Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32\times 32$ sites, as well as a highly frustrated $J_1-J_2$ model up to $24\times 24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.

The electron-phonon $\left(e\text{−}\mathrm{ph}\right)$ interaction remains of great interest in condensed matter physics and plays a vital role in realizing superconductors, charge density waves (CDW), and polarons. We study the two-dimensional Holstein model for $e\text{−}\mathrm{ph}$ coupling using determinant quantum Monte Carlo across a wide range of its phase diagram as a function of temperature, electron density, dimensionless $e\text{−}\mathrm{ph}$ coupling strength, and the adiabatic ratio of the phonon frequency to the Fermi energy. We describe the behavior of the CDW correlations, the competition between superconducting and CDW orders and polaron formation, the optimal conditions for superconductivity, and the transition from the weak-coupling regime to the strong-coupling regime. Superconductivity is optimized at intermediate $e\text{−}\mathrm{ph}$ coupling strength and intermediate electron density, and the superconducting correlations increase monotonically with phonon frequency. The global maximum for superconductivity in the Holstein model occurs at large phonon frequency, the limit where an attractive Hubbard model effectively describes the physics.

We present a temperature- and doping-dependent resonant inelastic x-ray scattering experiment at the V $L_{2,3}$ and O $K$ edges in ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x{\mathrm{VO}}_3$ thin films with $x=0$ and $x=0.1$. This material is a canonical example of a compound that exhibits a filling-control metal-insulator transition and undergoes orbital ordering and antiferromagnetic transitions at low temperature. Temperature-dependent measurements at the V $L_3$ edge reveal an intra-$t_{2g}$ excitation that blueshifts by 40 meV from room temperature to 30 K at a rate that differs between the para- and antiferromagnetic phases. The line shape can be partially explained by a purely local model using crystal field theory calculations. For the low Sr concentration $x=0.1$, the doping is shown to affect the local electronic structure primarily on the O sites, beyond a simple Mott-Hubbard picture. Furthermore, the presence of phonon overtone features at the O $K$ edge evidences that the low-energy part of the spectrum is dominated by phonon response.

The Majorana zero-energy modes (MZMs) residing at the boundary of topological superconductors have attracted a great deal of interest recently, as they provide a platform to explore fundamental physics such as non-Abelian statistics, as well as fault-tolerant quantum computation. Period doubling of Shapiro steps in a Josephson junction under microwave irradiation has been regarded as strong evidence for the emergence of the MZMs at the junction edges. However, questions remain as to how the Shapiro steps respond to the presence of a 4π-periodic Josephson current. In this study, we investigated the characteristic features of Shapiro steps with respect to the ratio ($\alpha$) of the 4π-periodic current to the topologically trivial 2π-periodic one, as well as the reduced microwave frequency $\left(\mathrm{\Omega }\right)$ and McCumber parameter $\left(\beta \right)$ of the junction. Our analysis reproduced Shapiro steps similar to those observed experimentally for specific parameter sets of $\alpha ,\mathrm{\Omega }\left(\lesssim 0.1\right)$, and $\beta (\gtrsim 1.0)$. Full suppression of the first lobe of the $n=1$ step guarantees the presence of a 4π-periodic Josephson current. In addition, we discuss the range of $\mathrm{\Omega }$ and $\beta$ needed for full suppression of the first lobe of the $n=1$ step, even for small $\alpha (<0.1)$. To observe “period-doubled Shapiro steps,” even with a small $\alpha$, the junction should have a large $I_c{R}_N$ product and sufficiently large junction capacitance.

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional ${\mathbb{Z}}_2$ topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry $G$ and subsystem symmetries of a normal subgroup $N$ along lower-dimensional subregions.

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to “spin $S$,” where the quantum numbers have an absolute value less than or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S=1$. The phase transition for $S=1$ is an infinite-order Gaussian transition with the same critical exponents $\delta$ and $\eta$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S\ge 2$. The essential singularity in the correlation length for $S=1$ is different from that for $S\ge 2$. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated ${\widehat{U}}^{\pm }=exp(\pm i\widehat{\theta })$ operators by the spin ladder operators ${\widehat{S}}^{\pm }$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S=1$, which allows us to extract the $\eta$ exponent with high accuracy.

