We propose an order parameter for the symmetry-protected topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT entanglement, is defined as the entanglement between $A$ and $B$, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region $C$ is measured and known, where $C$ is a connected region surrounded by $A, B$, and the boundaries of the system. In the case of one-dimensional systems we prove that in the limit where $A$ and $B$ are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the nontrivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.

We consider conductance mapping of the snake orbits confined along the $n-p$ junction defined in graphene by the electrostatic doping in the quantum Hall regime. We explain the periodicity of conductance oscillations at the magnetic field and the Fermi energy scales by the properties of the $n-p$ junction as a conducting channel. We evaluate the conductance maps for a floating gate scanning the surface of the device. In the quantum Hall conditions the currents flow near the edges of the sample and along the $n-p$ junction. The conductance mapping resolves only the $n-p$ junction and not the edges. The conductance oscillations along the junction are found in the maps with periodicity related to the cyclotron orbits of the scattering current. Stronger probe potentials provide support to localized resonances at one of the sides of the junction with current loops that interfere with the $n-p$ junction currents. The interference results in a series of narrow lines parallel to the junction with positions that strongly depend on the magnetic field through the Aharonov-Bohm effect. The consequences of a limited transparency of finite-width $n-p$ junctions are also discussed.

We theoretically investigate the effective $g$ factor in the black phosphorus (BP) thin films (TFs) based on a multiband $\mathbf{k}·\mathbf{p}$ theory. We demonstrate that the effective single particle $g$ factor in pristine BP TF is anisotropic arising from its anisotropic band structure with $g_{xx}^{*}\approx g_{yy}^{*}\approx 2.0$ and $g_{zz}^{*}$ sensitively depending on the interband coupling and the band gap. The $g_{zz}^{*}$ approaches 2.0 with increasing hole doping density and gate electric field since both of them minish the interband coupling by reducing the overlap integral between the electron and hole wave functions. We also estimate the exchange interaction enhancement on the effective single particle $g$ factor by using the screened Hartree-Fock approximation. The exchange interaction enhanced $g$ factor ($g_{\text{ex}}$) shows maxima (minima) at odd (even) filling factors. The effective $g$ factor ($g^{*}$) oscillates with the increase of magnetic field and sensitively depends on the Landau level broadening as well as the gate electric field since both of them affect the interband coupling and the electron-electron interactions.

We formulate the problem of numerical analytic continuation in a way that lets us draw meaningful conclusions about the properties of the spectral function based solely on the input data. Apart from ensuring consistency with the input data (within their error bars) and the a priori and a posteriori (conditional) constraints, it is crucial to reliably characterize the accuracy—or even ambiguity—of the output. We explain how these challenges can be met with two approaches: stochastic optimization with consistent constraints and the modified maximum entropy method. We perform illustrative tests for spectra with a double-peak structure, where we critically examine which spectral properties are accessible (second peak position and its spectral weight) and which ones are lost (second peak width/shape). For an important practical example, we apply our protocol to the Fermi polaron problem.

The conductivity of $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$ drops two orders of magnitude below the Verwey temperature $T_v$, known as the Verwey transition, due to the formation of charge ordering (CO). Here, we report the discovery of a large birefringence effect correlated with the CO in $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$ controlled by ultrafast-laser-assisted magnetic field cooling. The polarization rotation (PR) of the light reflected from a single crystalline $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$ film below $T_v$ shows a twofold symmetry as the cooling field (CF) rotates through ${360}^{\circ }$ within the film plane. The maximum PR occurs for the CF parallel to the cubic $\langle 110\rangle$ axes, and its amplitude depends on the sample orientation. These results are well interpreted by taking into account two CO patterns with orthogonal CO orientations, with their fractional areas determined by the ratio of the field components along the [110] and $\left[1\overline{1}0\right]$ axes. Our results indicate that application of the CF along $\langle 110\rangle$ axes may result in the single orientation CO state, which is highly desirable for unraveling the subtle CO structure to better understand the driving mechanism of the Verwey transition. In addition, ultrafast pump-probe measurements reveal a diminishment of the twofold PR at 0.8 ps due to fast melting of the CO state by the ultrafast laser pulse.

A Weyl semimetal denotes an electronic phase of solids in which two bands cross linearly. In this paper we study the effect of a spatially correlated disorder on such a phase. Using a renormalization-group analysis, we show that in three dimensions three scenarios are possible depending on the disorder correlations. A standard transition is recovered for short-range correlations. For disorder decaying slower than $1/r^2$, the Weyl semimetal is unstable to any weak disorder and no transition persists. In between, a new phase transition occurs. This transition still separates a disordered metal from a semimetal, but with a new critical behavior that we analyze to two-loop order.

