# Topological Edge States and Fractional Quantum Hall Effect from Umklapp
Scattering

Jelena Klinovaja  Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Daniel Loss  Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland

###### Abstract

We study anisotropic lattice strips in the presence of a magnetic field in the
quantum Hall effect regime. At specific magnetic fields, causing resonant
Umklapp scattering, the system is gapped in the bulk and supports chiral edge
states in close analogy to topological insulators. In electron gases with
stripes, these gaps result in plateaus for the Hall conductivity exactly at
the known fillings $n/m$ (both positive integers and $m$ odd) for the integer
and fractional quantum Hall effect. For double strips we find topological
phase transitions with phases that support midgap edge states with flat
dispersion. The topological effects predicted here could be tested directly in
optical lattices.

###### pacs:

71.10.Fd; 05.30.Pr; 71.10.Pm; 73.43.-f

Introduction. Condensed matter systems with topological properties have
attracted wide attention over the years. Wilczek ; Hasan_RMP ; Zhang_RMP ;
Alicea_2012  E.g., the integer and fractional quantum Hall effects (IQHE and
FQHE) Klitzing ; Tsui_82  find their origin in the topology of the system.
QHE_Review_Prange ; book_Jain ; Hofstadter ; Laughlin ; Streda ; Thouless ;
Halperin ; Laughlin_FQHI ; Halperin_crystall ; oded_2013 ; Kane_lines
Similarly, band insulators with topological properties have become of central
interest recently, Hasan_RMP ; Zhang_RMP ; Topological_class_Ludwig  as well
as exotic topological states like fractionally charged fermions Jackiw_Rebbi ;
FracCharge_Su ; FracCharge_Kivelson ; FracCharge_Chamon ; CDW ;
Two_field_Klinovaja_Stano_Loss_2012 ; Klinovaja_Loss_FF_1D ;
Frac_graphene_2007 ; Franz_2009  or Majorana fermions. Read_2000 ; Nayak ; fu
; Nagaosa_2009 ; Sato ; lutchyn_majorana_wire_2010 ; oreg_majorana_wire_2010 ;
Klinovaja_CNT ; Sticlet_2012 ; bilayer_MF_2012 ; MF_nanoribbon

Here, we study two-dimensional (2D) strips in magnetic fields, both
analytically and numerically, modeled by an anisotropic tight-binding lattice.
We identify a striking mechanism by which the magnetic field induces resonant
Umklapp scattering (across Brillouin zones) that opens a gap in the bulk
spectrum and results in chiral edge states in analogy to topological
insulators. Quite remarkably, the resonant scattering occurs at well-known
filling factors for the IQHE Klitzing  and FQHE Tsui_82  $\nu=n/m$, where
$n,m$ are positive integers and $m$ odd. We argue below that this mechanism
could shed new light on the QHE for 2D electron gases as well, where the
formation e.g. of a periodic structure (energetically favored also by a
Peierls transition) might support the periodic structure needed for the
Umklapp scattering.

Finally, we consider a double strip of spinless fermions, or, equivalently, a
single strip with spinful fermions. Here, we find two topological phase
transitions accompanied by a closing and reopening of the bulk gap, and, as a
result, three distinct phases. The trivial phase is without edge states. The
first topological phase is similar to the one discussed above and carries two
propagating chiral modes at each edge for $\nu=1$. The second topological
phase has only one state at each edge. Quite remarkably, its dispersion is
flat throughout the Brillouin zone, making this phase an attractive playground
for studying interaction effects.

Figure 1: (a) Strip: two-dimensional lattice of width in $x$-direction, $W$,
with unit cell defined by the lattice constants $a_{x}$ and $a_{y}$. The
hopping amplitudes in $y$-direction, $t_{y1}$ and $t_{y2}$, carry the phase
$\phi$, arising from a perpendicular magnetic field ${\bf B}$, and we assume
$t_{x}\gg t_{y1},t_{y2}$. (b) Doubly degenerate spectrum of $H_{x}$ [see Eqs.
(1)] for the rows with $\sigma=1$ (upper blue) and with $\sigma=\bar{1}$
(lower green). The hoppings $t_{y1}$ (dashed line) and $t_{y2}$ (dotted line)
induce resonant scattering between right ($R_{\sigma}$) and left
($L_{\bar{\sigma}}$) movers, which open gaps at the Fermi wavevectors $\pm
k_{F}$ defined by the chemical potential $\mu$.

Anisotropic tight-binding model. We consider a 2D tight-binding model of a
strip that is of width $W$ in $x$\- and extended in $y$-direction, see Fig.
1a. The unit cell is composed of two lattice sites ($\sigma=\pm 1$) along $y$
that are distinguished by two hopping amplitudes, $t_{y1}$ and $t_{y2}$. Every
site is labeled by three indices $n,m$, and $\sigma$, where $n$ ($m$) denotes
the position of the unit cell along the $x$\- ($y$-) axis. The hopping along
$x$ is described by

$\displaystyle H_{x}=-t_{x}\sum_{n,m,\sigma}(c^{\dagger}_{n+1,m,\sigma}$
$\displaystyle c_{n,m,\sigma}+h.c.),$ (1)

where $t_{x}$ is the hopping amplitude in $x$-direction and $c_{n,m,\sigma}$
the annihilation operator acting on a spinless fermion at site $(n,m,\sigma)$,
and the sum runs over all sites. The hopping along $y$ is described by

$\displaystyle
H_{y}=\sum_{n,m}(t_{y1}e^{-in\phi}c^{\dagger}_{n,m,1}c_{n,m,\bar{1}}$
$\displaystyle\hskip
80.0pt+t_{y2}e^{in\phi}c^{\dagger}_{n,m+1,1}c_{n,m,\bar{1}}+h.c.),$ (2)

Without loss of generality, we consider $t_{y2}\geq t_{y1}\geq 0$. The phase
$\phi$ is generated by a uniform magnetic field ${\bf B}$ applied in
perpendicular $z$-direction, see Fig. 1. We choose the corresponding vector
potential $\bf A$, to be along the $y$-axis, ${\bf A}=(Bx){\bf e}_{y}$,
yielding the phase $\phi=eBa_{x}a_{y}/2\hbar c$. Here, $a_{x,y}$ are the
corresponding lattice constants.

Chiral edge states. Taking into account translational invariance of the system
in $y$-direction, we introduce the momentum $k_{y}$ via Fourier
transformation, see Appendix A. The Hamiltonians become diagonal in $k_{y}$,
i.e.,
${H}_{x}=-t_{x}\sum_{n,k_{y},\sigma}(c^{\dagger}_{n+1,k_{y},\sigma}c_{n,k_{y},\sigma}+h.c.)$,
and ${H}_{y}=\sum_{n,k_{y}}[(t_{y1}e^{-in\phi}+t_{y2}e^{i(n\phi-
k_{y}a_{y})})c^{\dagger}_{n,k_{y},1}c_{n,k_{y},\bar{1}}+h.c.]$. Thus, the
eigenfunctions of ${H}={H}_{x}+{H}_{y}$ factorize as
$e^{ik_{y}y}\psi_{k_{y}}(x)$, where we focus now on $\psi_{k_{y}}(x)$ and
treat $k_{y}$ as parameter.

Figure 2: Spectrum $E(k_{y})$ of left edge state (red line or dots)
propagating along $y$ for a strip ($t_{y1}/t_{x}=0.02$, $t_{y2}/t_{x}=0.1$) of
width $W/a_{x}=801$ and with phase $\phi=\pi/2$, obtained (a) analytically
[see Eq. (4)] and (b) numerically [see Eqs. (1) and (2)]. For
${\bar{k}}_{-}<k_{y}<{\bar{k}}_{+}$, there exists one edge state at each edge.
The left (red dots) and the right (blue dots) edge states are chiral and
propagate in opposite $y$-directions. For each $k_{y}$ [(c) $k_{y}a_{y}=\pi$,
(d) $k_{y}a_{y}=13\pi/12$] there is one left (red dots) and one right (blue
dots) edge state if Eq. (5) is satisfied. Here, $\epsilon(N)$ corresponds to
the $N$th energy level. The probability density $|\psi_{\sigma}|^{2}$ (e)
[(f)] of the left [right] localized state decays exponentially in agreement
with the analytical result, Eq. (14) in Appendix C.

