Theory of Magnetism and Magnetic Materials
Motivation. The micromagnetic structure of magnetic (ferromagnetic, antiferromagnetic, etc) materials shows patterns that appear and are observed at the meso- to nano-scale. These are studied in their fundamental physical and mathematical aspects. The academic studies are combined with the technological interest from a large industry on magnetic materials that includes magnetic storage and Magnetoresistive-RAM applications.
In the 60s and 70s magnetic bubbles were observed in ferromagnetic films with perpendicular anisotropy. These appeared as microscale cylindrical patches of magnetisation reversed with respect to their surroundings and their study led to the technology of bubble memory. They were the first examples of what are called "topological magnetic solitons" in ferromagnetic films. Magnetic bubbles could also be propagated in the film, and their dynamical (propagation) properties were recognised to be peculiar.
Magnetic bubbles in ferromagnetic films. Our earlier work [Physica D, 1996] was motivated by the extensive available data on the statics and dynamics of magnetic bubbles and it extended previous work. We studied the statics and dynamics of magnetic bubbles in a thin film. At the time, the subject was mostly of theoretical interest as bubble technology was not pursued any more. In a more general setting, however, the dynamical phenomena that we study are versatile in the area, underlying any transformation of the micromagnetic configuration, specifically, the recording and transmission of magnetic information. more...
Magnetic patterns such as bubbles are called topological solitons because they carry a topological number called the skyrmion number. This plays a crucial role in their existence and stability. Bubble motion is counter-intuitive and completely contradicts what a naive application of Newtonian mechanics would imply. The surprising dynamics was linked to the bubble’s topological number.
The framework of our studies is set by the Landau-Lifshitz equation and they are both field-theoretical and numerical. We elaborate on a direct link between topology and dynamics for solitons that has been rigorously established. This was based on the construction of anumbiguous conservation laws for the linear and angular momentum of topological solitons, such as magnetic bubbles. The conservation laws are given as suitable moments of the topological density thus providing the link between topology and dynamics. We calculated numerically magnetic bubbles and studied their properties for the case of a ferromagnetic film, that is, in a three-dimensional model where the finite film thickness is considered. This requires to take into account the long-range magnetostatic field. We extended the theory for the dynamics of topological magnetic solitons within this setting. [Physica D, 1996].
Magnetic bubbles in particles. more... The fabrication of small magnetic elements or particles (since the 1990s) with sub-micrometer size raised the question of the existence of stable magnetic patterns in these meso-scale elements. We have theoretically predicted the existence of a single magnetic bubble in meso-scale ferromagnetic elements [PRB, 2005], [PRB, 2006]. This was followed by experimental observation in the technologically important FePt alloy [PRB, 2007]. These ideas have now been significantly expanded by the recently established experimental group of Dr Christoforos Moutafis (University of Manchester) in several publications including [Nature Physics 11, 225228 (2015)].
Antiferromagnets. In [Nonlinearity, 1998] we set a framework for the study of antiferromagnets in a continuum approximation. Antiferromagnets are materials central in phenomena such as high temperature superconductivity. However, they typically attract far less attention than ferromagnets due to the lack of a net magnetisation and the ensuing difficulty for experimental observations. more...
We have systematically derived a contimuum model for a planar (2D) antiferromagnet, starting from the discrete (Heisenberg) model. This is a type of σ-model. Its static sector is similar to that for ferromagnetets. But, when the model is then applied in order to study vortex dynamics in the antiferromagnetic continuum, they present fundamental differences from their ferromagnetic counterparts. When we apply an external magnetic field, however, then the dynamics of antiferromagnetic solitons changes dramatically and it acquirs a link to a topological number. Interestingly, this is not the same topological number as in ferromagnets.
This field is potentially widely open and promising, especially, given the recent (2011) development of techniques for experimental observation of antiferromagnetic structures. However, the progress is not yet at a satisfactory level.
Magnetic vortices. Magnetic vortices had been theoretically predicted early on but no experimental realisation was achieved in ferromagnetic films, mainly because their energy diverges logarithmically with the sample size. The situation was suddenly reversed in the last two decades following the fabrication of mesoscopic ferromagnetic elements (< 1 μm). Since vortices were observed (in 2000) as stable and robust structures (indeed, as ground states) in mesoscopic magnetic elements, renewed interest was sparked and a very extensive experimental activity in academic and industrial laboratories has started.Vortex-antivortex pairs. more... We have described theoretically the dynamics of vortex-antivortex dipoles, that is, a vortex together with an antivortex with opposite polarities. By exploiting a link of their topology to dynamics, within a hamiltonian formalism, we find that magnetic vortex-antivortex dipoles undergo rotation, in stark contrast to vortex-antivortex pairs in fluid dynamics. [Phys. Rev. Lett. 2007].
