*4.3. Generalized Duality*

Harmonic duality can be extended to a two-dimensional resistive sheet with a spatially inhomogeneous sheet conductance *<sup>G</sup>*(*<sup>x</sup>*, *y*). Keller proved the duality for a system composed of two different conductances [6], and Dykhne generalized it to a system with an arbitrary scalar function *<sup>G</sup>*(*<sup>x</sup>*, *y*) [7]. The extended duality is often called *Keller–Dykhne duality*. Furthermore, Mendelson generalized the duality for a tensor *<sup>G</sup>*(*<sup>x</sup>*, *y*) [8].

Here, we review the derivation of the duality. As with circuit duality, we set a reference resistance *<sup>R</sup>*ref(= 1/*G*ref). Referring to a 90◦ rotation in Equation (40), we introduce the following dual fields on *z* = 0:

$$E^\* = R\_{\rm rot} I \mathbf{K} = R\_{\rm rot} \mathbf{e}\_z \times \mathbf{K},\tag{43}$$

$$\mathbf{K}^{\star} = \mathbf{G}\_{\text{ref}} / \mathbf{E} = \mathbf{G}\_{\text{ref}} \mathbf{e}\_{z} \times \mathbf{E},\tag{44}$$

with the unit vector *ez* along the *z* direction. The 90◦ rotation interchanges divergence and rotation of a two-dimensional vector field *<sup>v</sup>*(*<sup>x</sup>*, *y*) as

$$
\nabla \times (\mathbf{e}\_z \times \mathbf{v}) = \mathbf{e}\_z \nabla \cdot \mathbf{v},
\tag{45}
$$

$$
\nabla \cdot (\mathbf{e}\_z \times \mathbf{v}) = -\mathbf{e}\_z \cdot (\nabla \times \mathbf{v}),
\tag{46}
$$

where we used ∇ × (*A* × *B*) = *A*(∇ · *B*) − *B*(∇ · *A*)+(*B* · ∇)*A* − (*A* · ∇)*B* and ∇ · (*A* × *B*) = *B* · (∇ × *A*) − *A* · (∇ × *<sup>B</sup>*). Thus, KVL and KCL for *E* and *K* automatically follow: ∇ × *E* = 0 and ∇ · *K* = 0. In addition, Ohm's law in Equation (30) is converted to

$$\mathbf{K}^{\star} = \mathbf{G}^{\star} E^{\star} \tag{47}$$

with

$$\mathbf{G}^{\star} = (\mathbf{G}\_{\text{ref}})^2 \mathbf{J} \mathbf{G}^{-1} \mathbf{J}^{-1}. \tag{48}$$

For *R* = *G*−<sup>1</sup> and *R* = (*G*)−1, we obtain

$$(R^\star f R f)^{-1} = (R\_{n\ell})^2. \tag{49}$$

Therefore, *E* and *K* give a solution for the sheet with *G* under interchanging the Dirichlet and Neumann boundary conditions. Note that *G* can be a tensor. For a scalar *G*, we simply obtain *GG* = (*<sup>G</sup>*ref)2.

#### *4.4. Effective Response and Duality*

Duality can relate the effective response of an original sheet with its dual counterpart. To see this statement, consider a resistive sheet with two terminals as shown in Figure 27a. Between electrodes, we have the voltage *V* = − *c*1 *E* · d*r* and current flow *I* = *c*2 *K* · *n*2 d*<sup>r</sup>*. The effective conductance between the terminals is defined as *G*eff = *I*/*V*. The dual system is also shown in Figure 27b, and the voltage and current are represented by *V* = − *c*2 *E* · d*r* and *I* = *c*1 *K* · *<sup>n</sup>*1d*<sup>r</sup>*, respectively. Using *A* · (*B* × *C*) = *B* · (*C* × *A*) = *C* · (*A* × *<sup>B</sup>*), we obtain

$$V^\star = R\_{n\ell} I\_\prime \tag{50}$$

$$I^\star = G\_{\text{ref}} V.\tag{51}$$

Therefore, the effective conductance *<sup>G</sup>*efffor the dual system is given by

$$\mathcal{G}\_{\text{off}}^{\star} \mathcal{G}\_{\text{off}} = (\mathcal{G}\_{\text{ref}})^2. \tag{52}$$

If the system is self-dual and passive, which means that there is no active element with a negative resistance, then *G*eff = *<sup>G</sup>*eff = *G*ref automatically follow. Thus, self-duality determines the effective response regardless of the structural geometry.

