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Beekman, Aron J.; Nissinen, Jaakko; Wu, Kai; Zaanen, Jan Dual Gauge Field Theory of Quantum Liquid Crystals in Three Dimensions

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Beekman, Aron J.; Nissinen, Jaakko; Wu, Kai; Zaanen, Jan Dual Gauge Field Theory of Quantum Liquid Crystals in Three Dimensions This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Beekman, Aron J.; Nissinen, Jaakko; Wu, Kai; Zaanen, Jan Dual gauge field theory of quantum liquid crystals in three dimensions Published in: Physical Review B DOI: 10.1103/PhysRevB.96.165115 Published: 09/10/2017 Document Version Publisher's PDF, also known as Version of record Please cite the original version: Beekman, A. J., Nissinen, J., Wu, K., & Zaanen, J. (2017). Dual gauge field theory of quantum liquid crystals in three dimensions. Physical Review B, 96(16), 1-50. [165115]. https://doi.org/10.1103/PhysRevB.96.165115 This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) PHYSICAL REVIEW B 96, 165115 (2017) Dual gauge field theory of quantum liquid crystals in three dimensions Aron J. Beekman,1,* Jaakko Nissinen,2,3 Kai Wu,4 and Jan Zaanen2 1Department of Physics, and Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 2Institute-Lorentz for Theoretical Physics, Leiden University, PO Box 9506, NL-2300 RA Leiden, The Netherlands 3Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland 4Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94025, USA (Received 19 June 2017; revised manuscript received 12 September 2017; published 9 October 2017) The dislocation-mediated quantum melting of solids into quantum liquid crystals is extended from two to three spatial dimensions, using a generalization of boson-vortex or Abelian-Higgs duality. Dislocations are now Burgers-vector-valued strings that trace out worldsheets in space-time while the phonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. We propose an effective dual Higgs potential that allows for restoring translational symmetry in either one, two, or three directions, leading to the quantum analogues of columnar, smectic, or nematic liquid crystals. In these phases, transverse phonons turn into gapped, propagating modes, while compressional stress remains massless. Rotational Goldstone modes emerge whenever translational symmetry is restored. We also consider the effective electromagnetic response of electrically charged quantum liquid crystals, and find among other things that as a hard principle only two out of the possible three rotational Goldstone modes are observable using propagating electromagnetic fields. DOI: 10.1103/PhysRevB.96.165115 I. INTRODUCTION superconductors, could feature symmetry properties anal- ogous to classical liquid crystals. The stripe “crystalline” A. Quantum liquid crystals: The context order is now destroyed at zero temperature by quantum Liquid crystals are “mesophases” of matter with a “ves- fluctuations in the form of proliferating dislocations, such tigial” pattern of spontaneous symmetry breaking arising at that on macroscopic length scales the system forms a nematic intermediate temperatures or coupling: rotational symmetry is quantum fluid (superconductor), which maintains, however, broken while translational invariance partially or completely the orientational preference of the stripe electronic crystal. persists. Classical liquid crystals are formed from highly This constituted the birth of the subject of quantum liquid anisotropic molecular constituents which, upon cooling from crystals. Quite some empirical support was found since then the liquid phase, can order their respective orientations while affirming the existence of such forms of quantum liquid maintaining translational freedom. Only at lower temperatures crystals. This includes direct evidences for the existence of does crystallization set in. These forms of matter have been quantum nematic order in underdoped cuprates, likely related known for about a century, and their theoretical description to the original context of fluctuating stripes [12–19]. A similar was established by De Gennes and many others [1–3]. Starting mechanism could be at hand in so-called pair density waves, from the opposite side, it was long realized that dislocations which combines charge, spin, and superconducting orderings (the topological defects associated with translational order) [20–26]. This theme later flourished in the context of the are responsible for material degradation and even melting iron superconductors where quite some evidence surfaced of solids [4]. Berezinskii, Kosterlitz, and Thouless (BKT) in for the prominent role of orientational symmetry breaking their landmark papers already suggested that unbinding of driven by the electron system as being central to their physics dislocations and disclinations (rotational topological defects) [27–29]. An ambiguity in these condensed matter systems will lead to the disordering of two-dimensional solids [5–7], is that the crystal formed by the atoms is already breaking the theory of which was further developed and refined by space translations and rotations while the electron and ion Nelson, Halperin, and Young [8–10]. We will refer to the systems are coupled. The quasi-two-dimensional electron topological melting driven by dislocation unbinding as the systems in the iron and copper superconductors are typically KTNHY transition. Here, it was also predicted that an inter- realized in tetragonal square lattices where the rotational mediate phase exists as a result of the exclusive proliferation symmetry is broken to a point group characterized by a fourfold of dislocations in a triangular 2D crystal, dubbed the hexatic axis. This fourfold symmetry is spontaneously broken to an liquid crystal. Translational symmetry is fully restored but orthorhombic crystal structure characterized by a twofold the rotational symmetry remains broken down to the C6 point rotational symmetry C4 → C2, dubbed the “Ising-nematic group characterizing the triangular crystal. phase.” Given that symmetrywise the purely electronic and Almost two decades later, Kivelson, Fradkin, and Emery crystalline tendencies to lower the point group symmetry are [11] proposed that the spatial ordering of electrons in strongly indistinguishable, one does face a degree of ambiguity that correlated electron systems, as realized in underdoped high-Tc cannot be avoided, giving rise to ongoing debates about the origin of the electronic nematicity in these materials [29]. Inspired by the initial suggestion by Kivelson et al. one of *[email protected] the authors (J.Z.) initiated a program to extend the KTNHY 2469-9950/2017/96(16)/165115(50) 165115-1 ©2017 American Physical Society BEEKMAN, NISSINEN, WU, AND ZAANEN PHYSICAL REVIEW B 96, 165115 (2017) topological melting ideas to the quantum realms, initially in this theory to three spatial dimensions and we recommend the two space dimensions. The emphasis has been here all along novice to the subject to have a close look at QLC2D first. We on the fundamental, theoretical side based on the symmetries will often refer back to those results, and we do not hesitate to and associated defects. The main restriction is that it only skip derivations and explanations provided there when these deals with matter formed from bosons: the constructions rest are representative for the way things work in 3+1D as well. on the machinery of statistical physics being mobilized in We also refer the reader to the introduction of QLC2D for the D + 1-dimensional Euclidean space-time, turning into the more background on the history of and the physical interest in quantum physics of bosons after Wick rotation. This matter quantum liquid crystals. lives in the Galilean continuum and the point of departure is the spontaneous breaking of space translations and rotations into a crystal. The KTNHY transition is just one particular B. From two to three dimensions: Weak-strong duality example of a Kramers-Wannier (or weak-strong) duality and and the string condensate it was found out in the 1980s how to extend this to three Our universe has three spatial dimensions and therefore dimensions when dealing with Abelian symmetries. In the the most natural quantum states of matter are formed in 3+1 context of crystalline elasticity one can rest on strain-stress dimensions. The generalization of the theory to 3+1D has duality, where phonon degrees of freedom are mapped to been quite an ordeal—we are even not completely confident dual stress gauge fields. This amounts to a generalization that the solution we present here is really watertight. Wherein of the famous vortex-boson or Abelian-Higgs duality, as lies the difficulty? This is rooted in the fundamentals of pioneered by Kleinert [30]. Using the well-known mapping of Abelian weak-strong dualities, which are very well under- a D-dimensional quantum to a D + 1-dimensional classical stood in both 1+1D/2D (BKT topological melting) and system, the 2+1D quantum liquid crystals were investigated 2+1D/3D (Abelian-Higgs duality [36,43–53]), while it is by strain-stress duality starting with Ref. [31]. The procedure much less settled in 3+1D for quite deep reasons. At the is essentially the same as the KTNHY case in two dimensions. heart of these dualities is the notion that given a particular One first establishes the structure of the weak-strong duality form of spontaneous symmetry breaking, the unique agents by focusing on the minimal U(1) case associated with vortex associated with restoring the symmetry are the topological melting, to then extend it to the richer theater of the space excitations. groups underlying the crystalline symmetry breaking, profiting Let us first consider a broken global U(1) symmetry, from the fact that the restoration of translational invariance by where the vortex is the topological workhorse.
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