Networks of quantum wire junctions: a system with quantized integer Hall resistance without vanishing longitudinal resistivity
NNetworks of quantum wire junctions: a system with quantized integer Hall resistancewithout vanishing longitudinal resistivity
Jaime Medina,
1, 2
Dmitry Green ∗ , and Claudio Chamon Facultad de Ciencias, Universidad Aut´onoma de Madrid, 28049 Cantoblanco, Madrid, Spain Physics Department, Boston University, Boston, MA 02215, USA
170 East 83rd Street, New York, NY 10028, USA (Dated: November 2, 2018)We consider a honeycomb network built of quantum wires, with each node of the network havinga Y-junction of three wires with a ring through which flux can be inserted. The junctions are thebasic circuit elements for the network, and they are characterized by 3 × R xy = h/e , although thelongitudinal resistivity is non-vanishing. We show that these results are robust against disorder, inthis case non-homogeneous interaction parameters g for the different wires in the network. PACS numbers:
I. INTRODUCTION
The transport properties of junctions of quantum wiresare of interest both seen from basic and applied perspec-tives. From the basic physics aspect, quantum wires pro-vide experimentally realizable ways for studying inter-acting electrons in one-dimensional geometries, and inparticular junctions where 3 or more wires meet can dis-play rather rich behaviors. Theoretically, the problem ofquantum wire junctions is related to dissipative quantummechanics in two or higher dimensions, and to bound-ary conformal field theory.
It also has a mathematicalconnection to certain aspects of open string theory ina background magnetic field.
From a practical view-point, junctions of quantum wires should serve as im-portant building blocks for the integration of quantumcircuits, as they are the natural element to split electricsignals and serve as interconnects.Junctions of quantum wires have been the subject ofmany recent studies, which have uncovered manyinteresting transport properties as function of interac-tion strength. Quantum wires with few transport chan-nels, at low energies, can be described as Tomonaga-Luttinger liquids, characterized by a Luttinger parameter g which encodes the electron-electron interactions. The transport properties of a given junction depends onthe Luttinger parameters for each wire. At low energies,the conductance properties of the junctions of n wires areencoded in an n × n conductance tensor or matrix G jk that relate the incoming currents to the applied voltageson the wires via I j = (cid:80) k G jk V k . At low voltages andlow temperatures, the tensor takes universal forms dic- ∗ [email protected] tated by the nature of the infrared stable fixed pointsin the renormalization group (RG) sense. These fixedpoints have been categorized for the case of Y-junctions( n = 3) of spinless and spinful electrons as function ofthe interaction parameter g when all the wires are iden-tical, and more recently in the case when the wires arenot identical and have different values g i . FIG. 1: (a) Scheme of a grid showing the flow of the currentand the boundary conditions. External currents are fixed, aswell as the potential on the node on the upper right corner.(b) Building block of the grid: junction of three quantumwires with a magnetic flux threading the ring. The V , , arethe voltages applied on each wire, and the I , , the currentsarriving at the junction from each of the three wires. In this paper we investigate the transport propertiesof networks constructed using Y-junctions of quantumwires as building blocks. Fig. 1a depicts an example of anetwork shaped in the form of a rectangle, and Fig. 1bshows the individual Y-junctions used in each node. Weconsider a simplified model where the 3 × a r X i v : . [ c ond - m a t . s t r- e l ] F e b the junction itself, for example the size of a ring as shownin Fig. 1b, should be smaller than the dephasing lengthso that the junction is treated quantum mechanically.The case when the full system is treated quantum me-chanically is extremely difficult to analyze, because it isan interacting problem. For instance, a lattice versionof the problem would essentially be an example of a two-dimensional interacting lattice model with a fermion signproblem.We find rather