Phase diagram of Kondo-Heisenberg model on honeycomb lattice with geometrical frustration

We calculated the phase diagram of the Kondo-Heisenberg model on two-dimensional honeycomb lattice with both nearest-neighbor and next-nearest-neighbor antiferromagnetic spin exchanges, to investigate the interplay between RKKY and Kondo interactions at presence of magnetic frustration. Within a mean-field decoupling technology in slave-fermion representation, we derived the zero-temperature phase diagram as a function of Kondo coupling $J_k$ and frustration strength $Q$. The geometrical frustration can destroy the magnetic order, driving the original antiferromagnetic (AF) phase to non-magnetic valence bond state (VBS). In addition, we found two distinct VBS. As $J_k$ is increased, a phase transition from AF to Kondo paramagnetic (KP) phase occurs, without the intermediate phase coexisting AF order with Kondo screening found in square lattice systems. In the KP phase, the enhancement of frustration weakens the Kondo screening effect, resulting in a phase transition from KP to VBS. We also found a process to recover the AF order from VBS by increasing $J_k$ in a wide range of frustration strength. Our work may provide deeper understanding for the phase transitions in heavy-fermion materials, particularly for those exhibiting triangular frustration

On Weighted Graph Sparsification by Linear Sketching

A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21]. All these linear sketching algorithms, however, only work on unweighted graphs. In this paper, we initiate the study of weighted graph sparsification by linear sketching by investigating a natural class of linear sketches that we call incidence sketches, in which each measurement is a linear combination of the weights of edges incident on a single vertex. Our results are: 1. Weighted cut sparsification: We give an algorithm that computes a $(1 + \epsilon)$-cut sparsifier using $\tilde{O}(n \epsilon^{-3})$ linear measurements, which is nearly optimal. 2. Weighted spectral sparsification: We give an algorithm that computes a $(1 + \epsilon)$-spectral sparsifier using $\tilde{O}(n^{6/5} \epsilon^{-4})$ linear measurements. Complementing our algorithm, we then prove a superlinear lower bound of $\Omega(n^{21/20-o(1)})$ measurements for computing some $O(1)$-spectral sparsifier using incidence sketches. 3. Weighted spanner computation: We focus on graphs whose largest/smallest edge weights differ by an $O(1)$ factor, and prove that, for incidence sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are optimal up to an $n^{o(1)}$ factor