We develop a method of variational optimization of the infinite projected entangled pair states on the honeycomb lattice. The method is based on the automatic differentiation of the honeycomb-lattice corner transfer matrix renormalization group. We apply the approach to the antiferromagnetic Heisenberg spin-1/2 and ferromagnetic Kitaev models on the honeycomb lattice. The developed formalism gives quantitatively accurate results for the main physical observables and has a necessary potential for further extensions.
Search for higher-order topological insulators, characterized by topologically protected gapless boundary states of codimension higher than one, in bosonic systems has attracted growing interest. Here, we establish twisted bilayer honeycomb magnets as a new platform for hosting second-order topological magnon insulators (SOTMIs) without fine-tuning. We employ a simple, minimal Heisenberg spin model to describe misaligned bilayer sheets of honeycomb ferromagnetic magnets with a large commensurate twist angle. We found that the higher-order topology in this bilayer system shows a significant dep…
In this work, a long-range resonating valence bond state is proposed as a variational wave function for the ground state of the $S=1/2$ antiferromagnetic Heisenberg model on the honeycomb lattice. Employing Variational Monte Carlo (VMC) method, we show that the ground state energy obtained from such RVB wave function, lies well below the energy of the Néel state and compares very well to the energies evaluated from spin-wave theory and series expansion method. We also obtain the spin-spin correlation function along zig-zag and armchair directions and find that the two correlations are differen…
Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al, Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory (LFWT) suggested a dimerized, color ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called …
We show that the honeycomb Heisenberg antiferromagnet with $J_1/2=J_2=J_3$, where $J_{1/2/3}$ are first-, second- and third-neighbour couplings respectively, forms a classical spin liquid with pinch-point singularities in the structure factor at the Brillouin zone corners. Upon dilution with non-magnetic ions, fractionalised degrees of freedom carrying $1/3$ of the free moment emerge. Their effective description in the limit of low-temperature is that of spins randomly located on a triangular lattice, with a frustrated interaction of long-ranged logarithmic form. The XY version of this magnet …
In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is furt…
We report the structures and physical properties of two new iridates, NaIrO$_3$ and Sr$_3$CaIr$_2$O$_9$, both of which contain continuous two-dimensional honeycomb connectivity. NaIrO3 is produced by room temperature oxidative deintercalation of sodium from Na$_2$IrO$_3$, and contains edge-sharing IrO6 octahedra that form a planar honeycomb lattice. Sr$_3$CaIr$_2$O$_9$, produced via conventional solid-state synthesis, hosts a buckled honeycomb lattice with novel corner-sharing connectivity between IrO6 octahedra. Both of these new compounds are comprised of Ir$^{5+}$ (5$d^4$) and exhibit negli…
Motivated by the possible charge ordering in the recently discovered superconductor Na_xCoO_2 yH_2O, which at filling x=1/3 would correspond to a half-filled honeycomb lattice, we investigate the single-hole dynamics of the t-J model on this lattice. Using exact diagonalization, series expansion and the self-consistent Born approximation, we calculate the quasi-particle dispersion, bandwidth and residues and compare our findings with the well established results for the square lattice. Given the similarities between both lattices, it is not surprising that we find qualitatively similar feature…
It is shown that the unique sign structure of the ground state of the Hubbard model on honeycomb lattice, which is shown to be insensitive to the trapped $Z_{2}$ gauge flux when the system is defined on a torus, may cause the absence of topological degeneracy on this bipartite system. Examples of variational Mott insulating state on the honeycomb lattice are given to illustrate the close relation between the sign structure of the ground state and the (absence of) topological degeneracy.
We propose a new principle to realize flatbands which are robust in real materials, based on a network superstructure of one-dimensional segments. This mechanism is naturally realized in the nearly commensurate charge-density wave of 1T-TaS${}_2$ with the honeycomb network of conducting domain walls, and the resulting flatband can naturally explain the enhanced superconductivity. We also show that corner states, which are a hallmark of the higher-order topological insulators, appear in the network superstructure.
