# Observation of a topological edge state stabilized by dissipation

Helene Wetter  Physikalisches Institut, Universität Bonn, Nussallee 12, 53115
Bonn, Germany  Michael Fleischhauer  Department of Physics and Research Center
OPTIMAS, RPTU Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Stefan
Linden  Physikalisches Institut, Universität Bonn, Nussallee 12, 53115 Bonn,
Germany  Julian Schmitt<EMAIL_ADDRESS>Institut für Angewandte
Physik, Universität Bonn, Wegelerstr. 8, 53115 Bonn, Germany

###### Abstract

Robust states emerging at the boundary of a system constitute a hallmark for
topological band structures. Other than in closed systems, topologically
protected states can occur even in systems with a trivial band structure, if
exposed to suitably modulated losses. Here, we study the dissipation-induced
emergence of a topological band structure in a non-Hermitian one-dimensional
lattice system, realized by arrays of plasmonic waveguides with tailored loss.
We obtain direct evidence for a topological edge state that resides in the
center of the band gap. By tuning dissipation and hopping, the formation and
breakdown of an interface state between topologically distinct regions is
demonstrated.

###### pacs:

03.65.Vf,42.82.Et,42.70.Qs

Topology is an important paradigm for our understanding of phases of matter
[1], with the quantum Hall effect constituting a prominent example of a
topological system isolated from the environment [2]. Interfacing materials
with distinct topological properties has remarkable implications leading to
localized edge states at the boundary, which due to their robustness against
disorder are considered as valuable resource states for quantum technologies
[3]. Conceptually, the robustness results from the existence of global
integer-valued invariants, which can only change in a phase transition
associated with the closing of a gap. Inspired by solid-state systems,
topological states in closed Hermitian systems have been experimentally
realized in a wide range of platforms, such as ultracold atoms or photonics
[4, 5, 6].

Exploring topological phenomena in open systems presents a complementary
approach to realize robust edge states, where the coupling between the system
and the environment (_e.g._ by pumping or dissipation of particles) acts as a
resource rather than a limitation. Starting from the prediction of topological
transitions in non-Hermitian quantum walk [7], conceptual questions about the
classification of open-system topological phases for non-Hermitian and
Lindbladian settings [8, 9, 10, 11, 12], the role of topological invariants
and edge states [13, 14, 15, 16, 17], and the band theory [18, 19] have been
addressed theoretically. Experimentally, non-Hermitian systems have been
realized in photonics, where driven-dissipative effects can be engineered
[20]. Combining topologically nontrivial photonic crystals with gain or loss,
this has allowed for the observation of topological (lasing) states in
waveguides [21, 22], resonator arrays [23, 24] and exciton-polaritons [25,
26]. Topological protection is here, however, inherited from the photonic band
structure, and not from a coupling to reservoirs. The implementation of
topological phases that solely arise from non-Hermiticity and lack a Hermitian
counterpart, as proposed in refs. [8, 9, 27, 28] and realized with mechanical
metamaterials [29], acoustic cavities [30], and electrical circuits [31], has
so far remained elusive for optical systems.

Fig. 1: Experimental scheme. (a) Lattice system with nontrivial topology
induced solely by dissipation (top) and experimental realization with DLSPP
waveguides spaced by distance $d$, where losses are induced by Chromium
stripes of width $w$ (bottom). (b) Complex-valued energy spectrum for 40
lattice sites with $g_{1}=|g_{2}|$. For $g_{2}<0$, the system is topologically
trivial and the probability density $|\Psi|^{2}$ is concentrated in the first
two lattice sites (top left). In the topologically nontrivial regime
($g_{2}>0$), $|\Psi|^{2}$ is exponentially localized at the edges (top right)
and associated with midgap states at zero energy (red lines). (c) Left:
Waveguide sample with Chromium stripes (dark gray) arranged to realize trivial
(topological) domains in the left (right) sample half. The Chromium-free
region on top is used to excite the waveguides by grating coupling of laser
light. Right: Measured dissipation in a single waveguide $\mathrm{Im}\beta$
versus $w$ and hopping $J$ between two waveguides versus $d$, along with fits
(lines).

In this Letter, we report measurements of light-matter states with nontrivial
topological properties, solely induced by tailored dissipation. Using surface
plasmon polaritons (SPPs) confined in waveguide arrays, we obtain signatures
for open-system topological edge states by identifying zero-energy modes
localized at the boundary of the sample. The underlying one-dimensional (1D)
lattice with 4-site unit cell realizes a non-Hermitian extension of the
paradigmatic SSH model [32], despite uniform hopping throughout the lattice.
By tuning dissipation and hopping, the birth and death of a non-Hermitian
topologically protected edge state is demonstrated.

The basic principle of our topological system, see Fig. 1(a), relies on a 1D
lattice with spatially uniform nearest-neighbor hopping $J$ and spatially
varying dissipation at the lattice sites [27, 28]. The unit cell consists of 4
sites spaced by $d$, which are subject to a gain-loss pattern
$(ig_{1},-ig_{2},-ig_{1},ig_{2})$ with real-valued dimensionless amplitudes
$g_{1,2}$. The Bloch Hamiltonian of the open system at wave vector $k_{x}$

$\hat{H}_{k_{x}}=J\begin{pmatrix}ig_{1}&1&0&e^{-4ik_{x}d}\\\ 1&-ig_{2}&1&0\\\
0&1&-ig_{1}&1\\\
e^{4ik_{x}d}&0&1&ig_{2}\end{pmatrix}-iJg_{0}1\kern-2.5pt\text{l}$ (1)

is a non-Hermitian matrix, _i.e._ , $\hat{H}\neq\hat{H}^{\dagger}$, with
complex energy eigenvalues $E$. Here, $g_{0}>0$ accounts for a loss at all
lattice sites, which governs our implementation using purely dissipative
waveguides, see Fig. 1(a) (bottom). Due to the global loss the steady-state is
here the trivial vacuum. Nevertheless, as we will demonstrate in our
experiments, the system possesses a nontrivial _non-equilibrium_ topology,
protected by the symmetries of the Liouvillian $\cal L$ that governs the
dynamics of the density matrix $\rho$, according to $\dot{\rho}={\cal L}\rho$
[12]. The dynamical generators fulfill time reversal ($\sf T$), charge-
conjugation ($\sf C$), and chiral ($\sf S$) symmetries [33], characterizing
the class BDI [1], which can have a nontrivial topology in 1D.

The topological character of the 4-site model becomes apparent when
considering the energy band structure and eigenstates in a finite-length
lattice for different loss patterns, as shown in Fig. 1(b); for simplicity, we
consider the symmetric case $g_{1}=|g_{2}|$ and $g_{0}=|g_{2}|$ with $g_{2}$
as the control parameter. In the different parameter regimes,
$\mathrm{diag}(\hat{H})=(0,0,0,0)$ for $g_{2}=0$ (lossless trivial, I),
$\mathrm{diag}(\hat{H})=-2iJ|g_{2}|(0,0,1,1)$ for $g_{2}<0$ (dissipative
trivial, II), and $\mathrm{diag}(\hat{H})=-2iJg_{2}(0,1,1,0)$ for $g_{2}>0$
(topologically nontrivial, III), respectively (blue and red circles in Fig.
1(b)). Phase (I) exhibits a metal-like gapless band structure with probability
densities $|\Psi|^{2}$ delocalized in the bulk, while the band structure in
phase (II) is gapped. In striking contrast, phase (III) features two midgap
states at zero energy in $\mathrm{Re}E$ (red lines), which are localized at
the boundary of the system and decay exponentially into the bulk. Due to the
dissipative nature of the system, $\mathrm{Im}E<0$ for all states. Note that
while there is not yet a general understanding of non-equilibrium invariants
for density matrices, a topological number for the non-Hermitian system has
been identified theoretically as a global Berry phase, which takes discrete
values $W=0$ or $1$ for $g_{1}g_{2}<0$ or $g_{1}g_{2}>0$ in phases (II) or
(III), respectively [14, 27]. Based on the non-equilibrium symmetries and the
above mentioned spectral and spatial signatures, topological states solely
induced by dissipation rather than Hermitian band engineering are
theoretically expected in our system.

