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- Photonics Research
- Vol. 6, Issue 4, A10 (2018)

Abstract

1. INTRODUCTION

A flat band, as the name suggests, is a dispersionless band that extends in the whole Brillouin zone. Systems that exhibit flat bands have attracted considerable interest in the past few years, including optical ^{[1,2]} and photonic lattices ^{[3–6]}, graphene ^{[7,8]}, superconductors ^{[9–12]}, fractional quantum Hall systems ^{[13–15]}, and exciton–polariton condensates ^{[16,17]}. The flatness of the band leads to a zero group velocity, which has important implications on the dynamic and localization properties, including the inverse Anderson transition ^{[18]}, localization with unconventional critical exponents and multifractal behavior ^{[19]}, mobility edges with algebraic singularities ^{[20]}, and unusual scaling behaviors ^{[21–23]}.

For a Hermitian lattice, a completely flat band in the entire Brillouin zone is formed when there exists a Wannier function that is an eigenstate of the whole system. To understand this relation, we only need to resort to the definition of the Wannier function itself, which we denote by

The simplest way to find such a Wannier function is in a frustrated lattice ^{[24]}, where quantum tunnelings from the edges of the Wannier function to the neighboring unit cells interfere destructively and are completely cancelled, hence isolating the Wannier function from the rest of the lattice. Take the 1D Lieb lattice, for example [see Fig.

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Figure 1.(a) Band structure of a Hermitian Lieb lattice. The flat band is shown by the thick line. Inset: schematic of the Lieb lattice, where

Unlike condensed matter systems, the realization of flat bands in optics involves parallel waveguides or cavities that are coupled evanescently. As such, a new degree of freedom can be introduced to manipulate the forming and properties of flat bands in these systems, i.e., non-Hermiticity brought forth by optical gain and loss. As we shall see later, this additional tuning knob enables more flexible control of band structures, with which a completely different approach can be employed to generate a flat band. Here, it is worth pointing out that, in a non-Hermitian system, the band structure is complex-valued in general, and we define a flat band by requiring that its real part is ^{[25–27]}, where optical gain and loss are arranged in a judicious way to satisfy the parity-time symmetry ^{[28]}. More recently, several studies probed the existence of non-Hermitian flat bands using either gain and loss modulations or complex-valued couplings ^{[29–32]}.

The goal of this paper is to provide a unified view of how non-Hermitian flat bands, as defined above, can be constructed. More specifically, we discuss systematically three such approaches, namely, using spontaneous restoration of non-Hermitian particle-hole (NHPH) symmetry (“Approach 1” ^{[32]}), a persisting flat band from the underlying Hermitian system (“Approach 2”), and a compact Wannier function that is an eigenstate of the entire system (“Approach 3” ^{[29–31]}). For Approach 3 in particular, we give the simplest lattice structure where this approach can be applied, which contains only two lattice sites in a unit cell. We further identify a special case of such a flat band where every point in the Brillouin zone is an exceptional point (EP) of order 3 ^{[25,33]}, and a localized excitation in this “EP3 flat band” can display either a conserved power, quadratic power increase, or even quartic power increase, depending on whether the compact Wannier function or one of the two generalized eigenvectors is initially excited. Nevertheless, the asymptotic wave function in the long time limit is always given by an eigenstate, in this case, the compact Wannier function or its superposition in two or more unit cells.

2. CONSTRUCTING NON-HERMITIAN FLAT BANDS

A. Approach 1

Approach 1, as mentioned above, was first suggested in Ref. ^{[32]}, where the flat band is a result of spontaneously restored NHPH symmetry ^{[34–36]} for all modes in the flat band. NHPH symmetry requires that the Hamiltonian of the system anticommutes with an antilinear operator, i.e., ^{[36]}. NHPH symmetry leads to a symmetric spectrum satisfying ^{[37,38]}. Therefore, this flat band actually contains at least two bands with degenerate real parts, which are distinguished only by their different imaginary parts.

Here we exemplify a two-dimensional (2D) lattice with a flat band formed via this approach. We consider a rectangular lattice with identical lattice sites and with lattice constants

Figure 2.Band structure of a 2D rectangular lattice with loss introduced to the A sites. (a) Schematics of the rectangular lattice and its reciprocal lattice. The unit cell is highlighted by the rounded box.

The band structure of this rectangular lattice in the Hermitian limit (

Figure 3.Same as Fig.

As

If we lift the NHPH symmetry of the system, e.g., by introducing a detuning ^{[4,5]}; the latter is similar to its 1D counterpart, as shown in Fig.

Figure 4.Band structure of the 2D square lattice shown in Fig.

Figure 5.Persisting and nonpersisting Hermitian flat bands. (a) Band structure of a Hermitian quasi-1D edge-centered square lattice. Inset:

B. Approach 2

Another and a more intuitive approach (“Approach 2”) to construct a non-Hermitian flat band is to start from a Hermitian system that has a flat band. The aim is then to maintain this Hermitian flat band (at least its real part) after the introduction of gain and loss. For the 1D Lieb lattice mentioned in the introduction, if the same ^{[25]} persists with a parity-time symmetric perturbation ^{[37,39–45]} largely due to this reason. Surprisingly, even when an arbitrary gain and loss configuration is imposed in the unit cell of the Lieb lattice shown in Fig.

