Langevin simulations of interacting vortex lines in dirty high-temperature superconductors established that the vortex glass phase ages: within the accessible pre-asymptotic window, two-time correlations and linear responses obey multiplicative power-law scaling, and the fluctuation-dissipation theorem (FDT) is violated in a simple linear manner with an effective temperature close to the vortex-liquid-to-vortex-glass crossover temperature. Completing this characterization is the open problem: determine the asymptotic out-of-equilibrium dynamics of the vortex glass — whether the two-time scaling functions survive or are modified when the growing length crosses over to logarithmic creep growth beyond a time t*, whether the effective temperature is asymptotically universal (temperature-independent and equal to the crossover temperature), where inter-vortex interactions and caging qualitatively change the aging class, and whether the predicted linear FDT violation and effective temperature can be verified experimentally in real superconductors via simultaneous noise-and-response protocols.

Background

In a type-II superconductor placed in a magnetic field, magnetic flux penetrates as quantized vortex lines (flux lines), each carrying one flux quantum. In clean samples these lines form a regular Abrikosov lattice; in dirty (strongly disordered) samples, random point defects pin the lines, and at low temperature the vortex system is described as a vortex glass (VG): a pinned, disordered configuration with extremely slow, history-dependent relaxation, separated from the flowing vortex liquid (VL) by a glassy crossover at a temperature TgT_g. In high-temperature superconductors this glassy vortex state governs the decay of persistent currents (thermally activated flux creep) and hence the current-carrying capacity of the material.

Glassy systems relax out of equilibrium through aging: two-time observables such as the correlation C(t,tw)C(t, t_w) between the waiting time twt_w (the age since the quench) and a later time tt never become stationary but keep slowing down as twt_w grows. Two aging classes are distinguished. In additive aging the correlator separates into a stationary part depending on the time difference t−twt - t_w plus an aging part depending on the ratio t/twt/t_w; in multiplicative aging the entire two-time function scales as tw−bfc(t/tw)t_w^{-b} f_c(t/t_w), the form typical of critical dynamics. In equilibrium, the fluctuation-dissipation theorem (FDT) fixes the linear response of an observable from its spontaneous fluctuations through the bath temperature TT; out of equilibrium the FDT is generally violated, and in several glassy systems the violation takes a simple two-slope form in a parametric plot of the integrated response against the correlation, permitting the definition of an effective temperature TeffT_{\mathrm{eff}} that characterizes the slow (aging) modes.

Bustingorry, Cugliandolo and Domínguez (Phys. Rev. B 75, 024506 (2007)) simulated the out-of-equilibrium dynamics of the VG with a London-Langevin model — interacting elastic vortex lines, discretized into L = 50 transverse planes, moving through a continuous random pinning potential under Langevin thermal noise. After a quench into the VG phase they found that (i) the two-time density-density correlation and mean-squared-displacement correlation age with multiplicative power-law scaling; (ii) the associated linear responses also age, and the FDT violation is simple and linear, defining an effective temperature of the same order as the dynamic crossover temperature TgT_g; and (iii) at the simulated vortex density the out-of-equilibrium properties are essentially those of a single elastic line in a random environment (a directed-polymer-type problem), the elastic and pinning energies dominating over the vortex-vortex repulsion.

These conclusions hold only in the pre-asymptotic window accessible with short lines. In the single-line problem the growing correlation length L(t)L(t) is known to cross over at a time t∗t^* from power-law growth to logarithmic-in-time growth — the creep regime, in which barriers grow with length so that relaxation slows down exponentially — and a similar crossover is expected in the interacting VG, beyond the reach of the original simulations. Related results exist on either side of the problem: in driven vortex lattices a moving lattice exhibits a measurable transverse effective temperature obtained from a generalized FDT, and experiments have documented aging and memory effects in vortex matter (magnetization creep, memory and rejuvenation in transport) as well as ratio-based (t/twt/t_w) aging of the response of a driven Bragg glass (a weakly disordered vortex lattice that retains quasi-long-range positional order) in the superconductor 2H-NbSe2_2. What does not exist, on either the simulation or the experimental side, is the asymptotic characterization of the VG itself and any measurement combining spontaneous fluctuations with the linear response needed to test the predicted linear FDT violation and extract TeffT_{\mathrm{eff}} in the VG phase.