In this work, we study nonlinear aspects of the deterministic spin dynamics of an anisotropic single-domain magnetic particle at finite temperature modeled by the Landau-Lifshitz-Bloch equation. The magnetic field has two components: a constant term and a term involving a harmonic time modulation. The dynamical behavior of the system is characterized with the Lyapunov exponents and by means of bifurcation diagrams and Fourier spectra. In particular, we explore the effects of the magnitude and frequency of the applied magnetic field, finding that the system presents multiple transitions between regular and chaotic states when varying the control parameters. We also address the temperature dependence and evidence that it plays an important role in these transitions, almost suppressing the chaotic behavior close to the Curie temperature. Finally, we find that the system has hyperchaotic states for specific values of field and temperature.

We study the thermodynamics of a spin-1/2 XYZ Heisenberg chain with a Dzyaloshinskii-Moriya interaction. This model describes the low-energy behaviors of a one-dimensional two-component bosonic model with a synthetic spin-orbit coupling in the deep insulating region. In the limit $U^{\prime }/U\to \infty$, where $U$ is the strength of the onsite intracomponent repulsion and $U^{\prime }$ is the intercomponent one, we solve our model exactly by Jordan-Wigner transformation, and thus provide a benchmark for our following numerical approach. In other cases, we calculate the entropy and the specific heat numerically by the transfer-matrix renormalization-group method. Their low-temperature behaviors depend crucially on the properties of the zero-temperature phases. A refined ground-state phase diagram is then deduced from their low-temperature behaviors. Our findings offer an alternative way to detect those distinguishable phases experimentally.

Momentum and energy conservation are fundamental tenets of physics, which valid physical theories have to satisfy. In the reversible transformation between superconducting and normal phases in the presence of a magnetic field, the mechanical momentum of the supercurrent has to be transferred to the body as a whole and vice versa, the kinetic energy of the supercurrent stays in the electronic degrees of freedom, and no energy is dissipated nor entropy is generated in the process. We argue on general grounds that to explain these processes it is necessary that the electromagnetic field mediates the transfer of momentum between electrons and the body as a whole, and this requires that when the phase boundary between normal and superconducting phases is displaced, a flow and counterflow of charge occurs in a direction perpendicular to the phase boundary. This flow and counterflow does not occur according to the conventional BCS-London theory of superconductivity, therefore we argue that within BCS-London theory the Meissner transition is a “forbidden transition.” Furthermore, to explain the phase transformation in a way that is consistent with the experimental observations, requires that (i) the wave function and charge distribution of superconducting electrons near the phase boundary extend into the normal phase, and (ii) that the charge carriers in the normal state have holelike character. The conventional theory of superconductivity does not have these physical elements, the theory of hole superconductivity does.

The new hexagonal aluminous (NAL) phase, chemical formula $A{B}_2{C}_6{\mathrm{O}}_{12}$ ($A= {\mathrm{Na}}^{+}, {\mathrm{K}}^{+}, {\mathrm{Ca}}^{2+}$; $B= {\mathrm{Mg}}^{2+}, {\mathrm{Fe}}^{2+}, {\mathrm{Fe}}^{3+}$; $C= {\mathrm{Al}}^{3+}, {\mathrm{Si}}^{4+}, {\mathrm{Fe}}^{3+}$), is considered a major component ($\sim 20$ vol%) of mid-ocean ridge basalt (MORB) under the lower-mantle condition. As MORB can be transported back into the Earth's lower mantle via subduction, a thorough knowledge of the NAL phase is essential to fully understand the fate of subducted MORB and its role in mantle dynamics and heterogeneity. In this Rapid Communication, the complicated spin crossover of the Fe-bearing NAL phase is revealed by a series of local density approximation + self-consistent Hubbard $U$ (LDA+$U_{sc}$) calculations. Only the ferric iron (${\mathrm{Fe}}^{3+}$) substituting Al/Si in the octahedral ($C$) site undergoes a crossover from the high-spin (HS) to the low-spin (LS) state at $\sim 40$ GPa, while iron substituting Mg in the trigonal-prismatic ($B$) site remains in the HS state, regardless of its oxidation state (${\mathrm{Fe}}^{2+}$ or ${\mathrm{Fe}}^{3+}$). The volume/elastic anomalies and the iron nuclear quadrupole splittings determined by calculations are in great agreement with room-temperature experiments. The calculations further predict that the HS-LS transition pressure of the NAL phase barely increases with temperature due to the three nearly degenerate LS states of ${\mathrm{Fe}}^{3+}$, suggesting that the elastic anomalies of this mineral can occur at the top lower mantle.