**Figure 27.** (**a**) resistive sheet with a sheet conductance *<sup>G</sup>*(*<sup>x</sup>*, *y*) and two terminals; (**b**) corresponding counterpart with *G* = (*<sup>G</sup>*ref)<sup>2</sup> *JG*−<sup>1</sup> *J*−1. Unit normal vectors are denoted by *n*1 and *n*2 for curves *c*1 and *c*2, respectively.

#### *4.5. Self-Duality and Singularity*

The self-dual effective response can sometimes predict a critical behavior of the system. To see how critical behavior appears in self-dual systems, we consider an ideal checkerboard with different admittances *Y*1 and *Y*2, as shown in Figure 28a. Here, the AC response with an angular frequency *ω* is discussed for the checkerboard. For a capacitive admittance *Y*1 = j*ωC* and an inductive admittance *Y*2 = (j*ω<sup>L</sup>*)−1, the checkerboard is self-dual with respect to a reference conductance *G*ref = √*<sup>Y</sup>*1*Y*2 = √*C*/*L*. Then, we obtain a real effective admittance *Y*eff = *G*ref due to the self-duality. The positive real admittance indicates that the system is lossy. However, the system is lossless because the effective admittance composed only of capacitors and inductors must be purely imaginary. Therefore, we do not have a physical solution for such a checkerboard composed of capacitors and inductors. Thus, the lossless ideal checkerboard is *singular*.

The above observation can be also interpreted from a branch cut for the self-dual admittance *<sup>Y</sup>*eff(*<sup>Y</sup>*1,*Y*2) = √*<sup>Y</sup>*1*Y*2 [77]. We fix *Y*2 at a point of the negative imaginary axis and gradually displace *Y*1 from *Y*2 as shown in Figure 28b. When we consider *<sup>Y</sup>*eff(*<sup>Y</sup>*1) = √*<sup>Y</sup>*1*Y*2 as a function of *Y*1, *<sup>Y</sup>*eff(*<sup>Y</sup>*1) must have a branch cut in the complex plane. The previous discussion clearly shows that the branch cut is located along the positive imaginary axis. This result indicates that two approaches from Re(*<sup>Y</sup>*1) > 0 and Re(*<sup>Y</sup>*1) < 0 regions to a point at the singular branch lead to different values of *Y*eff. The effective admittance on the branch is critically sensitive to loss or gain of the system.

The above result can be generalized for arbitrary *Y*2. Using linearity

$$\mathcal{Y}\_{\text{eff}}(\text{s}\mathcal{Y}\_1, \text{s}\mathcal{Y}\_2) = \text{s}\mathcal{Y}\_{\text{eff}}(\mathcal{Y}\_1, \mathcal{Y}\_2) \tag{53}$$

for a scalar *s*, we can write *Y*eff as *Y*eff = *<sup>Y</sup>*2*R*ref*Y*eff(*ηG*ref, *<sup>G</sup>*ref) with *η* = *Y*1/*Y*2. Therefore, all system characteristics are derived from *y*eff(*η*) := *<sup>R</sup>*ref*Y*eff(*ηG*ref, *<sup>G</sup>*ref) satisfying *y*eff(*η*) = *η <sup>y</sup>*eff(*η*<sup>−</sup><sup>1</sup>) due to the self-duality *<sup>Y</sup>*eff(*<sup>Y</sup>*1,*Y*2) = *<sup>Y</sup>*eff(*<sup>Y</sup>*2,*Y*1). Because singular *Y*1 is represented by *Y*1 = *sY*2(*<sup>s</sup>* < 0) in the previous discussion, self-dual *y*eff(*η*) has a branch cut along the negative real axis of *η*.

