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Sci Rep. 2015; 5: 9810.
Published online 2015 May 11. doi:  10.1038/srep09810
PMCID: PMC5386193

Phase transitions of the ionic Hubbard model on the honeycomb lattice

Abstract

Many-body problem on the honeycomb lattice systems have been the subject of considerable experimental and theoretical interest. Here we investigate the phase transitions of the ionic Hubbard model on the honeycomb lattice with an alternate ionic potential for the half filling and hole doping cases by means of cellular dynamical mean field theory combining with continue time quantum Monte Carlo as an impurity solver. At half filling, as the increase of the interaction at a fixed ionic potential, we find the single particle gap decreases firstly, reaches a minimum at a critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m1.jpg, then increases upturn. At An external file that holds a picture, illustration, etc.
Object name is srep09810-m2.jpg, there is a band insulator to Mott insulator transition accompanying with the presence of the antiferromagnetic order. Away from half filing, the system shows three phases for the different values of hole density and interaction, paramagnetic metal, antiferromagnetic metal and ferromagnetic metal. Further, we present the staggered particle number, the double occupancy, the staggered magnetization, the uniform magnetization and the single particle spectral properties, which exhibit characteristic features for those phases.

The correlation effects in the honeycomb lattice systems have been extensively studied, which result in a number of exotic phenomena in both theory and experiment, such as the correlated electrons in the graphene1,2 and Silicene3,4,5, topological Mott insulator6 and quantum spin liquid7,8. Most of those studies are based on the standard Hubbard model, one of the most popular models in the strongly correlated system. For half-filling case, the electrons on the honeycomb lattice are described by a non-interacting massless Dirac fermion model with linear low energy dispersion relation. The system is semimetal, in which the Fermi surface are only six isolated points at the corners of the Brillouin zone. For the peculiar nature of the Fermi surface, the interaction effects can be suppressed by the low density of states in the Fermi level9,10,11,12. Away from the half-filling, the different behavior will arise in this system13. For example, at the An external file that holds a picture, illustration, etc.
Object name is srep09810-m3.jpg or An external file that holds a picture, illustration, etc.
Object name is srep09810-m4.jpg filling, the system shows many weak coupling instabilities to various ordered states, including spin density waves14, Pomeranchuk metal15, and p/d-wave superconductors16,17,18.

Recently, a new class of two dimensional materials An external file that holds a picture, illustration, etc.
Object name is srep09810-m5.jpg An external file that holds a picture, illustration, etc.
Object name is srep09810-m6.jpg has been found19,20,21, which is formed on a single layer honeycomb lattice consisting of alternating “M” and “N” orbitals with a level offset. Experimental results show that An external file that holds a picture, illustration, etc.
Object name is srep09810-m7.jpg supports a unconventional superconductor19,22. The origin of superconductivity can be revealed based on the ionic-Hubbard model on the honeycomb lattice with the staggered lattice potential. The ionic Hubbard model, an extended version of the Hubbard model, was proposed to explain the neutral-ionic transition in the quasi-one-dimensional charge-transfer organic materials23. It has also been proposed to investigate the band insulator to Mott insulator transition, such as the one dimension system24,25 and two dimension square lattice system26. However, the charge dynamics with spins and the phase diagram of this model on the honeycomb lattice have not been studied. Moreover, the ionic Hubbard model on the honeycomb lattice can also be realized by cold atoms loaded in the optical lattices27,28,29,30,31,32, in which the on-site interaction, hopping amplitude, doping, and temperature can be fully controlled using Feshbach resonances, changing the lattice depth, changing the number of fermions, and varying the cooling time.

The dynamical mean field theory (DMFT)33 and its cluster extensions34,35 are powerful method to investigate the strongly correlated system, due to the efficient description of the quantum fluctuations. The cellular dynamical mean field theory (CDMFT)35 is one of the cluster extensions of DMFT, and the cluster is constructed in real lattice space. In contrast to a single site is chosen to construct the self-consistent equation in DMFT, the CDMFT picks up a cluster. This makes it is possible to include short range spatial fluctuations inside the cluster, which are important in the low dimensional systems. This method have been used to study the correlation effects on the honeycomb lattice and square lattice, such as Mott transition36,37, topological phase transition38 and charge order insulator transition39,40.