remarkable results for the transportcharacteristics of the network of Y-junctions, even whenthe role of quantum mechanics is just to select the RGstable fixed point conductances of the elementary build-ing blocks. When the conductance is controlled by thechiral fixed points χ ± , we find that the whole networkbehaves as a Hall bar, with a Hall resistance that is quan-tized to R xy = ± h/e , like in the integer quantum Halleffect, with the sign given by the particular chirality ofthe fixed points χ + or χ − . However, the longitudinalresistivity ρ xx (cid:54) = 0, unlike in the case of the quantizedHall effect where ρ xx vanishes. The quantization of R xy is a manifestation of the universal fixed point conduc-tances. The chiral fixed points are stable for a range ofLuttinger parameters 1 < g <
3, and which of χ + or χ − is selected depends on the flux threading the ring inthe Y-junction. The flux breaks time-reversal symme-try, but it does not need to be quantized at any givenvalue; because of interactions, the conductance of the Y-junction flows to fixed point values for a range of fluxes.The quantization of R xy = ± h/e for the network asa whole is independent of the value of g in the wires,as long as they are in the range of stability of the chi-ral fixed points. Moreover, we show that the quanti-zation R xy = ± h/e is stable against disorder in thewire parameters. Specifically, we show that the quan-tization of R xy remains even when the values of g fordifferent wires are not uniform but disordered, i.e. , theyare randomly distributed around some average value g with some spread δg .The paper is organized as follows. In Sec. II we brieflyreview the results for the conductance characteristics ofsingle quantum Y-junctions, which are the elementarybuilding blocks for the honeycomb wire-networks. InSec. III we present analytical results from which one canunderstand the origin of the quantization of R xy whenthe conductance tensor of each of the Y-junctions in thenetwork is associated to a chiral fixed point. In Sec. IV wepresent numerical studies confirming the analytical find-ings by analyzing grids with different values of the inter-action parameter g , different geometries and sizes, andextrapolate these results to the thermodynamic limit.These numerical calculations are of much value for thenext step, taken in Sec. V, where we discuss the robust-ness of the quantization of the Hall resistance in the casewhen the wires each have different Luttinger parametersdistributed randomly. The Appendix contains a detaileddescription of the numerical method to solve our networkof Y-junctions. II. SINGLE Y-JUNCTION AS ELEMENTARYCIRCUIT ELEMENT
Each of these Y-junctions in the network consists ofthree wires that are connected to a ring which can bethreaded by a magnetic flux, as shown in Fig. 1b. Thisflux breaks time-reversal symmetry, and the currents inthe junction will depend on the potential at its extremesand the magnetic flux inside the junction.The current-voltage response of each Y-junction is de-termined by its conductance tensor G jk . Within linearresponse theory, the total current I j flowing into the junc-tion from wire j is related to the voltage V k applied towire k by I j = (cid:88) k G jk V k (2.1)where j, k = 1 , ,
3. Two sum rules apply to the con-ductance tensor because of conservation of current andbecause the currents are unchanged if the voltages are allshifted by a constant: (cid:88) j G jk = (cid:88) k G jk = 0 . (2.2)The G jk reach universal values at low temperaturesand low bias voltages. These universal values are dictatedby the RG stable fixed point that is reached for givenvalues of the Luttinger parameters in the wires. Here weshall focus on the case where all the three wires have thesame parameter g . In Sec. V we will consider the moregeneral case of network of wires where the three wires foreach Y-junction have different g ’s.When the three wires have the same g , the fixed pointconductance tensor has a Z symmetry and takes theform G jk = G S δ jk −
