# Measuring $\mathbf{CP}$ violating phase in beauty baryon decays

Rahul Sinha<EMAIL_ADDRESS>The Institute of Mathematical Sciences,
Taramani, Chennai 600113, India  Homi Bhabha National Institute, Training
School Complex, Anushaktinagar, Mumbai 400094, India  Shibasis Roy
<EMAIL_ADDRESS>Department of Physics, University of Calcutta, 92
Acharya Prafulla Chandra Road, Kolkata 700009, India  N. G. Deshpande
<EMAIL_ADDRESS>Institute for Fundamental Science, University of Oregon,
Eugene, Oregon 94703, USA

###### Abstract

One of the outstanding problems in physics is to explain the baryon-anti-
baryon asymmetry observed in nature. According to the well-known Sakharov
criterion for explaining the observed asymmetry, it is essential that $C\\!P$
violation exist. Even though $C\\!P$ violation has been observed in meson
decays and is an integral part of the standard model (SM), measurements in
meson decays indicate that, $C\\!P$ violation in the SM is insufficient to
explain the observed baryon-anti-baryon asymmetry. SM predicts the existence
of yet to be observed $C\\!P$ violation in baryon decays. A critical test of
the SM requires that $C\\!P$ violation be measured in baryon decays as well,
in order to verify that it agrees with the measurement using meson decays. In
this letter we propose a new method to measure $C\\!P$ violating phase in
$b$-baryons, using interference arising implicitly due to Bose symmetry
considerations of the decaying amplitudes.

The proposal to observe $C\\!P$ violation in $\Sigma$-baryon decays [1] was
first made by Okubo in 1957, much before the first observation of $C\\!P$
violation in 1964 by Fitch and Cronin using neutral $K$-mesons. $C\\!P$
violation has since been observed in $B$-meson and $D$-mesons as well. $C\\!P$
violating phases have been measured accurately in $B-$meson decays and the
measurements are found consistent within the standard model (SM) and in
agreement the Kobayashi-Maskawa hypothesis.

One of the well-known Sakharov conditions [2, 3] to explain baryon-anti-baryon
asymmetry in the universe, is the existence of $C\\!P$ violation. The
observation of $C\\!P$ -violation in baryon decays is expected within the SM
and thus, of great interest. It would provide a direct probe of matter-
antimatter asymmetry, arising from $C\\!P$ violating sector of the SM. This is
essential since measurements in meson decays indicate that, $C\\!P$ violation
in the SM is by itself insufficient to explain the observed baryon-anti-baryon
asymmetry [4]. However, $C\\!P$ violation has only been observed in mesons
decays and is yet to be convincingly demonstrated [5, 6, 7, 8, 9, 10, 11, 12,
13, 14] in baryon decays. It is well-known that the largest weak phases within
the SM appear in $b\to u$ and $t\to d$ transitions. It is thus obvious that it
would be easiest to observe $C\\!P$ violation in $b$-baryons and one must
explore the weak decays of the anti-triplet ($\bar{3}$) beauty baryons.

While there is no reason to expect a disagreement between $C\\!P$ violating
phases measured using baryons and that using mesons, within the SM, it
nevertheless remains imperative, that a clean measurement of the weak phase
also be performed in baryons decays. A comparison of the measurements of
$C\\!P$ violation in meson and baryon decays would provide a critical probe of
physics beyond the SM. Unfortunately, measurement of $C\\!P$ violation in
baryon decays is not straight forward. It is well-known that for $C\\!P$
violation to be observed the amplitude must have two contributions with both
having different strong phases and weak phases. However, in order to measure
$C\\!P$ violating phase using the decay(s) of any particle(s) one must satisfy
certain further conditions. One of the critical requirements to measure any
phase is that amplitudes interfere. In the case of neutral $B$-mesons mixing
between particle and antiparticle allows for two distinct amplitudes to
interfere, with one amplitude corresponding to direct decay and the other to
decay via mixing. This results in the well-known time dependent $C\\!P$
violation. However, baryon number conservation forbids oscillations between
baryons and anti-baryons disallowing such time dependent mixing and consequent
interference of two amplitudes. It is also essential that at least one of the
decay amplitudes be re-parametrization invariant [15, 16] such that the
$C\\!P$ violating phase to be measured can be uniquely defined [17]. Inability
to satisfy this condition results in fewer independent observables than
necessary to measure the weak phase. The methods to measure the phase $\beta$
and $\alpha$ without approximations, in $B$-mesons satisfy all these
conditions.

