We study the quantum phases and phase transitions of the Kane-Mele Hubbard (KMH) model on a zigzag ribbon of honeycomb lattice at a finite size via the weak-coupling renormalization group (RG) approach. In the non-interacting limit, the KM model is known to support topological edge states where electrons show helical property with orientations of the spin and momentum being locked. The effective inter-edge hopping terms are generated due to finite-size effect. In the presence of an on-site Coulomb repulsive interaction and the inter-edge hoppings, special focus is put on the stability of the topological edge states (TI phase) in the KMH model against (i) the charge and spin gaped (II) phase, (ii) the charge gaped but spin gapless (IC) phase and (iii) the spin gaped but charge gapless (CI) phase depending on the number (even/odd) of the zigzag ribbons, doping level (electron filling factor) and the ratio of the Coulomb interaction to the inter-edge tunneling. We discuss different phase diagrams for even and odd numbers of zigzag ribbons. We find the TI-CI, II-IC, and II-CI quantum phase transitions are of the Kosterlitz-Thouless (KT) type. By computing various correlation functions, we further analyze the nature and leading instabilities of these phases.
Motivated by recent numerical results, we study the phase diagram of the Kane-Mele-Hubbard (KHM) model, especially the nature of its quantum critical points. The phase diagram of the Kane-Mele-Hubbard model can be understood by breaking the SO(4) symmetry of our previous work down to U(1)_spin x U(1)_charge x PH symmetry. The vortices of the inplane Neel phase carry charge, and the proliferation of the charged magnetic vortex drives the transition between the inplane Neel phase and the QSH insulator phase; this transition belongs to the 3d XY universality class. The transition between the liqu…
The dimerized Kane-Mele model with/without the strong interaction is studied using analytical methods. The boundary of the topological phase transition of the model without strong interaction is obtained. Our results show that the occurrence of the transition only depends on dimerized parameter . From the one-particle spectrum, we obtain the completed phase diagram including the quantum spin Hall (QSH) state and the topologically trivial insulator. Then, using different mean-field methods, we investigate the Mott transition and the magnetic transition of the strongly correlated dimerized Kane-…
The phase diagram of Kane-Mele-Heisenberg (KMH) model in classical limit~\cite{zare}, contains disordered regions in the coupling space, as the result of to competition among different terms in the Hamiltonian, leading to frustration in finding a unique ground state. In this work we explore the nature of these phase in the quantum limit, for a $S=1/2$. Employing exact diagonalization (ED) in $S_z$ and nearest neighbor valence bond (NNVB) bases, bond and plaquette valence bond mean field theories, We show that the disordered regions are divided into ordered quantum states in the form of plaquet…
Topological insulators present a bulk gap, but allow for dissipationless spin transport along the edges. These exotic states are characterized by the $Z_2$ topological invariant and are protected by time-reversal symmetry. The Kane-Mele model is one model to realize this topological class in two dimensions, also called the quantum spin Hall state. In this review, we provide a pedagogical introduction to the influence of correlation effects in the quantum spin Hall states, with special focus on the half-filled Kane-Mele-Hubbard model, solved by means of unbiased determinant quantum Monte Carlo …
We investigate the new quantum phases on the extended Kane-Mele-Hubbard model of honeycomb lattice in the Hofstadter regime. In this regime, orbital motion of the electrons can induce various topological phases with spontaneously broken symmetries when the spin orbit coupling and electron correlations coexist. Here, we consider the interaction effects in the Kane-Mele model and discuss possible phases in the presence of magnetic field at integer fillings of electrons. In particular, focusing on 2π/3 magnetic flux per plaquette, the realization of numerous quantum phases are discussed within th…
The description of interactions in strongly-correlated topological phases of matter remains a challenge. Here, we develop a stochastic functional approach for interacting topological insulators including both charge and spin channels. We find that the Mott transition of the Kane-Mele-Hubbard model may be described by the variational principle with one equation. We present different views of this equation from the electron Green's function, the free-energy and the Hellmann-Feynman theorem. The band gap remains finite at the transition and the Mott phase is characterized by antiferromagnetism in…