**Figure 28.** (**a**) ideal checkerboard sheet with sheet admittances *Y*1 and *Y*2; (**b**) domain of definition for the effective admittance *<sup>Y</sup>*eff(*<sup>Y</sup>*1) = √*<sup>Y</sup>*1*Y*2. The branch cut along the positive imaginary axis is indicated by a wavy line.

#### *4.6. Differential-Form Approach for Duality*

Although Keller–Dykhne duality was formulated through vector analysis, the essence of the duality can be vividly extracted by differential forms. Furthermore, differential forms are suitable to discretize continuous fields to circuit systems. Discretizing differential forms, we bridge between Keller–Dykhne duality and circuit duality.

Here, let us introduce differential forms and the exterior derivative in plain terms. To this end, we only focus on Cartesian coordinates. For more technical details of differential forms, see [59,78].

#### - Covector and 1-Form

First, consider an electric field at a point (*<sup>x</sup>*0, *y*0). The Cartesian basis is denoted by {*<sup>e</sup>x*, *<sup>e</sup>y*}. A displacement Δ*r* = Δ*xex* + <sup>Δ</sup>*y<sup>e</sup>y* and the electric potential difference Δ*ϕ* are related through

$$-\Delta \boldsymbol{\varphi} = E\_x(\mathbf{x}\_{0\prime} \boldsymbol{y}\_0) \Delta \mathbf{x} + E\_y(\mathbf{x}\_{0\prime} \boldsymbol{y}\_0) \Delta \mathbf{y}.\tag{54}$$

It is possible to consider an electric field *E* as a linear function as *E* : Δ*r* → <sup>−</sup>Δ*ϕ*. Physically, we may understand the electric field as an apparatus that measures the (minus) electric potential difference for a displacement Δ*<sup>r</sup>*. Such a linear map from a vector to a scalar is called a *covector*. Introducing a dual basis {d*<sup>x</sup>*, <sup>d</sup>*y*} with respect to the Cartesian basis {*<sup>e</sup>x*, *<sup>e</sup>y*}, we can write the covector as

$$E(\mathbf{x}0, y0) = E\_x(\mathbf{x}0, y0)\mathbf{d}\mathbf{x} + E\_y(\mathbf{x}0, y0)\mathbf{d}y. \tag{55}$$

The action of the *interior product* between *<sup>E</sup>*(*<sup>x</sup>*0, *y*0) and a displacement vector Δ*r* is written as

$$
\Delta \mathbf{r}\_{\perp} E(\mathbf{x}\_{0\prime} y\_0) = E\_x(\mathbf{x}\_{0\prime} y\_0) \Delta \mathbf{x} + E\_y(\mathbf{x}\_{0\prime} y\_0) \Delta y. \tag{56}
$$

From the definition, we have *<sup>e</sup>x*d*<sup>x</sup>* = 1, *<sup>e</sup>y*d*<sup>x</sup>* = 0, *<sup>e</sup>y*d*<sup>y</sup>* = 1, and *<sup>e</sup>x*d*y* = 0. A covector *α* should be depicted as a set of parallel lines with an outer orientation as shown in Figure 29a. As the covector becomes stronger, the lines become denser. The interior product *v α* gives the signed number of lines that the vector *v* pierces. Note that we consider a limit operation in the strict sense as follows: we set a small resolution value *ε* and define a set of lines {*Lm*|*m* ∈ Z} with *Lm* = {*r*|*<sup>r</sup> α* = *<sup>m</sup><sup>ε</sup>*}. The number of the lines of {*Lm*} pierced by *v* is denoted as *<sup>N</sup>*(*ε*). Then, we obtain *v α* = lim*ε*→<sup>0</sup> *<sup>ε</sup><sup>N</sup>*(*ε*).