1) + G A (cid:15) jk , (2.3)where (cid:15) ij = δ i,j − − δ i,j +1 with i + 3 ≡ i and we separatethe symmetric and anti-symmetric components of thetensor, whose magnitudes are encoded in the scalar con-ductances G S and G A . G A vanishes when time-reversalsymmetry is not broken, for instance in the absence ofmagnetic flux through the ring.The fixed point values of G S and G A depend on thestrength of electron-electron interactions, encoded in theLuttinger parameter g . We will focus on the chiral fixedpoints χ ± , which are stable in the range 1 < g < In the chiral cases, the conductances are given by G S = G χ = e h g g and G A = ± g G χ . Thus the chiral conduc-tance tensors are: G ± jk = G χ δ jk − ± g(cid:15) jk ] . (2.4)We shall work in units where the quantum of conductance e /h is set to 1.The Y-junctions are then assembled into a network asshown in Fig. 1a. We consider a regular hexagonal gridof Y-junctions with 2 c external connections on both thetop and bottom sides and r on both the right and leftside. Parametrized in such a way and with wires of unitlength, the dimensions of the grid as a function of r and c are L x = 6 cL y = √ r + 1) . (2.5)In this grid we shall fix the current flow along the x -axis from left to right and we shall fix the currents flowinginto the top and the bottom to zero, as shown in Fig.1a. Given the conductance tensors at every node of thenetwork, we compute the potentials and the currents onthe links of the grid. The resistances and resistivities ofthe networks are studied for different orientations andsystems sizes, and for different values of g . In appendixA we present details of the method used to numericallycompute the response of the networks. III. ANALYTICAL RESULTS
We will measure the longitudinal and transverse re-sponses in the framework of the classical Hall problemby injecting a transverse current along the x -axis andimposing a zero current boundary condition along thetwo edges that are parallel to the x -axis. This approachsuggests that we solve for the potential in the bulk as afunction of the external current. In other words we needto invert the fundamental equation (2.1) for I and V foreach junction in the bulk.While the full network problem is not tractable analyt-ically, we can still gain some insight from a combinationof analytics and heuristics. In particular we will be ableto prove quantization of the transverse resistivity ana-lytically, even with some forms of disorder. Similarly wewill derive the general form of the longitudinal resistiv-ity. We will confirm these results numerically in latersections. Let us start with the unit cell of the hexagonallattice. There are two vertices (nodes) in each cell andcurrent is directed along the bonds (wires) as shown inFig. 2. Looking at the right-hand node first, the poten-tials on the external wires, V and V , and the potentialon the internal wire V are defined only up to an additiveconstant. This means that Eq. (2.1) is not invertible.However, by setting V = 0, or equivalently shifting allpotentials in the two nodes by a constant V i → V i − V ,the gauge is fixed and we obtain, using Eq. (2.4), thefollowing: (cid:18) V − V V − V (cid:19) = 12 g (cid:18) ∓ g ± g (cid:19) (cid:18) I I (cid:19) . (3.1)The solution in the left node is similar but with the per-mutation ( V , V ) → ( V (cid:48) , V (cid:48) ) and ( I , I ) → − ( I (cid:48) , I (cid:48) ), V VV
VA DCBII I II
23 1 32
FIG. 2: Unit cell of the hexagonal network. Currents areassumed to be positive when directed along the arrows in thewires. Dotted lines denote the boundary of the unit cell. Therectangular region
ABCD shown is used for computing theresistances and resistivities of the network. which follows from rotational symmetry and the orienta-tion that we have chosen for the currents.Now consider the potential gradient in the x − and y − directions. It is straightforward to derive the changein potential per unit cell, ∆ V x and ∆ V y , directly fromEq. (3.1) as follows:∆ x V = V (cid:48) − V = 12 g [2 I − I − I (cid:48) ∓ g ( I (cid:48) − I )]∆ y V = V − V = 12 g [ ± gI + I − I ] . (3.2)It is instructive to consider a simple case. We will gen-eralize this result below, but for now consider a uniformcurrent in the bulk in the x − direction (or “armchair”configuration to borrow nomenclature from graphene).Each horizontal wire in each unit cell has a current I = I . By symmetry the other wires split the currentequally: I = I = I (cid:48) = I (cid:48) = − I/
