Landau Levels and Electronic States for Pseudospin-1 Lattices with a Bandgap: Application to a Lieb Lattice

Abstract

We have carried out detailed theoretical and numerical calculations and developed a physics-based model for quantitatively describing the Landau levels of several pseudospin-1 structures with a flat band and a finite bandgap in their electronic-energy spectrum under a strong and uniform magnetic field. We have investigated the Landau-level-based dynamics, as well as the corresponding eigenstates, for gapped graphene, a dice lattice with both a zero and finite bandgap and, eventually, for the Lieb lattice, which represents a separate type of square lattice with a very special non-symmetric (elevated) location of the flat band which intersects the conduction band at its lowest point. Exact analytical consideration of Landau-level states has been performed and explained when dealing with all types of considered lattices. Our model could be further generalized for treating cases with an arbitrary position for the flat band between the valence and conduction bands. Our current results have direct implications for a deep-level investigation of the quantum Hall effect, as well as other magnetic and topological properties of these novel materials.

1. Introduction

Since the groundbreaking transport measurements were reported on graphene in 2004 [1], all novel low-dimensional materials have received tremendous attention from researchers in condensed matter physics and relevant materials science fields. Later, some other structures with tunable spin-orbit bandgaps, such as buckled honeycomb lattices, were discovered. Their magneto-optical properties [2] were also studied thoroughly, including general collective-excitation modes [3,4,5,6,7,8] and massive hyperbolic plasmons [9]. Furthermore, other novel electron dispersions have been founded in both Kekule-distorted graphene [10,11,12,13] and semi-Dirac materials. Meanwhile, their optical properties [14,15] and topological electronic behaviors [16] were investigated extensively. Specifically, the optical conductivity of all these materials [12,17,18,19], including anisotropic and tilted [20] Dirac cones [21,22,23,24], were fully explored both theoretically and experimentally.

Among these recently discovered two-dimensional materials, considerable attention has been given to a unique group with a flat (or nondispersive) band in their low-energy band structure [25,26]. This flat band, which could be located arbitrarily between an upper-conduction and a lower-valance band, gives rise to infinite degeneracy for energies coinciding with those of the flat-band electrons and makes the electronic properties of this class of materials feature-rich [27]. Some typical examples of such flat-band materials include the dice, Lieb [28,29,30] and kagome lattices [31,32,33], as well as a few other important materials [34,35]. The latter materials also exhibit unique weak antilocalization and ferromagnetism in single crystalline magnetic Weyl semimetals, as well as quantum oscillation for the nodal line semimetal [36,37].

The $\alpha$-${\mathcal{T}}_3$ model represents a very special type of two-dimensional structure which also exhibits a flat band in their low-energy spectrum [38,39]. The flat band in the $\alpha$-${\mathcal{T}}_3$ model originates from the existence of an additional hub atom at the center of each hexagon of a graphene-like two-dimensional lattice. The hopping coefficients between the hub and rim (regular) atoms in the $\alpha$-${\mathcal{T}}_3$ model could be parameterized by a chosen number $\alpha$ within the range $0<\alpha <1$. Consequently, we obtain a physical model, which realistically describes a variety of materials with a flat band, and simultaneously, receives tremendous attention from researchers. Specifically, electronic structures [40,41,42], transport [43,44,45], collective optical [4], magnetic [46] and electronic phenomena in $\alpha$-${\mathcal{T}}_3$ rings [47,48] and the properties of other two-dimensional lattice materials [27,49] have been extensively investigated.

Interestingly, the main characteristics of the electronic dispersions, such as bandgaps, group velocities and the anisotropy of graphene and other two-dimensional materials, will change significantly by applying an off-resonance electromagnetic dressing field [50,51,52,53]. In such a case, the modification of the energy spectrum depends strongly on the polarization of incident irradiation, in addition to the field frequency and strength. In fact, there has already been a number of crucial publications aiming to study the electronic, collective and transport properties of materials with a Dirac cone deformed by an optical-dressing field [54,55,56].

As a different and impactful issue, the magnetic properties of these emergent two-dimensional materials, as well as their corresponding electron dynamics under magnetic fields, have been explored extensively. These have received encouraging responses from researchers working in various fields. This includes the calculation and detailed analysis of Landau-level dispersion and electronic states under a magnetic field in graphene [57,58,59], silicene [60,61] and transition-metal dichalcogenides [62,63], covering their magneto-optical [64,65] and magneto-transport [66,67,68,69,70] properties as well. Meanwhile, various magnetic-field responses of the $\alpha$-${\mathcal{T}}_3$ material were also studied [61]. Additionally, these research efforts further extend to magneto-plasmons and other collective behaviors in graphene materials [71] with a tunable spin-orbit bandgap [2,61,72,73] and in $\alpha$-${\mathcal{T}}_3$ model [74,75] as well. The Hamiltonian, Landau-level spectrum and zero-mode degeneracies in pseudospin-1 systems in Ref. [76] have a direct connection with the lattice topology [77], which represents one of the first demonstrations of an $\alpha -{\mathcal{T}}_3$ lattice. The low-energy spin-1 Hamiltonian and Landau levels in a Lieb lattice were initially addressed in Ref. [78]. The electronic states and the Landau levels for the magnetic field are the key building blocks for calculating the most important collective and transport properties of a two-dimensional material, such as magneto-conductivity, magneto-optical conductivity, Boltzmann transport coefficients and the corresponding inverse relaxation time. Specifically, the polarization function, which is the key ingredient for calculating the dielectric function and conductivity, is obtained as the sum of all possible transitions between the unoccupied and occupied electronic state (in our case, the transitions between Landau levels). This function can be calculated and evaluated theoretically once you know the electronic states in the new material. At the same time, magneto-conductivity, magneto-optical conductivity and magneto-plasmons have been experimentally measured in most known Dirac cone materials [79,80,81]. The structure of the Landau levels directly determines the Hall plateau and the Hall plateau-to-plateau transitions in many Dirac materials which directly affect a number of crucial physical quantities such as magneto-conductivity and their temperature dependence [82].

