Quantum critical Bose gas in the two-dimensional limit in the honeycomb antiferromagnet YbCl3_3 under magnetic fields

BEC is a quantum phenomenon, where a macroscopic number of bosons occupy the lowest energy state and acquire coherence at low temperatures. It is realized not only in $^4$He and dilute atomic gases, but also in quantum magnets, where hardcore bosons, introduced by the Matsubara-Matsuda transformation of spins, condense. In 3D antiferromagnets, an XY-type long-range ordering (LRO) occurs near a magnetic-field-induced transition to a fully polarized state (FP) and has been successfully described as a BEC in the last few decades. An attractive extension of the BEC in 3D magnets is to make their 2D analogue. For a strictly 2D system, BEC cannot take place due to the presence of a finite density of states at zero energy, and a Berezinskii-Kosterlitz-Thouless (BKT) transition may instead emerge. In a realistic quasi-2D magnet consisting of stacked 2D magnets, a small but finite interlayer coupling stabilizes marginal LRO and BEC, but such that 2D physics, including BKT fluctuations, is still expected to dominate. A few systems were reported to show such 2D-limit BEC, but at very high magnetic fields that are difficult to access. The honeycomb $S$ = 1/2 Heisenberg antiferromagnet YbCl$_3$ with an intra-layer coupling $J\sim$ 5 K exhibits a transition to a FP state at a low in-plane magnetic field of $H_{\rm s}$ = 5.93 T. Here, we demonstrate that the LRO right below $H_{\rm s}$ is a BEC in the 2D-limit stabilized by an extremely small interlayer coupling $J_{\perp}$ of 10$^{-5}J$. At the quantum critical point Hs, we capture 2D-limit quantum fluctuations as the formation of a highly mobile, interacting 2D Bose gas in the dilute limit. A much-reduced effective boson-boson repulsion Ueff as compared with that of a prototypical 3D system indicates the presence of a logarithmic renormalization of interaction unique to 2D.Comment: 24 pages, 12 figure

Teaching and learning about whole numbers in primary school

This book offers a theory for the analysis of how children learn and are taught about whole numbers. Two meanings of numbers are distinguished – the analytical meaning, defined by the number system, and the representational meaning, identified by the use of numbers as conventional signs that stand for quantities. This framework makes it possible to compare different approaches to making numbers meaningful in the classroom and contrast the outcomes of these diverse aspects of teaching. The book identifies themes and trends in empirical research on the teaching and learning of whole numbers since the launch of the major journals in mathematics education research in the 1970s. It documents a shift in focus in the teaching of arithmetic from research about teaching written algorithms to teaching arithmetic in ways that result in flexible approaches to calculation. The analysis of studies on quantitative reasoning reveals classifications of problem types that are related to different cognitive demands and rates of success in both additive and multiplicative reasoning. Three different approaches to quantitative reasoning education illustrate current thinking on teaching problem solving: teaching reasoning before arithmetic, schema-based instruction, and the use of pre-designed diagrams. The book also includes a summary of contemporary approaches to the description of the knowledge of numbers and arithmetic that teachers need to be effective teachers of these aspects of mathematics in primary school. The concluding section includes a brief summary of the major themes addressed and the challenges for the future. The new theoretical framework presented offers researchers in mathematics education novel insights into the differences between empirical studies in this domain. At the same time the description of the two meanings of numbers helps teachers distinguish between the different aims of teaching about numbers supported by diverse methods used in primary school. The framework is a valuable tool for comparing the different methods and identifying the various assumptions about teaching and learning.</p