2. This configu-ration leads to a particularly simple potential gradient:∆ x V = 3 I/ g and ∆ y V = ± I/ ABCD in Fig. 2, with sides d AB = √ d AD = 2. In the transverse directionthe width of the rectangle is twice the distance betweenthe midpoint of the wires (with currents I and I ), andthe voltage drop V AB = 2 ∆ y V . The Hall resistance(which coincides with the Hall resistivity ρ xy ) is therefore R xy = V AB /I = 2 ∆ y V /I = ±
1. In other words the Hallresistance is independent of g and quantized to unity!Similarly, in the longitudinal direction the length ofthe rectangle is 4/3 the distance between the midpoint ofthe wires (with currents I (cid:48) and I ), and V AD = 4 / x V .The longitudinal resistance is R xx = 4 / x V /I = 2 /g .There is an additional geometric factor in the longitu-dinal resistivity given by ρ xx = ( d AB /d AD ) R xx , and itis thus given by ρ xx = √ /g . Hence the resistivity isnon-zero and there is dissipation unlike in the standardquantum Hall effect.Had we used an alternate (“zigzag”) configurationwhere the transverse current is zero I = 0 and the uni-form current is in the y − direction, I = I (cid:48) = − I = − I (cid:48) = I , we would have found a similar result, i.e. ,that the resistance in the x − direction is quantized to R xy = ± y − direction is ρ yy = √ /g .We find this result both unexpected and remarkable.By taking the classical conductivity limit for each wire wehave allowed decoherence along the wires. However wehave preserved the quantum coherence on each vertex, asthe chiral relation Eq. (2.4) is by nature a consequenceof quantum scattering. Nonetheless even after relaxinga portion of the coherence, some element of quantizationin the thermodynamic limit has survived in the form ofan integer quantized Hall resistivity. On the other hand,decoherence has destroyed the zero longitudinal resistiv-ity of the quantum Hall effect, and so we are left witha hybrid quantum-classical Hall effect. Note also thatthe simple uniform solution above suggests robustnessagainst disorder, another element of the integer quantumHall effect. As the transverse gradient of V is indepen-dent of g in the uniform bulk, suppose that g is allowedto vary slowly from vertex to vertex, more slowly thanthe current. In this regime we would expect quantizationto persist, and indeed we will confirm that numericallylater in this paper.We will substantiate the assumptions and findingsabove numerically in the next section. IV. NUMERICAL RESULTS
In this section we shall present numerical results for thevoltages and currents in the wires of the network. Thesenumerical studies serve first as a check of the analyti-cal results presented in the previous section III for thecase where all the interaction parameters are the samefor all wires. Second, and more importantly, they serveas a stepping stone to the case of non-homogeneous (dis-ordered) interaction parameters in the wires, which willbe considered in Sec. V. The method used to solve forthe voltages and currents in the grid is presented in Ap-pendix A.Let us focus on the armchair layout of Fig. 1a (simi-lar results follow in the case of the zigzag case). Also,without loss of generality, we consider below only the χ + fixed point. Current is injected and collected uniformlyinto the wires on the right and on the left of the network,respectively. More precisely, there are r wires servingas connections to the outside on each side of the grid,and current I = I x /r is injected in and collected out ofthese external wires. The total current flowing along the horizontal or x -direction is therefore I x .The distribution of the currents in the inner parts ofthe grid that follow from this uniform injection of exter-nal currents is shown in Fig. 3. We find a close to uni-form distribution, with slightly larger currents closer tothe edges. This distribution is independent of the valueof g . These patterns of current flow in the inner wires ofthe grid are in agreement with the current distributionsdiscussed in the analytical studies of the previous section. FIG. 3: Currents flowing through the Y-junctions that liealong a vertical line in the middle of the bar ( x = L x /
2) as afunction of vertical position y/L y . Note that for y values awayfrom the edges the currents tend to I = 1 and I , = 1 /
2, aspredicted analytically for the asymptotic limit.
