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1. Universal features of higher-form symmetries at phase transitions

   https://doi.org/10.21468/SciPostPhys.11.2.033  

   Wu, Xiao-Chuan; Jian, Chao-Ming; Xu, Cenke (January 2021, SciPost Physics)

   We investigate the behavior of higher-form symmetries at variousquantum phase transitions. We consider discrete 1-form symmetries, whichcan be either part of the generalized concept “categorical symmetry”(labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) )introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. We demonstrate that for many quantum phase transitionsinvolving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O 𝒞 ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P ⟨ ( log O 𝒞 ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 𝒞 is an operator defined associated with loop \mathcal{C} 𝒞 (or its interior \mathcal{A} 𝒜 ),which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. P P is the perimeter of \mathcal{C} 𝒞 ,and the b \log P b log P term arises from the sharp corners of the loop \mathcal{C} 𝒞 ,which is consistent with recent numerics on a particular example. b b is a universal microscopic-independent number, which in (2+1)d ( 2 + 1 ) d is related to the universal conductivity at the quantum phasetransition. b b can be computed exactly for certain transitions using the dualitiesbetween (2+1)d ( 2 + 1 ) d conformal field theories developed in recent years. We also compute the"strange correlator" of O_\mathcal{C} O 𝒞 : S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle S 𝒞 = ⟨ 0 | O 𝒞 | 1 ⟩ / ⟨ 0 | 1 ⟩ where |0\rangle | 0 ⟩ and |1\rangle | 1 ⟩ are many-body states with different topological nature. 

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2. Chiral π-Conjugated Double Helical Aminyl Diradical with the Triplet Ground State

   https://doi.org/10.1055/a-2379-9406  

   Guo, Haoxin; Rajca, Andrzej (February 2025, Synlett)

   Abstract We describe effective development of the highly diastereoselective synthesis of double helical tetraamine 2-H2-C2 and propose a mechanism for its formation. The resolution of 2-H2-C2 is facilitated by a high racemization barrier of 43 kcal mol–1 and it is implemented via either a chiral auxiliary or preparative supercritical fluid chromatography. This enables preparation of the first high-spin neutral diradical, with spin density delocalized within an enantiomeric double helical π-system. The presence of two effective 3-electron C–N bonds in the diradical leads to: (1) the triplet (S = 1) high-spin ground state with a singlet-triplet energy gap of 0.4 kcal mol–1 and (2) the long half-life of up to 6 days in 2-MeTHF at room temperature. The diradical possesses a racemization barrier of at least 26 kcal mol–1 in 2-MeTHF at 293 K and chiroptical properties, with an absorption anisotropy factor |g| ≈ 0.005 at 548 nm. These unique magnetic and optical properties of our diradical form the basis for the development of next-generation spintronic devices. 1 Introduction 2 Synthesis and Resolution of the C 2-Symmetric Double Helical Tetraamine 2-H2-C 2 3 Synthesis and Characterization of Neutral High-Spin Aminyl Diradical 22• -C 2 4 Conclusion 

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3. Tight Bounds for ℓ ₁ Oblivious Subspace Embeddings

   https://doi.org/10.1145/3477537  

   Wang, Ruosong; Woodruff, David P. (January 2022, ACM Transactions on Algorithms)

   An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications. 

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4. Stretched exponential decay for subcritical parking times on

   https://doi.org/10.1002/rsa.21001  

   Damron, Michael; Lyu, Hanbaek; Sivakoff, David (February 2021, Random Structures & Algorithms)

   null (Ed.)

   In the parking model on ℤd, each vertex is initially occupied by a car (with probability p) or by a vacant parking spot (with probability 1−p). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that p=1/2 is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when p<1/2, and by infinitely many distinct cars when p≥1/2. Furthermore, any given car a.s. eventually parks for p≤1/2 and with positive probability does not park for p>1/2. We study the subcritical phase and prove that the tail of the parking time τ of the car initially at the origin obeys the bounds exp(−C1tdd+2)≤ℙp(τ>t)≤exp(−c2tdd+2) for p>0 sufficiently small. For d=1, we prove these inequalities for all p∈[0,1/2). This result presents an asymmetry with the supercritical phase (p>1/2), where methods of Bramson--Lebowitz imply that for d=1 the corresponding tail of the parking time of the parking spot of the origin decays like e−ct√. Our exponent d/(d+2) also differs from those previously obtained in the case of moving obstacles. 

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5. Search for highly-ionizing particles in pp collisions at the LHC’s Run-1 using the prototype MoEDAL detector

   https://doi.org/10.1140/epjc/s10052-022-10608-2  

   Acharya, B.; Alexandre, J.; Benes, P.; Bergmann, B.; Bertolucci, S.; Bevan, A.; Bhattacharyya, R.; Branzas, H.; Burian, P.; Campbell, M.; et al (August 2022, The European Physical Journal C)

   Abstract A search for highly electrically charged objects (HECOs) and magnetic monopoles is presented using 2.2 $$\hbox {fb}{^{-1}}$$ fb - 1 of $$p-p$$ p - p collision data taken at a centre of mass energy (E $$_{CM}$$ CM ) of 8 TeV by the MoEDAL detector during LHC’s Run-1. The data were collected using MoEDAL’s prototype Nuclear Track Detectord array and the Trapping Detector array. The results are interpreted in terms of Drell–Yan pair production of stable HECO and monopole pairs with three spin hypotheses (0, 1/2 and 1). The search provides constraints on the direct production of magnetic monopoles carrying one to four Dirac magnetic charges and with mass limits ranging from 590 GeV/c $$^{2}$$ 2 to 1 TeV/c $$^{2}$$ 2 . Additionally, mass limits are placed on HECOs with charge in the range 10 e to 180 e , where e is the charge of an electron, for masses between 30 GeV/c $$^{2}$$ 2 and 1 TeV/c $$^{2}$$ 2 . 

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