Topological properties of linear circuit lattices

Victor V. Albert,∗ Leonid I. Glazman, and Liang Jiang†Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA

(Dated: October 9, 2014)

Motivated by the topologically insulating (TI) circuit of capacitors and inductors proposed and tested inarXiv:1309.0878, we present a related circuit with less elements per site. The normal mode frequency matrix ofour circuit is unitarily equivalent to the tight-binding matrix of a quantum spin Hall insulator (QSHI). Spinfulfermionic time-reversal symmetry manifests itself in the TI circuit context as a result of a discrete symmetry ofthe circuit; elastic backscattering between edge modes does not occur whenever a circuit perturbation is invariantunder such a symmetry. We make such testable predictions with regards to backscattering for both circuits. Theidea behind these models is generalized, providing a platform to simulate tunable and locally accessible latticeswith arbitrary complex spin-orbit hopping of any range. A simulation of a non-Abelian Aharonov-Bohm effectusing such linear circuit designs is discussed.

PACS numbers: 42.70.Qs, 03.65.Vf, 78.67.Pt

The realization that electrons propagating on edges of two-dimensional topological insulators at zero temperature areprotected from certain disorder [1–4] has spurred researchsimulating these and similar edge effects in artificial materials[5–8] (reviewed in [9]). The existence of edge modes whoseenergies lie within a given bulk gap of a quadratic fermionicHamiltonian can be traced to a feature of its tight-binding ma-trix [10]. Namely, a topologically nontrivial tight-binding ma-trix is characterized by having a nontrivial value of some topo-logical invariant at that bulk gap. Therefore, the problem ofengineering edge modes in artificial systems can be reduced tomaking sure that time evolution is governed by some topolog-ically nontrivial matrix. Many efforts emulate the electronicsystems that inspired us, but over time we should be able toconstruct a wider variety of systems than those readily avail-able in nature (e.g. [11]). While edge mode protection inartificial materials is often not as intrinsic or robust, these di-rections should nevertheless advance understanding and couldoffer novel applications of the materials in question.

In this letter, we discuss topologically insulating (TI) cir-cuits [12] – lattices of inductors and capacitors whose normalmode frequency matrix Ω2 mimics a topologically nontriv-ial tight-binding matrix of an electronic system. Topologicalphotonics is a rapidly emerging field and there are many pro-posals [5, 6]; here we study only inductors and capacitors withthe goal of providing the simplest building blocks that canlead to topological nontriviality. We discuss a minimal exam-ple whose Ω2 is (unitarily) equivalent to the tight-binding ma-trix of a spinful 2D electron gas in a magnetic field (see Sec.5.2 in [13]), i.e., a spin-doubled Azbel-Hofstadter model [14](deemed the time-reversal invariant (TRI) Hofstadter model[15]). Our example simulates 1/3 magnetic flux per plaquette.Such a model is (topologically) similar to the spin-doubledHaldane model lattice [16] (see Sec. 9.1.2 in [13]) that is fea-tured in the more general Kane-Mele Z2 topological insulator[1, 2]. We determine how features of such models carry overto the circuit context, summarized in a table at the end of thearticle. The first TI circuit, proposed and tested in Ref. [12],is a simple extension of our example. We further general-

ize the recipe and provide a method to construct Ω2 equiva-lent to the tight-binding matrix of a lattice of spins with arbi-trary complex spin-orbit hoppings. Notably, we show how tosimulate any U(1) hopping with a smaller circuit than that of[12], which simulated a specific U(1) hopping. This providesa platform to synthesize background gauge fields using lin-ear circuits in parallel to studies with more complex elements[17] and to intense investigations using ultracold atoms (e.g.[18–21] and refs. therein).

