Geometric characterization of anomalous Landau levels of isolated flat bands

The original Onsager’s semiclassical approach predicts IFBs inert under external magnetic field, and thus it cannot explain the LLS of IFBs. On the other hand, the modified semiclassical approach developed by M.-C. Chang and Q. Niu Chang and Niu (1996) can resolve this problem. Contrary to the Onsager’s approach, where the band structure at zero magnetic field $\varepsilon_{n}({{\mathbb{k}}})$ is used to define the closed semiclassical orbits and the corresponding area $S_{0}(\epsilon)$, the modified semiclassical approach employs the modified band structure given by

where ${\mathbb{B}}=B\hat{{\mathbb{z}}}$ is the magnetic field, $n$ is the band index, and $\mu_{n}({{\mathbb{k}}})$ is the orbital magnetic moment of the $n$-th magnetic band in the $z$-direction arising from the self-rotation of the corresponding wave packet Chang and Niu (1996). The explicit form of $\mu_{n}({{\mathbb{k}}})$ is

where $H({{\mathbb{k}}})$ is the Hamiltonian in momentum space and $\partial_{i}=\partial_{k_{i}}$ $(i=x,y)$. Hence, the second term on the right-hand side of Eq. (3) indicates the leading energy correction from the orbital magnetic moment coupled to the magnetic field. In usual dispersive bands, the $B$-linear quantum correction is negligibly small in weak magnetic field limit compared to the zero-field band-width. This is the reason why the original Onsager’s semiclassical scheme in Eq. (1) works well.

In the case of a flat band with zero band-width, on the other hand, the $B$-linear quantum correction always dominates the modified band structure $E_{n,B}({{\mathbb{k}}})$ in Eq. (3) even in weak magnetic field limit. Moreover, the modified band dispersion of an IFB is generally dispersive so that the relevant semiclassical orbits can be defined unambiguously. As a result, one can obtain the Landau levels of the IFB in the adjacent gapped regions by applying the semiclassical quantization rule to $E_{n,B}({{\mathbb{k}}})$, which naturally explains the LLS of the IFB. Especially, around the band edges of $E_{n,B}({{\mathbb{k}}})$, one can define the effective mass $m^{*}$, which is inversely proportional to $B$, from which the Onsager’s scheme predicts Landau levels with a spacing $\hbar eB/m^{*}\propto B^{2}$. The resulting Landau spectrum is bounded by the upper and lower band edges of $E_{n,B}({{\mathbb{k}}})$. The total magnitude $\Delta$ of the LLS is determined by the difference between the maximum and the minimum values of $E_{n,B}({{\mathbb{k}}})$, namely, $\Delta={\rm max}\,E_{n,B}({{\mathbb{k}}})-{\rm min}\,E_{n,B}({{\mathbb{k}}})$. This result is valid as long as the band gap $E_{\rm gap}$ between the IFB and its neighboring band at zero magnetic field is large enough, i.e., $E_{\rm gap}\gg{\rm max}|E_{n,B}({{\mathbb{k}}})|$. The generic behavior of an IFB under magnetic field is schematically described in Fig. 1 where one can clearly observe that the Landau levels of the IFB spread into the gaps at zero-field above and below the IFB.

Interestingly, we find that the LLS of IFBs is a manifestation of the non-trivial wave-function geometry of the flat band arising from inter-band couplings. One can show that the modified band dispersion of the IFB is given by

in which

where $\phi_{0}=h/e$, $\phi=BA_{0}$ is the magnetic flux per unit cell, and $A_{0}$ is the unit cell area assumed to be $A_{0}=1$. Here, we assume that the $n$-th band is the IFB at the zero energy without loss of generality so that $\varepsilon_{m}({{\mathbb{k}}})$ in Eq. (5) should be interpreted as the energy of the $m$-th band with respect to the flat band energy. We note that $A^{nm}_{i}({\mathbb{k}})$ indicates the cross-gap Berry connection between the $n$-th and $m$-th bands ($n\neq m$) defined in Eq. (2), and $\chi^{nm}_{ij}({\mathbb{k}})$ is the corresponding fidelity tensor that describes the transition amplitude between the $n$-th and $m$-th bands as discussed below. See Supplementary Note 1 for the detailed derivation of Eq. (5). Hence, Eq. (5) indicates that the modified band dispersion of the IFB is given by the summation of the transition amplitudes $\chi^{nm}_{xy}({{\mathbb{k}}})$ between the IFB and the $m$-th band weighted by the energy $\varepsilon_{m}({{\mathbb{k}}})$ of the $m$-th band as illustrated in Fig. 1. This means that the immobile carriers with infinite effective mass in an IFB can respond to external magnetic field through the inter-band coupling, characterized by the cross-gap Berry connection, to dispersive bands. The geometric character of the LLS is evident in our interpretation based on Eq. (5).

