On the Topological Structure of Nonlocal Continuum Field Theories
2. Preliminary Considerations
2.1. What Is Nonlocality?
2.2. Key Contributions and Motivations in the Present Work
- Infinitesimal interactions: this characterizes local field theories, e.g., local electromagnetism, where all operators are differential operators.
- Non-infinitesimal but local interactions: here, nonlocal operators, such as integral operators, may be present. In this type of theory, interactions are extended into small topological neighborhoods around the source/observation point.
2.3. An Outline of the Present Work
3. The Nonlocal Continuum Response Model
3.1. A Generic Nonlocal Response Model in Inhomogeneous Continua
On the other hand, if the medium is local, then the material response function can be written asIn nonlocal continuum field theories, knowledge of the field response at a specific point requires knowledge of the cause (excitation field) on an entire topological neighborhood set .
3.2. Spatial Dispersion in Homogeneous Nonlocal Material Domains
3.3. Preliminary Remarks on the Existence of Multiple Topological Scales in Nonlocal Continuum Field-Theoretic Structures
- The second is the case captured by the inset in the right hand side in Figure 1. Fine “microscopic” cells, each homogeneous and, hence, describable by a response function of the form , can be combined to build up a complex effective nonlocal response tensor over its topologically global domain . Such juxtaposition at the microscopically local level that effectively leads to the emergence of a global behavior is a classic example of multiscale physics. However, note that it even acquires a higher importance in the present context due to the fact that both of the constituent cell level (rectangular “bricks” in the inset of Figure 1) and the global domain level already belong to the physically, e.g., electromagnetically, nonlocal dimension of the relevant nonlocal continuum field theory.
- Finally, the third directly observable topological scale is that connected to what we termed “topological holes” in Figure 1. These are arbitrarily-shaped gaps, such as holes, vias, etchings, etc., which are intentionally introduced in order to influence the electromagnetic response by modifying the topology of the three-dimensional material manifolds .
- Physics-based local/non-local distinction: this is where basically physical considerations are at stakes. We distinguish between:
- Physics-based non-local level: this includes how the response of the material continuum depends on locations not infinitesimally close to the point where the excitation field is applied. That is, is nonzero but it is also not a differential. (On infinitesimal domains, see Remark 1.)
- Physics-based local level: this is the physical regime whose essence is captured by local constitutive relations of the form (15).
- Topology-based local/non-local distinction: mathematical considerations dominate at this level. We have:
- Topology-based non-local level: this is the topologically global level, e.g., the entire topological manifold in contrast to the local description applicable only to a coordinate patch , and so on. At this level, the non-local-as-global is an emerging structure based on gluing together “smaller pieces” of the total manifold. We will see examples of processes occurring basically at this level when we use partition of unity methods.
4. The Microscopic Topological Structure of Physics-Based Nonlocal Domains
- Initially, in the present Section, we introduce the rudiments of the main physics-based microtopological structure associated with nonlocality in continuum field theories, but without delving into considerable mathematical details. The aim is to familiarize ourselves with the minimal necessary physical setting and how it naturally gives rise to a more refined picture of the nonlocal material domain compared with the traditional (and much simpler) topological structure of local electromagnetism based on spacetime points.
- In the second stage, treated in Section 5, a more careful mathematical picture is developed using the theory of topological fiber bundles. We eventually show (Section 5.3) that the physics-based (in this case the electromagnetic) nonlocal operator (20) can be reformulated as a Banach bundle map (homomorphism) over the three-dimensional space of the material domain under consideration. Some computational examples and applications are provided in the later Sections, e.g., see Section 7.
4.2. The Concept of Topological Microdomains in Nonlocal Continuum Field Theories
4.3. Construction of Excitation Field Function Spaces on the Topological Microdomains of Nonlocal Media
4.4. The Global Topological Structure of Nonlocal Electromagnetic Material Domains: First Look
- Each open domain in will by assigned a distribution of open sets , i.e., the physics-based nonlocality microdomains topology defined in Section 4.2, see in particular Definition 1 and Remark 5. Physically, it expresses the fine microtopological structure of nonlocal continua, e.g., electromagnetic material nonlocality.
- The structure is solely determined by the physics of field–matter interaction. A concrete example explicitly illustrating how the detailed physical content of the underlying process contributes to the construction of will be given in Section 7.
- We further emphasize that the various sets constitute an open cover of D, that is, we have
- The decomposition of the material domain D into smaller building blocks exemplified by (32) is fundamental for computational topological models of nonlocal MTMs. For example, in Section 7 we will exploit this expansion in order to construct a topological coarse-grained model for inhomogeneous nonlocal semiconductor metamaterials.
