r,t) 400 400 1000 1000 9500 9500 10 -2 10 -2 10 -4 10 -4 0 4 8 15 0 4 8 15 r/r 0 r/r 0 Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Stretched exponential decay of spatial correlation in IWM • G s ( r , t ) ∼ e − r 2 / c for small r ∀ t . Observation: • G s ( r , t ) shows complex tail (large t). ( r , t ) ∼ e − lr k for r > r c ( r , t ) ∼ e − r 2 / c for r ≤ r c and G large G small Postulate: s s • Optimal r c and other parameters ( including k ) determined by minimizing total χ 2 . (a) V Ir (b) V Cr Conf Conf 2.0 k 1.0 0.6 T=0.002 2.0 2.0 1.0 T=0.006 0.4 1.0 T=0.075 T=0.020 T=0.100 T=0.030 k T=0.020 k T=0.250 T=0.050 0.3 0.3 T=0.030 10 -1 10 0 10 1 t 10 2 10 3 10 4 10 -1 10 0 10 1 t 10 2 10 3 10 4 10 0 10 1 10 2 10 -1 10 0 t 10 1 10 2 t • Small t , All T : k ≃ 2, (Gaussian tail). • Large t , Low T : (IWM + CWM) k ∼ 1 (exponential tail) [P. Chaudhuri et al. , PRL (2007)] • Large t , High T : • (CWM) 1 ≥ k ≤ 2, ( stretched Gaussian tail); Expt: [He et al. ACS Nano (’13)] • (IWM) k < 1, T-dependent Stretched exponential tail of spatial correlation! Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Time scales: α − relaxation time from overlap Function 1 1 10 2 10 2 τ α 10 1 τ α 10 1 0.8 0.8 10 0 N 10 0 • Q ( t ) = 1 � 0 0.015 0.03 0.045 W ( | � r i ( t ) − � r i (0) | ) 0 0.015 0.03 0.045 T 0.6 0.6 N T Q(t) Q(t) (b) V Cr i =1 T=0.002 Conf T=0.006 0.4 0.4 T=0.010 where W ( r i ) = 1 if r i < r cut , & (a) V Ir T=0.020 Conf W ( r i ) = 0 if r i > r cut (satisfied once, 0.2 0.2 T=0.030 T=0.050 only on first passage). 0 0 10 -1 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 [Kob et al. (’12); Karmakar et al. (’14)] t t 1.2 10 1 (a) V Ir 10 1 Conf (b) V Cr Conf τ x • α − relaxation time ( τ α ) from Q ( t ): 0.8 τ x 10 0 T=0.002 10 0 T=0.006 0.9 Q ( τ α ) = e − 1 T 0 0.02 0.04 T 0 0.02 0.04 T=0.010 0.6 log τ α Χ 4 (t) N=75 T=0.020 2 2 N=150 Χ 4 (t) log τ α N=500 T=0.030 • χ 4 ( t ) = 1 0.6 0 0 N [ � Q 2 ( t ) � − � Q ( t ) � 2 ] T=0.050 0.4 -2 -2 30 90 150 30 90 150 1/T 1/T 0.3 0.2 [Karmakar et.al. PNAS, (’08)] 0 0 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 t t • χ 4 ( t ) measures extent of dynamic heterogeneity (spatial correlations in particles’ dynamics). • τ x ( T ) is the time-scale when dynamic heterogeneity is maximum at the given T . Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
Dynamical Correlations • When particles’ cages rearrange, the 1 0.8 system relaxes & particles diffuse. 0.6 ⇒ corresponding structural change T=0.002 T=0.006 characterized by a cage correlation (CC) 0.4 C g (t) T=0.010 T=0.020 function. T=0.030 T=0.050 Cage correlation function: 0.2 (a) V Ir (b) V Cr C g ( t ) = � L ( i ) ( t ) · L ( i ) ( 0 ) � Conf N=150 Conf N=150 [Rabani et.al. PRL’99] � L ( i ) 2 ( 0 ) � 0.1 10 -1 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 t t • C g ( t ) ∼ exp [ − ( t /τ g ) c ]; c ∼ 0 . 5 for irregular, and c ∼ 0 . 6 for circular • Persistence time ( τ p , solid line): a traps. particle displaced beyond a cut-off for the first time. (a) V Ir τ p , T=0.020 (b) V Cr Conf Conf τ e , T=0.020 • Exchange time ( τ e , dotted line): 0.6 N=500 N =500 T=0.030 time required for subsequent passage by T=0.050 0.5 P(log( τ )) T=0.100 cut-off distance. [Hedges et. al. , J. Chem. 0.4 Phys.(2007)] 0.3 0.2 • Two distributions decouple for 0.1 Irregular confinement but signature of 0 decoupling is weaker for Circular -1 0 1 2 3 4 5 6 7 -1 0 1 2 3 4 5 6 7 log( τ ) log( τ ) confinement. Amit Ghosal Spatio-temporal correlations across the melting of 2 D Wigner m
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