We study the effects of short-ranged interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in the Kane-Mele model at half-filling with staggered potentials which explicitly breaks the discrete particle-hole symmetry. Within Hartree-Fock mean-field analysis, we conclude that the on-site repulsive interactions help stabilize the topological phase (quantum spin Hall) against the staggered potentials by enlarging the regime of the topological phase along the axis of the ratio of the staggered potential strength and the spin-orbit coupling. In sharp contr…
We construct a topological ladder model, one-dimensional, following the steps which lead to the Kane-Mele model in two dimensions. Starting with a Creutz ladder we modify it so that the gap closure points can occur at either $k = π/ 2$ or $-π/2$. We then couple two such models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We derive the relevant $2\mathbb{Z}$ topological index, calculate the phase diagram and demonstrate the existence of edge states. We also give the …
By using the constrained-phase quantum Monte Carlo method, we performed a systematic study of the pairing correlations in the ground state of the doped Kane-Mele-Hubbard model on a honeycomb lattice. We find that pairing correlations with $d+id$ symmetry dominate close to half filling, but pairing correlations with $p+ip$ symmetry dominate as hole doping moves the system below three-quarters filling. We correlate these behaviors of the pairing correlations with the topology of the Fermi surfaces of the non-interacting problem. We also find that the effective pairing correlation is enhanced gre…
We study a generalized Kane-Mele-Hubbard model with third-neighbor hopping, an interacting two-dimensional model with a topological phase transition as a function of third-neighbor hopping, by means of the determinant projector Quantum Monte Carlo (QMC) method. This technique is essentially numerically exact on models without a fermion sign problem, such as the one we consider. We determine the interaction-dependence of the Z2 topological insulator/trivial insulator phase boundary by calculating the Z2 invariants directly from the single-particle Green's function. The interactions push the pha…
The interacting Kane-Mele model with a long-range hopping is studied using analytical method. The original Kane-Mele model is defined on a honeycomb lattice. In the work, we introduce a four-lattice-constant range hopping and the on-site Hubbard interaction into the model and keep its lattice structure unchanged. From the single-particle energy spectrum, we obtain the critical strength of the long-range hopping $t_L$ at which the topological transition occurs in the non-interacting limit of the model and our results show that it is independent of the spin-orbit coupling. After introducing the …
We consider the Kane-Mele-Hubbard model with a magnetic $π$ flux threading each honeycomb plaquette. The resulting model has remarkably rich physical properties. In each spin sector, the noninteracting band structure is characterized by a total Chern number $C=\pm 2$. Fine-tuning of the intrinsic spin-orbit coupling $λ$ leads to a quadratic band crossing point associated with a topological phase transition. At this point, quantum Monte Carlo simulations reveal a magnetically ordered phase which extends to weak coupling. Although the spinful model has two Kramers doublets at each edge and is ex…
The classical phase diagram of the Kane-Mele-Heisenberg model is obtained by three complementary methods: Luttinger-Tisza, variational minimization, and the iterative minimization method. Six distinct phases were obtained in the space of the couplings. Three phases are commensurate with long-range ordering, planar N{é}el states in horizontal plane (phase.I), planar states in the plane vertical to the horizontal plane (phase.VI) and collinear states normal to the horizontal plane (phase.II). However the other three, are infinitely degenerate due to the frustrating competition between the coupli…
The antiferromagnetic Chern insulator phase in the Kane-Mele-Hubbard model
Lanzhou University（兰州大学）；Nanjing University of Science and Technology（南京理工大学）；Tokyo University of Science（东京理科大学）；Beijing Computational Science Research Center（北京计算科学研究中心）
Spin-orbit (SO) coupling is the crucial parameter to drive topological insulating phases in electronic band models. In particular, the generic emergence of SO coupling involves the Rashba term which fully breaks the SU(2) spin symmetry. As soon as interactions are taken into account, however, many theoretical studies have to content themselves with the analysis of a simplified U(1) conserving SO term without Rashba coupling. We intend to fill this gap by studying the Kane-Mele-Hubbard (KMH) model in the presence of Rashba SO coupling and present the first systematic analysis of the effect of R…
We investigate the magnetic response in the quantum spin Hall phase of the layered Kane-Mele model with Hubbard interaction, and argue a condition to obtain the Meissner effect. The effect of Rashba spin orbit coupling is also discussed.
