In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of $N_A$ out of $N$ degrees of freedom is given by ${\langle S_A\rangle }_{\mathrm{G}}=(N-\frac{1}{2})\mathrm{\Psi }\left(2N\right)+(\frac{1}{4}-N_A)\mathrm{\Psi }\left(N\right)+(\frac{1}{2}+N_A-N)\mathrm{\Psi }(2N-2N_A)-\frac{1}{4}\mathrm{\Psi }(N-N_A)-N_A$, where $\mathrm{\Psi }$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\langle S_A\rangle }_{\mathrm{G}}=N(log2-1)f+N(f-1)log(1-f)+\frac{1}{2}f+\frac{1}{4}log(1-f)+O(1/N)$, where $f=N_A/N\le 1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Łydżba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\to \infty }{\left(\mathrm{\Delta }{S}_A\right)}_{\mathrm{G}}^2=\frac{1}{2}[f+f^2+log(1-f)]$.

Drawing inspiration from bilayer graphene, this paper introduces its photonic analog comprising two stacked graphenelike photonic crystals that are coupled in the near field through spoof surface plasmons. Beyond the twist degree of freedom that can radically alter the band structure of the bilayer photonic graphene, the photonic dispersion can be also tailored via the interlayer coupling which exhibits an exponential dependence on the distance between the two photonic crystals. We theoretically, numerically, and experimentally characterize the band structures of $\mathit{AA}$- and $\mathit{AB}$-stacked bilayer photonic graphene, as well as for twisted bilayer photonic graphene with even and odd sublattice exchange symmetries. Furthermore, we numerically predict the existence of magic angles in bilayer photonic graphene, which are associated with ultraflat bands resulting from interlayer hybridization. Finally, we demonstrate that the bilayer photonic graphene at a particular twist angle satisfying even sublattice exchange symmetry is a high-order photonic topological insulator. The proposed bilayer photonic graphene could constitute a useful platform for identifying new quantum materials and inspiring next-generation photonic devices with new degrees of freedom and emerging functionality.

While the Heisenberg model for magnetic Mott insulators on planar lattice structures is comparatively well understood in the case of transition metal ions, the intrinsic spin-orbit entanglement of $4f$ magnetic ions on such lattices shows fascinating new physics largely due to corresponding strong anisotropies both in their single-ion and their exchange properties. We show here that the Yb delafossites, containing perfect magnetic ${\mathrm{Yb}}^{3+}$ triangular lattice planes with pseudospin $s=\frac{1}{2}$ at low temperatures, are an ideal platform to study these phenomena. Competing frustrated interactions may lead to an absence of magnetic order associated to a gapless spin liquid ground state with a huge linear specific heat exceeding that of many heavy fermions, whereas the application of a magnetic field induces anisotropic magnetic order with successive transitions into different long-range ordered structures. In this comparative study, we discuss our experimental findings in terms of a unified crystal-field and exchange model. We combine electron paramagnetic resonance (EPR) experiments and results from neutron scattering with measurements of the magnetic susceptibility, isothermal magnetization up to full polarization, and specific heat to determine the relevant model parameters. The impact of the crystal field is discussed as well as the symmetry-compatible form of the exchange tensor, and we give explicit expressions for the anisotropic $g$ factor, the temperature dependence of the susceptibility, the exchange-narrowed EPR linewidth, and the saturation field.

We use series expansion methods to investigate the zero-temperature phase diagram of a spin-$\frac{1}{2}$ Heisenberg frustrated bilayer model. For different parameter regions the model shows two related antiferromagnetic phases as well as a nonmagnetic dimerized phase. The phase boundaries are located to high precision and agree very well with a previous calculation using different methods. The triplon spectrum in the dimerized phase is also studied.

We investigate the thermal evolution of magnetic correlations of the Co-based honeycomb lattice ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ with ${}^{23}\mathrm{Na}$ nuclear magnetic resonance and static magnetic susceptibility $\chi \left(T\right)$. The studied compound shows three-dimensional (3D) long-range magnetic ordering at $T_{\mathrm{N}}=26$ K. On cooling through $T^{*}\approx 110$ K, a simple paramagnetic state undergoes a crossover to a correlated paramagnetic state featuring a power-law dependence of the nuclear spin-lattice $(1/T_1)$ and spin-spin $(1/T_2)$ relaxation rates as well as of the out-of-plane $\chi \left(T\right)$. The magnetic-field-direction dependence of $1/T_1$, $1/T_2$, and $\chi \left(T\right)$ uncovers anisotropic spin-spin correlations of a two-dimensional (2D) renormalized classical character. In a magnetically ordered state, we are able to identify four successive transitions or crossovers occurring at $T_{\mathrm{N}}=26$, $T_{\mathrm{N}1}=16$, $T_{\mathrm{N}2}=7$, and $T_{\mathrm{N}3}=3.5$ K. The multiple transitions and crossovers are associated with the coexistence of 2D and 3D magnetic orders or reorientation of the ordered spins. Our results suggest the presence of various types of frustrating interactions and their energy hierarchy that control complex magnetic structures and anisotropic magnetism.

