Nonperturbative functional renormalization-group approach to transport in the vicinity of a $(2+1)$-dimensional O($N$)-symmetric quantum critical point
Nonperturbative functional renormalization-group approach to transport in the vicinity of a $(2+1)$-dimensional O($N$)-symmetric quantum critical point
复制标题
$(2 1)$ 维 O($N$) 对称量子临界点附近输运的非微扰函数重正化群方法
DOI：
10.1103/physrevb.95.014513
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
N. Dupuis
中科院分区：
文献类型：
--
作者：
F. Rose;N. Dupuis
用非微扰泛函重整化群方法研究了二维量子O($N$)模型，计算了零温电导率在量子临界点附近的低频极限。我们的结果是在外部(即非动力学)非阿贝尔规范场存在的情况下，从依赖于标度的有效作用量的导数展开到二阶。在无序相中，电导率张量是对角的，而在有序相中，电导张量是由两个独立的元素定义的，它们分别与SO($N$)转动有关，它们改变和不改变有序参数的方向。当$N=2$时，有序相的电导率降低为单一组分$\sigma_{\rMA}(\omega)$。我们证明了$Lim_{\omega\to 0}\sigma(\omega，\Delta)\sigma_{\rMA}(\omega，-\Delta)/\sigma_q^2$是一个普适数，它是我们计算的作为$N$的函数($\Delta$测量到量子临界点的距离，$q$是电荷，$\sigma_q=q^2/h$电导量子)。另一方面，我们认为在整个有序相中，σ_(RmB)(omega\to 0)/sigma_q$是普适的，与$N无关，当$N\to\inty$时，等于量子临界点的普适电导率$\sigma^*/\sigma_q$。
Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum O($N$) model, we compute the low-frequency limit $\omega\to 0$ of the zero-temperature conductivity in the vicinity of the quantum critical point. Our results are obtained from a derivative expansion to second order of a scale-dependent effective action in the presence of an external (i.e., non-dynamical) non-Abelian gauge field. While in the disordered phase the conductivity tensor $\sigma(\omega)$ is diagonal, in the ordered phase it is defined, when $N\geq 3$, by two independent elements, $\sigma_{\rm A}(\omega)$ and $\sigma_{\rm B}(\omega)$, respectively associated to SO($N$) rotations which do and do not change the direction of the order parameter. For $N=2$, the conductivity in the ordered phase reduces to a single component $\sigma_{\rm A}(\omega)$. We show that $\lim_{\omega\to 0}\sigma(\omega,\delta)\sigma_{\rm A}(\omega,-\delta)/\sigma_q^2$ is a universal number which we compute as a function of $N$ ($\delta$ measures the distance to the quantum critical point, $q$ is the charge and $\sigma_q=q^2/h$ the quantum of conductance). On the other hand we argue that the ratio $\sigma_{\rm B}(\omega\to 0)/\sigma_q$ is universal in the whole ordered phase, independent of $N$ and, when $N\to\infty$, equal to the universal conductivity $\sigma^*/\sigma_q$ at the quantum critical point.