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Topological quantum computing holding global anti-interference ability is realized by braiding some anyons, such as well-known Fibonacci anyon. Here, based on F-matrices and R-symbols and fusion rules of metaplectic anyon model, we obtained two anyon models V112_2 and V133_1. We study systematically the compilation of the metaplectic anyon model through elementary braiding matrices obtained analytically. For one-qubit case, the classical H- and T-gate can be well constructed using the genetic algorithm enhanced Solovay-Kitaev algorithm (GA-enhanced SKA) by V112_2/V133_1, and the obtained accuracy of the H/T-gate is slightly inferior to the corresponding gates of Fibonacci anyon model, but it also can meet the requirements of fault-tolerant quantum computing. For two-qubit case, we use the exhaustive method for short length and the GA for long length to obtain braidword for both V112_2 and V133_1 models. The resulting matrix for the CNOT-gate can approximate the local equivalence class of the CNOT-gate well and demonstrates a much smaller error than the Fibonacci model especially for V133_1.

Quantum compiling trying to approximate the target qubit gate by finding an optimal sequence (braid word) of basic braid operations is a fundamental problem in quantum computing. We develop a genetic algorithm (GA) enhanced Solovay-Kitaev algorithm (SKA) to approximate single qubit gates with four basic braid matrices of Fibonacci anyons. The GA-enhanced SKA demonstrates that the algorithm performs strongly and can easily find the ideal braid word from an exponentially large space. The resulting precision of the approximate single-qubit quantum gate is superior to that of the Monte Carlo (MC) enhanced SKA, as well as comparable to that of the deep reinforcement learning (RL) for the length of braid word greater than 25. The 2(3)-order approximation of GA-enhanced SKA for basic braiding length l0=50(30) leads to an optimal braid word at a distance of 5.9*10-7, which is sufficient for most cases of quantum computing. Our work provides an alternative approach to solving and optimizing quantum compilation of non-Abelian anyon quantum gates and is useful for realizing topological quantum computation in the future.

Massive Dirac fermions, which are essential for realizing novel topological phenomena, are expected to be generated from massless Dirac fermions by breaking the related symmetry, such as time-reversal symmetry (TRS) in topological insulators or crystal symmetry in topological crystalline insulators. Here, we report scanning tunneling microscopy and angle-resolved photoemission spectroscopy studies of {\alpha}-Bi4I4, which reveals the realization of massive Dirac fermions in the (100) surface states without breaking the TRS. Combined with first-principle calculations, our experimental results indicate that the spontaneous symmetry breaking engenders two nondegenerate edges states at the opposite sides of monolayer Bi4I4 after the structural phase transition, imparting mass to the Dirac fermions after taking the interlayer coupling into account. Our results not only demonstrate the formation of the massive Dirac fermions by spontaneous symmetry breaking, but also imply the potential for the engineering of Dirac fermions for device applications.

Flat-band in twisted graphene bilayer has garnered widespread attention, and whether flat-bands can be realized in carbon monolayer is an interesting topic worth exploring in condensed matter physics. In this work, we demonstrate that, based on the theory of compact localized states, a series of two-dimensional carbon allotropes with flat-bands can be achieved. Two of them named as 191-8-66-C-r567x-1 and 191-10-90-C-r567x-1 are confirmed to be dynamically stable carbon phases with isolated or weakly overlapped flat-bands at the Fermi-level. The maximum Fermi velocities of the flat-band electrons are evaluated to be 1x10^4 m/s and 0.786x10^4 m/s, both of which are lower than the Fermi velocity of the flat-band electrons in magic-angle graphene (4x10^4 m/s). Furthermore, 191-8-66-C-r567x-1 has been confirmed to be a flat-band related magnetic half-metal with a magnetic moment of 1.854 miuB per cell, while 191-10-90-C-r567x-1 is a flat-band related magnetic normal metal with a magnetic moment of 1.663 miuB per cell. These results not only show that flat-bands can be constructed in carbon monolayer, but also indicate the potential for achieving metal-free magnetic materials with light elements based on flat-band theory., Comment: 10 pages, 3 figures

