Quenching to field-stabilized Adam Iaizzi* Postdoctoral Fellow - PowerPoint PPT Presentation

J24.00013 Quenching to field-stabilized Adam Iaizzi* Postdoctoral Fellow magnetization plateaus in the National Taiwan University unfrustrated Ising antiferromagnet 臺灣⼤學 Better title: Stable frozen states without disorder *iaizzi@bu.edu 1

Adam Iaizzi - iaizzi@bu.edu 2D Ising AFM 1 Exact, T=0 0.9 H = J ∑ σ i σ j − h ∑ σ i 0.8 ❖ 0.7 ⟨ i , j ⟩ i 0.6 ❖ Square lattice m 0.5 0.4 ❖ Antiferromagnet 0.3 0.2 ❖ 2-fold degenerate GS 0.1 ❖ T c = 2.26… 0 0 1 2 3 4 5 h ❖ Simplest model with PT ( 0 h < 4 ❖ With field: poorly studied m ( T = 0 , h ) = 1 h > 4 2

Adam Iaizzi - iaizzi@bu.edu Dynamics ❖ Metropolis Monte Carlo 1 ❖ Single-spin-flip updates Exact, T=0 0.9 0.8 ❖ Choose spin at random 0.7 ❖ Flip with probability 0.6 P = min [ 1, e −Δ E / T ] m 0.5 0.4 ❖ Quench: 0.3 0.2 ❖ Start from (totally T = ∞ 0.1 random) state 0 0 1 2 3 4 5 h ❖ Instant quench to T ❖ What happens? 3

Adam Iaizzi - iaizzi@bu.edu Actual behavior ❖ Instantaneous quench to 1.0 = 2 finite T<T c = 4 0.8 = 8 ❖ High T: EQ = 16 0.6 = 64 ❖ Low T: Non-ergodic m 0.4 ❖ Plateaus ❖ Stable frozen states 0.2 ❖ No intrinsic disorder 0.0 0 1 2 3 4 5 ❖ Valleys of ergodicity h 4

Adam Iaizzi - iaizzi@bu.edu Zero temperature magnetization 1.0 8 0 h = 0 , = 2 > > 0 . 057 0 < h < 2 J, > > = 4 > 0.8 > > 0 h = 2 J, = 8 < m ( T = 0 , h ) = = 16 0 . 282 2 J < h < 4 J, > 0.6 = 64 > > 0 . 55 h = h s = 4 J, > > m > > : 1 h > h s . 0.4 0.2 ❖ Ergodic for h = 0, ± 2, ± 4 0.0 0 1 2 3 4 5 ❖ From now on: T = 0 h 5

Adam Iaizzi - iaizzi@bu.edu What is happening? Second plateau: h = 3 First plateau: h = 1 View animations online: http://bit.ly/iaizziTPS 6

n e d fi n Adam Iaizzi - iaizzi@bu.edu o Freezing Mechanism C x = +1 x = -1 + + h>4 _ x = σ i = ± 1 + + y = +4 + + + h<4 + + X y = σ j = 0 , ± 2 , ± 4 + + _ h>2 j + + + + + y = +2 _ _ h<2 h 1 , e � ( y � h ) ∆ x/T i + + P = min h>0 _ y = 0 + + + h=0 + + _ + y = -2 ❖ 10 local spin configurations ❖ 5 pairs _ y = -4 + 7

Adam Iaizzi - iaizzi@bu.edu Zero temperature dynamics x = +1 x = -1 Δ E = ( y − h ) Δ x + + h>4 _ + + y = +4 + + + h<4 + + dynamics: + + T = 0 ❖ _ h>2 + + + + + y = +2 _ _ ❖ Accept if h<2 Δ E ≤ 0 + + ❖ Reject if Δ E > 0 h>0 _ y = 0 + + + h=0 : reversible update Δ E = 0 ❖ + + _ + y = -2 ❖ Reversible updates when h = y = 0, ± 2, ± 4 _ y = -4 + ❖ Valleys of ergodicity 8

Adam Iaizzi - iaizzi@bu.edu h = 0 ❖ Maps onto ferromagnet ❖ Bulk domains and straight domain walls stable quench, stuck in stripe T = 0 ❖ FIG. 13. Relaxation of a stripe state in two dimensions at small state w/ P = 0.3390... nonzero temperature: � a � nucleation of a dent � freely flippable spins are indicated � ; � b � diffusive growth of the dent; � c � dent reaches the system size and hence the domain wall steps to the left. This overall ❖ Connection to critical process ultimately leads to the disappearance of the stripe. percolation theory ❖ Otherwise reach G.S. Spirin, Krapivsky, Redner, PRE 65 016119 (2001) Barros, Krapivsky, Redner, PRE 80 040101(R) (2009) 9