We calculate the zero-temperature differential conductance $dI/dV$ of a voltage-biased one-dimensional junction between a nontopological and a topological superconductor for arbitrary junction transparency using the scattering matrix formalism. We consider two representative models for the topological superconductors: (i) spinful $p$-wave and (ii) $s$-wave with spin-orbit coupling and spin splitting. We verify that in the tunneling limit (small junction transparencies) where only single Andreev reflections contribute to the current, the conductance for voltages below the nontopological superconductor gap ${\mathrm{\Delta }}_s$ is zero and there are two symmetric conductance peaks appearing at $eV=\pm {\mathrm{\Delta }}_s$ with the quantized value $(4-\pi )2e^2/h$ due to resonant Andreev reflection from the Majorana zero mode. However, when the junction transparency is not small, there is a finite conductance for $e\left|V\right|<{\mathrm{\Delta }}_s$ arising from multiple Andreev reflections. The conductance at $eV=\pm {\mathrm{\Delta }}_s$ in this case is no longer quantized. In general, the conductance is particle-hole asymmetric except for sufficiently small transparencies. We further show that, for certain values of parameters, the tunneling conductance from a zero-energy conventional Andreev bound state can be made to mimic the conductance from a true Majorana mode.

Recent experiments performed in current-driven nanocontacts with strong perpendicular anisotropy have shown that spin-transfer torque can drive self-localized spin waves [W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E. W. Bauer, and C. Serpico, Phys. Rev. B 81, 014426 (2010); S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, and J. Åkerman, Phys. Status Solidi RRL 5, 432 (2011)], that above a certain intensity threshold can condense into a nanosized and highly nonlinear dynamic state known as a magnetic droplet soliton [S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. Åkerman, Science 339, 1295 (2013)]. Here we demonstrate analytically, numerically, and experimentally that at sufficiently large driving currents and for a spin polarization direction tilted away from the normal to a nanocontact plane, the circular droplet soliton can become unstable against the excitations in the form of periodic deformations of its perimeter. We also show that these perimeter excitation modes (PEMs) can be excited parametrically when the fundamental droplet soliton precession frequency is close to the double frequency of one of the PEMs. As a consequence, with increasing magnitude of a bias magnetic field the PEMs with progressively higher indices and frequencies can be excited. Full qualitative and partly quantitative agreement with experiment confirm the presented theoretical picture.

In one-dimensional electronic systems with strong repulsive interactions, charge excitations propagate much faster than spin excitations. Such systems therefore have an intermediate temperature range [termed the “spin-incoherent Luttinger liquid” (SILL) regime] where charge excitations are “cold” (i.e., have low entropy) whereas spin excitations are “hot.” We explore the effects of charge-sector disorder in the SILL regime in the absence of external sources of equilibration. We argue that the disorder localizes all charge-sector excitations; however, spin excitations are protected against full localization, and act as a heat bath facilitating charge and energy transport on asymptotically long time scales. The charge, spin, and energy conductivities are widely separated from one another. The dominant carriers of energy in much of the SILL regime are neither charge nor spin excitations, but neutral “phonon” modes, which undergo an unconventional form of hopping transport that we discuss. We comment on the applicability of these ideas to experiments and numerical simulations.

We report simultaneous radiotracer diffusion experiments of Co-57 and of Zr-95 in binary $\mathrm{Z}{\mathrm{r}}_{64}\mathrm{N}{\mathrm{i}}_{36}, \mathrm{Z}{\mathrm{r}}_{36}\mathrm{N}{\mathrm{i}}_{64}$, and ternary $\mathrm{Z}{\mathrm{r}}_{60}\mathrm{N}{\mathrm{i}}_{25}\mathrm{A}{\mathrm{l}}_{15}$ alloys above but near the liquidus temperature ($T_l$). In contrast to the multicomponent $\mathrm{Z}{\mathrm{r}}_{46.75}\mathrm{T}{\mathrm{i}}_{8.25}\mathrm{C}{\mathrm{u}}_{7.5}\mathrm{N}{\mathrm{i}}_{10}\mathrm{B}{\mathrm{e}}_{27.5}$ (Vit4), where a significant component decoupling at the $T_l$ is observed, the ratio between Zr and Co self-diffusion coefficients is smaller than two in the alloys with fewer components. The difference in the degree of decoupling compared to Vit4 can be explained in terms of different $T_l$'s. Moreover, owing to the high accuracy of the simultaneous tracer diffusion technique, we are able to resolve a small but notable composition dependence of the ratio $D_{\mathrm{Co}}/D_{\mathrm{Zr}}$, which decreases with increasing Zr content for all glass forming alloys reported here. In contrast to a hard sphere (HS)-like mixture, where decoupling is controlled only by atomic sizes, this indicates a coupling of the Co/Ni and Zr diffusion, due to the strong chemical affinity between the diffusing components. Our results are in very good agreement with recent simulations in $\mathrm{Z}{\mathrm{r}}_{64}\mathrm{N}{\mathrm{i}}_{36}$ based on mode coupling theory (MCT).

