We systematically analyze the representability of toric code ground states by a restricted Boltzmann machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the restricted Boltzmann machine to accommodate arbitrary ground states by introducing essential nonlocal connections efficiently. This model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from $Z_2$ to $Z_n$ toric code and discuss future directions.

We use several techniques to probe the wave functions proposed to describe the $\mathcal{A}$ phases by Das, Das, and Mandal [Phys. Rev. Lett. 131, 056202 (2023); Phys. Rev. Lett. 132, 106501 (2024); Phys. Rev. B 110, L121303 (2024)]. As opposed to representing fractional quantum Hall liquids, we find these wave functions to describe states that clearly display strong phase separation. In the process of exploring these wave functions, we have also constructed several methods for diagnosing phase separation and generating such wave functions numerically. Finally, we uncover a property of entanglement spectra that can be used as a check for the accuracy of numerics.

Atomic-scale spatial resolution was achieved in the mapping and spectroscopy of polygonal grain boundaries (GBs) on atomically flat highly oriented pyrolytic graphite using scanning tunneling microscopy (STM). These GBs are long-range ordered one-dimensional periodic structures comprising pentagon-heptagon pairs. A comprehensive study combining local electronic, valleytronic, mechanical, and topological properties is conducted on this GB which reveals the interplay between these properties. On the atoms and bonds of individual GB polygons, spatially localized conductance states were probed, in addition to edge states. The spatial extent of these states was observed on GB by bias-dependent imaging. The electron scattering angle at the GB edges was also modulated with a sample bias which exhibited valley flipping and unique quantum interference effects such as backscattering and intervalley scattering. In situ strain-induced mechanical and electronic modifications were observed through surface deformation caused by the STM tip, revealing flattened electronic energy band dispersion and shifts in carrier doping.

We study exciton energy spectra and their propagation in moiré superlattices formed in transition-metal dichalcogenide heterobilayers. In such structures, as a result of weak interlayer interaction, an effective, moiré, potential acting on excitons arises. Usually, excitons are considered to be localized in such a potential. Here we demonstrate that the coupling of optically active excitons with an induced electromagnetic field produces energy dispersion that is linear in the wave vector even if the quantum-mechanical tunneling between the localization sites is suppressed. The effect can be described as a result of the processes of virtual generation-recombination of excitons at the localization sites that results in the $r^{-3}$ dependence of the transfer-matrix element on the intersite distance $r$. Based on the calculated energy spectrum, we study exciton propagation in moiré superlattices with allowance for the light-exciton interaction. We consider semiclassical diffusion of excitons, and we take into account exciton-phonon and exciton-static defect scattering. For these mechanisms, the diffusion coefficient decreases with an increase of the temperature. We also analyze the hopping propagation regime, and we demonstrate that the temperature dependence of the exciton diffusion coefficient is described by the power law rather than by an exponential function of the temperature.

We study the formation and the subsequent dynamics of shock waves in repulsive one-dimensional Bose gases during the free expansion of a density hump. By building coherent Fermi states for interacting Bethe fermions, we define a quantum fluctuating initial state expressed in terms of universal quantities, namely the density and the Luttinger parameter. In the integrable case, this fluctuating state is then evolved by generalized hydrodynamics (GHD) and, differently from nonfluctuating initial states, it develops density ripples on top of the hydrodynamic mean value. Our analysis gives a general theory of quantum ripples and wave breaking in integrable and quasi-integrable one-dimensional liquids and clarifies the role of the interaction strength. In particular, for strongly/intermediately interacting bosons, we find quantum ripples originating from low-energy modes at the Fermi surface interfering when transported by GHD. In the low-coupling limit, near the quasicondensate regime, we find instead that density ripples have a semiclassical nature, and their description requires information on the curvature of the Fermi surface.

High harmonic generation (HHG) in solids is sensitively dependent on the band dispersions of crystals, which involves intraband and interband processes in conventional semiconducting materials. Here, we numerically simulate the HHG from a two-dimensional (2D) $\mathrm{NbOC}{\mathrm{l}}_2$ material of a flat valence band in strong midinfrared laser fields, which shows special harmonic signals that are predominantly originated from interband polarization. The suppressed intraband HHG is mainly attributed to its intrinsic heavy carrier with large effective mass. By bending the flat band under strain engineering, the HHG spectra of strained 2D $\mathrm{NbOC}{\mathrm{l}}_2$ demonstrate typical features with below-gap harmonics and high-energy harmonics dominated by intraband and interband processes, respectively. In addition, 2D $\mathrm{NbOC}{\mathrm{l}}_2$ exhibits low exfoliation energy, excellent dynamical and thermal stability, and desirable optical property, which endow it with more experimental feasibilities.