Adam Iaizzi - iaizzi@bu.edu First plateau - + - + - + - + + - + - + - + - + + - + + - + + - + - + - + - + - + - + - - + - - + - - + - + - + - + - + - + - + + - + + - + + - + - + - + - + - + - + - - + - - + - - + - + - + - + - + - + - + + - + 0 < h < 2 J ❖ + - + + - + - + + - + + - + - + - + + - - + + - + - + - - + - + + - + - + - - + + - - + - + - + + - + - + + - + - + + - ❖ Corner domain walls now - + + - + - + - - + - + - - + - + + - + + - - + - + - + + - + - + + - + + - + - stable - + + - + - + - + + - + - - + - - + - + + - + + - + - + - - + - + + - + + - + - - + - + - + + - + + - + - + - + - + - + ❖ Corners host excess + spin - + + - + - - + + - + - + - + - + - + + + - + + - + + - - + - + - + + - + - + - - + - + - + - + + - + - + + - + - + - + ❖ Net magnetization 0.057 + - + - + - + + - + - + - - + - + - + - - + - + - + + - + - + - + + - + - + - + + + - + - + - + - + + - + - + - + - + - + - + - + - + - + + - + - + - + - + + - 10

Adam Iaizzi - iaizzi@bu.edu Second plateau - + + - + - + + - + - + - + - + + - + + + + - + - + + - + - + + + - + + - + + - + - + - + - + + - + + - + + - + + + - + - + - + - + - + + - + + + - + + - + + + + - + - + - + + - + - + - + + - + + - + ❖ Straight line DW not stable + + - + + + + - + - + - + - + + + - + - - + + + - + + + - + - + - + - + + + - + + + - + + + - + + + + - + - + + - + + - - + + + + - + + + - + + - + - + + - + + ❖ Corners/diagonal DW stable + - + - + + + - + + + - + - + - + + + - - + - + + + - + - + - + + + - + - + - + + - + + - + + + + - + + - + + - + - + - ❖ Excess + spin along DW - + + - + - + - + + + - + + - + + + - + + + - + + + + + + - + + - + + + - + + - + - + + - + - + - + + - + - + - + + - + ❖ Net magnetization 0.282 + + - + + - + - + + - + - + - + - + + - - + + - + + - + - + + - + - + - + - + + + + - + - + + - + - + + - + + + - + - + + + + - + - + + - + - + + + - + + - + - + - + + - + + - + - + - + - + - + + - + 11

Adam Iaizzi - iaizzi@bu.edu h = h s ❖ Does not freeze + + - + + + - + + - + + + + - + + - + - + - + + + + + - + + + + + - + + - + - + + + + + + + - + - + + - + + + + + - + - - + + + + - + - + + - + - + + + - + + + ❖ Vanishing dependence T + + + - + + - + + + + + + - + - + + + - - + - + - + + + + - + + + + - + + + + + + + + + + + + + + + - + + - + + + - + + ❖ Coexistence of + + + - + - + + - + + + + + - + + + + + - + + + + + + + + + + - + + + - + + - + + - + + + + - + - + + + + + - + + + + - + + - + - + + + + + - + + + + - + - + + ❖ 2x AFM G.S. + + + - + - + + + + + - + + + + + + + + + - + + - + + + - + + + + - + - + - + - + + + + + + + - + - + + + + - + - + - + ❖ Fully-polarized state + - + + + - + + - + + - + - + - + + + + - + - + - + + + + + - + + + + + + + + + + - + + + + + - + - + + - + + + + - + + ❖ Equivalent to hard-squares + + + - + - + + - + + - + + + + - + - + + + - + + + + - + + + + - + + + + - + - + - + - + + + + + + + + + + + + + + + + problem 12

Conclusions 1.0 = 2 ❖ Instantaneous quenches in Ising AFM + field = 4 0.8 = 8 = 16 ❖ Stable plateaus without disorder 0.6 = 64 m ❖ In plateau: all spins unflippable 0.4 ❖ Describe in terms of stable local configurations 0.2 ❖ Ergodicity restored when updates available Δ E = 0 0.0 0 1 2 3 4 5 h ❖ Useful for understanding when MC fails ❖ Monte Carlo dynamics physical dynamics ≠ ❖ We choose the dynamics ❖ Plateau states are local energy minima under these dynamics ❖ Choosing dynamics rearranges energy surface 13

Open Questions ❖ Enumerate plateau states? ❖ Connect to percolation theory? ❖ Derive magnetization in plateaus ❖ What is the finite-T scaling of “valleys of ergodicity”? ❖ Other lattices? ❖ Other dynamics?

Thanks for your attention! Contact: Preprint available: Adam Iaizzi arXiv:2001.09268 email: iaizzi@bu.edu or scan QR code web: www.iaizzi.me 15