Recent advancements underscore the critical need to develop ferroelectric materials compatible with silicon. We systematically explore possible ferroelectric silicon quantum films and discover a low-energy variant (hex-OR-2*2-P) with energy just 1 meV/atom above the ground state (hex-OR-2*2). Both hex-OR-2*2 and hex-OR-2*2-P are confirmed to be dynamically and mechanically stable semiconductors with indirect gaps of 1.323 eV and 1.311 eV, respectively. The ferroelectric hex-OR-2*2-P exhibits remarkable in-plane spontaneous polarization up to 120 Pc/m and is protected by a potential barrier (13.33 meV/atom) from spontaneously transitioning to hex-OR-22. To simulate the switching ferroelectricity in electric fields of the single-element silicon bilayer, we develop a method that simultaneously learns interatomic potentials and Born effective charges (BEC) in a single equivariant model with a physically informed loss. Our method demonstrates good performance on several ferroelectrics. Simulations of hex-OR-2*2-P silicon suggest a depolarization temperature of approximately 300 K and a coercive field of about 0.05 V/{\AA}. These results indicate that silicon-based ferroelectric devices are feasible, and the ground state phase of the silicon bilayer (hex-OR-2*2) is an ideal system. Our findings highlight the promise of pure silicon ferroelectric materials for future experimental synthesis and applications in memory devices, sensors, and energy converters., Comment: 18 pages, 3 figures

Flat bands and Dirac cones in materials are at the source of the exotic electronic and topological properties. The Lieb lattice is expected to host these electronic structures, arising from quantum destructive interference. Nevertheless, the experimental realization of a two-dimensional Lieb lattice remained challenging to date due to its intrinsic structural instability. After computationally designing a Platinum-Phosphorus (Pt-P) Lieb lattice, we have successfully overcome its structural instability and synthesized it on a gold substrate via molecular beam epitaxy. Low-temperature scanning tunneling microscopy and spectroscopy verified the Lieb lattice's morphology and electronic flat bands. Furthermore, topological Dirac edge states stemming from pronounced spin-orbit coupling induced by heavy Pt atoms have been predicted. These findings convincingly open perspectives for creating metal-inorganic framework-based atomic lattices, offering prospects for strongly correlated phases interplayed with topology., Comment: 24 pages,11 figures

NbSe2, which simultaneously exhibits superconductivity and spin-orbit coupling, is anticipated to pave the way for topological superconductivity and unconventional electron pairing. In this paper, we systematically study topological superconducting (TSC) phases in monolayer NbSe2 through mixing on-site s-wave pairing (ps) with nearest-neighbor pairing (psA1) based on a tight-binding model. We observe rich phases with both fixed and sensitive Chern numbers (CNs) depending on the chemical potential ({\mu}) and out-of-plane magnetic field (Vz). As the psA1 increases, the TSC phase manifests matching and mismatching features according to whether there is a bulk-boundary correspondence (BBC). Strikingly, the introduction of mixed wave pairing significantly reduces the critical Vz to form TSC phases compared with the pure s-wave paring. Moreover, the TSC phase can be modulated even at Vz=0 under appropriate {\mu} and psA1, which is identified by the robust topological edge states (TESs) of ribbons. Additionally, the mixed pairing influences the hybridization of bulk and edge states, resulting in a matching/mismatching BBC with localized/oscillating TESs on the ribbon. Our finding is helpful for the realization of TSC states in experiment, as well as designing and regulating TSC materials., Comment: 12 pages. 5 figures

Uncorrelated disorder in generalized 3D Lieb models gives rise to the existence of bounded mobility edges, destroys the macroscopic degeneracy of the flat bands and breaks their compactly-localized states. We now introduce a mix of order and disorder such that this degeneracy remains and the compactly-localized states are preserved. We obtain the energy-disorder phase diagrams and identify mobility edges. Intriguingly, for large disorder the survival of the compactly-localized states induces the existence of delocalized eigenstates close to the original flat band energies -- yielding seemingly divergent mobility edges. For small disorder, however, a change from extended to localized behavior can be found upon decreasing disorder -- leading to an unconventional ``inverse Anderson" behavior. We show that transfer matrix methods, computing the localization lengths, as well as sparse-matrix diagonalization, using spectral gap-ratio energy-level statistics, are in excellent quantitative agreement. The preservation of the compactly-localized states even in the presence of this disorder might be useful for envisaged storage applications.

