Quantum spin liquids.

BACKGROUND Years ago, Lev Landau taught us how to think about distinct phases of matter through an order parameter that characterizes the symmetry-broken state relative to the symmetry-preserving state from which it emerges. More recently, however, it has been realized that not all phases of matter are captured by this paradigm. This was spectacularly demonstrated by the discovery of fractional quantum Hall states in the 1980s. Over the years, it has been elucidated that these states, along with their exotic excitations—quasiparticles carrying a rational fraction of the elementary charge of an electron—are the consequence of topological properties of ground state wave functions with a special type of long-range quantum entanglement. One might wonder whether analogous phenomena occur for spins. Whether these “quantum spin liquids” actually exist in nature has been the subject of much investigation. ADVANCES Since Philip Anderson contemplated the idea of quantum spin liquids in 1973, there has been a lot of research to establish what they are and how they can be characterized. Of particular note was the realization that an effective low-energy theory inevitably resembles the gauge theory treatments also invoked in high-energy physics. However, these gauge fields are “emergent” in the sense that they reflect important structure of the many-particle state. Specifically, they describe excitations that carry a fraction of the quantum of spin in terms of emergent quasiparticles with gauge charge and/or gauge flux, analogous to the electric charge and magnetic flux in electrodynamics. One consequence is that these quasiparticle excitations can have nontrivial statistical interactions when they are braided around each other. Although most studies have focused on gapped spin liquids, equally intriguing are gapless versions—for instance, ones where the quasiparticle (“spinon”) spectrum is that of relativistic electrons described by the Dirac equation. Much work has been done to address specific models and connect them to experimental analogs. This has involved a combination of analytically solvable models, as well as the development of new numerical methods that provide approximate solutions given a microscopic (lattice scale) Hamiltonian. Perhaps most excitingly, there has been an increasingly promising effort to identify quantum spin liquids in nature. Much of the work has focused on materials where the magnetic ions reside on lattices that frustrate classical magnetic order. Examples include the triangular, kagome, hyperkagome, and pyrochlore lattices. Several candidate materials have been discovered, including organic salts, where molecular dimers realize spin-½ degrees of freedom on a distorted triangular lattice; herbertsmithite, where spin-½ copper ions form a kagome lattice; and α-RuCl3, where j =1/2 ruthenium ions form a honeycomb lattice and that is thought to be proximate to the famous Kitaev model. All of these materials have properties reminiscent of spin liquids, though their documented fidelity as model systems is limited by disorder, subleading interactions, or lack of experimental information. OUTLOOK Given the infinite variety of potential materials and the many research groups now exploring this space, we are optimistic that a pristine materials realization of a quantum spin liquid will be discovered in the coming years. Perhaps even now a spin liquid exists in a long-forgotten drawer of a museum. Efforts to achieve ultrahigh-quality samples and new experiments designed to determine whether fractionalization and long-range entanglement occur in such materials will be key. In addition to tantalizing clues based on such techniques as thermal Hall conductivity, nuclear magnetic resonance, and inelastic neutron scattering, future methods may involve looking for spin currents to prove fractionalization, as has been done for charge degrees of freedom in the fractional quantum Hall case, or probing the range and character of quantum entanglement, as previously done in ultracold gases. Moreover, if quasiparticle excitations can be isolated and then manipulated, the prospect of a new form of topologically protected quantum computation also exists. Finally, chemically doped versions of spin liquids have been predicted to provide an unconventional route to superconductivity. The search for such phases will undoubtedly be an exciting undertaking.