Figure 1. (color online) (a) Circuit diagram of a TI circuit lattice,whose normal mode frequency matrix Ω2 is equivalent to the tight-binding matrix of the spin-doubled Hofstadter model in the Landaugauge with respective ±1/3 magnetic flux per plaquette. All induc-tors (capacitors) have uniform inductance (capacitance), so colors areused for visual aid only. The lattice consists of triangular sites φφφm,n(shaded grey), each consisting of three nodes φ(µ)

m,n (µ = 0, 1, 2).The vertical inductive connection is dependent on the horizontal in-dex m and generated by the cyclic wiring permutation Vy in Eq. (1).(b) Band structure of Ω2 simulating a semi-infinite sample, i.e., avertical strip 59 sites wide with the left edge consisting of (Vy)0 per-mutations and right edge mode bands removed. Bands for the spin up(down) component of the TRI Hofstadter model are in red (blue). Foreach gap, the spin Chern number Csc [Eq. (4)] is written inside thatgap (green). The edge modes below the lowest bulk band arise be-cause of circuit edge effects [22] and are not topologically protectedbecause they do not traverse a gap.

Minimal example.—We distill the idea from [12] in theform of a simplified example [Fig. 1(a)]. The circuit con-

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sists of a lattice of sites (grey), each site consisting of threenodes. Inductors link sites to each other while capacitors cou-ple the nodes within a site. We stress that no external flux isthreaded through any loop of the circuit and the magnetic fluxof the Hofstadter model is simulated via the intersite inductivewiring. Transforming the real normal mode frequency ma-trix Ω2 into Hofstadter tight-binding form consists of group-ing degrees of freedom into vectors and performing a trans-formation to complex variables. In an ungrounded circuit,each node m,n, µ (with µ = 0, 1, 2 labeling the degrees offreedom of the site) has a time-integrated voltage φ(µ)m,n ≡´ t−∞ v

(µ)m,n(t′)dt′ associated with it [23]. Although we can just

as well use voltages since the circuit is linear, such degrees offreedom are circuit QED standard and allow for easy inclu-sion of Josephson junctions [24]. We now group the nodesat each site m,n into a vector φφφTm,n = 〈φ(0)m,n, φ(1)m,n, φ(2)m,n〉to eventually disentangle in the index µ and remove redun-dant degrees of freedom. For example, the Lagrangian con-tribution of the link between site m,n and m,n + 1 [seeFig. 1(a)] is then organized into a (kinetic) capacitive part12

∑δ=0,1 φφφ

Tm,n+δC0φφφm,n+δ and a (potential) inductive part

12 (∑

δ=0,1

φφφTm,n+δI3φφφm,n+δ−φφφTm,nVyφφφm,n+1−φφφTm,n+1VTy φφφm,n)

with In n×n identity and respective onsite/intersite couplings

C0 =1

3

2 −1 −1−1 2 −1−1 −1 2

and Vy =

0 1 00 0 11 0 0

. (1)

Above, the colors match those of the corresponding elementsfrom Fig. 1(a) and we have set a uniform capacitance of athird (for normalization) and inductance of one. The equationof motion (EOM) for φφφm,n in the lattice from Fig. 1(a) is

C0φφφm,n = −4φφφm,n + Vxφφφm+1,n + V Tx φφφm−1,n (2)

+(Vy)mφφφm,n+1 + (V Ty )mφφφm,n−1,

where Vx = I3 and 4 is the number of nearest neighbors for asite in the bulk. The three distinct powers of Vy [(Vy)3 = I3]correspond to three vertical inductive wiring permutations andmimic the Hofstadter model in the Landau gauge.

To disentangle the µ index, one can apply the discreteFourier transform F to the three nodes of each site m,n:ζζζm,n = Fφφφm,n or ζ(µ)m,n = 1√

3ei

2π3 µνφ

(ν)m,n (µ = 0, 1, 2 and

repeated indices summed). This site-preserving transforma-tion to a complex vector ζζζTm,n = 〈ζ(0)m,n, ζ

(1)m,n, ζ

(2)m,n〉 block-

diagonalizes Ω2 in µ at the expense of introducing complexnumbers. In the ζζζ basis, the simultaneously diagonal capaci-tive and inductive coupling matrices are C0 = diag(0, 1, 1),Vy = diag(1, ei

2π3 , e−i

2π3 ), and Vx = Vx. The trans-

formed circuit Lagrangian does not contain ζ(0)m,n terms (since(C0)00 = 0), so the ζ(0)m,n ≡

∑µ φ

(µ)m,n components for site

m,n are each “half” of a degree of freedom (akin to a clas-sical harmonic oscillator in the limit of zero mass) and can

be thought of as ordinary normal modes in the limit of zerocapacitance. The EOM for ζ(1)m,n, ζ

(1)?m,n = ζ

(2)m,n, treated as

independent full degrees of freedom, is

ζ(j)m,n = −4ζ(j)m,n + ζ(j)m+1,n + ζ

(j)m−1,n (3)

+ei2π3 mjζ

(j)m,n+1 + e−i

2π3 mjζ

(j)m,n−1.

