Critical regions in a one-dimensional flat band lattice with a quasi-periodic potential

Abstract

In our previous work, the concept of critical region in a generalized Aubry–André model (Ganeshan–Pixley–Das Sarma’s model) has been established. In this work, we find that the critical region can be realized in a one-dimensional flat band lattice with a quasi-periodic potential. It is found that the above flat band lattice model can be reduced to an effective Ganeshan–Pixley–Das Sarma’s model. Depending on various parameter ranges, the effective quasi-periodic potential may be bounded or unbounded. In these two cases, the Lyapunov exponent, mobility edge, and critical indices of localized length are obtained exactly. In this flat band model, the localized-extended, localized-critical and critical-extended transitions can coexist. Furthermore, we find that near the transitions between the bound and unbounded cases, the derivative of Lyapunov exponent of localized states with respect to energy is discontinuous. At the end, the localized states in bounded and unbounded cases can be distinguished from each other by Avila’s acceleration.

Introduction

A lot of novel physical phenomena, for example, the existence of localized flat band states^1,2,3,4, ferro-magnetism transition^5,6, super-Klein tunneling^7,8,9,10, preformed pairs¹¹, strange metal¹², high $T_c$ superconductivity/superfluidity^{13,14,15,16,17,18,19,20,21,22,23}, etc., can appear in a flat band system. Due to the infinitely large density of states of a flat band, a short-ranged potential can result in an infinite number of bound states, even a hydrogen atom-like energy spectrum, i.e., $E_n\propto 1/n^2,n=1,2,3,...$^24,25. Furthermore, a long-ranged Coulomb potential can destroy completely the flat band^26,27. In addition, a long-ranged Coulomb potential can result in wave function collapse^28,29, and 1/n energy spectrum³⁰, even the bound states in a continuous spectrum (BIC)³¹.

In the past decades, Anderson localization and the existence of mobility edges in one-dimensional lattice model with quasi-periodic potentials have attracted great interests^{32,33,34,35,36,37,38,39,40,41}. A famous example where the localized-extended transition can occur is the Aubry–André lattice model (AA model)⁴², i.e.,

$$\begin{aligned} t[\psi (n+1)+\psi (n-1)]+2\lambda \cos (2\pi \beta n+\phi )\psi (n)=E\psi (n). \end{aligned}$$

(1)

where t is hopping, $n\in Z$ is lattice site index, $2\lambda$ describes the quasi-periodic potential strength, $\beta$ is an irrational number, $\phi$ is a phase. When the quasi-periodic potential is weak, all the eigenstates are extended states. While when the potential strength is sufficiently large, all the eigenstates become localized states. In this model, there are no mobility edges. The non-existence of mobility edges originates from the Aubry–Andrè self-duality of this model. However, the breaking of the self-duality would result in the appearance of mobility edges^{32,33,34,35,37,43,44,45,46,47,48}.

A generalized Aubry–Andre model (GAA model) that can have exact mobility edges has been proposed by Ganeshan, Pixley and Das Sarma⁴⁹. The GAA model is

$$\begin{aligned} t[\psi (n+1)+\psi (n-1)]+\frac{2\lambda \cos (2\pi \beta n+\phi )}{1-\alpha \cos (2\pi \beta n+\phi )}\psi (n)=E\psi (n). \end{aligned}$$

(2)

In comparison with the AA model, there is an additional real parameter $\alpha$ in the denominator of quasi-periodic potential $\frac{2\lambda cos(2\pi \beta n+\phi )}{1-\alpha cos(2\pi \beta n+\phi )}$. Interestingly, the mobility edges can be exactly obtained with a generalized self-duality transformation. Recently, a mosaic lattice model with mobility edges has been proposed⁵⁰. The mobility edges can also be exactly determined with Avila’s theory^40,51,52.

In most of the previous studies of the GAA model, $\alpha$ is mainly limited to $|\alpha |<1$ due to concern of possible appearances of divergences of periodic potential (see Eq. 2). Then, the quasi-periodic potential is bounded. Very recently, the localization problem for the unbounded ($|\alpha |\ge 1$) case has been investigated by the present author and the co-author⁵³. It is found that when $|\alpha |\ge 1$, the quasi-periodic potential and energy spectrum would be unbounded. In addition, there exists a critical region which consists of critical states in a parameter plane. As the energy approaches the localized-critical transition point (known as the mobility edge), the critical index of localized length $\nu =1/2$, which is different from $\nu =1$ of bounded case ($|\alpha |<1$). In addition, it is found that Avila’s acceleration for the unbound case is also quantized. The systems with different E can be classified by the Lyapunov exponent and Avila’s acceleration.

