Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($\partial \mathrm{TRG}$) that can be applied to improve various TRG methods, in an automatic fashion. $\partial \mathrm{TRG}$ systematically extends the essential concept of second renormalization [Phys. Rev. Lett. 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration. Given the forward TRG process, $\partial \mathrm{TRG}$ automatically finds the gradient of local tensors through backpropagation, with which one can deeply “train” the tensor networks. We benchmark $\partial \mathrm{TRG}$ in solving the square-lattice Ising model, and we demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The global optimization as well as GPU acceleration renders $\partial \mathrm{TRG}$ a highly efficient and accurate many-body computation approach.

Weyl semimetals (WSMs) have unusual optical responses originating from unique topological properties of their bulk and surface electron states. Their third-order optical nonlinearity is expected to be very strong, especially at long wavelengths, due to linear dispersion and high Fermi velocity of three-dimensional Weyl fermions. Here we derive the third-order nonlinear optical conductivity of WSMs in the long-wavelength limit and calculate the intensity of the nonlinear four-wave mixing signal as it is transmitted through the WSM film or propagates away from the surface of the material in the reflection geometry. All results are analytic and show the scaling of the signal intensity with variation of all relevant parameters. The nonlinear generation efficiency turns out to be surprisingly high for a lossy material, of the order of several mW per ${\mathrm{W}}^3$ of the incident pump power. Optimal conditions for maximizing the nonlinear signal are realized in the vicinity of bulk plasma resonance. This indicates that ultrathin WSM films of the order of skin depth in thickness could find applications in compact optoelectronic devices.

The anharmonicity of the acoustic phonon dispersion of graphene has been studied by the harmonic linear response (HLR) approach at finite temperature. This is a nonperturbative method based on the linear response of the system to applied forces, as derived from equilibrium computer simulations. Anharmonic shifts are analyzed in the long-wavelength limit at room temperature, with emphasis on the effect of applied tensile or compressive in-plane stress. The simulation results are compared with available analytical models, based either on first-order perturbation theory or on a description by anomalous exponents. The simulations show better agreement to the expectations of the perturbational approach. The effect of temperature and zero-point vibrations on the acoustic out-of-plane anharmonic shifts of graphene are briefly reviewed.

We develop a real-space theory of the voltage bias driven transition from a Mott insulator to a correlated metal. Within our Keldysh mean-field approach the problem reduces to a self-consistency scheme for the charge and spin profiles in this open system. We solve this problem for a two dimensional antiferromagnetic Mott insulator at zero temperature. The charge and spin magnitude is uniform over the system at zero bias, but a bias $V$ leads to spatial modulation over a length scale $\xi \left(V\right)$ near the edges. $\xi \left(V\right)$ grows rapidly and becomes comparable to system size as $V$ increases towards a threshold scale $V_c$. The linear response conductance of the insulator is zero with the current being exponentially small for $V\ll V_c$. The current increases rapidly as $V\to V_c$. Beyond $V_c$, we observe an inhomogeneous low moment antiferromagnetic metal, and at even larger bias, a current saturated paramagnetic metal. We suggest an approximate scheme for the spectral features of this nonequilibrium system.

Plutonium is a critically important material as the behavior of its $5f$ electrons stands midway between the metalliclike itinerant character of the light actinides and localized atomic-core-like character of the heavy actinides. The $\delta$ phase of plutonium ($\delta$-Pu), whereas still itinerant, has a large coherent Kondo peak and strong electronic correlations coming from its near-localized character. Using sophisticated Gutzwiller wave function and dynamical mean-field theory correlated theories, we study the Fermi surface and associated mass renormalizations of $\delta$-Pu together with calculations of the de Haas–van Alphen frequencies. We find a large ($\sim 200\%$) correlation induced volume expansion in both the hole and the electron pockets of the Fermi surface in addition to an intermediate mass enhancement. All of the correlated electron theories predict, approximately, the same hole pocket placement in the Brillouin zone, which is different from that obtained in conventional density-functional band-structure theory, whereas the electron pockets from all theories are in, roughly, the same place.

