Learning from insulators: New trends in the study of conductivity of metals

One of the early successes of quantum mechanics was the derivation of transport properties in crystalline solids from first principles. It gave a mathematical explanation for why some materials are conductors, i.e., allow for the propagation of electrons, while others are insulators. Thanks to advances in mathematics and the availability of computers, our understanding has progressed within the last 90 years from simple models for e.g., a single particle moving in a periodic potential to more realistic models that include external fields, disorder and many-body effects. And those advances gave birth to semiconductor electronics, which are comprised of materials that can transition between an insulating and a metallic state depending on external control parameters.

In the last few decades a lot of research was dedicated to the study of insulators. There were two major driving forces: the discovery of topological insulators with the Quantum Hall Effect uncovered the surprising role that topology and symmetries can play. The second revelation was that disorder can suppress conductivity, a phenomenon known as Anderson localization. These two lines of investigation have matured into two very active fields in theoretical and experimental condensed matter physics, and more recently, also in the field of classical waves. First-principles explanations required the development of new mathematical approaches that draw upon an incredible number of disciplines, particularly topology, functional analysis, semiclassics, probability theory, and noncommutative geometry.

An early attempt to systematically investigate electronic transport in metals (i.e., conductors) was proposed by Lifshitz in the 1950s: because there was no way at that time to solve the Schrödinger equation numerically, he proposed to replace the quantum evolution equation by semiclassical Hamiltonian equations of motion; these describe a single electron moving in a perfectly periodic crystal. In this framework, a central role is played by the Fermi surface, the energy level set of the Hamiltonian equations to the so-called Fermi energy; the Fermi energy in turn characterizes the state of the system. The particular topology and geometry of the Fermi surface have deep consequences on the conduction properties of the system: when the Fermi energy lies in a spectral gap, the Fermi surface trivially reduces to the empty set, which corresponds to the insulating phase of the material. In the case where the Fermi surface is not trivial, the Hamiltonian equations can be seen as a dynamical system whose trajectories are constrained to the Fermi surface, and these constraints are expected to influence the conductivity of the material. In a nutshell, this was the core idea for the theory of galvanomagnetic effects in metals proposed by Lifshitz in 1957.

The advancements in recent years in the field of (topological) insulators, operator theory and classical mechanics suggest that it is time to look again at conductors, and for this, to extend the notion of the Fermi surface to more realistic models for metals and semiconductors. In all, the Fermi surface seems to be the key in understanding transport properties in solids: only states near the Fermi energy can be excited and therefore those states dominate many aspects of the physics of metals. However, as soon as perturbations, disorder or electron-electron interactions are present, even its very definition has so far eluded mathematical physicists. Solving this will require input from a broad range of complementary fields and was the main motivation for the workshop. In bringing together experts in different fields coming from classical mechanics, semiclassical analysis and quantum systems, we wanted to create an interdisciplinary environment suitable to collaboratively tackle what we believe is a promising approach to the study of conductivity properties in metals and semiconductors: developing a mathematically rigorous and physically motivated definition of the Fermi surface and study its properties.

The special issue collects lecture notes, surveys and research articles that stemmed or where inspired by the workshop. The breadth of the contributions, spanning from classical mechanics to operator theory and disordered systems, reflects the interdisciplinary nature of the workshop. We hope that they will help to provide the necessary background and to act as a reference and an inspiration for future research in the field.

The workshop included four mini-courses on different aspects of the classical and quantum theory of transport:

• Classical Mechanics and Transport by Andreas Knauf, whose lecture notes are published as part of this special issue,⁸ 

• Semiclassical Approximations for Solid State Systems by Max Lein, available on YouTube,¹¹ 

• Disordered Quantum Systems by Christian Sadel available on YouTube,¹⁶ and

• Adiabatic Quantum Transport by Sven Bachmann, available on YouTube.¹ 

In addition to the comprehensive review on the classical mechanics of transport, this issue begins with the article of Peter Kuchment⁹ which summarizes the state-of-the-art of Fermi surfaces of periodic operators, and closes with current conjectures and open problems. These results form the starting point for generalizations that apply to more realistic models for charge transport in metals that may include perturbations or disorder.

The paper by Maltsev and Novikov¹³ reviews the developments in the study of the galvanomagnetic effect and its relations to the topology of the Fermi surface via Novikov’s problem: classifying of non-closed trajectories for particles with an arbitrary dispersion relation in a presence of a magnetic field. The paper also introduces a generalization of the problem, studying the general topology of electron trajectories on the Fermi surface and the corresponding physically observable properties.

