Non-Hermitian lattices with a flat band and
polynomial power increase [Invited]
LI GE
1,2
1
Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA
2
The Graduate Center, CUNY, New York, New York 10016, USA (li.ge@csi.cuny.edu)
Received 4 August 2017; revised 6 November 2017; accepted 12 November 2017; posted 15 November 2017 (Doc. ID 302868);
published 5 March 2018
In this work, we first discuss systematically three general approaches to construct a non-Hermitian flat band,
defined by its dispersionless real part. These approaches resort to, respectively, spontaneo us restoration of
non-Hermitian particle-hole symmetry, a persisting flat band from the underlying Hermitian system, and a com-
pact Wannier function that is an eigenstate of the entire system. For the last approach in particular, we show the
simplest lattice structure where it can be applied, and we further identify a special case of such a flat band where
every point in the Brillouin zone is an exceptional point of order 3. 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 localized eigenstate or one of the two generalized eigenvectors is initially excited. Nevertheless, the
asymptotic wave function in the long time limit is always given by the eigenstate, in this case, the compact
Wannier function or its superposition in two or more unit cells.
© 2018 Chinese Laser Press
OCIS codes: (130.2790) Guided waves; (230.7370) Waveguides; (080.6755) Systems with special symmetry; (160.5293) Photonic
bandgap materials.
https://doi.org/10.1364/PRJ.6.000A10
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 sys-
tems [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 multi-
fractal behavior [19], mobility edges with algebraic singularities
[20], and unusual scaling behaviors [21–23].
For a Hermitian lattice, a completely flat band in the en-
tire Brillouin zone is formed when there exists a Wannier
function that is an eigenstate of the whole system. To under-
stand this relation, we only need to resort to the definition of
the Wannier function itself, which we denote by W
n
x − ja
in one dimension (1D). Here, n is the band index, a is
the lattice constant, and j is the unit cell index. The Bloch
wave function with wave vector k in the nth band can be
written as
Ψ
n
x; k
X
j
e
ikaj
W
n
x − ja; (1)
and it satisfies H
0
Ψ
n
x; kω
n
kΨ
n
x; k, where H
0
is the
Hamiltonian of the entire system instead of the Bloch
Hamiltonian H k of the unit cell. Now if H
0
W
n
x − ja
ω
w
W
n
x − ja, i.e., there exists a Wannier function that is an
eigenstate of the whole system with eigenvalue ω
w
; then, we
immediately find ω
n
kω
w
, which is k-independent.
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 iso-
lating the Wannier function from the rest of the lattice. Take
the 1D Lieb lattice, for example [see Fig. 1(a)]. It has an
L-shaped unit cell, with decorated lattice sites (A) coupling
to every other site (B) on the main lattice. In the tight-binding
model, it is captured by the Bloch Hamiltonian
Hk
2
6
4
ω
A
G 0
G ω
B
J1 e
−ika
0 J1 e
ika
ω
C
3
7
5
; (2)
where G;J ∈ R are the vertical and horizontal coupling co-
efficients. When there is no detuning between the A sites and the
C sites on the main lattice (i.e., ω
A
ω
C
≡ ω
0
), a V-shaped
Wannier function exists, which spans two unit cells. This
Wannier function has a nonzero amplitude only at the two A
A10
Vol. 6, No. 4 / April 2018 / Photonics Research
Research Article
2327-9125/18/040A10-08 Journal © 2018 Chinese Laser Press
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