Interestingly, the phase $\gamma$ can be measured, without the need for meson
mixing. Interference of two decay amplitudes arises from decay to the same
final state via intermediate $D^{0}$ and $\bar{D}^{0}$ decays. The only
proposal [18] to measure weak phase using baryons is a straightforward
extension of this method used in mesons, to achieve interference of two
amplitudes. A proposal to measure $C\\!P$ violating weak phase in baryon
decays that is not a straightforward extension of the approach used in meson
decays and one that relies on processes specific to baryons is still missing.

In this letter we propose a new approach to measure $C\\!P$ violating weak
phase in baryon decays using interference arising due to Bose symmetry
considerations of the decaying amplitudes. We start by exploring the weak
decays of the anti-triplet $\bar{3}$ beauty baryons which have been studied
earlier in Ref. [19, 20] and where large $C\\!P$ violation may be expected. As
mentioned earlier for weak-phases to be measurable, the additional criterion
of re-parametrization invariance must also hold. This re-parametrization
condition is severely restrictive for the weak decay of
$\bar{3}\leavevmode\nobreak\ b$-baryon and we find that even when the
interference criterion due to Bose symmetry holds, measurement of the weak
phase is not possible in most cases since re-parametrization invariance of the
amplitude does not hold. Considering all the weak decay modes of the $\bar{3}$
$b$-baryons we find one mode namely, $\Xi_{b}\to\Sigma^{\prime}(1385)\pi$,
where it is possible to measure the $C\\!P$ violating weak phase.

Figure 1: Feynman diagrams contributing to the quasi two body decay
$\Xi_{b}^{-}(q)\to\Lambda^{0}(q_{1})\pi^{-}(q_{2})\pi^{0}(q_{3})$ via the
$J^{P}=\tfrac{3}{2}^{+}$, $\Sigma^{\prime}(1385)$ baryon resonance.

We show in detail how the weak phase can be measured using the decay
$\Xi_{b}\to\Sigma^{\prime}\pi$, where $\Sigma^{\prime}$ is the
$J^{P}=\tfrac{3}{2}^{+}$ decuplet baryon resonance corresponding to
$\Sigma^{\prime}(1385)$. We consider the decays $\Xi_{b}^{-}\to\Sigma^{\prime
0}(1385)\pi^{-}$ and $\Xi_{b}^{-}\to\Sigma^{\prime-}(1385)\pi^{0}$ depicted in
Fig. 1. Note that $\Xi_{b}^{-}$ is an isospin $1/2$ state of $bsd$ quarks. The
subsequent strong-decays of the $\Sigma^{\prime 0}\to\Lambda^{0}\pi^{0}$ and
$\Sigma^{\prime-}\to\Lambda^{0}\pi^{-}$ result in the identical final state
$\Xi_{b}^{-}\to\Lambda^{0}\pi^{-}\pi^{0}$ for both the decay modes. The
$\Lambda^{0}\pi^{-}\pi^{0}$ Dalitz plot would depict both these decays and
have special properties under exchange of the two-pions which are identical
bosons under isospin symmetry. The interference between these two modes would
arise implicitly from Bose symmetry correlations even though the two decay
modes do not effectively interfere on the Dalitz plot. The $\pi^{-}\pi^{0}$
can either be in a $\left|1,-1\right>$ or $\left|2,-1\right>$ isospin state.
Since, the two $\pi$-bosons are identical under isospin their total wave-
function must be symmetric. This necessitates that the two pions in the odd
isospin state $\left|1,-1\right>$ must be anti-symmetric under spatial
exchange, whereas, the two pions of even isospin state $\left|2,-1\right>$
must be symmetric under spatial exchange. The $\Lambda^{0}$ is an isospin
$\left|0,0\right>$ state, hence, the combined isospin of the final
$\Lambda^{0}\pi^{-}\pi^{0}$ state must be identical to the isospin of the two
pion state, which can either be a $\left|2,-1\right>$ or $\left|1,-1\right>$.
Isolating the $\pi^{-}\pi^{0}$ symmetric state which has isospin
$\left|2,-1\right>$, would thus be equivalent to isolating the $\Delta I=3/2$
changing contribution to the decay, which cannot arise from the penguin
diagram and must have a pure $b\to u$ weak phase.

Figure 2: Topological diagrams contributing to decay
$\Xi_{b}^{-}\to\Sigma^{\prime}\pi$ decays. The blob on the $b\to d$ transition
in the diagram on the left corresponds to the $b\to d$ penguin. The red line
straddling between the $s$ and $d$ quarks of the initial $\Xi_{b}$ baryon
indicates anti-symmetry of quarks in the initial $\bar{3}$ state. The
symmetric wave-function of the $J^{P}=\tfrac{3}{2}$ nucleon $\Sigma^{\prime}$
allows only the above topologies.  Figure 3: Jackson frame in which
$\Xi_{b}^{-}(q)$ decays to three bodies $\Lambda^{0}(q_{1})$, $\pi^{-}(q_{2})$
and $\pi^{0}(q_{3})$ .