These variables are linear superpositions of bosonic modes (ifthe circuit were quantized [23]) and their hopping propertiesresemble the TRI Hofstadter model in the Landau gauge, i.e.,they acquire a (simulated) Peierls phase upon a vertical hop-ping. Thus, the block-diagonal normal mode frequency matrixΩ2 =

⊕µ Ω2

µ consists of the trivial mode matrix Ω20 and the

matrices Ω21,2 forming the spin-doubled Hofstadter model.

Topological invariant.—In Fig. 1(b), the band structure ofΩ2

1 (Ω22) is plotted in red (blue), depicting slightly distorted

[22] counterpropagating edge modes. Since the Hofstadtermodels are decoupled, the pseudo-spin 〈ζ(1), ζ(2)〉 is con-served. Therefore, for a given gap, the spin-doubled Hofs-tadter model is characterized by the Z spin Chern number [4]

Csc = 12 (C1 − C2). (4)

Given an edge, the Chern numbers Cj are simply the numberof times the edge modes of Ω2

j wind around a horizontal linedrawn in the gap (Secs. 5.3.1 and 6.4 in [13]). Moreover, thequantity C = Cscmod2 determines whether there is an even orodd number of pairs of counterpropagating edge modes (thisis the invariant of the more general Z2 TI [2], a QSHI withno spin conservation). The invariant C can be explained byKramers degeneracy, which prohibits elastic backscatteringbetween counterpropagating edge modes only for odd num-bers of edge mode pairs per edge [25]. Both our example and[12] contain one gapless edge mode pair per edge and, sincepseudo-spin is conserved, constitute a QSHI.

Due to the invariants established above, there must existsome operator in the circuit context that mimics the antiuni-tary electronic time-reversal operator iσ2K (with Ki = −iKand σ1,2,3 the usual Pauli matrices), squares to −I2, and gen-erates a Kramers degeneracy (a similar observation has beenmade [8] with photonic topological insulators [7]). Such anoperator does indeed exist and comes about from a symmetryof the circuit. In the φφφ basis, the coupling matrix Vy , a cyclicpermutation of all nodes in each site, commutes with Ω2 andgenerates the symmetry group C3 ≈ I3, Vy, V †y . A genericlinear commuting operator (with identity components in thedimensions indexed by m,n) can be expressed as cµ(Vy)µ

for some cµ=0,1,2 ∈ C. Since Vy and K commute, all an-tilinear extensions of the above operators can be expressed ascµ(Vy)µK. In the ζζζ basis,

K → K = F †KF = F †F ?K = (1⊕ σ1)K,

which squares to 1. However, the operator S [such thatS = (1 ⊕ σ2)K and S2 = 1 ⊕ (−I2)] is also in the spanof (Vy)µK. Thus, electronic time-reversal symmetry in the

3

tight-binding context maps to a combination of ordinary time-reversal and cyclic permutations in the circuit context. It isworth noting that Σ = SK = 1 ⊕ (−iσ3) characterizes theconserved pseudo-spin for the two Hofstadter copies.