Due to the divergence of the unbounded quasi-periodic potential in the original GAA model, a natural question arises: is it possible to realize the critical region without any divergences in the energy spectrum? Additionally, for a given quasi-periodic potential strength, it is desirable to be able to realize both bounded and unbounded quasi-periodic potentials in a simple model. Furthermore, one may continue to ask a question: can the localized-extended transition and localized-critical transition coexist?

In this study, we have discovered that a simple spin-1 flat band lattice model with a quasi-periodic potential can achieve the objectives mentioned above. This lattice model can be simplified into an effective GAA model, and even in the unbounded case, there are no divergences in the energy spectrum. Our findings reveal that in the one-dimensional flat band lattice model with a quasi-periodic potential, localized, extended, and critical states can coexist. Furthermore, the two distinct transitions, i.e., localized-extended and localized-critical transitions also appear in the potential strength $V_0$-E (energy) plane. All these findings are crucial for understanding the localization problem of quasi-periodic potentials in flat band system.

The work is organized as follows. First, the model Hamiltonian and its three energy bands are given. Next, we investigate the localization problem of quasi-periodic potential of type III and type II . Finally, a summary is given.

Figure 1

The one-dimensional flat band lattice (top) and diamond lattice (bottom) are shown in the figure. The lattice is composed of two sets of diamond lattices intersecting each other. The solid and dashed lines represent the hopping between the two sets of diamond lattices. The sites within the ellipses can be considered as unit cells in the bottom diamond lattice.

Figure 2

The three energy bands of free particle Hamiltonian $H_0$. There exists a flat (middle) band ($E_{0,k}$) in between the upper band ($E_{+,k}$) and the lower band ($E_{-,k}$).

Results

A spin-1 lattice model with a flat band

In this work, we consider a tightly binding lattice Hamiltonian consisting of three sublattices A, B, and C, i.e.,

$$\begin{aligned}{}&H=H_0+V_p\nonumber \\&H_0=-\frac{it}{\sqrt{2}}\sum _{n\in Z}[a^{\dag }_{n-1}b_{n}+b^{\dag }_{n-1}a_{n}+b^{\dag }_{n-1}c_{n}+c^{\dag }_{n-1}b_{n}]+h.c. +m\sum _{n\in Z}[a^{\dag }_{n}a_{n}-c^{\dag }_{n}c_{n}], \end{aligned}$$

(3)

where $V_p$ is potential energy, integer n is the unit cell’s index, $H_0$ is the free-particle Hamiltonian, $t>0$ is hopping parameter, and $m>0$ is energy gap parameter. $a(b/c)_{n}$ are the annihilation operators for states at sublattices A(B/C), respectively. The flat band lattice structure is depicted in Fig. 1. It is composed of two intersecting diamond lattices, as shown in the figure.

When potential $V_p=0$, applying a Fourier transform, the free particle Hamiltonian $H_0$ can be written as

$$\begin{aligned}{}&H_0=\sqrt{2}t\sum _{-\pi \le k\le \pi }\sin (kd)[a^{\dag }_{k}b_{k}+b^{\dag }_{k}a_{k}+b^{\dag }_{k}c_{k}+c^{\dag }_{k}b_{k}]+m\sum _{-\pi \le k\le \pi }[a^{\dag }_{k}a_{k}-c^{\dag }_{k}c_{k}], \end{aligned}$$

(4)

where d is lattice constant. In the whole manuscript, we would set $d=1$ for simplifications.

Furthermore, in the above Hamiltonian $H_0$, we can identify the above three sublattices A, B, C as three spin components 1, 2, 3. In the spin basis $|1,2,3\rangle$, the three eigenstates and the eigenenergies are

