Interplay of Multiple Symmetries, Emerging Exotic States and Fields, and the New Quantum Complexity in Condensed Matter Physics

Dear Colleagues,

Symmetry is the strongest concept in condensed matter physics, which is the laboratory of ideas for other fields of physics as well. The breaking of specific symmetries allowed for classifying materials according to their corresponding quantum ordered states. Details of their structure and interactions have almost become irrelevant in the so-called low energy sector, so that just symmetry breaking matters.

In the last decades, a novel form of quantum complexity has emerged. Exotic new kinds of quantum states have been discovered in real systems. They often appear in proximity to a potential quantum critical point, or competing frustrated interactions or a crossover between dimensionalities. In the complex phase, diagrams of the relevant systems these states sometimes define domes around expected quantum critical points. Sometimes, these kinds of states appear only at interfaces or other nanostructures.  The common characteristic of all of these exotic states is that the fields and/or order parameters that characterize them involve multiple symmetries, of which the non-trivial interplay plays a key role not only for their emergence, but also for the complex, extraordinary, and, in many cases, still enigmatic transitions that they exhibit. Even a real space phase separation has been observed for some of them.

The study of unconventional superconductors, those for which additional symmetries to the usual gauge symmetry are involved, provided an early entry into this word of quantum complexity. It was first observed in the antiferromagnetic superconductors family, which include systems as diverse and important as cuprates, heavy fermions, pnictides, organics, and others, but now it seems to be present in all families of unconventional superconductors. Unconventional particle-hole condensates, which include all kinds of unconventional density waves and magnetic states, as well as various so called Pomeranchuk instabilities, including electronic nematic or electronic smectic states, etc., are similar examples of exotic states that are non-superconducting, yet they are abundant in similar materials and may have a non-trivial interplay with superconductivity as well. Multiferroics are another typical host of exotic states from the interplay of inversion symmetry, breaking with the symmetries involved in the magnetic particle-hole condensates of various kinds, while the somewhat analogous non-centrosymmetric superconductors are involved in the interplay of the broken inversion symmetry with various symmetries in the particle–particle channel.

The interplay of symmetry with topology is another open exciting issue, but when multiple symmetries and some of the emerging exotic states or fields are involved, this becomes even more exciting. For example, in a two dimensional tetragonal system, the commensurate d_x2-y2 CDW named orbital antiferromagnet coexisting with a d_xy CDW produces a chiral insulating state that is a typical Chern insulator, exhibiting the anomalous quantum Hall effect among other extraordinary properties, like the giant Nernst effect or a Meissner effect for vertical fields. If both order parameters are SDW instead of CDW, nothing changes in their topology, but if one of the two is CDW and the other is SDW, then a helical state is produced, which is a typical so-called topological insulator exhibiting the spin quantum Hall effect. Needless to say, effectively spinless p_x + ip_y exotic superconducting states are arguably the best supports of Majorana fermions and topological qubits, and the interplay of multiple symmetries is at the basis of the paths towards the engineering of such states in nanostructures. The analogous p-wave particle-hole condensates are equally exciting states that exhibit topological transitions and extraordinary phenomena as well. The exotic induced states that appear at the interfaces of some insulators may also be regarded as resulting from an engineered interplay between multiple symmetries. Moreover, some versions of all the above exotic states could potentially be produced in artificial lattices of fermions, and very significant achievements in this direction have happened recently. In some cases, the interplay of multiple symmetries produces emergent exotic fields and no accompanying stable states, and quantum complexity may thus show up in the dynamic behavior of such systems. Artificial spin–orbit coupling is a typical example where the interplay of multiple symmetries may lead to the emergence of an SOC field.

The expanding published information for the wide range of related phenomena not listed here, usually mixes potentially irrelevant for the low energy sector detailed characteristics of each system with the much stronger generic symmetry aspects that become thus less transparent. It is both timely and important to have a Special Issue in which this common element of multiple symmetry interplay will be the main focus for all related phenomena. This will eventually help with the identification of universal rules and the mechanisms of the multiple symmetry interplay, ideally achieving a simplified global understanding of this novel quantum complexity.

I therefore invite both experimental and theoretical contributions, which may be either original research or short reviews of related phenomena, and should focus more on the identification of the multiple symmetries and their interplay behind specific exotic states and related transitions, and possibly some contributions with generic ideas about the mechanisms that govern this new quantum complexity.

Prof. Dr. Varelogiannis Georgios
Guest Editor

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• Coexistence and competition of multiple order parameters and fields
• Hidden symmetries and hidden order parameters
• Unconventional Superconductors
• Unconventional particle-hole condensates
• Unconventional Density Waves
• Unconventional magnetism
• Pomeranchuk instabilities
• Symmetry and topology
• Engineering of Majorana fermions
• Induced phases and fields at interfaces
• Multiferroics and related transitions
• Inhomogeneous quantum ordered states
• Artificial spin–orbit coupling
• Quantum complexity in artificial lattices