The Landau-level structure could be measured with the use of different techniques, including exploiting the Hall plateaus, which in the simplest approximation, represent each step at the energy of each consequent level. Integer or half-integer structures of the Hall plateaus are easily detectable in most infrared transmission measurements [83]. One can also directly measure the density-of-states in a new material using the so-called capacitance measurements [84].

The remainder of this paper is organized as follows: In Section 2, we present a general formalism for Dirac electrons in magnetic fields, including their energy eigenvalues (Landau levels) as well as their wave functions. Section 3 is devoted to presenting several novel derivations for some known results associated with Landau levels within gapped graphene, including computing wave functions for a case with a finite gap ${\Delta }_0$. Importantly, our most crucial results likewise stem from considering the finite bandgaps in several Dirac cone materials. In Section 4, we consider Landau quantization of electron energies and corresponding electronic wave functions within a dice lattice with both zero and finite bandgaps, accompanied by a flat band located midway between the valence and conduction bands. Following this, we derive in Section 5 the Landau levels and corresponding eigenstates of electrons within a Lieb lattice with an elevated flat band. Our final remarks and conclusions are presented in Section 6.

2. General Formalism for Landau Quantization

We begin with a brief review of the general mathematical formalism employed in calculating the energy eigenstates of free electrons under a spatially uniform magnetic field B, usually referred to as Landau levels. We shall employ the Landau gauge for the vector potential $\mathit{A}=\left\{A_x,A_y,A_z\right\}$ so that $A_x=B_{0}y$ while $A_y=A_z=0$. Therefore, the magnetic field in this case is given by $\mathit{B}=\mathbf{\nabla }\times \mathit{A}=B_0\widehat{z}$, which is uniform in space domain.

Formally, the effect of a magnetic field B can be included by using a so-called Peierls substitution in any considered Hamiltonian, i.e.,

$$k_{x,y}⟶k_{x,y}+{\displaystyle \frac{e}{\hslash }}{A}_{x,y}.$$

(1)

In the case of graphene, its Hamiltonian operator acquires the following explicit form [85]

$${\widehat{\mathcal{H}}}_g=\sqrt{2}{\displaystyle \frac{\hslash v_F}{{\ell }_B}}\left[\begin{array}{cc}0& \widehat{a}\\ {\widehat{a}}^{†}& 0\end{array}\right],$$

(2)

where ${\ell }_B=\sqrt{\hslash /e{B}_0}$ is magnetic length, and two operators are defined by

$$\begin{array}{ccc}\hfill & & \widehat{a}={\ell }_B{k}_x+{\displaystyle \frac{y}{{\ell }_B}}+i{\ell }_B{k}_y,\hfill \\ & & {\widehat{a}}^{†}={\ell }_B{k}_x+{\displaystyle \frac{y}{{\ell }_B}}-i{\ell }_B{k}_y,\hfill \end{array}$$

(3)

which are known as the creation and annihilation operators, respectively. Using the expression in Equation (3), one can further verify the well-known commutation relation $\left[\widehat{a},{\widehat{a}}^{†}\right]=1$ and meanwhile obtain the results for their actions on the electronic states in the Fock space, given by

$$\begin{array}{ccc}\hfill & & \widehat{a}|n\rangle =\sqrt{n}|n-1\rangle ,\hfill \\ & & {\widehat{a}}^{†}|n\rangle =\sqrt{n+1}|n+1\rangle .\hfill \end{array}$$

(4)

Here, the results in Equations (1)–(4) are all the relations required to calculate the actual Landau levels and their corresponding wave functions in the Dirac cone materials considered in this study.

3. Magneto-Energy Levels of Gapped Graphene

To start, we consider the case of gapped graphene described by the following low-energy Hamiltonian

$${\widehat{\mathcal{H}}}_g^{\left(0\right)}={\widehat{\Sigma }}_{x,y}^{\left(2\right)}·k+{\Delta }_0{\widehat{\Sigma }}_z^{\left(2\right)}=\left[\begin{array}{cc}{\Delta }_0& \hslash v_F(k_x-i{k}_y)\\ \hslash v_F(k_x+i{k}_y)& -{\Delta }_0\end{array}\right],$$

(5)

which differs from well-known graphene by a finite gap $2{\Delta }_0$ included as a ${\widehat{\Sigma }}_z^{\left(2\right)}$ term. Here, ${\widehat{\Sigma }}_{x,y,z}^{\left(2\right)}$ are regular $2\times 2$ Pauli matrices.

