Optical nonlinearities usually appear for large intensities, but discrete transitions allow for giant nonlinearities operating at the single-photon level. This has been demonstrated in the last decade for a single optical mode with cold atomic gases, or single two-level systems coupled to light via a tailored photonic environment. Here, we demonstrate a two-mode giant nonlinearity with a single semiconductor quantum dot (QD) embedded in a photonic wire antenna. We exploit two detuned optical transitions associated with the exciton-biexciton QD level scheme. Owing to the broadband waveguide antenna, the two transitions are efficiently interfaced with two free-space laser beams. The reflection of one laser beam is then controlled by the other beam, with a threshold power as low as 10 photons per exciton lifetime ($1.6\mathrm{nW}$). Such a two-color nonlinearity opens appealing perspectives for the realization of ultralow-power logical gates and optical quantum gates, and could also be implemented in an integrated photonic circuit based on planar waveguides.

Recently, it has been established that two-dimensional bosonic symmetry-protected topological (SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology $H^3(G,\mathrm{U}\left(1\right))$. Later, group supercohomology was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the algebraic theory of symmetry and defects in two-dimensional topological phases. We reproduce the partial classifications given by group supercohomology, and we also show that with an additional $H^1(G,{\mathbb{Z}}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems with a total symmetry group $G\times {\mathbb{Z}}_2^f$ is obtained. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

We apply the artificial neural network in a supervised manner to map out the quantum phase diagram of disordered topological superconductors in class DIII. Given the disorder that keeps the discrete symmetries of the ensemble as a whole, translational symmetry which is broken in the quasiparticle distribution individually is recovered statistically by taking an ensemble average. By using this, we classify the phases by the artificial neural network that learned the quasiparticle distribution in the clean limit and show that the result is totally consistent with the calculation by the transfer matrix method or noncommutative geometry approach. If all three phases, namely the ${\mathbb{Z}}_2$, trivial, and thermal metal phases, appear in the clean limit, the machine can classify them with high confidence over the entire phase diagram. If only the former two phases are present, we find that the machine remains confused in a certain region, leading us to conclude the detection of the unknown phase which is eventually identified as the thermal metal phase.

We present a state-of-the-art study of the optical properties of free-standing silicene and of single-layer Si one- and two-dimensional (1D and 2D) nanostructures supported on Ag(110) and Ag(111) substrates. Ab initio simulations of reflectance anisotropy spectroscopy (RAS) and surface differential reflectivity spectroscopy (SDRS) applied to the clean Ag surface and Si/Ag interfaces are compared with new measurements. For Si/Ag(110), we confirm a pentagonal nanoribbon geometry, strongly bonded to the substrate, and rule out competing zigzag chain and silicenelike models. For Si/Ag(111), we reproduce the main experimental features and isolate the optical signal of the epitaxial silicene overlayer. The absorption spectrum of a silicene sheet computed including excitonic and local field effects is found to be quite similar to that calculated within an independent particle approximation and shows strong modifications when adsorbed on a Ag substrate. Important details of the computational approach are examined and the origins of the RAS and SDRS signals are explained in terms of the interface and substrate response functions. Our study does not find any evidence for Si adlayers that retain the properties of freestanding silicene.

Historically, comprehensive studies of dilute ferromagnetic semiconductors, e.g., $p$-type (Cd,Mn)Te and (Ga,Mn)As, paved the way for a quantitative theoretical description of effects associated with spin-orbit interactions in solids, such as crystalline magnetic anisotropy. In particular, the theory was successful in explaining uniaxial magnetic anisotropies associated with biaxial strain and nonrandom formation of magnetic dimers in epitaxial (Ga,Mn)As layers. However, the situation appears much less settled in the case of the cubic term: the theory predicts switchings of the easy axis between in-plane $\langle 100\rangle$ and $\langle 110\rangle$ directions as a function of the hole concentration, whereas only the $\langle 100\rangle$ orientation has been found experimentally. Here, we report on the observation of such switchings by magnetization and ferromagnetic resonance studies on a series of high-crystalline quality (Ga,Mn)As films. We describe our findings by the mean-field $p\text{−}d$ Zener model augmented with three new ingredients. The first one is a scattering broadening of the hole density of states, which reduces significantly the amplitude of the alternating carrier-induced contribution. This opens the way for the two other ingredients, namely the so-far disregarded single-ion magnetic anisotropy and disorder-driven nonuniformities of the carrier density, both favoring the $\langle 100\rangle$ direction of the apparent easy axis. However, according to our results, when the disorder gets reduced, a switching to the $\langle 110\rangle$ orientation is possible in a certain temperature and hole concentration range.

