Experimental realization of non-Abelian gauge ﬁled in circuit system

Synthetic gauge ﬁeld, especially the non-Abelian gauge ﬁeld, has emerged as a new way to 1 explore exotic physics in a wide range of materials and platforms. Here we present the build- 2 ing blocks, consisting of capacitors and inductors, to implement the non-Abelian tunneling 3 matrices and show that circuit system is an appropriate choice to realize the non-Abelian 4 gauge ﬁeld. To demonstrate the novel physics enabled by the non-Abelian gauge ﬁeld, we 5 provide a simple and modular scheme to design the Rashba-Dresselhaus spin-orbit interac- 6 tion and topological Chern state in circuits. By measuring the spin texture and chiral edge 7 states of the resonant frequency band structures, we conﬁrm the spin-orbit effect and topo- 8 logical Chern state in circuits. Our schemes open a broad avenue to study non-Abelian gauge 9 ﬁeld and related physics in circuit platform. 10

playground to investigate the non-Abelian gauge field and related remarkable physics. 48 We start with the Yang-Mills Hamiltonian with non-Abelian gauge field 28 where m is the effective mass of carrier and the gauge filed component A x and A y are Hermitian 50 matrices. Assuming that the vector potentials take the form of A x = (α + β )σ x and A y = 51 (α − β )σ y , which do not commute with each other. Substituting A x(y) into Hamiltonian (1), we 52 get the standard Rashba-Dresselhaus spin-orbit interaction (SOI) 29, 30 53 H = p 2 2m + α(p x σ x + p y σ y ) + β(p x σ x − p y σ y ) + const., where σ x(y) are Pauli matrices acting on the pseudo spin, p 2 = p 2 x +p 2 y . α = α /2m and β = β /2m 54 are the Rashba and Dresselhaus SOI constants, respectively. Above derivations explicitly show the 55 mapping between non-Abelian gauge field and SOI. By applying a pseudo-spin rotation σ x → −σ y 56 and σ y → σ x , eq. (2) is equivalent to the Rashba SOI H R = −α(p x σ y − p y σ x ) and the Dresselhaus 57 2 SOI H D = −β(p x σ y + p y σ x ) that have been used in much literature. In this paper, we investigate the implementation of the non-Abelian gauge field of the Rashba-Dresselhaus SOI type in the 59 circuit. 60 We now outline the experimental scheme, based on capacitors and inductors, which generates where ω is the frequency of the AC signal, j = √ −1, m and n are cell indexes, unit matrix, P is the permutation matrix for permuting 1 to 2, 2 to 3, .... , N − 1 to N , and N to 1.

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Matrix C mn describes the connection configuration of capacitors in C cells.

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The current conservation condition requires the sum of currents entering each node in the 74 circuit is zero, i.e. i m(n)τ = 0. Therefore, eq. (3) can be formulated in the form of a tight-binding 75 Hamiltonian (see Supplementary Information) where matrix U is chosen to diagonalize the matrices P and L, Λ = U LU † ,ṽ m,n = U v m,n , and the , and Λ = wires, which is difficult to be achieved in rigid materials. For the two-dimensional circuit network 95 given in fig. (2 a), using similar method applied to derive eq. (3) and eq. (4), we obtain Hamiltonian where ⊕ stands for matrix direct sum, h 1 (k) = 2(cos k 1 + 3 cos k 2 + cos 2k 1 + cos 2k 2 ) − 13,  This scheme can be easily generalized to 1D or 3D lattice for the study of other types of SOI and 113 related novel physics. 114 We now turn to discuss the implementation of topological circuit with a non-zero Chern num- with the requirement of opening a topological gap, we design a circuit illustrated in fig. (3 a, b).

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The main device we use here is the integrator operational amplifier, which is connected in series 123 with a capacitor and then in parallel with an inductor. The circuit depicted in fig. (3 a, b) leads to 124 a similar Hamiltonian as given in eq. (5), with h 1 (k) = 9j/(2CRω) + 8 cos k 1 + 12 cos k 2 − 23 125 and a new term d 3 (k)σ 3 in the two-by-two block Hamiltonian (see Supplementary Information)  fig. (1). (b) Schematic of a cyan cell in a. The integrator operational amplifier is marked by a green dotted box. The integrator is serially connected with a red capacitor (the capacitance is three times the capacitance of the black capacitor in the integrator) and then connected in parallel with the inductor L. Each node has a grounded resistor, which has twice the resistance compared to the resistors in the integrator. (c) Printed circuit board layout of one unit cell of the topological Chern circuit. The devices in the blue dashed box correspond to L module given in fig. (b). The green (yellow) dashed box correspond to the fig. (a). The data that support the findings of this study are available from the corresponding author upon 234 reasonable request.

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Code availability 236 The codes that support the findings of this study are available from the corresponding author upon 237 reasonable request.