Symmetry protection.—Mirroring topological protection inQSHIs and Z2 TIs, counterpropagating edge modes of a TIcircuit must also be “protected” to some degree. Emulatingone-particle elastic scattering processes in TRI electronic sys-tems [25], a crossing between edge modes on the same edgeat time-reversal invariant points k = 0, π in the Brillouin zonewill not be lifted by inductance or capacitance perturbationsthat commute with S (which is now in the φφφ basis). For thisexample, such perturbations are all those which do not breakthe C3 symmetry, i.e., commute with Vy (with the examplefrom [12] proving more interesting). Specifically, let a genericinductive link between sites m,n and p, q be parametrized by

φφφTm,nM11φφφm,n+φφφTp,qM22φφφp,q+φφφTm,nM12φφφp,q+φφφ

Tp,qM

T12φφφm,n,

(5)where real 3× 3 matrices Mjj (j = 1, 2) are onsite couplingsat the two respective sites and M12 is the intersite coupling.Such a perturbation will not cause elastic backscattering be-tween edge modes whenever [Mjj′ , S] = 0. For example, anidentical simultaneous perturbation of all three inductances inany given link [Mjj ∝ I3, M12 ∝ (Vy)µ] or an onsite per-turbation (Mjj′ ∝ δj1δj′1[(Vy)µ + (V T

y )µ]) will not mix edgemodes. A similar scheme holds for capacitive perturbations.Since Mjj′ are real, the set of perturbations that commuteswith S also commutes with conserved pseudo-spin Σ = SK.This design is thus similar to optical resonator designs [6] inthat both are robust against disorder that does not induce flipsof pseudo-spin (as has been noted in [9] for the case of [6]).

In short, the protection of the edge modes is not intrinsiclike in the electronic context, but is rather a byproduct of thecircuit symmetry Vy and spinless time-reversal K. Therefore,although this system is mathematically equivalent to a QSHI,it is physically reminiscent of topological crystalline insula-tors (TCIs) [26]. In this case however, the necessary sym-metry is a local permutation of nodes as opposed to a globalspatial reflection. As a result, edge modes should exist for allsurface terminations vs. only those respecting the reflectionsymmetry. Another difference is that, in a circuit TI, a pertur-bation may break the permutation symmetry without causingbackscattering. Such perturbations exist in the original TI cir-cuit [12].

Comparison with [12].—We digress to comment that [12]is similar to the above example, except that the number ofnodes per site d = 4 instead of 3 and the role of inductancesand capacitances is switched. With these differences, Ω2 willbe equivalent to two sets of zero-frequency modes (ζ(0)n,m andζ(1)m,n, akin to classical harmonic oscillators in the limit of zero

spring constant) and the inverse of the TRI Hofstadter tight-binding matrix with one-fourth magnetic flux per plaquette(ζ(2)n,m and ζ(3)m,n). Instead of Vy generating C3, the 4 × 4 in-tersite capacitive coupling Uy generates C4. The antilinearoperator S [with S = (I2 ⊕ σ2)K and S2 = (I2 ⊕ −I2)]

lies in the span of (Uy)µK (with µ = 0, 1, 2, 3), revealinga Kramers degeneracy for the conjugate Hofstadter models.Perturbations, of the form of Eq. (5) but this time with 4 × 4matricesMjj′ , do not cause edge modes to backscatter as longas [Mjj′ , S] = 0. The set of real matrices commuting withS is spanned by six elements, which include the four pow-ers of the symmetry generator Uy . The two additional pertur-bations, rather contrived when written in the φφφ basis, do notcommute with Uy and can be explained by the fact that thezero-frequency components ζ(0)m,n, ζ

(1)m,n can be coupled with-

out affecting ζ(2)m,n, ζ(3)m,n.

Generalizations.—The above d = 3, 4 circuits are easy togeneralize to models emulating a TRI Hofstadter model with1/d magnetic flux using d nodes per site. However, represen-tation theory reveals much more compact designs for both TIcircuits and more general systems.

First, an arbitrary complex hopping can be achieved usingonly three nodes per site. For simplicity, we first focus on onelink. Instead of having one wiring permutation (e.g. Vy inFig. 1), one can implement all three permutations (Vy)µ in alinear superposition [Fig. 2(a)]. In this case, each permuta-tion gains its own degree of freedom. The intersite inductivecoupling matrix is then Vy → VA = `

(µ)inv (Vy)µ, where `(µ)inv

is the inverse inductance of permutation µ. In the ζζζ basis, thecoupling is diagonal with (VA)µν = `