In this paper, we study the phase transitions of the ionic Hubbard model on the honeycomb lattice as a function of the hole doping and temperature. We adopt the CDMFT combined with the continuous time quantum Monte Carlo method (CTQMC)41,42. In order to determine the phase diagram, we calculate the staggered particle number, the double occupancy, the staggered magnetization, the uniform magnetization and the single particle spectral properties. At half filling, the system goes from a paramagnetic band insulator phase to an antiferromagnetic Mott insulator phase with the increase of the interaction. At small hole doping, the system has two phases, a paramagnetic metal for weak interaction and an antiferromangetic metal for large interaction. For finite hole doping above a critical value, the system shows three phases, a paramagnetic metal at weak interaction region, a antiferromagnetic metal at intermediate interaction region, then a ferromagnetic metal at strong interaction region.

Results

The strongly correlated honeycomb lattice with staggered potential

We consider the ionic Hubbard model on the honeycomb lattice (see inset in Fig. 1). The system is composed of two alternating sublattices An external file that holds a picture, illustration, etc.
Object name is srep09810-m8.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m9.jpg. The Hamiltonian can be written as

An external file that holds a picture, illustration, etc.
Object name is srep09810-m10.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep09810-m11.jpg creates (destroys) an electron with spin An external file that holds a picture, illustration, etc.
Object name is srep09810-m12.jpg at site An external file that holds a picture, illustration, etc.
Object name is srep09810-m13.jpg. An external file that holds a picture, illustration, etc.
Object name is srep09810-m14.jpg is the hopping amplitudes of fermions over nearest-neighbor sites, and we set An external file that holds a picture, illustration, etc.
Object name is srep09810-m15.jpg as the unit energy. An external file that holds a picture, illustration, etc.
Object name is srep09810-m16.jpg An external file that holds a picture, illustration, etc.
Object name is srep09810-m17.jpg is the amplitude of the on-site repulsive interaction, and An external file that holds a picture, illustration, etc.
Object name is srep09810-m18.jpg is a staggered one-body potential on the two sublattices in each unit cell, which is also called the “ionic" potential. The last term, the chemical potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m19.jpg is fixed so that the average occupancy is An external file that holds a picture, illustration, etc.
Object name is srep09810-m20.jpg, where An external file that holds a picture, illustration, etc.
Object name is srep09810-m21.jpg is the hole density.

Figure 1
Phase diagram of the inonic Hubbard model on the honeycomb lattice.

We begin with the tight-binding Hamiltonian with staggered potential on the honeycomb lattice, corresponding to that the interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m22.jpg in the ionic Hubbard model. After the fourier transformation, we can get the dispersion of the free electrons,

An external file that holds a picture, illustration, etc.
Object name is srep09810-m23.jpg

where

An external file that holds a picture, illustration, etc.
Object name is srep09810-m24.jpg

In this system, there are two bands, and the energy gap of the two bands An external file that holds a picture, illustration, etc.
Object name is srep09810-m25.jpg. From the tight binding model analysis above, we can learn that the system can be adjusted to various phases: a semimetal when staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m26.jpg and hole doping density An external file that holds a picture, illustration, etc.
Object name is srep09810-m27.jpg, a band insulator when staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m28.jpg and hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m29.jpg, and a normal metal when staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m30.jpg (or An external file that holds a picture, illustration, etc.
Object name is srep09810-m31.jpg) and hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m32.jpg. In this paper, we mainly study the correlation effects in the band insulator and hole doping band insulator.