We study the optical effect of chiral topological superconductors in two dimensions. The linear optical response from chiral Bogoliubov edge modes in clean superconductors has in-gap resonances, which originated from the transitions within the edge particle-hole pair bands. Interestingly, the number of resonance peaks is determined by the Bogoliubov–de Gennes Chern number of topological superconductors. Such a sharply distinctive feature in optical absorption offers a simple way to distinguish topological superconductors with different Chern numbers. We further demonstrate that linear optical effect could probe different topological phases in quantum anomalous Hall insulator-superconductor junction devices. This finding provides an applicable method to detect chiral Bogoliubov edge states and is distinguishable from collective modes in superconductors and possible trivial explanations.

Charge density wave (CDW) and their interplay with correlated and topological quantum states are at the forefront of condensed matter research. The $4H_b\text{−}{\mathrm{TaS}}_2$ is a CDW ordered quantum heterostructure that is formed by alternative stacking of Mott insulating 1T-${\mathrm{TaS}}_2$ and Ising superconducting 1H-${\mathrm{TaS}}_2$. While the $\sqrt{13}\times \sqrt{13}$ and $3\times 3$ CDWs have been respectively observed in the bulk 1T-${\mathrm{TaS}}_2$ and 2H-${\mathrm{TaS}}_2$, the CDWs and their pivotal role for unconventional superconductivity in the $4H_b\text{−}{\mathrm{TaS}}_2$ remain unsolved. In this Letter, we reveal the two-dimensional (2D) $\sqrt{13}\times \sqrt{13}$ chiral CDW in the 1T layers and intraunit cell coupled 2D $2\times 2$ CDW in the 1H and 1H' layers of $4H_b\text{−}{\mathrm{TaS}}_2$. Our results establish $4H_b\text{−}{\mathrm{TaS}}_2$ a 2.5D quantum heterostructure, where 2D quantum states emerge from 3D crystalline structure.

${\mathrm{LiV}}_2{\mathrm{O}}_4$ is a member of the so-called $3d$ heavy fermion (HF) compounds, with effective electron mass exceeding 60 times the free electron mass, comparable with $4f$ HF compounds. The origin of the strong electron correlation in combination with its metallic character have been a subject of intense theoretical and experimental discussion, with Kondo-like and Mott physics being suggested as its physical origin. Here, we report state-of-the art ab initio density functional theory $+$ dynamical mean-field theory calculations for ${\mathrm{LiV}}_2{\mathrm{O}}_4$ for the full three-orbital V $t_{2g}$ manifold and present temperature-dependent spectral properties. We map out the phase diagram for a representative three-orbital model system as a function of doping and interaction strength, which indicates that ${\mathrm{LiV}}_2{\mathrm{O}}_4$ is located between two orbital-selective Mott phases, giving rise to a robust strongly correlated Hund's metal behavior. At low temperature, we find the emergence of a strongly renormalized sharp quasiparticle peak a few meV above the Fermi level of V $a_{1g}$ character, in agreement with experimental reports.

We revised normal-state and superconducting properties of an alloy superconductor $\mathrm{N}{\mathrm{b}}_4\mathrm{R}{\mathrm{h}}_2{\mathrm{C}}_{1-\delta }$ through electrical transport, magnetic susceptibility, and specific heat. Our results reveal that it is a Pauli paramagnetic bad metal, a strongly coupled superconductor with specific heat jump $\mathrm{\Delta }{C}_{\mathrm{e}}/{\gamma }_n{T}_{\mathrm{c}}\sim 1.65$ and double $s$-wave gap symmetry. The evolution of superconducting transition temperature $T_{\mathrm{c}}$ was investigated by high-pressure electrical transport. It was found that $T_{\mathrm{c}}$ exhibits a nonmonotonic pressure dependence with a maximum $T_{\mathrm{c}}^{\max }\sim 11$ K, which is a record for the η-carbide alloy superconductors. Zero-temperature upper critical field $B_{\mathrm{c}2}\left(0\right)$ reduces smoothly from beyond to below the Pauli limit in 40–50 GPa. In situ synchrotron x-ray diffraction reveals a relatively large bulk modulus $B_0$ ∼ 278.6 GPa in comparison with other alloy superconductors, which can explain the small-pressure coefficient $\mathrm{d}{T}_{\mathrm{c}}\text{/d}P$ ∼ −0.045 K/GPa above 20 GPa. The observations of high $T_{\mathrm{c}}$ and large $B_0$ make $\mathrm{N}{\mathrm{b}}_4\mathrm{R}{\mathrm{h}}_2{\mathrm{C}}_{1-\delta }$ a promising superhard alloy superconductor. The pressure evolution of $B_{\mathrm{c}2}\left(0\right)$ is discussed in terms of Fermi surface reconstructions and the ratio of superconducting coherence length and electron mean free path of polycrystalline superconductors.

