Hopfions in Condensed Matter and Field Theory

Abstract: Nontrivial topological defects such as knotted solitons called hopfions have been observed in a variety of materials including chiral magnets, nematic liquid crystals and even in ferroelectrics as well as studied in other physical contexts such as Bose-Einstein condensates. These topological entities can be modeled using the relevant physical variable, e.g., magnetization, polarization or the director field. Specifically, we find exact static soliton solutions for the unit spin vector field of an inhomogeneous, anisotropic three-dimensional (3D) Heisenberg ferromagnet and calculate the corresponding Hopf invariant H analytically and obtain an integer, demonstrating that these solitons are indeed hopfions [1]. H is a product of two integers, the first being the usual winding number of a skyrmion in two dimensions, while the second encodes the periodicity in the third dimension. We also study the underlying geometry of H, by mapping the 3D unit vector field to tangent vectors of three appropriately defined space curves. Our analysis shows that a certain intrinsic twist is necessary to yield a nontrivial topological invariant: linking number [2]. Finally, we focus on the formation energy of hopfions to study their properties for potential applications.

Short bio: Avadh Saxena is former Group Leader of the Condensed Matter and Complex Systems group (T-4) at Los Alamos National Lab, New Mexico, USA where he has been since 1990. He is also an affiliate of the Center for Nonlinear Studies at Los Alamos. His main research interests include phase transitions, optical, electronic, vibrational, transport and magnetic properties of functional materials, device physics, soft condensed matter, non-Hermitian quantum mechanics, geometry, topology and nonlinear phenomena & materials harboring topological defects such as solitons, polarons, excitons, breathers, skyrmions and hopfions. He recently completed a book on “Phase Transitions from a Materials Perspective” (Cambridge University Press, 2024). He is an Affiliate Professor at the Royal Institute of Technology (KTH), Stockholm, Sweden and holds adjunct professor positions at the University of Barcelona, Spain, University of Crete, Greece, Virginia Tech and the University of Arizona, Tucson. He is Scientific Advisor to National Institute for Materials Science (NIMS), Tsukuba, Japan. He is a Fellow of Los Alamos National Lab, a Fellow of the American Physical Society (APS), a Fellow of the Japan Society for the Promotion of Science (JSPS) and a member of the Sigma Xi Scientific Research Society, APS and American Ceramic Society (ACerS).