In the presence of a magnetic field $\mathit{B}=B_0\widehat{z}$, the canonical or Peierls substitution in Equation (1) can be employed. This leads to the following new Hamiltonian

$${\widehat{\mathcal{H}}}_g\left(B\right)=\left[\begin{array}{cc}{\Delta }_0& {\gamma }_B\widehat{a}\\ \\ {\gamma }_B{\widehat{a}}^{†}& -{\Delta }_0\end{array}\right],$$

(6)

where ${\gamma }_B=\sqrt{2}\hslash v_F/{\ell }_B$. Mathematically speaking, the simplest way to find energy eigenvalues of the Hamiltonian in Equation (6) is by simply taking the square on both sides of the eigenvalue equation. As a result, we get a new diagonal Hamiltonian matrix, written as

$${\widehat{\mathcal{H}}}_g^2\left(B\right)=2{\left({\displaystyle \frac{\hslash v_F}{{\ell }_B}}\right)}^2\left[\begin{array}{cc}{\gamma }_B({\widehat{a}}^{†}\widehat{a}+1)+{\Delta }_0^2& 0\\ \\ 0& {\gamma }_B{\widehat{a}}^{†}\widehat{a}+{\Delta }_0^2\end{array}\right].$$

(7)

By using the harmonic-oscillator relation ${\widehat{a}}^{†}\widehat{a}|n\rangle =n|n\rangle$, as well as the following eigenvalue relation

$${\widehat{\mathcal{H}}}_g^2|{\Psi }_n\rangle ={\epsilon }_n^2|{\Psi }_n\rangle ,$$

(8)

we immediately find ${\epsilon }_n=\pm \sqrt{2n{\gamma }_B^2+{\Delta }_0^2}$.

In a similar way, we will look for a solutions in the following general form

$$|{\Psi }_{n_1,n_2}^{\left(G\right)}\rangle =\left[\begin{array}{c}{c}_1|n_1\rangle \\ \\ c_2|n_2\rangle \end{array}\right],$$

(9)

then the corresponding eigenvalue equations become

$$\begin{array}{ccc}\hfill {\Delta }_0{c}_1|n_1\rangle +{\gamma }_B{c}_2\widehat{a}|n_2\rangle & =& \epsilon c_1|n_1\rangle ,\hfill \\ \hfill \\ \hfill {\gamma }_B{c}_1{\widehat{a}}^{†}|n_1\rangle -{\Delta }_0{c}_2|n_2\rangle & =& \epsilon c_2|n_2\rangle .\hfill \end{array}$$

(10)

Equation (10) can be equivalently written as

$$\begin{array}{ccc}\hfill {\Delta }_0{c}_1|n_1\rangle +{\gamma }_B{c}_2\sqrt{n_2}|n_2-1\rangle & =& \epsilon c_1|n_1\rangle ,\hfill \\ \hfill \\ \hfill {\gamma }_B{c}_1\sqrt{n_1+1}|n_1+1\rangle -{\Delta }_0{c}_2|n_2\rangle & =& \epsilon c_2|n_2\rangle .\hfill \end{array}$$

(11)

From Equation (11), we easily find that its solution in the Fock space exists only for $n_1=n_2-1$, or equivalently $n_2=n_1+1$. Now, let us assume $n_2=n$ and $n_1=n-1$. By following Equation (11), we acquire

$$\begin{array}{ccc}\hfill {\Delta }_0{c}_1|n-1\rangle +{\gamma }_B{c}_2\sqrt{n}|n-1\rangle & =& \epsilon c_1|n-1\rangle ,\hfill \\ \hfill \\ \hfill {\gamma }_B{c}_1\sqrt{n}|n\rangle -{\Delta }_0{c}_2|n\rangle & =& \epsilon c_2|n\rangle .\hfill \end{array}$$

(12)

The presence of a physical solution to Equation (12) requires that its coefficient determinant must be zero. This leads to the following dispersion relation

$$2v_F^{2}eB\hslash n-{\epsilon }_n^2+{\Delta }_0^2=0,$$

(13)

which produces ${\epsilon }_n=\pm \sqrt{2v_F^{2}eB\hslash n+{\Delta }_0^2}$, and it is equivalent to a spectrum equation for the Landau levels of a gapped graphene. Additionally, its wave function for $n\ge 1$ is found to be

$$|{\Psi }_n\rangle \left({\Delta }_0\right)=\left[\begin{array}{c}{\gamma }_B\sqrt{n}|n-1\rangle \\ \\ \pm \left(\sqrt{{\gamma }_B^{2}n+{\Delta }_0^2}-{\Delta }_0\right)|n\rangle \end{array}\right].$$

(14)

If ${\Delta }_0\to 0$, on the other hand, the result of Equation (14) immediately turns into

$$|{\Psi }_{n>0}\rangle ({\Delta }_0\to 0)={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}|n-1\rangle \\ \\ \pm |n\rangle \end{array}\right],$$

(15)

which is reduced in the case of free-standing (zero-gap) graphene. In particular, for $n=0$, the wave function (14) reduces to

$$|{\Psi }_{n=0}\rangle =\left[\begin{array}{c}0\\ \\ |0\rangle \end{array}\right].$$

(16)

Here, we have finished the calculation of a closed-form analytical expression for both the electronic states and Landau energy levels of electrons within a gapped graphene under an external perpendicular magnetic field.