We present a systematic angle-resolved photoemission spectroscopy study of the superconducting gap in FeSe. The gap function is determined in a full Brillouin zone including all Fermi surfaces and $k_z$ dependence. We find significant anisotropy of the superconducting gap in all momentum directions. While the in-plane anisotropy can be explained by both nematicity-induced pairing anisotropy and orbital-selective pairing, the $k_z$ anisotropy requires an additional refinement of the theoretical approaches.

We consider a quantum dot with $\mathcal{K}\ge 2$ orbital levels occupied by two electrons connected to two electric terminals. The generic model is given by a multilevel Anderson Hamiltonian. The weak-coupling theory at the particle-hole symmetric point is governed by a two-channel $S=1$ Kondo model characterized by intrinsic channels asymmetry. Based on a conformal field theory approach we derived an effective Hamiltonian at a strong-coupling fixed point. The Hamiltonian capturing the low-energy physics of a two-stage Kondo screening represents the quantum impurity by a two-color local Fermi liquid. Using nonequilibrium (Keldysh) perturbation theory around the strong-coupling fixed point we analyze the transport properties of the model at finite temperature, Zeeman magnetic field, and source-drain voltage applied across the quantum dot. We compute the Fermi-liquid transport constants and discuss different universality classes associated with emergent symmetries.

We establish a unified framework for Majorana-based fault-tolerant quantum computation with Majorana surface codes and Majorana color codes. All logical Clifford gates are implemented with zero-time overhead. This is done by introducing a protocol for Pauli product measurements with tetrons and hexons which only requires local 4-Majorana parity measurements. An analogous protocol is used in the fault-tolerant setting, where tetrons and hexons are replaced by Majorana surface code patches, and parity measurements are replaced by lattice surgery, still only requiring local few-Majorana parity measurements. To this end, we discuss twist defects in Majorana fermion surface codes and adapt the technique of twist-based lattice surgery to fermionic codes. Moreover, we propose a family of codes that we refer to as Majorana color codes, which are obtained by concatenating Majorana surface codes with small Majorana fermion codes. Majorana surface and color codes can be used to decrease the space overhead and stabilizer weight compared to their bosonic counterparts.

The rich dynamics and phase structure of driven systems include the recently described phenomenon of the “discrete time crystal” (DTC), a robust phase which spontaneously breaks the discrete time translation symmetry of its driving Hamiltonian. Experiments in trapped ions and diamond nitrogen vacancy centers have recently shown evidence for this DTC order. Here, we show nuclear magnetic resonance (NMR) data of DTC behavior in a third, strikingly different, system: a highly ordered spatial crystal in three dimensions. We devise a DTC echo experiment to probe the coherence of the driven system. We examine potential decay mechanisms for the DTC oscillations, and demonstrate the important effect of the internal Hamiltonian during nonzero duration pulses.

${\mathrm{Cd}}_3{\mathrm{As}}_2$ is widely considered among the few materials realizing the three-dimensional (3D) Dirac semimetal phase. Linearly dispersing states, responsible for the ultrahigh charge mobility, have been reported by several angle-resolved photoelectron spectroscopy (ARPES) investigations. However, in spite of the general agreement between these studies, some details are at odds. From scanning tunneling microscopy and optical experiments under magnetic field, a puzzling scenario emerges in which multiple states show linear dispersion at different energy scales. Here, we solve this apparent controversy by reinvestigating the electronic properties of the (112) surface of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ by combining ARPES and theoretical calculations. We disentangle the presence of massive and massless metallic bulk and surface states, characterized by different symmetries. Our systematic experimental and theoretical study clarifies the complex band dispersion of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ by extending the simplistic 3D Dirac semimetal model to account for multiple bulk and surface states crossing the Fermi level, and thus contributing to the unique material transport properties.

