The superconducting proximity effect leads to strong modifications of the local density of states in diffusive or chaotic cavity Josephson junctions, which displays a phase-dependent energy gap around the Fermi energy. The so-called minigap of the order of the Thouless energy $E_{\mathrm{Th}}$ is related to the inverse dwell time in the diffusive region in the limit $E_{\mathrm{Th}}\ll \Delta$, where $\Delta$ is the superconducting energy gap. In the opposite limit of a large Thouless energy $E_{\mathrm{Th}}\gg \Delta$, a small new feature has recently attracted attention, namely, the appearance of a further secondary gap, which is around two orders of magnitude smaller compared to the usual superconducting gap. It appears in a chaotic cavity just below the superconducting gap edge $\Delta$ and vanishes for some value of the phase difference between the superconductors. We extend previous theory restricted to a normal cavity connected to two superconductors through ballistic contacts to a wider range of contact types. We show that the existence of the secondary gap is not limited to ballistic contacts, but is a more general property of such systems. Furthermore, we derive a criterion which directly relates the existence of a secondary gap to the presence of small transmission eigenvalues of the contacts. For generic continuous distributions of transmission eigenvalues of the contacts, no secondary gap exists, although we observe a singular behavior of the density of states at $\Delta$. Finally, we provide a simple one-dimensional scattering model which is able to explain the characteristic “smile” shape of the secondary gap.

Toroidal order, given by a composite of electric and magnetic orders, manifests itself not only in unusual magnetism but also in anomalous transport and magnetoelectric effects. We report our theoretical results on the influence and stability of toroidal order in metals on the basis of a microscopic model. We consider an effective single-band Hubbard-type model with a site-dependent antisymmetric spin-orbit coupling, which is derived from a four-band tight-binding model including atomic spin-orbit coupling, off-site hybridization between orbitals with different parities, and an odd-parity crystalline electric field. For this single-band model on a layered honeycomb lattice, we investigate the electronic structure, magnetotransport, and magnetoelectric effect in a toroidal ordered state with a vortexlike magnetic structure. The ferroic order of the microscopic toroidal moments acts as an effective gauge field for electrons, which modulates the electronic band structure with a shift of the band bottom in momentum space. In addition, the site-dependent antisymmetric spin-orbit coupling gives rise to highly anisotropic Hall responses. The most salient feature is two different types of magnetoelectric response: one is a magnetic order with net toroidal magnetization induced by an electric current perpendicular to the planes, and the other is a uniform transverse magnetization induced by an electric current within the planes. We examine the ground state of the effective model by the mean-field approximation, and show that the toroidal order is stabilized by strong electron correlations at low electron density. We also discuss the temperature dependence of the magnetoelectric effects associated with spontaneous toroidal ordering. Implications for experiments are also presented.