A new set of lattice-models based on the hexagonal $\sqrt{N}\times\sqrt{N}$ super-cells of the well-known honeycomb lattice with single-hole defect (HL-D-1/2N) are proposed to realize the nontrivial isolated flat-bands. Through performing both tight-binding and density functional theory calculations, we demonstrate that the experimentally realized graphitic carbon nitride (Adv. Mater., 22, 1004, 2010; Nat. Commun., 9, 3366, 2018), the HL-D-1/8 based C$_4$N$_3$, is a perfect system to host such flat bands. For the flat high-energy P-6m2 C$_4$N$_3$ structure, it displays the ferromagnetic half-metallicity which is not related to the isolated flat bands. However, the P-6m2 C$_4$N$_3$ structure is dynamically unstable. Using a structure searching method based on group and graph theory, we find that a new corrugated Pca21 C4N3 structure has the lowest energy among all known C$_4$N$_3$ structures. This Pca21 C$_4$N$_3$ structure is an intrinsic ferromagnetic half-semiconductor (Tc$\approx$241 K) with one semiconducting spin-channel (1.75 eV) and one insulating spin-channel (3.64 eV), which is quite rare in the two-dimensional (2D) systems. Its ferromagnetic semiconducting property originates from the isolated p$_z$-state flat-band as the corrugation shift the flat band upward to the Fermi level. Interestingly, this Pca21 C$_4$N$_3$ structure is found to be piezoelectric and ferroelectric, which makes C$_4$N$_3$ an unusual transition-metal-free 2D multiferroic., Comment: 6 pages, 4 fingures

Searching for new quantum spin Hall insulators with large fully opened energy gap to overcome the thermal disturbance at room temperature has attracted tremendous attention due to the one-dimensional (1D) spin-momentum locked topological edge states serving as dissipationless channels for the practical applications in low consumption electronics and high performance spintronics. Here, we report the investigation of topological nature of monolayer Bi4Br4 by the techniques of scanning tunneling microscopy and angle-resolved photoemission spectroscopy (ARPES). The topological non-triviality of 1D edge state integrals within the large bulk energy gap (~ 0.2 eV) is revealed by the first-principle calculations. The ARPES measurements at different temperature show a temperature-induced Lifshitz transition, corresponding to the resistivity anomaly caused by the shift of chemical potential. The connection between the emergency of superconductivity and the Lifshitz transition is discussed.

Based on first-principles calculations, the ground state configuration (Cmma-CH) of hydrogenated Biphenylene sheet (Science, 372, 852, 2021) is carefully identified from hundreds of possible candidates generated by RG2 code (Phys. Rev. B., 97, 014104, 2018). Cmma-CH contains four benzene molecules in its crystalline cell and all of them are inequivalent due to its Cmma symmetry. The hydrogen atoms in Cmma-CH bond to carbon atoms in each benzene with a boat-like (boat-1:DDUDDU) up/down sequence and reversed boat-1 (UUDUUD) sequence in adjacent benzene rings. It is energetically less stable than the previously proposed allotropes (chair, tricycle, stirrup, boat-1, boat-2 and twist-boat) of hydrogenated graphene, but its formation energy from hydrogenating Biphenylene sheet is remarkably lower than those for hydrogenating graphene to graphane. Our results confirm that Cmma-CH is mechanically and dynamically stable 2D hydrocarbon phase which is expectable to be experimentally realized by hydrogenating the synthesized Biphenylene sheet. The HSE06 based band structures show that Cmma-CH is an indirect band gap insulators with a gap of 4.645 eV., Comment: 6 page; 5 figures