4. Dice Lattice with Zero Bandgap

As we mentioned previously in this paper, our present study focuses on investigating pseudospin-1 Dirac materials with a flat band in their low-energy spectrum. The schematics of energy dispersions in both gapped dice and Lieb lattices, as depicted in Figure 1, contain a flat band, which makes these materials and their electronic properties similar to each other. Interestingly, different positions of this flat band inside a bandgap result in a significant difference between the two. In the case of a gapped dice lattice, it sits exactly in the middle between the valence and conduction bands, separated equally from band edges by a bandgap parameter ${\Delta }_0$, and this makes the whole energy spectrum very symmetric. In another case, i.e., the Lieb lattice, the flat band is located at an elevated position, intersecting the bottom of a conduction band. Such an overlap of the flat band with the bottom of a conduction band has a lot of implications for the collective and many-body properties of electrons in these types of materials, e.g., in the computed polarization function, where all electronic transitions and various energy separations will be taken into account. Here, we focus on the role which the location of the flat-band plays in the presence of a perpendicular and spatially uniform magnetic field.

The low-energy Hamiltonian for a dice lattice with a zero-gap parameter ${\Delta }_0=0$ (the actual bandgap is $2{\Delta }_0$) can be written as

$${\widehat{\mathcal{H}}}^{\left(D\right)}\left(k\right)={\displaystyle \frac{\hslash v_F}{\sqrt{2}}}\left[\begin{array}{ccc}0& k{e}^{-i{\Theta }_{\mathbf{k}}}& 0\\ k{e}^{i{\Theta }_{\mathbf{k}}}& 0& k{e}^{-i{\Theta }_{\mathbf{k}}}\\ 0& k{e}^{i{\Theta }_{\mathbf{k}}}& 0\end{array}\right].$$

(17)

However, in the presence of a magnetic field, the Hamiltonian in Equation (17) is transformed into

$${\widehat{\mathcal{H}}}^{\left(D\right)}\left(k\right)={\gamma }_B^{\left(D\right)}\left[\begin{array}{ccc}0& \widehat{a}& 0\\ \widehat{a^{†}}& 0& \widehat{a}\\ 0& {\widehat{a}}^{†}& 0\end{array}\right],$$

(18)

where $\widehat{a}$ and ${\widehat{a}}^{†}$ represent annihilation and creation operators, respectively, and ${\gamma }_B^{\left(D\right)}={\gamma }_B^{\left(G\right)}/\sqrt{2}=\hslash v_F/{\ell }_B$. From now on, we will calculate the energy levels (Landau levels) of a dice lattice in cases with a zero bandgap ${\Delta }_0=0$.

Here, we first write down the wave function in the most general form as

$$|{\Psi }_{n_1,n_2,n_3}^{\left(D\right)}\rangle =\left[\begin{array}{c}{c}_1|n_1\rangle \\ c_2|n_2\rangle \\ c_3|n_3\rangle \end{array}\right],$$

(19)

where $c_1,c_2,c_3$ are three coefficients to be determined. Inserting the wave function in Equation (19) into a Schrödinger equation with its Hamiltonian given by Equation (18), one obtains an eigenvalue equation as follows:

$$\begin{array}{ccc}\hfill & & {\gamma }_B^{\left(D\right)}\widehat{a}{c}_2|n_2\rangle -\epsilon c_1|n_1\rangle =0,\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}{\widehat{a}}^{†}{c}_1|n_1\rangle +{\gamma }_B\widehat{a}{c}_3|n_3\rangle -\epsilon c_2|n_2\rangle =0,\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}{\widehat{a}}^{†}{c}_2|n_2\rangle -\epsilon c_3|n_3\rangle =0,\hfill \end{array}$$

(20)

where parameter $\epsilon$ stands for the eigenvalue to be determined. After analyzing Equation (20), we know that the conditions of $n_1=n_2-1$ and $n_2=n_3-1$ must be satisfied in order to acquire a nonzero (non-trivial) solution for these coupled equations. Specifically, by writing $n_3=n$, $n_2=n-1$ and $n_1=n-2$, we find from Equation (20) that

$$\begin{array}{ccc}\hfill & & -\epsilon c_1+{\gamma }_B^{\left(D\right)}\sqrt{n-1}{c}_2=0,\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}\sqrt{n-1}{c}_1-\epsilon c_2+{\gamma }_B^{\left(D\right)}\sqrt{n}{c}_3=0,\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}\sqrt{n}{c}_2-\epsilon c_3=0.\hfill \end{array}$$

(21)

After an analysis of Equation (21), it becomes clear that this system can support the nonzero solution ${[c_1,c_2,c_3]}^T\ne 0$ only if its coefficient determinant is zero. In this way, we find the energy dispersions of a dice lattice, given by ${\epsilon }_n=0$ for a flat band, as well as by another nonzero solution

$${\epsilon }_n=\pm {\gamma }_B^{\left(D\right)}\sqrt{2n-1}.$$

(22)