The electric field is given by a covector field

$$E(\mathbf{x}, y) = E\_\mathbf{x}(\mathbf{x}, y)\mathbf{d}\mathbf{x} + E\_\mathbf{y}(\mathbf{x}, y)\mathbf{d}y,\tag{57}$$

which smoothly depends on (*<sup>x</sup>*, *y*). Covector fields are generally called *1-forms*. An example of a 1-form is shown in Figure 29b. For a general 1-form *α*, *v α* is evaluated at the tangent space to which *v* belongs: *v α* = *v <sup>α</sup>*(*<sup>x</sup>*0, *y*0) for *v* whose starting point is (*<sup>x</sup>*0, *y*0). The 1-form *α* can be integrated along a curve *c* as

$$\int\_{c} \mathfrak{a} := \lim\_{N \to \infty} \sum\_{i=0}^{N-1} \Delta r\_{i \cdot \mathbb{J}} \mathfrak{a}\_{\prime} \tag{58}$$

where curve *c*(*t*) : [0, 1] → E2 is discretized as Δ*<sup>r</sup>i* = *<sup>c</sup>*(*ti*+<sup>1</sup>) − *c*(*ti*) with *ti* = *i*/*N* (E2: two-dimensional Euclidean space). Note that Δ*<sup>r</sup>i* is considered to be in the tangent space of *<sup>c</sup>*(*ti*). This definition can be linearly extended for any 1-chain *c*, and a 1-form is considered as a 1-cochain. By integrating, we can express a voltage difference as *V* = − *cE*.

**Figure 29.** (**a**) illustration of a covector *α*. The signed number of lines that a vector *v* pierces is given by *v α*. The positive direction of *α* is depicted by carets; (**b**) covector field (1-form); (**c**) discretized curve *c* with displaced vectors Δ*<sup>r</sup>i*to define the integral of a 1-form *α* along *c*.


Similar to a covector, a (covariant) tensor maps *p* input vectors to a scalar output: *<sup>T</sup>*(*<sup>v</sup>*1, *v*2, ··· , *<sup>v</sup>p*), which has linearity in each slot. The interior product of a vector *v* and a tensor *T* can be defined as *v T* = *<sup>T</sup>*(*<sup>v</sup>*, !, ···), which indicates that the first slot is filled with *v* and the other slots are left waiting for inputs (!).

To obtain a higher-order tensor, other products are introduced. First, the tensor product of two covectors *α* and *β* is defined as follows: *α* ⊗ *β*(*<sup>u</sup>*, *v*) = *α*(*u*)*β*(*v*) for *u*, *v* in a vector space *U*. For 1-forms, the tensor product operates on each tangent space.

Second, the wedge product for two 1-forms *α* and *β* is defined as an antisymmetrized tensor

$$
\mathfrak{a} \wedge \mathfrak{k} = \frac{1}{2} (\mathfrak{a} \otimes \mathfrak{k} - \mathfrak{k} \otimes \mathfrak{a}).\tag{59}
$$

The wedge product of two 1-forms satisfies

$$
\alpha \wedge \beta = -\beta \wedge \mathfrak{a}\_\prime \tag{60}
$$

$$
\mathfrak{a} \wedge \mathfrak{a} = 0.\tag{61}
$$

As d*x* ∧ d*x* = d*y* ∧ d*y* = 0, only d*x* ∧ d*y* plays an important role. For *u* = *<sup>u</sup>xex* + *<sup>u</sup>yey* and *v* = *<sup>v</sup>xex* + *<sup>v</sup>yey*, we have

$$\operatorname{div} \wedge \operatorname{d} y(\mathfrak{u}, \mathfrak{v}) = \frac{1}{2} \det \begin{bmatrix} \mathfrak{u}^{\mathfrak{x}} & \mathfrak{v}^{\mathfrak{x}} \\ \mathfrak{u}^{\mathfrak{y}} & \mathfrak{v}^{\mathfrak{y}} \end{bmatrix}. \tag{62}$$