Modification of interatomic distances due to high pressure leads to exotic phenomena, including metallicity, superconductivity and magnetism, observed in materials not showing such properties in normal conditions. In two-dimensional crystals, such as graphene, atomic bond lengths can be modified by more that 10 percent by applying in-plane strain, i.e., without generating high pressure in the bulk. In this work, we study the strain-induced Mott transition on a honeycomb lattice by using computationally inexpensive techniques, including Gutzwiller Wave Function (GWF) and different variants of G…
The SU(4) Heisenberg model can serve as a low energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric, or in systems of alkaline-earth atoms on optical lattice. Recently, it has been argued that on the honeycomb lattice the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to…
In addition to low-energy spin fluctuations, which distinguish them from band insulators, Mott insulators often possess orbital degrees of freedom when crystal-field levels are partially filled. While in most situations spins and orbitals develop long-range order, the possibility for the ground state to be a quantum liquid opens new perspectives. In this paper, we provide clear evidence that the SU(4) symmetric Kugel-Khomskii model on the honeycomb lattice is a quantum spin-orbital liquid. The absence of any form of symmetry breaking - lattice or SU(N) - is supported by a combination of semicl…
Parafermions are a natural generalization of Majorana fermions. We consider a breathing Kagome lattice with complex hoppings by imposing $\mathbb{Z}_{3}$ clock symmetry in the complex energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We demonstrate that the topological corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric model. It is also shown that the model is realized in electric circuits properly designed, where the parafermion corner states are observed by impe…
Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
The possible emergence of a spin liquid phase in the half-filled Hubbard model on the honeycomb lattice; a simple model of graphene, is studied using the variational cluster approximation. We found that the critical interaction strength of a magnetic transition is slightly lower than that of the non-magnetic metal-to-insulator transition. Thus a semi-metallic state becomes a magnetic insulator as the interaction strength increases and a non-magnetic insulating state does not occur. Therefore a spin liquid state characterized by an insulating state without long range order, is not realized in t…
Magnetic phases in the $J_{1}$-$J_{2}$ antiferromagnetic XY model on the honeycomb lattice
Akhiezer Institute for Theoretical Physics, NSC KIPT（阿基谢尔理论物理研究所）；Haiqu, Inc.（海奎公司）；Education and Research Institute “School of Physics and Technology”, Karazin Kharkiv National University（哈尔科夫国立大学物理与技术教育研究所）
We study the ground-state phase diagram of the quantum $J_1-J_2$ model on the honeycomb lattice by means of an entangled-plaquette variational ansatz. Values of energy and relevant order parameters are computed in the range $0\le {J_2}/{J_1} \le 1$. The system displays classical order for ${J_2}/{J_1}\lesssim 0.2$ (Néel), and for $J_2/J_1 \gtrsim 0.4$ (collinear). In the intermediate region, the ground-state is disordered. Our results show that the reduction of the half-filled Hubbard model to the model studied here does not yield accurate predictions.