The Hall voltage is the potential drop V y along the ver-tical or y -direction. We note that the potential drop V y is computed by looking at the potentials for two points atthe same horizontal position ( i.e. , the same x position),one at the top and one at the bottom of the network.We show in Fig. 4 the potentials measured at the topand at bottom of the (rectangular shaped) grid. No-tice that the potentials drop linearly with the horizontaldirection, but that the difference between the two poten-tials, V y , is constant.The Hall resistance is computed as follows. Let ¯ V y bethe average over the horizontal positions x of the Hallvoltage drop. (Since in this case without disorder V y isconstant, the average is actually unnecessary here.) Thenthe Hall resistance is given by R xy = ¯ V y /I x . We findnumerically that R xy = 1 as expected from the analyticalarguments. Recall that we are working in units where e /h = 1, so indeed we have R xy = he , (4.1)which we find is independent of the value of g . We remarkthat we find that this quantization holds independent ofthe aspect ratio, orientation (armchair vs. zigzag) or sizeof the grid.We also computed the potential difference betweenpoints on the left and on the right sides of the grid, V x ,as a function of the vertical direction y . In this casewe find that the horizontal potential difference is almostconstant as function of y (as opposed to the case of thevertical drop V y , which is exactly independent of x ). Thedifference is bigger, by an amount of order 1 /L y , when y is in the middle of the grid as compared to when y is atthe edges. We define ¯ V x as the y -position averaged volt-age difference between the left and right sides of the grid.The longitudinal resistance is given by R xx = ¯ V x /I x , andthe longitudinal resistivity by ρ xx = L y /L x ¯ V x /I x .We find that the longitudinal resistance is non-zero, inagreement with Sec. III. We find numerically, however,that there are finite system size corrections to the ana-lytical predictions. We find that R xx ( g, L x , L y ) = √ g L x L y − A ( L x , L y ) , (4.2)where A is a factor of order 1 that corrects for finite sizes.We find numerically that in the thermodynamic limit A → A = 0independent of system size in the zigzag case. Therefore,in the thermodynamic limit we obtain ρ xx = lim L x ,L y →∞ L y L x R xx ( g, L x , L y ) = √ g , (4.3)in agreement with the result in Sec. III.The Hall angle θ H is given by tan θ H = ρ xy /ρ xx , andwe naturally find, given the agreement with the resultsfor ρ xx and ρ xy above, thattan θ H = g √ g ; this isno longer the case when disorder is introduced in the nextSec. V. V. ROBUSTNESS AGAINST DISORDER
In this section we will generalize the wire networks tothe case when the interaction parameters g for each ofthe wires in the network are not uniform, but insteadare drawn independently from a distribution. We shallconsider a distribution in which g in each of the wires inthe network takes a value between (¯ g − δg, ¯ g + δg ), withuniform probability. Because the interaction parametershould be positive, δg < ¯ g . FIG. 4: Voltages at the top and and bottom edges as functionof horizontal position x/L x when the node at the top rightcorner is grounded. The grid size is r = 50, c = 60 and g = √
3. Notice that the difference between the voltages atthe top and bottom edges for a given x/L x is exactly 1 innatural units.FIG. 5: Density plot for the voltages on the grid nodes, for asystem with r = 50, c = 60 and g = √
3. Notice the constantslope of the equipotential lines, which is related to the Hallangle θ H . The Hall angle depends on the interaction strengthand is given by Eq. 4.4. When the wires connecting to a given Y-junction havedifferent values of g , the conductance tensor G ij for achiral fixed point is no longer given by Eq. (2.4), butinstead it takes the form (see Ref. 28) G jk = 2 g j ( g + g + g ) δ jk + g j g k ( ± g m (cid:15) jkm − g g g + g + g + g . (5.1)Using this conductance tensor, one can compute numeri-cally (using the method of Appendix A) the voltages andcurrents in all wires of the network for a given realizationof the disorder.We shall show below that the quantization R xy = 1of the Hall conductance that we found in the clean limitremains , in the thermodynamic limit, even in the pres-ence of disorder. For a finite lattice, as one should ex-pect, there are fluctuations that we quantify below forthe armchair configuration.We compute Hall resistance R xy (defined as the aver-age of the voltage differences between top and bottom ofthe network, divided by the injected current) for severalrealizations of disorder and system sizes. For a fixed sys-tem size, we then find the disorder average R xy and stan-dard deviation ∆ R xy = (cid:113) R xy − R xy of R xy . We findthat R xy → R xy → L increases (weuse lattices with r = c = L ). We show in Fig. 6 the finitesize scaling of the ∆ R xy . That ∆ R xy → R xy → FIG. 6: Standard deviation of the Hall resistance for 100simulations with ¯ g = √ δg = ¯ g/
10 as a function of 1 /L for a grid with r = c = L (which fixes the aspect ratio). Itscales to zero in the large L limit, implying that the system isself-averaging and the Hall resistance R xy → We have also checked the effects of disorder for thezigzag configuration, reaching similar conclusions thatdisorder does not alter the quantization of the conduc-tance in the thermodynamic limit.In summary, we find that, in the thermodynamic limit,the general results of the previous sections hold even inthe presence of disorder.