To experimentally investigate the topological properties of the non-Hermitian
lattice system, we utilize SPPs confined in evanescently coupled arrays of
dielectric loaded SPP waveguides (DLSPPW) with tailored losses. The samples
are fabricated by two-step electron beam lithography [33, 34]. Figure 1(a)
(bottom) outlines a typical waveguide structure, realized by depositing PMMA
ridges of about 200nm width and center-to-center spacing $d$ on top of a glass
substrate, previously coated with a low-absorption 60nm-thin Gold layer to
host the plasmonic part of the polaritons. Losses at individual lattice sites
are introduced and controlled by adding Chromium stripes of variable width $w$
below the ridges.

Fig. 2: SPP evolution for metal-like (left, I), dissipative topologically
trivial (middle, II) and dissipation-induced topological domains (right, III).
(a) Real-space intensity distribution for injection at the edge (top) and at
the first site of a unit cell in the bulk (bottom), for a sample of 48
waveguides spaced by $d=$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
corresponding to $J=0.045(3)$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$.
Solid and dashed lines indicate sample boundaries and unit cells. For the used
$w=$0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in (II) and (III),
$\mathrm{Im}\beta=$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ and
$g_{1}=|g_{2}|=1.1$. (b) Simulated intensity evolution in the arrays from (a)
with $J=$0.045\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$\text{Im}\beta=$0.1\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
($0.01\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$) for the lossy [red in
(a)] and low-loss [blue in (a)] waveguides, in good agreement with the
experimental results. (c) Momentum-resolved energy spectra with $\cos$-shaped
band for bulk excitation, band gap in the trivial and flat band in the
topological domains, respectively, for edge excitation, along with theory
prediction (top).

An exemplary sample containing two different loss patterns in the unit cells,
corresponding to the topologically trivial (II) and nontrivial (III) region,
is shown in Fig. 1(c). The SPP evolution in the array is excited with a 980nm
laser at $15\text{\,}\mathrm{\SIUnitSymbolMicro W}$ optical power and
characterized by leakage radiation microscopy [6, 33]. By varying $w$ and $d$,
both the additional absorption $\mathrm{Im}\beta$ from Chromium and the
hopping $J$ (both in units $$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$)
can be accurately controlled, see Fig. 1(c) (right); here $\beta$ denotes the
complex-valued propagation constant in a single waveguide [6]. In terms of eq.
(1), the dissipation parameters follow as $g_{2}=\mathrm{Im}\beta/(2J)$. The
shown data has been obtained by measuring the decay of the intensity in a
single waveguide and by recording the coherent oscillation between two
adjacent waveguides in separate samples, respectively.

First, we study the evolution of the SPPs upon injecting a wave packet at the
edge and in the bulk of the waveguide arrays, respectively, for three loss
patterns according to phases (I) (metal-like), (II) (dissipative trivial), and
(III) (dissipative topological). Figure 2(a) shows the real-space SPP
intensity distributions obtained by imaging the leakage radiation. For phase
(I) (Fig. 2(a), left column), the SPP evolution mimics a two-state quantum
walk of a particle in a periodic potential. This metal-like behaviour is
highlighted by a conical diffusion of the signal along with a characteristic
interference pattern when injecting the wave packet in the bulk; upon
injection at the edge, the wave packet simply propagates in the $-x$
direction. With losses as in the topologically trivial phase (II) (Fig. 2(a),
middle), the diffusion is inhibited and an oscillation of the intensity
between two neighboring low-loss waveguides is observed for both excitation
protocols. In contrast, with losses as in the topologically nontrivial phase
(III) (Fig. 2(a), right), the edge excitation reveals apart from the overall
damping a quasi-stationary intensity evolution that remains locked to the
outermost waveguide. This localization occurs at the edge only, as understood
from the corresponding probability density $|\Psi|^{2}$ shown in Fig. 1(b)
(top), while excitation in the bulk reveals the same oscillatory dynamics as
in phase (II), except for a phase shift due to the neighboring low-loss
waveguide now lying above the excited one. The phenomenology observed in the
measured real-space intensity is in good agreement with numerical simulations
based on coupled mode theory [33], see Fig. 2(b), giving conclusive evidence
that the non-Hermitian model is well-captured by our DLSPPW platform.

Figure 2(c) shows the momentum-resolved occupation of the energy bands within
the first two Brillouin zones from $k_{x}=-2\pi/d$ to $2\pi/d$, as obtained by
recording the leakage radiation in the back-focal Fourier plane of the
microscope objective. In the metal-like phase (I) and for bulk excitation, the
spectrum matches the expected $\cos$-shaped energy band that complements the
independently observed ballistic transport in real space. The spectrum agrees
with the simulated one in Fig. 2(c), except for a circular segment at
$k_{z}\lesssim$6.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ that is well
understood to arise from unconfined SPP propagation outside of the array, also
visible in the gray shaded area in Fig. 2(a). In the dissipative topologically
trivial phase (II) and for edge excitation, the momentum distribution
considerably changes. Two energy bands separated by a gap near
$k_{z}=6.56(2)~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ and visible at
$k_{x}\approx-0.5\pi/d$ are observed. In the topologically nontrivial phase
(III), on the other hand, the momentum distribution exhibits only a single
flat energy band centered at
$k_{z}=6.59(2)~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$, with a
spectral width determined by losses and residual transport in the bulk. By
comparing with Fig. 1(b), the data gives evidence for a topological zero-state
in the band gap.

