Electrical circuit simulation of nonreciprocal Aubry-André models

Cheng En-Hong; Lang Li-Jun

doi:10.7498/aps.71.20220219

Abstract

The introduction of non-Hermiticity into traditional Hermitian quantum systems generalizes their basic notions and brings about many novel phenomena, e.g., the non-Hermitian skin effect that is exclusive to non-Hermitian systems, attracting enormous attention from almost all branches of physics. Contrary to the quantum platforms, classical systems have the advantages of low cost and mature techniques under room temperature. Among them, the classical electrical circuits are more flexible on simulating quantum tight-binding models in principle with any range of hopping under any boundary conditions in any dimension, and have become a powerful platform for the simulation of quantum matters. In this paper, by constructing an electrical circuit, we simulate by SPICE the static properties of a prototypical non-Hermitian model—the nonreciprocal Aubry-André (AA) model that has the nonreciprocal hopping and on-site quasiperiodic potentials. The paper is organized as follows: Following the introduction, in Sec. II we review in detail the Laplacian formalism of electrical circuits and the mapping to the quantum tight-binding model. Then, in Sec. III, an electrical circuit is proposed with resistors, capacitors, inductors, and the negative impedance converters with current inversion (INICs), establishing a mapping between the circuit's Laplacian and the non-reciprocal AA model's Hamiltonian under periodic boundary conditions (PBCs) or open boundary conditions (OBCs). Especially, the nonreciprocity, the key of this model, is realized by INICs. In Sec IV, based on the mapping, for the proposed circuit under PBCs, we reconstruct the circuit's Laplacian via SPICE by measuring voltage responses of an AC current input at each node. The complex spectrum and its winding number $\nu$ can be calculated by the measured Laplacian, which are consistent with the theoretical prediction, showing $\nu=\pm 1$ for non-Hermitian topological regimes with complex eigenenergies and extended eigenstates, and $\nu=0$ for topologically trivial regimes with real eigenenergies and localized eigenstates. In Sec V, for the circuit under OBCs, a similar method is used for measuring the node distribution of voltage response, which simulates the competition of non-Hermitian skin effects and the Anderson localization, depending on the strength of quasiperiodic potentials; the phase transition points also appear in the inverse participation ratios of voltage responses. During the design process, the parameters of auxiliary resistors and capacitors are evaluated for obtaining stable responses, because the complex eigenfrequecies of the circuits are inevitable under PBCs. Our detailed scheme can directly instruct further potential experiments, and the designing method of the electrical circuit is universal and can in principle be applied to the simulation for other quantum tight-binding models.

Keywords:

Authors and contacts

Cheng En-Hong¹,
Lang Li-Jun^1,2

1.
Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
2.
Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China

Corresponding author: Lang Li-Jun, ljlang@scnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11904109) and the Basic and Applied Basic Research Foundation of Guangdong Province, China (Grant No. 2019A1515111101)