Problem Statement

Determine the complete out-of-equilibrium (aging) dynamics of the vortex glass phase of dirty high-temperature superconductors, extending the pre-asymptotic characterization of Bustingorry, Cugliandolo and Domínguez (2007) to its asymptotic, universal, and experimentally verifiable content. The common objective uniting the requested results is a full characterization of how the vortex glass ages after a quench, expressed through two-time correlations, the associated linear responses, the growing correlation length, and the effective temperature. Concretely, within the London-Langevin description of interacting elastic vortex lines in a random pinning potential — a class that includes both continuous random potentials and attractive localized point pins — or an equivalent elastic-line model with an established mapping to it:

  1. Asymptotic scaling: establish the growth law of the growing correlation length L(t)L(t) — the dynamic length extracted from the two-time correlations and mean-squared displacements (not to be confused with the number of discretization planes, also denoted LL); different observables may possess different growing lengths — and the asymptotic two-time scaling form of the correlations and responses, in particular whether the multiplicative power-law aging scaling persists beyond the crossover time t∗t^* at which L(t)L(t) crosses from power-law to logarithmic (creep) growth, and how the scaling functions are modified in the creep regime.

  2. Universality of the effective temperature: determine whether the FDT violation in the aging regime is asymptotically linear with a temperature-independent effective temperature equal to the vortex-liquid-to-vortex-glass (VL-VG) crossover temperature TgT_g, or whether a more complex dependence on bath temperature, waiting time, or vortex density appears at long times.

  3. Interaction-dominated regime: delineate the region of vortex density and pinning strength in which inter-vortex interactions (many-body caging, in which each vortex is trapped by the repulsive cage formed by its neighbors, as in structural glasses) qualitatively change the aging behavior — for example from ratio-based multiplicative scaling toward two-step relaxation — given that single-line relaxation dominates at the density simulated originally.

  4. Experimental verification: test the predictions in real superconductors — the aging of two-time quantities and the simple linear FDT violation with a measurable effective temperature in the vortex glass phase — through simultaneous noise (voltage and/or magnetic) and time-dependent response measurements, or equivalent tracer or vortex-shear protocols. This deliverable is completed by design and analysis rather than by carrying out the measurement: a concrete, experimentally realizable protocol, together with its expected signal and an error-and-feasibility analysis (required sensitivity, bandwidth, and accessible temperature and field ranges), suffices.

These four deliverables jointly support one objective: the complete characterization of how the vortex glass ages after a quench. Deliverables 1 and 2 extend the open questions explicitly left unresolved by the source study — the asymptotic scaling beyond the crossover time t∗t^* and the strict universality of the effective temperature; deliverable 3 addresses the high-density interaction-dominated (caging) regime that the source flags as relevant beyond the simulated vortex density; and deliverable 4 concerns the noise-and-response, vortex-shear, and tracer measurements that the source proposes as welcome experimental tests of the predicted linear FDT violation in real superconductors. A complete characterization requires all four deliverables; no single one resolves the objective on its own. Each deliverable is nonetheless independently assessable, and a result that resolves one of them, or that rigorously establishes a limit or component of it (a single-line limit, a particular parameter set, or a protocol feasibility study), resolves that part of the objective while leaving the remaining deliverables open.