One way to understand this behavior is again by using NHPH symmetry. This system has NHPH symmetry even with an arbitrary gain and loss modulation, and the flat band modes stay in the symmetric phase of the NHPH symmetry where

As it turns out, this persisting flat band can also be understood as the result of (i) a well-known mathematical theorem: one root of a cubic equation with real coefficients is always real; and (ii) the underlying Hermitian flat band is at the identical on-site energy, i.e.,

In fact, this property holds for many frustrated lattices with an odd number (

Figure 6.Persisting flat band in a 2D non-Hermitian Lieb lattice with

C. Approach 3

The third approach (“Approach 3”) to construct a non-Hermitian flat band is to construct a localized Wannier function that is an eigenstate of the whole system. Similar to the Hermitian case mentioned in the introduction, this approach can be applied to a frustrated lattice, and the resulting flat band has a ^{[30]} for a triangle lattice, and two other examples are given in Ref. ^{[31]}. The difference in these two similar studies is that the former considered a lattice that does not have a flat band without gain or loss, while the latter used lattices that do have a flat band in the Hermitian limit. In this sense, the latter is similar to Approach 2 in spirit, with a non-Hermitian perturbation that still makes the Wannier function an eigenstate of the whole system.

We should mention that two non-Hermitian flat bands similar in construction were also found numerically in Ref. ^{[29]}, but the authors there did not discuss the origin of the flatness or the existence of a compact Wannier function in the non-Hermitian case for either flat band. We point out here that the Wannier function for these two flat bands spans one and two unit cells, respectively. More specifically, the model considered in Ref. ^{[29]} can be written as

Figure 7.(a, b) Two compact Wannier functions (partially transparent dots) for the cross-stitched lattice shown. Couplings are represented by dashed lines (

All the three references mentioned above considered a unit cell with three lattice sites, which, however, is not the simplest lattice structure that can have a Wannier function with the aforementioned property; the simplest one is the saw lattice shown in Fig.

3 PRESENCE OF EPS AND POLYNOMIAL POWER INCREASE

An EP is a non-Hermitian degenerate point where not only the eigenvalues but also the wave functions of two or more eigenstates coalesce ^{[46–54]}. EPs are found in both Refs. ^{[30,31]}, and the formation of the non-Hermitian flat band is attributed to these EPs, to some extent ^{[30]}. However, from our discussion of Approach 3 and especially the example given by Eq. (

When all system parameters are fixed and the wave vector ^{[30]} found that a non-Hermitian flat band can have two EPs at two different values of ^{[31]} identified a scenario that every possible state in the flat band corresponds to an EP of order 2. Here, we first point out that, in fact, a higher-order EP can be found for every wave vector in the Brillouin zone, without increasing the size of the unit cell (i.e., three lattice sites). In addition, we unveil a polynomial power increase in such an “EP3 flat band,” which can display either quadratic or quartic behaviors.

To exemplify this EP3 flat band, we turn to the lattice described by Eq. (^{[25,33]}, including the collapsed and identical Wannier function for all values of

An initial excitation of this Wannier function or its superposition in more than one unit cell leads to a conserved energy in this EP3 flat band because the corresponding eigenvalue is real [see Fig. ^{[55]} to investigate the dynamics in the system. Let

Figure 8.Polynomial power dependence for a localized initial excitation in an EP3 flat band. The excitation of (a) a Wannier function in a single unit cell, (b) a C site, and (c) an A site are shown by the red arrows. Note the different scales of the vertical axis in these three panels, even though the initial amplitudes of the excitation all equal 1. (d) Fixed amplitude

Similarly, we define the second generalized eigenvector

4. DISCUSSION AND CONCLUSION

In conclusion, we have discussed systematically three approaches to achieve a non-Hermitian flat band. Their relation is summarized in Fig.

Figure 9.Relation between the three general approaches to construct a non-Hermitian flat band.

Even though Approaches 1 and 2 do not overlap, they still have a delicate relationship from the perspective of NHPH symmetry. When the couplings in Approach 2 are allowed to be complexified, the system itself does not need to have NHPH symmetry, as the last case in Section ^{[36]}. For the case with a nonpersisting Hermitian flat band, as considered in Fig.

Although only the last approach, i.e., constructing a Wannier function that is an eigenstate of the entire system, leads to a flat imaginary part of the dispersion relation as well, the existence of localized defect states with a slight perturbation does not rely on this additional property ^{[32]}. Because all implementations of flat bands in a photonic structure has defects, whether using laser-written waveguides ^{[3–5]} or microcavities fabricated by various etching methods ^{[6]}, the dispersive imaginary part does not affect the non-Hermitian flat bands from revealing this key manifestation of their Hermitian counterpart. If for some applications a vanished imaginary part of the entire flat band is preferred, Approach 3 should be adopted. For example, the two flat bands shown in Fig. ^{[56]}.

For simplicity, we have discussed mainly quasi-1D lattices, but the results presented here can be easily generalized to 2D lattices, as we have exemplified in Section ^{[57]}. Experimentally preparing a non-Hermitian system at an EP remains to be difficult, but recent successes of demonstrating EP-based sensing schemes ^{[58,59]} have proven that such a challenge can be overcome with finely tuned optical systems.

Acknowledgment

**Acknowledgment**. The author acknowledges support from NSF.

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Li Ge. Non-Hermitian lattices with a flat band and polynomial power increase [Invited][J]. Photonics Research, 2018, 6(4): A10

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