Skyrme Crystal in bilayer and multilayer graphene
Abstract
The ground state of the twodimensional electron systems in Bernal bilayer and ABCstacked multilayer graphenes in the presence of a strong magnetic field is investigated with the HartreeFock approximation. Phase diagrams of the systems are obtained, focusing on charge density wave states including states with vortices of valley pseudospins (called a Skyrme crystal). The singleelectron states in these stacked graphenes are given by twocomponent wave functions. That of the first excited Landau level has the same component as the lowest Landau level of the ordinary twodimensional electrons. Because of this localized wave functions, the Skyrme crystal has low energy in this first excited level up to four layers of graphene, when the interlayer distance is assumed to be infinitesimal. At the same time, bubble crystals are suppressed, so the phase diagram is different from that of a singlelayer graphene.
pacs:
73.21.b, 73.22.Gk, 73.22.PrI introduction
Over the past decades, the conventional twodimensional electron system (2DES) in semiconductor heterostructures in a strong magnetic field has been studied. It is found that electronelectron interaction brings various phases [e.g., fractional quantum Hall effect (FQHE) states PhysRevLett.48.1559 (); PhysRevLett.50.1395 () and charge density wave (CDW) states PhysRevB.19.5211 ()]. Similarly, graphene, RevModPhys.81.109 () a flat sheet of carbons with a honeycomb lattice exfoliated from a graphite in 2004, Sci.306.666 () is under intensive investigation as a new 2DES whose electron has a valley degree of freedom () and a linear dispersion. Exhibition of FQHE PhysRevLett.97.126801 (); JPSJ.78.104708 () and CDW PhysRevB.75.245414 (); PhysRevB.78.085309 () states in the new 2DES has been expected; FQHE was recently observed in a singlelayer graphene (SLG). nat462.192 (); nat462.196 () Furthermore, fewstacked graphenes have recently attracted attention due to their intriguing properties: a band structure tunable by the number of layers and their stacking sequence, and a band gap controllable by a perpendicular electrical field. PhysRevB.77.155416 (); ProgTheorPhysSuppl.176.227 ()
We focus on Bernal bilayer and ABCstacked multilayer graphenes, which have chiral electrons concentrated on outer layers in lowenergy states. In this paper, the CDW ground state of the 2DES in bilayer graphene (BLG) and multilayer graphene (LG, ) in a strong magnetic field is studied with the HartreeFock approximation. As candidates of the ground state, the electron (hole) Wigner crystal, electron (hole) bubble crystal, and Skyrme crystal are considered. A Skyrme crystal is the state with a topological texture of valley pseudospins. It has the lowest energy around a filling in SLG. PhysRevB.78.085309 () Our meanfield analysis predicts that the valley Skyrme crystal also has low energy in BLG and LG () at the first excited Landau level when the interlayer distance is assumed to be infinitesimal.
This paper is organized as follows. In the next section, we set up a lowenergy effective Hamiltonian in the presence of a magnetic field for SLG, BLG, and LG. Then, the HartreeFock approximation is applied to the system of interacting electrons and selfconsistent equations are derived. In Sec. III, CDW states are introduced, including Skyrme crystals (more properly, the meron and meron pair crystal). As a preliminary calculation, the energy of a skyrmion (antiskyrmion) pair excitation at a ferromagnetic state is evaluated and compared with that of a separated particlehole excitation to find the condition where the Skyrme crystal has low energy. In Sec. IV, numerical results are presented at the Landau level where the Skyrme crystal is expected. In Sec. V, the validity of the results and a relation to experiments are discussed.
Ii Model
ii.1 Singleparticle state in a magnetic field
For a singlelayer graphene (SLG), the lowenergy effective Hamiltonian around the valley is given by
(1) 
which has an electron with a linear dispersion. RevModPhys.81.109 (); PhysRev.71.622 (); PhysRev.109.272 (); PhysRevLett.53.2449 (); PhysRevLett.61.2015 (); JPSJ.74.777 (); PhysRevB.73.125411 () Here, is Fermi velocity, and and are Pauli matrices acting on sublattice (A, B) space. The Hamiltonian for the valley is . In a perpendicular magnetic field , using a magnetic ladder operator (), the singleparticle Hamiltonian is written as
(2) 
where is magnetic length.