Fig. 3: Dissipation-induced emergence of interface state between distinct
topological domains. (a) Interface between phase-(II) and phase-(III) domains
realized by different loss patterns, and real-space intensity distribution
$I(x,z)$ upon excitation of the interface waveguide (red) for increasing
losses
$\mathrm{Im}\beta=\\{0,0.06,0.09,0.1\\}$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-1}$ from larger Chromium widths $w$. Losses correspond to
$g_{1}=|g_{2}|=\\{0,0.7,1,1.1\\}$ and, as before,
$J=0.045(3)$\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$. (b) Line profiles
of the intensity averaged around $z=$50\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ for low (top), intermediate (center) and high losses (bottom). (c) Decay
length $\ell$ of intensity in interface waveguide for cases in (a). Despite
increased Chromium absorption, an enhanced propagation distance is observed,
providing evidence for a topologically robust state. Shaded areas show
simulations for the interface (red) and bulk (gray), where the enhanced $\ell$
indicates the robustness of the interface beyond $g_{2}\approx 0.7$. Fig. 4:
Breaking the topological protection at an exceptional point. (a) Real-space
intensity distribution for increasing hopping $J$, realized by reduced
waveguide spacings
$d=\\{1.8,1.6,1.4,1.2,1.0\\}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ at
$w=$0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
($\mathrm{Im}\beta=$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$). For
excitation at the trivial edge (left), the population oscillates with
increasing frequency as $J$ becomes larger. Upon injecting light into the
boundary between domains (III) and (II) (middle), a quasi-stationary interface
state is observed, which at $J\approx$0.09\text{\,}\mathrm{\SIUnitSymbolMicro
m}$^{-1}$ becomes delocalized in the bulk, in agreement with simulations
(right). (b) Frequency $k_{z}$ fitted to the evolution at the excited
waveguide with $f(z)=a_{1}\cos(k_{z}z+\phi)e^{-z/\ell}+a_{0}$. Solid line and
vertical bars give numerical results, and error bars denote uncertainties of
fit parameter. (c) Left: Calculated probability density $|\Psi|^{2}$ at the
boundaries in (a), labelled by A and B, along with bulk modes (gray). A
delocalization of the interface mode (red) is visible for too large $J$,
indicating the loss of topological character. Right: Complex eigenenergies of
the two topological (colored) and 38 bulk (gray) states, as $J$ is varied. For
$J\approx$0.095\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$, the eigenvalues
coalesce at an exceptional point (yellow), closing the imaginary and opening
the real band gap.

A unique feature of the investigated system lies in the fact that topological
properties emerge as a consequence of dissipation alone, in a lattice which
would be otherwise topologically trivial. To systematically test this
dissipation-induced birth of topological order, we focus on an interface
between two distinct domains prepared in phases (II) and (III), respectively,
and gradually increase the loss $\mathrm{Im}\beta$. Figure 3(a) shows the
real-space SPP evolution for increasing Chromium widths $w$, after
consistently exciting the same low-loss waveguide, which is located at the
interface. The conical intensity spread into the bulk for $w=0$, previously
seen in Fig. 2, is gradually transformed into a quasi-stationary, _i.e._ ,
transversely localized occupation of the interface waveguide. Remarkably,
despite larger absorption $\mathrm{Im}\beta$, the SPP propagation length is
significantly enhanced. This is understood from the increase of $\mathrm{Im}E$
for the topological zero-states for large enough $g_{2}\gtrsim 0.7$ [see Fig.
1(b) (bottom panel, red line)], marking a clear distinction point from its
more lossy topologically trivial counterpart in a phase-II system for
$g_{2}<0$ [Fig. 1(b) (bottom panel, gray line)]. Thus, the intensity becomes
more localized and long-lived at the interface due to the presence of
topologically distinct domains. The extended propagation distance is visually
more striking in the line profiles at
$z=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in Fig. 3(b). Quantitatively,
Fig. 3(c) shows the fitted $1/e$ decay length of the intensity at the
interface as a function of $w$, along with the theoretically expected decay
length at the interface waveguide and in the bulk, confirming the genuine
dissipation-enhanced topological robustness of the interface state.

Conversely to the discussed formation of an interface state upon introducing
dissipation, we next focus on _breaking_ the topological protection at the
interface by increasing the hopping $J$ in the presence of dissipation. For
this, we have reduced the waveguide spacing $d$, while keeping the losses in
the interfaced phases (II) and (III) fixed at
$\mathrm{Im}\beta=$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ [with
$w=$0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ as in Fig. 2]. Figure 4(a)
shows the real-space SPP evolution after injection of a wave packet at the
boundary between a trivial phase-(II) domain and the vacuum (left). Not
surprisingly, the population oscillates between the two low-loss waveguides as
before, but now with a larger wave vector $k_{z}$ as $J$ is increased; see the
blue data in Fig. 4(b). At the phase (II)-(III) interface shown in Fig. 4(a)
(right), on the contrary, the SPP population stays localized at the interface
waveguide without any appreciable transverse transport or oscillation; see the
red data in Fig. 4(b). Eventually, at
$J\approx$0.09\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$, the tunnel
coupling becomes so large that the SPPs diffuse away from the interface into
the bulk, lifting the topological protection of the interface state.

The diminished topological robustness is understood to result from a
delocalization of the interface mode with increased $J$, as shown in Fig.
4(c). The calculated probability density $|\Psi|^{2}$ is shown for the trivial
edge (label A) and interface (B) modes, along with bulk modes. For larger $J$,
we find the transverse localization length of the interface mode to extend
further, revealing an exponential decay of $|\Psi|^{2}$ into the bulk with a
maximum probability that consistently occurs at the interface waveguide. For
the trivial edge, on the other hand, the maximum of the probability soon
shifts away from the boundary of the sample and generally does not exhibit an
exponential decay towards the bulk region. This phenomenology shares a close
analogy with the topological edge states encountered in the SSH model [32],
where—despite being a Hermitian system with a two-site unit cell and
alternating hopping —the probability density of the dimerized edge modes
decays exponentially. In the non-Hermitian 4-site model, the alternating
hopping is replaced by an effective (de-)coupling of pairs of gain-gain or
loss-loss (mixed gain-loss) lattice sites [27], and a dimerization with two-
and four-site periodicity occurs in $\mathrm{Re}\Psi$ and $\mathrm{Im}\Psi$,
respectively. This notion of exponential localization supports our claim that
a topologically protected state is indeed observed.

Finally, the spectrum of the complex energy eigenvalues in Fig. 4(c) provides
a physical picture about the broken topological protection of the interface
state (colored circles; energies of bulk modes are shown in gray) near
$J\approx$0.095\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$. For small $J$,
the eigenvalues of the zero-states with $\mathrm{Re}E=0$ fall in the band gap
and are separated in $\mathrm{Im}E$ (blue, green circles); note that due to
$\mathrm{Im}E\neq 0$ the non-Hermitian system is not
$\mathcal{P}\mathcal{T}$-symmetric [20]. As $J$ increases, the imaginary gap
closes and the eigenvalues coalesce at an exceptional point (yellow circle),
followed by an opening of a real energy gap, which lifts the edge state
degeneracy and merges both with the bulk bands, visible in Fig. 4(c) (red
circles). At the exceptional point, the system has lost its topological
character.

In conclusion, we have experimentally demonstrated open-system topological
states induced by dissipation alone, using SPP waveguide arrays with uniform
hopping and spatially-distributed loss. Evidence for the topological nature of
the non-Hermitian system is obtained from a localized midgap edge state
between distinct topological domains. By independently tuning dissipation and
hopping, both the emergence and breaking of topological order is observed. For
the future, lowering the SPP losses may enable direct measurements of the
topological invariant by interferometry [35] and give access to non-Hermitian
Floquet engineering by modulated loss and hopping [36, 34]. An intriguing
perspective lies in the implementation of open-system topological states with
optical quantum gases within optically-active microcavities [37, 38], opening
ways to study the fate of topological order in the presence of fluctuations in
one and two dimensions [39].

We thank V. Zimmermann and Z. Fedorova for discussions. S.L., M.F. and J.S.
acknowledge support from the DFG within SFB/TR 185 (277625399). J.S.
acknowledges support by the EU (ERC, TopoGrand, 101040409), and by DFG within
the Cluster of Excellence ML4Q (EXC 2004/1–390534769).