Let us discuss the geometric character of the fidelity tensor $\chi_{xy}^{nm}({{\mathbb{k}}})$ more explicitly. In general, the geometry of the quantum state $u_{n}({{\mathbb{k}}})$ can be derived from the Hilbert-Schmidt quantum distance Bužek and Hillery (1996); Dodonov et al. (2000); Berry (1989) defined as

which measures the similarity between $u_{n}({{\mathbb{k}}})$ and $u_{n}({{\mathbb{k}}}^{\prime})$. For ${{\mathbb{k}}}^{\prime}={{\mathbb{k}}}+d{{\mathbb{k}}}$, we obtain

where $\mathfrak{G}_{ij}^{n}({{\mathbb{k}}})$ indicates the quantum geometric tensor Provost and Vallee (1980); Zanardi et al. (2007); Ma et al. (2010) whose explicit form is

which shows that the quantum geometric tensor $\mathfrak{G}_{ij}^{n}({{\mathbb{k}}})$ of the $n$-th band is given by the summation of the fidelity tensor $\chi_{ij}^{nm}({{\mathbb{k}}})$ over all $m\neq n$. We note that $\chi_{ij}^{nm}({{\mathbb{k}}})$ itself cannot define a distance as the triangle inequality is not satisfied. However, it is related to the transition probability or the fidelity $F\left(u_{n}({{\mathbb{k}}}),u_{m}({{\mathbb{k}}}^{\prime})\right)$ between the $n$-th and $m$-th bands Jozsa (1994) through the following relations:

Thus, the geometric interpretation based on Eqs. (5) and (11) clearly shows that the LLS originates from the inter-band coupling.

The LLS of an IFB is strongly constrained by symmetry. First, we consider a generic symmetry $\sigma$ whose action on the Hamiltonian is given by

where $s\in\{0,1\}$, $p\in\{-1,1\}$, and $U_{\sigma}({{\mathbb{k}}})$ and $O_{\sigma}$ are a unitary and orthogonal matrices representing $\sigma$, respectively. $\overline{x}^{s=1}$ denotes the complex conjugation of $x$ while $\overline{x}^{s=0}=x$. Note that $s=0$ and $1$ are relevant to the unitary and anti-unitary symmetries, respectively, while $p=-1$ and $+1$ correspond to anti-symmetry and symmetry, respectively.

Among all possible symmetries of the form in Eq. (12), we find that the modified band dispersion $E_{n,B}({{\mathbb{k}}})$ vanishes when the system respects the chiral $C$ or space-time-inversion $I_{\rm ST}$ symmetries in the zero magnetic flux (see Methods and Supplementary Notes 2 and 3 for the detailed derivation). $C$ and $I_{\rm ST}$ are characterized by $(O_{\sigma},s,p)=(\mathds{1},0,-1)$ and $(\mathds{1},1,1)$, respectively, where $\mathds{1}$ is the identity matrix. In the following, we demonstrate that the LLS is proportional to $B^{2}$ for a flat-band system with $I_{\rm ST}$ symmetry in the zero magnetic field, while the LLS is forbidden in the presence of chiral symmetry. Interestingly, although $I_{\rm ST}$ symmetry would be broken as the magnetic field is turned on, the LLS is strongly constrained by $I_{\rm ST}$ symmetry.

We further find that ${\rm max}\,E_{n,B}({{\mathbb{k}}})=-{\rm min}\,E_{n,B}({{\mathbb{k}}})$ when the system respects a symmetry satisfying $(-1)^{s}p\,{\rm Det}O_{\sigma}=-1$ and $O_{\sigma}\neq\mathds{1}$, such as time-reversal $T$ or reflection $R$ symmetry, at the zero magnetic field (see Methods and Supplementary Note 2 for detailed derivations). This implies that the minimum and maximum values of the LLS have the same magnitude but with the opposite signs. The relevant tight-binding models are shown in the Supplementary Note 4.