The eigenenergies are
(3) 
where a positive (negative) sign is taken as the electron (hole) state. The corresponding eigenstates have the form
(4) 
for the valley. In the Landau gauge, is given by
(5) 
where is the length of the system, is the Hermite polynomial, and is the guiding center coordinate. A macroscopic number of states with different are degenerated at one Landau level. A singleparticle density on each sublattice becomes broader as index increases. The states with broad density induce a bubble CDW and lose the profit to form skyrmions (see Sec. III).
Bernalstacked bilayer graphene (BLG) and ABCstacked multilayer graphene note1 () (LG) have lowenergy quasiparticles with a dispersion , where ( ) is the number of layers. PhysRevB.77.155416 (); ProgTheorPhysSuppl.176.227 (); PhysRevB.73.214418 (); PhysRevLett.96.086805 () In two adjacent layers, BLG and LG have interlayer hoppings between the sublattice in the lower layer and the sublattice in the upper layer. In a magnetic field, the effective singleparticle Hamiltonian for BLG and LG, which act on the outermost layer (top, bottom) space, have the same form
(6) 
where offdiagonal components represent times hopping via highenergy states in the inner layers. Its eigenenergy is
(7) 
where . At zero energy, Landau levels are degenerated. The corresponding eigenstate has the form
(8) 
for the valley. Notice that in the first excited state realized at , the upper component of the wave function is given by , which is the groundstate wave function of the ordinary 2 electrons.
At Landau level , the valley degree of freedom coincides with the layer degrees of freedom. If the layer degree of freedom in the LG model is regarded as the sublattice degree of freedom, Eq. (8) is formally identical to that of the SLG model when . In the following, the case of () represents the model of SLG (BLG).
ii.2 HartreeFock Hamiltonian
Although the electron state is mainly considered in the following, the hole state can be treated in the same way. In the present case of strong magnetic fields, electronic spins are completely polarized and the gap around the neutrality point is sufficiently large due to exchange enhancement; thus we take only one spin component and ignore the effect of Landaulevel transitions.
The interaction between the electrons in different layers is weakened by an interlayer distance ( Å for BLG). When T, is quite small compared to magnetic length Å, so we consider the vanishing limit of interlayer distance as an approximate model.
Apart from a constant kinetic term, the Hamiltonian for interacting electrons in LG () is given by
(9) 
where the field operator is represented by
(10) 
and Coulomb interaction is written as
(11) 
Here, is the dielectric constant, represents valley and , respectively, is an annihilation operator of the electron at valley with guiding center . A valley scattering term is relatively small when the cutoff is used, PhysRevB.75.245414 (); PhysRevB.78.085309 (); PhysRevB.74.161407 () since ( Å is the lattice constant), so we can ignore it as far as the CDW with magnetic length scale is concerned. Then, the Hamiltonian in space is written as
(12) 
(13) 
(14) 
where is the Laguerre polynomial, and is defined by and . When Landaulevel index is smaller than the number of layers , the valley degree of freedom coincides with that of the layers (sublattices for SLG), and and and are twice the amount of Eq. (13).
For the HartreeFock decoupling of , we assume the following order parameters:
(15) 
where is a reciprocal lattice vector of the CDW state. Then the HartreeFock Hamiltonian is given by PhysRevLett.65.2662 (); PhysRevB.44.8759 (); PhysRevB.46.10239 (); PhysRevB.45.11054 ()
(16) 
where . The HartreeFock potential consists of a direct term
(17) 
(18) 
and an exchange term
(19) 
(20) 
Here, is a Bessel function of the first kind.
The realspace density of electrons at valley and layer (sublattice) is given by
(21) 
The filling factor at Landau level is defined as
(22) 
where ( is a area of the system) is the degeneracy of Landau orbitals and is the number of electrons. The factor 2 comes from the valley degree of freedom.
ii.3 Green’s function method
To determine the order parameters selfconsistently, the Green’s function method is used. PhysRevB.75.245414 (); PhysRevB.78.085309 (); PhysRevB.44.8759 (); PhysRevB.46.10239 () The singleparticle Matsubara Green’s function is defined by
(23) 
The Fourier transformation
(24) 
relates to the order parameters by
(25) 
The equation of motion for is derived as
(26) 
from the Heisenberg equation. Here, is the Matsubara frequency for fermions, and the selfenergy is given by
(27) 
To solve this selfconsistent equation, we diagonalize the selfenergy matrix
(28) 
where is the th eigenvector with eigenvalue . The order parameters are obtained from the eigenvectors and eigenvalues
(29) 
where is the FermiDirac distribution function. The chemical potential is determined from
(30) 
Selfconsistent equations are numerically calculated to yield the order parameters and the HartreeFock energy per particle for several CDW states introduced in Sec. III.
