Neutron Scattering and Symmetry in Condensed Matter Physics

As condensed matter physicists, we think of crystals as beautifully ‘symmetric’ objects with periodic properties in space and are familiar with the translational and rotational symmetries of their lattices. In general, in physics, when we speak of symmetry, we refer to ‘the property of remaining invariant under certain mathematical transformation’ of, for example, the sign of electric charge, parity, direction of time flow, or orientation in space. Such mathematical transformations often also help with simplification of numerical calculations of physical laws. Symmetry concepts, however, offer more than just mathematical methodologies and simplified calculations. The importance of symmetry in condensed matter stems from the fact that there is an intimate connection between the symmetry of the Hamiltonian that describes a system and its properties (energy levels and their degeneracy). Since it is completely independent from the exact form of the Hamiltonian, the symmetry of the system imposes restrictions (selection rules for transitions between different states) on the possible solutions, and it is possible to determine solutions entirely by symmetry arguments. The symmetry properties of a physical system can provide powerful insight into its nature and offer predictions about the values of its measurable physical quantities. For example, one can determine whether a specific property is allowed under certain symmetry conditions and degrees of freedom required to describe it, entirely through symmetry arguments. Furthermore, from any observed regularities in measured quantities, one can trace back the symmetry of a system. Physical systems can thus be classified based on their symmetry. Such classification of physical systems, naturally, then links very different physical systems but with similar symmetry (even bridging classical and quantum physics systems).

Symmetry concepts also play an important role in the study of phase transitions. For example, a paramagnetic to ferromagnetic second-order phase transition involves a change in symmetry: the magnetization (order parameter) goes from zero value in the high-symmetry phase to a nonzero value in the low-symmetry phase. For phase transitions that are described by Landau phenomenological theory, a generalized phase diagram can be constructed again entirely based on symmetry arguments. Symmetry concepts are not only crucial for studying conventional phases of matter such as crystals, magnetic materials, and traditional superconductors; they are also key in topological insulators with a subtler interplay between symmetry and topology, resulting in a more complex phenomenology.

The second half of the last century witnessed the development of many experimental techniques that have enabled scientists to make precise measurements of microscopic properties of condensed matter systems. One notable technique is neutron scattering, revealing information about the sample that is often not attainable through other techniques. The unique properties of neutrons are the reason neutron scattering is proven to be one of the most valuable probes in condensed matter physics. Neutrons interact with condensed matter systems in two ways: nuclear and magnetic. Since thermal neutrons have a wavelength which is much larger than the nuclear interaction range, their scattering is described by s-wave scattering and characterized by a scattering length parameter. Neutrons also have a magnetic moment. Thus, they interact magnetically with electrons’ spin or orbital moment. In a neutron scattering experiment, a beam of well-characterized neurons are incident onto the sample and are detected after they interact with it. The properties of detected neutrons (orientation, energy, spin state) are then used to infer the sample properties. Two distinct types of neutron experiments include: elastic, where static properties (lattice and magnetic structures) are revealed, and inelastic, where dynamical properties (phonons, spin-waves, and spin excitations) are determined. Since the original work of Clifford Shull at Oak Ridge National Lab and Bertram Brockhouse at Chalk Ricer Laboratories in the 1940s and 1950s, for which they received a joint Nobel Prize in Physics in 1994, there have been tremendous advances in all aspects of this technique, from sophisticated instruments and extreme sample environments, to data analysis and visualization. All these have led to significant contributions by neutron scattering in all emerging themes of condensed matter physics such as quantum materials, unconventional superconductors (cuprate, iron–arsenide, and heavy fermions), low-dimensional quantum magnets, and topological insulators. Many novel theoretical models and their predictions such as Haldane gap and topological skyrmions were validated and further advanced by neutron scattering experiments.

Here, we present a selection of neutron scattering articles that showcase the power of this technique and how the usage of symmetry-based methods for the analysis of the ensuing data has led to many gains in understanding the physical behavior of quantum materials.

Dr. Zahra Yamani
Guest Editor

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• symmetry
• neutron scattering
• condensed matter physics
• magnetism
• superconductivity
• lattice structure
• lattice vibrations
• phonons
• magnetization
• magnetic order
• spin excitations
• crystal field effects