- Finally, the topology induces the “function superspace” (30) defined as a class of function spaces , where each vector field acts on one microdomain element chosen from the topology .
4.5. A Reformulation of the Nonlocal Continuum Response Function
5. The Fiber Bundle Superspace Formalism in the Field Theory of Generic Nonlocal Continua
5.1. Preparatory Step: Promoting the Material Domain D to a Manifold
- It provides a natural and obvious generalization of the basic structure (31) from the mathematical perspective.
- Engineers often need to insert metamaterials into specific device settings, hence the shape of the material becomes highly restricted. It is therefore important to develop efficient tools to deal with variations of geometric and topological degrees of freedom and how they could possibly impact the design process.
- Applied scientists and engineers are often interested in deriving fundamental limitation on metamaterials, e.g., what are the ultimate allowable response–excitation relations or constitutive response functions possible given this material domain topology?
- Sophisticated full-wave electromagnetic numerical solvers prefer working with local coordinates in order to handle complicated shapes, even if a global coordinate system is sometimes available, making the deployment of the three-manifold structures for describing the material domain D useful.
- In topological photonics and materials , most applications seem to focus on lower-dimensional states of matter like those associated with quantum Hall effects and edge states (surface waves).14 There, new phenomena appear at material structures where the base space (material domain D) is a two-surface, which is best described mathematically as a differential two-manifold.
5.2. Attaching Fibers to Generic Points in the Nonlocal Material Manifold D
The above technical problem will be solved in Section 5.3 by using the technique of partition of unity borrowed from differential topology [26,57,62]. It will allow us to split up each full microdomain into several suitable sub-microdomains (details below), which can be later joined up together in order to give back the original EM nonlocality microdomain .Since the differential structure associated with chartscan be fixed by essentially mathematical considerations alone, while the collection of microdomainsis solely determined by the physics of electromagnetic nonlocality (See Remark 3 and Section 4), there is no direct and simple way to determine and express the vector transformationbecause several different coordinate patches other than and , belonging to the differential three-manifold atlas, might be involved in geometrically building up the microdomain .
- and each function is .16
- Since the open cover is locally finite, at each point , only a finite number of will intersect .
- Let the set of indices of those intersecting be . Then we require that
- Initially, the physics-based collection of sets
- Introduce a differential atlas
- Finally, the same atlas is linked to a set of functions (partition of unity) that can be recruited as “topological bases” in order to expand any differentiable field excitation function into sum of individual sub-fields defined on open subsets of the material domain (see Section 5.3).
- Step I: Construct a tailored fiber bundle based on the partition of unity charts introduced above.
- Step II: the original physical structure (31) is recovered by gluing together various sub-microdomain of each EM nonlocal microdomain .
5.3. Direct Construction of Bundle Homomorphism as Generalization of Linear Operators in Electromagnetic Theory
5.3.1. The Basic Definition of the Nonlocal Material, (or Continuum, Metamaterial (MTM), etc.), Banach (Fiber) Bundle Superspace
5.3.2. The Nonlocal Material Continuum Fiber Bundle Homomorphism
- The continuum itself is mathematically modeled as a Banach bundle superspace instead of its conventional differential manifold representation . The response of the medium is to be sought at some point .
- The bundle superspace encompasses an additional structure compared to , namely a distinct copy of a linear function space attached at each point . This is nothing but the fiber , which is a Banach space of functions defined on the region . This function space can be intuitively understood as a rigorous and exact model of the excitation field when the latter is restricted to (topologically localized at) the physics-based nonlocal domain .
- It should be noted that in local continuum field theory, e.g., conventional electromagnetism in normal temporally dispersive media, each one of the subdomains is essentially one point . Therefore, in the case of local continua, the excitation field F is there found to be preferably defined as acting on the conventional space instead of being a section of a Banach bundle superspace .
- A vector bundle homomorphism (to be formalized in Definition 3) will map one element of this fiber function space, namely, the particular excitation field to its value in the range vector bundle . For the case of electromagnetic field theory, the latter may be taken as a vector space fiber isomorphic to with a copy of this fiber attached to each .
5.3.3. Computing Global Data Starting from Local Data
As mentioned before, it is the partition of unity , what will make this expansion of the topological formulation technically feasible.How can we extend the description of the nonlocal continuum’s response operators starting from excitation fields defined locally to excitation fields applied on the entire physical cluster of nonlocal microdomains ?
- Region ;
- Vector field acting on .
