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Topological gapless systems, as the connection of the different topological quantum phases, have received much attention. Topological nonmediocre nodes are typically observed in two- or three-dimensional gapless systems. In this paper, we demonstrate that the topological nonmediocre nodes are existent in a model that lies between one dimension and two dimensions. Superconducting circuits, as essential all-solid state quantum devices, have offered a promising platform for studying the macro-controlling quantum effects. Recently, experimental achievements have enabled the realization of tunable coupling strengths between transmon qubits and the implementation of a one-dimensional Su-Schrieffer-Heeger (SSH) model [Li X et al.
2018 Phys. Rev. Appl. 10 054009 ]. According to this work, herein we present a two-leg SSH model implemented in superconducting circuits and demonstrate the existence of topological nonmediocre nodes. Firstly, two-leg superconducting circuit with transmon qubits which are coupled with their nearest-neighbor sites by capacitors is designed. To construct the two-leg SSH model, we introduce two alternating-current magnetic fluxes to drive each transmon qubit. We discover two types of phase boundaries in the SSH model and obtain the corresponding energy spectra and phase diagram. We identify two distinct topological insulating phases characterized by winding number ±1, and the corresponding edge states exhibit distinct characteristics. Moreover, we discuss the topological properties of the two phase boundaries. By representing the Bloch states as a vector field in k space, we demonstrate the existence of two kinks of nonmediocre nodes with first-type phase boundaries. These two nonmediocrenodes possess distinct topological charges of 1 and –1, respectively. On the other hand, the nonmediocre nodes with the second-type phase boundaries are topologically trivial. These results open the way for exploring novel topological states, ladder physical systems, and nodal point topological semimetals.[1] Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004
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图 1 双链SSH模型 (a) 两条transmon比特链分别标记为A和B. 每个transmon比特都与其最近邻比特两两耦合. 这里所有的耦合器均为电容.
$ Q_{\nu j} $ 表示的是第ν条链上的第j个比特.$ C_{\nu j} $ 和$E_{\nu j}^{{\rm{J}}}$ 分别是第ν条链上, 第j个比特的有效电容和约瑟夫森能.$ C_{\nu ij} $ 是耦合第ν条链上, 第i和第j个比特的电容.$ C_{ABj} $ 表示A, B两条链上第j个比特之间耦合的电容.$ \phi _{\nu j} $ 是第ν条链上, 第j个比特的约瑟夫森结的相位. 本文中transmon比特的约瑟夫森结由超导量子干涉仪(SQUID)形成,$E_{\nu j}^{{\rm{J}}0}$ 是SQUID中每个约瑟夫森结的能量. 每个比特都受到两个外加磁通$ \varPhi_{\nu j}^{(1)}(t) $ 和$ \varPhi_{\nu j}^{(2)}(t) $ 的调制. (b) 双链SSH模型示意图, 图中红色和蓝色的实心球表示SSH模型中一个原胞的两种比特Figure 1. Two-leg SSH model: (a) Two-leg (labeled respectively by A and B) superconducting circuits with transmon qubits. The qubits are coupled with their nearest-neighbor sites. All couplers are capacitors.
