Coh et al. presented in [Phys. Rev. B 88, 121106(R) (2013)] a systematic search of the simplest so-called “canonical” structures allowing isotropic magnetoelectric response, and reported a total of 30 such magnetic configurations. Using magnetic symmetry we show in this Comment that this listing is severely incomplete, and 14 additional distinct cases satisfying the same conditions should be added. The complete list of these elementary magnetic arrangements is then presented in a short and efficient form as distinct Wyckoff positions of some cubic magnetic space groups.

We agree with the corrections that Perez-Mato et al. [Phys. Rev. B 90, 167101 (2014).] present in their Comment. However, we argue that the representation-theory approach, as used in our Rapid Communication, is more general than the magnetic-symmetry point of view adopted in the Comment.

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.

The phonon dispersions and scattering rates of the thermoelectric material ${\mathrm{Ag}}_{1-x}{\mathrm{Sb}}_{1+x}{\mathrm{Te}}_{2+x}(x=0,0.1,0.2)$ were measured with inelastic neutron scattering, as function of both temperature $T$ and off stoichiometry $x$. In addition, detailed measurements of diffuse scattering were performed with both neutron and synchrotron x-ray diffraction. The results show that phonon scattering rates are large and weakly dependent on $T$ or $x$, and the lattice thermal conductivity calculated from these scattering rates and group velocities is in good agreement with bulk transport measurements. We also find that the scattering rates and their temperature dependence cannot be accounted for with common models of phonon scattering by anharmonicity or point defects. The diffuse scattering measurements show a pervasive, complex signal, with several distinct components. In particular, broad superstructure reflections indicate a short-range ordering of the Ag and Sb cations on their sublattice. Single-crystal Bragg peak intensities also reveal large static atomic displacements, compatible with results from Rietveld refinement of neutron powder diffraction data. Our results indicate that a complex nanostructure, arising from multiple variants of nanoscale anisotropic superstructures of cations, and large atomic displacements, is likely responsible for the strong phonon scattering.

We study the gapped phase of Kitaev's honeycomb model (a $Z_2$ spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a $Z_2$ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a $Z_2$ vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes. To facilitate these theoretical developments, we recast the degenerate perturbation theory for spins in the language of Majorana fermions.

We report a detailed nuclear magnetic resonance (NMR) study by combined ${}^{23}\mathrm{Na}$ and ${}^{75}\mathrm{As}$ measurements over a broad range of doping to map the phase diagram of ${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$. In the underdoped regime $\left(x\le 0.017\right)$, we find a magnetic phase with robust antiferromagnetic (AFM) order, which we denote the $s$-AFM phase, cohabiting with a phase of weak and possibly proximity-induced AFM order $(w$-AFM) whose volume fraction $V\simeq 8$% is approximately constant. Near optimal doping, at $x=0.0175$, we observe a phase separation between static antiferromagnetism related to the $s$-AFM phase and a paramagnetic (PM) phase related to $w$-AFM. The volume fraction of AFM phase increases upon cooling, but both the Néel temperature and the volume fraction can be suppressed systematically by applying a $c$-axis magnetic field. On cooling below $T_c$, superconductivity occupies the PM region and its volume fraction grows at the expense of the AFM phase, demonstrating a phase separation of the two types of order based on volume exclusion. At higher dopings, static antiferromagnetism and even critical AFM fluctuations are completely suppressed by superconductivity. Thus the phase diagram we establish contains two distinct types of phase separation and reflects a strong competition between AFM and superconducting phases both in real space and in momentum space. We suggest that both this strict mutual exclusion and the robustness of superconductivity against magnetism are consequences of the extreme two-dimensionality of NaFeAs.

A strong spin Nernst effect with nontrivial dependences on the carrier concentration and electric field applied is expected in silicene and other low-buckled Dirac materials. These Dirac materials can be considered as being made of two independent electron subsystems of the two-component gapped Dirac fermions. For each subsystem, the gap breaks a time-reversal symmetry and thus plays the role of an effective magnetic field. Accordingly, the standard Kubo formalism has to be altered by including the effective magnetization in order to satisfy the third law of thermodynamics. We explicitly demonstrate this by calculating the magnetization and showing how the correct thermoelectric coefficient emerges.