We examine the stability of classical states with a generic incommensurate spiral order against quantum fluctuations. Specifically, we focus on the frustrated spin-1/2 $XY$ and Heisenberg models on the honeycomb lattice with nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic couplings. Our variational approach is based on the Jastrow wave functions, which include quantum correlations on top of classical spin waves. We perform a systematic optimization of wave vectors and Jastrow pseudo-potentials within this class of variational states and find that quantum fluctuations f…
The band structure of noninteracting fermions in the honeycomb lattice exhibits the Dirac cones at the corners of the Brillouin zone. As a consequence, fermions in this lattice manifest a semiconducting behavior below some critical value of the onsite attraction, $U_{c}$. However, above $U_{c}$, the superconducting phase can occur. We discuss an interplay between the semiconductor--superconductor transition and the possibility of realization of the spin-polarized superconductivity (the so-called Sarma phase). We show that the critical interaction can be tuned by the next-nearest-neighbor (NNN)…
We present a thorough tight-binding analysis of the band structure of a wide variety of lattices belonging to the class of honeycomb and Kagome systems including several mixed forms combining both lattices. The band structure of these systems are made of a combination of dispersive and flat bands. The dispersive bands possess Dirac cones (linear dispersion) at the six corners (K points) of the Brillouin zone although in peculiar cases Dirac cones at the center of the zone $(Γ$ point) appear. The flat bands can be of different nature. Most of them are tangent to the dispersive bands at the cent…
We show that unconventional boundary phenomena occur in the $3Q$ charge bond orders on the hexagonal lattices. At the Van Hove singularity with three nesting momenta, $3Q$ orders can trigger a $\text{C}_6$-symmetric insulator under bond modulations. On the kagome lattice, in-gap corner states appear in the energy spectrum and carry fractional corner charge $-2e/3$. Such corner phenomena originate from the corner filling anomaly and indicate a higher-order topological insulator. The in-gap corner states are also observed on the triangular lattice. The honeycomb lattice does not support fraction…
Quantum phase transitions in the Hubbard model on the honeycomb lattice are investigated in the variational cluster approximation. The critical interaction for the paramagnetic to antiferromagnetic phase transition is found to be in remarkable agreement with a recent large-scale quantum Monte Carlo simulation. Calculated staggered magnetization increases continuously with $U$ and thus we find the phase transition is of a second order. We also find that the semimetal-insulator transition occurs at infinitesimally small interaction and thus a paramagnetic insulating state appears in a wide inter…
The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and…
The non-interacting band structure of spinless fermions in a two-dimensional ($d=2$) $p$-band honeycomb lattice exhibits two quadratic band touching points (QBTPs), which lie at the Fermi levels of filling $ν=1/4$ and its particle-hole conjugated filling $ν=3/4$. A weak Hubbard interaction $U$ spontaneously breaks the time-reversal symmetry and removes the QBTP, rendering the system into a quantum anomalous Hall insulator (QAHI). The first-order topological nature of QAHI is characterized by a nontrivial Chern number and supports ($d-1$)-dimensional chiral edge modes. With increasing the inter…
We study the phase diagram of the frustrated XY model on the honeycomb lattice by using accurate correlated wave functions and variational Monte Carlo simulations. Our results suggest that a spin-liquid state is energetically favorable in the region of intermediate frustration, intervening between two magnetically ordered phases. The latter ones are represented by classically ordered states supplemented with a long-range Jastrow factor, which includes relevant correlations and dramatically improves the description provided by the purely classical solution of the model. The construction of the …
Motivated by superconductivity (SC) in layered nitrides, we study an ionic-Hubbard model on a honeycomb lattice, which consists of two sublattices with an energy-level offset, by using an optimization variational Monte Carlo method. Parameter values for the Coulomb interaction and the energy-level offset, to realize a band insulator in non-doped state, are evaluated, and the SC state is investigated in a carrier-doped band insulator. It is found that, in a weakly doped band insulator, spin fluctuation remains and the spin-singlet d-wave SC state is realized. Present results support an unconven…
In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($Γ$) couplings. We investigate the quantum ($S = 1/2$) $K$-$J$-$Γ$ model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic f…