According to the theory of group representations, the types of band degeneracy can be divided into accidental degeneracy and essential degeneracy. The essentially degenerate nodal lines (NLs) are typically resided on the high-symmetry lines of the Brillouin zone. Here, we propose a type of NL in two dimensions that is essentially degenerate but is hidden within the high-symmetry planes, making it less observable, dubbed a hidden-essential nodal line (HENL). The existence of HENL is guaranteed as long as the system hosts a horizontal glide-mirror symmetry, hence such NLs can be widely found in both nonmagnetic and magnetic systems. We perform an exhaustive search over all 528 magnetic layer groups (MLGs) for HENL that can be enforced by glide-mirror symmetry with both spinless and spinfull systems. We find that 122 candidate MLGs host spinless HENL, while 63 candidate MLGs demonstrate spinful HENL. In addition, we reveal that horizontal mirror and time-reversal symmetry in type-II and type-IV MLGs with spin-orbital coupling can enforce HENL formed. The 15 corresponding candidate MLGs have also been presented. Furthermore, we derive a few typical lattice models to characterize the existence for the HENL. For specific electronic fillings in real materials, namely $4N+2$ in spinless systems (and $2N+1$ in spinful systems), the presence of the HENLs in candidate MLGs is required regardless of the details of the systems. Using ab initio calculations, we further identify possible material candidates that realize spinless and spinful HENL. Moreover, spinful HENLs exhibit a persistent spin texture with the characteristic of momentum-independent spin configuration. Our findings uncover a type of topological semimetal state and offer an ideal platform to study the related physics of HENLs.

We have previously shown that the Stoner instability two-dimensional has unconventional behavior: it is strongly first order but features a susceptibility which diverges at the transition point. Here, we analyze the Stoner transition for two-dimensional systems with spin and valley degrees of freedom, beyond mean field. At low density, we show that in one-valley and isotropic two-valley systems the leading effect beyond mean-field theory is suppression of the Stoner instability. In anisotropic two-valley systems, we show that, for larger anisotropy, the transition remains mean-field-like and retains its unconventional properties. We discuss applications to AlAs quantum wells.

We conduct a comprehensive ab initio investigation of electron-electron interactions within the pyrochlore structures of $R_2{\mathrm{Ru}}_2{\mathrm{O}}_7, R_2{\mathrm{Ir}}_2{\mathrm{O}}_7, {\mathrm{Ca}}_2{\mathrm{Ru}}_2{\mathrm{O}}_7$, and ${\mathrm{Cd}}_2{\mathrm{Ru}}_2{\mathrm{O}}_7$, where $R$ denotes a rare-earth element. Utilizing a multiorbital Hubbard model, we systematically explore the effects of various rare-earth elements and applied high pressure on the correlation strength in these compounds. Our calculations on the Coulomb interaction parameter $U$ and the bandwidth $W$ reveal that the chemical pressure for $R_2{\mathrm{Ru}}_2{\mathrm{O}}_7$ and $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ leads to an unusual increase in $U/W$ ratio, hence, an increase in correlation strength. Contrary to conventional understanding of bandwidth control, our study identifies that the Hubbard $U$ is more influential than the bandwidth $W$ behind the metal-insulator landscape of $R_2{\mathrm{Ru}}_2{\mathrm{O}}_7$ and $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, leading to an interaction-controlled metal-insulator transition. We also find unexpected behavior in physical pressure. Whereas physical pressure leads to a decrease in the correlation strength $U/W$ as usual in $R_2{\mathrm{Ru}}_2{\mathrm{O}}_7$, the effect is notably small in ${\mathrm{Ca}}_2{\mathrm{Ru}}_2{\mathrm{O}}_7$ and ${\mathrm{Cd}}_2{\mathrm{Ru}}_2{\mathrm{O}}_7$, which provides an important clue to understanding the unusual pressure-induced metal-insulator transition observed experimentally.