We analyze charge order in hole-doped cuprates within the the spin-fermion model. We show that a magnetically mediated interaction, which is known to give rise to $d$-wave superconductivity and charge order with momentum along zone diagonal, also gives rise to charge order with momenta $Q_x=(2Q,0)$ and $Q_y=(0,2Q)$ consistent with the experiments. We show that an instability towards ${\Delta }_k^Q=\langle c_{\mathbf{k}+\mathbf{Q}}^{†}{c}_{\mathbf{k}-\mathbf{Q}}\rangle$ with $\mathbf{Q}=Q_x$ or $Q_y$ is a threshold phenomenon, but the dimensionless spin-fermion coupling is above the threshold, if the magnetic correlation length $\xi$ exceeds a certain critical value. At a critical $\xi$, the onset temperature for the charge order terminates at a quantum-critical point distant from the magnetic one. We argue that the charge order with $Q_x$ or $Q_y$ changes sign under $\mathbf{k}\to \mathbf{k}+(\pi ,\pi )$, but $|{\Delta }_k^Q|\ne |{\Delta }_{k+(\pi ,\pi )}^Q|$. In real space, such an order has both bond and site components; the bond one is larger. We further argue that ${\Delta }_k^Q$ and ${\Delta }_{-k}^Q$ are not equivalent, and their symmetric and antisymmetric combinations describe, in real space, incommensurate density modulations and incommensurate bond current, respectively. We derive the Ginzburg-Landau functional for four-component $U\left(1\right)$ order parameters ${\Delta }_{\pm k}^Q$ with $\mathbf{Q}=Q_x$ or $Q_y$ and analyze it first in mean-field theory and then beyond mean field. Within mean field we find two types of charge-density-wave (CDW) states, I and II, depending on system parameters. In state I, density and current modulations emerge with the same $\mathbf{Q}=Q_x$ or $Q_y$, breaking $Z_2$ lattice rotational symmetry, and differ in phase by $\pm \pi /2$. The selection of $\pi /2$ or $-\pi /2$ additionally breaks $Z_2$ time-reversal symmetry, such that the total order parameter manifold is $U\left(1\right)\times Z_2\times Z_2$. In state II, density and current modulations emerge with different $\mathbf{Q}$ and the order parameter manifold is $U\left(1\right)\times U\left(1\right)\times Z_2$, where in the two realizations of state II, $Z_2$ corresponds to either lattice rotational or time-reversal symmetry breaking. We extend the analysis beyond mean field and argue that discrete symmetries get broken before long-range charge order sets in. For state I, which, we argue, is related to hole-doped cuprates, we show that, upon lowering the temperature, the system first breaks $Z_2$ lattice rotational symmetry ($C_4\to C_2$) at $T=T_n$ and develops a nematic order, then breaks $Z_2$ time-reversal symmetry at $T_t<T_n$ and locks the relative phase between density and current fluctuations, and finally breaks $U\left(1\right)$ symmetry of a common phase of even and odd components of ${\Delta }_k^Q$ at $T=T_{\mathrm{CDW}}<T_t<T_n$ and develops a true charge order. We argue that at a mean field, $T_{\mathrm{CDW}}$ is smaller than superconducting $T_{\mathrm{SC}}$, but preemptive composite order lifts $T_{\mathrm{CDW}}$ and reduces $T_{\mathrm{SC}}$ such that at large $\xi$ charge order develops prior to superconductivity. We obtain the full phase diagram and present quantitative comparison of our results with ARPES data for hole-doped cuprates.

We investigate the fine structure splitting (FSS) of excitons confined in strain-free, highly symmetric GaAs/AlGaAs quantum dots (QDs) as a function of size. QDs with similar geometry are created by filling nanoholes on an AlGaAs surface with different amounts of GaAs. In turn, nanoholes of regular reversed-cone shape are obtained by Al-droplet etching of an AlGaAs layer. The resulting QDs have high optical quality, as witnessed by emission linewidths smaller than 20 μeV under nonresonant excitation. The average FSS decreases from 12.5 ± 6.4 to 3.6 ± 1.7 μeV as the QD height increases from 5.4 ± 0.3 to 8.0 ± 0.2 nm, following the expected 1/volume dependence. Theoretical calculations using atomic force microscopy data of QD structures as input show that the geometrical similarity of QDs is reflected in invariant aspect ratio and lateral elongation of the wave functions and reproduce the experimental FSS energies.

Optical spectroscopies are most often used to probe dynamical correlations in materials, but they are also a probe of symmetry. Polarization anisotropies are of course sensitive to structural anisotropies, but have been much less used as a probe of more exotic symmetry breakings in ordered states. In this paper, a Jones transfer matrix formalism is discussed to infer the existence of exotic broken symmetry states of matter from their electrodynamic response for a full complement of possible broken symmetries including reflection, rotation, rotation reflection, inversion, and time reversal. A specific condition to distinguish the case of macroscopic time-reversal symmetry breaking is particularly important as in a dynamical experiment like optics, one must distinguish $\mathit{reciprocity}$ from time-reversal symmetry as dissipation violates strict time-reversal symmetry of an experiment. Different forms of reciprocity can be distinguished, but only one is a sufficient (but not necessary) condition for macroscopic time-reversal symmetry breaking. I show the constraints that a Jones matrix develops under the presence or absence of such symmetries. These constraints typically appear in the form of an algebra relating matrix elements or overall constraints (transposition, unitarity, hermiticity, normality, etc.) on the form of the Jones matrix. I work out a number of examples including the trivial case of a ferromagnet and the less trivial cases of magnetoelectrics and vector and scalar spin “chiral” states. I show that the formalism can be used to demonstrate that Kerr rotation must be absent in time-reversal symmetric chiral materials. The formalism here is discussed with an eye towards its use in time-domain terahetrz spectroscopy in transmission, but with small modifications it is more generally applicable.

