Mean-field calculation of doping-induced quantum spin Hall state to superconductor
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摘要:
$t-\lambda$ 模型是一个基于二维狄拉克费米子体系实现量子自旋霍尔绝缘体和超导的 晶格模型，特别重要的是它在费米子系统实现了超越朗道范式的解禁闭量子临界性．本文给出对易面$t-\lambda$ 模型的详细平均场理论和计算结果，特别是平均场相图, 掺杂纯的量子自旋霍尔态有2种结果：（1）相互作用强度较小时，量子自旋霍尔态和超导态之间发生一级相变；（2）深入量子自旋霍尔态掺杂时，随着掺杂的增大，先经过一连续相变进入两相共存区，再发生量子自旋霍尔态消失的一级相变．计算表明平均场理论可以抓住对称破缺相的特征，但是忽略涨落导致得到了量子自旋霍尔态与超导态共存的错误结论，更无法正确描述拓扑激发导致的相变．Abstract: The designer$t-\lambda$ lattice model is based on a two-dimensional Dirac fermion system, and it can realize quantum spin Hall (QSH) and s-wave superconductivity (SSC)． In particular, it realizes the deconfined quantum criticality between QSH and SSC, which goes beyond the Landau paradigm． This paper gives the mean-field theory and simulation result of this model with anisotropy and the mean-field phase diagram． There are two mean-field scenarios for doping the pure QSH state: (1) a first-order QSH-SSC transition occurs in the small interaction region; (2) a continuous QSH to QSH-SSC co-existence transition followed by a first-order QSH-vanishing transition upon doping deeply inside the QSH state． The calculation shows that the mean-field theory captures the features for symmetry-broken phases but gives the wrong conclusion of QSH-SSC co-existence due to ignoring fluctuation． Besides, it is unable to characterize transitions with topological excitation. -
图 1 蜂巢晶格上
$ t-\lambda $ 模型序参量示意注：黑（蓝）色圆点表示元胞内的A（B）子晶格；红色虚线表示自旋流算符$ \hat{ {\boldsymbol{J}} }_{{\boldsymbol{r}} + {\boldsymbol{\delta}}_n,{\boldsymbol{r}} + {\boldsymbol{\eta}}_n} $，粉色虚线圈表示超导算符$ \hat{\eta}^{\dagger}_{{\boldsymbol{r}},{\boldsymbol{\tilde{\delta}}_a}} $.
图 3 相互作用强度较小时观测量随化学势的变化
注：$ \Delta=0.5 $，$ \lambda=0.1 $；a 序参量，b 掺杂因子；在$ \mu=0.6 $处发生从QSH到SSC的一级相变.
图 4 相互作用较大时观测量随化学势的变化
注：$ \Delta=0.5 $，$ \lambda=0.2 $；a 序参量，b 掺杂因子；当$ \mu $接近0.8时，系统进入QSH和SSC共存相; 当$ \mu $增大到1时，系统离开QSH，处于单纯的SSC相，此相变为一级相变.
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