We demonstrate that the finite-difference time-domain (FDTD) method can be used to study the formation of laser-induced periodic surface structures (LIPSSs) under oblique incidence and arbitrary polarization states. The inhomogeneous energy absorption below a rough surface at oblique incident angle is studied by the FDTD method and compared to the analytical efficacy theory developed by Sipe et al. [Phys. Rev. B 27, 1141 (1983)]. The two approaches show excellent agreement. The surface topographies of both low-spatial-frequency LIPSSs (LSFLs) and high-spatial-frequency LIPSSs (HSFLs) at oblique incidence and various polarization states (linear, circular, radial, and azimuthal) are simulated by taking into account topography-driven interpulse feedback effects. For $s$- and mixed $s$- and $p$-polarized beams at large incident angles, the simulated LSFLs show orientations that are neither perpendicular nor parallel to the laser polarization. Their physical origin is explained by a nonzero angle between the in-plane wave vector of the incident light and the scattered light. By using circularly-polarized beams, the simulated surface topographies are further shown to be in excellent agreement with the triangular LSFLs and the fine-dot nanoscaled structures reported in the literature. In addition, the simulation of HSFLs suggests that their periods have almost no dependence on the incident angle nor the polarization angle, while their surface topographies resemble that of blazed gratings at large incident angles for $p$-polarized and mixed $s$- and $p$-polarized beams. The origin is explained by the appearance of asymmetry in the scattered near-field light. The extension to oblique incidence and arbitrary polarization states demonstrates that the FDTD approach on LIPSSs is a highly versatile and powerful technique which has the potential to be one of the foundations for a complete modeling and understanding of LIPSSs under various irradiation conditions.

We theoretically study the quench dynamics of induced anisotropy of a large-spin magnetic molecule coupled to spin-polarized ferromagnetic leads. The real-time evolution is calculated by means of the time-dependent density-matrix numerical renormalization group method implemented within the matrix product states framework, which takes into account all correlations in a very accurate manner. We determine the system response to a quench in the spin-dependent coupling to ferromagnetic leads. In particular, we focus on the transient dynamics associated with crossing from the weak- to the strong-coupling regime, where the Kondo correlations become important. The dynamics is examined by calculating the time-dependent expectation values of the spin-quadrupole moment and the associated spin operators. We identify the relevant timescales describing the quench dynamics and determine the influence of the effective exchange coupling of molecules and spin polarization of the leads on the dynamical behavior of the system. Furthermore, the generalization of our predictions for large spin values of molecules is considered. Finally, we analyze the effect of finite temperature and show that it gives rise to a reduction of magnetic anisotropy by strong suppression of the time-dependent spin-quadrupole moment due to thermal fluctuations.

We investigate the influence of chirality on the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phase of $\mathcal{PT}$-symmetric chiral systems. Starting from the point that transverse magnetic (TM) and transverse electric (TE) waves have different exceptional points, we show that with circularly polarized waves (which are linear combinations of TM and TE waves) mixed $\mathcal{PT}$-symmetric phases can be realized and the extent of these phases can be highly controlled by either or both the chirality and the angle of incidence. Additionally, while the transmission of both TM and TE waves in nonchiral $\mathcal{PT}$-symmetric systems is the same for forward and backward propagation, we show that with chirality this symmetry can be broken. As a result, it is possible to realize asymmetric, i.e., side-dependent, rotation, and ellipticity in the polarization state of the transmitted wave. Our results constitute a simple example of a chiral $\mathcal{PT}$-symmetric optical system in which the various phases (full $\mathcal{PT}$, mixed, broken) and the asymmetric effects can be easily tuned by adjusting the chirality parameter and/or the angle of incidence.

Out-of-time-ordered correlators (OTOCs) are of crucial importance for studying a wide variety of fundamental phenomena in quantum physics, ranging from information scrambling to quantum chaos and many-body localization. However, apart from a few special cases, they are notoriously difficult to compute even numerically due to the exponential complexity of generic quantum many-body systems. In this paper, we introduce a machine learning approach to OTOCs based on the restricted-Boltzmann-machine architecture, which features wide applicability and could work for arbitrary-dimensional systems with massive entanglement. We show, through a concrete example involving a two-dimensional transverse field Ising model, that our method is capable of computing early-time OTOCs with respect to random pure quantum states or infinite-temperature thermal ensembles. Our results showcase the great potential for machine learning techniques in computing OTOCs, which open up numerous directions for future studies related to similar physical quantities.

Antiferromagnetically coupled ferromagnetic thin films have recently attracted significant attention in magnonics because of the possibility to tune the spin-wave dispersion by altering the interlayer exchange coupling. To implement such coupled films in magnonic devices, a detailed understanding of the precessional dynamics of magnetization in such systems is required. Here, we present a systematic characterization of the precessional dynamics for systems with the layer magnetization going from nearly antiparallel to parallel alignment in a magnetic field. Experimentally, we have measured the ultrafast-laser-induced magnetization precession in samples with different interlayer exchange coupling strengths using the time-resolved magneto-optical Kerr effect. In our measurements, in addition to the acoustic and optical modes, an extra mode is observed that is due to the laser-induced decoupling of the two ferromagnetic layers. The observed precessional dynamics is in good agreement with our theoretical model based on the Landau-Lifshitz-Gilbert equation, and can be separated into three different field regions determined by the relationship between the Zeeman energy and the energy associated with the interlayer exchange coupling. This systematic paper, which gives a detailed description of how the interlayer exchange coupling and Zeeman energy influence the precessional dynamics, provides an important guide to implement antiferromagnetically coupled films in functional magnonic devices.