(τ)inv e

i 2π3 τνδµν (no sumover ν). Parameterizing the µ = 1 component in terms of anamplitude/phase obtains (VA)11 = tAe

iθA with

tA =

√[`(0)inv − 1

2 (`(1)inv + `

(2)inv )]2 + 3

4 (`(1)inv − `

(2)inv )2

θA = tan−1( √

3(`(1)inv − `

(2)inv )

2`(0)inv − (`

(1)inv + `

(2)inv )

). (6)

Naturally, (VA)00 =∑µ `

(µ)inv ≡ λA and (VA)22 = tAe

−iθA .Additionally, there is a diagonal inductance contribution of12λAζζζ

†ζζζ to both of the connected sites (which reduces toλA = 1 for one permutation with unit inductance). Thus, thehopping and diagonal terms tA, θA, λA can be tuned using`(µ)inv 2µ=0 with the constraint λA ≥ tA since `(µ)inv ≥ 0. Thesymmetry protection still holds for this design since all (Vy)µ

are elements of C3.Second, non-Abelian couplings can straightforwardly be

implemented in this scheme. Instead of using the permu-tations (Vy)µ, three other permutations P (Vy)µ [with P =1 ⊕ σ1 and [P, Vy] 6= 0; see Fig. 2(b)] can be superimposedto give an inverse inductance coupling matrix Vy → VNA =

`(µ)inv P (Vy)µ. Nonzero entries of VNA include an off-diagonal

hopping (VNA)12 = `(µ)inv e

−i 2π3 µ ≡ tNAeiθNA and a diagonal

contribution (VNA)00 =∑µ `

(µ)inv ≡ λNA. Similar to VA, the

hopping and diagonal terms tNA, θNA, λNA of VNA canbe tuned using `(µ)inv 2µ=0. As an example, one can alreadyrealize the Hofstadter moth model [19], a non-Abelian gen-eralization of the Hofstadter model, by letting the horizontalcoupling Vx = I3 → P for the lattice in Fig. 1(a).

4

Figure 2. (color online) (a) Generalized linear superposition of threedifferent wiring permutations (Vy)µ and their respective inverse in-ductances `(µ)inv , µ = 0, 1, 2 (solid, dashed, dotted respectively). Sucha combination achieves any U(1) hopping in the ζζζ basis. (b) Threeadditional wiring permutations P (Vy)µ which create U(2) hoppingterms in the ζζζ basis. (c) A circuit to simulate the Aharonov-Bohm(AB) effect. A vector signal φφφin enters from the left, propagatesthrough N sites via two different paths A and B, and produces twooutputs, φφφA,B . One can measure the interference between these out-puts [Eq. (8)] and observe oscillations for evenN since permutationsVy and P do not commute.

The above design allows one to create a lattice withspatially nonuniforn noncommuting unitary hoppings be-tween sites [e.g. tm,n exp(iθm,n) using either (Vy)µ orP (Vy)µ] while still maintaining identical diagonal contribu-tions (λm,n ≡ λ). Despite this apparent flexibility, one can-not create arbitrary U(2) hoppings using only three nodes persite (assuming that diagonal contributions are to remain iden-tical). The reason for this is that linear superpositions of thesix permutations [(Vy)µ and P (Vy)µ] multiplied by nonneg-ative real coefficients (since our variables are inverse induc-tances) do not span the space of unitary 2 × 2 matrices act-ing on 〈ζ(1), ζ(2)〉. More permutations are needed, meaningthat one needs more nodes per site to generate them. Findingthis minimal number of nodes maps to an open problem fromgroup theory [27, 28], and we have numerically determined[29] that one needs 8 (9, 16, 25, 13) nodes per site in order tosimulate unitary hoppings of dimension 2 (3, 4, 5, 6).