Phase diagram of the ionic Hubbard model

In this section, we summarize our main results of the ionic Hubbard model on the honeycomb lattice, deferring the details of how they were obtained to the following sections. The phase diagram as a function of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m33.jpg for half filling and hole doping at staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m34.jpg and temperature An external file that holds a picture, illustration, etc.
Object name is srep09810-m35.jpg obtained from the analysis using 6-site cluster is shown in Fig. 1. The results obtained using 8-site cluster are also shown to quantitatively see the cluster-size dependence. In the noninteracting limit An external file that holds a picture, illustration, etc.
Object name is srep09810-m36.jpg, the system is band insulator and normal metal at half filling and hole doping cases, respectively. With the increase of the interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m37.jpg, the system shows two phases for the half filling case, corresponding to the band insulator and the antiferromagnetic Mott insulator, and the two phases separate at the critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m38.jpg. Below the critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m39.jpg, the energy gap in the band insulator are the same for both spin components and decrease as the interaction increasing. In the Mott insulator phase, the single particle energy gap are different for both spin components, such as An external file that holds a picture, illustration, etc.
Object name is srep09810-m40.jpg. And the Mott gap increase monotonously with the increase of the interaction.

For the small hole doping case, the system goes a phase transition from paramagnetic metal to antiferromagnetic metal when changing the interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m41.jpg. At the hole density An external file that holds a picture, illustration, etc.
Object name is srep09810-m42.jpg, the phase transition occurs at critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m43.jpg. For finite hole doping above a critical value, there are three phases at different interaction, corresponding to paramagnetic metal, antiferromagnetic metal, and ferromagnetic metal. For example, at hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m44.jpg, the system is paramagnetic metal below a critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m45.jpg, ferromagnetic metal above another critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m46.jpg, and antiferromagnetic metal between those two interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m47.jpg.

In Fig. 1, we also present the results for the 8-site cluster. In this case, the properties of this system are qualitatively same, but the phase boundary shifts a little, such as, in An external file that holds a picture, illustration, etc.
Object name is srep09810-m48.jpg, the phase transition of paramagnetic metal to antiferromagnetic metal is at An external file that holds a picture, illustration, etc.
Object name is srep09810-m49.jpg (An external file that holds a picture, illustration, etc.
Object name is srep09810-m50.jpg for 6-site cluster), and the antiferromagnetic metal to ferromagnetic metal is at An external file that holds a picture, illustration, etc.
Object name is srep09810-m51.jpg (An external file that holds a picture, illustration, etc.
Object name is srep09810-m52.jpg for 6-site cluster). We describe below in details of the spectral and magnetic properties that lead to this diagram.

Local quantities and spectral properties for the half filling case An external file that holds a picture, illustration, etc.
Object name is srep09810-m53.jpg

In this section, we firstly try to understand the correlation effects on the band insulator on the honeycomb lattice. We concentrate on the half-filling case An external file that holds a picture, illustration, etc.
Object name is srep09810-m54.jpg for different values of the staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m55.jpg with the average occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m56.jpg. In the noninteracting limit, the system prefers a band insulator phase, in which most of the electrons stay on a sublattice with lower potential, resulting in zero density of states in the Fermi surface. When the local interaction is turned on, the band insulator competes with the Mott insulator with one electron per lattice site. In Fig. 1, the phase diagram give us the results of the band insulator to Mott insulator at An external file that holds a picture, illustration, etc.
Object name is srep09810-m57.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m58.jpg. Here we will give more detailed description on the results for different values of staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m59.jpg.

In order to examine how the system evolves with the variation of the local interaction, we firstly calculate the four local quantities: staggered charge density An external file that holds a picture, illustration, etc.
Object name is srep09810-m60.jpg, double occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m61.jpg, staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m62.jpg, and uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m63.jpg. The staggered charge density and double occupancy are related with the charge fluctuation, the staggered magnetization and the uniform magnetization give us the information about the spin fluctuation. The staggered charge density is defined by the difference between the particle number densities at two sublattices,

An external file that holds a picture, illustration, etc.
Object name is srep09810-m64.jpg

where the sublattice number densities can be calculated as An external file that holds a picture, illustration, etc.
Object name is srep09810-m65.jpg for An external file that holds a picture, illustration, etc.
Object name is srep09810-m66.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m67.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m68.jpg is the site numbers of the cluster. We also calculate the double occupancy defined by

An external file that holds a picture, illustration, etc.
Object name is srep09810-m69.jpg

The staggered magnetization and uniform magnetization are defined as

An external file that holds a picture, illustration, etc.
Object name is srep09810-m70.jpg

and

An external file that holds a picture, illustration, etc.
Object name is srep09810-m71.jpg

respectively, where the sublattice magnetization is calculated as An external file that holds a picture, illustration, etc.
Object name is srep09810-m72.jpg for An external file that holds a picture, illustration, etc.
Object name is srep09810-m73.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m74.jpg.