We numerically study the dynamics after a parameter quench in the one-dimensional transverse-field Ising model with long-range interactions ($\propto 1/r^{\alpha }$ with distance $r$), for finite chains and also directly in the thermodynamic limit. In nonequilibrium, i.e., before the system settles into a thermal state, we find a long-lived regime that is characterized by a prethermal value of the magnetization, which in general differs from its thermal value. We find that the ferromagnetic phase is stabilized dynamically: as a function of the quench parameter, the prethermal magnetization shows a transition between a symmetry-broken and a symmetric phase, even for those values of $\alpha$ for which no finite-temperature transition occurs in equilibrium. The dynamical critical point is shifted with respect to the equilibrium one, and the shift is found to depend on $\alpha$ as well as on the quench parameters.

Using ab initio methods, we investigate the modification of the magnetic properties of the $m=2$ member of the strontium iridates Ruddlesden–Popper series ${\mathrm{Sr}}_{m+1}{\mathrm{Ir}}_m{\mathrm{O}}_{3m+1}$, bilayer ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$, induced by epitaxial strain and oxygen vacancies. Unlike the single-layer compound ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, which exhibits a robust in-plane magnetic order, the energy difference between in-plane and out-of-plane magnetic orderings in ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ is much smaller and it is expected that small external perturbations could induce magnetic transitions. Our results indicate that epitaxial strain yields a spin-flop transition, which is driven by the crossover between the intralayer $J_1$ and interlayer $J_2$ magnetic exchange interactions upon compressive strain. While $J_1$ is essentially insensitive to strain effects, the strength of $J_2$ changes by one order of magnitude for tensile strains $\ge 3\%$. In addition, our study clarifies that the unusual in-plane magnetic response observed in ${\mathrm{Sr}}_3{\mathrm{Ir}}_2{\mathrm{O}}_7$ upon the application of an external magnetic field originates from the canting of the local magnetic moments due to oxygen vacancies, which locally destroy the octahedral networks, thereby allowing for noncollinear spin configurations.

We study experimentally and theoretically a coherently driven strongly coupled quantum dot–microcavity system. Our focus is on physics of the unexplored intermediate excitation regime where the resonant laser field dresses a strongly coupled single exciton-photon (polariton) system resulting in a ladder of laser-dressed Jaynes-Cummings states. In that case, both the coupling of the emitter to the confined light field of the microcavity and to the light field of the external laser are equally important, as proved by observation of injection pulling of the polariton branches by an external laser. This intermediate interaction regime is of particular interest since it connects the purely quantum mechanical Jaynes-Cummings ladder and the semiclassical Autler-Townes ladder. Exploring the driving strength dependence of the mutually coupled system we establish the maximum in the resonance fluorescence signal to be a robust fingerprint of the intermediate regime and observe signatures indicating the laser-dressed Jaynes-Cummings ladder. In order to address the underlying physics we excite the coupled system via the matter component of fermionic nature undergoing saturation—in contrast to commonly used cavity-mediated excitation.

The insulating state of matter can be probed by means of a ground state geometrical marker, which is closely related to the modern theory of polarization (based on a Berry phase). In the present work we show that this marker can be applied to determine the metal-insulator transition in disordered systems. In particular, for noninteracting systems the geometrical marker can be obtained from the configurational average of the norm-squared one-body density matrix, which can be calculated within open as well as periodic boundary conditions. This is in sharp contrast to a classification based on the static conductivity, which is only sensible within periodic boundary conditions. We exemplify the method by considering a simple lattice model, known to have a metal-insulator transition as a function of the disorder strength, and demonstrate that the transition point can be obtained accurately from the one-body density matrix. The approach has a general ab initio formulation and could in principle be applied to realistic disordered materials by standard electronic structure methods.