$$\begin{aligned}{}&\langle x|-, k\rangle =\frac{e^{ikx}}{2\sqrt{4t^2\sin ^2(k)+m^2}}\left( \begin{array}{ccc} \sqrt{4t^2\sin ^2(k)+m^2}-m\\ -2\sqrt{2}t\sin (k)\\ \sqrt{4t^2\sin ^2(k)+m^2}+m \end{array}\right) ,\nonumber \\&E_{-, k}=-\sqrt{4t^2\sin ^2(k)+m^2};\nonumber \\&\langle x|0, k\rangle =\frac{e^{ikx}}{\sqrt{4t^2\sin ^2(k)+m^2}}\left( \begin{array}{ccc} -\sqrt{2}t\sin (k)\\ m\\ \sqrt{2}t\sin (k) \end{array}\right) ,\nonumber \\&E_{0, k}=0;\nonumber \\&\langle x|+, k\rangle =\frac{e^{ikx}}{2\sqrt{4t^2\sin ^2(k)+m^2}}\left( \begin{array}{ccc} \sqrt{4t^2\sin ^2(k)+m^2}+m\\ -2\sqrt{2}t\sin (k)\\ \sqrt{4t^2\sin ^2(k)+m^2}-m \end{array}\right) ,\nonumber \\&E_{+, k}=\sqrt{4t^2\sin ^2(k)+m^2}, \end{aligned}$$

(5)

where $|-(0/ +), k\rangle$ denote the eigenstates of lower, middle (flat) and upper bands, respectively and $E_{-( 0/+), k}$ represent the corresponding three energy bands. It is found that a flat band with zero energy ($E_{0, k}=0$) appears in between upper and lower bands (see Fig. 2). It is noted that in the vicinity of momentum $k = 0$, the three-band Hamiltonian described above can be approximated by a continuous spin-1 Dirac model with a flat band. The bound state problems of this continuous spin-1 Hamiltonian have been extensively studied with various types of potentials^{24,25,29,30,31}. It is found that a short-ranged potential can induce in an infinite number of bound states and the bound state in continuum (BIC) can appear .

Figure 3

Lyapunov exponents for potential strength $V_0/m=1,3,5$. The discrete points are the numerical results for eigenenergies. The solid lines are given by Eq. (12). The mobility edges for $V_0/m=1,5$ are indicated by blue arrows and black arrows, respectively. Near mobility edges of the localized-extended transition (e.g., $E_c\simeq -3.30$ m,-0.62 m, and 1.62 m for $V_0/m=1$ and $|\alpha |<1$), the Lyapunov exponent $\gamma (E)\propto |E-E_c|$ approaches zero. The critical index of the localized length $\nu$ is 1 for $|\alpha |<1$. While E is near the localized-critical transition (e.g., $E_c=-2m$, $-m$, 0 and m for $V_0/m=5$ and $|\alpha |>1$), the Lyapunov exponent $\gamma (E)\propto |E-E_c|^{1/2}$ (as $E\rightarrow E_c$), and the critical index of the localized length $\nu =1/2$. The transition points ($|\alpha |=1$) $E=E_0\equiv \pm V_0/2=\pm 2.5$ m between localized states of bounded and unbounded cases for $V_0=5$ m are also indicated by red arrows. It shows that the derivative of Lyapunov exponent with respect to E is discontinuous near the transition points. In the whole paper, we take $t=m$, irrational number $\beta =(\sqrt{5}-1)/2$ and phase $\phi =0$.

Figure 4

Several typical wave functions for extended, localized, and critical states.

Localized-extended and localized-critical transitions in a quasi-periodic potential of type III

In the following, we assume the potential energy $V_p$ has following form in spin basis $|1,2,3\rangle$, namely,

$$\begin{aligned} V_p=V_{11}(n)\bigotimes |1\rangle \langle 1| \end{aligned}$$

(6)

where quasi-periodic potential

$$\begin{aligned} V_{11}(n)=V_0\cos (2\pi \beta n+\phi ) \end{aligned}$$

with potential strength $V_0$. In the whole manuscript, we would refer to such a kind of potential as potential of “type III”³⁰. The bound state problems with potential of “type I, II and III” have been investigated^24,29,30,31.

The Schrödinger equation ($H\psi =E\psi$) can be expressed using three component wave functions. These are represented by the equations

$$\begin{aligned}{}&\frac{-it}{\sqrt{2}}[\psi _{2}(n+1)-\psi _{2}(n-1)]=[E-m-V_{11}(n)]\psi _{1}(n),\nonumber \\&\frac{-it}{\sqrt{2}}[\psi _{1}(n+1)-\psi _{1}(n-1)+\psi _{3}(n+1)-\psi _{3}(n-1)]=E\psi _{2}(n),\nonumber \\&\frac{-it}{\sqrt{2}}[\psi _{2}(n+1)-\psi _{2}(n-1)]=[E+m]\psi _{3}(n). \end{aligned}$$

(7)