The results for the staggered charge density An external file that holds a picture, illustration, etc.
Object name is srep09810-m75.jpg and the double occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m76.jpg as a function of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m77.jpg for temperature An external file that holds a picture, illustration, etc.
Object name is srep09810-m78.jpg are shown in Figs. 2(a) and 2(b). Due to the staggered on-site potential, An external file that holds a picture, illustration, etc.
Object name is srep09810-m79.jpg is always nonzero, even thought the Hubbard An external file that holds a picture, illustration, etc.
Object name is srep09810-m80.jpg tries to suppress it. An external file that holds a picture, illustration, etc.
Object name is srep09810-m81.jpg decreases monotonically as a function of An external file that holds a picture, illustration, etc.
Object name is srep09810-m82.jpg, and shows no discontinuity at An external file that holds a picture, illustration, etc.
Object name is srep09810-m83.jpg. In the weak interaction region, the electrons prefer to gather on the lower potential sublattice An external file that holds a picture, illustration, etc.
Object name is srep09810-m84.jpg. The system experiences an imbalance between the two sublattice, resulting in higher double occupancy, compared with the Hubbard model when An external file that holds a picture, illustration, etc.
Object name is srep09810-m85.jpg and a nonzero staggered charge density. Such tendencies become stronger as An external file that holds a picture, illustration, etc.
Object name is srep09810-m86.jpg grows. In the ionic limit An external file that holds a picture, illustration, etc.
Object name is srep09810-m87.jpg, it is energetically favorable that all the electrons are in the sublattice An external file that holds a picture, illustration, etc.
Object name is srep09810-m88.jpg, producing unity of the staggered charge. As An external file that holds a picture, illustration, etc.
Object name is srep09810-m89.jpg increasing, the energy cost of two electrons to stay in the same site becomes large, both the double occupancy and the staggered charge density decrease monotonically with the imbalance between the two sublattices become weaker. In the strong coupling limit, the staggered charge density is close to 0.

Figure 2
Four local quantities as a function of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m237.jpg for the half filling case.

In Figs. 2(c) and 2(d) we plot the staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m90.jpg and uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m91.jpg as a function of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m92.jpg for temperature An external file that holds a picture, illustration, etc.
Object name is srep09810-m93.jpg respectively. For a given An external file that holds a picture, illustration, etc.
Object name is srep09810-m94.jpg, there exists a threshold value An external file that holds a picture, illustration, etc.
Object name is srep09810-m95.jpg at which the staggered magnetization turns on with a jump. Both the value of the An external file that holds a picture, illustration, etc.
Object name is srep09810-m96.jpg and the amplitude of the jump in An external file that holds a picture, illustration, etc.
Object name is srep09810-m97.jpg are decreasing functions of An external file that holds a picture, illustration, etc.
Object name is srep09810-m98.jpg. In the half filling case An external file that holds a picture, illustration, etc.
Object name is srep09810-m99.jpg, the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m100.jpg is almost zero, independent of the staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m101.jpg and interaction strength An external file that holds a picture, illustration, etc.
Object name is srep09810-m102.jpg.

The local density of states provide more detailed information on the single particle properties. The spin-resolved single particle density of states are computed as follows