The solved wave functions associated with Equation (22) are

$$|{\Psi }_n^{\left(D\right)}\rangle ={\displaystyle \frac{1}{\sqrt{2(2n-1)}}}\left[\begin{array}{c}\sqrt{n-1}|n-2\rangle \\ \pm |n-1\rangle \\ \sqrt{n}|n\rangle \end{array}\right],$$

(23)

where $n=1,2,3,\dots$. By choosing $n=1$ in Equation (23) as an example, we have

$$|{\Psi }_{n=1}^{\left(D\right)}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}0\\ \pm |0\rangle \\ |1\rangle \end{array}\right].$$

(24)

For the flat band ${\epsilon }_n=0$, on the other hand, its wave function is

$$|{\Psi }_n^{\left(D\right)}\rangle ={\displaystyle \frac{1}{\sqrt{(2n-1)}}}\left[\begin{array}{c}\sqrt{n}|n-2\rangle \\ 0\\ -\sqrt{n-1}|n\rangle \end{array}\right],$$

(25)

where $n=2,3,4,\dots$. However, as $n=1$, the wave function is modified to

$$|{\Psi }_{n=1}^{\left(D\right)}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}0\\ 0\\ -|0\rangle \end{array}\right].$$

(26)

The Landau levels for a zero-bandgap dice lattice, as shown in Figure 2, are actually a specific limiting case of the energy eigenstates of $\alpha$-${\mathcal{T}}_3$ materials, which were discussed in Ref. [46]. In the presence of a magnetic field, the eigen-energies of valence and conduction bands reduce to an infinite set of quantized energy levels $\backsim \pm \sqrt{2n-1}$, but the flat band ${\epsilon }_n=0$ remains dispersionless with an infinite degeneracy. Consequently, we conclude that such a configuration of Landau levels displays both similarities and distinctions (a constant energy shift) in comparison with the energy dispersion ${\epsilon }_n\backsim \sqrt{2n}$ of graphene.

5. Gapped Dice Lattice

A dice lattice with a finite bandgap ${\Delta }_0$ is obtained in terms of a $3\times 3$ generalization of the Pauli matrices ${\widehat{\Sigma }}_{x,y,z}^{\left(3\right)}$ in the following way:

$${\widehat{\mathcal{H}}}^{\left(D\right)}\left(k\right)={\widehat{\Sigma }}_{x,y}^{\left(3\right)}·k+{\Delta }_0{\widehat{\Sigma }}_z^{\left(3\right)}={\displaystyle \frac{\hslash v_F}{\sqrt{2}}}\left[\begin{array}{ccc}{\Delta }_0& k{e}^{-i{\Theta }_{\mathbf{k}}}& 0\\ k{e}^{i{\Theta }_{\mathbf{k}}}& 0& k{e}^{-i{\Theta }_{\mathbf{k}}}\\ 0& k{e}^{i{\Theta }_{\mathbf{k}}}& -{\Delta }_0\end{array}\right],$$

(27)

where

$${\widehat{\Sigma }}_x^{\left(3\right)}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right],$$

(28)

$${\widehat{\Sigma }}_y^{\left(3\right)}=\left[\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right],$$

(29)

which are supplemented by an extra bandgap-related matrix, i.e.,

$${\widehat{\Sigma }}_z^{\left(3\right)}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right].$$

(30)

In the presence of a magnetic field, the Hamiltonian in Equation (27) is transformed into a form similar to the expression in Equation (18), and written as

$${\widehat{\mathcal{H}}}^{\left(D\right)}\left(k\right)={\gamma }_B^{\left(D\right)}\left[\begin{array}{ccc}{\Delta }_0& {\widehat{a}}^{†}& 0\\ {\widehat{a}}^{†}& 0& \widehat{a}\\ 0& {\widehat{a}}^{†}& -{\Delta }_0\end{array}\right].$$

(31)

In correspondence with Equation (21) having ${\Delta }_0=0$, for the current case, we find

$$\begin{array}{ccc}\hfill & & -({\Delta }_0+\epsilon )c_1+{\gamma }_B^{\left(D\right)}\sqrt{n-1}{c}_2=0\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}\sqrt{n-1}{c}_1-\epsilon c_2+{\gamma }_B^{\left(D\right)}\sqrt{n}{c}_3=0\hfill \\ \hfill & & {\gamma }_B^{\left(D\right)}\sqrt{n}{c}_2+({\Delta }_0-\epsilon )c_3=0.\hfill \end{array}$$

(32)

Compared with the case of ${\Delta }_0=0$, the finite value of ${\Delta }_0$ here results in a quite different asymmetric dispersion equation, i.e.,

$${\left({\gamma }_B^{\left(D\right)}\right)}^2{\Delta }_0-{\left({\gamma }_B^{\left(D\right)}\right)}^2{\epsilon }_n+2n{\left({\gamma }_B^{\left(D\right)}\right)}^2{\epsilon }_n+{\Delta }_0^2{\epsilon }_n-{\epsilon }_n^3=0,$$

(33)

which cannot be solved in any straightforward way. However, it could be solved by utilizing an analogy to a trigonometrical expansion formula, as discussed in Ref. [42].

Equation (33) could also be solved approximately by using a perturbative approach. The comparatively basic problem with a completely analytical solution is a case of zero bandgap. Therefore, the perturbation equations result in expressions for the energy eigenvalues with good precision for the small bandgap. Specifically, it should be small compared to the main characteristic energy given by the coefficient ${\gamma }_B^{\left(L\right)}$ for the specific material under consideration.