We first consider the spin-orbit-coupled (SOC) Lieb model Goldman et al. (2011) as an example of generic flat-band systems. The lattice structure for this model is shown in Fig. 2a. The model consists of the nearest neighbor hopping with the amplitude 1 and the spin-orbit coupling between the next nearest neighbor sites, which are denoted as green solid and dashed arrows, respectively, in Fig. 2a. The tight-binding Hamiltonian in momentum space is given by

where $\lambda_{\rm soc}$ denotes the strength of spin-orbit coupling. The flat band’s energy is zero, i.e., $\varepsilon_{\rm socL,fb}({{\mathbb{k}}})=0$, and the energies of the other two bands are

which are plotted in Fig. 2b for $\lambda_{\rm soc}=0.2$. The band gap between the IFB and its neighboring bands is given by $4|\lambda_{\rm soc}|$ if $|\lambda_{\rm soc}|<1/2$, and $2$ if $|\lambda_{\rm soc}|\geq 1/2$, thus the flat band is decoupled from other bands for non-zero $\lambda_{\rm soc}$.

The analytic form of the fidelity tensor $\chi_{xy}^{nm}({{\mathbb{k}}})$ is given by

where $f(k_{x},k_{y})=4\lambda_{\rm soc}\sin\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\left(\cos^{2}\frac{k_{x}}{2}+1\right)+i\varepsilon_{{\rm socL},+}({\mathbb{k}})\cos\frac{k_{x}}{2}\sin\frac{k_{y}}{2}$. Then, from Eq. (5), the modified band dispersion for the flat band is given by

In Fig. 2c and d, ${\rm Im}\,\chi^{{\rm fb},-}_{{\rm socL},xy}({\mathbb{k}})$ and $E^{\rm socL}_{{\rm fb},B}({\mathbb{k}})$ are shown. We note that

These minimum and maximum values of $E^{\rm socL}_{{\rm fb},B}({\mathbb{k}})$ correspond to the lower and upper bounds for the LLS of the IFB as illustrated by red lines in Fig. 2e and f. Interestingly, the fidelity tensors $\chi^{{\rm fb},+}_{{\rm socL},xy}({\mathbb{k}})$ and $\chi^{{\rm fb},-}_{{\rm socL},xy}({\mathbb{k}})$ are conjugate of each other. This originates from the anti-unitary symmetry $C\circ I_{\rm ST}$, a combination of chiral $C$ and space-time-inversion $I_{\rm ST}$ symmetries, present in the system (see Supplementary Note 2 for the details.)

We construct a chiral-symmetric Lieb (c-Lieb) model as a representative example for chiral-symmetric IFB systems. The c-Lieb is defined on the same Lieb lattice as the SOC-Lieb model, but with different hoppings. As shown in Fig. 3a, this model consists only of the nearest neighbor hoppings, denoted by green arrows. The hopping parameter from a B-site to a C-site is $t_{1}$ for the rightward hopping, and 1 for the leftward hopping. On the other hand, the hopping parameter from a B-site to an A-site is $t_{2}$ for the upward hopping, and 1 for the downward hopping. The corresponding tight-binding Hamiltonian in momentum space is given by

with energy eigenvalues $\varepsilon_{\rm cL,fb}({{\mathbb{k}}})=0$ and $\varepsilon_{{\rm cL},\pm}({{\mathbb{k}}})=\pm\sqrt{2+t_{1}^{2}+t_{2}^{2}+2t_{1}\cos k_{x}+2t_{2}\cos k_{y}}$. The chiral symmetry operator $C$ is given by $C={\rm Diag}(1,-1,1)$ which gives a symmetry relation,

Note that the wave function of the flat band is also a simultaneous eigenstate of the chiral symmetry having a definite chiral charge $c=+1$:

Also, we obtain the fidelity tensor $\chi_{xy}^{{\rm fb},\pm}$, expressed by

The band structure and $\rm Im\,\chi^{{\rm fb},-}_{{\rm cL},xy}({{\mathbb{k}}})$ are shown in Fig. 3b and c. Equation (22) indicates that the modified band dispersion $E_{{\rm fb},B}({{\mathbb{k}}})$ vanishes for all ${{\mathbb{k}}}$ because $\varepsilon_{{\rm cL},+}({{\mathbb{k}}})=-\varepsilon_{{\rm cL},-}({{\mathbb{k}}})$, which means that there is no LLS in the weak magnetic field. Also, we calculate the Hofstadter spectrum Hofstadter (1976) for the c-Lieb model. Interestingly, we find that the LLS is absent even in the strong magnetic field, as shown in Fig. 3d. The existence of such zero-energy flat bands in the finite magnetic flux is guaranteed by chiral symmetry $C$. As explained in Supplementary Note 3, the minimal number of zero-energy flat bands is given by $|{\rm Tr}[C]|$ at the zero magnetic flux. Moreover, when the system has the $|{\rm Tr}[C]|(>0)$ number of zero-energy flat bands at the zero magnetic flux, the LLS of flat band(s) is forbidden unless a gap closes at zero energy $E=0$ as the magnetic flux increases (see Supplementary Note 3). In the c-Lieb model, such a gap closing at $E=0$ does not occur at any magnetic flux. Hence, there is no LLS in all range of magnetic flux. On the other hand, when a gap closes at $E=0$ as the magnetic flux increases, the LLS is forbidden only in a finite range of magnetic flux. As an example, in Supplementary Note 4, we show the Hofstadter spectrum of the ten-band model for twisted-bilayer graphene proposed in Ref. Po et al., 2019.

The LLS of an IFB is weakly dependent on magnetic field when the system respects space-time-inversion $I_{\rm ST}$ symmetry at zero magnetic field. We consider spinless fermions on the checkerboard lattice shown in Fig. 4a, which is sometimes called the Tasaki or decorated square lattice Tasaki (1992); Aoki et al. (1996); Misumi and Aoki (2017). This model respects both time-reversal $T$ and inversion $I$ symmetries. Hence, a combined symmetry, space-time-inversion symmetry $I_{\rm ST}=I\circ T$, exists. We note that the following discussion holds even if $T$ and $I$ are broken as long as $I_{\rm ST}$ is not broken. The tight-binding Hamiltonian consists of the hopping processes up to the third nearest neighbor hopping. In momentum space, the Hamiltonian is written as

where $k_{\pm}=k_{x}\pm k_{y}$. For $t=1.0$, the band structure is shown in Fig. 4b. This system hosts a flat band with zero energy and a dispersive band with positive energy. The energy eigenvalues are given by $\varepsilon_{I_{\rm ST},\uparrow}({{\mathbb{k}}})=1+(\cos\frac{k_{+}}{2}+2t\cos\frac{k_{-}}{2})^{2}$ and $\varepsilon_{I_{\rm ST},{\rm fb}}({{\mathbb{k}}})=0$.

n this system, $I_{\rm ST}$ is simply given by the complex conjugation, i.e., $I_{\rm ST}=\mathcal{K}$ and

Also, explicit calculations show ${\rm Im}\,\chi^{{\rm fb},\uparrow}_{I_{\rm ST},xy}({{\mathbb{k}}})=0$ and the vanishing modified band dispersion for the flat band $E_{n,B}({{\mathbb{k}}})=0$, which results from Eq. (25). In Supplementary Note 2, we have proved that space-time inversion $I_{\rm ST}$ imposes $E_{n,B}({{\mathbb{k}}})=0$ in general. We also note that $E_{n,B}({{\mathbb{k}}})=0$ is consistent with the fact that the orbital angular momentum, which is proportional to the orbital magnetic moment, is constrained to be zero in $I_{\rm ST}$-symmetric systems. Although the LLS is negligible in the weak magnetic field, it becomes considerably large in the strong magnetic field as shown in the Hofstadter spectrum in Fig. 4c. As shown in Fig. 4c, the Landau levels of the flat band acquire or lose their energy as the magnetic flux increases from 0 to some finite value much less than 1. This implies that the higher-order corrections of the magnetic field must be considered. Although it is out of the scope of this work, we present a fitting of the highest and lowest Landau levels of the flat band with respect to the magnetic flux:

which is plotted in Fig. 4d where one can observe the dominant quadratic dependence on the magnetic field.