Adopting a similar procedure as Ref.³⁰, eliminating wave functions of 2th and 3th components in Eq. (7), we get an effective equation for $\psi _1(n)$, i.e.,

$$\begin{aligned}{}&t^2[\frac{E-V_{11}(n+2)/2}{E+m}\psi _{1}(n+2)-2\frac{E-V_{11}(n)/2}{E+m}\psi _{1}(n)+\frac{E-V_{11}(n-2)/2}{E+m}\psi _{1}(n-2)]=-E[E-m-V_{11}(n)]\psi _{1}(n). \end{aligned}$$

(8)

Further we introduce an auxiliary wave function $\psi (n)\equiv \frac{E-V_{11}(n)/2}{E+m}\psi _{1}(n)$, an effective hopping $\tilde{t}$, an effective total energy $\tilde{E}$, an effective potential strength $\lambda$ and an effective parameter $\alpha$, i.e.,

$$\begin{aligned}{}&\tilde{t}\equiv t^2,\nonumber \\&\tilde{E}\equiv -E^2+m^2+2\tilde{t}=-E^2+m^2+2t^2,\nonumber \\&\lambda \equiv -\frac{V_0(E+m)^2}{4E},\nonumber \\&\alpha \equiv \frac{V_0}{2E}, \end{aligned}$$

(9)

we get an equation for $\psi (n)$

$$\begin{aligned}{}&\tilde{t}[\psi (n+2)+\psi (n-2)]+\frac{2\lambda \cos (2\pi \beta n+\phi )}{1-\alpha \cos (2\pi \beta n+\phi )}\psi (n)=\tilde{E}\psi (n). \end{aligned}$$

(10)

Comparing it with Eq. (2), it is found that this is an effective generalized Aubry–André model whose effective lattice constant is two times of the original lattice constant, i.e., $\tilde{d}=2d=2$. In the whole manuscript, we would refer to this model Eq. (10) as the effective GAA model. It is expected that the critical region and localized-critical transition would appear in this model⁵³.

We should note that when the effective parameter $|\alpha |\ge 1$, the effective GAA model in Eq. (10) appears to have an unbounded quasi-periodic potential. However, unlike the original unbounded GAA model⁵³, the energy spectrum in this model is still bounded. This is due to the fact that, for a finite potential strength $V_0$, $|\alpha |\ge 1$ implies that

$$\begin{aligned} |\alpha |=|\frac{V_0}{2E}|\ge 1\Rightarrow |E|\le \frac{|V_0|}{2}. \end{aligned}$$

(11)

This means that, for a given potential strength $V_0$, the energy E will always be bounded. This can also be understood by considering that the quasi-periodic potential $V_{11}$ in the original three-component lattice model Eq. (7) does not have any singularities or divergences, thus the energy spectrum must be bounded.

In the following text, we will demonstrate that while the energy spectrum is bounded, other intriguing physical phenomena associated with unbounded quasi-periodic potentials, such as the presence of a critical region, a localized-critical transition, and a distinct critical index, can still manifest in the effective GAA model described by Eq. (10). Throughout the manuscript, we will continue to refer to the case where $|\alpha |=|\frac{V_0}{2E}|\ge 1$ as the unbounded case.

The localized properties of eigenstates can be characterized by Lyapunov exponent. This exponent can either be zero or positive, depending on the type of state. Extended and critical states have a Lyapunov exponent of zero, while localized states have a positive exponent. Avila’s theory^40,51,52 has provided exact calculations of Lyapunov exponents for both bounded ($|\alpha |<1$) and unbounded ($|\alpha |\ge 1$) cases in a recent study⁵³, i.e.,

$$\begin{aligned}{}&\gamma (E)= \left\{ \begin{array}{cccc} \frac{1}{2}Max\{0,\log (\frac{|P|+\sqrt{P^2-4\alpha ^2}}{2|1+\sqrt{1-\alpha ^2}|})\}, \ |\alpha |<1 \ \& \ P^2>4\alpha ^2\\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\alpha |<1 \ \& \ P^2<4\alpha ^2\\ \frac{1}{2}\log (\frac{|P|+\sqrt{P^2-4\alpha ^2}}{2|\alpha |}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\alpha |\ge 1 \ \& \ P^2>4\alpha ^2\\ 0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\alpha |\ge 1 \ \& \ P^2<4\alpha ^2, \end{array}\right. \end{aligned}$$

(12)

where

$$\begin{aligned} P=\frac{\alpha \tilde{E}+2\lambda }{\tilde{t}}=\frac{\alpha \tilde{E}+2\lambda }{t^2}. \end{aligned}$$

(13)

It has been observed that in comparison to Eq. (26) in Ref.⁵³, there is an additional factor of 1/2 in Eq. (12) due to the lattice constant being twice the original size.