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References

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[2]	Daley A J 2014 Adv. Phys. 63 77 Google Scholar
[3]	Moiseyev N 2011 Non-Hermitian Quantum Mechanics (1st Ed.) (Cambridge: Cambridge University Press)
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[5]	Guo A, Salamo G J, Duchesne D, et al. 2009 Phys. Rev. Lett. 103 093902 Google Scholar
[6]	Peng B, Ozdemir S K, Lei F, et al. 2014 Nat. Phys. 10 394 Google Scholar
[7]	Poli C, Bellec M, Kuhl U, Mortessagne F, Schomerus H 2015 Nat. Commun. 6 6710 Google Scholar
[8]	Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2019 Nat. Commun. 10 855 Google Scholar
[9]	Ren Z, Liu D, Zhao E, He C, Pak K K, Li J, Jo G B 2021 arXiv: 2106.04874
[10]	Miri M A, Alù A 2019 Science 363 eaar7709 Google Scholar
[11]	Bergholtz E J, Budich J C, Kunst F K 2021 Rev. Mod. Phys. 93 015005 Google Scholar
[12]	Zhang X L, Jiang T, Chan C T 2019 Light Sci. Appl. 8 88
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[14]	Leykam D, Bliokh K Y, Huang C, Chong Y, Nori F 2017 Phys. Rev. Lett. 118 040401 Google Scholar
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[29]	Zeng Q B, Yang Y B, Xu Y 2020 Phys. Rev. B 101 020201
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[31]	Xu Z H, Xia X, Chen S 2021 Sci. China Phys., Mech. Astron. 65 227211
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[41]	Zeuner J M, Rechtsman M C, Plotnik Y, et al. 2015 Phys. Rev. Lett. 115 040402 Google Scholar
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[44]	Cerjan A, Huang S, Wang M, Chen K P, Chong Y, Rechtsman M C 2019 Nat. Photonics 13 623 Google Scholar
[45]	Wang K, Dutt A, Yang K Y, Wojcik C C, Vuč kovic J, Fan S 2021 Science 371 1240 Google Scholar
[46]	Brandenbourger M, Locsin X, Lerner E, Coulais C 2019 Nat. Commun. 10 4608 Google Scholar
[47]	Ghatak A, Brandenbourger M, van Wezel J, Coulais C 2020 Proc. Natl. Acad. Sci. U.S.A. 117 29561 Google Scholar
[48]	Wu F Y 2004 J. Phys. A: Math. Gen. 37 6653 Google Scholar
[49]	Schindler J, Lin Z, Lee J M, Ramezani H, Ellis F M, Kottos T 2012 J. Phys. A: Math. Theor. 45 444029 Google Scholar
[50]	Helbig T, Hofmann T, Imhof S, et al. 2020 Nat. Phys. 16 747 Google Scholar
[51]	Lang L J, Weng Y, Zhang Y, Cheng E, Liang Q 2021 Phys. Rev. B 103 014302 Google Scholar
[52]	Ezawa M 2019 Phys. Rev. B 100 081401 Google Scholar
[53]	Hofmann T, Helbig T, Lee C H, Greiter M, Thomale R 2019 Phys. Rev. Lett. 122 247702 Google Scholar
[54]	Ezawa M 2019 Phys. Rev. B 99 121411 Google Scholar
[55]	Zou D, Chen T, He W, Bao J, Lee C H, Sun H, Zhang X 2021 Nat. Commun. 12 7201 Google Scholar
[56]	Zhang X X, Franz M 2020 Phys. Rev. Lett. 124 046401 Google Scholar
[57]	Rafi-Ul-Islam S M, Siu Z B, Jalil M B A 2021 arXiv: 2102.03727
[58]	Tzeng W J, Wu F Y 2006 J. Phys. A: Math. Gen. 39 8579 Google Scholar
[59]	Lee C H, Imhof S, Berger C, Bayer F, Brehm J, Molenkamp L W, Kiessling T, Thomale R 2018 Commun. Phys. 1 39 Google Scholar
[60]	Hadad Y, Soric J C, Khanikaev A B, Alù A 2018 Nat. Electron. 1 178 Google Scholar
[61]	Wang Y, Lang L J, Lee C H, Zhang B, Chong Y D 2019 Nat. Commun. 10 1102 Google Scholar
[62]	Pozar D M 2012 Microwave Engineering (4th Ed.) (Hoboken: John Wiley & Sons, Inc.)
[63]	Helbig T, Hofmann T, Lee C H, Thomale R, Imhof S, Molenkamp L W, Kiessling T 2019 Phys. Rev. B 99 161114 Google Scholar
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Cited By

图 1 (a) 上图: 非互易AA模型的电路模拟示意图, 包含$ N $个有效电压节点$ V_n\; (n = 1, \cdots, N) $, 节点间元件$ C_0 $和${\rm{INIC}}_{\rm (b)}$模拟格点间的耦合, 其中${\rm{INIC}}_{\rm(b)}$用于实现关键的非互易耦合, 其定义见图(b); 接地元件$(L_0, R_0, C_{0, {\rm{r}}, n})$模拟格点的在位势; X相关模块和开关控制对不同边界条件的模拟. 下图: X相关模块的定义. (b) $ {\rm{INIC}} $元件的内部电路图, 由理想放大器、阻抗$ Z_{\pm} $和目标元件$C_{\rm{I}}$(没有$X_{\rm{b}}$)构成, 可以实现$ V_{{\rm{l, r}}} $两端不同方向的导纳不同; ${\rm{INIC}}_{\rm b}$仅需将INIC中的目标元件$C_{\rm{I}}$再并联一个$X_{\rm{b }}$即可. (c) 负阻抗模块^[49], 左右图分别实现对地单端口和自由两端口间的等效负阻抗$ -Z $, 其中理想放大器上的标记表示输出电压与输入电压的关系. 各元件的具体功能描述详见正文.