Quantum spin Hall insulators have one-dimensional (1D) spin-momentum locked topological edge states (ES) inside the bulk band gap, which can serve as dissipationless channels for the practical applications in low consumption electronics and high performance spintronics. However, the clean and atomically sharp ES serving as ideal 1D conducting channels are still lack. Here, we report the formation of the quasi-1D Bi4I4 nanoribbons on the surface of Bi(111) with the support of the graphene-terminated 6H-SiC(0001) and the direct observations of the topological ES at the step edge by scanning tunneling microscopy and spectroscopic-imaging results. The ES reside surround the edge of Bi4I4 nanoribbons and exhibits remarkable robustness against non time reversal symmetry perturbations. The theoretical simulations verify the topological non-trivial character of 1D ES, which is retained after considering the presence of the underlying Bi(111). Our study supports the existence of topological ES in Bi4I4 nanoribbons, paving the way to engineer the novel topological features by using the nanoribbons as the 1D building block.

Two-dimensional materials with layered structures, appropriate bandgap, and high carrier mobilities have gathered tremendous interests due to their potential applications in optoelectronic and photovoltaic devices. Here, we report the growth of BiI3 thin film with controllable atomic thickness on graphene-terminated 6H-SiC(0001) substrate by molecular beam epitaxy (MBE) method. The growth kinetic processes and crystalline properties of the BiI3 film are studied by scanning tunneling microscopy (STM). The scanning tunneling spectroscopy (STS) reveals a bandgap of 2.8 eV for monolayer BiI3 with a weak dependence on film thickness for few-layer BiI3, which greatly exceeds the previous reported values identified by macroscopic optical measurements. This discrepancy originates from the edge effect of BiI3 that renders the bandgap downshift to 1.5 - 1.6 eV, as identified by the STS curves and the further confirmed by density functional theory (DFT) calculations. Our work provides a method to fabricate high-quality monolayer BiI3 film and resolves its intrinsic bandgap as well as the edge effect on reduction of bandgap, benefitting not only to fundamental researches but also to nanoelectronic and optoelectronic applications.

General solutions to the quantum Rabi model involve subspaces with unbounded number of photons. However, for the multiqubit multimode case, we find special solutions with at most one photon for arbitrary number of qubits and photon modes. Unlike the Juddian solution, ours exists for arbitrary single qubit-photon coupling strength with constant eigenenergy. This corresponds to a horizontal line in the spectrum, while still being a qubit-photon entangled state. As a possible application, we propose an adiabatic scheme for the fast generation of arbitrary single-photon multimode W states with nonadiabatic error less than 1%. Finally, we propose a superconducting circuit design, showing the experimental feasibility of the multimode multiqubit Rabi model., Comment: 6 pages, 5 figures plus Supplemental Materials

We study the localization properties of generalized, two- and three-dimensional Lieb lattices, $\mathcal{L}_2(n)$ and $\mathcal{L}_3(n)$, $n= 1, 2, 3$ and $4$, at energies corresponding to flat and dispersive bands using the transfer matrix method (TMM) and finite size scaling (FSS). We find that the scaling properties of the flat bands are different from scaling in dispersive bands for all $\mathcal{L}_d(n)$. For the $d=3$ dimensional case, states are extended for disorders $W$ down to $W=0.01 t$ at the flat bands, indicating that the disorder can lift the degeneracy of the flat bands quickly. The phase diagram with periodic boundary condition for $\mathcal{L}_3(1)$ looks similar to the one for hard boundaries. We present the critical disorder $W_c$ at energy $E=0$ and find a decreasing $W_c$ for increasing $n$ for $\mathcal{L}_3(n)$, up to $n=3$. Last, we show a table of FSS parameters including so-called irrelevant variables; but the results indicate that the accuracy is too low to determine these reliably. \end{abstract}

