Continuum of Bound States in a Non-Hermitian Model

2025-04-22 0 0 9.45MB 13 页 10玖币

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Qiang Wang,1Changyan Zhu,1Xu Zheng,1Haoran Xue,1Baile Zhang,1, 2, ∗and Y. D. Chong1, 2, †

1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,

Nanyang Technological University, Singapore 637371, Singapore

2Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore, 637371, Singapore

In a Hermitian system, bound states must have quantized energies, whereas free states can form

a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing

non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector

potential. The eigenstates, which we call “continuum Landau modes” (CLMs), have gaussian spatial

envelopes and form a continuum ﬁlling the complex energy plane. We present experimentally-

realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have

other features matching the continuous model. One of these lattices can serve as a rainbow trap,

whereby the response to an excitation is concentrated at a position proportional to the frequency.

Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a

wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin eﬀect,

this requires a simple lattice design with reciprocal couplings.

The bound states of a quantum particle in an inﬁ-

nite continuous space have energies that are quantized

[1, 2]. This stems from a theorem that compact Her-

mitian Hamiltonians have pure point spectra [3–5], and

accounts for the energy quantization in standard mod-

els such as the harmonic oscillator and potential well [6].

The principle also applies to Anderson localized states

in random potentials, whose eigenenergies are dense but

countable [7, 8], and to the long-wavelength regime of

discrete (e.g., tight-binding) lattices [9–14]. Free states,

by contrast, are spatially extended and form continuous

spectra. It is interesting to ask whether any physical

system could behave diﬀerently, such as having an un-

countable set of bound states. How might such a Hamil-

tonian be realized, and what interesting properties might

its eigenstates have?

Over the past two decades, a large literature has devel-

oped around the study of non-Hermitian Hamiltonians

[15–22], catalyzed by the realization that such Hamil-

tonians can be implemented on synthetic classical wave

structures such as photonic resonators and waveguide ar-

rays [23–27]. Non-Hermitian systems have been found to

exhibit various interesting and useful features with no

Hermitian counterparts. For instance, their spectra can

contain exceptional points corresponding to the coales-

cence of multiple eigenstates [28–32], which can be used

to enhance optical sensing [33–35]. Another example is

the non-Hermitian skin eﬀect, whereby a non-Hermitian

lattice’s eigenstates condense onto its boundaries [36–45],

with possible applications including light funneling [46]

and the stabilization of laser modes [47, 48].

This raises the possibility of using non-Hermitian sys-

tems to violate the standard distinctions between quan-

tized bound states and continuous free states, which were

derived under the assumption of Hermiticity [49]. Here,

we investigate a non-Hermitian Hamiltonian that has

spatially localized energy eigenstates, which we call “con-

tinuum Landau modes” (CLMs), at every complex en-

ergy E. The Hamiltonian features a ﬁrst-order imaginary

dependence on momentum, along with a Landau-type

vector potential [6]; its eigenstates, the CLMs, map to the

zero modes of a continuous family of Hermitian 2D Dirac

models [50–55]. In two dimensions (2D), the CLM’s cen-

ter position r0varies linearly and continuously with the

complex plane coordinates of E. By contrast, previous

studies of non-Hermitian models with vector potentials

found only quantized bound states, similar to the Her-

mitian case [56–58]. We moreover show that the desired

Hamiltonian arises in the long-wavelength limit of a 2D

lattice with nonuniform complex mass and nonreciprocal

hoppings [40, 43–45], which can be realized experimen-

tally with photonic structures [59, 60] or other classical

wave metamaterials [46, 61–67]. If the lattice size is ﬁ-

nite, the CLMs become countable but retain other key

properties like the dependence between r0and E.

One dimensional (1D) versions of the model can be re-

alized in lattices with non-uniform real mass and nonre-

ciprocal hoppings, with the CLM positions proportional

to Re(E); or non-uniform imaginary mass and reciprocal

hoppings, with CLM positions proportional to Im(E).

The ﬁrst type of lattice can act as a non-Hermitian rain-

bow trap [68–71], in which excitations induce intensity

peaks at positions proportional to the frequency. Com-

pared to a recent proposal for rainbow trapping using

topological states within a bandgap [70, 71], the CLM-

based rainbow trapping scheme has the potential to op-

erate over a wide frequency bandwidth. The second type

of 1D lattice acts as a wave funnel [46]: the response

to an excitation is concentrated at one boundary. This

is similar to the funneling caused by the non-Hermitian

skin eﬀect [46], but requires only the placement of on-

site gain/loss without nonreciprocal couplings, and may

therefore be easier to implement [72–75].

We begin by reviewing Landau quantization in 2D Her-

mitian systems. As shown in Fig. 1(a), a free nonrela-

tivistic particle has a continuum of extended (free) states

arXiv:2210.03738v2 [quant-ph] 8 Jan 2023

FIG. 1. Eﬀects of a uniform magnetic ﬁeld on the spectra of

2D models. (a) For a nonrelativistic particle with quadratic

dispersion, the spectrum collapses into discrete Landau levels.