The linear term of the perturbation expansion is obtained as follows:

$${\epsilon }_n={\epsilon }_n^{\left(0\right)}+{\epsilon }_n^{\left(1\right)}{\Delta }_0+{\epsilon }_n^{\left(2\right)}{\Delta }_0^2+\dots ,$$

(34)

where the bandgap ${\Delta }_0$ is regarded as a small parameter for this expansion. Interestingly, for the flat band with ${\epsilon }_n^{\left(0\right)}=0$, one obtains ${\epsilon }_n^{\left(1\right)}=-1/(2n-1)$, and thus

$${\epsilon }_n={\displaystyle \frac{-{\Delta }_0}{2n-1}}.$$

(35)

We have obtained numerically a set of discrete Landau levels which are presented in Figure 3. We first notice that the valence and conduction bands are not exactly symmetric with respect to zero energy and no simple analytical formula or approximate expression exists for describing the energy locations of these levels. For the high-energy states with $n\gg 1$, the energy eigenstates are well approximated by expressions $\backsim \pm \sqrt{2n-1}$, implying that the effect due to bandgap ${\Delta }_0$ is decreased. This further supports the validity of a simple but efficient application of the WKB approximation for this case. Here, the flat band is no longer dispersionless and its degeneracy is lifted. On the other hand, we find an infinite set of separated non-degenerate Landau levels within a negative-energy ${\epsilon }_n<0$ region. For large $n\gg 1$, the effect of an energy gap becomes negligible and these levels approach the ${\epsilon }_n=0$ level.

We obtained a precise numerical solution and also applied perturbation theory to validate our closed-form analytical expression which, of course, had limited precision. We note that this analytical formula reproduces the most important features of the obtained exact results: it is negative (lies below the zero energy level) and is proportional to the gap. This means that it would be equal to zero for ${\Delta }_0\to 0$.

Next, we focus on the dependence of energy levels for a gapped dice lattice on the gap parameter ${\Delta }_0$, and present various numerically computed results in Figure 4 for a full comparison among Landau levels in the conduction, valence and flat bands all together. As found from Figure 4, these numerical results once again agree with our previous findings that the high-energy Landau levels with a large level index n ($n\gg 1$) will be much less sensitive to the size of bandgap ${\Delta }_0$, which can be easily verified by the fact that large-n Landau levels would like to group together as ${\Delta }_0$ increases in the flat-band case. In particular, the $n=1$ Landau level in panel (b) demonstrates a negative linear dependence on ${\Delta }_0$, which fully agrees with our perturbation-based solution in Equation (35).

6. Modeling Lieb Lattice with Elevated Flat Band

We now turn our attention to the major issue of this paper, i.e., modeling the effect of a magnetic field on a Lieb lattice with a non-symmetric elevated flat band, as depicted in Figure 1b. In this case, the low-energy Hamiltonian is found to be [24]

$${\widehat{\mathcal{H}}}^{\left(L\right)}(\mathit{k}|k_{\Delta })=\hslash v_F\left[\begin{array}{ccc}{k}_{\Delta }& k_x& 0\\ k_x& -k_{\Delta }& k_y\\ 0& k_y& k_{\Delta }\end{array}\right],$$

(36)

where the following substitution

$$k_{x,y}\to {\displaystyle \frac{\pi }{a_0}}+k_{x,y}$$

(37)

is required to obtain the low-energy spectrum for a Lieb lattice. Here, $a_0$ represents the lattice parameter and before making the substitution in Equation (37) into Equation (36), the three energy dispersions can be found as eigenvalues of the Hamiltonian in Equation (36) as ${\epsilon }_{\gamma =\pm 1}^{\left(L\right)}(\mathit{k}|k_{\Delta })/\hslash v_F=\gamma \sqrt{k_{\Delta }^2+k_x^2+k_y^2}$ and ${\epsilon }_{\gamma =0}^{\left(L\right)}(\mathit{k}|k_{\Delta })/\hslash v_F=k_{\Delta }$, which could be combined into a single expression, yielding

$${\epsilon }_{\gamma }^{\left(L\right)}(\mathit{k}|k_{\Delta })=\hslash v_F\left[{\delta }_{\gamma ,0}{k}_{\Delta }+\gamma (1-{\delta }_{\gamma ,0})\sqrt{k_{\Delta }^2+k^2}\right],$$

(38)

where $\gamma =0,\pm 1$ and ${\delta }_{\gamma ,0}$ is the Kronecker delta symbol. Its nonzero values demonstrate one of the two types of energy eigenvalues: dispersive valence and conduction bands for $\gamma =\pm 1$ and the flat band for $\gamma =0$.