An external file that holds a picture, illustration, etc.
Object name is srep09810-m103.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep09810-m104.jpg is the spin, An external file that holds a picture, illustration, etc.
Object name is srep09810-m105.jpg, and An external file that holds a picture, illustration, etc.
Object name is srep09810-m106.jpg is measured from the chemical potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m107.jpg. The density of states are derived from the imaginary time Green’s function An external file that holds a picture, illustration, etc.
Object name is srep09810-m108.jpg using maximum entropy method43. The local density of states are shown in Fig. 3 for several values of An external file that holds a picture, illustration, etc.
Object name is srep09810-m109.jpg at staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m110.jpg. For a quantitative analysis of the gap around a Fermi level, we investigate the spectral gap An external file that holds a picture, illustration, etc.
Object name is srep09810-m111.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m112.jpg for both spin components which are defined as the energy difference between the highest filled and lowest empty levels in the local density of states. Fig. 4 shows the spectral gap as a function of the interaction strength An external file that holds a picture, illustration, etc.
Object name is srep09810-m113.jpg for An external file that holds a picture, illustration, etc.
Object name is srep09810-m114.jpg at temperature An external file that holds a picture, illustration, etc.
Object name is srep09810-m115.jpg. In the noninteracting system An external file that holds a picture, illustration, etc.
Object name is srep09810-m116.jpg, the local density of states can be computed analytically and is composed of two bands which are separated by a band gap An external file that holds a picture, illustration, etc.
Object name is srep09810-m117.jpg due to the staggered potential. For weak interaction, the local density of states are the same for both spin components. However the band gap around a Fermi level decreases monotonically with the increase of interaction in this region. For example, the density of states for two spin components at interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m118.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m119.jpg are shown in Figs. 3(a1)(a2) and Figs. 3(b1)(b2) respectively, the band gap at An external file that holds a picture, illustration, etc.
Object name is srep09810-m120.jpg is smaller than An external file that holds a picture, illustration, etc.
Object name is srep09810-m121.jpg. The local density of states displays a minimum spectral gap at a critical value of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m122.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m123.jpg for up-spin component and An external file that holds a picture, illustration, etc.
Object name is srep09810-m124.jpg for down-spin component. When An external file that holds a picture, illustration, etc.
Object name is srep09810-m125.jpg above the critical value, the gap between the two bands in turn is enlarged, and the system goes to the Mott insulator phase. In this region, the local density of states for up-spin component and down-spin component show different behaviors (Figs. 3(c1)(c2). The Mott gap of the up-spin component is bigger than the down-spin component.

Figure 3
Spin-resolved single particle density of states An external file that holds a picture, illustration, etc.
Object name is srep09810-m245.jpg for the half filling case.
Figure 4
Spectral gaps of the two spin components An external file that holds a picture, illustration, etc.
Object name is srep09810-m252.jpg.

Local quantities and spectral properties for the hole doping case An external file that holds a picture, illustration, etc.
Object name is srep09810-m126.jpg

In this section, we now turn to study the phase transitions of the hole doping (An external file that holds a picture, illustration, etc.
Object name is srep09810-m127.jpg) band insulator as a function of the interaction on the honeycomb lattice. The average occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m128.jpg, resulting in finite density of states in the Fermi level. The system is a metal, and the low energy physics can be described by the Fermi liquid theory. When the interaction is turned on, there are many instabilities for the Fermi liquids, which is an enduring theme research in condensed matter physics. When the staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m129.jpg and the temperature An external file that holds a picture, illustration, etc.
Object name is srep09810-m130.jpg, the phase diagram for different values of hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m131.jpg is displayed in Fig. 1. For small hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m132.jpg, the system goes a phase transition from paramagnetic metal to antiferromagnetic metal at a critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m133.jpg. At finite hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m134.jpg above a critical value, the system has three phases as the interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m135.jpg varying, paramagnetic metal when An external file that holds a picture, illustration, etc.
Object name is srep09810-m136.jpg, antiferromagnetic metal when An external file that holds a picture, illustration, etc.
Object name is srep09810-m137.jpg, and ferromagnetic metal when An external file that holds a picture, illustration, etc.
Object name is srep09810-m138.jpg. Now we discuss how we use the single particle spectral and other local qualities to determine the phase boundaries An external file that holds a picture, illustration, etc.
Object name is srep09810-m139.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m140.jpg.

Firstly, let us study the single particle density of states as a function of interaction for various hole doping. Fig. 5 shows the single particle density of states An external file that holds a picture, illustration, etc.
Object name is srep09810-m141.jpg for both of the spin components at hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m142.jpg and staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m143.jpg$. The single particle density of states An external file that holds a picture, illustration, etc.
Object name is srep09810-m144.jpg are obtained from Eq.(8). With the increase of the interaction, the spectral weight is continuously transferred to the higher energy states, the chemical potential lies inside the lower band for both spin components all the time. When An external file that holds a picture, illustration, etc.
Object name is srep09810-m145.jpg, in the paramagnetic metal region, the density of states for both spins are the same, and there are two the spectral peaks above and below the Fermi level (Figs. 5(a1) and 5(a2)). When An external file that holds a picture, illustration, etc.
Object name is srep09810-m146.jpg, corresponding to the antiferromagnetic metal, the antiferromagnetic order sets in, making the density of states and gaps a little different for the two spin components (Figs. 5(b1) and 5(b2)). When An external file that holds a picture, illustration, etc.
Object name is srep09810-m147.jpg, in the ferromangnetic metal region, the density of states for the two spin components are renormalized much and very different (Figs. 5(c1) and 5(c2)). In both the antiferromagnetic metal and ferromagnetic metal, one of the spectral peaks above the Fermi level is suppressed.

Figure 5
Spin-resolved single particle density of states An external file that holds a picture, illustration, etc.
Object name is srep09810-m259.jpg for the hole doping case.

Besides the changes of the local density of states, the interaction will influence the momentum-resolved spectral density in the Fermi level An external file that holds a picture, illustration, etc.
Object name is srep09810-m148.jpg very much. The An external file that holds a picture, illustration, etc.
Object name is srep09810-m149.jpg-resolved spectral weight can be defined as

An external file that holds a picture, illustration, etc.
Object name is srep09810-m150.jpg

which are the maxima of the spectral weight at zero temperature as a function of An external file that holds a picture, illustration, etc.
Object name is srep09810-m151.jpg. In Fig. 6 we present An external file that holds a picture, illustration, etc.
Object name is srep09810-m152.jpg for the three different phases at hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m153.jpg and staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m154.jpg. In the hole doping case, the Fermi surface An external file that holds a picture, illustration, etc.
Object name is srep09810-m155.jpg are six rings in the An external file that holds a picture, illustration, etc.
Object name is srep09810-m156.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m157.jpg points located at the corners of the hexagon. When the interaction is small An external file that holds a picture, illustration, etc.
Object name is srep09810-m158.jpg, corresponding to paramagnetic metal, the Fermi surface is only weakly renormalized compared to the case of the interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m159.jpg (Figs. 6(a1)(a2)). In the intermediate interaction region An external file that holds a picture, illustration, etc.
Object name is srep09810-m160.jpg, corresponding to antiferromagnetic metal, the distribution of quasi-particle spectral near the An external file that holds a picture, illustration, etc.
Object name is srep09810-m161.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m162.jpg points became anisotropic in different directions in each corners (Figs. 6(b1)(b2)). In the large interaction region An external file that holds a picture, illustration, etc.
Object name is srep09810-m163.jpg, corresponding to ferromagnetic metal, the Fermi surface are strongly renormalized at each corners, the peak of An external file that holds a picture, illustration, etc.
Object name is srep09810-m164.jpg are broadened (Figs. 6(c1)(c2)).

Figure 6
The distribution of low energy spectral weight An external file that holds a picture, illustration, etc.
Object name is srep09810-m268.jpg in An external file that holds a picture, illustration, etc.
Object name is srep09810-m269.jpg space.