$ C_{\nu j} $ and$E_{\nu j}^{{\rm{J}}}$ are the effective capacitance and the Josephson energy of the qubit at the jth site on the$ \nu {\rm{th}} $ leg.$ C_{\nu ij} $ and$ C_{ABj} $ are the capacitors to couple the qubits at the jth site on the$ \nu {\rm{th}}$ leg with its nearest-neighbor sites along each leg and between the legs, respectively.$ \phi _{\nu j} $ is the phase of the Josephson junction of the qubit at the jth site on the$ \nu {\rm{th}} $ leg. The Josephson junction of the transmon qubit is a superconducting quantum interference device(SQUID).$E_{\nu j}^{{\rm{J}}0}$ is the Josephson energy of SQUID. Each qubit is modulate by two external magnetic fluxes$\varPhi_{\nu j}^{(1)}(t)$ and$\varPhi_{\nu j}^{(2)}(t)$ 
图 2 双链SSH模型的能谱图 (a)
$ t_{1}=t_{2}=1.5 $ ; (b)$ t_{1}=-1.5 $ 和$ t_{2}=0.5 $ ; (c)$ t_{1}=t_{2}=-0.5 $ Figure 2. Energy bands of SSH model: (a)
$ t_{1}=t_{2}=1.5 $ ; (b)$ t_{1}=-1.5 $ and$ t_{2}=0.5 $ ; (c)$ t_{1}=t_{2}=-0.5 $ 
图 3 双链SSH模型在参数空间
$t_{1}\text-t_{2}$ 中的拓扑相图. 深绿色的区域是拓扑数$ N=-1 $ 的区域, 浅咖色的区域是拓扑数$N=1$ 的区域, 浅绿色的区域是拓扑数$ N=0 $ 的区域; 红色的线标记的是第一类相边界, 黑色的线标记的是第二类相边界Figure 3. Topological phase diagram in the
$t_{1}\text-t_{2}$ plane. The bottle green, light coffee color and pale green areas indicate the areas with$ N=-1 $ ,$ N=1 $ and$ N=0 $ respectively. N denotes the winding number. The red and black lines indicate the first and second phase boundaries respectively
图 4 开边界能带和边界态 (
$ a_{1} $ )$ t_{1}=1.5 $ 和$ t_{2}=0.5 $ ; ($ a_{2} $ )$ t_{1}=0.5 $ 和$ t_{2}=1.5 $ ; ($ a_{3} $ )$ t_{1}=0.5 $ 和$ t_{2}=0.3 $ . ($ b_{1} $ )—($ b_{3} $ )是($ a_{1} $ )—($ a_{3} $ )中第20个能带的波函数Figure 4. Energy bands with open boundary condition with (
$ a_{1} $ )$ t_{1}=1.5 $ and$ t_{2}=0.5 $ ; ($ a_{2} $ )$ t_{1}=0.5 $ and$ t_{2}=1.5 $ ; ($ a_{3} $ )$ t_{1}= $ $ 0.5 $ and$ t_{2}=0.3 $ . The wave functions corresponding with the 20th energy bands of ($ a_{1} $ ), ($ a_{2} $ ), and ($ a_{3} $ ) are plotted in ($ b_{1} $ ), ($ b_{2} $ ), and ($ b_{3} $ ), respectively
图 5 不同参数下
$ {\boldsymbol{\varepsilon}} $ 在布洛赫球的$x\text{-}y$ 平面上的轨迹 (a)拓扑数$ N=1 $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=1.5 $ 和$ t_{2}=0.5 $ ; (b)拓扑数$ N=-1 $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=0.5 $ 和$ t_{2}=1.5 $ ; (c)拓扑数$ N=0 $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=1 $ 和$ t_{2}=-0.7 $ ; (d)参数满足第一类相边界$ t_{1}=t_{2} $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=t_{2}=1 $ ; (e)参数满足第二类相边界$ t_{1}+t_{2}=1 $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=0.8 $ 和$ t_{2}=0.2 $ ; (f)参数满足第二类相边界$ t_{1}+t_{2}=-1 $ 时$ {\boldsymbol{\varepsilon}} $ 的轨迹, 参数设定为$ t_{1}=-1.3 $ 和$ t_{2}=0.3 $ . 图中红点表示原点, 箭头表示轨迹的运动方向Figure 5. The curve of the vector
$ {\boldsymbol{\varepsilon}} $ in$x\text{-}y$ plane of the Bloch sphere with (a)$ t_{1}=1.5 $ and$ t_{2}=0.5 $ ; (b)$ t_{1}=0.5 $ and$ t_{2}=1.5 $ ; (c)$ t_{1}=1 $ and$ t_{2}=-0.7 $ ; (d)$ t_{1}=t_{2}=1 $ ; (e)$ t_{1}=0.8 $ and$ t_{2}=0.2 $ ; (f)$ t_{1}=-1.3 $ and$ t_{2}=0.3 $ . The red points and arrows indicate the origin points and direction of the curve respectively
图 6 参数设定为
$ t_{1}=t_{2}=1 $ (第一类相边界上)时矢量$ {\boldsymbol{F}}(k) $ 随波矢k的变化. 图中箭头表示自旋的方向,$ k_{\rm{c}} $ 为能级简并点处的波矢Figure 6. The variation of the vector
$ {\boldsymbol{F}}(k) $ as k changes with$ t_{1}=t_{2}=1 $ . The arrows and$ k_c $ indicate the direction of the spin and degenerate energy point respectively -
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