In spinels $A{\mathrm{Cr}}_2{\mathrm{O}}_4(A=\text{Mg},\text{Zn})$, realization of the classical pyrochlore Heisenberg antiferromagnet model is complicated by a strong spin-lattice coupling: the extensive degeneracy of the ground state is lifted by a magneto-structural transition at $T_N=12.5$ K. We study the resulting low-temperature low-symmetry crystal structure by synchrotron x-ray diffraction. The consistent features of x-ray low-temperature patterns are explained by the tetragonal model of Ehrenberg et al. [Pow. Diff. 17, 230 (2002)], while other features depend on sample or cooling protocol. A complex, partially ordered magnetic state is studied by neutron diffraction and spherical neutron polarimetry. Multiple magnetic domains of configuration arms of the propagation vectors ${\mathbf{k}}_{\mathbf{1}}=\left(\frac{1}{2}\frac{1}{2}0\right),{\mathbf{k}}_{\mathbf{2}}=\left(10\frac{1}{2}\right)$ appear. The ordered moment reaches 1.94(3) ${\mu }_B/{\mathrm{Cr}}^{3+}$ for ${\mathbf{k}}_{\mathbf{1}}$ and 2.08(3) ${\mu }_B/{\mathrm{Cr}}^{3+}$ for ${\mathbf{k}}_{\mathbf{2}}$, if equal amount of the ${\mathbf{k}}_{\mathbf{1}}$ and ${\mathbf{k}}_{\mathbf{2}}$ phases is assumed. The magnetic arrangements have the dominant components along the [110] and $[1-10]$ diagonals and a smaller $c$ component. We use inelastic neutron scattering to investigate the spin excitations, which comprise a mixture of dispersive spin waves propagating from the magnetic Bragg peaks and resonance modes centered at equal energy steps of 4.5 meV. We interpret these as acoustic and optical spin wave branches, but show that the neutron scattering cross sections of transitions within a unit of two corner-sharing tetrahedra match the observed intensity distribution of the resonances. The distinctive fingerprint of clusterlike excitations in the optical spin wave branches suggests that propagating excitations are localized by the complex crystal structure and magnetic orders.

We introduce a new quantity that we term recoverable information, defined for stabilizer Hamiltonians. For such models, the recoverable information provides a measure of the topological information as well as a physical interpretation, which is complementary to topological entanglement entropy. We discuss three different ways to calculate the recoverable information and prove their equivalence. To demonstrate its utility, we compute recoverable information for fracton models using all three methods where appropriate. From the recoverable information, we deduce the existence of emergent $Z_2$ Gauss-law-type constraints, which in turn imply emergent $Z_2$ conservation laws for pointlike quasiparticle excitations of an underlying topologically ordered phase.

The Bohr-Sommerfeld quantization rule lies at the heart of the semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and de Haas–van Alphen oscillations for conventional metals, as well as for a host of topological metals which have emerged in the recent intercourse between band theory, crystalline symmetries, and topology. The essential ingredients in any quantization rule are connection formulas that match the semiclassical (WKB) wave function across regions of strong quantum fluctuations. Here, we propose (a) a multicomponent WKB wave function that describes transport within degenerate-band subspaces, and (b) the requisite connection formulas for saddle points and type-II Dirac points, where tunneling respectively occurs within the same band, and between distinct bands. (a) and (b) extend previous works by incorporating phase corrections that are subleading in powers of the field; these corrections include the geometric Berry phase, and account for the orbital magnetic moment and the Zeeman coupling. A comprehensive symmetry analysis is performed for such phase corrections occurring in closed orbits, which is applicable to solids in any (magnetic) space group. We have further formulated a graph-theoretic description of semiclassical orbits. This allows us to systematize the construction of quantization rules for a large class of closed orbits (with or without tunneling), as well as to formulate the notion of a topological invariant in semiclassical magnetotransport—as a quantity that is invariant under continuous deformations of the graph. Landau levels in the presence of tunneling are generically quasirandom, i.e., disordered on the scale of nearest-neighbor level spacings but having longer-ranged correlations; we develop a perturbative theory to determine Landau levels in such quasirandom spectra.

We show that the mixing between spin and valley degrees of freedom in a silicon quantum bit (qubit) can be controlled by a static electric field acting on the valley splitting $\mathrm{\Delta }$. Thanks to spin-orbit coupling, the qubit can be continuously switched between a spin mode (where the quantum information is encoded into the spin) and a valley mode (where the quantum information is encoded into the valley). In the spin mode, the qubit is more robust with respect to inelastic relaxation and decoherence but is hardly addressable electrically. It can, however, be brought into the valley mode for electrical manipulation, then back to the spin mode. This opens new possibilities for the development of robust and scalable, electrically addressable spin qubits on silicon. We illustrate this with tight-binding simulations on a so-called “corner dot” in a silicon-on-insulator device for which the confinement and valley splitting can be independently tailored by front and back gates.

