Evidence of an odd-parity hidden order in a strongly spin-orbit coupled correlated iridate Contents: S1. RA-SHG data for Sin-Pout and Sin-Sout geometries above and below TΩ show that the crystal structure of Sr2IrO4 belongs to the centrosymmetric tetragonal 4/m point group as opposed to the previously accepted 4/mmm point group 4,5 . Given the presence of inversion symmetry, the leading order contribution to SHG is the non-local term of electricquadrupole type, which can be expressed as an effective nonlinear polarization aswhere is the electric-quadrupole susceptibility tensor. By enforcing 4/m point group symmetry, is reduced to having 21 non-zero independent elements 6 :With the four additional constraints from degenerate SHG { = , = , = , = }, the number of non-zero independent tensor elements is further reduced to 17. The rotation of the crystal by an angle φ about the c-axis is carried out mathematically by applying a basis transformation on the reduced tensor from the original (primed) to rotated (unprimed) reference frame usingwhere ( ) is the rotation matrix about the c-axis. Finally, the expression that is used to fit theA is a constant determined by the experimental geometry, ( ) is the intensity of the incident 4 beam and ̂ is the polarization of the incoming or outgoing light, which we select to be either linearly P or S polarized. We note that previous work has already shown that there is no evidence of a surface electric-dipole contribution SHG 3 and that the crystallographic symmetry of the surface remains unchanged across TΩ 7 .ii) Fitting RA-SHG data for T < TThe low temperature RA-SHG data are fit to a coherent sum of the electric-quadrupole term described above and a hidden order induced electric-dipole term. The electric-dipole contribution is expressed as a nonlinear polarization (2 ) ∝ ( ) ( ). By enforcing 2/m magnetic point group symmetry, is reduced to having 14 non-zero independent elements:We only discuss the results using a 2/m magnetic point group here although the same procedure was applied to all of the magnetic point groups we surveyed. The additional constraints from degenerate SHG { = , = , = , = } leaves 10 non-zero independent tensor elements remaining. A basis transformation was then carried out on using ( ) = ′ ′ ′ ′ ′ ′ and the expression used to fit the RA-SHG data at Invariance of the magnetic structure under the elements of 2/m1 is explicitly shown in Figs S2 b-e. We note that the 2/m1 magnetic point group assignment does not rely on the magnitude of the magnetic moments on the two structural sub-lattices being equal, even though experimentally they are found to be so. S4. Loop-current order in cuprates versus iridatesLoop-current ordered phases were initially proposed in a three-band CuO2 model of the copperoxide planes where the charge currents are emergent complex hopping terms between oxygen and copper sites 8,9 . In this section, we examine how this model would be modified for Sr2IrO4.Our intention is not to elaborate on any of the details of loop-curr...

The development of collective long-range order by means of phase transitions occurs by the spontaneous breaking of fundamental symmetries. Magnetism is a consequence of broken time-reversal symmetry, whereas superfluidity results from broken gauge invariance. The broken symmetry that develops below 17.5 kelvin in the heavy-fermion compound URu(2)Si(2) has long eluded such identification. Here we show that the recent observation of Ising quasiparticles in URu(2)Si(2) results from a spinor order parameter that breaks double time-reversal symmetry, mixing states of integer and half-integer spin. Such 'hastatic' order hybridizes uranium-atom conduction electrons with Ising 5f(2) states to produce Ising quasiparticles; it accounts for the large entropy of condensation and the magnetic anomaly observed in torque magnetometry. Hastatic order predicts a tiny transverse moment in the conduction-electron 'sea', a colossal Ising anisotropy in the nonlinear susceptibility anomaly and a resonant, energy-dependent nematicity in the tunnelling density of states.