We propose that the odd-frequency $s$ wave ($s^{\text{odd}}$ wave) superconducting gap function, which is usually unstable in the bulk, naturally emerges at the edge of $d$ wave superconductors. This prediction is based on the surface spin fluctuation pairing mechanism owing to the zero-energy surface Andreev bound state. The interference between bulk and edge gap functions triggers the $d+s^{\text{odd}}$ state, and the generated spin current is a useful signal uncovering the “hidden” odd-frequency gap. In addition, the edge $s^{\text{odd}}$ gap can be determined via the proximity effect on the diffusive normal metal. Furthermore, this study provides a decisive validation of the “Hermite odd-frequency gap function,” which has been an open fundamental challenge to this field.

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1>0$ between first neighbors and $J_2=\alpha J_1$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\Delta }\left(\alpha \right)>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\alpha =0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\alpha$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\alpha ,N)$ and magnetic susceptibility $\chi (T,\alpha ,N)$ of progressively larger systems up to $N=96$ or 152 spins. Convergence to the thermodynamic limit, $S(T,\alpha )$ or $\chi (T,\alpha )$, is demonstrated down to $T/J\sim 0.01$ in the sectors $\alpha <1$ and $\alpha >1$. $S(T,\alpha )$ yields the critical points between gapless phases with $S^{\prime }(0,\alpha )>0$ and gapped phases with $S^{\prime }(0,\alpha )=0$. The $S^{\prime }(T,\alpha )$ maximum at $T^{*}\left(\alpha \right)$ is obtained directly in chains with large $\mathrm{\Delta }\left(\alpha \right)$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\alpha )$ down to $T=0$ indicates power-law deviations $T^{-\gamma \left(\alpha \right)}$ from $exp[-\mathrm{\Delta }(\alpha )/T]$ with exponent $\gamma \left(\alpha \right)$ that increases with $\alpha$. The $\chi (T,\alpha )$ analysis also yields power-law deviations, but with exponent $\eta \left(\alpha \right)$ that decreases with $\alpha$. Spin correlation functions account for $S(T,\alpha )$ differences between frustration $\alpha <1$ within a chain and $\alpha >1$ between HAFs on sublattices. $S(T,\alpha )$ and the spin density $\rho (T,\alpha )=4T\chi (T,\alpha )$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\alpha )/\rho (T,\alpha )$ for $T<0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\mathrm{\Delta }\left(\alpha \right)$.

We consider the time evolution of Bloch electrons after a local excitation, like an interaction with a focused laser pulse that irradiates only a part of the sample. The disturbance caused by the excitation propagates along the sample towards detectors. We focus on the measurable time integral of the usually rapidly oscillating current. In the long time limit this integral is the total charge that is displaced by the excitation. We develop an analytic way for calculating this charge. The results are verified using an analytic example with quadratic dispersion in one dimension. Additionally, numerical calculations are also performed in order to visualize the relevant physical processes in a wide band gap material that is excited by a laser pulse.

We demonstrate that an off-resonant circularly polarized light-modulated double-barrier monolayer ${\mathrm{WSe}}_2$ junction can operate as a valley filter and a valley valve simultaneously. Spin-, valley-, and energy-dependent line-type resonant peaks appear in the transmission spectrum, and the splitting of the valley degree of freedom due to the coupling between the light-induced gap and the valley index is also present. By fixing the light on the first barrier, the valley switch can be controlled efficiently by the optical field on the second one. Furthermore, the gate voltage can adjust not only the conductance but also the operation window of the valley filter. The proposed junction provides insight into the manipulation of valley degree of freedom based on ${\mathrm{WSe}}_2$ and other transition metal dichalcogenides.