6. Interlude: The Nonlocal Continuum Fiber Bundle Superspace Algorithm—Summary and Transition to Applications
|Algorithm 1 The nonlocal continuum fiber bundle algorit|
7. Applications to Advanced Materials: Nonlocal Inhomogeneous Semiconductors
7.2. A Topological Coarse-Grained Model for Inhomogeneous Nonlocal Material Domains
- The proposed topological coarse-grained model utilizing the set of balls (left).
- The conventional paradigm where the unit cells are non-overlapping (right).
7.3. Resonant Nonlocal Semiconductor Domains and the Nonlocal Exciton-Polariton Model
7.4. Quantitative Estimation of the Electromagnetic Nonlocality Microdomain Structure in the Exciton-Polariton Dielectric Model
- Second, the presence of a spatially-decaying exponential factor of the form makes the Green function highly attenuating in spite of the fact that this attenuation is not mainly due to thermodynamic losses.
7.5. The Locally-Homogeneous Model of Nonlocal Semiconducting Domains
8. Application to Fundamental Theory: Electromagnetic Boundary Conditions in the Fiber Bundle Superspace Formalism
- The existence of extra or additional structures in the fiber bundle superspace approach to nonlocality in complex continua forces on us the need for introducing additional boundary conditions or information coming from the microscopic topological structure of the corresponding material superspaces.
- The fiber bundle superspace formalism of nonlocal metamaterials appears to be able to capture the intricate processes taking place inside and across various nonlocal material domains joined together through interfaces.
- This is achieved by providing an efficient apparatus to topologically encode some of rich and complex physics of field–matter interactions via the construction of appropriate infinite-dimensional function spaces (Banach space fibers) attached at each point of the materials’ base manifold.
- It is suggested that the relations between those additional fiber spaces are in fact what should be mainly taken into account while formulating boundary conditions for nonlocal continuum field theories, hence not merely the conventional relations involving only spatial interfaces between the material base manifolds as has been usually the practice in local field theories.
Conflicts of Interest
Appendix A.1. Survey of the Literature on Nonlocal Metamaterials
Appendix A.1.1. Introduction
Appendix A.1.2. Historically Important Examples
Appendix A.1.3. General Theories of Nonlocal Continua
Appendix A.1.4. Semiconductors, Metals, Plasma, Periodic Structures
Appendix A.1.5. Boundary Conditions in Nonlocal Metamaterials
Appendix A.1.6. Computational Techniques
Appendix A.1.7. Novel Systems and Devices with New Electromagnetic Behavior
Appendix A.1.8. Homogenization
Appendix A.1.9. Topological Materials and Photonics
Appendix A.2. On the History of Spatial Dispersion in Crystal and Plasma Physics
Appendix A.3. Some Further Engineering Applications of Nonlocal Metamaterials
Appendix A.3.1. Communications Systems and Information Transmission
Appendix A.3.2. Electromagnetic Metamaterials
Appendix A.3.3. Near-Field Engineering, Nonlocal Antennas, and Energy Applications
Appendix A.4. On the Concept of Superspace
Appendix A.5. Guide to the Mathematical Background
- Differential manifolds,
- Banach and Sobolev spaces,
- Vector bundles, and
- Partition of unity
Appendix A.5.1. Topology on Smooth Manifolds
Appendix A.5.2. Banach and Hilbert Spaces
Appendix A.5.3. Banach and Hilbert Manifolds
Appendix A.5.4. Sobolev Spaces
Appendix A.5.5. Vector Bundles
Appendix A.5.6. Additional Remarks on the Use of Sobolev Spaces in the Fiber Bundle Superspace Formalism
Appendix A.5.7. Partition of Unity Techniques
Appendix A.6. The General Electromagnetic Model of Nonlocal (Spatially-Dispersive) Isotropic Domains
- First, fundamental theory is deployed to derive analytical expressions for and .
- Afterwards, depending on the concrete values of the various physical parameters that enter into these expressions, e.g., frequency, temperature, molecular charge/mass/spin, density, etc., the obtained analytical expressions are expanded in power series with the proper number of terms.
- The expression of the dielectric tensor function is then put in the form of either a polynomial or rational polynomial in .
Appendix A.7. Origin of Electromagnetic Nonlocality in Excitonic Semiconductors
Appendix A.7.1. Review of the Semiconductor Physics of Excitons
- The electron mass must be replaced by the reduced exciton mass
- Due to the screening of Coulomb attraction by the dielectric medium, the effective electron charge should be replaced by , where is the static dielectric constant.