The order parameter sum rule at zero temperature, PhysRevB.27.4986 () extended to the case of valley degeneracy
(31) 
is easily derived from Eq. (29). Here, is the contribution from valley electrons to the partial filling factor . This relation is used to check convergence of the results.
Iii Charge Density Wave States
iii.1 Valley skyrmion
It is useful to map the valley degrees of freedom to a pseudospin. PhysRevB.78.085309 () In this language, the components of the pseudospin vector density are defined by
(32) 
(33) 
(34) 
In this paper, states with a topological pseudospin texture, which is called a skyrmion, SandI () are considered. A skyrmion is a kind of spin texture usually used to describe a magnetic order and was first introduced in hadron physics. NuclPhys.31.556 () For conventional and SLG 2DES, a state with aligned skyrmions, which is called a Skyrme crystal, has been shown theoretically to be the ground state around . PhysRevB.78.085309 (); SurfSci.361 ()
iii.2 Preliminary considerations
The excitation energy of a skyrmion in a ferromagnetic uniform state at is evaluated to find the conditions on which the Skyrme crystal is preferred. PhysRevB.51.5138 (); PhysRevB.74.075423 () When the interlayer distance , the energy of a skyrmion (antiskyrmion) pair excitation and the energy of a widely separated particlehole pair excitation are given by
(35) 
(36) 
where the form factor has the form
(37) 
The ratio of the two energies of the pair excitations is the same as that of singleparticle excitations, from particlehole symmetry. For conventional, SLG, BLG, and LG () 2DESs, these energies at Landau level are presented in Table 1. It shows that skyrmion excitation is favored (1) at in all systems as singleparticle wave functions being identical to conventional one, (2) at in SLG, and (3) at in LG (). Thus we can expect that the Skyrme crystal becomes the ground state in these situations around .
0  1  1/2  1  1/2  1  1/2  
1  0.75  0.875  0.6875  0.2188  0.75  0.875  
2  0.6406  1.1328  0.5664  0.3301  0.5977  0.3770  
3  0.5742  1.3418  0.5029  0.4097  0.5029  0.5151  
4  0.5279  1.5522  0.4608  0.4754  0.4528  0.6181  
5  0.4927  1.6834  0.4298  0.5328  0.4187  0.7048 
0  1  1/2  1  1/2  1  1/2  
1  0.75  0.875  0.75  0.875  0.75  0.875  
2  0.6406  1.1328  0.6406  1.1328  0.6406  1.1328  
3  0.5498  0.4448  0.5742  1.3418  0.5742  1.3418  
4  0.4660  0.5808  0.5285  0.4870  0.5279  1.5522  
5  0.4223  0.6829  0.4406  0.6284  0.4963  0.5390 
The reason why a skyrmion can be a lowenergy excitation is the following. QHE () The ground state at is a pseudospin ferromagnetic liquid state. When a hole is introduced in this state without flipping pseudospin of other electrons, the charge density of the hole is given by an eigenstate of the angular momentum, and is concentrated. On the other hand, when introduction of a hole is accompanied by pseudospinflip of other electrons, many states with the same angular momentum are connected by the Coulomb interaction, and the charge density of the hole has a wider distribution. This connected quantum state corresponds to a skyrmion. When the charge is confined like a wave function given by Eq. (5), the formation of a skyrmion reduces the charge locality. At high Landau level , however, a wave function is intrinsically broad, so the benefit to form skyrmions is lacking. In layered graphene, the spinor wave function has the localized component at Landau level , so it can drive the system to form skyrmions.
iii.3 Crystal structure
When Zeeman energy for valley pseudospins does not exist, a skyrmion splits into two merons (halfskyrmions). Four textures of a meron are possible from the two direction at center and the two vorticities.
Charge density wave states where electrons, holes, or merons form a triangular or square lattice structure are considered. The order parameters are defined at points
(38) 
(39) 
where and are integers. The following states are assumed:

Electron Wigner crystal (eWC) and electron bubble crystal (eBC): a triangular or square lattice with one or electrons per unit cell. The fundamental length in space is determined from the condition that the CDW has electrons in a unit cell: . Here, is the filling factor of electrons and is the area of a unit cell. The lowenergy state is pseudospin ferromagnetic because of the Pauli principle.

