Appendix A Appendix - Magnetic order and high-field magnetic phase transitions in hyper-honeycomb ฮฒโข-Na2โขPrO3

ยง A.1 Character tables

This appendix presents the irreducible representations, character tables, and ๐’’=0 magnetic basis vectors for the hyperhoneycomb Pr4+lattice in ฮฒโข-Na2โขPrO3for magnetic field applied parallel and perpendicular to the orthorhombic crystal axes ๐’‚,๐’ƒ,๐’„. These are used in the discussion of the calculated magnetic phase diagrams in Sectionย 5.7, especially in regards to Figureย 5.11. In all tables, a primed symmetry operation corresponds to the unprimed symmetry operation followed by time-reversal. For example 2yโ€ฒ denotes a 2-fold rotation around y followed by time-reversal. All coordinated are in orthorhombic setting with origin choice at 1ยฏ.

Table A.1: Irreducible representations (irreps), character table, and magnetic basis vectors for space group Fโขdโขdโขd with applied magnetic field parallel to crystallographic ๐’‚ axis. In character table โ€™+โ€™ indicates ฯ‡=1, โ€™-โ€™ indicates ฯ‡=โˆ’1.
(๐‘ฉโˆฅ๐’‚) E 2x 1ยฏ dx 2yโ€ฒ 2zโ€ฒ dyโ€ฒ dzโ€ฒ Basis Vectors
ฮ“1+โข(๐‘ฉโˆฅ๐’‚) + + + + + + + + Fx,Gy
ฮ“2+โข(๐‘ฉโˆฅ๐’‚) + - + - - + - + Fy,Gx
ฮ“3+โข(๐‘ฉโˆฅ๐’‚) + + + + - - - - Gz
ฮ“4+โข(๐‘ฉโˆฅ๐’‚) + - + - + - + - Fz
ฮ“1โˆ’โข(๐‘ฉโˆฅ๐’‚) + + - - + + - - Cx,Ay
ฮ“2โˆ’โข(๐‘ฉโˆฅ๐’‚) + - - + - + + - Cy,Ax
ฮ“3โˆ’โข(๐‘ฉโˆฅ๐’‚) + + - - - - + + Az
ฮ“4โˆ’โข(๐‘ฉโˆฅ๐’‚) + - - + + - - + Cz
Table A.2: Irreducible representations (irreps), character table, and magnetic basis vectors for space group Fโขdโขdโขd with applied magnetic field parallel to crystallographic ๐’ƒ axis. In character table โ€™+โ€™ indicates ฯ‡=1, โ€™-โ€™ indicates ฯ‡=โˆ’1.
(๐‘ฉโˆฅ๐’ƒ) E 2y 1ยฏ dy 2zโ€ฒ 2xโ€ฒ dzโ€ฒ dxโ€ฒ Basis Vectors
ฮ“1+โข(๐‘ฉโˆฅ๐’ƒ) + + + + + + + + Fy,Gx
ฮ“2+โข(๐‘ฉโˆฅ๐’ƒ) + - + - - + - + Fz
ฮ“3+โข(๐‘ฉโˆฅ๐’ƒ) + + + + - - - - Gz
ฮ“4+โข(๐‘ฉโˆฅ๐’ƒ) + - + - + - + - Fx,Gy
ฮ“1โˆ’โข(๐‘ฉโˆฅ๐’ƒ) + + - - + + - - Cy,Ax
ฮ“2โˆ’โข(๐‘ฉโˆฅ๐’ƒ) + - - + - + + - Cz
ฮ“3โˆ’โข(๐‘ฉโˆฅ๐’ƒ) + + - - - - + + Az
ฮ“4โˆ’โข(๐‘ฉโˆฅ๐’ƒ) + - - + + - - + Cx,Ay
Table A.3: Irreducible representations (irreps), character table, and magnetic basis vectors for space group Fโขdโขdโขd with applied magnetic field parallel to crystallographic ๐’„ axis. In character table โ€™+โ€™ indicates ฯ‡=1, โ€™-โ€™ indicates ฯ‡=โˆ’1.
(๐‘ฉโˆฅ๐’„) E 2z 1ยฏ dz 2xโ€ฒ 2yโ€ฒ dxโ€ฒ dyโ€ฒ Basis Vectors
ฮ“1+โข(๐‘ฉโˆฅ๐’„) + + + + + + + + Fz
ฮ“2+โข(๐‘ฉโˆฅ๐’„) + - + - - + - + Fx,Gy
ฮ“3+โข(๐‘ฉโˆฅ๐’„) + + + + - - - - Gz
ฮ“4+โข(๐‘ฉโˆฅ๐’„) + - + - + - + - Fy,Gx
ฮ“1โˆ’โข(๐‘ฉโˆฅ๐’„) + + - - + + - - Cz
ฮ“2โˆ’โข(๐‘ฉโˆฅ๐’„) + - - + - + + - Cx,Ay
ฮ“3โˆ’โข(๐‘ฉโˆฅ๐’„) + + - - - - + + Az
ฮ“4โˆ’โข(๐‘ฉโˆฅ๐’„) + - - + + - - + Cy,Ax
Table A.4: Irreducible representations (irreps), character table, and magnetic basis vectors for space group Fโขdโขdโขd with applied magnetic field perpendicular to crystallographic axis ๐’โˆˆ{๐’‚,๐’ƒ,๐’„}. In character table โ€™+โ€™ indicates ฯ‡=1, โ€™-โ€™ indicates ฯ‡=โˆ’1. Also present in text as Tableย 5.7.
E 1ยฏ 2๐’โ€ฒ d๐’โ€ฒ ๐’=๐’„ ๐’=๐’ƒ ๐’=๐’‚
ฮ“1+โข(๐‘ฉโŸ‚๐’) + + + + Gx,Gy,Fx,Fy Fx,Fz,Gy Gx,Fy,Fz
ฮ“2+โข(๐‘ฉโŸ‚๐’) + + - - Gz,Fz Gx,Gz,Fy Fx,Gy,Gz
ฮ“1โˆ’โข(๐‘ฉโŸ‚๐’) + - + - Ax,Ay,Cx,Cy Cx,Cz,Ay Ax,Cy,Cz
ฮ“2โˆ’โข(๐‘ฉโŸ‚๐’) + - - + Az,Cz Ax,Az,Cy Ay,Az,Cx
Table A.5: Irreducible representations (irreps), character table, and magnetic basis vectors for space group Fโขdโขdโขd with applied magnetic field in general orientation. In character table โ€™+โ€™ indicates ฯ‡=1, โ€™-โ€™ indicates ฯ‡=โˆ’1.
E 1ยฏ Basis Vectors
ฮ“+โข(๐‘ฉ) + + ๐‘ฎ,๐‘ญ
ฮ“โˆ’โข(๐‘ฉ) + - ๐‘ช,๐‘จ