We report a small-angle neutron scattering study of the helical and skyrmion lattice order in single-crystal ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ under quasihydrostatic helium gas pressures up to 5 kbar. By using helium gas as the pressure-transmitting medium (PTM) we ensure pressure application with improved hydrostaticity at cryogenic temperatures compared with previous reports where liquid PTMs were used. For 5-kbar He gas pressure we observe modest changes of the ambient pressure phase diagram; the critical temperature $T_c$ changes by $+2.8$(2)%, while in the low-$T$ limit the helical propagation vector $\left|q\right|$ changes by $-0.5\left(2\right)\%$, the lower critical field $H_{c1}$ changes by $+2.5$(1.0)%, and the upper critical field $H_{c2}$ remains unchanged within uncertainty. The skyrmion phase also changes little under pressure; its largest $T$ extent varies from $T_c-2.5\left(5\right)$ K at ambient pressure to $T_c-3.0\left(5\right)$ K at 5 kbar, and its location in the phase diagram follows the pressure-driven shift of $T_c$. The weak pressure dependences of the critical magnetic fields and skyrmion phase contrast strongly with much stronger pressure-driven changes reported from previous quasihydrostatic pressure studies. Taking into account the present results and those of other uniaxial pressure data, we suggest that the results of previous quasihydrostatic pressure studies were influenced by inadvertent directional stress pressure components. Overall, our study represents a high-pressure study of the chiral magnetism in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ under the most hydrostatic high-pressure conditions to date and serves also as a salient reminder of the sensitivity of chiral magnets to deviations from hydrostaticity in quasihydrostatic high-pressure studies.

The interplay between magnetoelectricity and magnetoelasticity (MEL) is studied in the context of voltage-controlled magnetic anisotropy (VCMA). Strain plays more than the role of changing lattice constants—that of the internal electric field in the heterostructure. As a prototype, FePt/MgO(001) is visited, where the behavior of two interfaces are drastically different: one exhibits switching, the other does not. Whether an external electric field ($E_{\text{ext}}$) is present or not, we found the VCMA coefficient larger than $1\mathrm{pJ}/\left(\mathrm{V}\mathrm{m}\right)$ as a consequence of the rearrangement of $d$ orbitals with $m=\pm 1$ and $\pm 2$ in response to an external electric field. In addition, magnetocrystalline anisotropy is analyzed with strain taken into account, where a nonlinear feature is presented, only accountable by invoking second-order MEL.

Reduced dimensionality offers a piece of crucial information in deciding the type of the quantum ground state in heavy fermion materials. Here we have examined the stoichiometric ${\mathrm{CeCo}}_2{\mathrm{Ga}}_8$ compound, which crystallizes in a quasi-one-dimensional crystal structure with Ga-Ce-Co chains along the $c$ axis. The low-temperature behavior of the magnetic susceptibility ($\chi \sim T^{-0.2}$), heat capacity ($C_p/T\sim -lnT$), and resistivity ($\rho \sim T^n$ with $n\sim 1$) strongly confirms the non-Fermi liquid ground state of ${\mathrm{CeCo}}_2{\mathrm{Ga}}_8$. We study the low-energy spin dynamics of the ${\mathrm{CeCo}}_2{\mathrm{Ga}}_8$ compound utilizing zero-field (ZF) and longitudinal-field muon spin relaxation ($\mu \mathrm{SR}$) measurements. ZF-$\mu \mathrm{SR}$ measurement reveals the absence of long-range magnetic ordering down to 70 mK. Interestingly, below 1 K the electronic relaxation rate rises sharply, suggesting the appearance of low-energy quantum spin fluctuations in ${\mathrm{CeCo}}_2{\mathrm{Ga}}_8$.

We report the magnetotransport properties of $T_d-\mathrm{MoT}{\mathrm{e}}_2$ based vertical junctions with NbN superconducting electrodes. Benefiting from the high critical magnetic field of NbN, the superconducting state is maintained up to 8 T. A sharp drop of resistance is observed at $T_c\sim 13\mathrm{K}$, corresponding to the critical temperature of NbN electrodes. The differential conductance spectra show that the zero-bias conductance is increased by magnetic fields with 41.6% increment at 14 T. Intriguingly, negative magnetoresistance (MR) ∼−48% is observed under 8 T at 3 K. The negative MR disappears above superconductivity critical temperature ∼13 K. The Weyl semimetal $\mathrm{MoT}{\mathrm{e}}_2$ combined with superconductor is promising for searching topological superconductivity and Majorana zero modes.