Hole Wigner crystal (hWC) and hole bubble crystal (hBC): a triangular or square lattice with one or holes per unit cell. The fundamental length in space is determined from the condition that the CDW has holes in a unit cell: . Here, is the filling factor of holes and is the area of a unit cell. The lowenergy state is pseudospin ferromagnetic because of the Pauli principle.

Meron crystal (MC): a square lattice with four merons of charge () or () per unit cell, equally spaced. A meron pair is equivalent to one skyrmion, so MC can be seen as a state with two skyrmions per unit cell. Thus is determined from the condition that the CDW has () skyrmions in a unit cell: . The component of pseudospin density in real space and the vorticity alternate from one site to the next (Figs. 2 and 2). The orientation of pseudospins has symmetry (Fig. 2). The density distribution in real space is bipartite in layers.

Meron pair crystal (MPC): a triangular lattice with four merons per unit cell. The merons are not equally spaced and bound into pairs. The is determined from the same way as MC. The energy of MPC is similar to MC, and has slightly lower energy in the low quasiparticle density regime (close to ) in general.
Iv Results
We use the partial filling factor at Landau level ; thus the total filling factor is given by for anylayered graphene. Assume that Zeeman splitting is sufficiently large, so the phase diagram for is identical to that for . Furthermore, the Hamiltonian has electronhole symmetry around , so the phase diagram for is caught by alternating particles for to antiparticles. Numerical calculation for MC and MPC are done at , since too many wave vectors are needed to get wellconverged solutions at . In the following results, the energies with an accuracy of are presented. Wigner and bubble crystals are calculated only in triangular symmetry; a square lattice generally has higher energy than a triangular one in a low quasiparticle density regime.
It is difficult to get MC solutions close to , so we extrapolate the order parameters of the MC solutions in . To execute the extrapolation, the quantity is fitted by a quadratic curve. The energies of the extrapolated MC states are represented in the following figures as a dotted line. It is noted that uncertainty remains in the extrapolation especially in the 4LG case. It comes from the relatively low validity of fitting the order parameters which have an inflection point near .
The energies of the valleyconcentrated hole Wigner crystal states which are quite accurately approximated by Gaussian form order parameters
(40) 
are represented as “GhWC” in the following figures. In the vanishing interlayer distance limit, the GhWC state with is degenerated to the hWC solutions with (Figs. 4, 6, 8).
In the following we show phase diagrams obtained by the present HF approximation in the whole range of . It should be remarked that the true phase diagram should contain regions of incompressible liquid states that cannot be obtained by the HF approximation. Thus the phase diagrams are partly incorrect. However, HF calculation gives qualitatively correct results when the fractional quantum Hall states do not appear. This is established from comparisons with the results by the exact diagonalization method PhysRevLett.50.1219 (); PhysRevB.29.6833 (); PhysRevLett.100.116802 () or the density matrix renormalization group method. PhysRevLett.86.5755 () In this paper we focus on the possibility of meron crystals near , where the liquid states are not expected, but only the chargeordered states compete. Therefore, the following discussion as to the realization of the meron crystal is reliable.
iv.1 Singlelayer graphene
The HartreeFock (HF) phase diagram of the 2DES in SLG has been obtained and compared with that of the conventional one. PhysRevB.75.245414 () It is shown that Skyrme crystals (MC and MPC) become the ground state around at Landau levels and . PhysRevB.78.085309 () Considering the excitation energies for SLG (Table 1), it is also possible for the Skyrme crystal phase to occur at and , but this has not been found yet in a meanfield calculation. Although the same HF calculation had been done for SLG, PhysRevB.75.245414 (); PhysRevB.78.085309 () we executed additional checks and investigated the higher filling regime.
At Landau level , the phase diagram is the following: eWC for , hWC for , MC for , and MPC for PhysRevB.78.085309 ()
At Landau level , the phase diagram is the following: eWC for , hWC for , MC for , and MPC for . PhysRevB.78.085309 () The range of a skyrmionic (MC or MPC) phase is narrower than that of the case.
At Landau level , the phase diagram is the following: eWC for , eBC2 for , hBC2 for , and hWC for . It is characteristic that 2electron (hole) bubble crystals exist around . The skyrmionic ground state is not seen in the range . The extrapolating analysis, however, suggests that the hWC and MC state are almost degenerated in .
At Landau level , the phase diagram is the following: eWC for , eBC2 for , eBC3 for , hBC3 for , hBC2 for , and hWC for . The skyrmionic state does not appear in .