The theoretical description of vortex-dipole dynamics was extended to include external probes, specifically, the effect of spin-polarized electrical current in the magnetic material. The latter is the standard method of probing magnetisation dynamics in current experimental work. We predict stable magnetization oscillations due to a totally surprising cooperation of (conservative) hamiltonian dynamics and of (non-conservative) spin-transfer torque due to the spin-polarized current or the spin-Hall effect [Europhys. Lett. 2012]. This system turned out to give interesting theoretical and also mathematical results, e.g., the asymptotic analysis of a rotating vortex pair [Physica D, 2015].
- We have reviewed magnetic vortex-antivortex pairs, a topic where we have made some of the main contributions.
- The vortex dipole system gives magnetisation oscillations and it thus is a microwave frequency generator at the nanoscale. Observations of these oscillations had motivated our work and were reported in [PRB 78, 174408 (2008)].
- We propose vortex- antivortex pairs as advantageous structures which can, at the same time, be used as stable units of stored information (due to their topological structure) and also as carriers of information which may perform logic operations (due to their dynamical behaviour) [Sci. Rep. 2015].
PresentationDynamics of point vortices. more... The equations of motion for point vortices in fluids were given by Helmholtz (1858) and by Kirchoff (1876) as a Hamiltonian system. We studied three point magnetic vortices using a method introduced by Groebli in his seminal papers in 1877 [Vierteljahrsschrift 22/1, [Vierteljahrsschrift 22/2]. We integrated completely the Hamiltonian system. We demonstrated that some of the solutions describe the transmutation of linear momentum into vortex position. This counter-intuitive physical process is demonstrated based on mathematically rigorous conservation laws [J. Math. Phys. 2010].
- "Groebli solution for magnetic vortices", NLQUGAS10, 13/4/2010, Ourense, Spain.
Skyrmions. Motivated by the recent identification and fabrication of Dzyaloshinskii-Moriya (DM) materials and the subsequent experimental observation of solitons (so-called “chiral skyrmions”) we have applied our theoretical formalism to this realistic system.
One could argue that DM materials are are, currently, the most interesting system because they support both, robust topological solitons (“chiral skyrmions”) and also topologically trivial (loosely called “nontopological”) skyrmions. Especially the dynamics of nontopological solitons was only partly described by the existing formalism - it was essentially left an open question. more...
We have taken this opportunity to revisit skyrmion dynamics. We find that the dynamics of topological and nontopological solitons differ in a profound way [Phys. Rev. B, 2015]. Newtonian dynamics is found for nontopological solitons, and we could rigorously attribute a mass to them. That seems to elevate magnetic solitons to particle-like objects. To complete the story, the dynamics of nontopological solitons turns relativistic for larger velocities. It is impressive that this complete set of results indicating the particle-like character of solitons, emerge from a fundamental field theory. Topological skyrmions, as was anticipated, undergo Hall motion under external forces. That is, non-Newtonian kinetics where the skyrmion moves in a direction perpendicular to the applied force.
We extended the theory to the case of a topological and a nontopological skyrmion under spin-transfer torque [Phys. Rev. B, 2015]. The Landau-Lifshitz equation (including damping and Spin-torque) can be brought to a form with known solutions in some cases. This gives a significant insight for the typically complicated behaviour of the skyrmions under external forces.
- Vortices and solitons in condensed matter, Materials Science department, University of Crete, 9/11/2015.
- Domain structure and dynamics in magnetic elements, Heraklion, Crete, April 8-11, 2013.
- Topological Patterns and Dynamics in Magnetic Elements and in Condensed Matter (Workshop and Seminar), Max-Planck Institute for the Physics of Complex Systems, Dresden, 27 June - 8 July 2016.
Nonlinear phenomena in atomic Bose-Einstein condensates
Motivation. Trapped atomic Bose-Einstein condensates is a system that is now widely used as a standard model of condensed matter. It is described, in a mean-field approximation, by a nonlinear Schrödinger equation. We typically include a trapping potential in the model which accounts for the trapping of the atomic cloud. The obtained equation is the Gross-Pitaevskii model.
The coherence inherent in condensates is what makes them a great model system for mathematical studies. We are interested in solitary waves in a confined geometry (quantized vortices, vortex rings and solitons) and we study nonlinear dynamics in this physical system.
Superfluid vortex rings. We conjectured the existence of traveling vortex rings in a trapped BEC and calculated them numerically [Phys. Rev. B, 2002]. Observation of static vortex rings was first reported in [Anderson et al, PRL 2001] and many experiments followed. more...
We found vortex ring modes (that is, energy and momentum of these nonlinear waves as a function of their velocity) and gave their connection to other nonlinear modes in the same system. Specifically, a soliton develops to become a vortex ring as the system turns from one-dimensional to three-dimensional. Hybrids of solitons and vortex rings exist in an intermediate region. We have shown the relation between solitons, vortex rings and also quantised vortices, thus giving the big picture for solitary waves in trapped BECs (within the related Gross-Pitaevskii model).