Polar kinks, as a kind of polarization soliton, play crucial roles in polarization switching, domain wall motion, and other dynamics in ferroelectric materials. However, the static and dynamic characteristics of polar kinks are poorly understood so far, hindering the potential device design based on such polar structure. Here, via dynamical phase-field simulations, we present a comprehensive study on the static and dynamic characteristics of polar kinks at the 180° domain walls of $\mathrm{BaTi}{\mathrm{O}}_3$ films under different electrical and mechanical conditions. Intriguing phenomena of polar kinks including generation and annihilation, solitonlike motion, kink-(anti)kink interaction, as well as topological transformation among kink, antikink, and flux closure are revealed. An extended generalized configurational force framework is proposed to obtain the driving force of the polar kinks. Under this framework, the equation of motion of polar kinks is derived by integrating the field configurations and can well capture the quasi-one-dimensional motion of polar kinks. Moreover, the concept of toroidal driving moment is adopted to understand the strain-induced topological transformation related to polar kinks. These results demonstrate the prominent and controllable soliton characteristics of polar kinks and provide useful insights for potential applications in logic devices.

Recently, ion exchange (IE) has been used to periodically modify the coercive field ($E_{\mathrm{c}}$) of the crystal prior to periodic poling, to fabricate fine-pitch domain structures in Rb-doped $\mathrm{KTiOP}{\mathrm{O}}_4$ (RKTP). Here, we use micro-Raman spectroscopy to understand the impact of IE on the vibrational modes related to the Rb/K lattice sites, $\mathrm{Ti}{\mathrm{O}}_6$ octahedra, and $\mathrm{P}{\mathrm{O}}_4$ tetrahedra, which all form the basis of the RKTP crystal structure. We analyze the Raman spectra of three different RKTP samples: (1) a RKTP sample that shows a poled domain grating only, (2) a RKTP sample that has an $E_{\mathrm{c}}$ grating only, and (3) a RKTP sample that has both an $E_{\mathrm{c}}$ and a domain grating of the nominally same spacing. This allows us to determine the impact of IE on the vibrational modes of RKTP. We characterize the changes in the lower Raman peaks related to the alkali-metal ions, as well as observe lattice modifications induced by the incorporation of $\mathrm{R}{\mathrm{b}}^{+}$ that extend further into the crystal bulk than the expected IE depth. Moreover, the influence of IE on the domain walls is also manifested in their Raman peak shift. We discuss our results in terms of the deformation of the $\mathrm{P}{\mathrm{O}}_4$ and $\mathrm{Ti}{\mathrm{O}}_6$ groups. Our results highlight the intricate impact of IE on the crystal structure and how it facilitates periodic poling, paving the way for further development of the $E_{\mathrm{c}}$-engineering technique.

The burgeoning field of rare-earth (RE)-based van der Waals (vdW) layered magnets is driven by their potential in novel magnetism and spintronics applications. These materials exhibit exceptional spin-lattice coupling and air stability, attributed to the strong spin-orbit coupling and substantial magnetic moments from the $4f$ electrons. This study investigates the structural and magnetic properties of dysprosium-based oxyhalide ternary compounds, $\mathrm{DyO}X$ $(X=\mathrm{F},\mathrm{Br})$, which are the prime candidates for multiphase transitions and tunable magnetic anisotropy through halogen substitution at the vdW layer edges. Utilizing complementary neutron diffraction, first-principles calculations, and magnetic measurements, we elucidate the intrinsic properties of these oxyfluoride compounds. DyOF exhibits an antiferromagnetic (AFM) ground state below the Néel temperature ($T_{\mathrm{N}}$) of 3.7 K, with a commensurate propagation vector k = (0, 0, 3/2) and magnetic moments aligned within the $ab$ plane. In contrast, DyOBr has a higher $T_{\mathrm{N}}=9.9$ K and displays more complex magnetic transitions. The large magnetic moment ($\sim 10 {\mu }_{\mathrm{B}}/\mathrm{Dy}$), strong magnetic anisotropy, and field-induced metamagnetic transitions observed in $\mathrm{DyO}X$ $(X=\mathrm{F},\mathrm{Br})$ magnetic insulators suggest their potential for realizing magnetism in the two-dimensional (2D) limit. The magnetocaloric effects are further investigated to manifest the properties of $\mathrm{DyO}X$ $(X=\mathrm{F},\mathrm{Br})$. Notably, DyOF demonstrates a superior magnetic refrigeration capacity, with a maximum magnetic entropy change $-\mathrm{\Delta }{S}_{\mathrm{M}}$ of $25.3 \mathrm{J}{\mathrm{K}}^{-1}\mathrm{k}{\mathrm{g}}^{-1}$ and a relative cooling power of 530.2 $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ at 7 T, surpassing most vdW layered magnets. The findings contribute to the understanding of $4f$ electron-related properties in 2D magnetism and pave the way for emerging applications in antiferromagnetic spintronics and cryomagnetic refrigeration.