Although the MC and MPC solutions are not found in for SLG at and , the Skyrme crystal is expected to have lower energy in the immediate vicinity of from the analysis in Sec. III.2.
iv.2 Landau level in bilayer graphene
The HF calculation for degenerated zeroenergy Landau levels in BLG suggests that the Skyrme crystal states of real spin or orbital pseudospin occur. PhysRevB.82.245307 ()
In what follows, the results for the first excited Landau level in BLG are presented. Figure 4 shows the energies per electron for several crystal structures. It shows the following sequence of ground states: eWC for , hWC for . The bubble state does not appear unlike the phase diagram at Landau level for SLG. Figure 4 shows the energies in the area close to . The extrapolated energy of the MC solutions have lower value than hWC for .
iv.3 Landau level in trilayer graphene
Figure 6 shows the energies per electron for several crystal structures at Landau level in triLG. It shows the following sequence of ground states: eWC for , eBC2 for , hBC2 for , hWC for . Although the bubble states are found around , its range is narrower than that of the case in SLG. The skyrmionic ground state is not seen in the range . Figure 6 shows the energies near . The extrapolated states of the MC solutions have an energy close to that of hWCs in the vicinity of .
iv.4 Landau level for tetralayer graphene
Figure 8 shows the energies per electron for various crystal structures at Landau level for tetraLG. It shows the following sequence of ground states: eWC for , eBC2 for , hBC2 for , and hWC for . The bubble states are found in the broad range . The skyrmionic ground state is not seen in the range . Figure 8 shows the energies near . The extrapolated states of the MC solutions have higher energy than that of hWCs in the vicinity of . As previously mentioned, however, the extrapolation method is no longer valid in tetraLG. According to the analysis in Sec. III.2, the Skyrme crystal is expected to have the lowest energy in the vicinity of .
V Discussion
The HartreeFock (HF) calculation suggests that the meron crystal (MC) phase appears at Landau level in BLG, and hole Wigner crystal (hWC) and MC states are degenerated around at Landau level in SLG, in triLG in the case of vanishing interlayer distance. Our calculation strongly suggests that meron pair crystals (MPCs) will have lower energy in the immediate vicinity of for its triangular symmetry.
Skyrme crystals (MC and MPC) have a charge distribution and collective mode different from that of hWC, so these states can be distinguished by transport properties PhysRevLett.65.2189 (); PhysRevLett.82.394 () and a microwave absorption spectrum. PhysRevB.78.085309 (); PhysRevLett.104.226801 () Furthermore, the CDWs exist in outer layers, so the local density of states (LDOS) can be measured in a spectroscopic manner. PhysRevB.80.195414 () In this paper, the unidirectional stripe phase is not considered. The stripe phase, however, also will appear around in BLG and LG, as SLG PhysRevB.75.245414 () and conventional 2DES if fractional quantum Hall states are not realized. Such a state, if exists, will be identified by anisotropic conduction. PhysRevLett.82.394 ()
Although we ignored the effects of disorder, a finite valley Zeeman energy, and Landaulevel transitions, it is unclear how these affect degeneracy of hWC and MC around . In particular, Landaulevel transitions in BLG and LG are larger than that of SLG under a magnetic field T. The gap near a charge neutrality point is for SLG, for BLG, for triLG, and for tetraLG. The typical Coulomb energy is . For SLG, the ratio is independent of field . In this case it is shown that the Landaulevel mixing does not change the CDW phase diagram (except the skyrmionic crystal). PhysRevB.77.205426 () For BLG and LG, high magnetic fields are needed to achieve a comparable ratio: for BLG, for triLG, and for tetraLG.
It has been pointed out that anisotropy in the pseudospin arises in the order of . PhysRevB.74.161407 (); PhysRevB.74.075422 (); PhysRevLett.98.196806 (); PhysRevLett.98.016803 (); RevModPhys.83.1193 () In the multilayer graphene, finite layer separation also brings anisotropy. This anisotropy is quite small, since , but may have some effect when the energies of two phases are quite close. We have done a calculation taking into account only the effect of as a preliminary investigation, and found that the finite is slightly unfavorable for the Skyrme crystal. However, to obtain a definite conclusion for the effect of finite , we need to take into account the effect of also. Such calculation is left for future investigation.
Acknowledgements.
Y.S. thanks R. Côté for helping him to find selfconsistent solutions to MC states. The numerical calculation was done by SR11000 at Information Technology Center, University of Tokyo.References
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