Assuming for the moment periodic boundary conditions also in $x$-direction, we
introduce a momentum $k_{x}$, see Appendix A. Immediately, the well-known
spectrum of $H_{x}$ follows, $\epsilon_{\sigma}=-2t_{x}\cos(k_{x}a_{x})$,
which is twofold degenerate in $\sigma$. The chemical potential $\mu$ is fixed
such that the Fermi wavevector $k_{F}$ is connected to the phase by
$\phi=2k_{F}a_{x}$. Next, we allow for hopping along $y$ as a small
perturbation to the $x$-hopping, i.e., $t_{x}\gg t_{y1},t_{y2}$, see Fig. 1b.
To obtain analytical solutions, it is most convenient to go to the continuum
description Two_field_Klinovaja_Stano_Loss_2012 ;
MF_wavefunction_klinovaja_2012 . The annihilation operator $\Psi(x)$ close to
the Fermi level can be represented in terms of slowly varying right
[$R_{\sigma}(x)$] and left [$L_{\sigma}(x)$] movers,
$\Psi(x)=\sum_{\sigma}R_{\sigma}(x)e^{ik_{F}x}+L_{\sigma}(x)e^{-ik_{F}x}$. The
corresponding Hamiltonian density $\mathcal{H}$ can be rewritten in terms of
the Pauli matrices $\tau_{i}$ ($\sigma_{i}$), acting on the right-left mover
(lattice) subspace (see Appendix B)
$\Psi=(R_{1},L_{1},R_{\bar{1}},L_{\bar{1}})$, as

$\displaystyle\mathcal{H}=\hbar\upsilon_{F}\hat{k}\tau_{3}+\frac{t_{y1}}{2}(\sigma_{1}\tau_{1}+\sigma_{2}\tau_{2})+\frac{t_{y2}}{2}\Big{[}(\sigma_{1}\tau_{1}-\sigma_{2}\tau_{2})$
$\displaystyle\hskip
10.0pt\times\cos(k_{y}a_{y})+(\sigma_{2}\tau_{1}+\sigma_{1}\tau_{2})\sin(k_{y}a_{y})\Big{]}.$
(3)

Here, $\hbar\hat{k}=-i\hbar\partial_{x}$ is the momentum operator with
eigenvalues $k$ taken from the corresponding Fermi points $\pm k_{F}$, and
$\upsilon_{F}=2(t_{x}/\hbar)a_{x}\sin(k_{F}a_{x})$ is the Fermi velocity. The
spectrum with periodic boundary conditions in $x$\- and $y$-directions is
given by $\epsilon_{l,\pm}=\pm\sqrt{(\hbar\upsilon_{F}k)^{2}+t_{yl}^{2}}$,
where $l=1,2$. This mechanism of opening a gap by oscillatory terms causing
resonant scattering between the Fermi points is similar to a Peierls
transition. Braunecker_Loss_Klin_2009  Next, we turn to a strip of finite
width $W$, see Fig. 1. We note that the bulk spectrum $\epsilon_{l,\pm}$ is
fully gapped, so states localized at the edges can potentially exist. To
explore this possibility we consider a semi-infinite nanowire ($x\geq 0$) and
follow the method developed in Refs. MF_wavefunction_klinovaja_2012, ;
Two_field_Klinovaja_Stano_Loss_2012, , assuming that the localization length
of bound states $\xi$ is much smaller than $W$. This allows us to impose
vanishing boundary conditions only at $x=0$,
$\psi_{k_{y}}(x)|_{x=0}\equiv(\psi_{1},\psi_{\bar{1}})|_{x=0}=0$. This
boundary condition is fulfilled only at one energy inside the gap
$|E|<t_{y1}$,

$E(k_{y})=\frac{t_{y1}t_{y2}\sin(k_{y}a_{y})}{\sqrt{t_{y1}^{2}+t_{y2}^{2}-2t_{y1}t_{y2}\cos(k_{y}a_{y})}},$
(4)

if the following condition is satisfied,

$t_{y1}>t_{y2}\cos(k_{y}a_{y}).$ (5)

The edge states exist for momenta $k_{y}\in({\bar{k}}_{-},{\bar{k}}_{+})$,
where ${\bar{k}}_{\pm}a_{y}=\pi\pm\arcsin(t_{y1}/t_{y2})$. An edge state
touches a boundary of the gap at ${\bar{k}}_{\pm}$ and afterwards disappears
in the bulk spectrum of the delocalized states, see Fig. 2. The only regime in
which the edge state exhibits all momenta corresponds to the uniform strip
with $t_{y1}=t_{y2}$. The localization length $\xi$ is determined by
$\xi=\hbar\upsilon_{F}/\sqrt{t_{y1}^{2}-E^{2}}$, with wavefunction given in
Appendix C. The edge state gets delocalized if its energy is close to the
boundary of the gap, so that $\xi$ becomes comparable to $W$. Similarly, we
can search for the solution decaying to the right, $x\leq 0$, and obtain Eq.
(4) with reversed sign, $E(k_{y})\to-E(k_{y})$.

We have confirmed above results by diagonalizing the tight-binding Hamiltonian
${H}$ (in $k_{y}$-representation) numerically, see Fig. 2. The spectrum
$E(k_{y})$ of the edge states localized along $x$ and propagating along $y$
shows that at any fixed energy inside the gap there can be only one edge state
at a given edge, see Fig. 2. Moreover, the edge states are chiral, as can be
seen from the velocity, $\upsilon=\partial E/\partial k_{y}$, which is
negative (positive) for the left (right) edge state. This means that transport
along a given edge of the strip can occur only in one direction determined by
the direction of the $\bf B$-field, see Fig. 1. Quite remarkably, the obtained
spectrum of edge states is of the same form as for topological insulators
Volkov_Pankratov ; Hasan_RMP  with a single Dirac cone consisting of two
crossing non-degenerate subgap modes. Due to the macroscopic separation of
opposite edges, $\xi\ll W$, these modes are protected from getting scattered
into each other by impurities, phonons or interaction effects, so that the
Dirac cone cannot be eliminated by perturbations that are local and smaller
than the gap. Thus, the edge states are topologically stable.

Umklapp scattering. We note that the system considered here is equivalent to a
2D system in the QHE regime. The above choice of magnetic field corresponds to
the IQHE with filling factor $\nu=1$, which is in agreement with one chiral
mode at each edge. To explore the possibility of inducing quantum Hall physics
at other filling factors, we fix the chemical potential $\mu=-\sqrt{2}t_{x}$,
so that the system has local particle-hole symmetry, and change the B-field.
Above, the phase $\phi$, generated by the magnetic field, was equal to $\pi/2$
for $k_{F}=\pi/4a_{x}$. However, this is not the only choice of phase leading
to the opening of a gap $\Delta_{g}$ at the Fermi level. Due to the
periodicity of the spectrum, resonant scattering between branches of
$\epsilon_{\sigma}$ occurs also via Umklapp scattering between different
Brillouin zones, with a phase $\phi=\pm\frac{p\pi}{2n}+2\pi q$, where $q$ is
an integer, $n$ a positive integer, and $p$ a positive odd integer with $p<2n$
and coprime to $n$. As a result, the Fermi level lies in the bulk gap for the
filling factors footnote_filling  $\nu=n/(4qn\pm p)$, which can be rewritten
as $\nu=n/m$, where $m>0$ is an odd integer. The size of the gap can be
estimated as $\Delta_{g}\propto t_{yl}(t_{yl}/t_{x})^{(n-1)}$ (assuming for
simplicity $t_{y1}=t_{y2}$). Finally, we remark that we checked numerically
that the gap $\Delta_{g}$ never closes for any finite ratio of $t_{x}$ and
$t_{yl}$ larger or smaller than one.

FQHE in 2D electron gas. We conjecture that the same mechanism of resonant
Umklapp scattering can also lead to the integer or fractional QHE in 2D
electron gases. At high magnetic fields interaction effects get strongly
enhanced and electrons tend to order themselves into periodic structures.
Wigner_Girvin ; Wigner_Kivelson ; Wigner_Jain ; book_Jain ; anisotropy_CDW ;
strips_Review ; anisotropy_West ; QHE_strips  In particular, we assume the
formation of stripes that are aligned along $x$ and periodically repeated in
$y$. While particles can hop between stripes, they move now continuously
inside them with quadratic dispersion. Thus, the perturbative solutions found
above in terms of right- and left-movers still apply. In addition, we assume
that the interaction generates a charge-density wave at wavevector $K$ inside
the stripe, providing an effective periodic potential in $x$, which will lead
to gaps. Thus, $K$ becomes the period of the Brillouin zone, and, at
$1/4$-filling of the lowest subband, we have $K=8k_{F}$. footnote:B_period

Again, the $B$-field leads to a gap at $k_{F}$ only if it results in phases
commensurable to $k_{F}$, i.e. $eBa_{y}/2\hbar c=\pm 2k_{F}\frac{p}{n}+qK$,
which is equivalent to $\nu=n/(4qn\pm p)\equiv n/m$. In this regime, there is
an additional energy gain due to a Peierls transition, favoring even more a
formation of periodic structures with gaps. Moreover, from this mean field
scenario it follows that the IQHE is more stable against disorder than the
FQHE since the latter requires Umklapp scattering through higher Brillouin
zones. The gap and the edge states can be tested in transport experiments. For
example, the Hall conductance $\sigma_{H}$ exhibits plateaus on the classical
dependence curve $\sigma_{H}\propto 1/B$, if the Fermi level lies in the gap.
This can be shown by using the Streda formula, Streda
$\sigma_{H}=ec\left(\frac{\partial{\bar{n}}}{\partial B}\right)_{\mu}$, where
${\bar{n}}$ is the bulk particle density which is uniquely determined by the
magnetic field via the relation $\nu eBa_{y}/2\hbar c=2k_{F}$ (for this it is
crucial that $K$ depends on $k_{F}$). If $\mu$ lies in the gap opened by the
Umklapp scattering, the change in the density for fixed $\mu$, $d{\bar{n}}$,
due to a change in the magnetic field, $dB$, is given by
$d{\bar{n}}=\frac{dk_{F}}{\pi a_{y}/2}=\nu\frac{e}{hc}dB$. Hence, the
conductance assumes the FQHE plateaus, $\sigma_{H}=\nu{e^{2}}/{hc}$, with
$\nu=n/m$ and independent of any lattice parameters. The width of the plateaus
is determined by the gap size $\Delta_{g}\propto t_{yl}(t_{yl}/\mu)^{(n-1)}$.
We note that the FQHE can be mapped back to the IQHE by redefining the charge
$e$ by $e^{\star}=e/m$ that allows us to keep all scattering events inside the
first Brillouin zone. Finally, the distance between stripes can be estimated
as ${a_{y}}/{2}={k_{F}}/{\pi\bar{n}}$.

Figure 3: Double strip for spinless fermions. The intra-strip couplings are
the same as in Fig. 1. The inter-strip coupling along $z$ is described by the
hopping amplitude $t_{z}$.
Figure 4: Spectrum of a double strip obtained by numerical diagonalization of
the tight-binding Hamiltonian $H_{2}=H_{x}+H_{y}+H_{z}$ for the same
parameters as in Fig. 2. (a) If $|t_{z}|<t_{y1}$ [$t_{z}/t_{x}=0.01$], there
are four edge states at any energy within the gap; two of them localized at
the left and two at the right edge. (b) If $t_{y1}<|t_{z}|<t_{y2}$
[$t_{z}/t_{x}=0.05$], there is one zero-energy edge state (i.e. with flat
dispersion) at each edge. (c) If $|t_{z}|>t_{y2}$ [$t_{z}/t_{x}=1.5$], there
are no edge states in the gap.

Double strip. Now we consider a double strip, consisting of two coupled strips
for spinless fermions, see Fig. 3. This system is equivalent to a single strip
but for spinful fermions. Below we focus on the double strip but we note that
one can identify the upper (lower) strip with spin up (down) state labeled by
$\eta=1$ ($\eta=-1$). The chemical potentials $\mu_{\eta}$ are opposite for
the two strips, $\mu_{1}=-\mu_{\bar{1}}=\sqrt{2}t_{x}$, and are chosen such
that the system is at half-filling. For the spinful strip the role of
$\mu_{\eta}$ is played by the Zeeman term, $\mu_{\eta}=\eta g\mu_{B}B$,
arising from the magnetic field $\bf B$ along $z$. Here, $g$ is the
$g$-factor, and $\mu_{B}$ is the Bohr magneton. The inter-strip hopping
amplitude $t_{z}$ is also accompanied by the phase $\phi^{z}$ arising from a
uniform magnetic field ${\bf B}_{2}$ applied along $y$,

$H_{z}=\sum_{n,m,\sigma,\eta}t_{z}e^{in\eta\phi^{z}}c_{n,m,\sigma,\eta}^{\dagger}c_{n,m,\sigma,\bar{\eta}}.$
(6)

The amplitude of ${\bf B}_{2}$ is chosen so that $\phi^{z}=(e/\hbar
c)B_{2}a_{x}a_{z}=\pi$. This amounts to apply a total field ${\bf
B}_{tot}={\bf B}+{\bf B}_{2}$ in the $yz$-plane. Moreover, the same $H_{z}$ is
generated in the spinful case by a $B_{2}$-field applied along $y$ with an
amplitude that oscillates in space along $x$ with period $2a_{x}$, or,
alternatively, by Rashba spin orbit interaction. Braunecker_Loss_Klin_2009

Again, we search for wavefunctions in terms of right and left mover fields
defined around two Fermi points, $k_{F1}=\pi/4a_{x}$ (upper strip) and
$k_{F\bar{1}}=3\pi/4a_{x}$ (lower strip). The linearized Hamiltonian density
for this extended model in terms of the Pauli matrices $\eta_{i}$ acting on
the upper/lower strip subspace is given by [see Eq. (3)]

$\displaystyle\mathcal{H}_{2}=\mathcal{H}(\sigma_{2}\to\sigma_{2}\eta_{3})+t_{z}\eta_{1}\tau_{1}\,.$
(7)

The resulting spectrum is
$\epsilon_{l,\pm,p}=\pm\sqrt{(\hbar\upsilon_{F}k)^{2}+(t_{yl}+pt_{z})^{2}}$,
with $p=\pm 1$. We note that the gap vanishes if $|t_{z}|=t_{y1}$ or
$|t_{z}|=t_{y2}$. The closing and reopening of a gap often signals a
topological phase transition. Indeed, imposing vanishing boundary conditions
at the edges, we find that there are two edge states (one at each edge) at
zero energy, $E=0$, if the following topological criterion is satisfied,
$t_{y1}<|t_{z}|<t_{y2}$, see Fig. 4. The wavefunction of the left edge state
for $t_{z}>0$ is given by (with $x=na_{x}$)

$\displaystyle\psi^{L}_{E=0}=(f(x),-if^{*}(x),i(-1)^{n}f(x),(-1)^{n+1}f^{*}(x)),$
$\displaystyle
f(x)=e^{-ik_{y}a_{y}/2}e^{-(k_{2-}+ik_{F1})x}-\cos(k_{y}a_{y}/2)$
$\displaystyle\hskip 10.0pt\times
e^{-(k_{1-}-ik_{F1})x}+i\sin(k_{y}a_{y}/2)e^{-(k_{1+}-ik_{F1})x}.$ (8)

The basis
$(\psi_{1,1},\psi_{1,\bar{1}},\psi_{\bar{1},1},\psi_{\bar{1},\bar{1}})$ is
composed of wavefunctions $\psi_{\eta,\sigma}$ defined at the $\sigma$-unit
lattice site of the $\eta$-strip. The smallest wavevectors
$k_{l,\pm}=|t_{l}\pm t_{z}|/\hbar\upsilon_{F}$ determine the localization
length of the edge state. We note that the probability densities
$|\psi_{\eta,\sigma}(x)|^{2}$ are uniform inside the unit cell.

