Rotation-reversal symmetries in crystals and handed structures

Journal name:
Nature Materials
Year published:
Published online


Symmetry is a powerful framework to perceive and predict the physical world. The structure of materials is described by a combination of rotations, rotation-inversions and translational symmetries. By recognizing the reversal of static structural rotations between clockwise and counterclockwise directions as a distinct symmetry operation, here we show that there are many more structural symmetries than are currently recognized in right- or left-handed helices, spirals, and in antidistorted structures composed equally of rotations of both handedness. For example, we show that many antidistorted perovskites possess twice the number of symmetry elements as conventionally identified. These new ‘roto’ symmetries predict new forms for ‘roto’ properties that relate to static rotations, such as rotoelectricity, piezorotation, and rotomagnetism. They enable a symmetry-based search for new phenomena, such as multiferroicity involving a coupling of spins, electric polarization and static rotations. This work is relevant to structure–property relationships in all materials and structures with static rotations.

At a glance


  1. Rotation-reversal symmetry and other antisymmetry operations.
    Figure 1: Rotation-reversal symmetry and other antisymmetry operations.

    Where the colours of atoms in a denote the orientation of magnetic spins, +M (blue, spin up, left-handed loop) and −M (orange, spin down, right-handed loop), the time-reversal symmetry operation 1 will switch them to −M and +M, respectively, as in b. c, Operations that reverse vectors (polar or axial, and time dependent or static) are shown. In addition to the spatial inversion, , and time reversal, 1′, required to complete this table, a new operation, namely the rotation-reversal symmetry operation, 1Φ, is required. If the colours of two sets of molecules (dashed squares), as in d, denote static rotations, +Φ and −Φ , the rotation-reversal symmetry operation 1Φ switches them to −Φ and +Φ, respectively, as in e.

  2. Roto groups and property classification.
    Figure 2: Roto groups and property classification.

    a, Based on the presence or absence of the three antisymmetry operations, 1Φ, , 1′ eight types of groups are defined: triple grey (TG), magneto (M), roto (R), magneto-roto (MR), polar (P), magneto-polar (MP), roto-polar (RP), and magneto-roto-polar (MRP). The number x of the MR groups is presently unknown. b, The number of point groups (PGs) that are invariant groups of a net (non-zero) spin, a net static rotation, and a net electric polarization, and combinations thereof, is given. A list of these point groups is given in the Supplementary Table S1.

  3. Symmetries in antidistorted cubic perovskite lattices.
    Figure 3: Symmetries in antidistorted cubic perovskite lattices.

    Antidistorted octahedra are shown disconnected for clarity. The Glazer rotation a0+a0+c0+ is depicted in a, where orange and aqua correspond to left- and right-handed rotations of octahedra, respectively, about the axes. Loops with arrows indicate the sense of rotation, and number of loops indicates the magnitude of rotation. The exploded symmetries in b reveal that whereas the conventional space group symmetry is Immm1′ , the complete roto space group symmetry is I4Φ/mmmΦ1′. Panel c shows a0+a0+c0+ with magnetic spins inside each octahedron, with an exploded view of its symmetries in d. The conventional space group symmetry is Immm , but the complete roto space group symmetry is I4Φ/mmmΦ.

  4. Symmetries in helices and spirals.
    Figure 4: Symmetries in helices and spirals.

    A solenoid can possess (a) static left-handed winding (orange), and carries a charge current (yellow arrows). The 1Φ symmetry only switches the static winding. The 1′ symmetry only switches the charge current. The 1′Φ symmetry switches both. A single infinite helix, (b) with a pitch of Λ, and a general infinite double helix, (c) with arbitrary shift between the two helices in the z direction, both have a roto point group of . A finite continuous helix with integral windings, (d) has a roto point group of mΦmΦ2 . A finite planar spiral, (e), has a roto point group of mmΦ2Φ . A single left-handed infinite helix, (f) with 32 screw axis with fractional atom positions at 0, 1/3, and 2/3 of the pitch is transformed by the 1Φ operation to right-handed 31 as in g. The complete roto point group is .


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  1. Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16803, USA

    • Venkatraman Gopalan
  2. Department of Physics, Eberly College of Science, The Pennsylvania State University, Penn State Berks, PO Box 7009, Reading, Pennsylvania 19610, USA

    • Daniel B. Litvin


V.G. conceived the idea of rotation-reversal symmetry, the roto groups, their influence on properties, and derived the symmetries listed using symmetry diagrams. D.B.L. critiqued these ideas, helped develop formal definitions for these concepts, and derived the symmetries listed using group theoretical methods. V.G. and D.B.L. co-wrote the article.

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