The influence of pressure on the multiferroicity in the perovskite metal-organic framework (MOF) $\left[{\left({\mathrm{CH}}_3\right)}_2{\mathrm{NH}}_2\right]\mathrm{Mn}{\left(\mathrm{HCOO}\right)}_3$ has been investigated by dielectric, pyroelectric, and magnetic measurements in a piston-cylinder cell. Under ambient pressure, the ordering of hydrogen bonds takes place at $T_{\mathrm{C}}=192\mathrm{K}$ and induces a first-order ferroelectric (FE) phase transition. With increasing pressure to P = 3.92 kbar, the phase transition shifts to lower temperatures and retains its first-order FE nature. However, when the pressure increases to 5.86 kbar, the phase transition splits into two branches: an antiferroelectric transition at a high temperature $T_2\sim 170\mathrm{K}$ and a FE transition at a low temperature $T_1\sim 163\mathrm{K}$, as evidenced by the dielectric permittivity and pyroelectric current measurements. With further increasing pressure, $T_2$ remains ∼170 K, whereas $T_1$ decreases greatly. These results imply that the compression of the perovskite framework yields two types of hydrogen bond ordering. The canted antiferromagnetic transition temperature increases from 8.4 K under ambient pressure to 9 K under 10.85 kbar, due to the enhanced exchange interaction between metal ions by compressing the MOF. In this paper, we suggest that the electrical properties of perovskite MOFs are more sensitive to external pressure than their magnetic properties. The evolution of multiple phase transitions with pressure is discussed in terms of the mixed crystalline phases under intermediate pressures.

Magnetic dopants in ferroelectric oxide host materials provide a platform for electric field control of isolated spins, facilitated by tuning of the magnetocrystalline anisotropy energy (MCAE). We present first-principles calculations of the MCAE experienced by isolated ${\mathrm{Fe}}^{3+}$ dopants in the tetragonal, orthorhombic, and rhombohedral phases of the prototypical ferroelectric ${\mathrm{BaTiO}}_3$. We identify an order-of-magnitude decrease in the MCAE in the rhombohedral phase relative to the tetragonal and orthorhombic phases. We explain this dramatic decrease, as well as the formation of a spin-easy plane in the tetragonal phase and spin-easy axes in the orthorhombic and rhombohedral phases, using crystal field theory arguments. Building a superposition model from crystal field theory, we show how a set of simple criteria based on crystalline environment can be used to estimate the MCAE. We suggest this as a route to rapidly screen candidate ferroelectric hosts and magnetic dopants that possess phases with spin-easy axes and maximal MCAE tunability.

Antiferromagnets are considered to be promising materials for future spintronic devices; the simulation and theoretical analysis are important to improve our understanding of the dynamics of antiferromagnetic materials, as relevant experiments are less convenient compared with ferromagnets. MnN is a collinear antiferromagnet noted for its large anisotropy, high Néel temperature, and potential for electrical switching in antiferromagnetic spintronics. In this work, the antiferromagnetic resonance of collinear MnN has been theoretically calculated using the macrospin model. The obtained results are then compared with our simulation performed at the natural atomic order of antiferromagnets, using parameters taken from reported first-principles calculations and experiments. The frequency-field dependence and chirality characteristics of all modes in antiferromagnetic resonance in the case of different external fields applied parallel or perpendicular to the easy axis are investigated in detail. The analytical results derived from the macrospin model show good agreement with the simulated data. In addition, the static magnetic properties of MnN obtained from atomistic simulation closely match the reported experimental or our analytical results. This work helps to understand the dynamics of collinear antiferromagnetic materials.