Topology in CMT I 對稱性保護拓撲序

09-18

來自專欄 Napkin for Physicist4 人贊了文章

拓撲物態自上世紀八十年代被提出以來，成為了凝聚態物理中最核心的問題之一。我現在所關心的幾個問題也屬於拓撲物態這個大方向。然而由於從來沒有機會系統地學習過拓撲物態的基本理論，研究過程中有時深感無力，所以想要借寫這個筆記的機會重新複習（預習）一遍。我對於拓撲物態大多來源於兩本書一節網課和一些零散的文獻，分別是Pályi等人所寫的A Short Course on Topological Insulators [1]，文小剛等人所寫的Quantum Information Meets Quantum Matter [2]，以及Delft University of Technology掛在edX上的Topology in Condensed Matter [3]。

在這個系列的第一篇文章里，我想簡單回顧一下Kitaev Chain模型來引入對稱性保護拓撲相（Symmetry Protected Topological States, SPT）的概念，主要內容來源於Topology in Condensed Matter [3]。

The motivation of Kitaev Chain comes from edge Majorana modes. So let me first introduce the Majorana operators.

There is a non-trivial way to decompose the fermion creation operators $c^{dagger}$ and annihilation operators $c$ .

$c^{dagger} = frac12 (gamma_1 + igamma_2), quad c = frac12 (gamma_1 - igamma_2)$

which results in the famous Majorana operators $gamma$ such that $gamma^2=1$ , $gamma^{dagger} = gamma$ and ${gamma_i,gamma_j}=2delta_{ij}$ . Majorana operators have to be paired to represent a physical degree of freedom. For example, the Hamiltonian always include some combination of $igamma_igamma_j$ .

$H = epsilon c^{dagger} c o frac12 epsilon (1-igamma_1gamma_2)$

Now consider a chain of $2N$ Majoranas, there are two ways to pair them.

Topologically trivial pairing with no unpaired Majoranas at two ends. Retrieved from [3].

Topologically non-trivial pairing with two unpaired Majoranas at two ends. Retrieved from [3].

The Hamiltonian can be written in the following way，

$H_{trivial} = - i sum_{n=1}^{N} gamma_{2n-1}gamma_{2n}, quad H_{non-trivial} = - i sum_{n=1}^{N-1} gamma_{2n}gamma_{2n+1}$

If you look closer, both of them have gaped bulk, while the non-trivial pairing results in two zero-energy Majorana modes at two ends since they are not included in the Hamiltonian.

Kitaev comes up with a nice way to make these two pairings two limits of a superconducting wire, which is now referred as the famous Kitaev Chain model [4].

$H_{Kitaev} = - mu sum_n c_n^{dagger}c_n - t sum_n (c_{n+1}^{dagger}c_n + h.c.) + Delta sum_n (c_nc_{n+1} + h.c.)$

where the two limits are $Delta = t eq 0, mu = 0$ for non-trivial pairing and $Delta = t = 0, mu eq 0$ for trivial pairing. This Hamiltonian can be written in the famous Bogoliubov-de Gennes (BdG) formalism with $H = frac12 C^{dagger} H_{BdG} C$ .

$H_{BdG} = - mu sum_n au^z | n angle langle n | - sum_n [(t au^z + i Delta au^y) | n angle langle n+1 | + h.c.]$

The basis are chosen to be $| n angle | au angle$ , where $| n angle$ indicates the site and $| au angle$ indicates whether it is an electron or a hole. The above Hamiltonian has particle-hole symmetry such that $PH_{BdG}P^{-1} = - H_{BdG}$ .

Energy spectrum and edge Majorana mode in the Kitaev Chain. Retrieved from [3].

The characterizing feature of the Kitaev Chain if the Majorana edge modes at two ends. The figure energy spectrum proves the existence of two edge modes in a wide parameter regime. More surprisingly, these two modes are robust under local perturbation! To see this, we need to remember that particle-hole symmetry results in symmetric spectrum around the zero energy. Since the bulk is gaped, the only way to lift the degeneracy of edge modes is to couple them together. However, the edge modes are far separated in space. Therefore, they would spread out and coupled only when the bulk gap closes, which makes the phase with edge modes topologically different from the one with out.

In this case, it is the particle-hole symmetry that protects these two edge modes. In fact, the topological phases that are protected by some symmetry is a much more general phenomenon. In particular, these topological phases are called Symmetry Protected Topological (SPT) phases. For example, the Su-Schrieffer-Heeger (SSH) Hamiltonian [1] is protected by the chiral symmetry and the cluster state Hamiltonian [2] is protected by the $mathbb{Z}_2 imes mathbb{Z}_2$ symmetry. The major difference between SPT and intrinsic topological phases, which I might introduce in the following notes, is that SPT only has short-range entanglement, while intrinsic topological phases has long-range entanglement.

It would be fun to talk about how to linearize the Kitaev Chain Hamiltonian in the momentum space and see how it is connected to the Jackiw-Rebbi mode. To be updated

PS: 題圖源於Flensberg等人繪製的Kitaev Chain模型示意圖 [5]。

Reference

[1] Asbo?th, J. K., Oroszla?ny, L., & Pa?lyi, A. (2016). A short course on topological insulators: Band structure and edge states in one and two dimensions. Cham: Springer.

[2] Zeng, B., Chen, X., Zhou, D., & Wen, X. (2018). Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems.arXiv:1508.02595 [cond-mat.str-el]

[3] Akhmerov, A., Sau, J., Heck, B. V., Irfan, M., Nijholt, B., & Rosdahl, T. ?. (n.d.). Topology in Condensed Matter: Tying Quantum Knots. Lecture. Retrieved from https://www.edx.org/course/topology-condensed-matter-tying-quantum-delftx-topocmx-0

[4] Kitaev, A. Y. (2001). Unpaired Majorana fermions in quantum wires.Physics-Uspekhi,44(10S), 131-136. doi:10.1070/1063-7869/44/10s/s29

[5] Leijnse, M., & Flensberg, K. (2012). Introduction to topological superconductivity and Majorana fermions.Semiconductor Science and Technology,27(12), 124003. doi:10.1088/0268-1242/27/12/124003