If $|t_{z}|<t_{y1}$, there are two edge states at each edge for $E$ inside the
gap; see Fig. 4. These states, propagating in $y$, have a momentum $k_{y}$
determined by $E$. This case is similar to one strip with spinless particles
discussed above. We note that the edge states found here are the higher-
dimensional extensions of the end bound states found in one-dimensional
nanowires Two_field_Klinovaja_Stano_Loss_2012 ; footnote1  and ladders
Klinovaja_Loss_FF_1D . For $|t_{z}|>t_{y2}$, there are no edge states; see
Fig. 4. Finally, the most interesting regime here is $t_{y1}<|t_{z}|<t_{y2}$,
where there is one zero-energy edge state at each edge, see Fig. 4. Such
states with flat dispersion are expected to be strongly affected by
interactions.

Conclusions. We have studied topological regimes of strips with modulated
hopping amplitudes in the presence of magnetic fields. We found topological
regimes with chiral edge states at filling factors that correspond to integer
and fractional QHE regimes. We showed that double strips sustain topological
phases with mid-gap edge states with flat dispersion. Optical lattices
Lewenstein  seem to be promising candidates for implementing directly the
anisotropic tight-binding models considered here.

This work is supported by the Swiss NSF, NCCR Nanoscience, and NCCR QSIT.

##  Appendix A Fourier transformation

We introduce the momentum $k_{y}$ via Fourier transformation,

$c_{n,m,\sigma}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{y}}e^{imk_{y}a_{y}}c_{n,k_{y},\sigma},$
(9)

where $N_{y}$ is the number of lattice sites in $y$-direction. By analogy, we
introduce a momentum $k_{x}$,

$c_{n,k_{y},\sigma}=(1/\sqrt{N_{x}})\sum_{k_{x}}e^{ink_{x}a_{x}}c_{k_{x}k_{y},\sigma},$
(10)

where $N_{x}$ is the number of lattice sites along $x$.

##  Appendix B Effective Hamiltonian

Here we derive the spectrum of $H=H_{x}+H_{y}$ in the continuum limit
following Refs. MF_wavefunction_klinovaja_2012, ;
Two_field_Klinovaja_Stano_Loss_2012, . The annihilation operator in position
space $\Psi(x)$ close to the Fermi level is expressed in terms of slowly
varying right [$R_{\sigma}(x)$] and left [$L_{\sigma}(x)$] movers as

$\Psi(x)=\sum_{\sigma}[R_{\sigma}(x)e^{ik_{F}x}+L_{\sigma}(x)e^{-ik_{F}x}].$
(11)

As a consequence, $H_{x}$ results in the kinetic term,

$\displaystyle H_{x}^{lin}=-i\hbar\upsilon_{F}\sum_{\sigma}\int dx\ $
$\displaystyle[R_{\sigma}^{\dagger}(x)\partial_{x}R_{\sigma}(x)$
$\displaystyle\hskip 30.0pt-
L_{\sigma}^{\dagger}(x)\partial_{x}L_{\sigma}(x)],$ (12)

and $H_{y}$ results in a term that couples right and left movers,

$\displaystyle H_{y}^{lin}=t_{y1}\int dx\
\big{[}L_{1}^{\dagger}(x)R_{\bar{1}}(x)+h.c.]$ $\displaystyle\hskip
30.0pt+t_{y2}\int dx\
[e^{-ik_{y}a_{y}}R_{1}^{\dagger}(x)L_{\bar{1}}(x)+h.c.\big{]}.$ (13)

Here, we used the specific choice of the parameters,
$2k_{F}a_{x}=\phi\in(0,\pi)$. It is this term that leads to resonant
scattering and opens the gaps at the Fermi points. The Fermi velocity
$\upsilon_{F}$ depends on the Fermi wavevector,
$\hbar\upsilon_{F}=2t_{x}a_{x}\sin(k_{F}a_{x})$. The Hamiltonian density
$\mathcal{H}$ corresponding to $H^{lin}=H_{x}^{lin}+H_{y}^{lin}=\int dx\
\Psi^{\dagger}\mathcal{H}\Psi$, can be rewritten in terms of the Pauli
matrices $\tau_{i}$ ($\sigma_{i}$), acting on the right-left mover (lattice)
subspace, $\Psi=(R_{1},L_{1},R_{\bar{1}},L_{\bar{1}})$, leading directly to
Eq. (3) in the main text.

##  Appendix C  Wavefunction of left edge state

The wavefunction of the state localized at the left edge of a spinless strip
is given by

$\displaystyle\psi_{k_{y}}(x)=\begin{pmatrix}e^{-k_{2}x-i(k_{F}x+\theta)}-e^{-k_{1}x+i(k_{F}x-\theta)}\\\
e^{-k_{2}x+ik_{F}x}-e^{-k_{1}x-ik_{F}x}\end{pmatrix},$ (14)

where we suppress the normalization factor. Here, we introduced the notations
$e^{i\theta}=(E+i\sqrt{t_{y1}^{2}-E^{2}})/t_{y1}$ and
$k_{l}=\sqrt{t_{yl}^{2}-E^{2}}/\hbar\upsilon_{F}$.

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Let $g_{x}(w)=(wf_{x}(w))^{\prime}$ be as defined in (A.4), and let

$\eta_{1}=t+\Delta_{i^{*}}\text{ and }\eta_{2}=\xi_{b,i}+\Delta.$

By Stein’s equation (A.1), one can write

$E_{1}=E_{11}+E_{12},$

where

$\displaystyle E_{11}$
$\displaystyle=\sum_{i=1}^{n}\int_{-1}^{1}\operatorname{\mathbb{E}}\Big{[}\int_{t+\Delta_{i^{*}}}^{\xi_{b,i}+\Delta}g_{x}(W_{b}^{(i)}+u)du\Big{]}k_{b,i}(t)dt$
$\displaystyle=\underbrace{\sum_{i=1}^{n}\int_{-1}^{1}\operatorname{\mathbb{E}}\Bigg{[}\int
g_{x}(W_{b}^{(i)}+u)I(\eta_{1}\leq
u\leq\eta_{2})du\Bigg{]}k_{b,i}(t)dt}_{E_{11.1}}$ $\displaystyle\hskip
28.45274pt-\underbrace{\sum_{i=1}^{n}\int_{-1}^{1}\operatorname{\mathbb{E}}\Bigg{[}\int
g_{x}(W_{b}^{(i)}+u)I(\eta_{2}\leq
u\leq\eta_{1})du\Bigg{]}k_{b,i}(t)dt}_{E_{11.2}}$

and

$\displaystyle E_{12}$
$\displaystyle=\sum_{i=1}^{n}\int_{-1}^{1}\Bigg{\\{}P(W_{b}+\Delta\leq
x)-P(W_{b}^{(i)}+\Delta_{i^{*}}+t\leq x)\Bigg{\\}}k_{b,i}(t)dt.$