We conducted atomic-scale scanning tunneling microscopy of a graphene nanosheet on graphite. In addition to a rhombus lattice representing the $\left(\sqrt{3}\times \sqrt{3}\right)R{30}^{\circ }$ superstructure, we resolved another quadrangle lattice similar to a rectangle in the sheet. Its lattice size was approximately $0.37\times 0.22\mathrm{n}{\mathrm{m}}^2$. To clarify the origin of this unique rectangular lattice, the overlap of the $\sqrt{3}\times \sqrt{3}$ superstructures along the direction of their long diagonals was theoretically examined using a simple model. The electron distribution with high energy in the occupied states of armchair-edged graphene nanoribbons (AGNRs) was calculated based on first principles. A rectangular lattice, resembling the one observed experimentally, was found to form on the AGNR under a specific width condition. This finding was also analyzed in terms of Clar's theory and the scattering of electron waves. We propose that wrinkles and adsorbates in graphene play a role similar to an armchair edge, resulting in the $\sqrt{3}\times \sqrt{3}$ phase. If these local defects are in close proximity, the rhombus phases interact to generate electronic structures predicted for AGNRs. This is probably the reason why a rectangular lattice was imaged on the graphene sheet that is not an ideal AGNR.

The dynamics of micrometer-sized magnetic domains in ultrathin ferromagnetic films is so dramatically slowed down by quenched disorder that the spontaneous elastic tension collapse becomes unobservable at ambient temperature. By magneto-optical imaging we show that a weak zero-bias ac magnetic field can assist such curvature-driven collapse, making the area of a bubble to reduce at a measurable rate, in spite of the negligible effect that the same curvature has on the average creep motion driven by a comparable dc field. An analytical model explains this phenomenon quantitatively.

Using first-principles calculations and low-energy effective model analysis, we propose that silver selenide-based ${\mathrm{RbAg}}_5{\mathrm{Se}}_3$ compound is an ideal type-II Dirac semimetal. There is a pair of Dirac points appearing at about 25.68 meV below the Fermi energy. The Dirac points existing on the $k_z$ axis are protected by $C_{4z}$ rotation symmetry in addition to $\mathcal{PT}$ symmetry. Meanwhile, the system is also characterized by a nontrivial ${\mathbb{Z}}_2$ topological invariant. The long clean Fermi arcs exist on the (010) surface, connecting the projections of the Dirac points. In addition, the Landau levels of the type-II Dirac points under magnetic field are also calculated. Our discovery not only adds one promising candidate into the type-II Dirac semimetals family but also provides an ideal platform for the study of topological phase transitions.

Classical nucleation theory describes the formation of the first solids from supercooled liquids and predicts an average waiting time for a system to freeze as it is cooled below the melting temperature. For systems at low to moderate undercooling, waiting times are too long for freezing to be observed via simulation. Here a system can be described by estimated thermodynamic properties, or by extrapolation from practical conditions where thermodynamic properties can be fit directly to simulations. In the case of crystallizing Earth's solid iron inner core, these thermodynamic parameters are not well known and waiting times from simulations must be extrapolated over approximately 60 orders of magnitude. In this work, we develop a new approach negating the need for freezing to be observed. We collect statistics on solidlike particles in molecular dynamic simulations of supercooled liquids at 320 GPa. This allows estimation of waiting times at temperatures closer to the melting point than is accessible to other techniques and without prior thermodynamic insight or assumption. Our method describes the behavior of nucleation at otherwise inaccessible conditions such that the nucleation of any system at small undercooling can be characterized alongside the thermodynamic quantities which define the first formed solids.

By developing models of increasing complexity, we show that a model without single-ion anisotropy (SIA) cannot explain the magnetic properties of $J_{\mathrm{eff}}=1/2 {\mathrm{Tb}}^{3+}$ moments in the orthorhombically distorted, honeycomb material ${\mathrm{Tb}}_2{\mathrm{Ir}}_3{\mathrm{Ga}}_9$. In four different models for the magnetization of a single honeycomb layer, the only sources of anisotropy are symmetric exchange interactions $J_{n\alpha \beta }=J_{n\beta \alpha }$ along three different bonds $n$, an anisotropic $\underline{g}$ tensor, and a Dzyalloshinskii-Moriya interaction (asymmetric exchange) that produces the observed canted moment along $\mathbf{b}$. With 21 parameters, the best such model yields ${\chi }^2=0.065$, which is substantially smaller than ${\chi }^2=0.112$ obtained using a Heisenberg model containing six parameters including easy-axis anisotropy. However, models without SIA fail to reproduce the linear dependence of the magnetization with a field perpendicular to the Ising axis while predicting a saturation magnetization that is far too low. Due to the complex crystal-field environments, we argue that SIA is necessary to study low-symmetry, three-dimensional $J_{\mathrm{eff}}=1/2$ materials containing ${\mathrm{Tb}}^{3+}$ ions.