In this work we explore the ultrafast magnetization dynamics induced by femtosecond laser pulses in a doped film of gadolinium iron garnet over a broad temperature range including the magnetization compensation point $T_M$. By exciting the phonon-assisted ${}^{6}S\to {}^{4}G$ and ${}^{6}S\to {}^{4}P$ electronic $d-d$ transitions simultaneously by one- and two-photon absorption processes, we find out that the transfer of heat energy from the lattice to the spin has, at a temperature slightly below $T_M$, a large influence on the magnetization dynamics. In particular, we show that the speed and the amplitude of the magnetization dynamics can be strongly increased when increasing either the external magnetic field or the laser energy density. The obtained results are explained by a magnetization reversal process across $T_M$. Furthermore, we find that the dynamics has unusual characteristics which can be understood by considering the weak spin-phonon coupling in magnetic garnets. These results open new perspectives for controlling the magnetic state of magnetic dielectrics using an ultrashort optically induced heat pulse.

Ferroic magnetic quadrupole order exhibiting macroscopic magnetoelectric activity is discovered in the novel compound $A\left(\mathrm{TiO}\right){\mathrm{Cu}}_4{\left({\mathrm{PO}}_4\right)}_4$ with $A$ = Pb, which is in contrast with antiferroic quadrupole order observed in the isostructural compounds with $A$ = Ba and Sr. Unlike the famous lone-pair stereochemical activity which often triggers ferroelectricity as in ${\mathrm{PbTiO}}_3$, the ${\mathrm{Pb}}^{2+}$ cation in $\mathrm{Pb}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4{\left({\mathrm{PO}}_4\right)}_4$ is stereochemically inactive but dramatically alters specific magnetic interactions and consequently switches the quadrupole order from antiferroic to ferroic. Our first-principles calculations uncover a positive correlation between the degree of $A$-O bond covalency and a stability of the ferroic quadrupole order.

We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound $\sigma$ orbital and the vacancy-induced bound $\pi$ state in an effective two-orbital single-impurity model. The local magnetic moments are stabilized by the Coulomb interaction as well as a significant ferromagnetic Hund's rule coupling between the orbitals predicted by a density functional theory calculation. A hybridization between the orbitals and the Dirac fermions is generated by the curvature of the graphene sheet in the vicinity of the vacancy. We present results for the local spectral function calculated using Wilson's numerical renormalization group approach for a realistic graphene band structure and find three different regimes depending on the filling, the controlling chemical potential, and the hybridization strength. These different regions are characterized by different magnetic properties. The calculated spectral functions qualitatively agree with recent scanning tunneling spectra on graphene vacancies.

Light interacting with metallic nanoparticles creates a strongly localized near-field around the particle that enhances inelastic light scattering by several orders of magnitude. Surface-enhanced Raman scattering describes the enhancement of the Raman intensity by plasmonic nanoparticles. We present an extensive Raman characterization of a plasmonic gold nanodimer covered with graphene. Its two-dimensional nature and energy-independent optical properties make graphene an excellent material for investigating local electromagnetic near-fields. We show the localization of the near-field of the plasmonic dimer by spatial Raman measurements. Energy- and polarization-dependent measurements reveal the local near-field resonance of the plasmonic system. To investigate the far-field resonance we perform dark-field spectroscopy and find that near-field and far-field resonance energies differ by 170 meV, much more than expected from the model of a damped oscillator (40 meV).

Experiments carried over the last years on the underdoped cuprates have revealed a variety of symmetry-breaking phenomena in the pseudogap state. Charge-density waves, breaking of $C_4$ rotational symmetry as well as time-reversal symmetry breaking have all been observed in several cuprate families. In this regard, theoretical models where multiple nonsuperconducting orders emerge are of particular interest. We consider the recently introduced [Volkov and Efetov, Phys. Rev. B 93, 085131 (2016)] spin-fermion model with overlapping ‘hot spots’ on the Fermi surface. Focusing on the particle-hole instabilities we obtain a rich phase diagram with the chemical potential relative to the dispersion at $(0,\pi );(\pi ,0)$ and the Fermi surface curvature in the antinodal regions being the control parameters. We find evidence for d-wave Pomeranchuk instability, d-form factor charge density waves, as well as commensurate and incommensurate staggered bond current phases similar to the d-density wave state. The current orders are found to be promoted by the curvature. Considering the appropriate parameter range for the hole-doped cuprates, we discuss the relation of our results to the pseudogap state and incommensurate magnetic phases of the cuprates.

The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy $S_{\mathrm{topo}}$ for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy $S_{\mathrm{nonlocal}}$ (a generalization of $S_{\mathrm{topo}}$). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type-I and type-II fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that $S_{\mathrm{nonlocal}}$ could obtain geometry-dependent values, and $S_{\mathrm{nonlocal}}$ is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed.