Magnetic properties on Chern insulator of checkerboard lattice with topological flat band

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Magnetic properties on Chern insulator of checkerboard lattice with topological flat band Shan Li a , Jing Yu b , Jing He a,∗ a b

Department of Physics, Hebei Normal University, Shijiazhuang 050024, China Faculty of Science, Liaoning Shihua University, Fushun 113001, China

a r t i c l e

i n f o

Article history: Received 6 November 2019 Received in revised form 9 March 2020 Accepted 19 March 2020 Available online xxxx Communicated by L. Ghivelder Keywords: Antiferromagnetic order Ferromagnetic order Chern insulator Spin waves Topological flat band Mean field theory

a b s t r a c t By means of the mean-field method and the random phase approximation, we study the magnetic properties of the correlated Chern insulator on a checkerboard lattice with topological flat band. The antiferromagnetic (AF) order is found to be more stable than the ferromagnetic (FM) order at half filling. While at quarter filling, the system becomes a FM-Chern insulator induced by the FM order. The phase diagram is more complex for other fillings. FM order is more stable than AF order for small doping due to the flatness of band structure, while FM and AF orders compete at large doping. © 2020 Elsevier B.V. All rights reserved.

1. Introduction The electronic bands with nonzero Chern number leads to lots of new topological phenomena in condensed matter physics [1,2]. Chern insulators, first proposed by Haldane in 1988 [3], are a class of topological materials with electronic bands of nonzero Chern number (or TKNN number [4]). The two-dimensional graphene-like lattice model breaks time-reversal symmetry and can be viewed as a lattice version of integral quantum Hall effect (IQHE) without magnetic field. Fractional quantum Hall effect (FQHE) [5–7] is a topological ordered state, which has attracted great attention since its discovery in two dimensional electron gas under the strong magnetic field. The key point to realize the FQHE without Landau levels is the topological flat band (TFB) [6,8,9], where strong correlation plays a more important role and FQHE emergence. Recently, researchers design a series of nearly-flat-band lattice models (checkerboard lattice, square lattice and honeycomb lattice) with nontrivial topological properties [6,8,9], where the hopping parameters, including the next-nearest-neighbor (NNN) hopping and the next-next-nearest-neighbor (NNNN) hopping, are finely tuned to reach a high flatness ratio of the band gap to bandwidth (about 20-50). Nonzero Chern number characterizes the nontrivial topological properties of the topological flat band.

*

Corresponding author. E-mail address: [email protected] (J. He).

https://doi.org/10.1016/j.physleta.2020.126425 0375-9601/© 2020 Elsevier B.V. All rights reserved.

Spin orders manifest magnetic properties of strongly correlated electronic systems [10]. According to Stoner’s criterion, Coulomb interaction breaks the band degeneracy and drives the system into a ferromagnetic ground state for the systems with flat or nearly flat bands [11,12]. Magnetic ordering has been studied extensively in various lattice models, such as honeycomb, square and checkerboard lattice models [13–15]. In these works, the authors use different approaches to identify magnetic phases or only consider the quarter filling case [14,15]. While in our paper, we study general cases of the checkerboard model with arbitrary doping, and obtain a much more rich phase diagram. In this paper, we consider a two-component fermionic TFB model with Coulomb repulsion interaction on the checkerboard lattice. We find that the AF order is more stable than the FM order at half filling. While at quarter filling, the system becomes a C = 1 FM Chern insulator, which is consistent with the results in Ref. [16], where it was rigorously proven that a certain (quasi-)one dimensional Hubbard model with TFB has a ferromagnetic ground state. Away from half filling, the phase diagram of our model becomes complex. For small doping, the FM order is more stable than the AF order due to the flatness of band structure. While for the large doping, there is a ground state energy competition between FM and AF order. The paper is organized as follows. In Sec. 2, we introduce the Hamiltonian of the TFB model with Coulomb repulsion interaction on a checkerboard lattice. In Sec. 3, we study the competition between ferromagnetic order and antiferromagnetic order, and then

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Fig. 1. (Color online) (a) The illustration of the checkerboard lattice. Red and blue dots denote the bipartite lattice sites. The purple arrow, black dashed (solid) line and green dashed line represent the NN hopping t, the NNN hopping t i j and the

NNNN hopping t  , respectively. (b) The energy√spectrums for free √ electron (U /t = 0) at the flat-band limit with t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4. The energy scale is t. The spin-up energy band and spin-down energy band are degenerate due to spin rotation symmetry. The bandwidth is much smaller than the energy gap, then there is a flat band. The inset shows the details of the flat band, where the flat band is slightly dispersive.

obtain the global phase diagram by using the mean-field method. Next, we discuss the magnetic and topological properties for general doping cases in Sec. 4. Finally, we conclude our discussions in Sec. 5.

We consider the spinfull Haldane model on a checkerboard lattice with topological flat band, whose Hamiltonian in real space is given by [6]

 i , j ,σ

+U





†

e i φi j cˆ i σ cˆ j σ −

nˆ i ↑ nˆ i ↓ − μ



i ,σ

t i j cˆ i σ cˆ j σ − t  †

i , j ,σ † cˆ i σ cˆ i σ

√



†

cˆ i σ cˆ j σ

i , j ,σ

+ h.c .

(1)

i ,σ

the hopping strength between NN, NNN and NNNN sites. t i j takes

the value t 1 (t 2 ) if the two sites are connected by a dashed (solid) line in Fig. 1(a). U is the onsite Coulomb repulsive interaction strength and μ is the chemical potential. It is obviously that the spin-rotation symmetry is respected, but time-reversal symmetry is broken for the system with φ = nπ (n ∈ Z ) at the noninteracting limit (U /t = 0). After the Fourier transformation, electronic annihilation operators on the two sublattices can be written as

1  ik· R i 1  ik· R i ˆ cˆ i ∈ A ,σ = √ e aˆ kσ , cˆ i ∈ B ,σ = √ e bk σ Ns Ns k

k

where N s is the number of unit cells. For the noninteracting case (U /t = 0), the Hamiltonian in momentum space is given by

H=

 k

†

k hk k . 

Here k = aˆ k↑ , bˆ k↑ , aˆ k↓ , bˆ k↓ †



†

†

†

†

 and



hk = − ( A + B ) Iσ + C σz − Re (ξk ) σx − Im (ξk ) σ y ⊗ Iτ ,

2), t  =

where ξk = E + i F , A = 4t  cos(kx) cos(ky ), B = (t 1 + t 2 )(cos(kx) + cos(ky )), C = (t 1 − t 2 )(cos(kx) − cos(ky )), E = 4t cos φ cos(kx/2) × cos(ky /2), F = 4t sin φ sin(kx/2) sin(ky /2). Iσ and σx/ y /z are unit matrix and Pauli matrices acting on the sublattice degrees of freedom. Iτ is the unit matrix acting on the spin degrees of freedom. The expression of energy spectrums are



|ξk |2 + C 2 .

When parameters in Eq. (1) are chosen as t 1 = −t 2 = t /(2 +

√

(2)

√

2),

t  = −t /(2 + 2 2) and φ = π /4, the checkerboard lattice has a flat band, as shown in Fig. 1(b). In this case, the energy bands are two-fold degenerates due to spin rotation symmetry, of which the lowest two are flat bands with flatness ratio (the band gap over bandwidth) about 30 (see Fig. 1(b)). The inset of Fig. 1(b) shows the details of the flat band, which is slightly dispersive. To characterize the topological properties of the flat band, we introduce Chern number defined as

†

where cˆ i σ (ˆc i σ ) represents fermion creation (annihilation) operator at site i with spin σ (↑ or ↓). i , j , i , j , and i , j  represent the nearest, next-nearest and next-next-nearest neighbor links, respectively. The nearest-neighbor (NN) hopping has a complex phase φi j = ±φ where the positive sign is determined by the direction of the purple arrow in Fig. 1(a). t, t i j and t  are

√

−t /(2 + 2 2) and φ = π /4. The energy scale is t.

E ± = −( A + B ) ±

2. Hamiltonian of checkerboard lattice model

H = −t

Fig. 2. The edge states of the checkerboard lattice with t 1 = −t 2 = t /(2 +

C=

1

2π2π dkx dk y F (kx , k y ),

(3)

∗

∗ ∂ ∂ ∂ ∂ − ), F (kx , k y ) = Im( | | ∂ k y ∂ kx ∂ kx ∂ k y

(4)

2π 0

0

where

and is the ground state wave function of the system. The formula of the integration in Eq. (3) can be written into a discrete summation as

C=

1  2π

F (kx , k y ) kx k y

(5)

kx ,k y

where

F (kx , k y ) kx k y  = Im ln( ∗ (kx , k y ) | (kx + kx , k y )  × ∗ (kx + kx , k y ) | (kx + kx , k y + k y )  × ∗ (kx + kx , k y + k y ) | (kx , k y + k y )  × ∗ (kx , k y + k y ) | (kx , k y ) ) After numerical calculations, the system is a Chern insulator with flat band and the Chern number is C = 2. In Fig. 2, we also present the edge states of the checkerboard lattice to further show the topological properties.

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3. Competition between ferromagnetic order and antiferromagnetic order In this section, we consider the competition between Ferromagnetic order and Antiferromagnetic order by mean-field ansatz in interacting case with U /t = 0. The mean-filed ansatz of the AF order is



†

 1 n + (−1)i σ M A F 2

cˆ i σ cˆ i σ =

(6)

and the mean-filed ansatz of FM order is



†

1

cˆ i σ cˆ i σ =

2

(n + σ M F )

(7)

where M A F is the staggered AF order parameter and M F is the FM order parameter, σ is the spin index with σ = 1 for spin up and σ = −1 for spin down. n = 1 − d is the average number of particles on each site, and d is the hole doping.

ξ2 =

3.1. Antiferromagnetic order Firstly, we consider the AF ordered state with M A F = 0 and M F = 0. Then, the Hamiltonian can be written as



H A F = −t

−t



μ−

i ,σ

t i j cˆ i σ cˆ j σ †

i , j ,σ



† cˆ i σ cˆ j σ

i , j ,σ

−



†

e i φi j cˆ i σ cˆ j σ −

i , j ,σ 

Un 2

−



(−1)

In Fig. 3(a), we show the dispersions of electrons with √ AF order √ for U /t = 6, t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4 at half-filling. Comparing Fig. 3(a) with Fig. 1(b), we find that the AF order changes the flat band into a dispersive one.

†

A cˆ i σ

σz cˆ iσ

(8)

Secondly, we consider the FM ordered state with M F = 0, M A F = 0. The Hamiltonian can be written as





†

cˆ i σ cˆ i σ + h.c .

H F = −t

−t

† cˆ i σ cˆ j σ

i , j ,σ

†

k h Ak k ,

(9)

−

k



μ−

i ,σ

where

†



Un

−



†

F cˆ i σ σz cˆ i σ

(12)

i ,σ †

cˆ i σ cˆ i σ + h.c .

2

Here F = U M F /2. After Fourier transformation, the Hamiltonian in momentum space is given by

h Ak = hk + A σz ⊗ τz − μeff Iσ ⊗ Iτ

μeff = μ −

t i j cˆ i σ cˆ j σ

i , j ,σ







†

e i φi j cˆ i σ cˆ j σ −

i , j ,σ

where A = U M A F /2. After Fourier transformation, the Hamiltonian is given by

H AF =

 |ξk |2 + (C − A )2

3.2. Ferromagnetic order i

i ,σ



Fig. 3. (Color online) (a) The energy spectrums of electron with AF order. We can see that the AF order changes the flat band into a dispersive band. (b) The energy spectrums of electron with FM order. We can find that the FM order didn’t change the flatness ratio of the flat band, just split into degenerate band. √ two parts of the √ The parameters are U /t = 6, t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4 at half-filling.

Un 2

is the effective chemical potential, and τz is the Pauli matrix acting on the spin degrees of freedom. The energy spectrums of AF order are

HF =



†

k h F k k ,

(13)

k

where

E A F (k) = E A ,k,α ,± − μeff , E A ,k,1,± = −( A + B ) + E A ,k,2,± = −( A + B ) −

 

α = 1 or 2

h F k = hk − F Iσ ⊗ τz − μeff Iσ ⊗ Iτ .

|ξk |2 + (C ± A )2

The expression of energy spectrums with the FM order are

|ξk |2 + (C ± A )2

E F (k) = E F ,k,α ,± − μeff ,

By minimizing the ground state energy, the self-consistent equations of AF order in the reduced Brillouin zone (BZ) are

1=

⎡

1 Ns M A F

C + A

E A ,k,2,+ <μeff

⎛

1−d= where

1 2N s

E F ,k,1,± = −( A + B ) − F ± and

×



⎣

α = 1, 2

⎝

2 ξ1

−

 E A ,k,2,+ <μeff

 ξ1 = |ξk |2 + (C + A )2



C − A

E A ,k,2,− <μeff

1+

E F ,k,2,± = −( A + B ) + F ±

⎤ 2 ξ2

 E A ,k,2,− <μeff

⎦



|ξk |2 + C 2

|ξk |2 + C 2

(10)

Similarly, by minimizing the ground state energy, the self-consistent equations of FM order in the reduced Brillouin zone (BZ) can be written as

(11)

1=

⎞ 1⎠



⎛ ⎝

1 Ns M F

×

 E F ,k,1,− <μeff

1 2

+

 E F ,k,1,+ <μeff

1 2

−

 E F ,k,2,− <μeff

⎞ 1 2

⎠,

(14)

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Fig. 4. (Color online) (a) The global phase diagram versus different doping. For the half-filling case with d = 0, there are three phases: C = 2 Chern insulator (C = 2 CI), C = 2 antiferromagnetic Chern insulator (C = 2 AF-CI) and normal antiferromagnetic (AF) insulator; for the quarter filling case with d = 0.5, there are two phases: normal metal and C = 1 ferromagnetic Chern insulator (C = 1 FM-CI); for other cases, there are three phases for the region of 0 < d < 0.5: normal metal, antiferromagnetic (AF) metal and ferromagnetic (FM) metal, and there are also three phases for the region of 0.5 < d < 1: normal metal, ferromagnetic (FM) metal and antiferromagnetic (AF) metal. (b) The per-site ground state energies for the AF order (blue line) and the FM order (red line). (U /t ) A F is the magnetic phase √transition point with √ the antiferromagnetic order. (U /t ) F M is the magnetic phase transition point with the ferromagnetic order. The parameters are set to t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4 at half-filling.

1−d=

⎛ ⎝

1 2N s

×

 E F ,k,1,− <μeff



1+

E F ,k,1,+ <μeff

1+



⎞ 1⎠ .

(15)

is about 50. In addition, since the difference of ground-state energies between AF-Metal phase and FM-Metal phase is not very large, the flatness of topological flat band may significantly affect the ground state of the system.

E F ,k,2,− <μeff

In Fig. 3(b), we also present the dispersion of electrons √ with FM √ order for U /t = 6, t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4 at half-filling. Comparing Fig. 3(b) with Fig. 1(b), we can find that the FM order didn’t change the flatness ratio of the flat band, just split the degenerate flat bands. 3.3. Global phase diagram By means of the mean field method, we obtain the global phase diagram under different doping conditions in √ Fig. 4(a) with pa√ rameters t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4. With the consideration of topological properties of the system, we find there are seven phases: normal metal, Antiferromagnetic (AF) metal, Ferromagnetic (FM) metal, C = 2 Chern insulator (C = 2 CI), C = 2 antiferromagnetic Chern insulator (C = 2 AF-CI), normal antiferromagnetic (AF) insulator and C = 1 ferromagnetic Chern insulator (C = 1 FM-CI). C = 2 antiferromagnetic Chern insulator is just the so-called A-type topological antiferromagnetic spin density wave (A-TSDW) in Ref. [17]. Due to the nonzero TKNN number (C = 2), there exists the IQHE with a quantized (charge) Hall conductivity σ H = 2e 2 /h. In the phase diagram of Fig. 4(a), at half-filling d = 0, the average number of particles on each site is n = 1, and the chemical potential is μ = U /2 (μeff = 0). We can see that there are three phases: C = 2 Chern insulator (C = 2 CI), C = 2 antiferromagnetic Chern insulator (C = 2 AF-CI) and normal antiferromagnetic (AF) insulator. For the case with weak coupling (U /t < 4.79), the ground state exhibits a Chern insulator with the flat band. With the increasing of the interaction (U /t > 4.79), the AF order emerges, and the ground state becomes the AF insulator (C = 2 AF Chern insulator and normal AF insulator). The reason for exhibiting AF order with strong coupling is that the ground state energy of the AF order is always lower than that of the FM order at halffilling, as shown in Fig. 4(b). Comparing with Ref. [13], the phase diagram have some changes. Above quarter filling case of d = 0.5, we have AF-Metal phase while Ref. [13] doesn’t. We speculate that the reason may be the flatness ratio (band gap over bandwidth) of different models. The checkerboard model’s flatness ratio is about 30, while Ref. [13]

4. Collective spin fluctuations by random phase approximation (RPA) method In this section, we go beyond the mean field theory to study the collective spin fluctuations by random phase approximation (RPA) method for the case with doping d = 0 and d = 0.5. 4.1. Half filling case: d = 0 Now, we consider the half filling case. In Fig. 5(a), we plot the staggered magnetization M A F versu the interaction U /t at halffilling, and we can see that there is a magnetic phase transition (U /t ) A F at U /t = 4.79. In Fig. 5(b), we give the energy gap at halffilling. In Fig. 5, there are two quantum phase transitions, one is magnetic phase transition (U /t ) A F where the AF order emerges, the other is topological quantum phase transition with gap closing like Fig. 5(b). When the AF order parameter is nonzero, the electronic band structure will be reconstructed, then the spectrums with AF order will become dispersive like Fig. 3(a). Next, we study collective spin fluctuations by RPA. The spin excitations are obtained from the poles of the transverse spin susceptibility tensor χ defined as

χi+− (q, i ω) = j

β

dτ e i w τ χi+− (q, τ ), j

0

where

χi+− (q, τ ) = j

1 Ns



T [ˆs+ (τ ), sˆ −j ,−q (0)] , i ,q

i , j = a, b label the two sublattices, and S q+ =



ˆk†−q,↓ cˆk,↑ kc



ˆk†+q,↑ cˆk,↓ , kc

− S− q =

denote the spin operators. Using the Wick’s theorem, we can derive the bare spin susceptibility as 0) χi+−( (q, i ω) = − j

1

β Ns

 k,ωn

ji

ij

G ↑ (k, i ωn )G ↓ (k − q, i ω + i ωn )

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Fig. 5. (Color online) (a) The staggered magnetization M A F versus the interaction strength U /t. (b) The energy gap of electrons. There are two quantum phase transitions: √ one is the magnetic phase transition (U /t ) A F , the other is the topological quantum phase transition with gap closing. The parameters are set to be t 1 = −t 2 = t /(2 + 2),

√

t  = −t /(2 + 2 2) and φ = π /4 at half-filling.

For the AF ordered state, the Matsubara Green’s functions are aa

G σ (i ωn , k) =

 12 (1 −

G ba σ (i ωn , k) =

2ξσ , j (k)

i ωn − E σ , j (k)

) ,

,

ξk∗

 j =±

,

ξk

 j =±

C +σ A ξσ , j (k)

i ωn − E σ , j (k)

j =±

G ab σ (i ωn , k) =

)

i ωn − E σ , j (k)

j =±

G bb σ (i ωn , k) =

C +σ A ξσ , j (k)

 12 (1 +

2ξσ , j (k)

i ωn − E σ , j (k)

where

E σ ,± (k) = −( A + B ) ±



,

|ξk |2 + (C + σ A )2 − μeff ,

ξσ ,± (k) = ±ξσ (k),  ξσ (k) = ± |ξk |2 + (C + σ A )2 . We may use the RPA approach to get the spin susceptibility tensor χ of the AF order from the Dyson equation

χ = χ 0 + U χ 0 χ ⇒ χ = [ ˆI − U χ 0 ]−1 χ 0 , where



χ=



+− χ +− χaa 0 ab +− +− , χ = χba χbb





+−(0) +−(0) χaa χab +−(0) +−(0) , χba χbb

and ˆI denotes the 2 × 2 identity matrix. The poles of the spin susceptibility tensor corresponding to the spin excitations are then obtained from the condition

Det [ ˆI − U χ 0 ] = 0. We note that the tensorial nature of the spin susceptibility is originating from two sublattices. The dispersions of spin collective modes with frequency ω and momentum q are determined by the above condition after performing the analytic continuation i ω → ω + i0+ . Finally, the spin wave dispersion determinant is given by +−(0)

D sp (k) = 1 − U χaa

+−(0)

+ U 2 χaa = 0.

+−(0)

− U χbb

+−(0) +−(0) +−(0) χbb − U 2 χab χba

Fig. 6. (Color online) The dispersions of the spin wave with AF order for the flat band at U /t = 4.8125 (blue line of C = 2 AF-CI) and√U /t = 7 (red line of√normal AF insulator). The parameters are t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4 at half-filling.

Then, we obtain the dispersions of spin excitations for the ground state in Fig. 6. From Fig. 6, we can see that in the presence of the AF order (U /t = 4.8125 in C = 2 AF-CI and U /t = 7 in normal AF insulator), there exit gapless Goldstone modes - spin waves, where the spin waves are dispersive and have linear behavior, ω (q) ∝ q around  point, and the spin waves do not exhibit anomalous behaviors for the flat-band case of both normal AF insulator and C = 2 AF-CI. 4.2. Quarter filling case: d = 0.5 In this part, we study the quarter filling with d = 0.5. In the global phase diagram of Fig. 4(a), we can find an important line whose ground state shows C = 1 Chern insulator induced by the FM order. We also see that there exists a quantum phase transition at U /t = 0.24 for d = 0.5, leading to two distinct phases: the normal metallic state and the C = 1 FM Chern insulator. For the weak interacting case (U /t < 0.24), the ground state exhibits the normal metallic state with flat band. The ground state will be transformed into a C = 1 FM Chern insulator with small interaction (U /t > 0.24). Therefore, we may expect that there may be only C = 1 FM Chern insulator and no quantum phase transition at the flat band limit (the flatness ratio is infinite). This result is consistent with the rigorous proof in Ref. [16], in which it was proved that the ground state of the Hubbard model with topological flat-bands is a ferromagnetic ordered state when the lowest flat band is half-filled (that corresponds to the quarter filling case here). In Fig. 7(a), we present the FM order versus the interaction U /t at quarter filling. It can be seen that the FM order suddenly jumps

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Fig. 7. (Color online) (a) The magnetization M F at the quarter filling. There are two phases: the normal metal and the C = 1 FM Chern insulator (C = 1 FM-CI). (U /t ) F M is the magnetic phase transition point with the Ferromagnetic system. In the FM-CI phase, the ferromagnetization saturates suddenly. (b) The electron spectrums at the quarter filling. The chemical potential surface (yellow surface) lies √ in the middle of the energy gap. The parameters are U /t = 7, t 1 = −t 2 = t /(2 + 2), t  = −t /(2 +

√

2 2) and φ = π /4.

Fig. 9. (Color online) The spin wave’s dispersions of the FM ordered state for the case of U /t = 3 and √ U /t = 6 at quarter √ filling, respectively. Common parameters are t 1 = −t 2 = t /(2 + 2), t  = −t /(2 + 2 2), φ = π /4.

5. Conclusion and analysis

Fig. 8. The illustration of the edge states at quarter filling with U /t = 3, t 1 = −t 2 =

√

√

t /(2 + 2), t  = −t /(2 + 2 2) and φ = π /4. The system is under periodic boundary conditions along y, but open boundary conditions along x.

to 0.5 and gets saturated for the C = 1 FM Chern insulator, where (U /t ) F M is the magnetic phase transition point. In Fig. 7(b), we draw the electron dispersion at U /t = 7 in the quarter filling. From Fig. 7(b), we can see that the chemical potential (yellow surface) lies in the middle of the energy gap. Then the lowest energy band is filled, and the rest bands are not be filled. Fig. 8 shows the edge state of ferromagnetism in quarter filling at U /t = 3. In Fig. 8, we can see that there is a gapless edge state indicating the topological property of the C = 1 FM Chern insulator. Next, we use the similar approach to calculate the spin susceptibility and the dispersion of spin collective modes of FM ordered states. In Fig. 9, we can see that although there exist flat bands of electrons, the collective spin modes are dispersive due to the dispersive Matsubara Green’s function. At the low-energy limit, the spin wave of the FM order is quite different from that of the AF order: around the  point, the spin wave’s dispersions of the FM order show quadratic behavior ω(q) ∝ q2 instead of a linear one in the AF ordered state. Finally, we give a brief discussion for other doping cases, like d = 0.6. From the global phase diagram of Fig. 4(a), it can be seen that for the weak coupling, the ground state exhibits the normal metal state. With the small interaction, the ground state becomes FM metal state. There is a first-order phase transition between normal metal M F = 0 and FM metal M F = 0. With the increasing of the interaction strength, the ground state energy of the AF ordered state is always lower than that of the FM ordered state. Then the ground state becomes a AF ordered metal. The electronic band structure changes when the phase transition occurs.

In this paper, we have studied the magnetic properties of the interacting checkerboard lattice with topological flat band by means of mean field method. Comparing with Ref. [13], the phase diagram has some changes. The emergence of AF-Metal phase above d = 0.5 may originate from the different flatness ratio of different models. In addition, the energy difference between AFMetal phase and FM-Metal phase is not very large, the flatness of topological flat band may significantly affect the properties of the ground state of the system. At half filling, we find that the ground state energy of AF order is more stable than that of FM order due to energy gaining from super-exchange effect. While far from half filling, the ground state energy of FM order is more stable than that of AF order due to the existence of the flat band. An important result is that at quarter filling d = 0.5, the system becomes a C = 1 FM Chern insulator. When the doping is larger than 0.5, small interaction will induce FM metal. With the increasing of interaction, the ground state becomes an AF metal due to that the ground state energy of AF ordered state is lower than that of FM ordered state. In addition, we also find that the degenerate flat band becomes a dispersive one with the AF order, and splits into two flat bands with the FM order. Finally, we calculate the dispersion of spin collective modes by the RPA approach, and find that the spin waves of FM state and AF state are dispersive. For AF ordered state, the dispersion of spin wave exhibits linear behavior, while in FM order, the dispersion of spin wave shows quadratic behavior. In the end, we give a brief discussion on the possible physical realizations on checkerboard lattice with TFB. Highly controllability and various detection techniques of cold atoms provide a powerful experimental platform to realize models in condensed matter physics. The Chern insulator described by Eq. (1) on the checkerboard lattice can be realized by fermionic cold atoms (such as 40 K) trapped in a spin-dependent optical lattice [18,19], where the phases of hopping are reached by applying laser-induced gauge potentials [20–22]. The on-site interaction can be realized via Feshbach resonance technique [23,24]. However, the real challenge is to design an optical lattice with big enough NNN and NNNN hoppings. Recently, using shaken optical lattice to realize strong further-neighbor hopping will be helpful to solve this problem [25]. For the Hubbard model, there is a big challenge to cool the system down to a lower temperature due to the heating [26]. Therefore, there is still a long way to reach the topological flatband Hubbard model and detect the related order in the experiment.

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Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by NSFC Grant No. 11404090, 11304136, the Science Foundation for Youth Top-notch Talent of Hebei Education Department (Grant No. BJ2018053), the third batch Youth Topnotch Talent of Hebei Province, and Distinguished Young Scholar fund of Hebei Normal University. References [1] [2] [3] [4]

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