We study a bipartite collective spin-1 model with exchange interaction between the spins. The bipartite nature of the model manifests itself by the spins being divided into two equal-sized subsystems; within each subsystem the spin-spin interactions are of the same strength, across the subsystems they are also equal, but the two coupling values within and across the subsystem are different. Such a setup is inspired by recent experiments with ultracold atoms. Using the $\mathrm{SU}\left(3\right)$ symmetry of the exchange interaction and the permutation symmetry within the subsystems, we can employ representation theoretic methods to diagonalize the Hamiltonian of the system in the entire parameter space of the two coupling strengths. These techniques then allow us to explicitly construct and explore the ground-state phase diagram. The phase diagram turns out to be rich containing both gapped and gapless phases. An interesting observation is that one of the five phases features a strong bipartite symmetry breaking, meaning that the two subsystems in the ground states are in different $\mathrm{SU}\left(3\right)$ representations.

We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau-type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.

The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O\left(N\right)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to \infty$, we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu ,\beta ,\theta$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\omega /\left(2v\right), v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.

Antiferromagnetic (AF) thin films typically exhibit a multidomain state, and control of the AF Néel vector is challenging, as AF materials are robust to magnetic perturbations. In this paper, uniaxial Néel vector control is demonstrated by relying on anisotropic strain engineering of epitaxial thin films of the prototypical AF material ${\mathrm{LaFeO}}_3$ (LFO). Orthorhombic (011)- and (101)-oriented ${\mathrm{DyScO}}_3$, ${\mathrm{GdScO}}_3$, and ${\mathrm{NdGaO}}_3$ substrates are used to engineer different anisotropic in-plane strain states. The anisotropic in-plane strain stabilizes structurally monodomain monoclinic LFO thin films. The uniaxial Néel vector is found along the tensile strained $b$ axis, contrary to bulk LFO having the Néel vector along the shorter $a$ axis, and no magnetic domains are found. Hence, anisotropic strain engineering is a viable tool for designing unique functional responses, further enabling AF materials for mesoscopic device technology.

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated 0 phase. It controls the critical behavior of the $0\text{−}\pi$ quantum transition and lifts the degeneracy in the $\pi$ phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in 0 and $\pi$ phases. While the magnetic susceptibility vanishes in the 0 phase it becomes divergent in the $\pi$ phase at zero temperature.

We demonstrate that electrostatic interactions between helical electrons at the edge of a quantum spin Hall insulator and a dynamical impurity can induce quasielastic backscattering. Modeling the impurity as a two-level system, we show that transitions between counterpropagating Kramers-degenerate electronic states can occur without breaking time-reversal symmetry, provided that the impurity also undergoes a transition. The associated electrical resistance has a weak temperature dependence down to a nonuniversal temperature scale. Our results extend the range of known backscattering mechanisms in helical edge modes to include scenarios where electron tunneling out of the system is absent.

We derive an expansion of the functional renormalization (fRG) equations that treats the frequency and momentum dependencies of the vertices in a systematic manner. The scheme extends the channel-decomposed fRG equations to the frequency domain and reformulates them as a series of linear integral equations in the particle-particle, particle-hole, and particle-hole exchange channels. We show that the linearity of the equations offers numerous computational advantages and leads to converged, stable solutions for a variety of Hamiltonians. As the expansion is in the coupling between channels, the truncations that are necessary to making the scheme computationally viable still lead to equations that treat contributions from all channels equally. As a first benchmark we apply the two-loop fRG equations to the single impurity Anderson model. We consider the sources of error within the fRG, the computational cost associated with each, and how the choice of regulator affects the flow of the fRG. We then use the optimal truncation scheme to study the extended Hubbard Hamiltonian in one and two dimensions. We find that in many cases of interest the fRG flow converges to a stable vertex and self-energy from which we can extract the various correlation functions and susceptibilities of interest.