We study the quantum many-body instabilities of the $t\text{−}{J}_{\mathrm{K}}\text{−}{J}_{\mathrm{H}}$ Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-$1/2$ model is believed to describe the magnetic properties of the layered transition-metal oxide ${\mathrm{Na}}_2$${\mathrm{IrO}}_3$. We determine the ground state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of the Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold $({\mathbb{Z}}_2$ quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet $p$-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling $\delta >1/4$, the $p$-wave pairing gives rise to a topological state with protected Majorana edge modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchanges, we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.

We study the Shannon mutual information in one-dimensional critical spin chains, following a recent conjecture [Alcaraz and Rajabpour, Phys. Rev. Lett. 111, 017201 (2013)], as well as Rényi generalizations of it. We combine conformal field theory (CFT) arguments with numerical computations in lattice discretizations with central charge $c=1$ and $c=1/2$. For a periodic system of length $L$ cut into two parts of length $\ell$ and $L-\ell$, all our results agree with the general shape dependence $I_n(\ell ,L)=(b_n/4)ln(\frac{L}{\pi }sin\frac{\pi \ell }{L})$, where $b_n$ is a universal coefficient. For the free boson CFT we show from general arguments that $b_n=c=1$. At $c=1/2$ we conjecture a result for $n>1$. We perform extensive numerical computations in Ising chains to confirm this, and also find $b_1\simeq 0.4801629\left(2\right)$, a nontrivial number which we do not understand analytically. Open chains at $c=1/2$ and $n=1$ are even more intriguing, with a shape-dependent logarithmic divergence of the Shannon mutual information.

We investigate the magnetization dynamics that arises when a thin-film ferromagnet is deposited on a topological insulator (TI), focusing in particular on domain-wall motion via current and the possibility of a spin-wave torque acting on the magnetization. We show analytically that the coupling between the magnetic domain wall and the TI removes the degeneracy of the wall profile with respect to its chirality and topological charge. Moreover, we find that the threshold for Walker breakdown of domain-wall motion is substantially increased and determined by the interaction with the TI, allowing for higher attainable wall velocities than in the conventional case where the hard-axis anisotropy determines the Walker threshold. Finally, we show that the allowed modes of spin-wave excitations and the ensuing magnetization dynamics in the presence of a TI coupling enable a magnonic torque acting even on homogeneous magnetization textures. Our results indicate that the TI-ferromagnet interaction has a similar effect on the magnetization dynamics as an intrinsic Dzyaloshinskii-Moriya interaction in ferromagnets.

We have embedded a voltage-biased Cooper-pair transistor (CPT) in a high-$Q$ superconducting microwave cavity. When the energy given to a tunneling Cooper pair by the voltage bias is equal to a multiple of the cavity photon frequency, the cavity is pumped to a strongly nonequilibrium state. The cavity photons act back on the CPT, allowing us to enter a regime of strongly correlated electronic-photonic transport. We directly observe the effects of photonic backaction on Cooper-pair transport, and see clear evidence for single-emitter lasing in the form of emission dominated by stimulated transport processes.

Eutectic ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$-Ru samples with $\mu$m-sized Ru-metal inclusions support inhomogeneous superconductivity above the bulk transition of ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$ in the so-called 3-Kelvin phase. In Pb/Ru/${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$ Josephson junctions as realized by Maeno et al., a Pb film is indirectly coupled to the superconductor ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$ mediated by the proximity-induced superconducting Ru inclusions, yielding an extended Josephson contact through the interface between Ru and ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$. Motivated by this experimental setup, we formulate a sine-Gordon model for the Josephson phase of the interface, assuming a simple cylindrical shape for the Ru inclusion hosting the proximity-induced $s$-wave superconducting phase. Considering the ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$ as a chiral $p$-wave superconductor, we discuss two types of Josephson junctions, a frustrated one due to the nature of the order parameter in ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$ and an unfrustrated one for the topologically trivial 3-Kelvin phase. While the latter situation displays standard junction behavior, the former yields an unusual limiting mechanism for the critical current, based on a pinning-depinning transition of a spontaneously induced magnetic flux driven by an externally applied current. We analyze different coupling limits and show that different critical currents can arise for the two topologies. This concept fits well to recent experimental data obtained for the above setup showing an anomalous temperature dependence of the critical current at the transition temperature $T_c$ of bulk ${\mathrm{Sr}}_2$${\mathrm{RuO}}_4$.