We first bound the integrand of $E_{11.1}$. Using the identity

$\displaystyle 1$ $\displaystyle=I(W_{b}^{(i)}+u\leq
x-1)+I(x-1<W_{b}^{(i)}+u,u\leq 3x/4)+I(x-1<W_{b}^{(i)}+u,u>3x/4)$
$\displaystyle\leq I(W_{b}^{(i)}+u\leq
x-1)+I(x-1<W_{b}^{(i)}+u,W_{b}^{(i)}+1>x/4)+(x-1<W_{b}^{(i)}+u,u>3x/4)$

and the bounds for $g_{x}(\cdot)$ in Lemma A.2, with
$|\Delta|\leq\frac{x|\bar{\mathfrak{D}}_{2}|}{2}+|\bar{D}_{1,x}|\leq\underbrace{\frac{2+\mathfrak{c}_{m}}{4}}_{<3/4}x$
and $1.6\bar{\Phi}(x)\leq xe^{1/2-x}$,

$\displaystyle\Bigg{|}\operatorname{\mathbb{E}}\Big{[}\int
g_{x}(W_{b}^{(i)}+u)I(\eta_{1}\leq u\leq\eta_{2})du\Big{]}\Bigg{|}$
$\displaystyle\leq
xe^{1/2-x}\|\eta_{2}-\eta_{1}\|_{1}+(x+2)\Big{\\{}\|I(W_{b}^{(i)}+1>x/4)(\eta_{2}-\eta_{1})\|_{1}+\|I(\eta_{2}>3x/4)(\eta_{2}-\eta_{1})\|_{1}\Big{\\}}$
$\displaystyle\leq
xe^{1/2-x}\|\eta_{2}-\eta_{1}\|_{1}+\frac{x+2}{e^{x/4-1}}\|e^{W_{b}^{(i)}}(\eta_{2}-\eta_{1})\|_{1}+\frac{x+2}{e^{3x/4}}\|e^{\xi_{b,i}+\Delta}(\eta_{2}-\eta_{1})\|_{1}$
$\displaystyle\leq\Bigg{(}xe^{1/2-x}+\frac{e(x+2)}{e^{(1-\mathfrak{c}_{m})x/4}}\Bigg{)}\|\eta_{2}-\eta_{1}\|_{1}+\frac{x+2}{e^{x/4-1}}\|e^{W_{b}^{(i)}}(\eta_{2}-\eta_{1})\|_{1}$
$\displaystyle\leq\frac{C(x+2)}{e^{(1-\mathfrak{c}_{m})x/4}}\Bigg{\\{}|t|+\|\Delta_{i^{*}}-\Delta+\xi_{b,i}\|_{1}+\|e^{W_{b}^{(i)}}(\Delta_{i^{*}}-\Delta+\xi_{b,i})\|_{1}\Bigg{\\}},$

where we have used the Bennett’s inequality (Lemma A.5) via
$\|e^{W_{b}^{(i)}}t\|_{1}\leq C|t|$ in the last line. Continuing,

$\displaystyle\Bigg{|}\operatorname{\mathbb{E}}\Big{[}\int
g_{x}(W_{b}^{(i)}+u)I(\eta_{1}\leq u\leq\eta_{2})du\Big{]}\Bigg{|}$
$\displaystyle\leq\frac{C(x+2)}{e^{(1-\mathfrak{c}_{m})x/4}}\Bigg{\\{}|t|+\Big{\|}x(\bar{\mathfrak{D}}_{2,i^{*}}-\bar{\mathfrak{D}}_{2})+(\bar{D}_{1,i^{*},x}-\bar{D}_{1,x})+\xi_{b,i}\Big{\|}_{1}$
$\displaystyle\hskip
113.81102pt+\Bigg{\|}e^{W_{b}^{(i)}}\Big{[}x(\bar{\mathfrak{D}}_{2,i^{*}}-\bar{\mathfrak{D}}_{2})+(\bar{D}_{1,i^{*},x}-\bar{D}_{1,x})+\xi_{b,i}\Big{]}\Bigg{\|}_{1}\Bigg{\\}}$
(C.30) $\displaystyle\leq
Ce^{-c(m)x}\Bigg{\\{}|t|+\|\xi_{b,i}\|_{2}+\|\bar{D}_{1,x}^{(i)}-\bar{D}_{1,x}\|_{2}+\|\bar{\mathfrak{D}}_{2}^{(i)}-\bar{\mathfrak{D}}_{2}\|_{3/2}\Bigg{\\}},$

where the last inequality uses
$\|e^{W_{b}^{(i)}}\|_{2}\vee\|e^{W_{b}^{(i)}}\|_{3}<C$ (by Bennett’s
inequality, Lemma A.5) and Property C.5. By a completely analogous argument,
we also have

(C.31) $\Bigg{|}\operatorname{\mathbb{E}}\Big{[}\int
g_{x}(W_{b}^{(i)}+u)I(\eta_{2}\leq u\leq\eta_{1})du\Big{]}\Bigg{|}\\\ \leq
Ce^{-c(m)x}\Bigg{\\{}|t|+\|\xi_{b,i}\|_{2}+\|\bar{D}_{1,x}^{(i)}-\bar{D}_{1,x}\|_{2}+\|\bar{\mathfrak{D}}_{2}^{(i)}-\bar{\mathfrak{D}}_{2}\|_{3/2}\Bigg{\\}}$

for the integrand of $E_{11.2}$. Combining (C.30) and (C.31) and integrating
over $t$, we have

(C.32) $|E_{11}|\leq
Ce^{-c(m)x}\Bigg{\\{}\sum_{i=1}^{n}\|\xi_{b,i}\|_{3}^{3}+\sum_{i=1}^{n}\|\xi_{b,i}\|_{2}\|\bar{D}_{1,x}^{(i)}-\bar{D}_{1,x}\|_{2}+\|\xi_{b,i}\|_{3}\|\bar{\mathfrak{D}}_{2}^{(i)}-\bar{\mathfrak{D}}_{2}\|_{3/2}\Bigg{\\}}$

where we have used (C.26) and $\|\xi_{b,i}\|_{2}^{3}\leq\|\xi_{b,i}\|^{3}_{3}$
and $\|\xi_{b,i}\|_{2}^{2}\leq\|\xi_{b,i}\|_{2}\leq\|\xi_{b,i}\|_{3}$.

For $E_{12}$, its integrand is bounded by

(C.33) $P(x-\Delta-\xi_{b,i}\leq W_{b}^{(i)}\leq
x-\Delta_{i^{*}}-t)+P(x-\Delta_{i^{*}}-t\leq W_{b}^{(i)}\leq
x-\Delta-\xi_{b,i})$

Since $0<\mathfrak{c}_{m}<1$ implies that

$\min(x-\Delta-\xi_{b,i},x-\Delta_{i^{*}}-t)\geq
x-\frac{(2+\mathfrak{c}_{m})x}{4}-1\geq\frac{x}{4}-1\quad\text{ for
}\quad|t|\leq 1,$

by defining

$W_{b}^{(i,j)}\equiv W_{b}-\xi_{b,i}-\xi_{b,j}\text{ and
}\Delta^{(j)}_{i^{*}}\equiv\bar{D}_{1,i^{*},x}^{(j)}-\frac{x\bar{\mathfrak{D}}_{2,i^{*}}^{(j)}}{2}\text{
for }1\leq i\neq j\leq n,$

we can apply the randomized concentration inequality (Lemma A.6) to bound
(C.33) as

