Loading field map
Quantum materials are defined by phenomena that cannot be explained by the single-particle band theory that once dominated condensed-matter physics. The central puzzle is how strong electron–electron interactions, geometric frustration, and topological constraints give rise to emergent properties such as high-temperature superconductivity, quantum spin liquids, and topologically protected edge states. Over the past century, five major conceptual frameworks have been developed to address this puzzle, each building on, reacting to, or coexisting with its predecessors.
Band theory of solids treats electrons as independent particles moving in a periodic potential created by the atomic lattice. This single-particle picture successfully explains why some materials are metals, others insulators, and still others semiconductors. It predicts band gaps, effective masses, and the basic transport properties of conventional metals and doped semiconductors. For decades, band theory was the default language of solid-state physics, and its successes—from transistors to integrated circuits—made it seem nearly universal.
Yet by the 1970s and 1980s, a growing list of materials resisted band-theory description. Heavy-fermion compounds, for example, showed effective electron masses hundreds of times larger than band theory predicted. The assumption that electron–electron interactions could be treated as a small perturbation began to crack. The decisive blow came in 1986 with the discovery of high-temperature superconductivity in copper-oxide ceramics (cuprates). Band theory could not explain why these materials became superconducting at temperatures far above the conventional limit, nor could it account for the strange metallic behavior in their normal state. A new framework was needed.
The strongly correlated electron paradigm emerged directly from the failure of band theory. Its central claim is that electron–electron interactions are not a correction but the dominant energy scale. The Hubbard model, which includes a term for the on-site repulsion between electrons, became the minimal theoretical tool. In this framework, materials such as cuprates and heavy fermions are understood as Mott insulators—materials that band theory predicts to be metals but that become insulating because of interactions. The paradigm introduced concepts such as spin–charge separation, where the electron’s spin and charge degrees of freedom behave as independent quasiparticles, and the resonating-valence-bond (RVB) state, a quantum superposition of paired spins.
Unlike band theory, which classifies materials by their band structure, the correlated paradigm classifies them by the strength and nature of interactions. Its methods include dynamical mean-field theory, exact diagonalization of small clusters, and experimental probes such as angle-resolved photoemission spectroscopy (ARPES) that directly measure the spectral function. The paradigm remains active today, especially in the search for the mechanism of high-temperature superconductivity, a problem that has resisted solution for nearly four decades.
The quantum magnetism and spin liquids paradigm is a narrower outgrowth of the correlated electron framework. While the strongly correlated paradigm covers a broad range of materials, this school focuses specifically on geometrically frustrated magnets—systems where the lattice geometry prevents the spins from ordering into a simple pattern, even at absolute zero. The classic example is the triangular lattice with antiferromagnetic interactions: no spin configuration can satisfy all bonds simultaneously.
This frustration can lead to a quantum spin liquid, a state where spins remain dynamic and entangled down to zero temperature, with no magnetic order. The RVB state proposed for cuprates was originally a spin-liquid wavefunction, and the paradigm has since expanded to include Kitaev spin liquids, quantum spin ice, and other fractionalized phases. The key experimental signature is the absence of magnetic order combined with exotic excitations such as spinons—neutral quasiparticles carrying spin but no charge. The paradigm shares many theoretical methods with the correlated framework—Hubbard-like models, slave-particle approaches—but its distinctive contribution is the emphasis on frustration as a mechanism for suppressing order and enabling fractionalization.
The topological matter paradigm introduced a fundamentally different way of classifying electronic phases. Instead of focusing on interactions or magnetic order, it classifies materials by the topology of their electronic band structure, captured by invariants such as the Chern number or the Z₂ index. These invariants are protected by symmetry and are robust against perturbations that do not close the band gap. The paradigm was catalyzed by the discovery of topological insulators in the mid-2000s—materials that are insulating in the bulk but conduct electricity on their surface via topologically protected edge states.
Initially, the topological paradigm developed in parallel with the correlated framework. Its early successes were for non-interacting or weakly interacting systems, where band theory could still be used to compute topological invariants. The methodological contrast is sharp: the correlated paradigm focuses on strong interactions and many-body wavefunctions, while the topological paradigm focuses on symmetry-protected invariants and single-particle band topology. However, the two frameworks have increasingly converged. The discovery of fractional quantum Hall states and, later, fractional Chern insulators showed that interactions can produce topological phases with fractionally charged excitations. Today, a major frontier is the search for correlated topological phases, where strong interactions and nontrivial band topology coexist.
The moiré heterostructures and twistronics paradigm represents a synthesis of the previous three frameworks. By stacking two layers of a two-dimensional material—most famously graphene—with a small relative twist angle, a moiré superlattice is created. This superlattice dramatically modifies the electronic band structure, producing flat bands near the Fermi level. Flat bands enhance the density of states, making even weak interactions effectively strong. In twisted bilayer graphene at the “magic angle” of about 1.1°, the system becomes a strongly correlated electron system, exhibiting Mott insulating states, unconventional superconductivity, and magnetic phases.
What makes the moiré paradigm a synthesis is that it combines band-structure engineering (a legacy of band theory) with strong correlations (the core of the correlated paradigm) and topological properties (the moiré minibands can carry nontrivial Chern numbers). Moreover, twisted layers can be designed to introduce geometric frustration, linking to the spin-liquids paradigm. The twist angle becomes a tunable parameter, allowing researchers to explore the phase diagram of correlated topological matter in a single device. This framework is the youngest but has already become a dominant experimental platform, testing ideas from all earlier paradigms.
Today, four of the five frameworks remain actively pursued: the strongly correlated electron paradigm, the quantum magnetism and spin liquids paradigm, the topological matter paradigm, and the moiré heterostructures paradigm. Band theory, while no longer a research frontier, continues to serve as an essential infrastructure for interpreting experiments and designing new materials. The leading frameworks agree that both interactions and topology are central to quantum materials, and that the most exciting phenomena arise where they intersect. They disagree on which aspects are most fundamental: the correlated community often sees topology as a consequence of interactions, while the topological community emphasizes symmetry protection as the primary organizing principle. The mechanism of high-temperature superconductivity remains the most prominent unresolved tension, with spin-liquid, topological, and conventional pairing scenarios all still in play. The moiré paradigm offers a controlled laboratory to adjudicate these debates, but it also raises new questions about the universality of the phases observed. The field is thus characterized by productive pluralism, with each framework providing a distinct lens on the same underlying quantum reality.