In order to characterize the properties of the eigenstates, we have also numerically solved Eq. (7). Specifically, we have taken a total of $N=500$ unit cells and constructed a $3N\times 3N$ matrix with open boundary conditions at the two end sites. This matrix was then diagonalized to obtain 1500 eigenenergies and eigenstates. The results are presented in Figs. 3, 4, 5, and 6.

In addition, with Eq. (10), the Lyapunov exponent can be calculated numerically with transfer matrix method⁵³, i.e.,

$$\begin{aligned}{}&\gamma (E)=\lim _{L \rightarrow \infty }\frac{\log (|\Psi (2L)|/|\Psi (0)|)}{2L}=\lim _{L\rightarrow \infty }\frac{\log (|T(2L)T(2L-2)...T(4)T(2)\Psi (0)|/|\Psi (0)|)}{2L} \end{aligned}$$

(14)

where L is a positive integer, transfer matrix

$$\begin{aligned} T(n)\equiv \left[ \begin{array}{ccc} \frac{\tilde{E}}{\tilde{t}}-\frac{2\lambda }{\tilde{t}} \cos (2\pi \beta n+\phi ) &{}-1 \\ 1&{}0\\ \end{array}\right] , \end{aligned}$$

(15)

$$\begin{aligned} \Psi (n)\equiv \left[ \begin{array}{ccc} \psi (n+2) \\ \psi (n)\\ \end{array}\right] , \end{aligned}$$

(16)

and

$$\begin{aligned} |\Psi (n)|=\sqrt{|\psi (n+2)|^2+|\psi (n)|^2}. \end{aligned}$$

(17)

To be more specific, we have set $t/m=1$ and $V_0/m=1,3,5$ and have numerically calculated the Lyapunov exponents for all the eigenenergies, as shown by the three sets of discrete points in Fig. 3. Our numerical calculation uses $L=100$, phase $\phi =0$, $\beta =(\sqrt{5}-1)/2$, $\psi (0)=0$, and $\psi (2)=1$ in Eq. (14). The solid lines in Fig. 3 correspond to Eq. (12) with the same parameters. It is evident that most of the discrete points fall on the solid lines. However, we also observe that there are some discrete points corresponding to localized states that do not fall on the solid lines. This is because these localized wave functions are too close to the left-hand boundary of the system. If one calculates their Lyapunov exponents from right-hand end with the initial condition $\psi (n=500)=0$ and $\psi (n=498)=1$ , then one can get the correct results.

It has been shown that, depending on whether $|\alpha |<1$ or $|\alpha |\ge 1$, there are two types of localized-delocalized transitions in the GAA model⁵³. If $|\alpha |<1$, the effective quasi-periodic potential in Eq. (10) is bounded, resulting in mobility edges that separate the localized states from the extended states, i.e., localized-extended transitions. On the other hand, when $|\alpha |\ge 1$, the quasi-periodic potential is unbounded, leading to the existence of critical regions that consist of critical states. In this case, the mobility edges separate the localized states from the critical states (see Ref.⁵³ and Fig. 5).

When $|\alpha |<1$, by Eqs. (12) and (9), the mobility edges are given by

$$\begin{aligned}{}&\gamma (E=E_c)=0\rightarrow |P|=2 \rightarrow |\frac{V_0}{E_c}-\frac{V_0(m+E_c)}{t^2}|=2. \end{aligned}$$

(18)

In parameter space, the extended state regions are given by (see Fig. 5)

$$\begin{aligned} |P|<2 \rightarrow |\frac{V_0}{E}-\frac{V_0(m+E)}{t^2}|<2. \end{aligned}$$

(19)

When $|\alpha |\ge 1$, by Eqs. (9) and (12), the mobility edges are given by

$$\begin{aligned}{}&\gamma (E=E_c)=0\rightarrow |P|=2|\alpha | \rightarrow |\frac{t^2-(m+E_c)E_c}{t^2}|=1. \end{aligned}$$

(20)

The critical regions are defined by (see Fig. 5)

$$\begin{aligned}{}&|P|<2|\alpha | \rightarrow |\frac{t^2-(m+E)E}{t^2}|<1. \end{aligned}$$

(21)

When the energy of localized state approaches the localized-delocalized (extended and critical) transition point (mobility edge), the Lyapunov exponent goes to zero according to the law of

$$\begin{aligned} \gamma (E)\propto |E-E_c|^\nu \rightarrow 0, \end{aligned}$$

(22)

where $\nu >0$ is critical index of localized length of localized states. Consequently, the localized length

$$\begin{aligned} \xi (E)\equiv 1/\gamma (E)\propto |E-E_c|^{-\nu }\rightarrow \infty \end{aligned}$$

(23)

becomes infinitely large. For a given potential strength $V_0$, near the mobility edges $E_c$ at which the localized-extended transition occurs, the localized length is⁵³

$$\begin{aligned} \xi (E)=1/\gamma (E)\propto |E-E_c|^{-1}, \end{aligned}$$

(24)

while for the localized-critical transition,

$$\begin{aligned} \xi (E)=1/\gamma (E)\propto |E-E_c|^{-1/2}. \end{aligned}$$

(25)

So the critical index is $\nu =1$ for $|\alpha |<1$, and $\nu =1/2$ for $|\alpha |\ge 1$ (see Fig. 3). We also note the existences of the critical states and the different mobility edges in an unbounded quasi-periodic potential lattice (similar to the GAA model) have been investigated in Ref.⁵⁴.

Several typical wave functions for the localized states, extended states, and critical states are shown in Fig. 4. It can be observed that the localized states only occupy a finite number of lattice sites, while the extended states occupy the entire lattice. The wave functions of critical states consist of several disconnected patches.

The phase diagram in the $V_0-E$ plane is presented in Fig. 5. From this figure, it can be seen that for a given $V_0$, the regions of localized states (labeled as $L_1$ and $L_2$), extended states (E), and critical states ($C_r$) can coexist. It should be noted that the coexistence of localized states, extended states, and critical states can also be observed in a spin-orbit coupled lattice with a one-dimensional quasi-periodic potential⁵⁵.

Figure 5

Phase diagram and standard deviations for quasi-periodic potential of type III. The extended state regions and the critical regions are labeled with E and $C_r$, respectively. The two types of localized state regions are denoted with $L_1$ (for $|\alpha |<1$) and $L_2$ (for $|\alpha |\ge 1$), respectively. (a) The boundaries between bounded and unbounded quasi-periodic potentials are defined by $|\alpha |=|\frac{V_0}{2E}|=1$ (the blue dashed lines). (b) Standard deviations are represented with different colors.

Figure 6

The standard deviations for localized, extended, and critical states are calculated with a potential strength of $V_0/m=3$. As the state index n increases from 1 to 1500, the energy E of the states also increases.

In order to distinguish between localized and extended states, as well as critical states, we use a similar approach as the original GAA model⁵³. We calculate the standard deviation of coordinates numerically, following the method described in⁴⁵

$$\begin{aligned}{}&\sigma =\sqrt{\sum _{\sigma =1,2,3;n}(n-\bar{n})^2|\psi _{\sigma }(n)|^2}, \end{aligned}$$

(26)

where the average value of coordinate is

$$\begin{aligned} \bar{n}=\sum _{\sigma =1,2,3;n}n|\psi _{\sigma }(n)|^2. \end{aligned}$$

(27)

The standard deviation of coordinates describes the spatial extent of wave functions. When the states are localized, the standard deviations of coordinates are small. For extended states, the standard deviations are much larger. When the states are critical, their standard deviations are in between the two (see Fig. 5). In comparison to localized and extended states, critical states also exhibit larger fluctuations in standard deviations (see Fig. 6 and Ref.⁵³).

Additionally, as $\alpha$ approaches the boundaries between bounded and unbounded quasi-periodic potentials, i.e. $\alpha =\pm 1$ (see the blue dashed lines in Fig. 5), there are critical-extended transitions in the phase diagram.

Finally, for bounded $|\alpha |<1$ and unbounded $|\alpha |\ge 1$ cases, there are two kinds of localized state regions which are denoted by $L_1$ and $L_2$ in Fig. 5. Near the localized ($L_1$)-localized ($L_2$) transitions, i.e, $|\alpha |=1$, we find that the derivative of Lyapunov exponents with respect to energy E, i.e., $\gamma '(E)\equiv \frac{d \gamma (E)}{dE}$ is discontinuous. For example, when $V_0=5m$, as the energy E approaches the localized-localized transition point ($E_0=\pm V_0/2=\pm 2.5$ m), the derivative of the Lyapunov exponent on the $L_2$ side is finite, while it diverges on the $L_1$ side, i.e, $\gamma '(E)\propto 1/\sqrt{|E-E_0|}\rightarrow \infty$ (see black line in Fig. 3).