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##  I Supplemental Material

###  I.1 Sample preparation

Our waveguide samples are fabricated in a multi-step process using electron
beam lithography (EBL), as described in ref. [34]. In a first step,
Poly(methyl methacrylate) (PMMA) is used as a positive tone resist at an
electron dose of $2.45\text{\,}\mathrm{C}\text{\,}{\mathrm{m}}^{-2}$ to create
a mask for the $6\text{\,}\mathrm{nm}$ chromium layer thermally evaporated on
top. A lift-off process reveals the chromium stripes, which introduce the
losses in the dissipative waveguide samples. In the second EBL step, the
waveguides are created by exposing the PMMA to an electron dose of
$7\text{\,}\mathrm{C}\text{\,}{\mathrm{m}}^{-2}$; here the PMMA acts a
negative tone resist. The width of each waveguide is approximately
$200\text{\,}\mathrm{nm}$. A scanning electron micrograph of an exemplary
waveguide sample equipped with Chromium stripes is shown in Fig. 1(c) of the
main text.

###  I.2 Experimental system

The surface plasmon polaritons (SPPs) are excited in a site-resolved way by
focusing a TM-polarized cw laser beam at $980\text{\,}\mathrm{nm}$ wavelength
and $15\text{\,}\mathrm{\SIUnitSymbolMicro W}$ optical power onto a grating
coupler placed on top of one waveguide, which diffracts the laser light into
the targeted waveguide in the $z$ direction. The leakage radiation emitted by
the SPPs is collected with an oil-immersion objective ($\times 60$
magnification, NA$=1.4$) and imaged onto an sCMOS camera (Andor Zyla) to
obtain the real-space intensity distribution. Leakage radiation microscopy
also enables recording the momentum-resolved Fourier space with a single
measurement by imaging the back-focal plane of the microscope objective. For
this purpose, the transmitted laser beam is filtered out by a knife edge in
the intermediate back-focal plane of the oil immersion objective to clean the
recorded image of residual scattered laser light.

###  I.3 Symmetry classification of topology

Generalized unitary and anti-unitary symmetries are a powerful tool to
classify topological properties of systems of non-interacting particles. The
ten fundamental symmetry classes of Hamiltonians [1] have lead to the famous
periodic table of topological insulators and superconductors [40, 41]. Out of
thermal equilibrium, a symmetry classification must include besides the
Hamiltonian also the generators describing dissipation and the associated
fluctuations [12, 11]. If the non-equilibrium dynamics is described by a
Lindblad master equation for the density operator $\rho$

$\partial_{t}\rho=-i[\hat{H},\rho]+\sum_{\mu}\Bigl{(}2\hat{L}_{\mu}\rho\hat{L}_{\mu}^{\dagger}-\\{\hat{L}_{\mu}^{\dagger}\hat{L}_{\mu},\rho\\}\Bigr{)},$
(2)

where the Hamiltonian $\hat{H}$ is bilinear in particle creation and
annihilation operators and the jump operators $\hat{L}_{\mu}$ are linear, a
complete classification of generalized symmetries can be given in terms of
$\hat{H}$ and the Hermitian operator

$\hat{M}=\sum_{\mu}\hat{L}_{\mu}^{\dagger}\hat{L}_{\mu}.$ (3)

We here follow the approach of Ref. [12] where non-interacting fermions have
been considered. Although the SPPs in our experiment are bosons, we can
directly apply the explicit formulation in terms of free fermions since the
physics in our non-interacting system is governed by single-particle physics
only. To this end we write $\hat{H}=\frac{1}{2}\hat{A}^{\dagger}\,{\sf
H}\hat{A}$ and $\hat{M}=\frac{1}{2}\hat{A}^{\dagger}\,{\sf M}\hat{A}$ in a
symmetric form, where
$\hat{A}=(\hat{c}_{1},\hat{c}_{1}^{\dagger},\hat{c}_{2},\hat{c}_{2}^{\dagger},\dots)^{\top}$
is a vector of fermion annihilation and creation operators
$\hat{c}_{i},\hat{c}_{i}^{\dagger}$ at lattice site $i$, respectively, with
$\sf H$ and $\sf M$ being Hermitian matrices. Furthermore, $\sf M$ is
decomposed as

$\displaystyle{\sf M}={\sf D}-$ $\displaystyle i{\sf P},$ (4)
$\displaystyle{\sf D}=\frac{1}{2}\left({\sf M}+\sigma_{x}{\sf
M}^{\top}\sigma_{x}\right),\quad$ $\displaystyle{\sf P}=\frac{i}{2}\left({\sf
M}-\sigma_{x}{\sf M}^{\top}\sigma_{x}\right),$ (5)

where the Pauli matrix $\sigma_{x}$ corresponds to the exchange of
annihilation and creation operators in $\hat{M}$.

In the experimental system the stationary state is the vacuum, and the
presence of edge states is solely a consequence of _non-equilibrium_ topology.
As shown in Ref. [12], the generator of the non-equilibrium dynamics of a
fermion lattice model is invariant under time reversal ${\sf T}$ if a unitary
matrix ${\sf U}_{T}$ exists such that in Bloch-momentum space with wave number
$k$ the following relations are fulfilled:

$\displaystyle{\sf U}_{T}\,{\sf H}^{*}(-k)\,{\sf U}_{T}^{-1}$
$\displaystyle=-{\sf H}(k),$ $\displaystyle{\sf U}_{T}\,{\sf D}^{*}(-k)\,{\sf
U}_{T}^{-1}$ $\displaystyle=+{\sf D}(k),$ (6) $\displaystyle{\sf U}_{T}\,{\sf
P}^{*}(-k)\,{\sf U}_{T}^{-1}$ $\displaystyle=-{\sf P}(k).$

Note the minus sign in the first equation, which is different from the
equilibrium case. Similarly, one finds that the non-equilibrium dynamics is
invariant under charge conjugation $\sf C$ if a unitary matrix ${\sf U}_{C}$
exists, such that

$\displaystyle{\sf U}_{C}\,{\sf H}^{\top}(-k)\,{\sf U}_{C}^{-1}$
$\displaystyle=-{\sf H}(k),$ $\displaystyle{\sf U}_{C}\,{\sf
D}^{\top}(-k)\,{\sf U}_{C}^{-1}$ $\displaystyle=+{\sf D}(k),$ (7)
$\displaystyle{\sf U}_{C}\,{\sf P}^{\top}(-k)\,{\sf U}_{C}^{-1}$
$\displaystyle=-{\sf P}(k).$

Finally, there is chiral symmetry $\sf S=\sf T\cdot\sf C$, if

$\displaystyle{\sf U}_{S}\,{\sf H}^{\dagger}(k)\,{\sf U}_{S}^{-1}$
$\displaystyle=+{\sf H}(k),$ $\displaystyle{\sf U}_{S}\,{\sf
D}^{\dagger}(k)\,{\sf U}_{S}^{-1}$ $\displaystyle=+{\sf D}(k),$ (8)
$\displaystyle{\sf U}_{S}\,{\sf P}^{\dagger}(k)\,{\sf U}_{S}^{-1}$
$\displaystyle=+{\sf P}(k).$

for a corresponding unitary matrix ${\sf U}_{S}$. Chiral symmetry is only a
nontrivial additional information if both $\sf T$ and $\sf C$ are broken.

