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1. Controlled creation and decay of singly-quantized vortices in a polar magnetic phase

   https://doi.org/10.1038/s42005-021-00554-y  

   Xiao, Y.; Borgh, M. O.; Weiss, L. S.; Blinova, A. A.; Ruostekoski, J.; Hall, D. S. (December 2021, Communications Physics)

   null (Ed.)

   Abstract Quantized vortices appear in physical systems from superfluids and superconductors to liquid crystals and high energy physics. Unlike their scalar cousins, superfluids with complex internal structure can exhibit rich dynamics of decay and even fractional vorticity. Here, we experimentally and theoretically explore the creation and time evolution of vortex lines in the polar magnetic phase of a trapped spin-1 87 Rb Bose–Einstein condensate. A process of phase-imprinting a nonsingular vortex, its decay into a pair of singular spinor vortices, and a rapid exchange of magnetic phases creates a pair of three-dimensional, singular singly-quantized vortex lines with core regions that are filled with atoms in the ferromagnetic phase. Atomic interactions guide the subsequent vortex dynamics, leading to core structures that suggest the decay of the singly-quantized vortices into half-quantum vortices. 

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2. Engineering and revealing Dirac strings in spinor condensates

   https://doi.org/10.1103/PhysRevResearch.6.023272  

   Xu, Gui-Sheng; Jain, Mudit; Zhou, Xiang-Fa; Guo, Guang-Can; Amin, Mustafa A; Pu, Han; Zhou, Zheng-Wei (June 2024, Physical Review Research)

   Artificial monopoles have been engineered in various systems, yet there has been no systematic study of the singular vector potentials associated with the monopole field. We show that the Dirac string, the line singularity of the vector potential, can be engineered, manipulated, and made manifest in a spinor atomic condensate. We elucidate the connection among spin, orbital degrees of freedom, and the artificial gauge, and show that there exists a mapping between the vortex filament and the Dirac string. We also devise a proposal where preparing initial spin states with relevant symmetries can result in different vortex patterns, revealing an underlying correspondence between the internal spin states and the spherical vortex structures. Such a mapping also leads to a new way of constructing spherical Landau levels, and monopole harmonics. Our observation provides insights into the behavior of quantum matter possessing internal symmetries in curved spaces. Published by the American Physical Society2024 

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3. Topological superfluid defects with discrete point group symmetries

   https://doi.org/10.1038/s41467-022-32362-5  

   Xiao, Y.; Borgh, M. O.; Blinova, A.; Ollikainen, T.; Ruostekoski, J.; Hall, D. S. (August 2022, Nature Communications)

   Abstract Discrete symmetries are spatially ubiquitous but are often hidden in internal states of systems where they can have especially profound consequences. In this work we create and verify exotic magnetic phases of atomic spinor Bose–Einstein condensates that, despite their continuous character and intrinsic spatial isotropy, exhibit complex discrete polytope symmetries in their topological defects. Using carefully tailored spinor rotations and microwave transitions, we engineer singular line defects whose quantization conditions, exchange statistics, and dynamics are fundamentally determined by these underlying symmetries. We show how filling the vortex line singularities with atoms in a variety of different phases leads to core structures that possess magnetic interfaces with rich combinations of discrete and continuous symmetries. Such defects, with their non-commutative properties, could provide unconventional realizations of quantum information and interferometry. 

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4. Contextures: Representations from Contexts

   Zhai, Runtian; Yang, Kai; Varici, Burak; Tsai, Che-Ping; Kolter, Zico; Ravikumar, Pradeep (July 2025, International Conference on Machine Learning (ICML))

   Despite the empirical success of foundation models, we do not have a systematic characterization of the representations that these models learn. In this paper, we establish the contexture theory. It shows that a large class of representation learning methods can be characterized as learning from the association between the input and a context variable. Specifically, we show that many popular methods aim to approximate the top-d singular functions of the expectation operator induced by the context, in which case we say that the representation learns the contexture. We demonstrate the generality of the contexture theory by proving that representation learning within various learning paradigms—supervised, self-supervised, and manifold learning—can all be studied from such a perspective. We also prove that the representations that learn the contexture are optimal on those tasks that are compatible with the context. One important implication of the contexture theory is that once the model is large enough to approximate the top singular functions, further scaling up the model size yields diminishing returns. Therefore, scaling is not all we need, and further improvement requires better contexts. To this end, we study how to evaluate the usefulness of a context without knowing the downstream tasks. We propose a metric and show by experiments that it correlates well with the actual performance of the encoder on many real datasets. 

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5. On Singular Vortex Patches, I: Well-posedness Issues

   https://doi.org/10.1090/memo/1400  

   Elgindi, Tarek; Jeong, In-Jee (March 2023, Memoirs of the American Mathematical Society)

   The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. m\geq 3. In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π 2 \frac {\pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 \frac {\pi }{2} or with corners which are only locally m m -fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on R 2 \mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time. 

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