We study the localization properties and the Anderson transition in the 3D Lieb lattice $\mathcal{L}_3(1)$ and its extensions $\mathcal{L}_3(n)$ in the presence of disorder. We compute the positions of the flat bands, the disorder-broadened density of states and the energy-disorder phase diagrams for up to 4 different such Lieb lattices. Via finite-size scaling, we obtain the critical properties such as critical disorders and energies as well as the universal localization lengths exponent $\nu$. We find that the critical disorder $W_c$ decreases from $\sim 16.5$ for the cubic lattice, to $\sim 8.6$ for $\mathcal{L}_3(1)$, $\sim 5.9$ for $\mathcal{L}_3(2)$ and $\sim 4.8$ for $\mathcal{L}_3(3)$. Nevertheless, the value of the critical exponent $\nu$ for all Lieb lattices studied here and across disorder and energy transitions agrees within error bars with the generally accepted universal value $\nu=1.590 (1.579,1.602)$.

Realization of deep learning with coherent optical field has attracted remarkably attentions presently, which benefits on the fact that optical matrix manipulation can be executed at speed of light with inherent parallel computation as well as low latency. Photonic neural network has a significant potential for prediction-oriented tasks. Yet, real-value Backpropagation behaves somewhat intractably for coherent photonic intelligent training. We develop a compatible learning protocol in complex space, of which nonlinear activation could be selected efficiently depending on the unveiled compatible condition. Compatibility indicates that matrix representation in complex space covers its real counterpart, which could enable a single channel mingled training in real and complex space as a unified model. The phase logical XOR gate with Mach-Zehnder interferometers and diffractive neural network with optical modulation mechanism, implementing intelligent weight learned from compatible learning, are presented to prove the availability. Compatible learning opens an envisaged window for deep photonic neural network., Comment: Original Manuscript

The widely used crystal structures for both heptazine-based and triazine-based two-dimensional (2D) graphitic carbon nitride (g-C$_3$N$_4$) are the flat P-6m2 configurations. However, the experimentally synthesized 2D g-C$_3$N$_4$ possess thickness ranging in 0.2-0.5 nm, indicating that the theoretically used flat P-6m2 configurations are not the correct ground states. In this work, we propose three new corrugated structures P321, P3m1 and Pca21 with energies of 66 (86), 77 (87) and 78 (89) meV/atom lower than that of the corresponding heptazine-based (triazine-based) g-C$_3$N$_4$ in flat P-6m2 configuration, respectively. These corrugated structures have very similar periodic patterns to the flat P-6m2 ones and they are difficult to be distinguished from each other according to their top-views. The optimized thicknesses of the three corrugated structures ranging in 1.347-3.142 {\AA} are in good agreement with the experimental results. The first-principles results show that these corrugated structural candidates are also semiconductors with band gaps slightly larger than those of the correspondingly flat P-6m2 ones. Furthermore, they possess also suitable band edge positions for sun-light-driven water-splitting at both $pH=0$ and $pH=7$ environments. Our results show that these three new structures are more promising candidates for the experimentally synthesized g-C$_3$N$_4$., Comment: 6 pages, 4 figures

Based on first-principles method we predict a new low-energy Stone-Wales graphene SW40, which has an orthorhombic lattice with Pbam symmetry and 40 carbon atoms in its crystalline cell forming well-arranged Stone-Wales patterns. The calculated total energy of SW40 is just about 133 meV higher than that of graphene, indicating its excellent stability exceeds all the previously proposed graphene allotropes. We find that SW40 processes intrinsic Type-III Dirac-cone (Phys. Rev. Lett., 120, 237403, 2018) formed by band-crossing of a local linear-band and a local flat-band, which can result in highly anisotropic Fermions in the system. Interestingly, such intrinsic type-III Dirac-cone can be effectively tuned by inner-layer strains and it will be transferred into Type-II and Type-I Dirac-cones under tensile and compressed strains, respectively. Finally, a general tight-binding model was constructed to understand the electronic properties nearby the Fermi-level in SW40. The results show that type-III Dirac-cone feature can be well understood by the $\pi$-electron interactions between adjacent Stone-Wales defects., Comment: 7 pages, 4 figures