Finally, we comment on the gap closing at $(\phi/\phi_{0},E)=(1,1.0)$ in the Hofstadter spectrum in Fig. 4c. At $(\phi/\phi_{0},E)=(1,1.0)$, the Landau levels related to the flat and dispersive bands show a closing of an indirect gap. We note that there is no closing of direct gaps in the Hofstadter Hamiltonian. Unlike the inevitable closing of the direct gap between topological bands in the finite magnetic flux reported before Lian et al. (2020); Herzog-Arbeitman et al. (2020), it is not necessary to close a direct gap in our system.

In order to derive the symmetry constraints on $E_{n,B}({{\mathbb{k}}})$ and $\chi^{nm}_{xy}({{\mathbb{k}}})$, let us consider a symmetry operation $\sigma$ acting on the Hamiltonian,

where $s\in\{0,1\}$, $p\in\{-1,1\}$, $U_{\sigma}$ indicates a unitary matrix representing the symmetry $\sigma$, and $\overline{x}=x^{*}$ means the complex conjugation of $x$. From now on, we use a compact notation $\mathfrak{g}_{\sigma}=(O_{\sigma},s,p)$ to describe the operation of the symmetry $\sigma$. For example, $\mathfrak{g}_{T}=(-\mathds{1}_{d},1,1)$ is used for time-reversal symmetry $T$ where $d$ and $\mathds{1}_{d}$ denote the dimensionality and the $d\times d$ identity matrix, respectively. The symmetry constraints on $E_{n,B}({{\mathbb{k}}})$ and $\chi_{ij}^{nm}({{\mathbb{k}}})$ derived from Eq. (28) are

where the band indices $m$ and $m_{\sigma}$ in Eq. (30) are chosen such that $\varepsilon_{m}({{\mathbb{k}}})=p\,\varepsilon_{m_{\sigma}}(O_{\sigma}{{\mathbb{k}}})$. Detailed derivation of Eqs. (29) and (30) and comments on the degenerate bands can be found in Supplementary Note 2. From Eq. (29), we obtain two symmetries that give vanishing modified band dispersion, $E_{n,B}({{\mathbb{k}}})=0$: $\mathfrak{g}_{C}=(\mathds{1}_{d},0,-1)$ and $\mathfrak{g}_{I_{\rm ST}}=(\mathds{1}_{d},1,1)$ which correspond to chiral symmetry $C$ and space-time-inversion symmetry $I_{\rm ST}$, respectively. On the other hand, when $(-1)^{s}p\,{\rm Det}O_{\sigma}=-1$ and ${\rm Det}O_{\sigma}\neq\mathds{1}$, the modified band dispersion satisfies $E_{n,B}(O_{\sigma}{{\mathbb{k}}})=-E_{n,B}({{\mathbb{k}}})$, which implies ${\rm max}\,E_{n,B}({{\mathbb{k}}})=-{\rm min}\,E_{n,B}({{\mathbb{k}}})$. Time-reversal $T$ and reflection $R$ symmetries belong to this case. Also, the contribution to the $E_{n,B}({{\mathbb{k}}})$ from each band via the inter-band coupling in Eq. (5) can be systematically understood by using Eq. (30) (see Supplementary Note 2 for details).

We calculate the Hofstadter spectrum by numerically implementing the Peierls substitution to the tight-binding Hamiltonian Hofstadter (1976).

To obtain the Landau fan diagram including disorder effect, we study a finite-size SOC Lieb model $H_{\rm socL}({{\mathbb{k}}})$ composed of 40 by 40 unit cells. Disorder is introduced by the Hamiltonian $H_{\rm dis}$ with components $\left(H_{\rm dis}\right)_{ij}=w_{i}\delta_{ij}$, where $i,j=1,\dots,4800$ denotes the unit cell index and $w_{i}\in[-W/2,W/2]$ follows a uniform probability distribution. By diagonalizing the disordered Hamiltonian $N_{\rm itr}=200$ times and averaging the results, the density of states (DOS) of Landau levels is obtained. Note that chiral edge states are found in the gap between flat and dispersive bands. It is because the two dispersive bands in the SOC Lieb model have the Chern number $\pm 1$ respectively depending on the sign of spin-orbit coupling, despite the topologically trivial middle flat band. However, the contribution of edge states to DOS is quantitatively negligible.