We show that repulsive local Coulomb interaction alone can drive valence-skipping charge disproportionation in the degenerate $d$-band, resulting in effective negative-$U$. This effect is shown to originate from anisotropic orbital-multipole scattering, and it occurs only for $d^1,d^4,d^6$, and $d^9$ fillings (and their immediate surroundings). Explicit boundaries for valence-skipping are derived, and the paramagnetic phase diagram for $d^4$ and $d^6$ is calculated. We also establish that the valence-skipping metal is very different, in terms of its local valence distribution, compared to the atomiclike Hund's metal. These findings explain why transition-metal compounds with the aforementioned $d$-band fillings are more prone to valence-skipping charge order and anomalous superconductivity.

We develop the first-order gradient correction to the exchange-correlation free energy of the homogeneous electron gas for use in finite-temperature density functional calculations. Based on this, we propose and implement a simple temperature-dependent extension for functionals beyond the local density approximation. These finite-temperature functionals show improvement over zero-temperature functionals, as compared to path-integral Monte Carlo calculations for deuterium equations of state, and perform without computational cost increase compared to zero-temperature functionals and so should be used for finite-temperature calculations. While the present functionals are valid at all temperatures including zero, non-negligible difference with zero-temperature functionals begins at temperatures above 10 000 K.

We attempt to clarify an ambiguity in the understanding of the electronic structure of an acceptor in a semiconductor. Instead of using only a single quantity acceptor binding energy $E_A$ as in the literature, there is in fact an impurity level $E_I$ that plays perhaps a more important role when dealing with the electronic transitions involving the acceptor. Together they determine an excitonic transition energy $E_A$ = $E_I-E_A$. We discuss the underlying physics and approximations of using three different approaches, all in the framework of a density-functional theory, for calculating the acceptor electronic structure, and point out the different meanings of the results yielded from these approaches and how they should be compared with experimental data to extract $E_I$ and $E_A$. Furthermore, the discussions provide a unified view of “deep” and “shallow” impurities: The characteristics of deep and shallow reflect two different aspects of a complete description of a typical impurity, either known as isoelectronic or acceptor (donor), rather than the spatial extension of the impurity potential.

Anderson localization of wave functions at zero energy in quasi-one-dimensional (1D) systems of $N$ disordered chains with interchain coupling $t$ is examined. Localization becomes weaker than for the 1D disordered chain ($t=0$) when $t$ is smaller than the longitudinal hopping $t^{\prime }=1$, and localization becomes usually much stronger when $t\gg t^{\prime }$. This is not so for all $N$. We find “immunity” to strong localization for open (periodic) lateral boundary conditions when $N$ is odd (a multiple of 4), with localization that is weaker than for $t=0$ and rather insensitive to $t$ when $t\gg t^{\prime }$. The peculiar $N$ dependence and a critical scaling with $N$ are explained by a perturbative treatment in $t^{\prime }/t$, and the correspondence to a weakly disordered effective chain is shown. Our results could be relevant for experimental studies of localization in photonic waveguide arrays.

${\mathrm{YTiO}}_3$, as a prototypical Mott insulator, has been the subject of numerous experimental investigations of its electronic structure. The onset of absorption in optical conductivity measurements has generally been interpreted to be due to interband transitions at the fundamental gap. Here we reexamine the electronic structure of ${\mathrm{YTiO}}_3$ using density functional theory with either a Hubbard correction $(\text{DFT}+U)$ or a hybrid functional. Interband transitions turn out to be much higher in energy than the observed onset of optical absorption. However, in the case of $p$-type doping, holes tend to become self-trapped in the form of small polarons, localized on individual Ti sites. Exciting electrons from the occupied lower Hubbard band to the small-polaron state then leads to broad infrared absorption, consistent with the onset in the experimental optical conductivity spectra.

Many-body calculations for multi-orbital systems at present typically employ Slater or Kanamori interactions which implicitly assume a full rotational invariance of the orbitals, whereas the real crystal has a lower symmetry. In cubic symmetry, the low-energy $t_{2g}$ orbitals have an on-site Kanamori interaction, albeit without the constraint $U=U^{\prime }+2J$ implied by spherical symmetry $(U$ is the intra-orbital interaction, $U^{\prime }$ is the interorbital interaction, $J$ is Hund's exchange). Using maximally localized Wannier functions we show that deviations from the standard, spherically symmetric interactions are indeed significant for $5d$ orbitals $(\sim 25\%$ for ${\mathrm{BaOsO}}_3$; $\sim 12\%$ if screening is included) but are less important for $3d$ orbitals $(\sim 6\%$ for ${\mathrm{SrVO}}_3$; $\sim 1\%$ if screened).