(b) For a Dirac particle, the spectrum forms an unbounded

sequence of Landau levels. (c) For the non-Hermitian Hamil-

tonian (1), the complex linear dispersion relation turns into a

continuum of bound states ﬁlling the complex energy plane.

with a quadratic energy dispersion, and when a uniform

magnetic ﬁeld is applied, the spectrum collapses into a

discrete set of Landau levels [6]. For a Dirac particle

with a linear dispersion relation, a magnetic ﬁeld like-

wise produces a discrete spectrum [50–52, 76], as shown

in Fig. 1(b). With appropriate gauge choices, the Landau

levels are spanned by normalizable eigenfunctions (bound

states) interpretable as cyclotron orbits.

Now consider the non-Hermitian 2D Hamiltonian

H=sx−i∂

∂x −By+isy−i∂

∂y +Bx,(1)

where sx,y =±1 (these are scalars, not matrices). For

B= 0, this kind of “non-Hermitian Dirac Hamiltonian”

has recently been analyzed in studies of non-Hermitian

band topology [77, 78]; its spectrum is given by E0(k) =

sxkx+isyky, as shown in Fig. 1(c). The B6= 0 case intro-

duces a symmetric-gauge vector potential corresponding

to a uniform out-of-plane magnetic ﬁeld 2Bˆz, via the

substitution −i∇→−i∇+A. The eigenstates of Hare

ψ(x, y) = Cexp −τ|r−r0|2+iq·r,(2)

r0(E, q) = 1

BIm[E−E0(q)]/sy

−Re[E−E0(q)]/sx,(3)

where Cis a normalization constant, τ=−sxsyB/2,

q= (qx, qy) is an arbitrary real vector, and Eis the

eigenenergy. If τ > 0, the wavefunctions are normalizable

on R2regardless of Eand q, with characteristic length

`∼B−1/2. The eigenenergies ﬁll the complex plane,

as shown in Fig. 1(c). For each E, there is a contin-

uum of bound states centered at diﬀerent r0, via Eq. (3);

also, states with the same qbut diﬀerent r0are non-

orthogonal. Note that such a continuum is not a generic

consequence of non-Hermiticity; other recently-studied

non-Hermitian models incorporating uniform magnetic

ﬁelds exhibit the usual quantized spectra [57, 58].

We call these eigenstates CLMs because they are

closely related to zeroth Landau level (0LL) modes of

massless 2D Dirac fermions [50–55]. CLMs with a given

energy Ehave a one-to-one map with the 0LL modes

of a given Hermitian Dirac Hamiltonian, whose gauge

is determined by E. The full set of CLM eigenstates

for Hthus maps to the 0LL modes of a family of Dirac

Hamiltonians with diﬀerent gauges, and the CLMs are

uncountable because the gauge can be continuously var-

ied. For details about this mapping, including the role of

gauge invariance, see the Supplemental Materials [49].

Due to the localization of the CLMs, wavefunctions

resist diﬀraction when undergoing time evolution with H.

For instance, a gaussian wavepacket maintains its width

under time evolution (even if the width diﬀers from that

of the CLMs); however, depending on the initial settings,

the wavepacket can move and undergo ampliﬁcation or

decay, as detailed in the Supplemental Materials [49].

CLMs can also be observed in the continuum limit

of discrete lattices. Take the 2D lattice depicted in

Fig. 2(a), whose Hamiltonian is

H=X

rhB(y−ix)a†

rar+txa†

r−ˆxar+ h.c.

+tya†

r−ˆyar−h.c.i,

(4)

where a†

r,arare creation and annihilation operators at

r= (x, y)∈Z2(the lattice constant is set to 1), and

tx, ty∈Rare hopping coeﬃcients. His non-Hermitian

due to the imaginary part of the mass and the nonre-

ciprocity of the yhoppings [40, 43–45]. Such nonrecipro-

cal hoppings can be realized on experimental platforms

such as circuit lattices, ﬁber loops, and ring resonator

lattices [46, 61–67], which have notably been used to

study the non-Hermitian skin eﬀect [36, 37, 40, 41, 43–

45]. Note, however, that this lattice does not exhibit the

skin eﬀect [49].

