© European Physical Society, EDP Sciences, 2025

Quantum materials

All materials are in a sense quantum materials. Because quantum mechanics, through the Schrödinger equation, determines the behavior of the electrons and thereby the physical properties of materials. Still, we can understand many materials using a classical, or at least semi-classical, picture. This includes metals, insulators, and semi-conductors. Left as quantum materials are those where no classical description exists. The quantum behavior is due to strong interactions between the electrons or other unique quantum effects.

Strong interactions often give rise to electronic ordering, resulting for example in superconductivity or different forms of magnetism. Other quantum effects are linked to strong intertwining between different degrees of freedom, such as charge, lattice, orbital, and spin, resulting in novel materials such as Dirac and Weyl semimetals [1] or topological insulators and other topological materials [2], where the topology of the quantum energy levels (bands) or wave functions determines the properties. Overall, the properties of quantum materials are often linked to emergence: properties that only emerge due to the vast number, 10²³, electrons in a material.

Quantum technology

Materials form a veritable playground to understand Nature, but also, importantly, underpins technological advancements. Today’s IT technology relies on manipulation of semiconductors, such as Si, to precisely conduct electricity down to the nanometer scale. Likewise, new quantum technology will require materials for deploying quantum behavior. Current quantum technology, spanning from quantum computing to quantum communication, primarily relies on already well-known quantum materials. For example, the qubits in many of today’s quantum computers are based on the quantum states in conventional super-conductors, such as Al or Nb. But conventional super-conductors are one of the few so far well-understood quantum materials. We can only imagine the possibilities left to unlock with other quantum materials, ranging from high-temperature superconductors and exotic magnetic structures, to using the non-local properties of quantum mechanics present in non-Abelian quasiparticles.

Theory and modeling

To advance our understanding of quantum materials to enable future quantum technology, theory and modelling is essential. Theory and modeling both predict new phenomena and materials and provide interpretations of existing experimental results, thereby working in tandem with experimental activities. However, since it is impossible to solve the Schrödinger equation exactly for all but a few electrons, our understanding of quantum materials requires elaborate efforts on multiple levels.

Quantum phenomena in materials can be treated on a conceptual level, where the key emergent phenomenon is captured using effective models. Mapping these results to real materials requires material-specific ab-initio modeling. Scaling up to real devices further necessitates identifying the essential features present at multiple different larger length scales using a multiscale approach. Device operation also requires knowledge of dynamical properties, spanning in addition multiple time scales. In the following we review some of the challenges and opportunities in these steps.

Conceptual understanding

Understanding intricate quantum phenomena requires conceptual low-energy models capturing the most important physics. Such models can consist of specifying an effective low-energy Hamiltonian (energy operator) for the essential physics and solving, exactly or numerically, for the energies and wave functions. Or starting from a known solution and understanding the effects of perturbations using diagrammatic field theory expansions. In particular, the ever-present Coulomb interaction between electrons causes a plethora of quantum phenomena, but so can interactions with the underlying lattice (atom nuclei) or light or other external fields.

As an example, conventional superconductivity is described within the celebrated BCS theory through an effective attraction between pairs of electrons (occurring due to Columb attraction between electrons and the atom nuclei). Other effective attractions may additionally be possible, generating unconventional superconductivity, also at much higher temperatures. However, the exact mechanism(s) behind unconventional superconductivity is in most cases yet unknown. Still, we can develop an understanding of unconventional superconductors using effective low-energy models.

Adding coupling to the electron spin through spin-orbit coupling, a relativistic effect, even a conventional super-conductor can present a topological band structure [2]. The topology guarantees that its surfaces contain effective Majorana fermions, particles that are their own antiparticles. The Majorana fermion can be viewed as half an electron, or more accurately, the wavefunction of an electron has split into two spatially separated parts, each forming a Majorana fermion as a new quasiparticle. Such non-locality is described by non-Abelian statistics for the quasiparticles and may be utilized for future robust quantum computing [3].

Combining different quantum materials into hybrid structures allows us to engineer even more possibilities, including combinations of single atomic layers in two-dimensional van der Waals structures. One example is two graphene layers with a small twist angle relative to each other, producing a moiré pattern. The moiré pattern can quench the kinetic energy of the electrons, thereby enhancing the effects of electron interactions, even to the limit where twisted bilayer graphene becomes superconducting [4]. Modeling such moiré structure requires making a choice between atomistic models, with many thousands of carbon atoms in a single moiré unit cell, or identifying more effective degrees of freedom to be able to use smaller models. Ultimately, the best choice is set by comparisons of the essential phenomena to experimental results.

Scaling up to real devices

As a prime example of quantum materials in our everyday life, not a single magnetic material would exist if not for quantum mechanical effects. Thus, while the concept of quantum materials is often associated with novel and exotic phases of matter, they have been a cornerstone of technological development ever since the first use of the compass, more than 2 000 years ago.

Magnetic materials also offer good examples of the challenges involved in the theoretical modeling required to harness the potential of quantum materials in technology. As an example, topological magnets such as magnetic skyrmions have been identified as promising for new technologies. Skyrmions are nanoscale, topologically protected magnetic structures that in theory exhibit remarkable stability and can be manipulated with ultralow energy, making them suitable candidates for future spintronic applications and unconventional computing [5]. But for this, both material-specific modeling and understanding spanning multiple length scales are needed, see Fig. 1 for an illustration.