The realization of the spin-Hall effect in quantum wells has led to a plethora of studies regarding the properties of the edge states of a 2D topological insulator. These edge states constitute a class of one-dimensional liquids, called the helical liquid, where an electron's spin quantization axis is tied to its momentum. In contrast to one dimensional conductors, magnetic impurities - below the Kondo temperature - cannot block transport and one expects the current to circumvent the impurity. To study this phenomenon, we consider the single impurity Anderson model embedded into an edge of a K…
We study the two-dimensional Kane-Mele-Hubbard model at half filling by means of quantum Monte Carlo simulations. We present a refined phase boundary for the quantum spin liquid. The topological insulator at finite Hubbard interaction strength is adiabatically connected to the groundstate of the Kane-Mele model. In the presence of spin-orbit coupling, magnetic order at large Hubbard U is restricted to the transverse direction. The transition from the topological band insulator to the antiferromagnetic Mott insulator is in the universality class of the three-dimensional XY model. The numerical …
We prove that the Kane-Mele-Hubbard model with purely imaginary next-nearest-neighbor hoppings has a particle-hole symmetry at half-filling. Such a symmetry has interesting consequences including the absence of charge and spin currents along open edges, and the absence of the sign problem in the determinant quantum Monte-Carlo simulations. Consequentially, the interplay between band topology and strong correlations can be studied at high numeric precisions. The process that the topological band insulator evolves into the antiferromagnetic Mott insulator as increasing interaction strength is st…
We determine the phase diagram of the Kane-Mele model with a long-range Coulomb interaction using an exact quantum Monte Carlo method. Long-range interactions are expected to play a role in honeycomb materials because the vanishing density of states in the semimetallic weak-coupling phase suppresses screening. According to our results, the Kane-Mele-Coulomb model supports the same phases as the Kane-Mele-Hubbard model. The nonlocal part of the interaction promotes short-range sublattice charge fluctuations, which compete with antiferromagnetic order driven by the onsite repulsion. Consequently…
Recently, the attractive Kane-Mele-Habbard (KMH) model on a honeycomb lattice at half filling has been studied in two papers: PRB 99, 184514 (2019) and PRB 94, 104508 (2016). The authors of the first one presented the phase diagram which interpolates the trivial and non-trivial topological states. However, the next-nearest-neighbor (NNN) hopping term has been neglected, although it is several orders of magnitude stronger than the internal spin-orbit coupling. We use the mean-field approximation to derive the phase diagram of the attractive KMH model with NNN hoping at half filling. The phase d…
We study the Kane-Mele-Hubbard model both at half-filling and away from half-filling using a slave-boson mean-field approach at zero temperature. We obtain a phase diagram at half-filling and discuss its connection to recent results from quantum Monte Carlo, slave-rotor, and $Z_2$ mean-field studies. In particular, we find a small window in parameter space where a spin liquid phase with gapped spin and charge excitations reside. Upon doping, we show the spin liquid state becomes a superconducting state by explicitly calculating the singlet pairing order parameters. Interestingly, we find an "o…
We investigate, firstly, the effects of the Rashba SOC on the band structrue of the Kane-Mele model. The competition between the Rashba SOC and the intrinsic SOC can lead to the rich phenomenology. The Rashba SOC can drive the indirect and direct energy gap to close successively, but maintain the band touching between the valence band and the conduction band when the Rashba SOC is large enough to dominant the competition. We find that these touching points are located at $K$ and $K^{\prime}$ or/and some $2π/3$ rotationally symmetric points around $K$ and $K^{\prime}$ in the Brillouin zone. The…
We theoretically study a spin Hamiltonian for spin-orbit-coupled ferromagnets on the honeycomb lattice. We find that the effective Hamiltonian for magnons, a quanta of spin-wave excitations from ordered states, is equivalent to the Haldane model for electrons in the honeycomb lattice, which indicates the nontrivial topology of the magnon bands and the existence of the associated edge state. We also take the Schwinger boson (or bosonic spinon) representations of spins that are applicable both to ordered and disordered phases. We show that the spinon bands within the mean-field treatment are top…
Recent experimental success in the realization of two-dimensional (2D) magnetism has stimulated the search for new magnetic 2D materials with strong magnetic anisotropy and high Curie temperature. One promising subgroup of 2D magnetic systems are Dirac half-metals (DHM) which have gained a lot of interest recently, as they host a high-temperature quantum anomalous Hall effect (QAHE). This article discusses predictions for intrinsic DHMs and identifies them as realizations of the Kane-Mele Hubbard model at quarter filling. This proposed unification contributes to a firmer understanding of these…
We investigate the dependence of band dispersion of the quantum spin Hall effect (QSHE) edge states in the Kane-Mele model on crystallographic orientation of the edges. Band structures of the one-dimensional honeycomb lattice ribbons show the presence of the QSHE edge states at all orientations of the edges given sufficiently strong spin-orbit interactions. We find that the Fermi velocities of the QSHE edge-state bands increase monotonically when the edge orientation changes from zigzag (chirality angle $θ= 0^\circ$) to armchair ($θ= 30^\circ$). We propose a simple analytical model to explain …
Layered quasi two dimensional systems have garnered huge interest both in the advancement of technology and in understanding emergent physics such as unconventional superconductivity, topological phases. In particular, the study of topological properties in some bilayer systems like transition metal chalcogenides and iridates has been the point of attraction due to comparatively strong spin orbit coupling of transition metal ions. In this paper, we analyze the topological phases induced by the interplay of electron correlation and spin orbit coupling in different stacking orders of bilayer hon…
Recently, the frustrated XY model for spins-1/2 on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultra-cold atoms setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using the bosonic dynamical mean-field theory. Focussing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do…
We present homotopy theoretic and geometric interpretations of the Kane-Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to nonequivariant homotopy. We prove a Mayer-Vietoris Theorem for manifolds with $\mathbb{Z}_2$-actions which intertwines Real and $\mathbb{Z}_2$-equivariant de Rham cohomology groups, and apply it to derive a new localisation formula for the Kane-Mele invariant…
The two-dimensional kagome lattice hosts Dirac fermions at its Brillouin zone corners K and K', analogous to the honeycomb lattice. In the density functional theory electronic structure of ferromagnetic kagome metal Fe$_3$Sn$_2$, without spin-orbit coupling we identify two energetically split helical nodal lines winding along $z$ in the vicinity of K and K' resulting from the trigonal stacking of the kagome layers. We find that hopping across A-A stacking introduces a layer splitting in energy while that across A-B stacking controls the momentum space amplitude of the helical nodal lines. The …
The time-dependent state of a classical spin locally exchange coupled to an edge site of a Kane-Mele model in the topologically non-trivial phase is studied numerically by solving the full set of coupled microscopic equations of motion for the spin and the electron system. Dynamics in the long-time limit is accessible thanks to dissipative boundary conditions, applied to all but the zigzag edge of interest. We study means to control the state of the spin via transport of a spin-polarization cloud through the helical edge states. The cloud is formed at a distant edge site using a local magnetic…
We explore the phase diagram of the strongly correlated Hubbard model with intrinsic spin orbit coupling on the honeycomb lattice. We obtain the low energy effective model describing the spin degree of freedom. We study the resulting model within the Schwinger boson and Schwinger fermion approaches. The Schwinger boson approach gives the boundary between the spin liquid phase and the magnetically ordered phases, Neel order and incommensurate Neel order. We find that increasing the strength of the spin orbit coupling, narrows the width of the spin liquid region. The Schwinger fermion approach s…
We systematically explore the ground-state phase diagram of the Kane-Mele-Kondo lattice model on the honeycomb lattice, in particular, we focus on its magnetic properties which has not been studied in the previous publication[Feng, Dai, Chung, and Si, Phys. Rev. Lett. \textbf{111}, 016402 (2013)]. Beside the Kondo insulator found in that paper, two kinds of antiferromagnetic spin-density-wave phases are identified. One is the normal antiferromagnetic spin-density-wave state and the other is a nontrivial topological antiferromagnetic spin-density-wave state with a quantized spin Hall conductanc…
We investigate a second-order topological quantum transition of a modified Kane-Mele model driven by electron-phonon interaction. The results show that the system parameters of the bare modified Kane-Mele model are renormalized by the electron-phonon interaction. Starting from the second-order topological phase for the bare model, the increasing electron-phonon coupling strength can drive the second-order topological insulator into a semimetal phase. Such a secondorder topological phase transition is characterized by the band-gap closing, discontinuity of averaged ferminoic number and topologi…
The discovery by Kane and Mele of a model of spinful electrons characterized by a $\mathcal{Z}_2$ topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the band-like excitations of magnetic insulators, and recently models supporting $\mathcal{Z}_2$ topological invariants have been proposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)] excitations. In both cases, magnetic excitations form time-…
The two-dimensional Hubbard model defined for topological band structures exhibiting a quantum spin Hall effect poses fundamental challenges in terms of phenomenological characterization and microscopic classification. In the limit of infinite coupling U at half filling, the spin model Hamiltonians resulting from a strong coupling expansion show various forms of magnetic ordering phenomena depending on the underlying spin-orbit coupling terms. We investigate the infinite U limit of the Kane-Mele Hubbard model with z-axis intrinsic spin-orbit coupling as well as its generalization to a generica…
In this paper, we study the effects of attractive correlation on the topological insulator ($TI$) with topological flat-bands using an extended attractive Kane-Mele-Hubbard model (KMHM). In the KMHM, we found a quantum phase transition from $TI$ to the superconductor ($SC$) state upon the increasing of the attractive Hubbard interaction $U$ at the mean field level. This type of $SC$ phase transition is different from the traditional $SC$ phase transition which develops from the gapless Fermi Liquid. Cooperon-type gapped excitations exist in the $TI$ side near this type of $SC$ phase transition…
By using the cluster perturbation theory, we investigate the effects of the local electron-phonon interaction in the quantum spin Hall topological insulator described by the half-filled Kane-Mele model on an honeycomb lattice. Starting from the topological non trivial phase, where the minimal gap is located at the two inequivalent Dirac points of the Graphene, $\text{K}$ and $\text{K}'$, we show that the coupling with quantum phonons induces a topological-trivial quantum phase transition through a gap closing and reopening in the $\text{M}$ point of the Brillouin zone. The average number of fe…
The spin Hall effect is investigated in a two-orbital tight-binding model on a honeycomb lattice. We show that the model exhibits three topologically-different insulating phases at half filling, which are distinguished by different quantized values of the spin Hall conductivity. We analytically determine the phase boundaries, where the valence and conduction bands touch with each other with forming the Dirac nodes at the Fermi level. The results are discussed in terms of the effective antisymmetric spin-orbit coupling. The relation to the Kane-Mele model and implications for a magnetoelectric …
We study how topological phases evolve in the Kane-Mele-Kondo lattice at finite temperatures and obtain the topological Doniach phase diagram. In particular, we find an intriguing crossover behavior induced by the interplay between the topological structure and electron correlations; the topological properties are restored by temperature effects. This restoration can be observed in the behavior of the bulk as well as the edge. In the bulk, we observe an increase of the spin-Hall conductivity at finite temperatures, while it is zero in the low temperature region. At the edge, we observe gapless…
We investigate the effect of spin-orbit coupling on the behavior of magnetic impurity at the edge of a zigzag graphene ribbon by means of quantum Monte Carlo simulations. A peculiar interplay of Kane-Mele type spin-orbit and impurity-host coupling is found to affect local properties such as the impurity magnetic moment and spectral densities. The special helical nature of the topological insulator on the edge is found to affect nonlocal quantities, such as the two-particle and spin-spin correlation functions linking electrons on the impurity with those in the conduction band.
Na$_2$IrO$_3$ was one of the first materials proposed to feature the Kane-Mele type topological insulator phase. Contemporaneously it was claimed that the very same material is in a Mott insulating phase which is described by the Kitaev-Heisenberg (KH) model. First experiments indeed revealed Mott insulating behavior in conjunction with antiferromagnetic long-range order. Further refined experiments established antiferromagnetic order of zigzag type which is not captured by the KH model. Since then several extensions and modifications of the KH model were proposed in order to describe the expe…
A definition of topological phases of density matrices is presented. The topological invariants in case of both noninteracting and interacting systems are extended to nonzero temperatures. Influence of electron interactions on topological insulators at finite temperatures is investigated. A correlated topological insulator is described by the Kane-Mele model, which is extended by the interaction term of the Falicov-Kimball type. Within the Hartree-Fock and the Hubbard I approximations the thermodynamic and topological phase diagrams are determined, where the long-range order is included. The r…
The Kane-Mele (KM) model is proposed to describe the quantum spin Hall effect of electrons on the two-dimensional honeycomb lattice. Here, we will show that, in a certain parameter region, the London equation is obtained from the effective field theory of the layered KM model with an electronic correlation.
A monolayer of transition metal trichalcogenides has received a lot of attention as potential two dimensional magnetic materials. The system has a honeycomb structure of transition metal ions, where both spin-orbit coupling and electron correlation effect play an important role. Here, motivated by these transition metal series with effective doping or mixed valence case, we propose the possible realization of magnetic Chern insulators at quarter filled honeycomb lattice. We show that the interplay of intrinsic spin-orbit coupling and electron correlation opens a wide region of ferromagnetic Ch…
The valence flat bands in transition metal dichalcogenide (TMD) heterobilayers are shown to exhibit strong intralayer spin-orbit coupling. This is reflected in a simple tight-binding model with spin-dependent complex hoppings based on the continuum model. A perpendicular electric field causes interlayer hybridization, such that the effective model is equivalent to the Kane-Mele model of topological insulators. The proposed model can be used as a starting point to understand interactions and the experimentally observed topological phases.
The study of symmetry-protected topological states in presence of electron correlations has recently aroused great interest as rich and exotic phenomena can emerge. Here, we report a concrete example by employing large-scale unbiased quantum Monte Carlo study of the Kane-Mele model with cluster charge interactions. The ground-state phase diagram for the model at half filling is established. Our simulation identifies the coexistence of a symmetry-protected topological order with a symmetry-breaking Kekul$\acute{e}$ valence bond order and shows that the spontaneous symmetry-breaking is accompani…
