1. Ajay Kumar Ghosh Jadavpur University Kolkata, India Vortex Line Ordering in the Driven 3-D Vortex Glass Vortex Wroc ł aw 2006 Stephen Teitel University of Rochester Rochester, NY USA Peter Olsson Umeå University Umeå, Sweden

2. Outline The problem: driven vortex lines with random point pins The model to simulate: frustrated XY model with RSJ dynamics Previous results Our results effects of thermal vortex rings on large drive melting importance of correlations parallel to the applied field Conclusions

3. Koshelev and Vinokur, PRL 73, 3580 (1994) motion averages disorder ⇒ shaking temperature ⇒ ordered driven state Giamarchi and Le Doussal, PRL 76, 3408 (1996) transverse periodicity ⇒ elastically coupled channels ⇒ moving Bragg glass Balents, Marchetti and Radzihovsky, PRL 78, 751 (1997); PRB 57, 7705 (1998) longitudinal random force remains ⇒ liquid channels ⇒ moving smectic Scheidl and Vinokur, PRE 57, 2574 (1998) Le Doussal and Giamarchi, PRB 57, 11356 (1998) We simulate 3D vortex lines at finite T > 0. Driven vortex lines with random point pinning For strong pinning, such that the vortex lattice is disordered in equilibrium, how do the vortex lines order when in a driven steady state moving at large velocity?

4. 3D Frustrated XY Model kinetic energy of flowing supercurrents on a discretized cubic grid coupling on bond i  phase of superconducting wavefunction magnetic vector potential density of magnetic flux quanta = vortex line density piercing plaquette  of the cubic grid uniform magnetic field along z direction magnetic field is quenched weakly coupled xy planes constant couplings between xy planes || magnetic field random uncorrelated couplings within xy planes disorder strength p

5. Equilibrium Behavior critical p c at low temperature p < p c ordered vortex lattice p > p c disordered vortex glass we will be investigating driven steady states for p > p c

6. Resistively-Shunted-Junction Dynamics Units current density: time: voltage/length: temperature: apply: current density I x response: voltage/length V x vortex line drift v y

7. twisted boundary conditions voltage/length new variable with pbc stochastic equations of motion RSJ details

8. Previous Simulations Domínguez, Grønbech-Jensen and Bishop - PRL 78, 2644 (1997) vortex density f = 1/6, 12 ≤ L ≤ 24, J z = J   weak disorder ?? moving Bragg glass - algebraic correlations both transverse and parallel to motion vortex lines very dense, system sizes small, lines stiff Chen and Hu - PRL 90, 117005 (2003) vortex density f = 1/20, L = 40, J z = J   weak disorder p ~ 1/2 p c at large drives moving Bragg glass with 1 st order transition to smectic single system size, single disorder realization, based on qualitative analysis of S(k) Nie, Luo, Chen and Hu - Intl. J. Mod. Phys. B 18, 2476 (2004) vortex density f = 1/20, L = 40, J z = J   strong disorder p ~ 3/2 p c at large drives moving Bragg glass with 1 st order transition to smectic single system size, single disorder realization, based on qualitative analysis of S(k) We re-examine the nature of the moving state for strong disorder, p > p c, using finite size analysis and averaging over many disorders

9. Parameters p = 0.15 > p c ~ 0.14 J z = J   L up to 96 vortex density f = 1/12 ground state p = 0 I x V x vortex line motion v y Quantities to Measure structural dynamic use measured voltage drops to infer vortex line displacements

10. ln S(k , k z =0) I x V x vortex line motion v y Phase Diagram

11. Previous results of Chen and Hu p ~ 1/2 p c weak disorder a, b - “moving Bragg glass” algebraic correlations both transverse and parallel to motion a´, b´ - “moving smectic” We will more carefully examine the phases on either side of the 1st order transition

12. Digression: thermally excited vortex rings melting of ordered state upon increasing current I is due to proliferation of thermally excited vortex rings I F(R) ~ RlnR - IR 2 ring expands when R > R c ~ 1/I from superfluid 4 He rings proliferate when F(R c ) ~ T ~ 1/I TmTm

13. Disordered state above 1st order melting T m TmTm ln S(k) vortex line motion v y When we increase the system size, the height of the peaks in S(k) along the k x axis do NOT increase ⇒ only short ranged translational order. ⇒ disordered state is anisotropic liquid and not a smectic I = 0.48, T = 0.13

14. Ordered state below 1st order melting T m ln S(k) vortex line motion v y C(x, y, z=0) Bragg peak at K 10 ⇒ vortex motion is in periodically spaced channels peak at K 11 sharp in k y direction ⇒ vortex lines periodic within each channel peak at K 11 broad in k x direction ⇒ short range correlations between channels ⇒ ordered state is a smectic not a moving Bragg glass I = 0.48, T = 0.09

15. Correlations between smectic channels short ranged translational correlations between smectic channels

16. Correlations within a smectic channel C(x, y, z=0) exp decay to const > 1/4 algebraic decay to 1/4 correlations within channel are either long ranged, or decay algebraically with a slow power law ~ 1/5 averaged over 40 random realizations

17. Correlations along the magnetic field snapshot of single channel  z ~ 9 As vortex lines thread the system along z, they wander in the direction of motion y a distance of order the inter-vortex spacing. Vortex lines in a given channel may have a net tilt.

18. Dynamics y 0 (t)y 3 (t)y 6 (t)y 9 (t) center of mass displacement of vortex lines in each channel Channels diffuse with respect to one another. Channels may have slightly different average velocities. Such effects lead to the short range correlations along x. 36 x 96 x 96

19. Dynamics and correlations along the field direction z See group of strongly correlated channels moving together. Smectic channels that move together are channels in which vortex lines do not wander much as they travel along the field direction z. Need lots of line diffusion along z to decouple smectic channels. As L z increases, all channels decouple. Only a few decoupled channels are needed to destroy correlations along x. smaller system: 48 x 48 x48

20. Behavior elsewhere in ordered driven state I=0.48, T=0.07, L=60 20 random realizations transverse correlations Many random realizations have short ranged correlations along x. These are realizations where some channels have strong wandering along z. Many random realizations have longer correlations along x;  x ~ L These are realizations where all channels have “straight” lines along z. More ordered state at low T? Or finite size effect? So far analysis was for I=0.48, T=0.09 just below peak in T m (I)

21. Conclusions For strong disorder p > p c (equilibrium is vortex glass) Driven system orders above a lower critical driving force Driven system melts above an upper critical force due to thermal vortex rings 1st order-like melting of driven smectic to driven anisotropic liquid Smectic channels have periodic (algebraic?) ordering in direction parallel to motion, short range order parallel to applied field; channels decouple (short range transverse order) Importance of vortex line wandering along field direction for decoupling of smectic channels Moving Bragg glass at lower temperature? or finite size effect?