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1. Numerical signatures of ultra-local criticality in a one dimensional Kondo lattice model

   https://doi.org/10.21468/SciPostPhys.17.2.034  

   Nikolaenko, Alexander; Zhang, Ya-Hui (January 2024, SciPost Physics)

   Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo couplingJ_K $J_K$at fixed doping x. At large positiveJ_K $J_K$, we confirm the expected conventional Luttinger liquid phase with2k_F=\frac{1+x}{2} $2k_F=\frac{1+x}{2}$(in units of2\pi $2\pi$), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In theJ_K ≤ 0 $J_K\le 0$side, our simulation finds the existence of a fractional Luttinger liquid (LL\star $\star$) phase with2k_F=\frac{x}{2} $2k_F=\frac{x}{2}$, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL\star $\star$) phase in higher dimensions. The LL\star $\star$phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positiveJ_K $J_K$. Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) ofJ_K $J_K$, the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around0.035 J $0.035J$) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponentz=+ $z=+$. The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference. 

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2. On Pisier Type Theorems

   https://doi.org/10.1007/s00493-024-00115-1  

   Nešetřil, Jaroslav; Rödl, Vojtěch; Sales, Marcelo (July 2024, Combinatorica)

   Abstract For any integer$$h\geqslant 2$$ $h⩾2$, a set of integers$$B=\{b_i\}_{i\in I}$$ $B={\left\{b_i\right\}}_{i\in I}$is a$$B_h$$ $B_h$-set if allh-sums$$b_{i_1}+\ldots +b_{i_h}$$ $b_{i_1}+\dots +b_{i_h}$with$$i_1<\ldots $i_1<\dots <i_h$are distinct. Answering a question of Alon and Erdős [2], for every$$h\geqslant 2$$ $h⩾2$we construct a set of integersXwhich is not a union of finitely many$$B_h$$ $B_h$-sets, yet any finite subset$$Y\subseteq X$$ $Y\subseteq X$contains an$$B_h$$ $B_h$-setZwith$$|Z|\geqslant \varepsilon |Y|$$ $\left|Z\right|⩾\epsilon \left|Y\right|$, where$$\varepsilon :=\varepsilon (h)$$ $\epsilon :=\epsilon \left(h\right)$. We also discuss questions related to a problem of Pisier about the existence of a setAwith similar properties when replacing$$B_h$$ $B_h$-sets by the requirement that all finite sums$$\sum _{j\in J}b_j$$ ${\sum }_{j\in J}{b}_j$are distinct. 

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3. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (December 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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4. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (June 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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5. Low-energy magnetic states of Tb adatom on graphene

   https://doi.org/10.1088/1361-648X/ad8fe9  

   Shaikh, Monirul; Klein, Alison; Wysocki, Aleksander L (November 2024, Journal of Physics: Condensed Matter)

   Abstract Electronic structure and magnetic interactions of a Tb adatom on graphene are investigated from first principles using combination of density functional theory and multiconfigurational quantum chemistry techniques including spin–orbit coupling (SOC) . We determine that the six-fold symmetry hollow site is the preferred adsorption site and investigate electronic spectrum for different adatom oxidation states including Tb³⁺, Tb²⁺, Tb¹⁺, and Tb⁰. For all charge states, the Tb $4f^8$configuration is retained with other adatom valence electrons being distributed over $5d_{xy}$, $5d_{x2+y2}$, and $6s/5d_0$single-electron orbitals. We find strong intra-site adatom exchange coupling that ensures that the $5d6s$spins are parallel to the4fspin. For Tb³⁺, the energy levels can be described by theJ = 6 multiplet split by the graphene crystal field (CF). For other oxidation states, the interaction of4felectrons with spin and orbital degrees of freedom of $6s5d$electrons in the presence of SOC results in the low-energy spectrum composed closely lying effective multiplets that are split by the graphene CF. Stable magnetic moment is predicted for Tb³⁺and Tb²⁺adatoms due to uniaxial magnetic anisotropy and effective anisotropy barrier around 440 cm⁻¹controlled by the temperature assisted quantum tunneling of magnetization through the third excited doublet. On the other hand, in-plane magnetic anisotropy is found for Tb¹⁺and Tb⁰adatoms. Our results indicate that the occupation of the $6s5d$orbitals can dramatically affect the magnetic anisotropy and magnetic moment stability of rare earth adatoms. 

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