Labelling the unit cell with index $j$ and the four sites within a unit cell
with $\mu\in\\{A,B,C,D\\}$, the Lindblad generators describing the gain and
loss processes for the dissipative topologically nontrivial case
$g_{1}=g_{2}=g_{0}\equiv g>0$ discussed in the main text are

$\displaystyle\hat{L}_{jA}$
$\displaystyle=\sqrt{gJ}\,\hat{c}_{j,A}^{\dagger},\qquad\hat{L}_{jB}=\sqrt{gJ}\,\hat{c}_{j,B},$
(9) $\displaystyle\hat{L}_{jC}$
$\displaystyle=\sqrt{gJ}\,\hat{c}_{j,C},\qquad\hat{L}_{jD}=\sqrt{gJ}\,\hat{c}_{j,D}^{\dagger}.$
(10)

For the overall loss, we have

$\hat{L}_{j\mu}^{(0)}=\sqrt{gJ}\,\hat{c}_{j,\mu}.$ (11)

With this, one finds the $8\times 8$ matrices

$\displaystyle{\sf H}(k)$
$\displaystyle=\left[\begin{array}[]{cccc}0&-\sigma_{z}&0&{\sf h}(k)\\\
-\sigma_{z}&0&-\sigma_{z}&0\\\ 0&-\sigma_{z}&0&-\sigma_{z}\\\ {\sf
h}^{*}(k)&0&-\sigma_{z}&0\end{array}\right],$ (16)

where we dropped the prefactor $J$. In eq. (16), $\sigma_{z}$ denotes one of
the Pauli matrices and

${\sf h}(k)=\left[\begin{array}[]{cc}-e^{-4ik}&0\\\
0&e^{4ik}\end{array}\right].$ (17)

Furthermore, for the matrices describing dissipation ($\sf D$) and
fluctuations ($\sf P$), we obtain

$\displaystyle{\sf D}=\left[\begin{array}[]{cccc}\mathbf{1}_{2}&&&\\\
&\mathbf{1}_{2}&&\\\ &&\mathbf{1}_{2}&\\\
&&&\mathbf{1}_{2}\end{array}\right],\ \ {\sf P}$
$\displaystyle=i\left[\begin{array}[]{cccc}\mathbf{0}_{2}&&&\\\
&\sigma_{z}&&\\\ &&\sigma_{z}&\\\ &&&\mathbf{0}_{2}\end{array}\right],$ (26)

where we dropped the common prefactor $\sqrt{gJ}$. Note that the dissipative
topologically trivial case $g_{1}=g_{0}\equiv g>0$, and $g_{2}=-g$ just
corresponds to an exchange of rows and columns in the matrix $\sf P$.

We find that $\sf H$, $\sf D$, and $\sf P$ possess time-reversal, charge
conjugation and thus also chiral symmetry, according to eqs. (6)-(8). The
corresponding unitary matrices read

${\sf U}_{T}=\left[\begin{array}[]{cccc}\mathbf{1}_{2}&&&\\\
&-\mathbf{1}_{2}&&\\\ &&\mathbf{1}_{2}&\\\
&&&-\mathbf{1}_{2}\end{array}\right],$ (27)

and

${\sf U}_{C}=\left[\begin{array}[]{cccc}&&&\sigma_{x}\\\ &&\sigma_{x}&\\\
&\sigma_{x}&&\\\ \sigma_{x}&&&\end{array}\right],$ (28)

and ${\sf U}_{S}={\sf U}_{T}\cdot{\sf U}_{C}$. This situation corresponds to
the symmetry class BDI, which according to the periodic table of topological
insulators allows in one spatial dimension for topologically distinct phases
characterized by a $\mathbb{Z}$-valued index [40, 41].

###  I.4 Numerical simulations

To validate the experimental results, we perform numerical simulations of the
SPP dynamics in the waveguide arrays, which yield both the real-space
evolution and the momentum-resolved spectral distributions. The numerical
method is based on coupled mode theory [42]. The evolution is governed by the
differential equation

$\frac{da_{j}}{dz}=iC_{j-1,j}a_{j-1}+i\beta_{j}a_{j}+iC_{j,j+1}a_{j+1},$ (29)

where $a_{j}\equiv a_{j}(z)$ denotes the field amplitude in the $j$-th
waveguide at position $z$. The spatially uniform coupling constants $C_{j,j\pm
1}=C$ between neighboring waveguides $j$, $j\pm 1$ are independent of $z$ and
have been measured in the fabricated waveguide structures, see the plot in
Fig. 1(c). The propagation constants $\beta_{j}=\mathrm{Re}\beta_{j}+i\
\mathrm{Im}\beta_{j}$ are composed of a globally constant
$\mathrm{Re}\beta_{j}=$6.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ and a
$j$-dependent $\mathrm{Im}\beta_{j}$; see Fig. 1(c) for its dependence on
different chromium widths. For all simulations, the initial conditions are
chosen such that the field at $z=0$ is concentrated in the first waveguide of
a unit cell, either (i) in the bulk, (ii) at the sample edge, or (iii) at the
interface between distinct topological domains. All simulations are performed
for system sizes of 40 waveguides and $g_{1}=|g_{2}|$.

Bulk excitation. We simulate the SPP evolution following an excitation of a
waveguide in the bulk of an array for two configurations, with and without
losses, as shown in Fig. 2(b). While the lossless case realizes phase (I), the
loss pattern in the dissipative case is tailored according to the
topologically trivial phase (II), which is equivalent to the bulk of the
topologically nontrivial phase (III). In phase (I), the simulation shows a
conical spread of the wave packet, resembling a two-state quantum random walk
with a $\cos$-shaped band structure in momentum space [6]. In phase (II) and
(III), the population oscillates between the excited lossless waveguide and
its lossless neighbor. For increased losses, the oscillation period becomes
larger and the transverse transport into neighboring unit cells is suppressed.
For $|g_{2}|>1$, the oscillation can be interpreted as a beating between the
eigenstates above and below the band gap with a period of $2\pi/\Delta k_{z}$,
where $\Delta k_{z}$ is the difference between the mean energies of the upper
and lower bands visible in Fig. 1(b). The momentum-resolved spectrum for the
topologically trivial phase (II) features population in the upper and lower
energy bands, respectively. Due to dissipation, the main contribution to the
SPP evolution occurs during the first half-period in the downward $-x$
direction; accordingly, the band structure appears slightly distorted and the
band gap is clearly pronounced only in a limited momentum range around
$k_{x}\approx$-1\text{\,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$, as visible in
the middle column of Fig. 2(c).

Edge excitation. To study the topological edge mode in phase (III), we excite
the outermost waveguide (i.e., at the interface to the vacuum), where the
expected probability density for the midgap state is largest. While for the
lossless case the wave packet spreads entirely into the bulk, for a
topologically nontrivial phase-(III) configuration the majority of the
population remains localized in the excited waveguide and only a small amount
spreads into the bulk. Accordingly, a flat band is visible in the momentum-
resolved spectrum indicating the zero-energy topological mode residing in the
band gap, see Fig. 2(c) (right column). For a phase-(II) configuration, the
lossy waveguides can be either the last or the first two ones within a unit
cell. Exciting the high-loss waveguide leads to a rapid population decrease
and a broadened spectral distribution. If the lossless waveguide is excited,
the intensity oscillates between the two outermost waveguides. Crucially, the
states that contribute to this oscillatory dynamics are not topologically
protected; neither resides in the band gap, exhibits an exponential decay
towards the bulk, or features a finite topological winding number [14, 27],
which here takes the value $W=0$.