VI. CONCLUSIONS
We investigated the transport properties of hexagonalnetworks whose nodes are Y-junctions of quantum wires.In our model the 3 × χ ± , (whenthe interaction parameter obeys 1 < g < R xy = ± h/e . Thisquantization is similar to that in the integer quantumHall effect. Further, the quantization is independent ofthe interaction parameter g even in the presence of disor-der in g . The quantization of the Hall resistance followsfrom the specific form of the conductance tensor at theRG stable chiral fixed point at each Y-junction. However,unlike in the quantized Hall effect, where the longitudinalresistivity vanishes, ρ xx is not zero: ρ xx = ( √ /g ) h/e .Dissipation in the longitudinal direction is a result of de-coherence within the wires. We emphasize that in ourmodel the wires are classical, but the nodes remain quan-tum mechanical and the form of the conductance tensor G at each junction is constrained by quantum scatteringeffects. The essential ingredient for the quantization ofthe Hall conductance is the value of the chiral fixed pointconductance of the individual junctions.Finally, let us comment on the finite temperature cor-rections to the value R xy = ± h/e in the network. Asopposed to the case of the quantum Hall effect where thequantization is exponentially accurate because of an en-ergy gap, the quantization in the networks has a powerlaw correction in T because the wire networks are gapless.The quantization should be as accurate as the conduc-tance tensor is close to that of the RG fixed point. Thecorrections to the conductance tensor scale as T ∆ , where∆ = 4 g/ (3 + g ) is the scaling dimension of the leadingirrelevant operator at the chiral fixed points. Notice that the temperature scaling of the conductivityabove should hold only under the assumption of decoher-ence within the wires. However, as temperature goes tozero, the coherence length increases, and therefore thereis an implicit assumption of order of limits for the re-sults in this paper to work as presented: the length ofthe wires should be taken to infinity before the limit of T = 0 is taken. But it is natural to wonder whether thequantization that we found in this work should persist ornot even if transport along the wires is always coherent.Indeed, one possibility is that in the coherent regime onemight have quantization of the Hall conductance withvanishing longitudinal resistivity. However, to addressthis regime one would need to tackle the fully interactingtwo-dimensional fermionic model, which is beyond thescope of this paper. One route to follow could be to con-sider a lattice model where the wires are described by atight binding model, with three wires coupled togetherat junctions by hopping matrix elements between them.One could possibly start with a non-interacting versionof the model, where the chiral conductances used in thispaper are obtained by fine tuning to the fixed point (sincethe non-interacting model is marginal and there is no RGflow). The problem then becomes one of electrons in asuperlattice, with the number of bands scaling with thenumber of sites describing the wires within a supercell.The Hall conductance for this tight-binding model couldbe obtained by computing the Chern number of the filledbands. If the Hall conductance does not vanish in thismodel, it is only protected algebraically in temperature,as there would be “mini gaps” separating bands that scaleinversely with the size of the wires, instead of true bandgaps. Analyzing such model may shine some light on theproblem of wire networks in the coherent regimes. Acknowledgments
This work was supported in part by the DOE GrantNo. DE-FG02- 06ER46316 (C.C.).