$\displaystyle
Ce^{-x/8}\Bigg{\\{}\beta_{2}+\beta_{3}+2\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\operatorname{\mathbb{E}}\Big{[}|\xi_{b,j}|e^{W_{b}^{(i,j)}/2}(|\Delta-\Delta^{(j)}|+|\Delta_{i^{*}}-\Delta_{i^{*}}^{(j)}|)\Big{]}$
$\displaystyle\hskip
85.35826pt+\operatorname{\mathbb{E}}\Big{[}|W_{b}^{(i)}|e^{W_{b}^{(i)}/2}\Big{(}|\Delta-\Delta_{i^{*}}|+|\xi_{b,i}|+|t|+\beta_{2}+\beta_{3}\Big{)}\Big{]}$
$\displaystyle\hskip 85.35826pt+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\Big{|}\operatorname{\mathbb{E}}[\xi_{b,j}]\Big{|}\operatorname{\mathbb{E}}\Big{[}e^{W_{b}^{(i,j)}/2}\underbrace{\Big{(}|t|+|\xi_{b,i}|+|\Delta^{(j)}-\Delta_{i^{*}}^{(j)}|+\beta_{2}+\beta_{3}\Big{)}}_{\leq
C(1+x)}\Big{]}\Bigg{\\}}$ $\displaystyle\leq
Ce^{-x/8}\Bigg{\\{}x\beta_{2}+\beta_{3}+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\Bigg{[}\|\xi_{b,j}\|_{2}(\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(j)}\|_{2})+x\|\xi_{b,j}\|_{3}(\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(j)}\|_{3/2})\Bigg{]}$
(C.34) $\displaystyle\hskip
85.35826pt+\|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}\|_{2}+x\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2}+\|\xi_{b,i}\|_{2}+|t|\Bigg{\\}}$
$\displaystyle\leq
Ce^{-x/8}\Bigg{\\{}x\beta_{2}+\beta_{3}+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\Bigg{[}\|\xi_{b,j}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(j)}\|_{2}+x\|\xi_{b,j}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(j)}\|_{3/2}\Bigg{]}$
(C.35) $\displaystyle\hskip
85.35826pt2\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+2x\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}+\|\xi_{b,i}\|_{2}+|t|\Bigg{\\}},$

where

  1. (i)

to attain (C.34), we have used that
$\Big{|}\operatorname{\mathbb{E}}[\xi_{b,i}]\Big{|}\leq\operatorname{\mathbb{E}}[\xi_{i}^{2}I(|\xi_{i}|>1)]$
from Lemma A.4, $|W_{b}^{(i)}|e^{W_{b}^{(i)}/2}\leq 2(1+e^{W_{b}^{(i)}})$, the
Bennett’s inequality (Lemma A.5) and applied Property C.4 on
$|\Delta_{i^{*}}-\Delta_{i^{*}}^{(j)}|$;

  2. (ii)

to attain (C.35), we have used Property C.5.

From (C.35), on integration with respect to $t$, for absolute constants
$C,c>0$,

(C.36) $|E_{12}|\leq
Ce^{-cx}\Bigg{\\{}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\sum_{i=1}^{n}\|\xi_{i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\sum_{i=1}^{n}\|\xi_{i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}\Bigg{\\}}$

by the properties of the K-function in (C.26),
$\|\xi_{b,i}\|_{2}^{3}\leq\|\xi_{b,i}\|^{3}_{3}$ and
$\|\xi_{b,i}\|_{2}^{2}\leq\|\xi_{b,i}\|_{2}\leq\|\xi_{b,i}\|_{3}$.

Lastly, combining (C.32) and (C.36), we obtain (C.27).

####  C.2.2. Proof of bounds for $E_{2}$, (C.28)

For $x\geq 1$, given
$|\Delta|\vee|\Delta_{i^{*}}|\leq(\frac{2+\mathfrak{c}_{m}}{4})x\leq\frac{3x}{4}$,
by (A.8) in Lemma A.2 and $|f^{\prime}_{x}|\leq 1$ (Lemma A.1),

$\displaystyle|f_{x}(W_{b}+\Delta_{i^{*}})-f_{x}(W_{b}+\Delta)|$
$\displaystyle\leq|f_{x}(W_{b}+\Delta_{i^{*}})-f_{x}(W_{b}+\Delta)|\Bigg{[}I\Bigg{(}W_{b}\leq\frac{x}{4}-1\Bigg{)}+I\Bigg{(}W_{b}>\frac{x}{4}-1\Bigg{)}\Bigg{]}$
$\displaystyle\leq
C\Big{(}e^{1/2-x}+I(W_{b}>x/4-1)\Big{)}\Big{(}|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}|+x|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}|\Big{)}$
$\displaystyle\leq
C\Big{(}e^{-x}+e^{-x/4}e^{W_{b}}\Big{)}\Big{(}|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}|+x|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}|\Big{)}.$

Hence,

$\displaystyle|E_{2}|$ $\displaystyle\leq
C_{1}e^{-x}\sum_{i=1}^{n}(\|\xi_{b,i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}\|_{2}+x\|\xi_{b,i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2})+$
$\displaystyle\hskip
56.9055ptC_{2}e^{-x/4}\sum_{i=1}^{n}(\|\xi_{b,i}e^{\xi_{b,i}}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}\|_{2}+x\|\xi_{b,i}e^{\xi_{b,i}}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2})$
$\displaystyle\leq
Ce^{-cx}\sum_{i=1}^{n}\Bigg{(}\|\xi_{b,i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\|\xi_{b,i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}\Bigg{)},$

where we have applied Bennett’s inequality (Lemma A.5) on $e^{W_{b}^{(i)}}$ in
the first inequality, and $e^{\xi_{b,i}}\leq e$, and Property C.5 in the
second. This establishes (C.28).

####  C.2.3. Proof of the bound for $E_{3}$, (C.29)

We will form bounds for each of

$\sum_{i=1}^{n}\mathbb{E}[\xi_{b,i}f_{x}(W_{b}^{(i)}+\Delta_{i^{*}})]\\\
\mathbb{E}[f_{x}^{\prime}(W_{b}+\Delta)]\sum_{i=1}^{n}\operatorname{\mathbb{E}}[(\xi_{i}^{2}-1)I(|\xi_{i}|>1)]\\\
\text{ and }\;\ \mathbb{E}[\Delta f_{x}(W_{b}+\Delta)],$

which can conclude (C.29).

Bounding the first two terms is relatively simple. For the first term, by the
independence between $\xi_{b,i}$ and $W_{b}^{(i)}+\Delta_{i^{*}}$, Bennett’s
inequality (Lemma A.5), $\Delta_{i^{*}}\leq 3x/4$, $0<f_{x}\leq 0.63$ in Lemma
A.1, and (A.7) in Lemma A.2,

$\displaystyle\sum_{i=1}^{n}|\mathbb{E}[\xi_{b,i}f_{x}(W_{b}^{(i)}+\Delta_{i^{*}})]|$
$\displaystyle\leq\sum_{i=1}^{n}\mathbb{E}\Big{[}(|\xi_{i}|-1)I(|\xi_{i}|>1)\Big{]}\mathbb{E}\left[f_{x}(W_{b}^{(i)}+\Delta_{i^{*}})\left(\underbrace{I(W_{b}^{(i)}\geq
x/4-1)}_{\leq e^{W_{b}^{(i)}+1}\cdot
e^{-x/4}}+I(W_{b}^{(i)}<x/4-1)\right)\right]$ (C.37)
$\displaystyle\leq\sum_{i=1}^{n}\mathbb{E}[\xi_{i}^{2}I(|\xi_{i}|>1)]\left(Ce^{-x/4}+1.7e^{-x}\right)\leq
Ce^{-x/4}\beta_{2}.$

For the second term, by Bennett’s inequality (Lemma A.5), that $\Delta\leq
3x/4$, $|f_{x}^{\prime}|\leq 1$ in Lemma A.1, and (A.8) in Lemma A.2,

$\displaystyle\left|\mathbb{E}[f_{x}^{\prime}(W_{b}+\Delta)]\sum_{i=1}^{n}\operatorname{\mathbb{E}}[(\xi_{i}^{2}-1)I(|\xi_{i}|>1)]\right|$
$\displaystyle\leq\beta_{2}\mathbb{E}\left[|f_{x}^{\prime}(W_{b}+\Delta)|\\{I(W_{b}<x/4-1)+I(W_{b}\geq
x/4-1)\\}\right]$ (C.38) $\displaystyle\leq\beta_{2}(e^{1/2-x}+Ce^{-x/4})\leq
Ce^{-x/4}\beta_{2}.$

Both (C.37) and (C.38) are less than
$Ce^{-cx}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]$, forming a
part of (C.29).