Appendix A.7.2. Simple Explanation of How Nonlocality Emerges in the Excitonic Semiconductor
Appendix A.8. An Alternative Intuitive Derivation of the Dielectric Model (90) and the Quantum Origin of Nonlocality in Excitonic Semiconductors
Appendix A.9. Computation of the Inverse Fourier Transform (114)
Appendix A.10. On Extending Definition 1 to Noncompact Regions
Appendix A.11. Possible Applications of the Superspace Formalism to Fundamental Methods in Metamaterials Research
Appendix A.11.1. Estimating Fundamental Limitations on Nonlocal Metamaterials
Appendix A.11.2. Numerical Methods
- The ability to resolve the issue of generalized boundary condition (already discussed in Section 8).
- Since every point belonging to a fiber superspace is in itself a smooth function defined on an entire material sub-microdomain, by building a new system of discretized recursive equations approximating the behavior of electromagnetic solutions living in the enlarged superspaces and one may anticipate arriving at a deeper understanding of the physics of nonlocality. The reason is that the topology of the nonlocal interaction regime is explicitly encoded into the geometry of the new expanded solution superspace itself. Characterizing this geometry is then possible through a suitable discrete approximation of the interior microtopological content of the superspace (fiber bundle) structure itself; i.e., not just at the “exterior” parts often found in the boundary conditions of classical local field continuum theories, but also “going inside” the problem space as such.
- It is also possible that such numerical methods may emerge as more computationally efficient and broader in applicability than the conventional methods rooted in local electromagnetism. One reason for this is that the Banach vector bundle formulation introduced in this paper is quite natural and appears to reflect the underlying physics of nonlocal metamaterials in a direct manner. From our general experience in numerical methods, “natural operations” tend to translate into numerical methods with better convergence, sensitivity, and robustness.
Appendix A.11.3. Topological Photonics
The author would like to thank one of the anonymous reviewers for suggesting this connection.
For a brief discussion of some possible engineering applications of metamaterials, see Section A.3.
Cf. Section 3.3.
If D asymptotically grows into an unbounded region, then the problem reduces to that of homogeneous unbounded domain (bulk media), well treated in the basic literature on spatial dispersion. Clearly, in this paper, we are not interested in such a topologically trivial problem.
Cf. Remark 3.
This compactness of the response kernel’s support cannot be proved in general, but is very plausible on physical grounds (causality considerations). Therefore, we posit such compactness as an axiomatic feature of all physically-realizable causal nonlocal continuum field theories. However, see Appendix A.10.
In this section and the one to follow, we do not worry much about the details of the electromagnetic model and for simplicity assume that only one vector field acts as excitation and one response field is induced. More complex media like bi-anisotropic domains and others  may also be treated within this formulation. For example, if two response fields are needed, the codomain in (33) can be simply changed to .
Cf. Remark 3.
See the discussion of nonlocal and topological metamaterials applications in Appendix A.3.
Cf. Appendix A.1.
See [59,68] for the full technical definition of subordinated cover. A collection of subsets of a topological space is said to be locally finite, if each point in the space has a neighborhood that intersects only finitely many of the sets in the collection. What we need here is that there exists some i and such that is inside , i.e., where .
See Section 7 for one possible method and examples.
Cf. Remark 3.
The discretization of the nonlocal MTM bundle homomorphism itself is outside the scope of the present work and will be addressed elsewhere.
For instance, by introducing holes into a simply-connected domain in order to make the latter disconnected.
The numerical value of may be different for transverse and longitudinal excitation fields.
In the effective-mass approximation, a simple way to estimate the exciton mass is via the relation , i.e., the sum of the effective electron and hole masses introduced in Appendix A.7. However, it must be noted that this relation is far from being universal, e.g., it should be modified when there are strong interactions [81,82]
It should be noted that there is no loss of generality here. The computational model to be presented shortly allows the estimation of the nonlocal microdomain topology based on a generic model of the form (90). If and are not identical, then the same procedure can be applied to each one of them separately.
There is a nice parallelism here with temporal dispersion where the latter is known to arise from the inertial effects of electrons in interaction with radiation fields .
This is more obvious in FEM or FDTD than MoM.
Cf. Section 5.2.
Cf. Section 8.
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|temporal||excitation field period|
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Mikki, S. On the Topological Structure of Nonlocal Continuum Field Theories. Foundations 2022, 2, 20-84. https://doi.org/10.3390/foundations2010003
Mikki S. On the Topological Structure of Nonlocal Continuum Field Theories. Foundations. 2022; 2(1):20-84. https://doi.org/10.3390/foundations2010003Chicago/Turabian Style
Mikki, Said. 2022. "On the Topological Structure of Nonlocal Continuum Field Theories" Foundations 2, no. 1: 20-84. https://doi.org/10.3390/foundations2010003