References

  1. S. Bustingorry, L. F. Cugliandolo, D. Domínguez, Langevin simulations of the out-of-equilibrium dynamics of vortex glasses in high-temperature superconductors, Phys. Rev. B 75, 024506 (2007), DOI 10.1103/PhysRevB.75.024506, arXiv:cond-mat/0608665, https://arxiv.org/abs/cond-mat/0608665
  2. S. Bustingorry, L. F. Cugliandolo, D. Domínguez, Out-of-equilibrium dynamics of the vortex glass in superconductors, Phys. Rev. Lett. 96, 027001 (2006), DOI 10.1103/PhysRevLett.96.027001, arXiv:cond-mat/0505736, https://arxiv.org/abs/cond-mat/0505736
  3. J. L. Iguain, S. Bustingorry, A. B. Kolton, L. F. Cugliandolo, Growing correlations and aging of an elastic line in a random potential, Phys. Rev. B 80, 094201 (2009), DOI 10.1103/PhysRevB.80.094201, arXiv:0903.4878, https://arxiv.org/abs/0903.4878
  4. S. Bustingorry, L. F. Cugliandolo, J. L. Iguain, Out-of-equilibrium relaxation of the Edwards-Wilkinson elastic line, J. Stat. Mech. (2007) P09008, DOI 10.1088/1742-5468/2007/09/P09008, arXiv:0705.3348, https://arxiv.org/abs/0705.3348
  5. S. Bustingorry, Aging dynamics of non-linear elastic interfaces: the Kardar-Parisi-Zhang equation, J. Stat. Mech. (2007) P10002, DOI 10.1088/1742-5468/2007/10/P10002, arXiv:0708.2615, https://arxiv.org/abs/0708.2615
  6. M. Pleimling, U. C. Täuber, Relaxation and glassy dynamics in disordered type-II superconductors, Phys. Rev. B 84, 174509 (2011), DOI 10.1103/PhysRevB.84.174509, arXiv:1106.1130, https://arxiv.org/abs/1106.1130
  7. U. Dobramysl, H. Assi, M. Pleimling, U. C. Täuber, Relaxation dynamics in type-II superconductors with point-like and correlated disorder, Eur. Phys. J. B 86, 228 (2013), DOI 10.1140/epjb/e2013-31101-x, arXiv:1211.6929, https://arxiv.org/abs/1211.6929
  8. M. Pleimling, U. C. Täuber, Characterization of relaxation processes in interacting vortex matter through a time-dependent correlation length, J. Stat. Mech. (2015) P09010, DOI 10.1088/1742-5468/2015/09/P09010, arXiv:1507.07283, https://arxiv.org/abs/1507.07283
  9. H. Assi, H. Chaturvedi, U. Dobramysl, M. Pleimling, Disordered vortex matter out of equilibrium: a Langevin molecular dynamics study, Mol. Simul. 42, 1401 (2016), DOI 10.1080/08927022.2015.1119826, arXiv:1509.02227, https://arxiv.org/abs/1509.02227
  10. H. Chaturvedi, H. Assi, U. Dobramysl, M. Pleimling, U. C. Täuber, Flux line relaxation kinetics following current quenches in disordered type-II superconductors, J. Stat. Mech. (2016) 083301, DOI 10.1088/1742-5468/2016/08/083301, arXiv:1606.06100, https://arxiv.org/abs/1606.06100
  11. X. Du, G. Li, E. Y. Andrei, M. Greenblatt, P. Shuk, Ageing memory and glassiness of a driven vortex system, Nature Physics 3, 111 (2007), DOI 10.1038/nphys512, arXiv:0705.0779, https://arxiv.org/abs/0705.0779
  12. M. Nicodemi, H. J. Jensen, Ageing and memory phenomena in magnetic and transport properties of vortex matter, J. Phys. A 34, 8425 (2001), DOI 10.1088/0305-4470/34/41/303, arXiv:cond-mat/0103070, https://arxiv.org/abs/cond-mat/0103070
  13. A. B. Kolton, R. Exartier, L. F. Cugliandolo, D. Domínguez, N. Grønbech-Jensen, Effective temperature in driven vortex lattices with random pinning, Phys. Rev. Lett. 89, 227001 (2002), DOI 10.1103/PhysRevLett.89.227001, arXiv:cond-mat/0206042, https://arxiv.org/abs/cond-mat/0206042
  14. S. Eley, A. Glatz, R. Willa, Perspective: Challenges and transformative opportunities in superconductor vortex physics, J. Appl. Phys. 130, 050901 (2021), DOI 10.1063/5.0055611, arXiv:2105.00055, https://arxiv.org/abs/2105.00055
  15. E. R. Louden, C. Rastovski, S. J. Kuhn, A. W. D. Leishman, L. DeBeer-Schmitt, C. D. Dewhurst, N. D. Zhigadlo, M. R. Eskildsen, Structural transition kinetics and activated behavior in the superconducting vortex lattice, Phys. Rev. B 99, 060502 (2019), DOI 10.1103/PhysRevB.99.060502, arXiv:1812.05970, https://arxiv.org/abs/1812.05970