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1. Codimension-2 defects and higher symmetries in (3+1)D topological phases

   https://doi.org/10.21468/SciPostPhys.14.4.065  

   Barkeshli, Maissam; Chen, Yu-An; Huang, Sheng-Jie; Kobayashi, Ryohei; Tantivasadakarn, Nathanan; Zhu, Guanyu (January 2023, SciPost Physics)

   (3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory. 

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2. Fractionalization of coset non-invertible symmetry and exotic Hall conductance

   https://doi.org/10.21468/SciPostPhys.17.3.095  

   Hsin, Po-Shen; Kobayashi, Ryohei; Zhang, Carolyn (January 2024, SciPost Physics)

   We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 $S_3$andO(2) $O\left(2\right)$gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified. 

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3. Gauging U(1) symmetry in (2+1)d topological phases

   https://doi.org/10.21468/SciPostPhys.12.6.202  

   Cheng, Meng; Jian, Chao-Ming (January 2022, SciPost Physics)

   We study the gauging of a global U(1) symmetry in a gapped system in(2+1)d. The gauging procedure has been well-understood for a finiteglobal symmetry group, which leads to a new gapped phase with emergentgauge structure and can be described algebraically using themathematical framework of modular tensor category (MTC). We develop acategorical description of U(1) gauging in a MTC, taking into accountthe dynamics of U(1) gauge field absent in the finite group case. Whenthe ungauged system has a non-zero Hall conductance, the gauged theoryremains gapped and we determine the complete set of anyon data for thegauged theory. On the other hand, when the Hall conductance vanishes, weargue that gauging has the same effect of condensing a special Abeliananyon nucleated by inserting 2\pi 2 π U(1) flux. We apply our procedure to theSU(2) _k k MTCs and derive the full MTC data for the \mathbb{Z}_k ℤ k parafermion MTCs. We also discuss a dual U(1) symmetry that emergesafter the original U(1) symmetry of an MTC is gauged. 

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4. Individual differences and long-term consequences of tDCS augmented cognitive training

   Katz, B.; Au, J.; Buschkuehl, M.; Abagis, T.; Zabel, C.; Jaeggi, S.; Jonides, J. (October 2017, Journal of cognitive neuroscience)

   A gr e at d e al of i nt er e st s urr o u n d s t h e u s e of tr a n s cr a ni al dir e ct c urr e nt sti m ul ati o n (t D C S) t o a u g m e nt c o g niti v e tr ai ni n g. H o w e v er, eff e ct s ar e i n c o n si st e nt a cr o s s st u di e s, a n d m et aa n al yti c e vi d e n c e i s mi x e d, e s p e ci all y f o r h e alt h y, y o u n g a d ult s. O n e m aj or s o ur c e of t hi s i n c o n si st e n c y i s i n di vi d u al diff er e n c e s a m o n g t h e p arti ci p a nt s, b ut t h e s e diff er e n c e s ar e r ar el y e x a mi n e d i n t h e c o nt e xt of c o m bi n e d tr ai ni n g/ sti m ul ati o n st u di e s. I n a d diti o n, it i s u n cl e ar h o w l o n g t h e eff e ct s of sti m ul ati o n l a st, e v e n i n s u c c e s sf ul i nt er v e nti o n s. S o m e st u di e s m a k e u s e of f oll o w- u p a s s e s s m e nt s, b ut v er y f e w h a v e m e a s ur e d p erf or m a n c e m or e t h a n a f e w m o nt hs aft er a n i nt er v e nti o n. H er e, w e utili z e d d at a fr o m a pr e vi o u s st u d y of t D C S a n d c o g niti v e tr ai ni n g [ A u, J., K at z, B., B u s c h k u e hl, M., B u n arj o, K., S e n g er, T., Z a b el, C., et al. E n h a n ci n g w or ki n g m e m or y tr ai ni n g wit h tr a n scr a ni al dir e ct c urr e nt sti m ul ati o n. J o u r n al of C o g niti v e N e u r os ci e n c e, 2 8, 1 4 1 9 – 1 4 3 2, 2 0 1 6] i n w hi c h p arti ci p a nts tr ai n e d o n a w or ki n g m e m or y t as k o v er 7 d a y s w hil e r e c ei vi n g a cti v e or s h a m t D C S. A n e w, l o n g er-t er m f oll o w- u p t o a ss es s l at er p erf or m a n c e w a s c o n d u ct e d, a n d a d diti o n al p arti ci p a nt s w er e a d d e d s o t h at t h e s h a m c o n diti o n w a s b ett er p o w er e d. W e a s s e s s e d b a s eli n e c o g niti v e a bilit y, g e n d er, tr ai ni n g sit e, a n d m oti v ati o n l e v el a n d f o u n d si g nifi c a nt i nt er a cti o ns b et w e e n b ot h b as eli n e a bilit y a n d m oti v ati o n wit h c o n diti o n ( a cti v e or s h a m) i n m o d els pr e di cti n g tr ai ni n g g ai n. I n a d diti o n, t h e i m pr o v e m e nt s i n t h e a cti v e c o nditi o n v er s u s s h a m c o n diti o n a p p e ar t o b e st a bl e e v e n a s l o n g a s a y e ar aft er t h e ori gi n al i nt er v e nti o n. ■ 

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5. 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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