In this paper, we show that spin waves, the elementary excitation of the Heisenberg magnetic system, obey a kind of intermediate statistics with a finite maximum occupation number n. We construct an operator realization for the intermediate statistics obeyed by magnons, the quantized spin waves, and then construct a corresponding intermediate-statistics realization for the angular momentum algebra in terms of the creation and annihilation operators of the magnons. In other words, instead of the Holstein-Primakoff representation, a bosonic representation subject to a constraint on the occupation number, we present an intermediate-statistics representation with no constraints. In this realization, the maximum occupation number is naturally embodied in the commutation relation of creation and annihilation operators, while the Holstein-Primakoff representation is a bosonic operator relation with an additional putting-in-by-hand restriction on the occupation number. We deduce the intermediate-statistics distribution function for magnons. On the basis of these results, we calculate the dispersion relations for ferromagnetic and antiferromagnetic spin waves. The relations between the intermediate statistics that magnons obey and the other two important kinds of intermediate statistics, Haldane-Wu statistics and the fractional statistics of anyons, are discussed. We also compare the spectrum of the intermediate-statistics spin wave with the exact solution of the one-dimensional s = 1/2 Heisenberg model, which is obtained by the Bethe ansatz method. For ferromagnets, we take the contributions from the interaction between magnons (the quartic contribution), the next-to-nearest neighbor interaction, and the dipolar interaction into account for comparison with the experiment.
Breakdown of intermediate one-half magnetization plateau of spin-1/2 Ising-Heisenberg and Heisenberg branched chains at triple and Kosterlitz-Thouless critical points
Condensation of an ideal gas with intermediate statistics on the horizon
我们考虑史瓦西和克尔黑洞的拉伸视界上的玻色子气体。结果表明，当系统的粒子数等于时空的量子比特数$ N \simeq {A}/{ł_{p}^{2}}$时，气体处于玻色-爱因斯坦凝聚态，其Hawking温度为$T_c=T_H$.气体的熵与视界的面积成正比，即$(A)$的结构。对于视界上量子自由度的更现实的模型，我们大概应该考虑相互作用玻色子（引力子）。具有中间统计的理想气体可以被认为是相互作用玻色子的有效理论。分析表明，只要适当地选择中间统计量中的参数，就可以得到正确的熵。
An intermediate state between the kagome-ice and the fully polarized state in Dy$_2$Ti$_2$O$_7$
Dy $_2$ Ti $_2$ O $_7$是目前最干净的自旋冰材料的例子。以前的理论和实验工作之间的kagome冰和全极化状态的一级转变已被作为偶极自旋冰模型的验证。在这里，我们进一步深入研究这一相变使用交流磁化率和直流磁化，并比较这一结果与蒙特卡罗模拟和以前的磁化和比热测量。我们发现的kagome冰和充分极化之间的中间状态的签名。这种特征在目前用于描述自旋冰材料的理论模型中是不存在的。