We develop a method for the calculation of charge-transfer statistics of persistent current in nanostructures in terms of the cumulant generating function (CGF) of transferred charge. We consider a simply connected one-dimensional system (a wire) and develop a procedure for the calculation of the CGF of persistent currents when the wire is closed into a ring via a weak link. For the noninteracting system we derive a general formula in terms of the two-particle Green's functions. We show that, contrary to the conventional tunneling contacts, the resulting cumulant generating function has a doubled periodicity as a function of the counting field. We apply our general formula to short tight-binding chains and show that the resulting CGF perfectly reproduces the known evidence for the persistent current. Its second cumulant turns out to be maximal at the switching points and vanishes identically at zero temperature. Furthermore, we apply our formalism for a computation of the charge-transfer statistics of genuinely interacting systems. First we consider a ring with an embedded Anderson impurity and employing a self-energy approximation find an overall suppression of persistent current as well as of its noise. Finally, we compute the charge-transfer statistics of a double-quantum-dot system in the deep Kondo limit using an exact analytical solution of the model at the Toulouse point. We analyze the behavior of the resulting cumulants and compare them with those of a noninteracting double-quantum-dot system and find several pronounced differences, which can be traced back to interaction effects.

We show that the Rashba spin-orbit interaction in $d$ electron solids, which originates from the broken inversion symmetry at surfaces or interfaces, is strongly dependent on the orbital characters of the bands involved. This is studied by developing a tight-binding model in the presence of a uniform perpendicular electric field and spin-orbit coupling. We argue that for valence electrons, the spin-orbit coupling strength scales only as the square of the atomic number. The electric field distorts the $d$ orbitals through the admixture of $p$ and $f$ states and also introduces intersite overlap parameters. Expressions for Rashba coefficients for the bands are obtained in both weak and strong spin-orbit interaction limits and are shown to be orbital dependent. The results are compared with first-principles calculations for model systems, showing good agreement. Our study demonstrates the orbital-dependent gate control of the Rashba effect for the purposes of oxide electronics.

${\mathrm{A}}^{\mathrm{IV}}{\mathrm{B}}^{\mathrm{VI}}$ crystals are believed to possess a rhombohedral (ferroelectric) structure at low temperature that changes to the rocksalt (paraelectric) structure above the Curie temperature. For GeTe it has been recently demonstrated that locally the structure retains the subsets of the shorter and longer bonds across the ferroelectric-to-paraelectric transition despite acquiring the cubic structure on average. Nothing is known about the existence of local distortions in SnTe, a prototypical topological crystalline insulator, where the crystal symmetry plays a crucial role. In this work we report the results of x-ray absorption measurements. We find that the structure is locally rhombohedrally distorted, and the distortions increase at $T>100\text{K}$, breaking the rocksalt average symmetry. Our density functional theory simulations performed at 0 K indicate that the role of spin-orbit coupling in the formation of the local structure of SnTe at low temperature is negligibly small. The small stochastic distortions do not affect the intrinsic band inversion of SnTe.

The $A{R}_2{\text{O}}_4$ family ($R=\text{rare}$ earth) has recently been attracting interest as a new series of frustrated magnets, with the magnetic $R$ atoms forming zigzag chains running along the $c$ axis. We have investigated polycrystalline ${\mathrm{BaNd}}_2{\mathrm{O}}_4$ with a combination of magnetization, heat-capacity, and neutron powder diffraction measurements. Magnetic Bragg peaks are observed below $T_N=1.7\mathrm{K}$, and they can be indexed with a propagation vector of $\vec{k}=(0,1/2,1/2)$. The signal from magnetic diffraction is well described by long-range ordering of only one of the two types of Nd zigzag chains, with collinear up-up-down-down intrachain spin configurations (double Néel state). Furthermore, low-temperature magnetization and heat-capacity measurements reveal two magnetic-field-induced spin transitions at 2.75 and 4 T for $T=0.46\mathrm{K}$. The high-field phase is paramagnetic, while the intermediate-field state may arise from a spin transition of the long-range ordered Nd chains. One possible candidate for the field-induced ordered state corresponds to an up-up-down intrachain spin configuration, as predicted for a classical $J_1-J_2$ Ising chain with a double Néel ground state in zero field.