In addition to the spectral properties, we also calculate four local quantities: the staggered charge density An external file that holds a picture, illustration, etc.
Object name is srep09810-m165.jpg, the double occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m166.jpg, the staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m167.jpg, and the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m168.jpg obtained from Eqs.(4), (5), (6) and (7), respectively. Figs. 7(a)(b) show the staggered charge density An external file that holds a picture, illustration, etc.
Object name is srep09810-m169.jpg and the double occupancy An external file that holds a picture, illustration, etc.
Object name is srep09810-m170.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is srep09810-m171.jpg for hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m172.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m173.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m174.jpg at staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m175.jpg. Although the system is metallic all the times, with the increase of the interaction, both of the two quantities decrease monotonically. Figs. 7(c)(d) show the staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m176.jpg and the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m177.jpg as a function of interaction for hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m178.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m179.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m180.jpg at staggered potential An external file that holds a picture, illustration, etc.
Object name is srep09810-m181.jpg. For small hole doping, such as An external file that holds a picture, illustration, etc.
Object name is srep09810-m182.jpg, the system shows a phase transition from paramagnetic metal to antiferromagnetic metal in which the staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m183.jpg turns on a finite value at An external file that holds a picture, illustration, etc.
Object name is srep09810-m184.jpg, and the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m185.jpg is zero all the time. When the hole doping above a critical value, such as An external file that holds a picture, illustration, etc.
Object name is srep09810-m186.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m187.jpg, the magnetic properties shows dramatic changes at the phase boundaries An external file that holds a picture, illustration, etc.
Object name is srep09810-m188.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m189.jpg. For small An external file that holds a picture, illustration, etc.
Object name is srep09810-m190.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m191.jpg, the magnetic order is not favored. For intermediate An external file that holds a picture, illustration, etc.
Object name is srep09810-m192.jpg, An external file that holds a picture, illustration, etc.
Object name is srep09810-m193.jpg, there is a nonzero staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m194.jpg. When the An external file that holds a picture, illustration, etc.
Object name is srep09810-m195.jpg is large, An external file that holds a picture, illustration, etc.
Object name is srep09810-m196.jpg, the nonzero staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m197.jpg is suppressed, the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m198.jpg becomes a nonezero value. Both of the staggered magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m199.jpg and the uniform magnetization An external file that holds a picture, illustration, etc.
Object name is srep09810-m200.jpg increase with the increase of the hole doping An external file that holds a picture, illustration, etc.
Object name is srep09810-m201.jpg.

Figure 7
Four local quantities as a function of interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m278.jpg for the hole doping case.

Discussion

In this work, we have investigated the effect of on-site interaction and staggered ionic potential in a band insulator and doped band insulator on the honeycomb lattice based on the ionic Hubbard model. By means of the cellular dynamical mean field theory combing with continue time Monte Carlo method, we construct a phase diagram as a function of interaction and hole doping. At half filling, although the single particle spectral functions always posses a energy gap, the system shows a band insulator to Mott insulator transition at a critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m202.jpg, with the single particle gap decreases firstly, reaches a minimum at a critical interaction An external file that holds a picture, illustration, etc.
Object name is srep09810-m203.jpg, then increases upturn, and the antiferromagnetic order gives a finite value above An external file that holds a picture, illustration, etc.
Object name is srep09810-m204.jpg. Away from half filing, many metallic phases with magnetic order are found, in order to exhibit characteristic features of the phases, the behavior of the staggered particle number, the double occupancy, the staggered magnetization, the uniform magnetization and the single particle spectral properties have bend studied. At small hole doping, the system goes a phase transition from a paramagnetic metal to an antiferromangetic metal with the increase of the interaction. For finite hole doping above a critical value, the system shows three phases, a paramagnetic metal at weak interaction region, a antiferromagnetic metal at intermediate interaction region, then a ferromagnetic metal at strong interaction region.

We get itinerant metals with spin density wave state which are an interesting class of materials where electrons show spin polarization or staggered spin polarization behavior. They have applications in spintronics as they can generate spin-polarized currents44,45,46. And the materials with the honeycomb lattice structure are very common, such as single layer graphene, silicene considered as the silicon-based counterpart of graphene, and monolayer molybdenum disulfide (ML-MDS), MoSAn external file that holds a picture, illustration, etc.
Object name is srep09810-m205.jpg, which play a vital role in nanoelectronics and nanospintronics. We hope that our study will motivate a research on along those lines and open up new possibilities in the area of spintronics. Moreover, with the development of the cold atom experiment, the honeycomb lattice have been simulated29,32,47,48, which can give us a platform to simulate and detect the phase transitions by loading ultracold atoms on the honeycomb optical lattices.