In the presence of a magnetic field, however, the two components of the wave vector are modified as $k_{x,y}\to k_{x,y}+r/\hslash A_{x,y}$. Therefore, we obtain that $2k_x=k_{+}+k_{-}$ and $2i{k}_y=k_{+}-k_{-}$ and we can present the Hamiltonian in Equation (36) in terms of the substitutions calculated earlier for quantities $k_{+}$ and $k_{-}$. As a result, in the presence of a magnetic field, we obtain

$${\widehat{\mathcal{H}}}^{\left(L\right)}(\mathit{k}|k_{\Delta })=\left[\begin{array}{ccc}{\Delta }_0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}}{2}}\left(\widehat{a}+{\widehat{a}}^{†}\right)& 0\\ {\displaystyle \frac{{\gamma }_B^{\left(L\right)}}{2}}\left(\widehat{a}+{\widehat{a}}^{†}\right)& -{\Delta }_0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}}{2i}}\left(\widehat{a}-{\widehat{a}}^{†}\right)\\ 0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}}{2i}}\left(\widehat{a}-{\widehat{a}}^{†}\right)& {\Delta }_0\end{array}\right].$$

(39)

Here, it is straightforward to verify that one cannot project this Hamiltonian onto the same Fock state $|n\rangle$ representation. However, by searching for the eigen-function in terms of Equation (39), we obtain one of its eigenvalue equations, given by

$${\Delta }_0{c}_1|n_1\rangle +{\displaystyle \frac{{\gamma }_B^{\left(L\right)}}{2}}\left[\sqrt{n_2}{c}_2|n_2-1\rangle +\sqrt{n_2+1}{c}_2|n_2+1\rangle \right]={\epsilon }_n{c}_1|n_1\rangle .$$

(40)

For the considered model, each of the three equations corresponding to the three components of the wave functions contain non-equivalent Fock states $|n_1\rangle$ and $|n_2\rangle$. Therefore, Equation (40) cannot be satisfied by any $|n_1\rangle$ and $|n_2\rangle$ states.

On the other hand, we are also able to model the Lieb lattice by an elevated flat band, as illustrated in Figure 1. For this purpose, let us first write down the Hamiltonian, using an elevated flat band, as

$${\widehat{\mathcal{H}}}^{\left(L\right)}(\mathit{k})=\left[\begin{array}{ccc}{\Delta }_0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}k}{2}}{e}^{-i{\Theta }_{\mathbf{k}}}& 0\\ {\displaystyle \frac{{\gamma }_B^{\left(L\right)}k}{2}}{e}^{i{\Theta }_{\mathbf{k}}}& -{\Delta }_0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}k}{2}}{e}^{-i{\Theta }_{\mathbf{k}}}\\ 0& {\displaystyle \frac{{\gamma }_B^{\left(L\right)}k}{2}}{e}^{i{\Theta }_{\mathbf{k}}}& {\Delta }_0\end{array}\right],$$

(41)

which results in two energy dispersions, given by ${\epsilon }_{\gamma }^{\left(L\right)}\left(k\right)={\Delta }_0$ as well as ${\epsilon }_{\gamma }^{\left(L\right)}\left(k\right)=\gamma \sqrt{k^2+{\Delta }_0^2}$. Furthermore, in the presence of a magnetic field, we find that the Hamiltonian in Equation (39) changes into

$${\widehat{\mathcal{H}}}^{\left(L\right)}\left(k\right)={\gamma }_B^{\left(L\right)}\left[\begin{array}{ccc}{\Delta }_0& \widehat{a}& 0\\ \widehat{a}& -{\Delta }_0& {\widehat{a}}^{†}\\ 0& {\widehat{a}}^{†}& {\Delta }_0\end{array}\right].$$

(42)

Our model Hamiltonian (42) still carries the most important features of the initially considered Hamiltonian of the Lieb lattice and faithfully reproduces the most crucial observables of the investigated quantum states. That is, the energy band structure and the density-of-states. This is directly verified by comparing both types of eigenvalue equations in k-space in the absence of a magnetic field.

Correspondingly, the previous Equation (21) for a dice lattice will be changed into

$$\begin{array}{ccc}\hfill & & ({\Delta }_0-\epsilon )c_1+{\gamma }_B^{\left(L\right)}\sqrt{n-1}{c}_2=0,\hfill \\ \hfill & & {\gamma }_B^{\left(L\right)}\sqrt{n-1}{c}_1-({\Delta }_0+\epsilon )c_2+{\gamma }_B^{\left(D\right)}\sqrt{n}{c}_3=0,\hfill \\ \hfill & & {\gamma }_B^{\left(L\right)}\sqrt{n}{c}_2+({\Delta }_0-\epsilon )c_3=0.\hfill \end{array}$$

(43)

Equation (43) leads to the eigenvalue equation as

$${\left({\gamma }_B^{\left(L\right)}\right)}^2{\Delta }_0-2n{\left({\gamma }_B^{\left(L\right)}\right)}^2{\Delta }_0-{\Delta }_0^3-{\left({\gamma }_B^{\left(L\right)}\right)}^2{\epsilon }_n+2n{\left({\gamma }_B^{\left(L\right)}\right)}^2{\epsilon }_n+{\Delta }_0^2{\epsilon }_n+{\Delta }_0{\epsilon }_n^2-{\epsilon }_n^3=0,$$

(44)

which gives rise to the energy levels by

$${\epsilon }_n={\Delta }_0,$$

(45)

$${\epsilon }_n=\pm \sqrt{{\left({\gamma }_B^{\left(L\right)}\right)}^2(2n-1)+{\Delta }_0^2}.$$

(46)