For B= 0, Hhas discrete translational symmetry and

dispersion relation E0

k= 2(txcos kx+itysin ky), where

kx,y ∈[−π, π]. Taking |ψki=Prexp(ik·r)Ψra†

r|∅i,

where |∅iis the vacumm state, the slowly-varying enve-

lope obeys HkΨr=EΨr, where [49]

Hk=E0

k−−iµk

∂

∂x −By+i−iνk

∂

∂y −Bx,(5)

µk= 2txsin kx, νk= 2tycos ky,(6)

to ﬁrst order in spatial derivatives. Note that the com-

plex masses in (4) produce the pseudo vector potential

[50–55]. For B/µk<0, B/νk>0, there exist CLMs simi-

lar to (2), with τreplaced by τx=−B/2µk,τy=B/2νk,

-30 0 30

0.5

-20 0 20

-10

-20 0 20 -30 0 30

0.5

-2 0 2

-2

-10 -5 0 5

-5

-10

-5

FIG. 2. Continuum Landau modes (CLMs) in a 2D lattice. (a) Schematic of a square lattice with reciprocal hoppings along x

(gray lines), nonreciprocal hoppings along y(black arrows), and onsite mass mx,y =B(y−ix) (size and darkness of the circles

indicate the real and imaginary parts). (b)–(c) Complex energy spectra for ﬁnite lattices with (b) B= 0.03 and (c) B= 0.3.

The color of each dot corresponds to the participation ratio (PR) of the eigenstate; a more localized state has lower PR. The

arrows on the color bar indicate the highest PR for the CLM ansatz, for each B. The dashed boxes are the bounds on CLM

eigenenergies derived from Eq. (7). (d) Plot of Re(E) versus hyi(left panel) and Im(E) versus hxi(right panel) for B= 0.3.

The black dashes and gray dotted lines respectively indicate the theoretical central trend line (corresponding to E0

k+q→0)

and bounding lines derived from Eq. (7). (e)–(f) Wavefunction amplitude |ψr|for the eigenstates marked by yellow stars in (b)

and (c) respectively. Hollow and ﬁlled circles respectively indicate the variation with xand y, along lines passing through the

center of each gaussian; solid curves show the CLM predictions. Insets show the distribution in the 2D plane. In (b)–(f), we

use tx=ty= 1 and a lattice size of 60 ×60, with open boundary conditions.

and Eq. (3) replaced by

r0(E, k,q) = 1

B−ImE−E0

k+q

ReE−E0

k+q+O(|q|2).(7)

As Ψris assumed to vary slowly in r, the solutions are

limited to the regime |q|  1. If the lattice is inﬁnite,

they form a continuous set spanning all E∈C. For a

ﬁnite lattice, the eigenstates are ﬁnite and hence count-

able, and the CLMs reduce to a band over a ﬁnite area

in the Eplane. In Fig. 2(b)–(c), we plot the spectra for

B= 0.03,0.3, each lattice having size Lx=Ly= 60,

open boundary conditions, and tx=ty= 1. By requir-

ing r0to lie in the lattice, Eq. (7) implies the bounds

|Re(E)|.BLy/2+2txand |Im(E)|.BLx/2+2ty. For

large Lx,y, the energy discretization is of order B.

All of the numerically obtained eigenstates are CLMs.

The color of each data point in Fig. 2(b)–(c) indicates

the participation ratio (PR), deﬁned for a wavefunction

der N2) corresponding to extended states [79]. We ﬁnd

that all eigenstates have PR consistent with the CLM

predictions, and well below N2(for each case, the max-

imum PR, attained when |µk|=|νk|= 1, is indicated

by an arrow in the color bar). Fig. 2(d) plots each eigen-

state’s energy against the position expectation values hxi

and hyi, for B= 0.3. This reveals the linear relation-

ship between Eand r0predicted in Eq. (7) (indicated by

dashes), and the upper and lower bounds introduced by

the E0

k+qterm (dotted lines). Fig. 2(e)–(f) compares the

spatial amplitude |ψr|for two arbitrarily-chosen numeri-

cal eigenstates to the CLM solutions, which are in good

agreement.

CLMs can also be realized in 1D lattices, which are

simpler to implement experimentally. We study two

kinds of lattices, which have diﬀerent behaviors.

The ﬁrst 1D lattice, shown in Fig. 3(a), has a real mass

gradient and nonreciprocal hoppings. Its Hamiltonian is

H=X

jhB(j−j0)a†

jaj+ta†

jaj−1−ta†

j−1aji,(8)

where B, t ∈R. For a ﬁnite lattice with 1 ≤j≤N, we

take j0=1

2(N+ 1). The nonreciprocal nearest neigh-

bor hoppings ±t∈Rare indicated by solid and dashed

arrows. For B= 0, the spectrum E0

k= 2it sin k(where

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ContinuumofBoundStatesinaNon-HermitianModelQiangWang,1ChangyanZhu,1XuZheng,1HaoranXue,1BaileZhang,1,2,andY.D.Chong1,2,y1DivisionofPhysicsandAppliedPhysics,SchoolofPhysicalandMathematicalSciences,NanyangTechnologicalUniversity,Singapore637371,Singapore2CentreforDisruptivePhotonicTechnologies,Nanyang...

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Continuum of Bound States in a Non-Hermitian Model

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