We study the localization properties of the two-dimensional Lieb lattice and its extensions in the presence of disorder using transfer matrix method and finite-size scaling. We find that all states in the Lieb lattice and its extensions are localized for $W \geq 1$. Clear differences in the localization properties between disordered flat band and disordered dispersive bands are identified. Our results complement previous experimental studies of clean photonic Lieb lattices and provide information about their stability with respect to disorder., Comment: 7 pages, 20 figures, 1 table

Two-dimensional materials such as transitional metal dichalcogenides exhibit unique optical and electrical properties. Here we report on the varying optical properties of CVD grown MoS2 monolayer flakes with different shapes. In particular, it is observed that the perimeter and the central region of the flakes have non-uniform photoluminescence (PL) energy and intensity. We quantified these effects systematically and propose that thermally induced strain during growth is the origin. The strain relaxation after transfer of the MoS2 flakes supports this explanation. Detailed investigations of the spatial distribution of the PL energy reveal that depending on the shape of the MoS2 flakes, the width of the strain field is different. Thus, our results help to elucidate the fundamental mechanisms responsible for the differences in PL and Raman signals between the perimeter region and the center region of monolayer MoS2 and suggest that the induced strain plays an important role in the growth of monolayer materials.

We prove, by means of a unified treatment, that the superradiant phase transitions of Dicke and classical oscillator limits of simple light-matter models are indeed of the same type. We show that the mean-field approximation is exact in both cases, and compute the structure and location of the transitions in parameter space. We extend this study to a fuller range of models, paying special attention to symmetry considerations. We uncover general features of the phase structure in the space of parameters of these models.

A two-dimensional carbon allotrope, Stone-Wales graphene, is identified in stochastic group and graph constrained searches and systematically investigated by first-principles calculations. Stone-Wales graphene consists of well-arranged Stone-Wales defects, and it can be constructed through a 90$^\circ$ bond-rotation in a $\sqrt{8}$$\times$$\sqrt{8}$ super-cell of graphene. Its calculated energy relative to graphene, +149 meV/atom, makes it more stable than the most competitive previously suggested graphene allotropes. We find that Stone-Wales graphene based on a $\sqrt{8}$ super-cell is more stable than those based on $\sqrt{9} \times \sqrt{9}$, $\sqrt{12} \times \sqrt{12}$ and $\sqrt{13} \times \sqrt{13}$ super-cells, and is a "magic size" that can be further understood through a simple "energy splitting and inversion" model. The calculated vibrational properties and molecular dynamics of SW-graphene confirm that it is dynamically stable. The electronic structure shows SW-graphene is a semimetal with distorted, strongly anisotropic Dirac cones., Comment: Accepted;5 pages;5 figures;53 references and 8 pages of supplementary file

The energy landscape of carbon is exceedingly complex, hosting diverse and important metastable phases, including diamond, fullerenes, nanotubes and graphene. Searching for structures, especially those with large unit cells, in this landscape is challenging. Here we use a combined stochastic search strategy employing two algorithms (AIRSS and RG2) to apply connectivity constraints to unit cells containing up to 100 carbon atoms. We uncover three low energy carbon polymorphs (Pbam-32, P6/mmm and I-43d) with new topologies, containing 32, 36 and 94 atoms in their primitive cells, respectively. Their energies relative to diamond are 96 meV/atom, 131 meV/atom and 112 meV/atom, respectively, which suggests potential metastability. These three carbon allotropes are mechanically and dynamically stable, insulating carbon crystals with superhard mechanical properties. The I43d structure possesses a direct band gap of 7.25 eV, which is the widest gap in the carbon allotrope family. Silicon, germanium and tin versions of Pbam-32, P6/mmm and I-43d also show energetic, dynamical and mechanical stability. The computed electronic properties show that they are potential materials for semiconductor and photovoltaic applications., Comment: Accepted; 6 pages; 3 figures; 11 pages of supplementary