The high tunneling electroresistance (TER) effect, generally caused by ferroelectric (FE)-modulated barrier height or width, is essential for the applications of multiferroic tunnel junctions in data storage. It is traditionally obtained by distinct electrical screening lengths of electrodes. Interface engineering can enhance the TER effect further. In this work, taking $\mathrm{Co}\text{−}{\left({\mathrm{TiO}}_2\text{−}\mathrm{BaO}\right)}_N\text{−}\mathrm{Co}$ tunnel junctions as examples, we demonstrate a distinct principle than the screening lengths for designing extraordinary TER effect. We reveal that when the interfacial FE displacement is much larger than that of the FE bulk, it will bend the barrier band near the interface violently, and the interfacial polarization direction pointing to or away from the interface determines whether the energy band rises or falls. The large interfacial Ba-O displacement and its corresponding polarization direction in $\mathrm{Co}\text{−}{\mathrm{BaTiO}}_3\text{−}\mathrm{Co}$ tunnel junctions can be significantly modulated by the direction of FE polarization, resulting in a metallic-insulating transition of the entire thin ${\mathrm{BaTiO}}_3$ barrier. For thick ${\mathrm{BaTiO}}_3$ barrier ($N=25$, $\sim 10$ nm), the effective tunnel barrier width shifts between about 2 nm and 6.5 nm as the polarization of ${\mathrm{BaTiO}}_3$ switches direction, which can dramatically modulate the tunneling efficiency. This effect shed light on a novel route for enhancing TER through the interface engineering.

By employing quasielastic neutron scattering, we studied the atomic-scale relaxation dynamics and transport mechanism of ${\mathrm{La}}_{50}{\mathrm{Ni}}_{15}{\mathrm{Al}}_{35}$ and ${\mathrm{Ce}}_{70}{\mathrm{Cu}}_{19}{\mathrm{Al}}_{11}$ metallic glass melts in the temperature range of $>200\mathrm{K}$ above their liquidus temperatures. The results show that both liquids exhibit stretched exponential relaxation and Arrhenius-type temperature dependence of the effective diffusion coefficient. The ${\mathrm{La}}_{50}{\mathrm{Ni}}_{15}{\mathrm{Al}}_{35}$ melt exhibits an activation energy of 0.545 ± 0.008 eV and a stretching exponent ∼0.77 to 0.86 in the studied temperature range; no change of activation energy, as suggested in previous reports, associated with liquid-liquid phase transition was observed. In contrast, the ${\mathrm{Ce}}_{70}{\mathrm{Cu}}_{19}{\mathrm{Al}}_{11}$ melt exhibits larger diffusivity with a much smaller activation energy of $0.201\pm 0.003\mathrm{eV}$ and a smaller stretching exponent ∼0.51 to 0.60, suggestive of more heterogeneous dynamics.

The pressure-induced spin crossover of iron in ferropericlase, the Fe-bearing MgO, is a key to understanding the seismological observations in the lower mantle. However, the experimentally measured spin-crossover pressure zone (SCPZ) shows large variations, which disagree with the theoretically predicted values that are generally less than half of experimental ones. Here, we resolve this outstanding controversy by revealing the critical role of Fe distribution in MgO in broadening the Fe SCPZ, using comprehensive first-principles calculations combined with cluster expansion approach and Monte Carlo simulations. By employing a large supercell containing up to $\sim {10}^6$ atoms, we derive the spin-crossover pressures as functions of Fe concentration for different Fe distributions. We determine a critical temperature of $T_c\sim 900\mathrm{K}$, below which Fe segregation (clustering) occurs, in accordance with the thermodynamic phase diagram. Above $T_c$, an entropy-driven randomized Fe distribution creates large variations in Fe local environments, which in turn broadens the SCPZ, such as 17.5 GPa for 25 mol % Fe-bearing MgO in good agreement with experiments. Therefore, the broad SCPZ, rendering a smooth change of ferropericlase mechanical properties during the spin crossover, should be mainly caused by entropy, consistent with the high-temperature state of the lower mantle.