Figure 1. (a) Upper panel: Sketch of an electrical circuit simulating the nonreciprocal AA model. It includes $ N $ voltage nodes $ V_n\; (n=1, \cdots, N) $ with elements $ C_0 $ and ${\rm{INIC}}_{\rm (b)}$ simulating the intersite couplings, where ${\rm{INIC}}_{\rm (b)}$ defined in panel (b) is the key element to realize the nonrecprocity, and the grounded elements $(L_0, R_0, C_{0, {\rm{r}}, n})$ simulating the on-site potentials; $\rm X$ modules and the switches control the simulation of boundary conditions. Lower panel: Definitions of X modules. (b) The internal circuit of the INIC, constructed by the ideal operational amplifier (opamp), impedance elements $ Z_\pm $, and the targeted element $C_{\rm{I}}$(without $X_{\rm{b}}$), which can realize unequal effective input inductances from the two different ports $ V_{{\rm{l, r}}} $; ${\rm{INIC}}_{\rm b}$ is defined by adding an extra $X_{\rm{b}}$ module in parallel with $C_{\rm{I}}$ in INIC. (c) Modules of negative impedance^[49]. The internal circuits of the grounded one-port and the floated two-port negative impedances $ -Z $ for the left and right panels, respectively, where the labels of the ideal opamps represent the relation of the output voltage to the input voltages. See relevant texts for the detailed description of each element.

DownLoad: Full-Size Img PowerPoint

图 2 (a) 非互易AA模型的参考相图, 相边界(红色实线)由文献[27]附录中的解析表达式(A6)计算缠绕数$ \nu $求得. 虚线用于图 3的计算. (b), (c) 分别对应图(a)中B, C两点的电路本征频率谱(取$ \gamma = 0 $时)(左图)以及矩阵$ \mathcal{{\boldsymbol{A}}}/C_0 $的本征谱(取$ \varPhi = 0 $时)(右图). 红色实点和蓝色实方块分别代表周期和开边界条件下的理论结果. 左图中的空心菱形标记驱动频率$ \omega $的位置, 右图中的空心菱形表示周期边界条件下SPICE的模拟结果. (d) 根据SPICE模拟得到的$ \mathcal{{\boldsymbol{A}}}(\varPhi) $计算得出$ \theta(\varPhi) = \ln{\det{[\mathcal{A}(\varPhi)/C_0]}} $随$ \varPhi $的变化. 左向三角、右向三角和菱形分别对应图(a)中A, B和C三点, 虚线为相应的理论值. 以上所有计算取有限尺寸$ N=21 $, 电路元件的具体设置参见相关正文.

Figure 2. (a) Referenced phase diagram of the non-reciprocal AA model, where phase boundaries (solid red lines) are obtained by calculating the winding number $ \nu $ with Eq. (B6) in the Appendix of Ref. [27]. The dashed lines are for the plots in Fig. 3. (b), (c) The eigenfrequencies of the circuit at $ \gamma = 0 $(left panels) and the eigenvalues of $ \mathcal{A}/C_0 $ at $ \varPhi = 0 $(right panels). Solid red dots and solid blue squares represents the theoretical results under PBCs and OBCs, respectively. The hollow diamonds in left panels label the driving frequency, and those in right panels are the simulated results under PBCs via SPICE. (d) $ \theta(\varPhi) = \ln{\det{[\mathcal{{\boldsymbol{A}}}(\varPhi)/C_0]}} $ versus $ \varPhi $ based on the simulated $ \mathcal{A}(\varPhi) $ via SPICE, where left-pointing triangles, right-pointing triangles, and solid diamonds represent points A, B, and C in panel (a), respectively. The dashed lines are the corresponding theoretical results. All figures are calculated in a finite size $ N=21 $. See relevant texts for the specific setting of the circuit's elements.

DownLoad: Full-Size Img PowerPoint

图 3 (a), (b)由SPICE模拟得到的电压响应(已经归一化)在节点上的分布, 分别对应图 2(a)中$ \alpha = 0.5 $和$ \lambda = 0.5 $的两条虚线. 频率为$ \omega = \omega_0/\sqrt{6} $的AC电流源接在第11个节点上. (c), (d)分别为由图(a)和 (b)中的电压分布根据(26)式计算的IPR, 其中菱形为模拟值, 虚线为理论值, 箭头指的是最小模拟值, 虚线标出的是理论相变值.