To finish proving (C.29), it remains to show the bound

(C.39) $\Big{|}\mathbb{E}[\Delta f_{x}(W_{b}+\Delta)]\Big{|}\leq
C(m)e^{-cx}\Bigg{(}\|\bar{D}_{1,x}\|_{2}+\operatorname{\mathbb{E}}[(1+e^{W_{b}})\mathfrak{D}_{2}^{2}]\Bigg{)}+x\Bigg{|}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]\Bigg{|},$

for the last term, which is more delicate to derive. We first write

(C.40) $\Big{|}\mathbb{E}[\Delta
f_{x}(W_{b}+\Delta)]\Big{|}=\Bigg{|}\Delta\int_{0}^{\Delta}f_{x}^{\prime}(W_{b}+t)dt+\mathbb{E}[\Delta
f_{x}(W_{b})]\Bigg{|}\\\
\leq\Bigg{|}\Delta\int_{0}^{\Delta}f_{x}^{\prime}(W_{b}+t)dt\Bigg{|}+\Bigg{|}\mathbb{E}[\Delta
f_{x}(W_{b})]\Bigg{|},$

and will control the two terms on the right hand side separately.

For the first right-hand-side term in (C.40), since $\Delta\leq 3x/4$, we have

$\displaystyle\Bigg{|}\Delta\int_{0}^{\Delta}f_{x}^{\prime}(W_{b}+t)dt\Bigg{|}$
$\displaystyle\leq
2e^{1/2-x}\operatorname{\mathbb{E}}[\bar{D}_{1,x}^{2}+x^{2}\bar{\mathfrak{D}}_{2}^{2}/4]+2\operatorname{\mathbb{E}}[\underbrace{I(W_{b}>x/4-1)}_{\leq
e^{W_{b}+1-x/4}}(\bar{D}_{1,x}^{2}+x^{2}\bar{\mathfrak{D}}_{2}^{2}/4)]$ by
(A.8) in Lemma A.2 and that $\Delta^{2}\leq
2(\bar{D}_{1,x}^{2}+x^{2}\bar{\mathfrak{D}}_{2}^{2}/4)$ $\displaystyle\leq
C_{1}e^{-x}(\operatorname{\mathbb{E}}[\bar{D}_{1,x}^{2}]+x^{2}\operatorname{\mathbb{E}}[\bar{\mathfrak{D}}_{2}^{2}])+C_{2}e^{-x/4}x^{2}\operatorname{\mathbb{E}}[e^{W_{b}}\bar{D}_{1,x}^{2}/x^{2}]+C_{3}e^{-x/4}x^{2}\operatorname{\mathbb{E}}[e^{W_{b}}\bar{\mathfrak{D}}_{2}^{2}]$
(C.41) $\displaystyle\leq
Ce^{-x/4}(x\|\bar{D}_{1,x}\|_{2}+x^{2}\operatorname{\mathbb{E}}[(1+e^{W_{b}})\bar{\mathfrak{D}}_{2}^{2}]),$

where (C.41) is true because, with Lemma A.5 and
$|\bar{D}_{1,x}|/x\leq\mathfrak{c}_{m}/4\leq 1/4$,

$\operatorname{\mathbb{E}}[e^{W_{b}}\bar{D}_{1,x}^{2}/x^{2}]\leq\|e^{W_{b}}\|_{2}\|\bar{D}_{1,x}^{2}/x^{2}\|_{2}=C\sqrt{\operatorname{\mathbb{E}}[\bar{D}_{1,x}^{4}/x^{4}]}\leq\frac{C\|\bar{D}_{1,x}\|_{2}}{x}.\
$

and

$\operatorname{\mathbb{E}}[\bar{D}_{1,x}^{2}]=(x/4)^{2}\operatorname{\mathbb{E}}\Bigg{[}\frac{\bar{D}_{1,x}^{2}}{(x/4)^{2}}\Bigg{]}\leq(x/4)\operatorname{\mathbb{E}}[\bar{D}_{1,x}]\leq
Cx\|\bar{D}_{1,x}\|_{2}$

For the second right-hand-side term in (C.40), using $0<f_{x}(w)\leq 0.63$
(Lemma A.1),

$\displaystyle|\mathbb{E}[\Delta f_{x}(W_{b})]|$
$\displaystyle\leq\operatorname{\mathbb{E}}[|\bar{D}_{1,x}f_{x}(W_{b})|]+\frac{x}{2}\Big{|}\operatorname{\mathbb{E}}[\bar{\mathfrak{D}}_{2}f_{x}(W_{b})]\Big{|}$
$\displaystyle\leq
0.63e^{1-x}\operatorname{\mathbb{E}}[|\bar{D}_{1,x}|e^{W_{b}}]+1.7e^{-x}\|\bar{D}_{1,x}\|_{2}+\frac{x}{2}\Big{|}\operatorname{\mathbb{E}}[\bar{\mathfrak{D}}_{2}f_{x}(W_{b})]\Big{|}$
by $0<f_{x}(w)\leq 0.63$, (A.7) from Lemmas A.1, A.2 and $I(W_{b}>x-1)\leq
e^{W_{b}+1-x}$ $\displaystyle\leq
Ce^{-x}\|\bar{D}_{1,x}\|_{2}+\frac{x}{2}\Big{|}\operatorname{\mathbb{E}}[\bar{\mathfrak{D}}_{2}f_{x}(W_{b})]\Big{|}\text{
by Bennett's inequality (Lemma~{}\ref{lem:bennett_censored})}$ (C.42)
$\displaystyle\leq
C(m)\Big{(}e^{-x}\|\bar{D}_{1,x}\|_{2}+xe^{-x}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}^{2}(1+e^{W_{b}})]\Big{)}+\frac{x}{2}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})],$

which can conclude (C.39) in combination with (C.40) and (C.41). The last
inequality (C.42) comes as follows: Write

$\operatorname{\mathbb{E}}[\bar{\mathfrak{D}}_{2}f_{x}(W_{b})]=\operatorname{\mathbb{E}}[(\bar{\mathfrak{D}}_{2}-\mathfrak{D}_{2})f_{x}(W_{b})]+\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})],$

Now, defining $\mathfrak{C}_{m}=1-\frac{9\mathfrak{c}_{m}^{2}}{16}$ (where
$0<\mathfrak{C}_{m}<1$),