The interplay between lattice, orbital, and spin degrees of freedom in iron chromite (spinel $[\mathrm{Fe}]\left\{\mathrm{C}{\mathrm{r}}_2\right\}{\mathrm{O}}_4)$ has been investigated to near megabar pressures. The cubic-to-tetragonal transition, from static Jahn-Teller distortions at the Fe locality, rises from $T_{\mathrm{J}\text{−}\mathrm{T}}\sim 135\mathrm{K}$ at ambient pressure to 300 K by $\sim 12\mathrm{GPa}$. The tetragonal distortion progressively increases and orbital moment quenching is triggered beyond $\sim 24\mathrm{GPa}$, as monitored by the magnetic hyperfine field $H_{hf}$ at Fe sites. In the range 30–60 GPa, original $\mathrm{F}{\mathrm{e}}^{2+}$ tetrahedral sites with an unquenched orbital moment $(H_{hf}\sim 20\mathrm{T})$ coexist with newly evolved $\mathrm{F}{\mathrm{e}}^{2+}$ at tetrahedral sites having the orbital moment quenched and a resultant large $H_{hf}\sim 35\mathrm{T}$. Additionally, new $\mathrm{F}{\mathrm{e}}^{2+}$ sites having distinguishable orbital moment quenching signatures $(H_{hf}\sim 42\mathrm{T})$ are discerned. Those sites also have other Fe nuclear hyperfine interaction parameter values typical of $\mathrm{F}{\mathrm{e}}^{2+}$ in octahedral coordination. There is a concurrent change to a steeper decrease of unit-cell volume as pressure rises above $\sim 30\mathrm{GPa}$. These electronic and lattice responses are interpreted as signatures of progressive partial spinel inversion from high-spin $\mathrm{Fe}\leftrightarrow \mathrm{Cr}$ tetrahedral/octahedral site exchange, triggered near $\sim 30\mathrm{GPa}$. Beyond 60 GPa a new diamagnetic low-spin Mössbauer spectral component emerges. This is preceded by an inflection and discontinuity in the pressure dependence of the resistance and tetragonal unit-cell volume, respectively. By $\sim 93\mathrm{GPa}$, half of the iron is low spin in octahedral sites from evolved tetragonal-spinel inversion processes. The remainder reside in tetrahedral high-spin sites with $H_{hf}\sim 30\mathrm{T}$. The charge gap from electron correlations does not close, despite an anticipated appreciable band broadening from a $\sim 30\%$ unit-cell volume reduction upon pressurization to $\sim 93\mathrm{GPa}$. This is attributable to an increase in effective on-site repulsion (Hubbard $U_{\mathrm{eff}}$) with increasing pressure or at spin crossover, specific to the $\mathrm{C}{\mathrm{r}}^{3+}\left(d^3\right)$ and $\mathrm{F}{\mathrm{e}}^{2+}\left(d^6\right)$ electronic configurations, respectively. Thus a $\sim 200\mathrm{meV}$ “Mott” gap prevails in the mixed spin-state of the partially inverted tetragonal phase $[\mathrm{F}{\mathrm{e}}_{1-x}\mathrm{C}{\mathrm{r}}_x]\left\{\mathrm{F}{\mathrm{e}}_x\mathrm{C}{\mathrm{r}}_{2-x}\right\}{\mathrm{O}}_4(x\sim 0.5)$ up to near a megabar (100 GPa) densification.

Interlayer electron transport through a graphene/hexagonal boron-nitride (h-BN)/graphene heterostructure is strongly affected by the misorientation angle $\theta$ of the h-BN with respect to the graphene layers with different physical mechanisms governing the transport in different regimes of angle, Fermi level, and bias. The different mechanisms and their resulting signatures in resistance and current are analyzed using two different models, a tight-binding, nonequilibrium Green function model and an effective continuum model, and the qualitative features resulting from the two different models compare well. In the large-angle regime $(\theta >4^{\circ })$, the change in the effective h-BN band gap seen by an electron at the $K$ point of the graphene causes the resistance to monotonically increase with angle by several orders of magnitude, reaching a maximum at $\theta ={30}^{\circ }$. It does not affect the peak-to-valley current ratios in devices that exhibit negative differential resistance. In the small-angle regime $(\theta <4^{\circ })$, umklapp processes open up new conductance channels that manifest themselves as nonmonotonic features in a plot of resistance versus Fermi level that can serve as experimental signatures of this effect. For small angles and high bias, the umklapp processes give rise to two new current peaks on either side of the direct tunneling peak.