The recent discovery of two heavy fermion materials PuCoGa_{5} and NpPd_{5}Al_{2} which transform directly from Curie paramagnets into superconductors, reveals a new class of superconductor where local moments quench directly into a superconducting condensate. A powerful tool in the description of heavy fermion metals is the large N expansion, which expands the physics in powers of 1/N about a solvable limit where particles carry a large number (N) of spin components. As it stands, this method is unable to jointly describe the spin quenching and superconductivity which develop in PuCoGa_{5} and NpPd_{5}Al_{2}. Here, we solve this problem with a new class of large N expansion that employs the symplectic symmetry of spin to protect the odd time-reversal parity of spin and sustain Cooper pairs as well-defined singlets. With this method we show that when a lattice of magnetic ions exchange spin with their metallic environment in two distinct symmetry channels, they are able to simultaneously satisfy both channels by forming a condensate of composite pairs between between local moments and electrons. In the tetragonal crystalline environment relevant to PuCoGa_{5} and NpPd_{5}Al_{2} the lattice structure selects a natural pair of spin exchange channels, giving rise to the prediction of a unique anisotropic paired state with g-wave symmetry. This pairing mechanism predicts a large upturn in the NMR relaxation rate above T_{c}, a strong enhancement of Andreev reflection in tunneling measurements and an enhanced superconducting transition temperature T_{c} in Pu doped Np_{1-x}Pu_{x}Pd_{5}Al_{2}.Comment: This is a substantially revised version of the original paper, focussing on the high temperature heavy electron superconductors PuCoGa_5 and NpPd_5Al_2. A substantially revised supplementary online material to this paper can be found in arXiv 0710.1128v

Identifying the time reversal symmetry of spins as a symplectic symmetry, we develop a large N approximation for quantum magnetism that embraces both antiferromagnetism and ferromagnetism. In SU (N ), N > 2, not all spins invert under time reversal, so we have introduced a new large N treatment which builds interactions exclusively out of the symplectic subgroup[SP (N )] of time reversing spins, a more stringent condition than the symplectic symmetry of previous SP (N ) large N treatments. As a result, we obtain a mean field theory that incorporates the energy cost of frustrated bonds. When applied to the frustrated square lattice, the ferromagnetic bonds restore the frustration dependence of the critical spin in the Néel phase, and recover the correct frustration dependence of the finite temperature Ising transition.

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We introduce the idea of emergent lattices, where a simple lattice decouples into two weakly coupled lattices as a way to stabilize spin liquids. In LiZn2Mo3O8, the disappearance of 2/3 of the spins at low temperatures suggests that its triangular lattice decouples into an emergent honeycomb lattice weakly coupled to the remaining spins, and we suggest several ways to test this proposal. We show that these orphan spins act to stabilize the spin liquid in the J1-J2 honeycomb model and also discuss a possible 3D analogue, Ba2MoYO6 that may form a "depleted fcc lattice."

We consider the internal structure of a d-wave heavy-fermion superconducting condensate, showing that it necessarily contains two components condensed in tandem: pairs of quasiparticles on neighboring sites and composite pairs consisting of two electrons bound to a single local moment. These two components draw upon the antiferromagnetic and Kondo interactions to cooperatively enhance the superconducting transition temperature. This tandem condensate is electrostatically active, with an electric quadrupole moment predicted to lead to a superconducting shift in the nuclear quadrupole resonance frequency.

The broken symmetry that develops below 17.5K in the heavy fermion compound URu2Si2 has long eluded identification. Here we argue that the recent observation of Ising quasiparticles in URu2Si2 results from a spinor hybridization order parameter that breaks double time-reversal symmetry by mixing states of integer and half-integer spin. Such "hastatic order" (hasta:[Latin]spear ) hybridizes Kramers conduction electrons with Ising, non-Kramers 5f 2 states of the uranium atoms to produce Ising quasiparticles. The development of a spinorial hybridization at 17.5K accounts for both the large entropy of condensation and the magnetic anomaly observed in torque magnetometry. This paper develops the theory of hastatic order in detail, providing the mathematical development of its key concepts. Hastatic order predicts a tiny transverse moment in the conduction sea, a collosal Ising anisotropy in the nonlinear susceptibility anomaly and a resonant energy-dependent nematicity in the tunneling density of states.