We find exponentially many exact quantum many-body scar states in a two-dimensional PXP model—an effective model for a two-dimensional Rydberg atom array in the nearest-neighbor blockade regime. Such scar states are remarkably simple valence bond solids despite being at an effectively infinite temperature, and thus strongly violate the eigenstate thermalization hypothesis. For a particular boundary condition, such eigenstates have integer-valued energies. Moreover, certain charge-density-wave initial states give rise to strong oscillations in the Rydberg excitation density after a quantum quench and towerlike structures in their overlaps with eigenstates.

Crystal and magnetic structures of the mineral centennialite ${\mathrm{CaCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2·0.6{\mathrm{H}}_2\mathrm{O}$ are investigated by means of synchrotron x-ray diffraction and neutron diffraction measurements complemented by density functional theory (DFT) and pseudofermion functional renormalization group (PFFRG) calculations. ${\mathrm{CaCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2·0.6{\mathrm{H}}_2\mathrm{O}$ crystallizes in the $P\overline{3}m1$ space group and ${\mathrm{Cu}}^{2+}$ ions form a geometrically perfect kagome network with antiferromagnetic $J_1$. No intersite disorder between ${\mathrm{Cu}}^{2+}$ and ${\mathrm{Ca}}^{2+}$ ions is detected. ${\mathrm{CaCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2·0.6{\mathrm{H}}_2\mathrm{O}$ enters a magnetic long-range ordered state below $T_{\text{N}}=7.2$ K, and the $\mathbf{q}=\mathbf{0}$ magnetic structure with negative vector spin chirality is obtained. The ordered moment at 0.3 K is suppressed to $0.58\left(2\right){\mu }_{\text{B}}$. Our DFT calculations indicate the presence of antiferromagnetic $J_2$ and ferromagnetic $J_d$ superexchange couplings of a strength which places the system at the crossroads of three magnetic orders (at the classical level) and a spin-$\frac{1}{2}$ PFFRG analysis shows a dominance of $\mathbf{q}=\mathbf{0}$ type magnetic correlations, consistent with and indicating proximity to the observed $\mathbf{q}=\mathbf{0}$ spin structure. The results suggest that this material is located close to a quantum critical point and is a good realization of a $J_1-J_2-J_d$ kagome antiferromagnet.

We study the motion of an interface separating two regions with different electronic orders following a short-duration pump that drives the system out of equilibrium. Using a generalized Ginzburg-Landau approach and assuming that the main effect of the nonequilibrium drive is to transiently heat the system we address the question of the direction of interface motion—in other words, which ordered region expands and which contracts after the pump. Our analysis includes the effects of differences in free-energy landscape and in order parameter dynamics and identifies circumstances in which the drive may act to increase the volume associated with the subdominant order, for example when the subdominant order has a second-order free-energy landscape while the dominant order has a first-order one.

We present an application of the extended stochastic Liouville–von Neumann equation (ESLN) method introduced earlier [G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 95, 125124 (2017); G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 97, 224310 (2018)], which describes the dynamics of an exactly thermalized open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalized at finite temperature using an imaginary-time evolution procedure before the application of real-time evolution. This initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open-system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher-order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under nonequilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are nonzero, initially thermalizing the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.

Absorption of ultrashort laser pulses in a metallic grating deposited on a transparent sample launches in the material both compression/dilatation (longitudinal) and shear coherent acoustic pulses in directions of different orders of acoustic diffraction. The propagation of the emitted acoustic pulses can be monitored by measuring the variation of the optical reflectivity of the time-delayed ultrashort probe laser pulses. The direction of probe light incidence and its polarization relative to the sample surface as well as the orientation of the metallic grating should be specifically chosen for efficient Brillouin scattering of the probe light from shear phonons propagating in the elastically isotropic materials. As theoretically predicted, the obtained experimental data contain multiple frequency components which are due to a variety of possible Brillouin scattering angles for both shear and compression/dilatation coherent acoustic waves. All these different frequency components are explained through multiplexing the propagation directions of probe light and coherent sound by the metallic diffraction grating. Our experimental scheme of time-domain Brillouin scattering with metallic gratings operating in reflection mode provides access to monitoring the shear acoustic waves launched in the direction of the first diffraction order by backward Brillouin scattering process. Applications include simultaneous determination of several different acoustic mode velocities and optical refractive index and, potentially, measurements of the acoustic dispersion of samples with a single direction of possible optical access.