$\displaystyle|\operatorname{\mathbb{E}}[(\bar{\mathfrak{D}}_{2}-\mathfrak{D}_{2})f_{x}(W_{b})]|$
$\displaystyle\leq\operatorname{\mathbb{E}}[|\bar{\mathfrak{D}}_{2}-\mathfrak{D}_{2}|f_{x}(W_{b})I(W_{b}\leq
x-1)]+\operatorname{\mathbb{E}}[|\bar{\mathfrak{D}}_{2}-\mathfrak{D}_{2}|f_{x}(W_{b})I(W_{b}>x-1)]$
$\displaystyle\leq
1.7e^{-x}\operatorname{\mathbb{E}}\Bigg{[}|\mathfrak{D}_{2}-\mathfrak{C}_{m}|I\Big{(}|\mathfrak{D}_{2}|>\mathfrak{C}_{m}\Big{)}\Bigg{]}+0.63e^{1-x}\operatorname{\mathbb{E}}\Bigg{[}|\mathfrak{D}_{2}-\mathfrak{C}_{m}|I\Big{(}|\mathfrak{D}_{2}|>\mathfrak{C}_{m}\Big{)}e^{W_{b}}\Bigg{]}$
by $\eqref{fxbdd}$ and $0<f_{x}(w)\leq 0.63$ from Lemmas A.1
$\displaystyle\leq
1.7e^{-x}\operatorname{\mathbb{E}}[|\mathfrak{D}_{2}|I(|\mathfrak{D}_{2}|>\mathfrak{C}_{m})]+0.63e^{1-x}\operatorname{\mathbb{E}}[|\mathfrak{D}_{2}|I(|\mathfrak{D}_{2}|>\mathfrak{C}_{m})e^{W_{b}}]$
$\displaystyle\leq
C(m)e^{-x}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}^{2}(1+e^{W_{b}})],$

where the last line uses that
$I(|\mathfrak{D}_{2}|>\mathfrak{C}_{m})\leq\mathfrak{C}_{m}^{-1}|\mathfrak{D}_{2}|$.

##  Appendix D Proof of Lemmas 4.3 and 4.4

###### Proof of Lemma 4.3.

As a useful fact, we first note that, for any $p\in(1,2]$,

(D.1) $e^{-s}\leq 1-s+s^{p}/p\text{ for }s\geq 0.$

This is because the derivative of $1-s+s^{p}/p-e^{-s}$ as a function in $s$
has the form

(D.2) $\frac{\partial}{\partial s}(1-s+s^{p}/p-e^{-s})=s^{p-1}+e^{-s}-1,$

which can be seen to be non-negative for all $s\in[0,\infty)$. (This is
obvious for $s\in(1,\infty)$ since $s^{p-1}>1>1-e^{-s}$ for $1\leq s<\infty$;
and it is also true for $s\in[0,1)$ since $1-e^{-s}\leq s\leq s^{p-1}$ for
$0\leq s\leq 1$.)

Using the trick by Hoeffding (1963, Section 5, Eqn. $(5.4)$), one can write

$U_{n}=\frac{1}{n!}\sum W(X_{i_{1}},\dots,X_{i_{n}}),$

where the summation is over all $n!$ permutation of $(i_{1},\dots,i_{n})$ of
$(1,\dots,n)$ and

$W(x_{1},\dots,x_{n})=\frac{h(x_{1},\dots,x_{m})+h(x_{m+1},\dots,x_{2m})+\cdots+h(x_{km-m+1},\dots,x_{km})}{\kappa},$

where $\kappa\equiv[n/m]$, the greatest integer $\leq n/m$. By the Chernoff
bounding technique and Jensen’s inequality, for any $t>0$,

$\displaystyle P(U_{n}\leq x)$ $\displaystyle\leq
e^{tx}\operatorname{\mathbb{E}}[e^{-tU_{n}}]$ $\displaystyle\leq
e^{tx}\operatorname{\mathbb{E}}[e^{-tW(X_{1},\dots,X_{n})}]=e^{tx}(\operatorname{\mathbb{E}}[e^{-th(X_{1},\dots,X_{m})/\kappa}])^{\kappa}.$

Using that $h(X_{1},\dots,X_{m})\geq 0$ and (D.1), we can continue and get
that

$\displaystyle P(U_{n}\leq x)$ $\displaystyle\leq
e^{tx}\left\\{1-\frac{t}{\kappa}\operatorname{\mathbb{E}}[h]+\frac{t^{p}}{p\kappa^{p}}\operatorname{\mathbb{E}}[h^{p}]\right\\}^{\kappa}$
$\displaystyle\leq\exp\left\\{t(x-\operatorname{\mathbb{E}}[h])+\frac{t^{p}}{p\kappa^{p-1}}\operatorname{\mathbb{E}}[h^{p}]\right\\},$

where the last inequality uses that $1+y\leq e^{y}$ for all
$y\in\operatorname{\mathbb{R}}$. By minimizing the right hand side with
respect to $t$, one can take
$t=\kappa\Big{(}\frac{\operatorname{\mathbb{E}}[h]-x}{\operatorname{\mathbb{E}}[h^{p}]}\Big{)}^{1/(p-1)}$
and obtain

$\displaystyle P(U_{n}\leq x)$
$\displaystyle\leq\exp\left(\frac{-\kappa(\operatorname{\mathbb{E}}[h]-x)^{p/(p-1)}}{(\operatorname{\mathbb{E}}[h^{p}])^{1/(p-1)}}+\frac{\kappa(\operatorname{\mathbb{E}}[h]-x)^{p/(p-1)}}{p(\operatorname{\mathbb{E}}[h^{p}])^{1/(p-1)}}\right)$
$\displaystyle=\exp\left(-\frac{(p-1)\kappa(\operatorname{\mathbb{E}}[h]-x)^{p/(p-1)}}{p(\operatorname{\mathbb{E}}[h^{p}])^{1/(p-1)}}\right)$

∎

###### Proof of Lemma 4.4.

Define the _canonical_ functions (Korolyuk and Borovskich, 2013, p.20-21)

$\displaystyle g_{1}(x_{1})$ $\displaystyle=h_{1}(x_{1})$ $\displaystyle
g_{2}(x_{1},x_{2})$
$\displaystyle=h_{2}(x_{1},x_{2})-g_{1}(x_{1})-g_{1}(x_{2})$
$\displaystyle\vdots$ $\displaystyle g_{m}(x_{1},\dots,x_{m})$
$\displaystyle=h_{m}(x_{1},\dots,x_{m})-\sum_{l=1}^{m}g_{1}(x_{l})-\sum_{1\leq
l_{1}<l_{2}\leq m}g_{2}(x_{l_{1}},x_{l_{2}})-$ $\displaystyle\hskip
85.35826pt\dots-\sum_{1\leq l_{1}<\dots<l_{m-1}\leq
m}g_{m-1}(x_{l_{1}},\dots,x_{l_{m-1}}).$

Note that $r$ can be alternatively defined as the first integer such that, as
functions,

$g_{k}(x_{1},\dots,x_{k})=0\text{ for }k=1,\dots,r-1,\text{ and }\quad
g_{r}(x_{1},\dots,x_{r})\neq 0;$

see the discussion in Korolyuk and Borovskich (2013, p.32) for instance. Then
Korolyuk and Borovskich (2013, Theorem 2.1.3 & 2.1.4) suggest that

$\operatorname{\mathbb{E}}[|U_{n}|^{p}]\leq\begin{cases}(m-r+1)^{p-1}\sum_{k=r}^{m}{m\choose
k}^{p}{n\choose
k}^{-p+1}\alpha_{p}^{k+1}\operatorname{\mathbb{E}}[|g_{k}|^{p}]&\text{if
}1\leq p\leq 2;\\\ (m-r+1)^{p-1}\sum_{k=r}^{m}{m\choose k}^{p}{n\choose
k}^{-p+1}n^{((p-2)k)/2}\gamma_{p}^{k+1}\operatorname{\mathbb{E}}[|g_{k}|^{p}]&\text{if
}p\geq 2,\end{cases}$

where $\alpha_{p}\equiv\sup_{x}(|x|^{-p}(|1+x|^{p}-1-px))\leq 2^{2-p}$ and
$\gamma_{p}\equiv\\{8(p-1)\max(1,2^{p-3})\\}^{p}$. The bound (4.42) is a
simple consequence of this based on (4.10).∎

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