Topological Doping and Superconductivity in Cuprates: An Experimental Perspective

Hole doping into a correlated antiferromagnet leads to topological stripe correlations, involving charge stripes that separate antiferromagnetic spin stripes of opposite phases. The topological spin stripe order causes the spin degrees of freedom within the charge stripes to feel a geometric frustration with their environment. In the case of cuprates, where the charge stripes have the character of a hole-doped two-leg spin ladder, with corresponding pairing correlations, anti-phase Josephson coupling across the spin stripes can lead to a pair-density-wave order in which the broken translation symmetry of the superconducting wave function is accommodated by pairs with finite momentum. This scenario is now experimentally verified by recently reported measurements on La2−xBaxCuO4 with x=1/8. While pair-density-wave order is not common as a cuprate ground state, it provides a basis for understanding the uniform d-wave order that is more typical in superconducting cuprates.

Geometric frustration in the myosin superlattice of vertebrate muscle

Geometric frustration results from an incompatibility between minimum energy arrangements and the geometry of a system, and gives rise to interesting and novel phenomena. Here, we report geometric frustration in a native biological macromolecular system---vertebrate muscle. We analyse the disorder in the myosin filament rotations in the myofibrils of vertebrate striated (skeletal and cardiac) muscle, as seen in thin-section electron micrographs, and show that the distribution of rotations corresponds to an archetypical geometrically frustrated system---the triangular Ising antiferromagnet. Spatial correlations are evident out to at least six lattice spacings. The results demonstrate that geometric frustration can drive the development of structure in complex biological systems, and may have implications for the nature of the actin--myosin interactions involved in muscle contraction. Identification of the distribution of myosin filament rotations with an Ising model allows the extensive results on the latter to be applied to this system. It shows how local interactions (between adjacent myosin filaments) can determine long-range order and, conversely, how observations of long-range order (such as patterns seen in electron micrographs) can be used to estimate the energetics of these local interactions. Furthermore, since diffraction by a disordered system is a function of the second-order statistics, the derived correlations allow more accurate diffraction calculations, which can aid in interpretation of X-ray diffraction data from muscle specimens for structural analysis.

Geometric frustration produces long-sought Bose metal phase of quantum matter

Two of the most prominent phases of bosonic matter are the superfluid with perfect flow and the insulator with no flow. A now decades-old mystery unexpectedly arose when experimental observations indicated that bosons could organize into the formation of an entirely different intervening third phase: the Bose metal with dissipative flow. The most viable theory for such a Bose metal to date invokes the use of the extrinsic property of impurity-based disorder; however, a generic intrinsic quantum Bose metal state is still lacking. We propose a universal homogeneous theory for a Bose metal in which geometric frustration confines the essential quantum coherence to a lower dimension. The result is a gapless insulator characterized by dissipative flow that vanishes in the low-energy limit. This failed insulator exemplifies a frustration-dominated regime that is only enhanced by additional scattering sources at low energy and therefore produces a Bose metal that thrives under realistic experimental conditions.

Director Deformations, Geometric Frustration, and Modulated Phases in Liquid Crystals

This article analyzes modulated phases in liquid crystals, from the long-established cholesteric and blue phases to the recently discovered twist-bend, splay-bend, and splay nematic phases, as well as the twist-grain-boundary (TGB) and helical nanofilament variations on smectic phases. The analysis uses the concept of four fundamental modes of director deformation: twist, bend, splay, and a fourth mode related to saddle-splay. Each mode is coupled to a specific type of molecular order: chirality, polarization perpendicular and parallel to the director, and octupolar order. When the liquid crystal develops one type of spontaneous order, the ideal local structure becomes nonuniform, with the corresponding director deformation. In general, the ideal local structure is frustrated; it cannot fill space. As a result, the liquid crystal must form a complex global phase, which may have a combination of deformation modes, and may have a periodic array of defects. Thus, the concept of an ideal local structure under geometric frustration provides a unified framework to understand the wide variety of modulated phases. Expected final online publication date for the Annual Review of Condensed Matter Physics, Volume 13 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Magnetic Frustration

Spins often prefer to anti-align with their neighbors in antiferromagnetic correlation. Materials with triangle lattices exhibit energetic degeneracy among the possible rearrangements of anti-aligned spins, which is denoted geometric frustration that is associated with strongly depressed transitions to magnetic order. Honeycomb iridates and ruthenates, pyrochlore systems, and double-perovskite iridates all feature triangular lattices as primary building blocks of their structures. Another frustration mechanism evolves from the Kitaev’s exact solution of a spin-liquid model on a honeycomb lattice with strong spin-orbit interactions. The protracted search for a Kitaev spin liquid has recently focused on the honeycomb itidates Na2IrO3 and Li2IrO3. A newer kind of quantum liquid has been identified in the magnetic insulator Ba4Ir3O10, where Ir3O12 trimers form an unfrustrated square lattice.

Effects of geometric frustration in Kitaev chains

We study the topological phase transitions of a Kitaev chain frustrated by the addition of a single long-range hopping. In order to study the topological properties of the resulting legged-ring geometry (Kitaev tie model), we generalize the transfer matrix approach through which the emergence of Majorana edge modes is analyzed. We find that geometric frustration gives rise to a topological phase diagram in which non-trivial phases alternate with trivial ones at varying the range of the hopping and the chemical potential. Robustness to disorder of non-trivial phases is also proven. Moreover, geometric frustration effects persist when translational invariance is restored by considering a multiple-tie system. These findings shed light on an entire class of experimentally realizable topological systems with long-range couplings.

Spin-ice physics in cadmium cyanide

Spin-ices are frustrated magnets that support a particularly rich variety of emergent physics. Typically, it is the interplay of magnetic dipole interactions, spin anisotropy, and geometric frustration on the pyrochlore lattice that drives spin-ice formation. The relevant physics occurs at temperatures commensurate with the magnetic interaction strength, which for most systems is 1–5 K. Here, we show that non-magnetic cadmium cyanide, Cd(CN)2, exhibits analogous behaviour to magnetic spin-ices, but does so on a temperature scale that is nearly two orders of magnitude greater. The electric dipole moments of cyanide ions in Cd(CN)2 assume the role of magnetic pseudospins, with the difference in energy scale reflecting the increased strength of electric vs magnetic dipolar interactions. As a result, spin-ice physics influences the structural behaviour of Cd(CN)2 even at room temperature.

Geometric frustration in polygons of polariton condensates creating vortices of varying topological charge

Vorticity is a key ingredient to a broad variety of fluid phenomena, and its quantised version is considered to be the hallmark of superfluidity. Circulating flows that correspond to vortices of a large topological charge, termed giant vortices, are notoriously difficult to realise and even when externally imprinted, they are unstable, breaking into many vortices of a single charge. In spite of many theoretical proposals on the formation and stabilisation of giant vortices in ultra-cold atomic Bose-Einstein condensates and other superfluid systems, their experimental realisation remains elusive. Polariton condensates stand out from other superfluid systems due to their particularly strong interparticle interactions combined with their non-equilibrium nature, and as such provide an alternative testbed for the study of vortices. Here, we non-resonantly excite an odd number of polariton condensates at the vertices of a regular polygon and we observe the formation of a stable discrete vortex state with a large topological charge as a consequence of antibonding frustration between nearest neighbouring condensates.