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...

We demonstrate evidence of a surface gap opening in topological insulator (TI) thin films of (Bi0.57Sb0.43)2Te3 below six quintuple layers through transport and scanning tunneling spectroscopy measurements. By effective tuning the Fermi level via gate-voltage control, we unveil a striking competition between weak localization and weak antilocalization at low magnetic fields in nonmagnetic ultrathin films, possibly owing to the change of the net Berry phase. Furthermore, when the Fermi level is swept into the surface gap of ultrathin samples, the overall unitary behaviors are revealed at higher magnetic fields, which are in contrast to the pure WAL signals obtained in thicker films. Our findings show an exotic phenomenon characterizing the gapped TI surface states and point to the future realization of quantum spin Hall effect and dissipationless TI-based applications.

In high-energy physics, the Higgs field couples to gauge bosons and fermions and gives mass to their elementary excitations. Experimentally, such couplings can be inferred from the decay product of the Higgs boson, i.e., the scalar (amplitude) excitation of the Higgs field. In superconductors, Cooper pairs bear a close analogy to the Higgs field. Interaction between the Cooper pairs and other degrees of freedom provides dissipation channels for the amplitude mode, which may reveal important information about the microscopic pairing mechanism. To this end, we investigate the Higgs (amplitude) mode of several cuprate thin films using phase-resolved terahertz third harmonic generation (THG). In addition to the heavily damped Higgs mode itself, we observe a universal jump in the phase of the driven Higgs oscillation as well as a non-vanishing THG above T c. These findings indicate coupling of the Higgs mode to other collective modes and potentially a nonzero pairing amplitude above T c .

The transition metal thiophosphates M PS3 (M = Mn, Fe, Ni) are a class of van der Waals stacked insulating antiferromagnets that can be exfoliated down to the ultrathin limit. MnPS3 is particularly interesting because its Néel ordered state breaks both spatial-inversion and timereversal symmetries, allowing for a linear magneto-electric phase that is rare among van der Waals materials. However, it is unknown whether this unique magnetic structure of bulk MnPS3 remains stable in the ultrathin limit. Using optical second harmonic generation rotational anisotropy, we show that long-range linear magneto-electric type Néel order in MnPS3 persists down to at least 5.3 nm thickness. However an unusual mirror symmetry breaking develops in ultrathin samples on SiO2 substrates that is absent in bulk materials, which is likely related to substrate induced strain.

We report a global structural distortion in Sr_{2}IrO_{4} using spatially resolved optical second and third harmonic generation rotational anisotropy measurements. A symmetry lowering from an I4_{1}/acd to I4_{1}/a space group is observed both above and below the Néel temperature that arises from a staggered tetragonal distortion of the oxygen octahedra. By studying an effective superexchange Hamiltonian that accounts for this lowered symmetry, we find that perfect locking between the octahedral rotation and magnetic moment canting angles can persist even in the presence of large noncubic local distortions. Our results explain the origin of the forbidden Bragg peaks recently observed in neutron diffraction experiments and reconcile the observations of strong tetragonal distortion and perfect magnetoelastic locking in Sr_{2}IrO_{4}.

S1. Fits of Sr

Scanning tunneling spectroscopic studies of Bi 2 Se 3 epitaxial films on Si (111) substrates reveal highly localized unitary impurity resonances associated with non-magnetic quantum impurities. The strength of the resonances depends on the energy difference between the Fermi level (E F ) and the Dirac point (E D ) and diverges as E F approaches E D . The Dirac-cone surface state of the host recovers within 2Å spatial distance from impurities, suggesting robust topological protection of the surface state of topological insulators against highdensity impurities that preserve time reversal symmetry. An exciting development in modern condensed matter physics is the beautiful manifestation of topological field theories in strongly correlated electronic systems, where topological field theories [1] are shown to provide a classification of order due to macroscopic entanglement that is independent of symmetry breaking [2]. The fractional quantum Hall (FQH) state is the first known example of such a quantum state that exhibits no spontaneous broken symmetry and has properties depending only on its topology rather than geometry [2]. Recently, a new class of time-reversal symmetry protected topological states known as the quantum spin Hall (QSH) states or the topological insulators (TI) has emerged and stimulated intense research activities [3,4].One of the novel properties associated with the TI is the presence of a Dirac spectrum of chiral low-energy excitations, which is a salient feature of the Dirac materials that exploits the mapping of electronic band structures and an embedded spin or pseudo-spin degree of freedom onto the relativistic Dirac equation [3][4][5][6][7][8][9]. These materials, including graphene [9] and the surface state (SS) of three dimensional (3D) strong topological insulators (STI) [3][4][5][6][7][8], have emerged as a new paradigm in condensed matter for investigating the topological phases of massless and massive Dirac fermions. In the case of 3D-STI, an odd number of massless Dirac cones in their SS is ensured by the Z 2 topological invariant of the fully gapped bulk [3][4][5][6][7][8]. Backscattering of Dirac fermions is suppressed due to topological protection that preserves the Dirac dispersion relation for any time-reversal invariant perturbation [3,4]. Thus, 3D-STI are promising materials for applications in areas of spintronics [3,4,10] and topological quantum computation [3,4,11] if their SS exhibit sufficient stability to impurities [12,13].While direct backscattering is prohibited in both the SS of 3D-STI and in graphene, sharp resonances are not excluded because Dirac fermions with a finite parallel momentum may be confined by potential barriers [9]. In fact, theoretical calculations for Dirac fermions in the presence of noninteracting impurities have predicted the occurrence of strong impurity resonances [12,13]. Nonetheless, no direct empirical observation of strong resonances has been demonstrated to date despite numerous reports of spectral evidences for quasiparticle inte...

Nonlinear optical generation from a crystalline material can reveal the symmetries of both its lattice structure and underlying ordered electronic phases and can therefore be exploited as a complementary technique to diffraction based scattering probes. Although this technique has been successfully used to study the lattice and magnetic structures of systems such as semiconductor surfaces, multiferroic crystals, magnetic thin films and multilayers, challenging technical requirements have prevented its application to the plethora of complex electronic phases found in strongly correlated electron systems. These requirements include an ability to probe small bulk single crystals at the micron length scale, a need for sensitivity to the entire nonlinear optical susceptibility tensor, oblique light incidence reflection geometry and incident light frequency tunability among others. These measurements are further complicated by the need for extreme sample environments such as ultra low temperatures, high magnetic fields or high pressures. In this review we present a novel experimental construction using a rotating light scattering plane that meets all the aforementioned requirements. We demonstrate the efficacy of our scheme by making symmetry measurements on a micron scale facet of a small bulk single crystal of Sr 2 IrO 4 using optical second and third harmonic generation.