Figure 3. (a), (b) Node distributions of voltages (normalized) simulated by SPICE along dashed lines of $ \alpha = 0.5 $ and $ \lambda = 0.5 $ in Fig. 2(a), respectively. The AC current source with $ \omega = \omega_0/\sqrt{6} $ is connected to the 11 th node. (c), (d) IPRs of the voltage distributions in panels (a) and (b), respectively, calculated by Eq. (26), where diamonds (dashed lines) are the simulation (theoretical) results. The arrows indicate the minima of simulated IPRs, while the dashed lines indicate the phase transition points in theory.

DownLoad: Full-Size Img PowerPoint

[1]	Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249
[2]	Daley A J 2014 Adv. Phys. 63 77 Google Scholar
[3]	Moiseyev N 2011 Non-Hermitian Quantum Mechanics (1st Ed.) (Cambridge: Cambridge University Press)
[4]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[5]	Guo A, Salamo G J, Duchesne D, et al. 2009 Phys. Rev. Lett. 103 093902 Google Scholar
[6]	Peng B, Ozdemir S K, Lei F, et al. 2014 Nat. Phys. 10 394 Google Scholar
[7]	Poli C, Bellec M, Kuhl U, Mortessagne F, Schomerus H 2015 Nat. Commun. 6 6710 Google Scholar
[8]	Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2019 Nat. Commun. 10 855 Google Scholar
[9]	Ren Z, Liu D, Zhao E, He C, Pak K K, Li J, Jo G B 2021 arXiv: 2106.04874
[10]	Miri M A, Alù A 2019 Science 363 eaar7709 Google Scholar
[11]	Bergholtz E J, Budich J C, Kunst F K 2021 Rev. Mod. Phys. 93 015005 Google Scholar
[12]	Zhang X L, Jiang T, Chan C T 2019 Light Sci. Appl. 8 88
[13]	Lee T E 2016 Phys. Rev. Lett. 116 133903 Google Scholar
[14]	Leykam D, Bliokh K Y, Huang C, Chong Y, Nori F 2017 Phys. Rev. Lett. 118 040401 Google Scholar
[15]	Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402 Google Scholar
[16]	Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803 Google Scholar
[17]	Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079
[18]	Xiong Y 2018 J. Phys. Commun. 2 035043 Google Scholar
[19]	Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808 Google Scholar
[20]	Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401(R Google Scholar
[21]	Yin C, Jiang H, Li L, Lü R, Chen S 2018 Phys. Rev. A 97 052115 Google Scholar
[22]	Jin L, Song Z 2019 Phys. Rev. B 99 081103 Google Scholar
[23]	Lee C H, Thomale R 2019 Phys. Rev. B 99 201103 Google Scholar
[24]	Zhang K, Yang Z, Fang C 2020 Phys. Rev. Lett. 125 126402 Google Scholar
[25]	Yang Z, Zhang K, Fang C, Hu J 2020 Phys. Rev. Lett. 125 226402 Google Scholar
[26]	Yokomizo K, Murakami S 2020 Prog. Theor. Exp. Phys. 2020 12A102 Google Scholar
[27]	Jiang H, Lang L J, Yang C, Zhu S L, Chen S 2019 Phys. Rev. B 100 054301 Google Scholar
[28]	Longhi S 2019 Phys. Rev. Lett. 122 237601 Google Scholar
[29]	Zeng Q B, Yang Y B, Xu Y 2020 Phys. Rev. B 101 020201
[30]	Zhang D W, Tang L Z, Lang L J, Yan H, Zhu S L 2020 Sci. China Phys., Mech. Astron. 63 267062
[31]	Xu Z H, Xia X, Chen S 2021 Sci. China Phys., Mech. Astron. 65 227211
[32]	Liu Y, Wang Y, Liu X J, Zhou Q, Chen S 2021 Phys. Rev. B 103 014203 Google Scholar