We studied theoretically the effect of a low concentration of adsorbed polar molecules on the optical conductivity of graphene, within the Kubo linear response approximation. Our analysis is based on a continuum model approximation that includes up to next-to-nearest neighbors in the pristine graphene effective Hamiltonian. Our results show that the conductivity can be expressed in terms of renormalized quasiparticle parameters ${\tilde{v}}_F, \tilde{M}$, and $\tilde{\mu }$ that include the effect of the molecular surface concentration $n_{\text{dip}}$ and dipolar moment $\mathcal{P}$, thus providing a quantum field theory approach to model a graphene-based chemical sensor.

Internal quantum efficiencies of the surface photoemission (SPE) and the volume photoemission (VPE) from metal nanoparticles into a semiconductor environment are found and compared with each other, taking into account the discontinuity in the electron effective mass on the metal-semiconductor interface. It is found that SPE is less sensitive to the discontinuity of the electron effective mass than VPE. This advantage, along with other advantages, makes SPE more efficient than VPE for generation of hot electrons from small metal nanoparticles. General formulas for quantum efficiencies and analytical expressions for quantum efficiency on the red photoeffect limit are presented. High efficiency of SPE from spherical gold nanoparticles with the step potential barrier on the metal-semiconductor interface is demonstrated.

In this paper, we have systematically studied the role of point defects in the recombination time of monolayer ${\mathrm{MoS}}_2$ using time-dependent ab initio nonadiabatic molecular dynamics simulations. Various types of point defects, such as S vacancy, S interstitial, Mo vacancy, and Mo interstitial have been considered. We show that defects strongly accelerate the electron-hole recombination, especially interstitial S atoms do that by three orders of magnitude higher compared to pristine ${\mathrm{MoS}}_2$. Mo defects (both vacancy and interstitial) introduce a multitude of de-excitation pathways via various defect levels in the energy gap. The results of this study provide some fundamental understanding of photoinduced de-excitation dynamics in presence of defects in highly technologically relevant 2D ${\mathrm{MoS}}_2$.

The thermoelectric performance of a topological Josephson nonlocal heat engine is thoroughly investigated. The nonlocal response is obtained by using a normal metal probe coupled to only one of the proximized helical edges in the middle of the junction. In this configuration, we investigate how the flux and phase biases trigger the nonlocal thermoelectric effects under the application of a thermal difference between the superconducting terminals. Possible experimental nonidealities such as asymmetric proximized superconducting gaps are considered, showing how the nonlocal response can be affected. The interplay between Doppler shift, which tends to close gaps, and Andreev interferometry, which affects particle-hole resonant transport, are clearly identified for different operating regimes. Finally, we discuss the power and the efficiency of the topological thermoelectric engine which reaches maximum power at maximal efficiency for a well-coupled normal probe. We find quite high nonlocal Seebeck coefficients of the order of tenths of $\mu \mathrm{V}$/K at a few Kelvin, a signal that would also be clearly detectable against any spurious local effects even with moderate asymmetry of the gaps.

Intrinsic and extrinsic valley Hall effects are predicted to emerge in graphene systems with uniform or spatially varying mass terms. Extrinsic mechanisms, mediated by the valley-dependent scattering of electrons at the Fermi surface, can be directly linked to quantum transport simulations. This is a promising route toward more complete experimental investigation of valleytronic phenomena in graphene, but a major obstacle is the difficulty in applying the sublattice-dependent potentials required. Here we show that strongly valley-dependent scattering also emerges from bilayer graphene quantum dots, where the gap size can be easily modulated using the interlayer potentials in dual-gated devices. Robust valley-dependent scattering and concomitant valley currents are observed for a range of systems, and we investigate the role of dot size, mass strength, and additional potential terms. Finally, we note that a strong valley splitting of electronic current also emerges when a biased bilayer dot is embedded in a single layer of graphene, but that the effect is less robust than for a bilayer host. Our findings suggest that bilayer graphene devices with custom mass profiles provide an excellent platform for future valleytronic exploration of two-dimensional materials.

A dynamics of the precession of coupled atomic moments in the tight-binding (TB) approximation is presented. By implementing an angular penalty functional in the energy that captures the magnetic effective fields self-consistently, the motion of the orientation of the local magnetic moments is observed faster than the variation of their magnitudes. This allows the computation of the effective atomic magnetic fields that are found consistent with the Heisenberg's exchange interaction, by comparison with classical atomistic spin dynamics on Fe, Co, and Ni magnetic clusters.