Furthermore, although both types of localized states for bounded $|\alpha |<1$ and unbounded $|\alpha |\ge 1$ cases have a positive Lyapunov exponent, $\gamma (E)>0$, they can still be distinguished by Avila’s acceleration, $\omega (E)$⁵³. For example, when the localized state is in region $L_1$, which corresponds to the bounded $|\alpha |<1$ case, the Avila’s acceleration is $\omega (E)=1/2$. On the other hand, when the localized state is in region $L_2$, which corresponds to the unbounded $|\alpha |\ge 1$ case, the Avila’s acceleration is $\omega (E)=0$. This can be expressed as:

$$\begin{aligned}{}&\omega (E)= \frac{1}{2}\times \left\{ \begin{array}{c} 1, \ for \ bound \ states \ of \ bounded \ case \\ \ \ \ \ 0, \ for \ bound \ states \ of \ unbounded \ case . \end{array}\right. \end{aligned}$$

(28)

It should be noted that, due to the two times of the original lattice constant here [see Eq. (10)], there is an extra factor of 1/2 in Eq. (28) when compared to the original GAA model (see Eqs. (45) and (47) of Ref.⁵³).

Localized-extended transitions and mobility edges in quasi-periodic potential of Type II

In this subsection, we will investigate the problem of Anderson localization in the flat band lattice model for a quasi-periodic potential of Type II, for the sake of completeness. We assume that the potential energy, denoted as $V_p$, takes the following form in the spin basis $|1,2,3\rangle$:

$$\begin{aligned}{}&V_p=V_{22}(n)\bigotimes |2\rangle \langle 2| =\left[ \begin{array}{ccc} 0 &{}0 &{} 0\\ 0&{}V_{22}(n)&{} 0\\ 0 &{}0 &{} 0 \end{array}\right] . \end{aligned}$$

(29)

Note that the potential only appears in the basis element $|2\rangle$ (or sublattice B). Throughout the manuscript, we will refer to this type of potential as “Type II” potential²⁴.

Furthermore, we assume $V_{22}$ is also a quasi-periodical potential, i.e.,

$$\begin{aligned} V_{22}(n)=V_0\cos (2\pi \beta n+\phi ), \end{aligned}$$

(30)

where $V_0$ is the potential strength, irrational number $\beta =(\sqrt{5}-1)/2$ determines the quasi-periodicity, and real number $\phi$ is a phase.

The Schrödinger equation ($H\psi =E\psi$) can be written as

$$\begin{aligned}{}&-it[\psi _{2}(n+1)-\psi _{2}(n-1)]/\sqrt{2}=[E-m]\psi _{1}(n),\nonumber \\&-it[\psi _{1}(n+1-\psi _{1}(n-1)+\psi _{3}(n+1)-\psi _{3}(n-1)]/\sqrt{2}=[E-V_{22}(n)]\psi _{2}(n),\nonumber \\&-it[\psi _{2}(n+1)-\psi _{2}(n-1)]/\sqrt{2}=[E+m]\psi _{3}(n). \end{aligned}$$

(31)

Adopting a similar procedure as Ref.²⁴ to eliminate wave functions for 1-th and 3-th components, we get an effective discrete Schrödinger equation for $\psi _2$

$$\begin{aligned} -t^2[\psi _{2}(n+2)-2\psi _{2}(n)+\psi _{2}(n-2)]+\frac{(E^2-m^2)V_{22}(n)}{E}\psi _{2}(n)=[E^2-m^2]\psi _{2}(n). \end{aligned}$$

(32)

Further introducing an effective hopping $\tilde{t}$, an effective energy $\tilde{E}$ and an effective potential strength $\lambda$

$$\begin{aligned}{}&\tilde{t}\equiv t^2,\nonumber \\&\tilde{E}\equiv -E^2+m^2+2t^2,\nonumber \\&\lambda \equiv V_{0}(-E^2+m^2)/(2E), \end{aligned}$$

(33)

Eq. (32) becomes the well-known Aubry-André (AA) model, i.e.,

$$\begin{aligned}{}&\tilde{t}[\psi _{2}(n+2)+\psi _{2}(n-2)]+2\lambda \cos (2\pi \beta n+\phi )\psi _{2}(n)=\tilde{E}\psi _{2}(n). \end{aligned}$$

(34)

According to the Aubry–André self-duality, when

$$\begin{aligned} \tilde{t}=|\lambda |\ \ \rightarrow t^2= |V_{0}(-E^2+m^2)/(2E)|, \end{aligned}$$

(35)

there exist localized-extended transitions. Due to the energy dependence of $\lambda$, the mobility edges would appear in the flat band system. The mobility edges are determined by Eq. (35). A similar mechanism of localized-extended transition in a flat band lattice model has also been investigated in Ref.⁵⁶.