Here, the most interesting thing is the wave function corresponding to the flat band with ${\epsilon }_n={\Delta }_0$. In fact, from Equation (43), we immediately conclude that $c_2=0$. Thus, this leaves us with

$$\sqrt{n-1}{c}_1=-\sqrt{n}{c}_3,$$

(47)

and the corresponding wave function is given by

$$|{\Psi }_n^{\left({\Delta }_0\right)}\rangle ={\displaystyle \frac{1}{\sqrt{(2n-1)}}}\left[\begin{array}{c}\sqrt{n}|n-2\rangle \\ 0\\ -\sqrt{n-1}|n\rangle \end{array}\right],$$

(48)

where $n=2,3,4,\dots$. For $n=1$, especially, this wave function is taken as

$$|{\Psi }_{n=1}^{\left({\Delta }_0\right)}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}0\\ 0\\ -|0\rangle \end{array}\right].$$

(49)

From Equations (48) and (49), we quickly find that they are identical to Equations (25) and (26) in the case of a zero-gap dice lattice.

Finally, we calculate and plot the Landau levels for our model for a Lieb lattice obtained by relatively simple analytical expressions in Equations (45) and (46). Our results are presented in Figure 5 and Figure 6. From these two figures, we find that these energy levels, corresponding to the valence and conduction bands, become symmetric with respect to the zero energy. Similarly to the previously considered case for a gapped dice lattice, the effect of an energy bandgap diminishes for higher Landau levels with $n\gg 1$. Meanwhile, the flat band remains dispersionless with a simple shift to the gap level ${\epsilon }_n={\Delta }_0$ and remains infinitely degenerate. The dependence of these energy levels on the gap ${\Delta }_0$, as shown in Figure 6, clearly demonstrates a linear dependence ${\epsilon }_n={\Delta }_0$, as well as a smooth and monotonic dependence of all energy levels, corresponding to the valence and conduction bands, on the gap parameter ${\Delta }_0$.

7. Summary and Remarks

In conclusion, we note that the main thrust of this paper is to present a thorough theoretical and numerical investigation of the quantized energy levels under a uniform external magnetic field (i.e., Landau levels) and the corresponding electronic states for several important Dirac cone materials with a flat band in their low-energy spectra. The key research focus of this paper has been an investigation of how the combination of a flat band and a finite bandgap affects the energy levels and electronic eigenstates in the presence of a strong magnetic field. Specifically, we were interested to find out how the position of the flat band affects the magnetic field properties of a two-dimensional lattice since in the absence of the gap a specific set of degenerate Landau levels are related to the flat band. However, the situation changes significantly in the presence of the gap and depends on where exactly this flat band is located inside the bandgap.

First, we considered a well-known case of graphene with an energy bandgap and provided new derivations for Landau quantization in this system, given by $\backsim \sqrt{{(\hslash v_{F}k)}^2+{\Delta }_0^2}$. Surprisingly, the square Hamiltonian of graphene with a finite bandgap under a magnetic field reduced to a diagonal matrix, which immediately gave rise to the corresponding energy eigenvalues, similarly to the case of an intrinsic graphene with a zero bandgap.

Next, we investigated the electronic states under a magnetic field in a dice lattice with a finite bandgap, which was shown as a limiting case of the general $\alpha$-${\mathcal{T}}_3$ model. However, the presence of a finite gap turned the eigenvalue equations into a non-trivial cubic equation for computing Landau levels in this system. This cubic equation could be solved either numerically or by using an analogy between this expression and a well-known trigonometric equation, or even a perturbation theory for a small bandgap parameter ${\Delta }_0$. The result is in stark contrast with regular energy dispersions (in the absence of a magnetic field), which could be easily found from a simple algebraic equation.

Finally, we investigated the Landau levels for a realistic model of a Lieb lattice with an elevated flat band intersecting the conduction band at its lowest point. This model has both common features and differences from the dice lattice and the $\alpha$-${\mathcal{T}}_3$ model. The main interest of such physics models is the presence of a flat band within the bandgap region, which greatly affects the electronic, magnetic and collective properties of the material. More importantly, the location of this flat band can be quite unique for the Lieb lattice, which results in substantially reduced symmetry of its low-energy spectrum. We have calculated the Landau levels of this type of energy dispersion and obtained a relatively simple analytical result $\backsim \sqrt{2n{\left({\gamma }_B^{\left(L\right)}\right)}^2+{\Delta }_0^2}$, which provides a quantitative description of its dynamical feature under a quantizing magnetic field. Meanwhile, we have also calculated the wave function corresponding to the flat band and compared its similarity with other wave functions in a dice lattice.

Generally speaking, magnetic quantization as well as electronic and collective properties under a strong magnetic field, including magneto-transport and magneto-plasmon properties, are crucial for understanding the fundamental physics of any new material, such as the quantum Hall effect. We have calculated new types of energy eigenvalues and corresponding wave functions. We strongly believe that these findings for several novel materials will inevitably open up a number of opportunities for researchers to investigate their various properties: electronic, optical, collective, magnetic and quantum transport. Therefore, we believe that our work and the obtained results have a lot of potential to be expanded upon in terms of their application to specific electronic states. Our theoretical model, novel closed-form analytical expressions and numerical results are believed to become a crucial advancement in developing next-level novel electronic nanodevices.