Fig. 1 Schematic of multiscale modeling of a skyrmion system. Solving the electronic structure on the DFT level (left), with the kohn-Sham equations to solve for the wave functions, provides parameters for the atomistic model, with a pairwise effective Heisenberg interaction to solve for the (atomic) magnetic moments (middle). The atomistic model is then used to describe the physics on the nanoscale or used for multi-scale modelling of devices using micromagnetic simulations, using micromagnetic energetics to solve for the magnetization vector field (right).

To understand and design skyrmion-hosting materials, we must begin at the sub-atomic level by describing their electronic structure. This is typically done through ab-initio density functional theory (DFT), a well-established method for electronic structure calculations [6]. The formation of skyrmions, and their stability, is intimately tied to interatomic magnetic interactions that depend on details of the electronic structure, involving also spin-orbit coupling effects. This requires a precise treatment of both electronic correlations and relativistic effects to ensure accurate modeling.

Once the relevant magnetic interactions are established, they can be mapped onto an extended Heisenberg Hamiltonian, which includes pairwise exchange interactions, such as Heisenberg exchange and Dzyaloshinskii-Moriya interactions, and anisotropy terms [7]. This Heisenberg Hamiltonian is an effective description, serving as the foundation for modelling at the next step, the atomistic level. Here, using Monte Carlo (MC) and Atomistic Spin Dynamics (ASD) we can explore the formation, dynamics, and stability of skyrmions under various external stimuli such as magnetic fields or electric currents [8].

The length scale of magnetic skyrmions can range from a few atoms to micrometer size. To address the macroscopic behavior of larger skyrmion-hosting systems, atomistic models must be coarse-grained into micromagnetic simulations, where the magnetization is treated as a continuous vector field. Such simulations enable us to study phenomena such as skyrmion motion, annihilation, and creation, as well as their interactions with defects or geometric boundaries on length scales approaching those of real devices. This multi-scale approach, starting from electronic wavefunctions and culminating in a vector-field description, provides a pathway towards device-scale simulations for skyrmion-based applications.

In short, a multi-scale approach allows us to bridge the gap between the quantum mechanical origins of topological magnets and their potential application in novel technologies. This paves the way for future energy-efficient, high-density information storage and computations.

Dynamics of operating devices

The operation of quantum devices is inherently governed by dynamics in the quantum mechanical system. Electrons at the Fermi energy of metals travel at speeds around a nanometer per femtosecond. Also involving electromagnetic fields to excite the system or to probe the response, requires time-resolution of the order of a few attoseconds to describe the quantum dynamics. In the other end lies the long-term stability of the quantum state. For example, magnetic storage applications require stability over many years, a difference of 25 orders of magnitude. Modeling this vast range of timescales demands a multi-scale approach also in time, beyond the length scale challenges described above.

To model device dynamics, we start with methods able to capture dynamics at the shortest timescales, such as time-dependent density functional theory (TDDFT). A scenario where the shortest timescales are intrinsically relevant is the search for energy efficient materials for information processing. For optimal control and to minimize heat dissipation when altering the information stored in a quantum system, the speed of operation needs to be faster than the time of thermal equilibration in the relevant degrees of freedom. This often requires tailored light-fields, with frequency matched to a particular resonance. The use of simulations can both guide in the design of light pulses and allow calculations of observables, also those not directly associated with the function, such as monitoring thermalization behavior.

TDDFT can in turn be used to estimate the susceptibility to perturbations. This indicates what degrees of freedom can be ignored by being integrated out, and what needs to be kept at longer time scales. One example is the extraction of pairwise exchange parameters for studying dynamics in the Heisenberg Hamiltonian mentioned above. In the description of real device dynamics, it is essential to construct reliable and effective models by projecting out degrees of freedom not relevant to the long-term dynamics.

Conclusions

Thanks to the enormity of choices in material physics, from the combinations of around 100 stable elements and the complexities of their electronic configurations, to the vast possibilities linked to emergence, new discoveries of quantum materials produce a seemingly never-ending source of new and novel quantum phenomena. Challenges for theoretical and computational physicists are both to find effective models that work on a unifying level, explaining whole classes of materials, and provide predictions for specific materials and devices. This requires a range of techniques, from low-energy effective models to advanced computational multiscale schemes capturing vast length and time scales.

About the Authors

Annica Black-Schaffer is a Professor and head of the Quantum Matter Theory research program in the Materials Theory division at Uppsala University.

Oscar Grånäs is a Senior lecturer in the Materials Theory division at Uppsala University.

Anders Bergman is a Senior lecturer in the Materials Theory division at Uppsala University.

References

All Figures

Fig. 1 Schematic of multiscale modeling of a skyrmion system. Solving the electronic structure on the DFT level (left), with the kohn-Sham equations to solve for the wave functions, provides parameters for the atomistic model, with a pairwise effective Heisenberg interaction to solve for the (atomic) magnetic moments (middle). The atomistic model is then used to describe the physics on the nanoscale or used for multi-scale modelling of devices using micromagnetic simulations, using micromagnetic energetics to solve for the magnetization vector field (right).

In the text