Interface excitation. To simulate the emergence of a topologically protected
edge mode at a ’nontrivial’ (i.e., not with the vacuum) interface between
distinct topological domains in phases (II) and (III), respectively, the first
waveguide of phase (III) unit cell is excited. For weak absorption
$\mathrm{Im}\beta$, the simulation shows a diffusion of the population into
the neighboring unit cells. For larger $\mathrm{Im}\beta$, diffusion into the
bulk is suppressed and a majority of the population remains at the interface
waveguide for longer propagation distances, see Fig. 3(c). The spread into the
bulk is understood from considering the eigenstates localized at the
interface. For the interface modes, the intensity on the 4th waveguide above
or below the interface (i.e., shifted by one unit cell) is larger as compared
to the neighbors, similar to the topological edge mode. Increasing absorption
leads to an enhanced intensity at the interface compared to the neighboring
unit cells. For the largest simulated $\mathrm{Im}\beta$, the population
spread is limited to the neighboring unit cell and, therefore, simulating two
unit cells would be sufficient.


# Universality of density waves in p-doped $\mathbf{La_{2}CuO_{4}}$ and
n-doped $\mathbf{Nd_{2}CuO_{4+y}}$

Manfred Bucher
Physics Department, California State University, Fresno, Fresno, California
93740-8031

(February 15, 2017)

###### Abstract

The contribution of $O^{2-}$ ions to antiferromagnetism in
$La_{2-x}Ae_{x}CuO_{4}$ ($Ae=Sr,Ba)$ is highly sensitive to doped holes. In
contrast, the contribution of $Cu^{2+}$ ions to antiferromagnetism in
$Nd_{2-x}Ce_{x}CuO_{4+y}$ is much less sensitive to doped electrons. The
difference causes the precarious and, respectively, robust antiferromagnetic
phase of these cuprates. The same sensitivities affect the doping dependence
of the incommensurability of density waves, $\delta(x)$. In the hole-doped
compounds this gives rise to a doping offset for magnetic and charge density
waves, $\delta_{m,c}^{p}(x)\propto\sqrt{x-x_{0p}^{N}}$. Here $x_{0p}^{N}$ is
the doping concentration where the Néel temperature vanishes,
$T_{N}(x_{0p}^{N})=0$. No such doping offset occurs for density waves in the
electron-doped compound. Instead, excess oxygen (necessary for stability in
crystal growth) of concentration $y$ causes a different doping offset in the
latter case, $\delta_{m,c}^{n}(x)\propto\sqrt{x-2y}$. The square-root formulas
result from the assumption of superlattice formation through partitioning of
the $CuO_{2}$ plane by pairs of itinerant charge carriers. Agreement of
observed incommensurability $\delta(x)$ with the formulas is very good for the
hole-doped compounds and reasonable for the electron-doped compound. The
deviation in the latter case may be caused by residual excess oxygen.

Keywords: High-temperature superconductors; Copper oxides; Stripes

##  I DENSITY WAVES IN HOLE-DOPED $\mathbf{La_{2}CuO_{4}}$

Doping $La_{2}CuO_{4}$ with divalent alkaline-earth, $Ae$, substitutes ionized
lanthanum atoms, $La\rightarrow La^{3+}+3e^{-}$, by ionized $Ae\rightarrow
Ae^{2+}+2e^{-}$ in the $LaO$ layers of the crystal. This causes electron
deficiency (hole doping) of concentration $p=x$ in $La_{2-x}Ae_{x}CuO_{4}$
($Ae=Sr,Ba$). Each missing electron at the dopand site is replaced by an
electron from an $O^{2-}$ ion, leaving an $O^{-}$ ion behind. As will become
clear, there is reason to assume that those $O^{-}$ ions reside in the
$CuO_{2}$ planes sandwiched by the $LaO$ layers. With respect to the parent
compound, the $O^{-}$ ions electronically act as holes. Phenomenologically, it
must be the doped holes that weaken, and rapidly destroy three-dimensional
antiferromagnetism (3D-AFM) in $La_{2-x}Ae_{x}CuO_{4}$, in contrast to the
robust 3D-AFM of electron-doped $Nd_{2-x}Ce_{x}CuO_{4}$.1  The mechanism for
the drastic difference of such dopand-dependent collapse of 3D-AFM in those
compounds are still unknown. (One proposed mechanism for the contribution of
$O^{2-}$ ions to 3D-AFM in high-transition-temperature copper oxides is in
terms of loop currents.2  However, that approach remains controversial.3 )

As experimentally observed at zero Néel temperature, $T_{N}(x_{0p}^{N})=0$,
the insulating 3D-AFM phase of $La_{2-x}Ae_{x}CuO_{4}$, collapses at a small
hole concentration $x_{0p}^{N}=x_{10}\equiv 2/10^{2}=0.02$ (see Fig. 1a) and a
metallic, so-called “pseudogap” phase ensues. While the small amount of
$O^{2-}\rightarrow O^{-}$ reduction destroys the 3D-AFM phase, the remaining
$O^{2-}$ ions, together with the unaffected $Cu^{2+}$ spin magnetic moments,
maintain 2D-AFM in the $CuO_{2}$ planes, sometimes called “spin glass.”3

Not only do the doped holes weaken and eventually destroy 3D-AFM, they also
give rise to an incommensurate hole superlattice in $La_{2-x}Ae_{x}CuO_{4}$,
as well as in $La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}$ and
$La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4}$. Thus the effect of hole doping on the
crystals’ magnetism is twofold: (i) a rapid collapse of their 3D-AFM and (ii)
the emergence of a 2D incommensurate magnetic pattern. The latter can be
regarded as an incommensurate magnetic dipole wave (MDW), also called spin
density wave (SDW).4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ;
17 ; 18  From an electric point of view, the hole superlattice can be
considered an incommensurate charge density wave (CDW).19 ; 20 ; 21 ; 22 ; 23
; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ; 34 ; 35  The MDWs and CDWs
are thus related to one another by the constituting holes. Their
unidirectional occurrence within domains is called “stripes.”36

With doping $x>x_{0p}^{N}$, the holes from the doping fraction $x_{0p}^{N}$
are essentially localized to keep 3D-AFM suppressed. The remainder of the
doping, $x-x_{0p}^{N}$, furnishes itinerant holes

FIG. 1. (a) Phase diagram of $La_{2-x}Sr_{x}CuO_{4}$ based on Refs. 7, 18, 24,
26, 38-43 with Néel temperature $T_{N}$, transition temperature of
superconductivity $T_{c}$, pseudogap temperature $T^{*}$ (extrapolated to
$T=0$) and temperatures $T_{CDW}$ and $T_{MDW}$ beneath which charge-density
waves and, respectively, magnetic dipole waves are detected. Special
concentrations $x_{n}=2/n^{2}$, marked on the ordinate axis, correspond to
commensurate doping of one $Sr$ atom per $na_{0}\times nb_{0}$ area in each
$LaO$ plane. (b) Incommensurability of MDWs, $\delta_{m}^{p}=\delta$, and of
CDWs $\delta_{c}^{p}=2\delta$, in $La_{2-x}Ae_{x}CuO_{4}$ due to doping with
$Ae=Sr$ or $Ba$, as well as in $La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}$ and
$La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4}$. Circles show data from neutron scattering
or X-ray diffraction (Refs. 4-35). The broken solid curve is a graph of Eq.
(1). Dotted slanted line $\delta=x$, dashed horizontal line at
$\delta=0.125=1/8$.