Non-Abelian Aharonov-Bohm effect.—We finish with a dis-cussion of applications. First we propose an experiment thatuses the φφφ-ζζζ duality to observe an electrical non-AbelianAharonov-Bohm (AB) effect [19, 20, 30]. An example of theAB effect is the interference exhibited by two wavefunctionsdue to the differing phases gained during traversal of their re-spective spatial paths. Since all circuit elements are reciprocalhere, it is the non-reciprocity of their permutations that leadsto interference effects. One can think of φφφ as the wavefunc-tions and sites n = 1, 2, ..., N as spatial positions [Fig. 2(c)].An incoming signal φφφTin = 〈φ(0)in , φ

(1)in , φ

(2)in 〉 is applied onto

paths A and B. Let

φ(µ)in =

√23 cos(ωt− 2π

3 µ), (7)

which is equivalent to ζζζTin = 1√2〈0, eiωt, e−iωt〉. Path A con-

tains N − 1 cyclic permutations Vy from Eq. (1) while pathB consists of N − 1 permutations P from Fig. 2(b) (with[Vy, P ] 6= 0). Remembering Eq. (3), we see that a phaseof ei

2π3 (e−i

2π3 ) is gained by ζ(1) (ζ(2)) as the signal “hops”

sites in path A. For path B, the ζ(1) and ζ(2) componentsare exchanged upon each application of P . One can super-impose the outputs φφφA and φφφB to observe their interference.For odd N , this interference is constant in time. For even N ,one should see oscillations due to a nontrivial path B:

|φφφA +φφφB |2 ∝ cos2ωt− 2π3 [(N − 1) mod3]. (8)

Since voltage is the derivative of φ, one can perform the aboveexperiment by applying voltage signals of the form ofφφφin fromEq. (7), measuring the six output signals at site N for pathsA and B, and superimposing them in the manner of Eq. (8)[31]. Since the AB effect is nonreciprocal, driving from rightto left (φφφin ↔ φφφA,B) should flip the sign of the phase gainedalong A.

Outlook.—This work has put some of the ideas from thefirst proposal of a topologically insulating (TI) circuit [12] ona more theoretical footing. We have outlined that the circuitnormal mode frequency matrix of [12] is unitarily equivalentto the tight-binding matrix of the time-reversal invariant (TRI)Hofstadter model [15] with 1/4 magnetic flux per plaquetteand presented a simplified example that simulates 1/3 fluxper plaquette. Since Hofstadter models posses edge modes,we have determined which perturbations will not cause edgemodes to backscatter. The analogy between the time-reversalinvariant Hofstadter model and the example here can be sum-marized in the following table:

TRI Hofstadter model TI circuitTight-binding matrix Normal mode frequency matrix Ω2

Fermionic mode cm,n ζ(1)m,n = ei

2π3νφ

(ν)m,n at site m,n

Peierls phase Intersite wiring permutationsKramers degeneracy S = (1⊕ σ2)K due to C3 symmetry

In the above table, φ(µ)m,n is the integrated voltage at nodem,n, µ as depicted in Fig. 1(a) and Ki = −iK.

The construction presented here allows normal mode ma-trices of linear circuits to emulate tight-binding Hamiltonianswith nontrivial values of topological invariants. We have ap-plied representation theory to determine the minimal circuitcomplexity required for non-Abelian background gauge fieldsin such systems. Besides proposing a way to simulate theAharonov-Bohm effect, we now speculate on other applica-tions of this circuit QED simulation tool [32]. A major flex-ibility is being able to construct and locally probe virtuallyany lattices (e.g. honeycomb [21] or Kagome [33]) and lat-tices with connections other than nearest neighbor at the samecost in complexity. Almost any physically relevant and exotic

5

geometry can be implemented [34]; a Mobius strip geometrywas tested in [12]. One can construct interfaces of lattices ofdifferent simulated magnetic fluxes per plaquette and observemixing of edge modes at the boundary, akin to graphene p-njunctions [35]. To simulate interactions, one can substitute in-ductors in the example here for Josephson junctions [24] orsubstitute one plate of each capacitor in the circuit from [12]for a mechanical oscillator [36]. These and other topics arecurrently under investigation.

The authors thank A. Dua, T. Morimoto, B. Elias, W. C.Smith, S. M. Girvin, M. H. Devoret, B. Bradlyn, Z. Minev, andA. Petrescu for fruitful discussions. This work was supported,in part, by NSF GRFP (VVA); NSF DMR Grant 1206612(LIG); and ARO, IARPA, AFOSR MURI, DARPA Quinessprogram, the Alfred P. Sloan Foundation, and the PackardFoundation (LJ).