A non-Fermi liquid state without time-reversal and parity symmetries arises when a chiral Fermi surface is coupled with a soft collective mode in two space dimensions. The full Fermi surface is described by a direct sum of chiral patch theories, which are decoupled from each other in the low-energy limit. Each patch includes low-energy excitations near a set of points on the Fermi surface with a common tangent vector. General patch theories are classified by the local shape of the Fermi surface, the dispersion of the critical boson, and the symmetry group, which form the data for distinct universality classes. We prove that a large class of chiral non-Fermi liquid states exists as stable critical states of matter. For this, we use a renormalization group scheme where low-energy excitations of the Fermi surface are interpreted as a collection of $(1+1)$-dimensional chiral fermions with a continuous flavor labeling the momentum along the Fermi surface. Due to chirality, the Wilsonian effective action is strictly UV finite. This allows one to extract the exact scaling exponents although the theories flow to strongly interacting field theories at low energies. In general, the low-energy effective theory of the full Fermi surface includes patch theories of more than one universality classes. As a result, physical responses include multiple universal components at low temperatures. We also point out that, in quantum field theories with extended Fermi surface, a noncommutative structure naturally emerges between a coordinate and a momentum which are orthogonal to each other. We show that the invalidity of patch description for Fermi liquid states is tied with the presence of UV/IR mixing associated with the emergent noncommutativity. On the other hand, UV/IR mixing is suppressed in non-Fermi liquid states due to UV insensitivity, and the patch description is valid.

A new method is presented for the analysis of small angle neutron scattering data from quasi-two-dimensional systems such as flux lattices, Skyrmion lattices, and aligned liquid crystals. A significant increase in signal to noise ratio and a natural application of the Lorentz factor can be achieved by taking advantage of the knowledge that all relevant scattering is centered on a plane in reciprocal space. The Bayesian form ensures that missing information is treated in a controlled way and can be subsequently included in the analysis. A simple algorithm based on Gaussian probability assumptions is provided which provides very satisfactory results. Finally, it is argued that a generalized model-independent Bayesian data analysis method would be highly advantageous for the processing of neutron and x-ray scattering data.

We use optical spectroscopy to investigate the electron-phonon interaction in ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, a well-known $5d$ transition metal oxide with spin-orbit entangled states. The temperature evolution in the optical spectra is well described by the Fröhlich polaron model, indicating a large electron-phonon interaction. We further find that electrons in different orbitals selectively couple with different phonon modes. While $J_{\mathrm{eff}}$ = 3/2 holes do not seem to couple with any phonons, $J_{\mathrm{eff}}$  = 1/2 and $3z^2-r^2$ electrons mainly couple with in-plane and out-of-plane Ir-O bending modes, respectively. The symmetries of the orbitals and phonons are consistent with our observations.

A long-standing issue, the magnetic structure of the so-called X phase of ${\mathrm{MnWO}}_4$, has been solved by single-crystal neutron diffraction. Emerging from the well-known ferroelectric phase which presents an electric polarization $P//b$ and which is induced by a cycloidal magnetic order called AF2, this phase is reached by application of a magnetic field along the $b$ axis and is characterized by a flop of the electric polarization towards the $a$ axis. We have identified this new phase as a reoriented cycloidal spin structure, AF2', and its evolution from the AF2 phase has been thoroughly described. Unlike for the usual magnetic-field-induced spin-flop transitions, this high-field flopped phase does not directly develop from the low-field phase but emerges via the nonpolar collinear commensurate AF1 phase, which creeps in between. This evolution is in good agreement with the observed field-induced change in $P_b$ and $P_a$ electric polarization as well as with magnetic susceptibility data.

We report on a direct observation of coherent light-matter interactions imprinted on a short pulse propagating along a room-temperature, electrically biased, quantum dash semiconductor optical amplifier. The principle of the observation is that we characterize the complex field (phase and amplitude) of the short optical pulse after the interaction. By comparing the measurements to a numerical simulation, we are able to decipher the time evolution of effective two-state Schrödinger wave functions representing the ensembles of electronic states along the propagation path. The imprinted signatures on the pulse phase and amplitude stem from the accumulated propagation and reveal systematically clear Rabi oscillations and self-induced transparency. The high sensitivity to the effective ensemble states is shown to result from the response of the refractive index.