The Fermi surface remains central also in the work by De Nittis and Polo Ojito.⁷ In this paper, the authors propose a new approach to define and study physical concepts from quantum mechanics in terms of algebraic operator theory, a direction of possible interest for the generalization of the analysis of transport properties.

In the presence of disorder, quantum systems are known to behave as insulators, under certain conditions. This phenomenon is known as Anderson localization, and in dimension 2 and above, disordered systems are expected to undergo a phase transition between extended and localized states. In Ref. 6 the authors introduce the Anderson model with decaying randomness as a toy model to study the metal-insulator transition from a spectral point of view: depending on the rate of decay of the randomness, this model exhibits pure point or absolutely continuous spectrum. In Ref. 14, Mashiko et al., study the asymptotic fluctuations of the density of states measure of this system and the critical value of the decay rate at which the fluctuations exhibit different asymptotic behaviors.

One of the main objectives of the workshop was to revisit the mathematical theory of topological insulators and discuss what current limitations we need to overcome to apply them to metallic transport. The appearance of localized metallic edge states protected by the bulk topological number is a well-known phenomenon in the physics of such materials called bulk-edge correspondence. The paper by Cornean et al.⁵ addresses this phenomenon for two-dimensional unbounded magnetic Dirac operators. Also in the realm of unbounded operators, the paper by Thiang¹⁷ provides a new constructive proof of a well-known idea from physics: a chiral-symmetric local Hamiltonian on a half-space has the same signed number of edge-localized states with energies in the bulk band gap as its bulk winding number. To classify topological edge states is convenient to have numerical topological invariant that provide their labeling. In Ref. 15, Monaco and Peluso, define a new $Z2$ topological invariant for a class of one-dimensional translation-invariant topological insulators.

Complementary to that were efforts to find efficient computational techniques to evaluate such topological invariants. The approach spearheaded by Loring, Schulz-Baldes, and co-workers was to restrict the infinite tight-binding Hamiltonian to a finite volume, thereby reducing an infinite operator to a matrix, and compute the invariants of the full operator from this matrix via the so-called spectral localizer. Cerjan et al.⁴ extend these ideas from selfadjoint tight-binding operators to non-selfadjoint operators with line gap.

Leinaas and Myrheim’s¹² theoretical prediction of quasi-particles, called anyons, with fractional statistics in two dimensions gained experimental support with the discovery of the fractional quantum Hall effect.^10,18 The contribution² by Bachmann et al. introduce explicit Hamiltonian lattice models of anyons with short-range interactions, potentially realizable in modern laboratories. This work presents a natural emergence of Abelian anyons as charge-flux pairs in lattice models, displaying quantized phase holonomy consistent with gauge and duality symmetries. This holonomy arises directly from the many-body Hamiltonian, negating the need for pre-assumed charge-flux pairs or tracer particles.

A semiclassical analysis of electron transport will lead to the study of classical Hamiltonian systems and their flow in a similar fashion as Novikov’s problem. In this respect better understanding of integrable systems and their perturbations is crucial. The paper by Brugués et al.³ sits in this context, providing a generalization of the notion of integrability and of the classification of semitoric systems—a well studied, physically relevant class of integrable systems—to a much broader class. In this paper, the authors introduce and characterize b-semitoric systems: four-dimensional systems whose singular points are non-degenerate and contain no hyperbolic components and that still satisfy a certain notion of integrability.

Overall, the workshop proved very successful from a scientific vantage point. It definitely succeeded in getting people to meet (virtually) and get them to see connections between their fields and others they were previously not aware of. The balance of speakers in terms of topics was spot-on, and the talks—as evidenced by the lively discussions—well-received.

It is clear that conduction properties in the metal regime are not yet a very developed field, with few results and many open questions. All in all, we hope that the material gathered during the workshop will provide a good resource for (young) researchers to get an overview of the subject and the current state-of-the-art, in order to be able to start working on it.

It is a pleasure to thank all the authors for accepting our invitation to contribute to this special issue as well as all the speakers and participants that contributed to make the workshop so lively and interesting. We would like to take this opportunity to than the Lorentz center for their hospitality and help in organizing the workshop. Finally, we are grateful to the Editor-in-Chief and the Editorial Manager of the Journal of Mathematical Physics for the privilege of guest-editing this issue and for their assistance in its preparation.

Giuseppe De Nittis: Writing – original draft (equal); Writing – review & editing (equal). Max Lein: Writing – original draft (equal); Writing – review & editing (equal). Constanza Rojas-Molina: Writing – original draft (equal); Writing – review & editing (equal). Marcello Seri: Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.