that segregate in the $CuO_{2}$ planes to an incommensurate superlattice of
spacing $\lambda_{c}^{p}(x)\propto(x-x_{0p}^{N})^{-1/2}$. It can be regarded
as a static CDW of incommensurable wave number
$\delta_{c}^{p}(x)=1/\lambda_{c}^{p}(x)$, first discovered in
$La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}$.4  The holes of the superlattice are located
at the nodes of the incommensurable MDW of wavelength
$\lambda_{m}^{p}(x)=2\lambda_{c}^{p}(x)$ based on defects in the nematic
2D-AFM pattern. In crystal domains large compared to the planar lattice
constants, the combined CDWs-MDWs of incommensurability
$\delta_{c}^{p}(x)=2\delta_{m}^{p}(x)$ are unidirectional (stripes). The
dependence of the incommensurability on hole doping, in reciprocal lattice
units (r.l.u.), is37

$\delta_{c,m}^{p}(x)=w_{c,m}\frac{\Omega^{\pm}}{{4}}\sqrt{x-{x_{0p}^{N}}}\;,$
(1)

with a wave-kind factor $w_{c}=2$ or $w_{m}=1$ and a stripe-orientation factor
$\Omega^{+}=\sqrt{2}$ for $x>x_{6}\equiv 2/6^{2}\simeq 0.056$ when stripes are
parallel to the $a$ or $b$ axis, but $\Omega^{-}=1$ for $x<x_{6}$ when stripes
are diagonal. The density waves emerge at the doping level $x_{0p}^{N}=0.02$
where 3D-AFM collapses. The derivation of Eq. (1) is based on a partition of
the $CuO_{2}$ plane by pairs of itinerant doped holes,37  incorporating the
observed stripe orientation, here in tetragonal approximation of the lateral
lattice constants, $a_{0}=b_{0}$. The formula agrees to high accuracy with
measured incommensurabilties of MDWs in hole-doped $La_{2-x}Ae_{x}CuO_{4}$,
$La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}$ and $La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4}$ by
inelastic neutron scattering4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ;
15 ; 16 ; 17 ; 18  and of CDWs by hard X-ray diffraction or resonant soft
X-ray scattering (RXS)19 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ;
30 ; 31 ; 32 ; 33 ; 34 ; 35  (see Fig. 1b).

To put Eq. (1) in perspective, several comments are in order. (i) The
experimentally observed relationship5  $\delta_{c}(x)=2\delta_{m}(x)$ is
indicative of the mutual dependence of CDWs and MDWs, caused by doped holes.
It implies that they _always_ exist together in copper oxides of 214
structure. (ii) The observed transition from diagonal to parallel stripe
orientation7  at doping $x_{6}$ coincides with the onset of superconductivity,
$x_{0}^{SC}=x_{6}$. The reason for this agreement is still unknown. (iii) The
sensitivity of detection of MDWs by neutron scattering and of CDWs by hard
X-ray diffraction or resonant soft X-ray scattering differs in several
respects. In the parallel regime, $x>x_{6}$, the temperature profiles beneath
which CDWs and MDWs are detected are dome-like with maxima at $x_{4}=1/8$,
ranking $T_{CDW}(x)>T_{c}(x)>T_{MDW}(x)$ (see Fig. 1a). A possible reason for
the dome-like shape of $T_{CDW}(x)$ and $T_{MDW}(x)$ could be, on the one
hand, relatively large domain sizes for $x$ near the commensurate value
$x_{4}$, antagonized by thermal agitation. On the other hand, the flanks of
the domes could result from a diminution of domain size by structural
instability due to proximity to the transition of diagonal/parallel stripes on
the low-doping (left) side and proximity to the pseudogap/strange-metal
transition on the higher-doping (right) side. In the diagonal regime,
$x<x_{6}$, only MDWs have been observed, no CDWs (but possibly indications21
for diagonal CDWs in $La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4}$). Concerning the
different features of $T_{CDW}(x)$ and $T_{MDW}(x)$, it is conceivable that
the diagonal pattern of 2D-AFM with wave vector
$\mathbf{q}_{AFM}=(\frac{1}{2},\frac{1}{2})$ favors the domain size of
diagonal MDWs. Conversely, the pattern of $Cu$-$O$ bonds parallel to the
lateral crystal axes would favor the domain size of parallel CDWs. (iv) The
excellent agreement of data with Eq. (1) lends support to the underlying
assumption that the doped holes reside in the $CuO_{2}$ plane. (v)
Historically, the notion has been put forth6  that the doping dependence of
the incommensurability of MDWs (and by implication of CDWs) in
$La_{2-x}Ae_{x}CuO_{4}$ is essentially linear, $\delta_{c,m}\sim x$, up to
$x\approx 0.12$, but levels off beyond to a value of
$\delta_{m}(x)=\frac{1}{2}\delta_{c}(x)\approx 1/8=0.125$. The notion of such
a ramp-like incommensurability profile, repeated many times in the literature,
was helpful in the early days but is now rather outdated. As Fig. 1b shows,
the data in the very low doping range indicate a parabolic (square-root like)
dependence. Because of the _break_ of the square-root function at $x_{6}$ the
branches of the solid curve skirt the dotted
$\delta_{m}^{/}(x)=\frac{1}{2}\delta_{c}^{/}(x)=x$ line in the $0.04<x<0.10$
range from beneath and above, giving the false _impression_ of a linear
dependence. The cluster of data near $x=1/8$ clearly shows that
$\delta_{m}(\frac{1}{8})=\frac{1}{2}\delta_{c}(\frac{1}{8})<1/8$. Beyond
$x\approx 0.12$ the data keep rising instead of leveling off. No _explanation_
for the putative ramp profile of $\delta_{c,m}(x)$ has ever been issued. In
contrast, the broken square-root dependence of $\delta_{c,m}(x)$, Eq. (1), has
been _derived_ by a partitioning of the $CuO_{2}$ planes by pairs of itinerant
doped holes, combined with the observed change of diagonal/parallel stripe
orientation at $x_{6}$. It is one of the very few simple relationships for the
pseudogap phase.

