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1. Boundary algebras of the Kitaev quantum double model

   https://doi.org/10.1063/5.0212164  

   Chuah, Chian Yeong; Hungar, Brett; Kawagoe, Kyle; Penneys, David; Tomba, Mario; Wallick, Daniel; Wei, Shuqi (October 2024, Journal of Mathematical Physics)

   The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group. 

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2. Small circle expansion for adjoint QCD2 with periodic boundary conditions

   https://doi.org/10.1007/JHEP11(2024)128  

   Dempsey, Ross; Klebanov, Igor R; Pufu, Silviu S; Søgaard, Benjamin T (November 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We study 1 + 1-dimensional SU(N) gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumferenceL. Using periodic boundary conditions, we derive the effective action for the quantum mechanics of the holonomy and the fermion zero modes in perturbation theory up to order (gL)³. When the adjoint fermion mass-squared is tuned tog²N/(2π), the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into theℤ_Ncenter symmetry sectors (universes) labeled byp= 0, . . . ,N– 1 and show that in one of the sectors the supersymmetry is unbroken, while in the others it is broken spontaneously. These results give us new insights into the (1, 1) supersymmetry of adjoint QCD₂, which has previously been established using light-cone quantization. When the adjoint mass is set to zero, our effective Hamiltonian does not depend on the fermions at all, so that there are 2^N−1degenerate sectors of the Hilbert space. This construction appears to provide an explicit realization of the extended symmetry of the massless model, where there are 2^2N−2operators that commute with the Hamiltonian. We also generalize our results to other gauge groupsG, for which supersymmetry is found at the adjoint mass-squaredg²h^∨/(2π), whereh^∨is the dual Coxeter number ofG. 

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3. From the quantum breakdown model to the lattice gauge theory

   https://doi.org/10.1007/s43673-024-00128-4  

   Hu, Yu-Min; Lian, Biao (August 2024, AAPPS Bulletin)

   Abstract The one-dimensional quantum breakdown model, which features spatially asymmetric fermionic interactions simulating the electrical breakdown phenomenon, exhibits an exponential U(1) symmetry and a variety of dynamical phases including many-body localization and quantum chaos with quantum scar states. We investigate the minimal quantum breakdown model with the minimal number of on-site fermion orbitals required for the interaction and identify a large number of local conserved charges in the model. We then reveal a mapping between the minimal quantum breakdown model in certain charge sectors and a quantum link model which simulates theU(1) lattice gauge theory and show that the local conserved charges map to the gauge symmetry generators. A special charge sector of the model further maps to the PXP model, which shows quantum many-body scars. This mapping unveils the rich dynamics in different Krylov subspaces characterized by different gauge configurations in the quantum breakdown model. 

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4. Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions

   https://doi.org/10.1103/PRXQuantum.5.030328  

   Liang, Zijian; Xu, Yijia; Iosue, Joseph T; Chen, Yu-An (August 2024, PRX Quantum)

   In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to ${\mathbb{Z}}_d$qudits, including instances where $d$is a nonprime number. This capability allows the identification of topological orders that differ from the ${\mathbb{Z}}_d$toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for ${\mathbb{Z}}_p$qudits (with $p$being a prime) are equivalent to finite copies of ${\mathbb{Z}}_p$toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society2024 

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5. The growth rate of multicolor Ramsey numbers of 3-graphs

   https://doi.org/10.1007/s40687-024-00463-w  

   Bradač, Domagoj; Fox, Jacob; Sudakov, Benny (September 2024, Research in the Mathematical Sciences)

   Abstract Theq-color Ramsey number of ak-uniform hypergraphG,  denotedr(G; q), is the minimum integerNsuch that any coloring of the edges of the completek-uniform hypergraph onNvertices contains a monochromatic copy ofG. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior ofr(G; q) for fixedGandqtending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height ofr(G; q) as a function ofq. More precisely, given a hypergraphG, we determine whenr(G; q) behaves polynomially, exponentially or double exponentially inq. This answers a question of Axenovich, Gyárfás, Liu and Mubayi. 

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