From one-dimensional charge conserving superconductors to the gapless Haldane phase
From one-dimensional charge conserving superconductors to the gapless Haldane phase
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从一维电荷守恒超导体到无间隙霍尔丹相
DOI：
10.1103/physrevb.98.214501
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发表时间：
2018
影响因子：
3.7
通讯作者：
P. Azaria
中科院分区：
文献类型：
--
作者：
A. Keselman;E. Berg;P. Azaria
我们发展了一个框架来分析一维拓扑超导体的电荷守恒。特别是，我们考虑模型与$N$口味的费米子和$（\mathbb{Z}_2）^N$对称性，与每个口味的费米子宇称守恒。对于一个单一的味道，我们恢复的结果是，一个独特的拓扑相位与指数本地化零模式不存在，由于缺乏一个间隙，以单个粒子在散装。然而，对于$N>1$，我们表明，系统的两端可以主机低能量，指数本地化模式。该分析可以很容易地推广到其他对称类的系统。为了说明这些想法，我们专注于与$SO\左（N\右）$对称相互作用的晶格模型，并研究平凡和拓扑无隙相位之间的相变，使用玻色化和弱耦合重整化群分析。作为一个具体的例子，我们详细研究了$N=3$的情况。我们表明，在这种情况下，拓扑非平凡的超导相对应于自旋为1的链中的Haldom相的无间隙模拟。在这一阶段，虽然散装是gapless单粒子激发，两端主机自旋1/2$自由度是指数本地化和保护的自旋间隙在散装。利用密度矩阵重整化群计算，我们得到了该模型的全相图。在这个模型中，我们把Andrei和Destri [Nucl. Phys. B，231（3），445-480（1984）]研究的自对偶线确定为无隙Haldom相和平凡无隙相之间的一阶跃迁线。这使我们能够识别的传播自旋1/2扭结在安德烈-Destri模型的拓扑端模存在于两个阶段之间的域壁。
We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with $N$ flavors of fermions and $(\mathbb{Z}_2)^N$ symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with $SO\left(N\right)$ symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of $N=3$. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-$1/2$ degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases.