Bilayer Honeycomb Research Articles

Bona fide interaction-driven topological phase transition in correlated symmetry-protected topological states

It is expected that the interplay between non-trivial band topology and strong electron correlation will lead to very rich physics. Thus a controlled study of the competition between topology and correlation is of great interest. Here, employing large-scale quantum Monte Carlo (QMC) simulations, we provide a concrete example of the Kane-Mele-Hubbard (KMH) model on an AA stacking bilayer honeycomb lattice with inter-layer antiferromagnetic interaction. Our simulation identified several different phases: a quantum spin-Hall insulator (QSH), a $xy$-plane antiferromagnetic Mott insulator ($xy$-AFM) and an inter-layer dimer-singlet insulator (dimer-singlet). Most importantly, a bona fide topological phase transition between the QSH and the dimer-singlet insulators, purely driven by the inter-layer antiferromagnetic interaction is found. At the transition, the spin and charge gap of the system close while the single-particle excitations remain gapped, which means that this transition has no mean field analogue and it can be viewed as a transition between bosonic SPT states. At one special point, this transition is described by a $(2+1)d$ $O(4)$ nonlinear sigma model (NLSM) with {\it exact} $SO(4)$ symmetry, and a topological term at {\it exactly} $\Theta = \pi$. Relevance of this work towards more general interacting SPT states is discussed.

Bosonic edge states in gapped honeycomb lattices

By quantum Monte Carlo simulations of bosons in gapped honeycomb lattices, we show the existence of bosonic edge states. For single layer honeycomb lattice, bosonic edge states can be controlled to appear, cross the gap and merge into bulk states by an on-site potential applied on the outmost sites of the boundary. On bilayer honeycomb lattice, bosonic edge state traversing the gap at half filling is demonstrated. The topological origin of the bosonic edge states is discussed with pseudo Berry curvature. The results will simulate experimental studies of these exotic bosonic edge states with ultracold bosons trapped in honeycomb optical lattices.

Dynamic phase transitions and the effects of frequency of oscillating magnetic field on the dynamic phase diagrams in the bilayer honeycomb lattice with AB stacking geometry

We study some dynamic properties of the bilayer honeycomb lattice with AB stacking geometry in the presence of a time-dependent oscillating external magnetic field. We employ the Glauber transition rates to construct the mean-field dynamical equations. First, we obtain dynamic phases in the system and observe the paramagnetic (p), ferromagnetic (f), compensated (c) antiferromagnetic (af), surface ferromagnetic (sf) and mixed (m) phases. Besides, coexistence phase regions also exist in the system. Second, we investigate the thermal behavior of the dynamic order parameters. From these study, the natures (first- or second-order) of the transitions are characterized and the dynamic phase transition (DPT) points are presented. The DPTs are obtained and the dynamic phase diagrams (DPD) are constructed plane of the temperature versus the amplitude of the magnetic field. We investigate the effect of the frequency of the oscillating external magnetic field on the DPD.

High-temperature quantum anomalous Hall effect in honeycomb bilayer consisting of Au atoms and single-vacancy graphene.

The quantum anomalous Hall effect (QAHE) is predicted to be realized at high temperature in a honeycomb bilayer consisting of Au atoms and single-vacancy graphene (Au2-SVG) based on the first-principles calculations. We demonstrate that the ferromagnetic state in the Au2-SVG can be maintained up to 380 K. The combination of spatial inversion symmetry and the strong SOC introduced by the Au atoms causes a topologically nontrivial band gap as large as 36 meV and a QAHE state with Chern number C = −2. The analysis of the binding energy proved that the honeycomb bilayer is stable and feasible to be fabricated in experiment. The QAHEs in Ta2-SVG and other TM2-SVGs are also discussed.

Magnetic hysteresis, compensation behaviors, and phase diagrams of bilayer honeycomb lattices

Magnetic behaviors of the Ising system with bilayer honeycomb lattice (BHL) structure are studied by using the effective-field theory (EFT) with correlations. The effects of the interaction parameters on the magnetic properties of the system such as the hysteresis and compensation behaviors as well as phase diagrams are investigated. Moreover, when the hysteresis behaviors of the system are examined, single and double hysteresis loops are observed for various values of the interaction parameters. We obtain the L-, Q-, P-, and S-type compensation behaviors in the system. We also observe that the phase diagrams only exhibit the second-order phase transition. Hence, the system does not show the tricritical point (TCP).

The Dynamic Hysteresis Curves and Compensation Types of Kinetic Bilayer Honeycomb Lattice System with AB Stacking Geometry

The dynamic magnetic properties in a kinetic spin-1/2 bilayer honeycomb lattice (BHL) system with AB stacking geometry under a time-dependent oscillating external magnetic field for both ferromagnetic and antiferromagnetic interactions are studied within the mean-field dynamical equations and the Glauber-type stochastic dynamics approach. First, we study the thermal behavior of the dynamic order parameters. Then, we investigate the temperature dependence of the dynamic total magnetization to ?nd the dynamic compensation points as well as to determine the type of behavior. We also examine the dynamic hysteresis behaviors of the system. Some characteristic phenomena are found depending on the ratio of the physical parameters and frequency of oscillating magnetic field. According to the values of Hamiltonian parameters P-, L-, S-, R-, Q- and N-type compensation behaviors and square-like and S-shaped single hysteresis loops as well as double and triple hysteresis loops exist in the system. Our results are in agreement with some experimental works in recent literature.

Exotic quantum phase transitions of strongly interacting topological insulators

Using determinant quantum Monte Carlo (d-QMC) simulations, we demonstrate that an extended Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin $S^z$ conservation and the previously mentioned strongly interacting fully gapped phase. At the latter quantum critical point the single particle excitations remain gapped, while spin and charge gap both close. We argue that the first quantum phase transition is related to the $\mathbb{Z}_{16}$ classification of the topological superconductor $^3\text{He-B}$ phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a $\Theta$-term.

Self consistent study of the quantum phases in a frustrated antiferromagnet on the bilayer honeycomb lattice

We study the frustrated Heisenberg model on the bilayer honeycomb lattice. The ground-state energy and spin gap are calculated, using different bosonic representations at mean field level and numerical calculations, to explore different sectors of the phase diagram. In particular we make use of a bond operator formalism and series expansion calculations to study the extent of dimer inter-layer phase. On the other hand we use the Schwinger boson method and exact diagonalization on small systems to analyze the evolution of on-layer phases. In this case we specifically observe a phase that presents a spin gap and short range Neel correlations that survives even in the presence of non-zero next-nearest-neighbor interaction and inter-layer coupling.

Electronic instabilities of the AA‐honeycomb bilayer

A functional renormalization group approach is used to study the instabilities due to electron‐electron interactions in a bilayer honeycomb lattice model with AA stacking, as it might be relevant for layered graphene with this structure. Starting with a tight‐binding description for the four π‐bands, the modes of the dispersion are integrated out by successively lowering an infrared cutoff and the leading tendencies in the effective interactions are determined. The antiferromagnetic spin‐density wave is an expected instability for dominant local repulsion among the electrons, but for nonlocal interaction terms also other instabilities occur. The phase diagrams depending on the model parameters are discussed. The results are compared to single‐layer graphene and the more common AB‐stacked bilayer, both qualitatively and quantitatively.

Layer anti-ferromagnetism on bilayer honeycomb lattice.

Bilayer honeycomb lattice, with inter-layer tunneling energy, has a parabolic dispersion relation, and the inter-layer hopping can cause the charge imbalance between two sublattices. Here, we investigate the metal-insulator and magnetic phase transitions on the strongly correlated bilayer honeycomb lattice by cellular dynamical mean-field theory combined with continuous time quantum Monte Carlo method. The procedures of magnetic spontaneous symmetry breaking on dimer and non-dimer sites are different, causing a novel phase transition between normal anti-ferromagnet and layer anti-ferromagnet. The whole phase diagrams about the magnetism, temperature, interaction and inter-layer hopping are obtained. Finally, we propose an experimental protocol to observe these phenomena in future optical lattice experiments.

Excitonic and superconducting orders from repulsive interaction on the doped honeycomb bilayer

Using a weak-coupling renormalization group formalism, we study competing ordered phases for repulsively interacting fermions on the bilayer honeycomb lattice away from half-filling, which is realized experimentally as doped bilayer graphene. As electrons are added to the system, excitonic order is suppressed, and unconventional superconductivity appears generically in its place. In general it is found that the maximum critical temperature for superconductivity appears directly adjacent to the dome of particle-hole order, illustrating the importance of fluctuations in these channels for the formation of unconventional superconductivity. We obtain the phase diagram showing characteristic ordering temperatures for both short- and long-ranged interactions, and show that the most likely superconducting instabilities occur in $d$-wave, $f$-wave, and pair density wave channels. The nature of and competition between these phases are further analyzed using both free energy expansion and self-consistent mean-field theory. The effects of finite temperature and trigonal warping due to further-neighbor hopping are studied, and implications for experiments on bilayer graphene are discussed.

Superconductivity on the brink of spin-charge order in a doped honeycomb bilayer.

Using a controlled weak-coupling renormalization group approach, we establish the mechanism of unconventional superconductivity in the vicinity of spin or charge ordered excitonic states for the case of electrons on the Bernal stacked bilayer honeycomb lattice. With one electron per site, this system, physically realized in bilayer graphene, is unstable towards a spontaneous symmetry breaking. Repulsive interactions favor excitonic order, such as a charge nematic and/or a layer antiferromagnet. We find that upon adding charge carriers to the system, the excitonic order is suppressed, and unconventional superconductivity appears in its place, before it is replaced by a Fermi liquid. We focus on firmly establishing this phenomenon using the renormalization group formalism within an idealized model with parabolic touching of conduction and valence bands.

Quantum phases in the frustrated Heisenberg model on the bilayer honeycomb lattice

We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger boson description of the spin operators followed by a mean field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling $J_{2}$ and the interlayer coupling $J_{\bot}$. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization and spin-spin correlations. We observe a phase with a spin gap and short range N\'eel correlations that survives for non-zero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of N\'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence bond crystal phase to an interlayer valence bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin wave approach to study the phase diagram as a function of the spin $S$, the frustration and the interlayer couplings.

Stepwise tuning of the substituent groups from mother BTB ligands to two hexaphenylbenzene based ligands for construction of diverse coordination polymers

Hexaphenylbenzene based tritopic carboxylate ligands H3L1 (1,3,5-tris(4-carboxyphenyl)-2,4,6-tris(4-tert-butylphenyl)benzene) and H3L2 (1,3,5-tris(4-carboxyphenyl)-2,4,6-triphenylbenzene) were applied to synthesize six coordination polymers (CPs) {[Zn4(L1)2(μ2-OH)2(DMA)4(H2O)]·(C2H6O)2(H2O)}n (1), {[Cd3(L1)2(μ2-H2O)2(H2O)4(DMF)2(C2H6O)]·(DMF)(H2O)(C2H6O)2}n (2), {[Cd(L1)(C2H6O)]·(H3O+)(DMF)(C2H6O)}n (3), {[Co3(L1)2(μ2-H2O)(H2O)6(DMF)2(C2H6O)]·(DMF)3}n (4), {[Mn3(L1)2(H2O)5(DMF)4(C2H6O)]·(C2H6O)}n (5), and {[Zn2(L2)Cl(H2O)(DMA)]·(DMA)2(H2O)}n (6) (DMF = N,N-dimethylformamide, DMA = N,N-dimethylacetamide). The CPs were characterized by single crystal X-ray diffraction, infrared spectra (IR), elemental analyses, powder X-ray diffraction (PXRD), and thermogravimetric analyses (TGA). 1 and 2 exhibit two-dimensional (2D) honeycomb bilayer structures based on tetranuclear zinc or trinuclear cadmium clusters. 3 is PH-induced supramolecular isomer with 2, featuring 2D corrugated honeycomb structure. In 4 and 5, only two carboxylate groups coordinate with trinuclear metal clusters to form 1D coordination chains. The chains are further connected through hydrogen bonds between the left carboxylate groups and coordination water molecules to form honeycomb bilayer sheets. 6 exhibits 2D planar honeycomb structure based on Zn2(COO)3 clusters. The different topological networks 1 and 6 are constructed from structurally similar ligands H3L1 and H3L2, showing that structural diversity could be realized by adjusting substituent groups on organic ligands. The photoluminescent behaviours of compounds 1, 2, and 6 and magnetic properties of 4 and 5 were also investigated.

D-wave superconductivity on the honeycomb bilayer

We introduce a microscopic model on the honeycomb bilayer, which in the small-momentum limit captures the usual (quadratic dispersion in the kinetic term) description of bilayer graphene. In the limit of strong interlayer hopping it reduces to an effective honeycomb monolayer model with also third-neighbor hopping. We study interaction effects in this effective model, focusing on possible superconducting instabilities. We find ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity in the strong-coupling limit of an effective $tJ$-model-like description that gradually transforms into $d+id$ time-reversal symmetry-breaking superconductivity at weak couplings. In this limit the small-momentum order-parameter expansion is ${({k}_{x}+i{k}_{y})}^{2}$ [or ${({k}_{x}\ensuremath{-}i{k}_{y})}^{2}$] in both valleys of the effective low-energy description. The relevance of our model and investigation for the physics of bilayer graphene is also discussed.

Excitonic instabilities and insulating states in bilayer graphene

The competing ground states of bilayer graphene are studied by applying renormalization group techniques to a bilayer honeycomb lattice with nearest neighbor hopping. In the absence of interactions, the Fermi surface of this model at half-filling consists of two nodal points with momenta $\mathbf{K}$, ${\mathbf{K}}^{\ensuremath{'}}$, where the conduction band and valence band touch each other, yielding a semimetal. Since near these two points the energy dispersion is quadratic with perfect particle-hole symmetry, excitonic instabilities are inevitable if interband interactions are present. Using a perturbative renormalization group analysis up to the one-loop level, we find different competing ordered ground states, including ferromagnetism, superconductivity, spin and charge density wave states with ordering vector $\mathbf{Q}=\mathbf{K}\ensuremath{-}{\mathbf{K}}^{\ensuremath{'}}$, and excitonic insulator states. In addition, two states with valley symmetry breaking are found in the excitonic insulating and ferromagnetic phases. This analysis strongly suggests that the ground state of bilayer graphene should be gapped, and with the exception of superconductivity, all other possible ground states are insulating.

An equivalence between monolayer and bilayer honeycomb lattices

In this brief report, we show the equivalence between the tight-binding descriptions of the monolayer and bilayer honeycomb lattices. With appropriate value of the third nearest neighbors coupling, the Hamiltonian for a monolayer is equivalent to the low energy effective Hamiltonian for bilayer in the presence of trigonal warping. A simple physical argument is provided to explain this correspondance.

Interacting electrons on trilayer honeycomb lattices

Few-layer graphene systems come in various stacking orders. Considering tight-binding models for electrons on stacked honeycomb layers, this gives rise to a variety of low-energy band structures near the charge neutrality point. Depending on the stacking order these band structures enhance or reduce the role of electron-electron interactions. Here, we investigate the instabilities of interacting electrons on honeycomb multilayers with a focus on trilayers with ABA and ABC stackings theoretically by means of the functional renormalization group. We find different types of competing instabilities and identify the leading ordering tendencies in the different regions of the phase diagram for a range of local and non-local short-ranged interactions. The dominant instabilities turn out to be toward an antiferromagnetic spin-density wave (SDW), a charge density wave and toward quantum spin Hall (QSH) order. Ab-initio values for the interaction parameters put the systems at the border between SDW and QSH regimes. Furthermore, we discuss the energy scales for the interaction-induced gaps of this model study and put them into context with the scales for single-layer and Bernal-stacked bilayer honeycomb lattices. This yields a comprehensive picture of the possible interaction-induced ground states of few-layer graphene.

Fermions on bilayer graphene: Symmetry breaking forB=0andν=0

We extend previous analyses of fermions on a honeycomb bilayer lattice via weak-coupling renormalization group (RG) methods with extremely short-range and extremely long-range interactions to the case of finite-range interactions. In particular, we consider different types of interactions including screened Coulomb interactions, much like those produced by a point charge placed either above a single infinite conducting plate or exactly halfway between two parallel infinite conducting plates. Our considerations are motivated by the fact that, in some recent experiments on bilayer graphene there is a single gate while in others, there are two gates, which can function as the conducting planes and which, we argue, can lead to distinct broken symmetry phases. We map out the leading instabilities of the system as its temperature is lowered as a function of the range of the interaction. We discover that the system is unstable towards an antiferromagnetic phase for short ranges of the interaction and towards a nematic phase at long ranges, in agreement with previous studies. While the antiferromagnetic phase results in a gap in the spectrum, the nematic phase is gapless, splitting the quadratic degeneracy points into two Dirac cones each. We also consider the effects of an applied magnetic field on the system in the antiferromagnetic phase via variational mean field theory. At low fields, we find that the antiferromagnetic order parameter, $\ensuremath{\Delta}(B)\ensuremath{-}\ensuremath{\Delta}(0)\ensuremath{\sim}{B}^{2}$. At higher fields, when ${\ensuremath{\omega}}_{c}\ensuremath{\gtrsim}2{\ensuremath{\Delta}}_{0}$, we find that $\ensuremath{\Delta}(B)\ensuremath{\approx}{\ensuremath{\omega}}_{c}/[\mathrm{ln}({\ensuremath{\omega}}_{c}/\ensuremath{\Delta}(0))+C]$, where $C\ensuremath{\approx}0.67$ and ${\ensuremath{\omega}}_{c}=eB/{m}^{*}c$. We also determine the energy gap for creating electron-hole excitations in the system, and, at high fields, we find it to be $a{\ensuremath{\omega}}_{c}+2\ensuremath{\Delta}(B)$, where $a$ is a nonuniversal, interaction-dependent, constant.

Antiferromagnetism in the Hubbard Model on the Bernal-Stacked Honeycomb Bilayer

Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin moments in the ordered ground state, with enhanced moments at the threefold coordinated sites. Furthermore, we find the antiferromagnetic ground state to be robust with respect to enhanced interlayer couplings and extended Coulomb interactions.