Figure 7

Lyapunov exponent for potential strength $V_0=4m$. The discrete points represent numerical results, while the solid line is obtained from Eq. (36). The mobility edges $E_c\simeq \pm 0.78m, \pm 1.28m$ for $V_0=4m$ are indicated by red arrows. Throughout this paper, we use $t=m$.

Figure 8

Phase diagram and standard deviations for quasi-periodic potential of type II. The localized state regions and the extended state regions are labeled in the figure. The standard deviations are indicated by different colors.

Figure 9

Standard deviations for localized and extended states. The energy E of states increases as state index n runs from 1 to 1500.

When the parameter E is an eigenenergy, the Lyapunov exponent for the AA model is given by⁵²

$$\begin{aligned}{}&\gamma (E)=\frac{1}{2}Max\{0,\log (|\frac{\lambda }{\tilde{t}}|)\}=\frac{1}{2}Max\{0,\log (|\frac{V_{0}(E^2-m^2)}{2Et^2}|)\}. \end{aligned}$$

(36)

It should be noted that due to the two times of the original lattice constant in Eq. (34), there is an additional factor of 1/2 in Eq. (36).

To further investigate this, we also performed numerical calculations for the Lyapunov exponent using Eq. (34). Specifically, we set $t/m=1$, $V_0/m=4$, and calculated the Lyapunov exponents for all eigenenergies (represented by the set of discrete points in Fig. 7). Our numerical calculations are performed with $L=100$, phase $\phi =0$, $\beta =(\sqrt{5}-1)/2$, $\psi _2(0)=0$, and $\psi _2(2)=1$. The solid line in Fig. 7 represents the values obtained from Eq. (36) using the same parameters. It can be seen that most of the discrete points fall onto the solid line, indicating good agreement between the analytical and numerical results.

We also calculated the standard deviations of the coordinates of the eigenstates using Eq. (26), and the results are shown in Figs. 8 and 9. The phase diagram and standard deviations are presented in Fig. 8, with different colors representing the different standard deviation values. It is evident that in the presence of a quasi-periodic potential of type II, the critical region does not appear, and only localized-extended transitions are observed. This is further supported by the data in Fig. 9, where it can be seen that for localized states, the standard deviations of the coordinates are very small, while for extended states, the standard deviations are much larger.

Summary

In conclusion, we investigate the Anderson localization problem in a one-dimensional flat band lattice model with a quasi-periodic potential. Our findings show that for type III potentials, localized states, extended states, and critical regions can coexist. We have observed transitions between localized-extended and localized-critical states as the energy varies. Near the mobility edges, the Lyapunov exponent goes to zero according to law $|E-E_c|$ for the localized-extended transition, while for the localized-critical transition, it approaches to zero by $|E-E_c|^{1/2}$. This results in a critical index of $\nu =1$ for the localized-extended transition and $\nu =1/2$ for the localized-critical transition. Additionally, we have identified localized ($L_1$)-localized ($L_2$) and critical-extended transitions in the phase diagram near the boundaries between bounded and unbounded quasi-periodic potentials. At these transitions, the derivative of the Lyapunov exponent with respect to energy is discontinuous. The localized states in $L_1$ and $L_2$ can be distinguished from each other using Avila’s acceleration.

In the presence of a type II quasi-periodic potential, the critical region does not appear and only localized-extended transitions occur, with a critical index of $\nu =1$.

Our results show that the flat band lattice model has much richer physics compared to the original Ganeshan–Pixley–Das Sarma’s GAA model. The most notable finding is that not only can localized states, extended states, and critical states coexist, but the flat band system also exhibits multiple transitions between these states, such as localized-extended, localized-critical, extended-critical, and localized-localized transitions.

While the unbounded quasi-periodic potential in the original GAA model may only have theoretical interest due to its divergences, the absence of any divergences in our model makes it more likely to be experimentally realized. Therefore, we expect that the localization physics of the flat band lattice with quasi-periodic potential can be observed experimentally.⁴¹.