Due to the antisymmetry d*x* ∧ <sup>d</sup>*y*(*<sup>u</sup>*, *v*) = −d*<sup>x</sup>* ∧ <sup>d</sup>*y*(*<sup>v</sup>*, *<sup>u</sup>*), d*x* ∧ d*y* measures a signed area of a triangle spanned by *u* and *v*. In Figure 30a, we illustrate d*x* ∧ d*y* as directed circles and d*x* ∧ <sup>d</sup>*y*(*<sup>u</sup>*, *v*) counts the number of circles inside the triangle spanned by *u* and *v*. If the direction from *u* to *v* is the same (opposite) as the direction of the circles, the circles are counted as positive (negative) numbers.

**Figure 30.** (**a**) d*x* ∧ d*y* as directed circles; (**b**) example of a 2-form; (**c**) discretization for integration.

#### - 2-Forms and Integration

We can also consider a field *ω* = *f*(*<sup>x</sup>*, *y*)d*x* ∧ d*y* with a scalar function *f*(*<sup>x</sup>*, *y*), where *ω* is called a *2-form*. An example of a 2-form is shown in Figure 30b, where the dense circle area has a stronger field than the sparse area. For integration, we discretize a directed 2-cell *S* as *S* = #*i Si* with disjoint small triangles *Si* (Figure 30c). A 2-form can be integrated on *S* as

$$\int\_{S} \omega = \lim\_{i} \sum\_{i} f(\mathbf{x}\_{i\prime} y\_{i}) \omega(\Delta \mathbf{u}\_{i\prime} \Delta \mathbf{v}\_{i}),\tag{63}$$

where Δ*<sup>u</sup>i* and Δ*<sup>v</sup>i* span a triangle *Si* with the same direction of *S* and (*xi*, *yi*) ∈ *Si*. The limit is taken for finer meshes. Generally, *S* can be an arbitrary 2-chain. Then, a 2-form can be considered as a 2-cochain. Note that integration of a general tensor *T* without antisymmetry cannot be defined and the antisymmetry is essential for integration. To define the integral, the integration over subdivided triangles should be the same as that of the original triangle. At least, *<sup>T</sup>*(*<sup>u</sup>*, *v*) = *T u*, *u*+*v* 2 + *T u*+*v* 2 , *v* is required for arbitrary *u* and *v*. Considering *u* = *v*, we obtain *<sup>T</sup>*(*<sup>u</sup>*, *u*) = 0 for arbitrary *u*. Then, the antisymmetry *<sup>T</sup>*(*<sup>u</sup>*, *v*) = <sup>−</sup>*<sup>T</sup>*(*<sup>v</sup>*, *u*) must hold due to *<sup>T</sup>*(*u* + *v*, *u* + *v*) = 0.

## - Exterior Derivative

The exterior derivative for a scalar function *f*(*<sup>x</sup>*, *y*) is defined as

$$\mathbf{d}f = \frac{\partial f}{\partial x}\mathbf{dx} + \frac{\partial f}{\partial y}\mathbf{d}y,\tag{64}$$

which corresponds to the gradient of the function *f* . For a 1-form *α* = *<sup>α</sup>x*d*<sup>x</sup>* + *<sup>α</sup>y*d*y*, we define the exterior derivative as

$$\mathbf{da} = (\mathbf{da}\_x) \wedge \mathbf{dx} + (\mathbf{da}\_y) \wedge \mathbf{dy}.\tag{65}$$

By direct calculation,

$$\mathbf{d}\boldsymbol{\alpha} = \left(\frac{\partial \boldsymbol{\alpha}\_y}{\partial \boldsymbol{x}} - \frac{\partial \boldsymbol{\alpha}\_x}{\partial \boldsymbol{y}}\right) \mathbf{d}\boldsymbol{x} \wedge \mathbf{d}y\_{\boldsymbol{\alpha}}$$

this corresponds to a rotation. By using the exterior derivative, KVL is represented as d*E* = 0.
