本文采用紧束缚模型研究三角晶格中的拓扑平带。在模型中引入净磁场为零的交错磁场,通过改变格点之间的次近邻跃迁强度和交错磁通的大小能够得到具有大带隙且拓扑非平庸的近平带结构。此拓扑平带的高陈数C = 2,进而可以产生稳定的整数量子霍尔效应。 We use the tight-bonding model to study the topological flat band in a triangular lattice. Under a staggered magnetic field with zero total flux through the unit cell, the topological nearly flat band with a large band gap and nonzero Chern number can be obtained by manipulating the next nearest neighbor hopping and staggered flux. This topological flat band carries a high Chern number C = 2, which can yield an integer quantum Hall effect.
拓扑平带,交错磁通,量子霍尔效应, Topological Flat Band Staggered Flux Quantum Hall Effect三角晶格中的拓扑平带和量子霍尔效应
H = ± t ∑ 〈 i j 〉 ( e i ϕ i j c i † c j + H . c . ) ± t ′ ∑ 〈 〈 i j 〉 〉 ( c i † c j + H . c . ) (1)
其中 〈 i j 〉 和 〈 〈 i j 〉 〉 分别表示最近邻和次近邻格点对, c i ( c i † ) 表示第 i 个格点上的消灭和产生算符,最近邻和次近邻格点上的磁通所产生的相位沿着黑色和红色箭头方向分别为 2 ϕ 和 ϕ , t ( − t ) 和 t ′ ( − t ′ ) 分别表示沿着实线(虚线)方向的最近邻和次近邻跃迁强度,如图1所示。可以通过久保公式计算霍尔电导 [16] :
σ x y ( E ) = i e 2 ℏ S ∑ ε m < E ∑ ε n > E 2 Im 〈 m | υ x | n 〉 〈 n | υ y | m 〉 ( ε m − ε n ) 2 (2)
其中 E 是费米能, S 是系统的面积, ε m 和 ε n 是本征态 | m 〉 和 | n 〉 所对应的本征能量, υ = ( υ x , υ y ) 为
速度算符。当费米能级 E 处于能隙之间时,霍尔电导表示为 σ x y ( E ) = e 2 h ∑ ε m < E C m ,其中 C m 是第 m 个
计算结果表明,在不同参数条件下三角晶格的能带中都含有三个能带。分别调节交错磁通和近邻跃迁强度,即调节交错磁通 ϕ 从0增加到 π ,同时使次近邻跃迁强度 t ′ 从 t ′ = − t 增加到 t ′ = t ,可以得到 Δ 12 / W 1 和 Δ 23 / W 2 的相图,如图2(a)和图2(b)所示。从图中容易得出,当交错磁通 ϕ = π / 3 时,带隙与能带的带宽比值出现最大。如图2(c)和图2(d)所示,得到了带隙与带宽的比值随次近邻跃迁强度的变化曲线。从图中可以发现,当交错磁通和次近邻跃迁取 ϕ = π / 3 和 t ′ = 0.14 t 时,带隙与最低能带的带宽比值最大, Δ 12 / W 1 ≈ 499 ;而当 ϕ = π / 3 和 t ′ = 0.2 t 时,带隙与中间能带的带宽比值最大, Δ 23 / W 2 ≈ 48 ,其中 Δ 12 、 Δ 23 、 W 1 、 W 2 分别表示中间能带和最低能带间的带隙、最高能带和中间能带间的带隙、最低能带的带宽、中间能带的带宽。此时,能带结构如图3(a)和图3(b)所示,可以看出,最低能带带宽较小,最高能带的带宽虽相对较大,一定程度上仍可认为是属于近平带结构。
由此可以得到,交错磁通 ϕ = π / 3 时,能带中出现了平带结构。通过计算可知此时的能带陈数不等于零,分别为 C lower = C middle = − 1 , C upper = 2 ,因此能带是拓扑非平庸的,这表明通过调控交错磁通可以得到具有陈数不等于零的拓扑非平庸能带。因此,在接下来的计算过程中选取交错磁通为 ϕ = π / 3 ,然后通过调节次近邻跃迁强度,去寻找拓扑非平庸的近平带结构。考虑到粒子的激发是从低能级向高能级跃迁,所以期望较低能级会具有拓扑平带结构。
如图3(a)和图3(b)所示,从能带和陈数的数值计算结果可知,当 ϕ = π / 3 和 t ′ = 0.14 t 以及 ϕ = π / 3 和 t ′ = 0.2 t 时,带隙与带宽比值最大,而此时各能级的陈数分别为 C lower = − 1 , C middle = − 1 , C upper = 2 ,这正是我们所寻找的拓扑近平带结构。如图3(c)所示,通计算系统的霍尔电导发现,带隙间有量子化的霍尔电导,电导的平台值分别为 σ x y = − e 2 / h ,和 − 2 e 2 / h ,这对应着整数量子霍尔态。因此可以在拓扑平带结构中实现零磁场的整数量子霍尔效应。由于该拓扑平带和二维朗道能级的相似性,在此系统中加上相互作用会有可能实现无磁场的分数激发,从而实现分数量子霍尔态。
通过调节这两个参数,我们得到了拓扑非平庸的近平带结构,此时能带具有高陈数C = 2,系统中可以存在稳定的整数量子霍尔效应,对应的霍尔电导台阶为 2 e 2 / h 。然而,当进一步考虑到粒子间相互作用时,系统中有望实现分数激发,进而出现分数量子霍尔效应。
基金项目
本项目得到了新疆大学大学生创新训练计划项目(201710755074)的支持。
文章引用
龚乐,魏浩,陶相如. 三角晶格中的拓扑平带和量子霍尔效应 Topological Flat Band and Quantum Hall Effect in the Triangular Lattice[J]. 凝聚态物理学进展, 2018, 07(01): 43-47. http://dx.doi.org/10.12677/CMP.2018.71006
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