Appendix A: Method
Our numerical approach consists of solving the fulllattice model exactly. In this section we describe ourmethodology in detail.Consider an arbitrary lattice with r external wires oneach side and 2 c external wires at the top and bottomedges. An equal current will be injected into each of the r wires on the left, and the 2 c edge wires will have acurrent of zero. This defines the boundary conditions. A2 × c “ghost nodes”, shownas dotted lines at the top edge, but they are only thereto facilitate the numbering scheme and no current willflow through them. Including the ghost nodes there area total of N = 4 c ( r + 1) nodes.The points on each wire that emanate from each nodeare governed by the equation V = GI where G is a3 × N degrees of freedom.However, starting in this way introduces many redun-dant variables in the bulk because in a classical wire thecurrent is the same everywhere along the wire and so isthe potential. We will unify the two points on each wirein the bulk by imposing a set of constraints. In generalthere are 6 cr + c − r such constraints, which equals thenumber of wires in the bulk.To write down the full network equation let us labeleach of the 3 N points by ( n, i ), where n = 1 , . . . , N isthe node index and i = 1 , , V ( n ) i and I ( n ) i , respectively. To illustrate this notation, in Fig. 7 FIG. 7: Example with r = 2, c = 2. Note the row of “ghostnodes” at the top edge. the constraint along the wire that connects nodes 6 and7 would be written as V (6)1 = V (7)1 and I (6)1 = − I (7)1 .Each node obeys the relation V ( n ) = G ( n ) I ( n ) where G ( n ) is the 3 × I (1) I (2) ... I ( N ) = G (1) · · · G (2) · · · · · · G ( N ) V (1) V (2) ... V ( N ) (A1)Next we impose the constraints to reduce the effec-tive dimensionality of the problem. Start with the set ofpoint pairs on each wire in the bulk { ( n, i ) , ( m, i ) } , where n and m are nearest neighbor nodes. The constraints are V ( n ) i = V ( m ) i and I ( n ) i = − I ( m ) i for each pair. We im-pose the constraint on voltages by adding the ( n, i )-thand ( m, i )-th columns together, removing the ( m, i )-thcolumn and removing V ( m ) i from the vector of potentialsin Eq. (A1). Similarly we impose the constraint on cur-rents by adding the ( n, i )-th and ( m, i )-th rows , deletingthe ( m, i )-th row and removing I ( m ) i from the vector ofcurrents. Also we replace the current I ( n ) i that has notbeen eliminated by zero because I ( n ) i + I ( m ) i = 0. There-fore each constraint is equivalent to removing one rowand one column and reduces the dimensionality of theoriginal problem by one. Furthermore, we have replacedeach current in the bulk by zero which is important be-cause the only currents that are left in Eq. (A1) are fullydetermined, being equal to either zero in the bulk or tothe boundary conditions.Eliminating the ghost nodes is straightforward – wesimply remove the ghost currents, potentials and theirassociated rows and columns in Eq. (A1). This reducesthe dimensionality further by 3 × c , which is the numberof wires emanating from the ghost nodes. The final stepis to fix the gauge. Since all potentials are determinedup to an overall constant, we pick an arbitrary potential,set it to zero, and remove the associated row and columnfrom Eq. (A1).To summarize, we started with 3 N = 12 c ( r + 1) re-dundant degrees of freedom and then through successivetransformations we imposed 6 cr + c − r constraints in the bulk, eliminated 6 c ghost points, and fixed one poten-tial to zero. The dimensionality has thus been reducedto 6 rc + 5 c + r − g is straight-forward. The derivation proceeds in exactly the sameway as we just described, but we start with non-uniform G ( n ) . C. Chamon, M. 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