模糊球上的玻色-爱因斯坦凝聚与超流性
Bose-Einstein condensation and superfluidity on a fuzzy sphere
浏览论文内容
中文总结 AI 辅助
本文研究模糊球上的BEC与超流性，发现非交换性可提升BEC临界温度，使正常流体分数随温度线性变化，相关特性与铜氧化物超导体的宇山定律类似。
中文摘要 AI 辅助
根据霍亨伯格定理，二维系统中温度T>0时不可能发生玻色-爱因斯坦凝聚（Bose-Einstein condensation, BEC）。相比之下，有限温度下二维系统中确实存在超流性，其产生源于伽利略不变性的破缺。本文研究紧致二维空间（非交换“模糊”球）上的BEC与超流性，其中标量玻色场被提升为N×N矩阵，维度N与空间的非交换参数相关，为系统引入额外尺度。研究发现，非交换性有利于有序相，从而增强BEC与超流性。我们分析了模糊球上理想和弱相互作用玻色气体的BEC，发现两种情况下的BEC临界温度均高于交换球S²上的临界温度。随后研究弱相互作用玻色系统的超流响应：为解释超流体中的涡旋，我们证明即使在普通球上，涡旋的集体坐标也会诱导非交换性；在此基础上，我们将涡旋缺陷的定义扩展到本文研究的固有非交换球（其中点的概念不成立）。非交换性被认为与BEC和超流性的实验相关，因为模糊球的热力学极限与平面上的不同，也与S²的情况不同。超流密度计算表明，在大球极限下，模糊球上的正常流体分数呈现与T线性相关的行为，而交换S²上则呈现常规的二维~T³行为；这种直接源于非交换性的线性依赖关系，让人联想到铜氧化物高温超导体中的宇山定律（Uemura's law）。
英文摘要
According to Hohenberg's theorem, Bose-Einstein condensation (BEC) in two dimensions is impossible for any temperature $T>0$. By contrast, superfluidity does occur in two dimensions at finite temperatures; it emerges due to the breaking of Galilei invariance. Here we consider BEC and superfluidity on a compact two-dimensional space taking the form of a non-commutative ("fuzzy") sphere, where the scalar bosonic fields are promoted to $N\times N$ matrices. The dimension $N$ is related to the non-commutativity parameter of space and introduces an additional scale into the system. We find that non-commutativity favors ordered phases and so enhances BEC and superfluidity. We analyze BEC in ideal and weakly interacting Bose gases on a fuzzy sphere, finding in each case that the critical temperature of BEC is greater compared to that found in the case of a commutative sphere $S^2$. Then we investigate the superfluid response of weakly interacting Bose systems. To account for vortices in a superfluid, we show that, even on an ordinary sphere, the collective coordinates of vortices induce non-commutativity. With this in mind, we extend the definition of vortex defects to an inherently non-commutative sphere studied here, where the notion of a point is untenable. The non-commutativity is expected to be experimentally relevant to BEC and superfluidity since the fuzzy sphere has a thermodynamic limit distinct from the one defined over a plane, unlike the $S^2$ case. The significance of this difference is illustrated by the superfluid density calculation indicating that, in the large sphere limit, the normal fluid fraction on the fuzzy sphere yields a linear in $T$ dependence, while on a commutative $S^2$ it exhibits the usual two-dimensional $\sim T^3$ behavior. This linear dependence, arising directly from non-commutativity, is reminiscent of Uemura's law in cuprate high-$T_c$ superconductors.