Methods

In order to study the ionic Hubbard model in honeycomb lattice which describes the correlation effects on the band insulator and the hole doped band insulator, the Cellular dynamical mean field theory are employed. The Cellular dynamical mean field theory is an extension of dynamical mean field theory, which is able to partially cure dynamical mean field theory’s spatial limitations. We replace the single site impurity by a cluster of impurities embedded in a self-consistent bath. The cluster-impurity problem embedded in a bath of free fermions can be written in a quadratic form,

An external file that holds a picture, illustration, etc.
Object name is srep09810-m206.jpg

where i and j are the coordinates inside the cluster-impurity, and the An external file that holds a picture, illustration, etc.
Object name is srep09810-m207.jpg is the Weiss field. The effective medium An external file that holds a picture, illustration, etc.
Object name is srep09810-m208.jpg is computed via the Dyson equation,

An external file that holds a picture, illustration, etc.
Object name is srep09810-m209.jpg

Within cellular dynamical mean field theory, the interacting lattice Green’s function in the cluster site basis is given by,

An external file that holds a picture, illustration, etc.
Object name is srep09810-m210.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep09810-m211.jpg are Matsubara frequencies, An external file that holds a picture, illustration, etc.
Object name is srep09810-m212.jpg is the chemical potential and An external file that holds a picture, illustration, etc.
Object name is srep09810-m213.jpg is the Fourier-transformed hopping matrix for the super lattice. In our analysis, the 6- and 8-site clusters in the inset of Fig. 1 are used to set up the cluster Hamiltonian. For the 6-site cluster case, the hopping matrix An external file that holds a picture, illustration, etc.
Object name is srep09810-m214.jpg of the cluster can be written as follows (An external file that holds a picture, illustration, etc.
Object name is srep09810-m215.jpg the reduced Brillouin-zone),

An external file that holds a picture, illustration, etc.
Object name is srep09810-m216.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep09810-m217.jpg are the nearest-neighbor super-lattice vectors, An external file that holds a picture, illustration, etc.
Object name is srep09810-m218.jpg is the lattice constant. In each iteration, in order to solve the effective cluster model and to calculate An external file that holds a picture, illustration, etc.
Object name is srep09810-m219.jpg, we use the weak coupling interaction expansion continuous time quantum Monte Carlo method.

The CDMFT iteration procedure is summary as follows. Given a cluster self-energy An external file that holds a picture, illustration, etc.
Object name is srep09810-m220.jpg, we can compute An external file that holds a picture, illustration, etc.
Object name is srep09810-m221.jpg via Eq.(5)(6), then solve the effective cluster model and to calculate a new An external file that holds a picture, illustration, etc.
Object name is srep09810-m222.jpg. Then us Eq.(5) again, we can get a new cluster self-energy An external file that holds a picture, illustration, etc.
Object name is srep09810-m223.jpg. Repeat the procedure until the results are convergence.

The weak coupling interaction expansion continuous time quantum Monte Carlo method is efficient method to treat the impurity model. The method employs same tricks, which used to derived Feynman perturbation theory, to stochastically generate the partition function An external file that holds a picture, illustration, etc.
Object name is srep09810-m224.jpg. In the interaction picture, An external file that holds a picture, illustration, etc.
Object name is srep09810-m225.jpg, where An external file that holds a picture, illustration, etc.
Object name is srep09810-m226.jpg is the time-ordering operator. The expansion of the partition function in power of An external file that holds a picture, illustration, etc.
Object name is srep09810-m227.jpg reads

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Object name is srep09810-m228.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep09810-m229.jpg and An external file that holds a picture, illustration, etc.
Object name is srep09810-m230.jpg is the number of the sites of the cluster. The observable expectation value An external file that holds a picture, illustration, etc.
Object name is srep09810-m231.jpg can be sampled during the Monte Carlo update. For example, the Green’s function

An external file that holds a picture, illustration, etc.
Object name is srep09810-m232.jpg

Acknowledgments

H. F. Lin acknowledges very helpful discussions with Y. H. Chen, F. J. Huang and Q. H. Chen. This work was supported by the NKBRSFC under Grants No. 2011CB921502, No. 2012CB821305, NSFC under Grants No. 61227902, No. 61378017, and No. 11434015, SKLQOQOD under Grant No. KF201403, SPRPCAS under Grant No. XDB01020300. We acknowledge the supercomputing center of CAS for kindly allocating computational resources.

Footnotes

Author Contributions H. F. L. performed calculations. H. F. L., H. D. L., H. S. T., W. M. L. analyzed numerical results. H. F. L., W. M. L. contributed in completing the paper.

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