∗ valbert4@gmail.com† liang.jiang@yale.edu

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Supplemental material: Topological properties of linear circuit lattices

Victor V. Albert, Leonid I. Glazman, and Liang JiangDepartments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA

(Dated: October 5, 2014)

Here we outline the problem of finding the minimal number of nodes per site (d) for circuit lattices such that one can simulateunitary hopping matrices of dimension n.

Returning to the n = 2 and d = 3 case, the coupling matrices of the six permutations (Vy)µ, P (Vy)µ (for µ = 0, 1, 2) in theφφφ basis form a permutation representation of the symmetric group S3, i.e., each matrix has one “1” in each row and column.(In general, the group of permutative connections between two sites, with each site containing d nodes, is the symmetric groupSd.) This permutation representation is reducible, i.e., all matrices can be simultaneously block diagonalized into matriceswith smaller nonzero blocks. This is exactly what is done by the Fourier transformation into the ζζζ basis. In other words, thepermutation representation of S3 reduces to a trivial one-dimensional and the desired two-dimensional irreducible representation(irrep). The one-dimensional part acts on ζ(0) while the two-dimensional part acts on the pseudo-spin 〈ζ(1), ζ(2)〉. Upondiagonalization, the onsite capacive coupling C0 isolates the two-dimensional irrep (since (C0)00 = 0).

On the other hand, if one uses only nonnegative coefficients (since our variables are inductances), an arbitrary U(2) matrix(in fact, any 2 × 2 matrix) can be written as a linear superposition of the 16 elements of a 2 × 2 irrep of the Pauli groupP1 ≈ iτI2, iτσ1, iτσ2, iτσ33τ=0. To realize arbitrary U(2) hoppings, one therefore needs to find the smallest symmetric groupSd that contains a permutation representation of P1 as its subgroup (formally, the minimal faithful permutation representation,or MFPR, of P1). This MFPR will then be reduced to the irreps of P1 via a transformation akin to that from the φφφ to the ζζζ bases.Finally, the onsite capacitive coupling can be tuned to select the desired irreps. A quick calculation [1] determines that one needsa circuit with d = 8 nodes per site in order to construct arbitrary U(2) hopping terms in some ζζζ basis.

By a similar procedure, one can realize U(n > 2). To construct arbitrary n-dimensional unitary matrices using only realnonnegative coefficients, one first expresses the desired n× n complex matrix M using the n3 elements of an n× n irrep of theGeneralized Pauli Group Πn [2]. In such an irrep, an element gjkl = wjXkZl ∈ Πn where j, k, l = 0, ..., n− 1; w = ei

2πn ;

X =

0 1 · · · 0... 0 1

...

0. . . 1

1 0 0 0

and Z =

w0

w1

. . .wn−1

.

To express M in terms of gjkl, notice that M can be expanded in powers of X and Z, M = cklg0kl, with complex coefficientsckl = 1

nTrg†0klM. Each ckl can then be expressed as ckl = djklwj with real djkl ≥ 0 such that M = djklgjkl; we note that

this decomposition is not unique. Each gjkl corresponds to a specific permutation of inductors and djkl is the inverse inductanceof all of the inductors involved. One then needs to determine the MFPR of Πn. This MFPR can then be reduced to irreps of Πn

and the capacitive coupling C0 among nodes in a site can be tuned to select the desired irreps, much like C0 had done for then = 2 case above Eq. (3) in the main text. We are fortunate in this case to not have any undesired nontrivial irreps since theirreps of Πn are either variants of the defining n × n irrep or a trivial 1 × 1 irrep [2]. Using an algorithm [3] implemented inMAGMA [4], we have determined the following dimensions d of the MFPRs of Πn.

n 3 4 5 6 7 8 9 10 11d 9 16 25 13 49 64 81 29 121

The results leads us to postulate that the upper bound for d is n2 (versus the usual bound of the group order, n3, due to Cayley’stheorem). This is in agreement with previous results (Sec. 4.2 of [3]), which found this bound for prime n.

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