We have studied the field and temperature dependencies of the magnetization of single crystals of ${\mathrm{Sr}}_3\mathrm{Ni}\mathrm{Ir}{\mathrm{O}}_6$. These measurements evidence the presence of an easy axis of anisotropy and two anomalies in the magnetic susceptibility. Neutron powder diffraction realized on a polycrystalline sample reveals the emergence of magnetic reflections below 75 $\mathrm{K}$ with magnetic propagation vector k $\sim$ (0, 0, 1), undetected in previous neutron studies [T. N. Nguyen and H.-C. zur Loye, J. Solid State Chem. 117, 300 (1995)]. The nature of the magnetic ground state, and the presence of two anomalies common to this family of material, are discussed on the basis of the results obtained by neutron diffraction, magnetization measurements, and symmetry arguments.

As a proof of principle, self-consistent Kohn-Sham calculations are performed with the exact exchange-correlation functional. Finding the exact functional for even one trial density requires solving the interacting Schrödinger equation many times. The density matrix renormalization group method makes this possible for one-dimensional, real-space systems of more than two interacting electrons. We illustrate and explore the convergence properties of the exact KS scheme for both weakly and strongly correlated systems. We also explore the spin-dependent generalization and densities for which the functional is ill defined.

We investigate infrared manifestations of the pseudogap in the prototypical cuprate and pnictide superconductors, ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_y$ and ${\mathrm{BaFe}}_2$${\mathrm{As}}_2$ (Ba122) systems. We find remarkable similarities between the spectroscopic features attributable to the pseudogap in these two classes of superconductors. The hallmarks of the pseudogap state in both systems include a weak absorption feature at about $500{\mathrm{cm}}^{-1}$ followed by a featureless continuum between 500 and $1500{\mathrm{cm}}^{-1}$ in the conductivity data and a significant suppression in the scattering rate below 700–900 ${\mathrm{cm}}^{-1}$. The latter result allows us to identify the energy scale associated with the pseudogap ${\Delta }_{\mathrm{PG}}$. We find that in the Ba122-based materials the superconductivity-induced changes of the infrared spectra occur in the frequency region below 100–200 ${\mathrm{cm}}^{-1}$, which is much lower than the energy scale of the pseudogap. We performed theoretical analysis of the scattering rate data of the two compounds using the same model, which accounts for the effects of the pseudogap and electron-boson coupling. We find that the scattering rate suppression in Ba122-based compounds below ${\Delta }_{\mathrm{PG}}$ is solely due to the pseudogap formation, whereas the impact of the electron-boson coupling effects is limited to lower frequencies. The magnetic resonance modes used as inputs in our modeling are found to evolve with the development of the pseudogap, suggesting an intimate correlation between the pseudogap and magnetism.

We discover novel topological pumps in the Josephson effects for superconductors with time-reversal symmetry. The phase difference, which is odd under the chiral symmetry defined by the product of time-reversal and particle-hole symmetries, acts as an anomalous adiabatic parameter. In contrast to topological pumps with conventional parameters, these pumps are characterized by $\mathbb{Z}\times \mathbb{Z}$ or ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ strong invariants. We determine the general classifications in class AIII, and those in class DIII with a single anomalous parameter. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological pumps in class DIII, the first ${\mathbb{Z}}_2$ invariant describes the coincidence of fermion parity and spin pumps whereas the second ${\mathbb{Z}}_2$ invariant reflects the non-Abelian statistics of Majorana Kramers pairs, leading to three distinct fractional Josephson effects.

Grazing-incidence x-ray diffraction (GID) was employed to probe the structure of atomically thin carbon layers on SiC(0001): a so-called buffer layer (BL) with a $6(\sqrt{3}\times \sqrt{3})R{30}^{\circ }$ periodicity, a monolayer graphene (MLG) on top of the BL, and a bilayer graphene (BLG). The GID analysis was complemented by Raman spectroscopy. The lattice parameter of each layer was measured with high precision by GID. The BL possesses a different lattice parameter and corrugation when it is uncovered or beneath MLG. Our results demonstrate that the interfacial BL is mainly responsible for the strain in MLG. By promoting its decoupling from the substrate via intercalation, it turns into graphene, leading to a simultaneous relaxation of the MLG and formation of a quasi-free-standing BLG.