Symmetry analysis of translational symmetry broken density waves: Application to hexagonal lattices in two dimensions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Venderbos, J. W. F. “Symmetry Analysis of Translational Symmetry Broken Density Waves: Application to Hexagonal Lattices in Two Dimensions.” Physical Review B 93, no. 11 (March 3, 2016). © 2016 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevB.93.115107 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/101605 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW B 93, 115107 (2016) Symmetry analysis of translational symmetry broken density waves: Application to hexagonal lattices in two dimensions J. W. F. Venderbos Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 17 December 2015; published 3 March 2016) In this work we introduce a symmetry classification for electronic density waves which break translational symmetry due to commensurate wave-vector modulations. The symmetry classification builds on the concept of extended point groups: symmetry groups which contain, in addition to the lattice point group, translations that do not map the enlarged unit cell of the density wave to itself, and become “nonsymmorphic”-like elements. Multidimensional representations of the extended point group are associated with degenerate wave vectors. Electronic properties such as (nodal) band degeneracies and topological character can be straightforwardly addressed, and often follow directly. To further flesh out the idea of symmetry, the classification is constructed so as to manifestly distinguish time-reversal invariant charge (i.e., site and bond) order, and time-reversal breaking flux order. For the purpose of this work, we particularize to spin-rotation invariant density waves. As a first example of the application of the classification we consider the density waves of a simple single- and two-orbital square lattice model. The main objective, however, is to apply the classification to two-dimensional (2D) hexagonal lattices, specifically the triangular and the honeycomb lattices. The multicomponent density waves corresponding to the commensurate M-point ordering vectors are worked out in detail. To show that our results generally apply to 2D hexagonal lattices, we develop a general low-energy SU(3) theory of (spinless) saddle-point electrons. DOI: 10.1103/PhysRevB.93.115107 I. INTRODUCTION [9–13]. In the context of density wave states, a number of proposals for realizing interaction-induced topological Interactions between electrons in solids are responsible particle-hole condensates have been made in recent years for a large variety of symmetry broken electronic phases. [14–18], which have set the stage for the rapidly growing and Unconventional superconductivity and antiferromagnetism evolving field of interacting topological phases. Motivated by are two canonical examples. From the perspective of weak the exciting possibility of electrons condensing into exotic col- coupling, the antiferromagnet is an example of a density wave: lective phases, possibly characterized by large nonzero angular a spatial modulation of electronic spin density characterized by momentum and spontaneously generated charge or spin cur- a finite (generally commensurate) propagation vector, breaking rents, in this paper we develop a symmetry analysis of density translational symmetry. In general, apart from spin density, waves with finite commensurate wave-vector modulation. other observables involving particle-hole pairs such as charge When particle-hole condensation occurs at finite wave density, orbital density, or current density can acquire finite vector, breaking translational symmetry, the crystal symmetry wave-vector modulations and break translational symmetry, group is reduced to the group of the wave vector. The giving rise to many possible unconventional and exotic density symmetry classification proposed in this paper naturally wave states. takes translational symmetry breaking into account. Instead A systematic way to study unconventional density waves of a reduced symmetry group, it is defined in terms of a is to classify them according to their symmetry properties. symmetry group called the extended point group. The essential Density wave states are condensates of particle-hole pairs and feature of the extended point group can be summarized by can be classified by specifying the angular momentum and spin noting that it treats a given set of ordering wave vectors of the particle-hole pair [1,2], in close analogy to unconven- on similar footing with angular momentum channels. As a tional superconductivity [3]. For instance, the familiar charge result, extended point groups provide a means to classify and spin density waves are both s-wave condensates. Distinct translational symmetry broken density waves, in much the angular momentum channels are labeled by representations same way as ordinary point groups provide a means to classify of the symmetry group of the crystal. In case of particle-hole different angular momentum channels [1–3]. In particular, condensates, contrary to a superconductor, spin and angular multidimensional representations signal degeneracies in both momentum are not tied together by Fermi statistics. cases. This property is particularly useful in case of inequiv- Knowledge of the symmetry of density wave states is alent but symmetry-related ordering wave vectors. Another essential for understanding the properties of materials, in way to think of extended point groups is to consider them as particular for connecting theory to experiment. In addition, the point groups of a mathematically enlarged unit cell, i.e., from a modern condensed matter perspective, determining a unit cell which supports the physically enlarged unit cell the symmetry of an electronic phase is particularly relevant, associated with condensation at given wave vector (or set of as symmetries can give rise to topological electronic states wave vectors). The extended point group contains translations [4–8]. The presence of a symmetry, such as time-reversal that do not map the enlarged unit cell to itself. These are the and/or parity symmetry, can protect or prohibit a topological translations broken by density wave formation. phase, which may be either gapped or gapless. A number of In the next section we start by introducing the symmetry works have specifically explored the role of lattice symmetries classification based on the notion of extended point groups. 2469-9950/2016/93(11)/115107(22) 115107-1 ©2016 American Physical Society J. W. F. VENDERBOS PHYSICAL REVIEW B 93, 115107 (2016) The classification of particle-hole condensates manifestly classification that takes this feature into account in a systematic distinguishes two types of order: time-reversal invariant charge way, we describe the system in terms of a mathematically order and time-reversal breaking flux order. The former class enlarged unit cell, chosen so as to support the physically ∼ˆ † ˆ enlarged unit cell of the density waves we want to study. The consists of diagonal site order of the form ψi ψi and ∼ ˆ † ˆ choice of the mathematically enlarged unit cell is determined off-diagonal bond order of the form Re ψi ψj , where i,j label lattice sites. The second class, flux order, corresponds to by the ordering wave vectors relevant to the particular physical dynamically generated orbital currents, i.e., order of the form system. All possible patterns of translational symmetry break- † ing can then be described within the new enlarged unit cell. ∼ Im ψˆ ψˆ . Importantly, instead of the imaginary part of i j The symmetry group of the (mathematically) enlarged unit bond expectation values, the symmetry classification is defined cell is the extended point group, which consists of all point in terms of fluxes: the sum of phases around a closed lattice group elements of the Bravais lattice supplemented with the plaquette. This preserves gauge invariance by construction. translations that do not map the enlarged unit cell onto itself, In this work, we only consider spin-singlet or spin-rotation i.e., the translations that can be broken in the density wave state. invariant density wave order. As a result, these are an extension of the ordinary point groups As a simple first example, we revisit the density wave with additional composite elements. Naturally, the extended states of the square lattice by an application of the symmetry point group depends on the set of ordering vectors. classification. As a second example, we apply the symmetry As an example, consider ordering at wave vector Q = classification to a two-orbital square lattice model, which can be viewed as a basic description of certain iron-pnictide ma- (π,π) of the square lattice. This breaks the translation T (a1) over the unit vector a1 and doubles the unit cell. Clearly, the terials [19,20]. Given these useful examples, our main goal is to apply the symmetry classification to lattices with hexagonal translation T (a1) does not map the doubled unit cell onto itself, but instead connects the two sites of the enlarged cell. The symmetry, in particular the triangular and honeycomb lattices. + − Hexagonal lattices have two sets of special