1. Phase Diagram of a Point Disordered Model Type-II Superconductor Peter Olsson Stephen Teitel Umeå University University of Rochester IVW-10 Mumbai, India

2. Vortex liquid Bragg glass Vortex glass Shiba et al., PRB 2002 Optimally doped untwinned YBCO Pal et al., Super. Sci. Tech 2002 What is the equilibrium phase diagram of a strongly fluctuating type-II superconductor? Bragg glass  vortex liquid vortex glass? vortex slush? critical end point? multicritical point? Experiments: point disorder

3. Hu and Nonomura, PRL 2001 Kierfeld and Vinokur, PRB 2004 Lindemann criterion XY model simulations Phase Diagram Theoretical:

4. Outline Introduction 3D XY model and parameters thermodynamic observables and order parameters Low disorder vortex lattice melting Large disorder vortex glass transition gauge glass and screening Intermediate disorder vortex slush? Conclusions the phase diagram!

5. 3D Frustrated XY Model phase of superconducting wavefunction magnetic vector potential kinetic energy of flowing supercurrents on a discretized cubic grid coupling on bond i  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

6. Parameters anisotropy system size ~ 80 vortex lines disorder strength varies vortex line density fixed ground state vortex configuration for disorder-free system increasing disorder strength p at fixed magnetic field f increasing magnetic field f at fixed disorder strength p exchange Monte Carlo method (parallel tempering) or systematically vary p to go from weak to strong disorder limit

7. Thermodynamic observables E - energy density Q - variable conjugate to the disorder strength p E and Q should in general change discontinuously at a 1st order phase transition E and Q must both be continuous at a 2nd order phase transition free energy F  = 1/k B T

8. Structure function  vortex lattice ordering parameter n z is vortex density in xy plane or K1K1 K2K2 real-space k-space kxkx kyky vortex liquid vortex solid

9. Helicity Modulus  phase coherence order parameter twisted boundary conditions twist dependent free energy phase coherent: F[   ] varies with   free energy sensitive to boundary phase incoherent: F[   ] independent of   free energy insensitive to boundary helicity modulus: (phase stiffness) evaluate at the twist   0 that minimizes the free energy F, or maximizes the histogram P twist histogram:measure by simulating in fluctuating twist ensemble   0 = 0 for a disorder-free system, but not necessarily with disorder p > 0

10. Low disorder  the vortex lattice melting transition Structure function p = 0.16 T = 0.1985 solid T = 0.2210 liquid solid Structure function indicates vortex solid to liquid transition

11. p = 0.16 Helicity modulus twist histograms normal superconducting

12. Plots of  S, U, E, or Q vs. T do not directly indicate the order of the melting transition. Need to look at histograms! Bimodal histogram indicates coexisting solid and liquid phases! 1st order melting transition vortex lattice ordering parameter  S

13. Use peaks in P(  S) histogram to deconvolve solid configurations from liquid configurations. Construct separate E and Q histograms for each phase to compute the jumps  E and  Q at the melting transition.

14. Melting phase diagram As disorder strength p increases,  E decreases to zero, but  Q remains finite. Transition remains 1st order, without weakening, along melting line. P. Olsson and S. Teitel, Phys. Rev. Lett. 87, 137001 (2001) f = 1/5

15. Large disorder  the vortex glass transition well above the melting transition line p = 0.40 No longer any vortex solid histograms of lattice ordering parameter P(  S) T = 0.90 below T g T = 0.221 above T g

16. Phase coherence Looking for a 2nd order vortex glass transition with critical scaling. In principal, scaling can be anisotropic since magnetic field singles out a particular direction. If anisotropic scaling, situation very difficult; need to simulate many aspect ratios L z /L . So assume scaling is isotropic,  = 1, and see if it works! (it does!) Use constant aspect ratio L z = L . P. Olsson, Phys. Rev. Lett. 91, 077002 (2003) Curves for different L all cross at t = 0, i.e. T = T g

17. Histograms of twist    for a particular realization of disorder twist histogram develops several local maxima as enter the vortex glass phase p = 0.40 well above the melting transition line

18. Helicity modulus p = 0.30, 0.40, 0.55 curves for a particular p cross at single T g averaged over 200  600 disorder realizations scaling collapse of data

19. Phase diagram for melting and glass transitions How do glass and melting transitions meet???

20. Vortex glass vs. gauge glass uniform random distribution gauge glass model: (Huse & Seung, 1990) gauge glass is intrinsically isotropic  average magnetic field vanishes (Katzgraber and Campbell, 2004) vortex glass model: magnetic field breaks isotropy although vortex glass is not isotropic, critical scaling is isotropic (Olsson, 2003) gauge glass and vortex glass are in the same universality class (also, Kawamura, 2003, Lidmar, 2003)

21. Screening (Bokil and Young, 1995, Wengel and Young, 1996) When include magnetic field fluctuations due to a finite, the gauge glass transition in 3D disappears, If gauge glass and vortex glass are in the same universality class, expect the same. Vortex glass ‘transition’ will survive only as a cross-over effect. Critical scaling will break down when one probes length scales Resistance in vortex glass will be linear at all T, for sufficiently small currents. how small? (Kawamura, 2003)

22. Phase diagram for melting and glass transitions How do glass and melting transitions meet??? Simulations get very slow and hard to equilibrate.

23. Intermediate disorder  still a vortex solid, but now two! p = 0.22  E and  Q consistent with values from lower p

24. Phase coherence p = 0.22

25. Intermediate solid phase “solid 1” p = 0.22 snapshot of vortex configurations for 4 successive layers Intermediate solid consists of coexisting regions of ordered and disordered vortices.

26. Intermediate solid phase “solid 1” p = 0.22 Some similarities to “vortex slush” of Nonomura and Hu. Does it survive as a distinct phase in larger systems? Only the orientation given by K 1 is coherent throughtout the thickness of the sample. Orientation K 2 may exist locally in individual layers, but without coherence from layer to layer. ---

27. Phase diagram of point disordered f = 1/5 3D XY model 1)Melting transition remains 1st order even where it meets glass transition 2)Glass transition becomes cross-over on large enough length scales 3)Possible intermediate solid? Needs more investigation 4)Lattice to glass transition at low T? Conclusions