Motivated by recent experimental observations in $α$-RuCl$_3$, we study the $K$-$Γ$ model on the honeycomb lattice in an external magnetic field. By a slave-particle representation and Variational Monte Carlo calculations, we reproduce the phase transition from zigzag magnetic order to a field-induced disordered phase. The nature of this state depends crucially on the field orientation. For particular field directions in the honeycomb plane, we find a gapless Dirac spin liquid, in agreement with recent experiments on $α$-RuCl$_3$. For a range of out-of-plane fields, we predict the existence of…
We study spontaneous symmetry breaking in a system of spinless fermions in the Honeycomb lattice paying special emphasis to the role of an enlarged unit cell on time reversal symmetry broken phases. We use a tight binding model with nearest neighbor hopping t and Hubbard interaction V1 and V2 and extract the phase diagram as a function of electron density and interaction within a mean field variational approach. The analysis completes the previous work done in Phys. Rev. Lett. 107, 106402 (2011) where phases with non--trivial topological properties were found with only a nearest neighbor inter…
We show the chiral d-wave RVB state on honeycomb lattice stands as a natural generalization of the staggered flux phase on square lattice. Although the state is generated from a time reversal symmetry broken mean field ansatz, it actually represents a fully symmetric spin liquid state with a positive definite wave function in the sense of Marshall sign rule for unfrustrated antiferromagnets. The evolution of the state with the parameter $Δ/χ$ follows exactly the same manner as that of the staggered flux phase on square lattice. The critical pairing strength corresponding to the $π$-flux phase …
The SU(4)-symmetric spin-orbital model on the honeycomb lattice was recently studied in connection to correlated insulators such as the $e_{g}$ Mott insulator Ba$_{3}$CuSb$_{2}$O$_{9}$ and the insulating phase of magic-angle twisted bilayer graphene at quarter filling. Here we provide a unified discussion of these systems by investigating an extended model that includes the effects of Hund's coupling and anisotropic, orbital-dependent exchange interactions. Using a combination of mean-field theory, linear flavor-wave theory, and variational Monte Carlo, we show that this model harbors a quantu…
Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor $J_1$ and second-neighbor $J_2$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. Néel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio $J_2/J_1$, we find strong evidence for a continuous transition to a nonmagnetic phase at $J_2/J_1 \approx 0.23$. C…
Motivated by the recently synthesized insulating nickelate Ni$_2$Mo$_3$O$_8$, which has been reported to have an unusual non-collinear magnetic order of Ni$^{2+}$ $S=1$ moments with a nontrivial angle between adjacent spins, we construct an effective spin-1 model on the honeycomb lattice, with the exchange parameters determined with the help of first principles electronic structure calculations. The resulting bilinear-biquadratic model, supplemented with the realistic crystal-field induced anisotropy, favors the collinear Néel state. We find that the crucial key to explaining the observed nonc…
Recent numerical work (Nature 464, 847 (2010)) indicates the existence of a spin liquid phase (SL) that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum $J_1-J_2$ spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model, finding a phase transition between antiferromagnet and an unusual $Z_…
We get an anomalous Hall metallic state in the Honeycomb lattice with nearest neighbors only arising as a spontaneously broken symmetry state from a local nearest neighbor Coulomb interaction V . The key ingredient is to enlarge the unit cell to host six atoms that permits Kekulé distortions and supports self-consistent currents creating non trivial magnetic configurations with total zero flux. We find within a variational mean field approach a metallic phase with broken time reversal symmetry (T) very close in parameter space to a Pomeranchuk instability. Within the T broken region the predom…
We explore the possibility of inducing a topological insulator phase in a honeycomb lattice lacking spin-orbit interaction using a metallic (or Fermi gas) environment. The lattice and the metallic environment interact through a density-density interaction without particle tunneling, and integrating out the metallic environment produces a honeycomb sheet with in-plane oscillating long-ranged interactions. We find the ground state of the interacting system in a variational mean-field method and show that the Fermi wave vector, kF, of the metal determines which phase occurs in the honeycomb latti…
Realization and design of topological insulators emerging from electron correlations, called topological Mott insulators (TMIs), is pursued by using mean-field approximations as well as multi-variable variational Monte Carlo (MVMC) methods for Dirac electrons on honeycomb lattices. The topological insulator phases predicted in the previous studies by the mean-field approximation for an extended Hubbard model on the honeycomb lattice turn out to disappear, when we consider the possibility of a long-period charge-density-wave (CDW) order taking over the TMI phase. Nevertheless, we further show t…