Solitons in Gross-Pitaevskii models have a non-standard linear momentum, as first noted by [Tsuzuki, 1971]. Vortex rings inherit this feature while they also present an unusual energy-momentum dispersion relation. We drew attention to the fact that static vortex rings have a nonzero linear momentum. As a result, the vortex ring modes present roton-like features. The overall features of vortex rings are unique. This is reflected in the fact that, despite their being quite complex mathematically, they have been occurring in experiments and studied continuously since their first observation in 2001. They occur in a variety of experimental contexts while efforts have been made to make connections to other known modes in quantum field theories.
Our work on vortex rings in trapped superfluids has been the basis for the understanding of vortex rings in currently available atomic BECs, e.g., [Ginsberg et al, PRL 2005]
Quantized vortices. One of the most remarkable characteristics of Bose-Einstein condensates is their response to rotation. Rotation in these superfluids occurs via the creation of quantized vortices. A few vortices or a lattice of vortices may be created, while even their fine features can be observed and studied.
We have extensively studied quantized vortices as isolated objects or forming lattices. These may be rotating in the superfluid, or they could have solitonic features.
Single vortices in trapped BEC. more... Motivated by the experiments reported in [PRL, 2002] we studied theoretically the single vortex excitations in spherical and elongated condensates as a function of the interaction strength. We solved numerically the Gross-Pitaevskii equation and we found two types of vortices, called S- and U-vortices. These have two different precession frequencies and angular momenta. The S vortex is related to the solitonic vortex, which is a nonlinear excitation in the nonrotating system and to the dark soliton. The U-vortex is a generalisation of the standard notion of a vortex filament. [PRA, 2005]. We further derived a class of virial relations for single vortex states in a confined Bose-Einstein condensate. [PRA, 2005].Vortex lattices. more... Fast rotation of a BEC results in the formation of a vortex lattice. The system is then in a new phase and it is interesting that it can undergo a type of phase transitions between vortex lattices with different symmetries.
One typically expects to find a standard Abrikosov lattice (triangular vortex lattice). We verified that in [PRA, 2004] but noted the surprising fact that the envelope for the condensate density follows a strong interactions profile (Thomas-Fermi profile) even at low condensate densities (i.e., low interactions).
For condensates where long-range dipolar interactions are present vortex lattices can appear in unusual forms. We find lattices of various symmetries: triangular and square vortex lattices, “stripe crystal” and “bubble crystal”. [PRA, 2007]
Non-abelian models. Our recent studies pertain to artificial non-Abelian potentials. These can be generated in BECs owing to the impressive control one has in ultra-cold atomic systems. Non-Abelian potentials transform BECs to laboratories of exotic many-body phenomena.
We studied vortex lattices generated by large effective non-Abelian magnetic fields and showed that the nature of the ground state changes dramatically, with structural changes of the vortex lattice.Presentation
- "Single vortices and vortex pairs in Bose-Einstein Condensates", Max-Planck Institute for the Physics of Complex Systems, Dresden, 2006.
Nonlinear phenomena in polariton condensates
Motivation. The study of Bose-Einstein condensates was more recently extended to non-atomic systems. For example, exciton-polariton condensates can be produced in semiconductors. Exciton-polaritons are matter-light quasiparticles that arise from the coupling between excitons and photon modes in a semiconductor microcavity and can form Bose-Einstein condensates (BEC) at relatively high temperatures.
Solitons: continuous and discontinuous. The system is described (in a mean-field approximation) by a system of two Schrödinger-type equations including a nonlinear term. We are interested in physical phenomena and mathematical results lying outside the parameter regime of validity of a single Gross-Pitaevskii model.
Polariton condensates emerge as a fertile ground for solitonic structures. Our work provides an understanding of these structures. We have analytically identified soliton solutions for the lossless system. One type of black soliton studied is of the standard type where the fields vanish at the soliton center. Beyond that, we identify a discontinuous soliton where the exciton field exhibits a jump at the soliton center, and therefore the exciton density does not vanish at any point. [PRB, 2015], [Physica D, 2016].Presentation
- Continuous and discontinuous solitons in polariton condensates, POLATOM 22/6/2015, Bad Honnef, Germany.
Localization and diffusion of energy in disordered nonlinear systems
While a linear disordered system shows energy localisation (Anderson localisation), it has been numerically observed that, an extra nonlinear term in the equation causes a wavepacket to spread apparently indefinitely. We study nonlinear chains with disorder and seek to understand the nature of the solutions of the nonlinear system, e.g., [PRL, 2008]
Non-topological solitons in Cosmology
Non-topological solitons (Q-balls and Q-rings) have been proposed to help in the explanation of dark matter in a cosmological setting. We have been interested in the dynamics of Q-balls. Simulations of Q-ball scattering showed that the right angle scattering effect observed for topological solitons in two dimensions persists also in the case of Q-balls. [PRD, 2000]