[33]	Lin Q, Li T, Xiao L, Wang K, Yi W, Xue P 2021 arXiv: 2112.15024
[34]	Mu S, Lee C H, Li L, Gong J 2020 Phys. Rev. B 102 081115 Google Scholar
[35]	Zhang D W, Chen Y L, Zhang G Q, Lang L J, Li Z, Zhu S L 2020 Phys. Rev. B 101 235150 Google Scholar
[36]	Xu Z, Chen S 2020 Phys. Rev. B 102 035153 Google Scholar
[37]	Lee E, Lee H, Yang B J 2020 Phys. Rev. B 101 121109 Google Scholar
[38]	Liu T, He J J, Yoshida T, Xiang Z L, Nori F 2020 Phys. Rev. B 102 235151 Google Scholar
[39]	Wang Z, Lang L J, He L 2022 Phys. Rev. B 105 054315 Google Scholar
[40]	Gou W, Chen T, Xie D, Xiao T, Deng T S, Gadway B, Yi W, Yan B 2020 Phys. Rev. Lett. 124 070402 Google Scholar
[41]	Zeuner J M, Rechtsman M C, Plotnik Y, et al. 2015 Phys. Rev. Lett. 115 040402 Google Scholar
[42]	Weimann S, Kremer M, Plotnik Y, et al. 2017 Nat. Mater. 16 433 Google Scholar
[43]	Zhu X Y, Gupta S K, Sun X C, He C, Li G X, Jiang J H, Lu M H, Liu X P, Chen Y F 2018 arXiv: 1801.10289
[44]	Cerjan A, Huang S, Wang M, Chen K P, Chong Y, Rechtsman M C 2019 Nat. Photonics 13 623 Google Scholar
[45]	Wang K, Dutt A, Yang K Y, Wojcik C C, Vuč kovic J, Fan S 2021 Science 371 1240 Google Scholar
[46]	Brandenbourger M, Locsin X, Lerner E, Coulais C 2019 Nat. Commun. 10 4608 Google Scholar
[47]	Ghatak A, Brandenbourger M, van Wezel J, Coulais C 2020 Proc. Natl. Acad. Sci. U.S.A. 117 29561 Google Scholar
[48]	Wu F Y 2004 J. Phys. A: Math. Gen. 37 6653 Google Scholar
[49]	Schindler J, Lin Z, Lee J M, Ramezani H, Ellis F M, Kottos T 2012 J. Phys. A: Math. Theor. 45 444029 Google Scholar
[50]	Helbig T, Hofmann T, Imhof S, et al. 2020 Nat. Phys. 16 747 Google Scholar
[51]	Lang L J, Weng Y, Zhang Y, Cheng E, Liang Q 2021 Phys. Rev. B 103 014302 Google Scholar
[52]	Ezawa M 2019 Phys. Rev. B 100 081401 Google Scholar
[53]	Hofmann T, Helbig T, Lee C H, Greiter M, Thomale R 2019 Phys. Rev. Lett. 122 247702 Google Scholar
[54]	Ezawa M 2019 Phys. Rev. B 99 121411 Google Scholar
[55]	Zou D, Chen T, He W, Bao J, Lee C H, Sun H, Zhang X 2021 Nat. Commun. 12 7201 Google Scholar
[56]	Zhang X X, Franz M 2020 Phys. Rev. Lett. 124 046401 Google Scholar
[57]	Rafi-Ul-Islam S M, Siu Z B, Jalil M B A 2021 arXiv: 2102.03727
[58]	Tzeng W J, Wu F Y 2006 J. Phys. A: Math. Gen. 39 8579 Google Scholar
[59]	Lee C H, Imhof S, Berger C, Bayer F, Brehm J, Molenkamp L W, Kiessling T, Thomale R 2018 Commun. Phys. 1 39 Google Scholar
[60]	Hadad Y, Soric J C, Khanikaev A B, Alù A 2018 Nat. Electron. 1 178 Google Scholar
[61]	Wang Y, Lang L J, Lee C H, Zhang B, Chong Y D 2019 Nat. Commun. 10 1102 Google Scholar
[62]	Pozar D M 2012 Microwave Engineering (4th Ed.) (Hoboken: John Wiley & Sons, Inc.)
[63]	Helbig T, Hofmann T, Lee C H, Thomale R, Imhof S, Molenkamp L W, Kiessling T 2019 Phys. Rev. B 99 161114 Google Scholar
[64]	Lang L J, Zhu S L, Chong Y D 2021 Phys. Rev. B 104 L020303 Google Scholar

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Vol. 71, Issue 16 - 2022-08-20

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