##  II CHARGE DENSITY WAVES IN ELECTRON-DOPED $\mathbf{Nd_{2}CuO_{4}}$

Pristine $Nd_{2}CuO_{4}$ has the same (214) crystal structure as
$La_{2}CuO_{4}$. A minor difference occurs in $Ce$-doped
$Nd_{2-x}Ce_{x}CuO_{4}$ for $x>x_{6}$ when oxygen ions in the $LnO$ layers
that bracket the $CuO_{2}$ planes switch from apical positions above or
beneath the $Cu^{2+}$ ions (T type) to positions apical to the $O^{2-}$ ions
of the $CuO_{2}$ plane (T’ type).1  It is noteworthy that the T $\rightarrow$
T’ transition at $x_{6}$ in the n-doped 214 copper oxides coincides with the
transition

FIG. 2. (a) Phase diagram of electron-doped $Nd_{2-x}Ce_{x}CuO_{4}$ after Ref.
44. (b) Observed incommensurability $\delta_{c}$ of CDWs in
$Ln_{2-x}Ce_{x}CuO_{4+y}$ ($Ln=Nd$, crystal, circles; $Ln=La$, film, square)
from Refs. 45 and 46. The curves are graphs of Eq. (2) without excess oxygen
($y=0$, dashed line) and with excess oxygen of $y=0.01$ (solid line). An
average error bar of the the data is shown to the right. The
incommensurability of the corresponding MDW curve,
$\delta_{m}(x)=\frac{1}{2}\delta_{c}(x)$, is not shown for sake of clarity.

from diagonal to parallel stripes at $x_{6}$ in the p-doped 214 compounds.
Doping the parent crystal $Nd_{2}CuO_{4}$ with _tetravalent_ cerium
substitutes ionized lanthanum atoms, $La\rightarrow La^{3+}+3e^{-}$, by
ionized $Ce\rightarrow Ce^{4+}+4e^{-}$, causing electron surplus (electron
doping) of concentration $n=x$ in $Nd_{2-x}Ce_{x}CuO_{4}$. The surplus
electrons attach themselves to copper ions in the $CuO_{2}$ plane,
$Cu^{2+}+\,e^{-}\rightarrow Cu^{+}$. Having a closed-shell electron
configuration, the $Cu^{+}$ ions have no electron-spin magnetic moment,
contrary to the unaffected $Cu^{2+}$ ions. This causes a weakening of AFM,
although (for reasons unknown) at a much lesser rate than for hole doping.
Collapse of 3D-AFM occurs44  in electron-doped $Nd_{2-x}Ce_{x}CuO_{4}$ at
$x_{0n}^{N}\simeq 0.134$ (see Fig. 2a), in stark contrast to $x_{0p}^{N}=0.02$
in hole-doped $La_{2-x}Ae_{x}CuO_{4}$, as mentioned.

The doped electrons in ${Nd_{2-x}Ce_{x}CuO_{4}}$ are free enough to segregate
in the $CuO_{2}$ plane to an incommensurate superlattice, which can be
regarded as a static CDW. Reducing a fraction, $n$, of $Cu^{2+}$ ions to spin-
free $Cu^{+}$, gives rise to a weak MDW superimposed on the strong AFM
background. The MDW is probably too weak to be detected with present
technology as no MDWs have been observed in ${Nd_{2-x}Ce_{x}CuO_{4}}$. Using
for those density waves the same assumption as for doped holes—here, formation
of a 2D superlattice that partitions the $CuO_{2}$ plane by pairs of doped
_electrons_ —combined with the same stripe orientation, should then give the
incommensurability $\delta_{c,m}^{n}(x)$ in square-root dependence on electron
doping.

However, two circumstances need to be considered. (i) Since doped electrons
don’t frustrate the contribution of $O^{2-}$ ions to AFM, no doped electrons
are exempted from CDWs to keep 3D-AFM suppressed—hence no doping offset
$\Delta x_{n}$ in $\delta_{c,m}^{n}(x)$ analogous to $\Delta x_{p}=x_{0p}^{N}$
in Eq. (1). (ii) It is well-known that electron-doped copper oxides need to be
grown, for reasons of stability, in an oxygen atmosphere.1  As-grown
$Ln_{2-x}Ce_{x}CuO_{4+y}$ ($Ln=Pr,Nd,Sm$) typically contain excess oxygen of
$y\approx 0.03$ which has to be removed subsequently by sufficient annealing
in an $Ar$ atmosphere47  This raises the possibility that the samples under
consideration may still contain residual excess oxygen. Upon ionization,
$O\rightarrow O^{2-}-2e^{-}$, excess oxygen ions reside in
$Ln_{2-x}Ce_{x}CuO_{4+y}$ interstitially at apical positions above or beneath
the $Cu^{2+}$ ions. The minus sign in the oxygen ionization balance accounts
for the taking of two doped electrons (if present) by an interstitial oxygen
atom (otherwise of two electrons from other atoms). This amounts, besides
electron doping via $Ce$, to additional hole doping, $p=2y$, which neutralizes
a fraction of the doped-electron density, $\Delta n=-2y$, below the
concentration of $Ce$, $n<x$. The consequence is a doping offset from excess
oxygen affecting the incommensurability in electron-doped compounds,

$\delta_{c,m}^{n}(x)=w_{c,m}\frac{\Omega^{\pm}}{{4}}\sqrt{x-2y}\;,$ (2)

with $w_{c,m}$ and $\Omega^{\pm}$ as in Eq. (1).

Figure 2b shows incommensurabilities of CDWs in ${Nd_{2-x}Ce_{x}CuO_{4+y}}$
and ${La_{1.92}Ce_{0.08}CuO_{4}}$, observed with resonant X-ray scattering
(RXS).44 ; 45  The data fall close to the dashed curve of Eq. (2) for the case
without excess oxygen ($y=0$) _only_ when $Ce$ doping is small or large,
whereas the data in the middle range of doping fall beneath the dashed curve
by about 15%. However, for the case with $y=0.01$ oxygenation the data closely
skirt the _solid_ curve. This raises the likelihood of residual excess oxygen
in the samples. For a resolution of this possibility, enhanced analysis of
excess oxygen would be valuable.

##  III SUMMARY AND REMARK

Leaving aside the deviation in Fig. 2b for clarification by future experiment,
the overall finding for 214 copper oxides is a universal _square-root_
dependence of density-wave incommensurability $\delta(x)$ on the doping of
charge carriers—be they holes, be they electrons—quantitatively expressed by
Eqs. (1) and (2). Their commonality is the underlying assumption that the
$CuO_{2}$ plane is partitioned by pairs of itinerant charge carriers. The
major difference between Eqs. (1) and (2) is the doping offset by $\Delta
x_{p}=x_{0p}^{N}$ with $Ae$ (hole) doping but _no_ corresponding offset with
$Ce$ (electron) doping. The difference is related on the one hand to the
precarious sensitivity of the $O^{2-}$ ions’ contribution to 3D-AFM when doped
by holes, together with the necessity to keep 3D-AFM suppressed by a hole
concentration $x_{0p}^{N}$. On the other hand it is related to the much lesser
sensitivity of the $Cu^{2+}$ ions’ contribution to 3D-AFM when doped by
electrons. A minor difference between Eqs. (1) and (2) concerns the influence
of excess oxygen—necessary for stability in crystal growth of
${Ln_{2-x}Ce_{x}CuO_{4+y}}$ ($Ln=Pr,Nd,Sm$)—by $\Delta x_{n}=-2y$.

As a refinement, the doping offset $\Delta x_{p}=x_{0p}^{N}$ in the radical of
Eq. (1) may be treated as doping dependent, $\Delta x_{p}(x)$. When for
$x\approx 1/8=0.125$ a value of $\Delta x_{p}(x)=0.01$ (instead of $\Delta
x_{p}(x)=x_{0p}^{N}=0.02$) is used, agreement with experiment improves from
very good (3% deviation) to excellent (1% deviation or less). In the framework
of the above interpretation this could be understood as a lesser need to
exempt holes from the CDW to keep 3D-AFM suppressed when overall more doped
holes are present.

ACKNOWLEDGMENTS

I thank Duane Siemens for discussions, suggestions, and critique. I also thank
Preston Jones for help with LaTeX.

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