The emergence of superconductivity in doped Mott insulators has been debated for decades. In this paper, we report the theoretical discovery of a time-reversal symmetry breaking superconducting ground state in the doped Mott insulator (described by the well known t-J model) on honeycomb lattice, based on a recently developed variational method: the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization and density-matrix renormalization methods to check our results on small clusters. We find systematic consistency for the ground-state energy as well as other phys…
By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic $S=1/2$ Heisenberg model on the honeycomb lattice, in presence of first-neighbor ($J_1$) and second-neighbor ($J_2$) couplings, for ${J_2 < 0.5 J_1}$. The ground state of the system shows long-range antiferromagnetic order for ${J_2/J_1 \lesssim 0.23}$, plaquette valence-bond order for ${0.23 \lesssim J_2/J_1 \lesssim 0.36}$, and columnar dimer order for ${J_2/J_1 \gtrsim 0.36}$. Within the antiferromagnetic state, a well-defined …
Spin liquids are novel states of matter with fractionalized excitations. A recent numerical study of Hubbard model on a honeycomb lattice\cite{Meng2010} indicates that a gapped spin liquid phase exists close to the Mott transition. Using Projective Symmetry Group, we classify all the possible spin liquid states by Schwinger fermion mean-field approach. We find there is only one fully gapped spin liquid candidate state: "Sublattice Pairing State" that can be realized up to the 3rd neighbor mean-field amplitudes, and is in the neighborhood of the Mott transition. We propose this state as the spi…
Recently two interesting candidate quantum phases --- the chiral spin density wave state featuring anomalous quantum Hall effect and the d+id superconductor --- were proposed for the Hubbard model on the honeycomb lattice at 1/4 doping. Using a combination of exact diagonalization, density matrix renormalization group, the variational Monte Carlo method and quantum field theories, we study the quantum phase diagrams of both the Hubbard model and t-J model on the honeycomb lattice at 1/4-doping. The main advantage of our approach is the use of symmetry quantum numbers of ground state wavefuncti…
We use density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-1/2 Heisenberg model on honeycomb lattice with first (J1) and second (J2) neighbor antiferromagnetic interactions, where a Z2 spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with width slightly over 15 lattice spacings and torus up to the size N=2*6*6. With increasing J2, we find a Neel phase with vanishing spin gap and a plaquette valence-bond (PVB) phase with non-zero spin gap. By extrapolating the…
Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $Γ'$ terms. To fully understand the effect of the $Γ'$ interactions, we investigate the quantum $K$-$Γ$-$Γ'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $Γ'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that…
We revisit the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and demonstrate that its performance can be substantially improved by determining the tensors using an eigenvalue solver as opposed to the power method used in CTMRG. We also generalize the variational uniform matrix product state (VUMPS) ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer matrices and discuss similarities with the corner methods. These two new algorithms will be crucial to improving the performance of v…
Honeycomb Crystal in a Mass-Asymmetric Electron-Hole Bilayer
Fudan University（复旦大学）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；The Hong Kong University of Science and Technology（香港科技大学）；Beijing National Laboratory for Condensed Matter Physics（北京凝聚态物理国家研究中心）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料科学与工程中心）；Hefei National Laboratory（合肥国家实验室）
In a recent Letter, Hassan and Sénéchal [1] discussed the existence of a spin-liquid phase of the half-filled Hubbard model on the honeycomb lattice. Using schemes, such as the variational cluster approximation (VCA) and the cluster dynamical mean field theory (CDMFT) in combination with exact diagonalization (ED), they argued that a single bath orbital per site of the six-atom unit cell is insufficient and leads to the erroneous conclusion that the system is gapped for all nonzero values of the onsite Coulomb interaction $U$. In contrast, we point out here that, in the case of the honeycomb l…
We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states (iPEPS), a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner transfer-matrix method. Benchmark results for challenging problems are presented, including the 2D Heisenberg model, the Shastry-Sutherland model, and the t-J model, which show that the variational scheme yields considerably more accurate resul…