The topological diagram contributions to the decays
$\Xi_{b}^{-}\to\Sigma^{\prime 0}(1385)\pi^{-}$ and
$\Xi_{b}^{-}\to\Sigma^{\prime-}(1385)\pi^{0}$ are depicted in Fig. 2. While
the decay $\Xi_{b}^{-}\to\Sigma^{\prime 0}(1385)\pi^{-}$ gets contributions
from both the tree-level and penguin diagrams with the corresponding
amplitudes denoted as $C$ and $P$ respectively, only the $P$ amplitude
contributes to the decay $\Xi_{b}^{-}\to\Sigma^{\prime-}(1385)\pi^{0}$. This
ensures that the weak phase cannot be reparametrizated and plays a crucial
role in the measurement of the $C\\!P$ violating weak phase. The $C\\!P$
violating weak phase measured using $b$-baryons with in our approach is free
from any hadronic uncertainty and relies only on reliable theoretical inputs
such as isospin and vanishingly small electro-weak penguin contributions in
$\Delta S=0$ $b\to d$ decays.

We consider the decay in the Gottfried-Jackson frame with $\Xi_{b}$ moving in
the $+\hat{z}$ axis, with the two pions going back to back with the
$\pi^{-}(q_{2})$ at an angle $\theta$ to the $\Lambda^{0}(q_{1})$. In this
frame (see Fig. 3) $\vec{q_{2}}+\vec{q_{3}}=0$. We define
$s\equiv(q_{2}+q_{3})^{2}=(q-q_{1})^{2}$,
$t\equiv(q_{1}+q_{3})^{2}=(q-q_{2})^{2}$ and
$u\equiv(q_{1}+q_{2})^{2}=(q-q_{3})^{2}$. $t$ and $u$ can be written as

$\displaystyle t$ $\displaystyle=x+y\cos\theta$ (1) $\displaystyle u$
$\displaystyle=x-y\cos\theta$ (2)

where

$\displaystyle x=\frac{M^{2}+m_{\Lambda}^{2}+2m_{\pi}^{2}-s}{2}$ (3)
$\displaystyle
y=\frac{\sqrt{s-4m_{\pi}^{2}}}{2\sqrt{s}}\lambda^{\nicefrac{{1}}{{2}}}(M^{2},m_{\Lambda}^{2},s),$
(4)

and
$\lambda(M^{2},m_{\Lambda}^{2},s)=(M^{4}+m_{\Lambda}^{4}+s^{2}-2M^{2}m_{\Lambda}^{2}-2M^{2}s-2m_{\Lambda}^{2}s)$.
$M$ is the mass of the $\Xi_{b}$ baryon, $m_{\Lambda}$ is the mass of the
$\Lambda^{0}$ and $m_{\pi}$ is the mass of the pions.

The matrix element for the weak-decay
$\Xi_{b}^{-}\to\Sigma^{\prime(-,0)}\pi^{(0,-)}$ is given by

${\cal
M}(\Xi_{b}\to\Sigma^{\prime}\pi)=-iq^{\pi}_{\mu}\bar{u}^{\mu}_{\scriptscriptstyle\Sigma^{\prime}}(a+b\gamma_{5})u_{\scriptscriptstyle\Xi_{b}},$
(5)

where, $u^{\mu}_{\scriptscriptstyle\Sigma^{\prime}}$ is the Rarita-Schwinger
spinor for the spin-$\nicefrac{{3}}{{2}}$ decuplet and hence a Lorentz index,
$q^{\pi}_{\mu}$ is the momentum of the $\pi$, and
$u_{\scriptscriptstyle\Xi_{b}}$ is the spinor of the $\Xi_{b}$. The two
coefficients $a$ and $b$ depend on the CKM elements and flavor structure
specific to the decay mode. It may be noted that $a$ and $b$ are related to
the $p$\- and $d$-wave decay amplitudes respectively. The $\Sigma^{\prime}$
baryon subsequently decays via strong-interaction to a $\Lambda^{0}$-baryon
and a pion with the matrix element for the decay being given by,

${\cal
M}(\Sigma^{\prime}\to\Lambda^{0}\pi)=ig_{\scriptscriptstyle\Sigma^{\prime}\Lambda\pi}q^{\pi}_{\mu}\bar{u}_{\Lambda}u^{\mu}_{\Sigma},$
(6)

where, $g_{\scriptscriptstyle\Sigma^{\prime}\Lambda\pi}$ is the invariant
coupling for the decay. The decay for the conjugate modes,
$\overline{\Xi}_{b}^{+}\to\overline{\Sigma}^{\prime(+,0)}\pi^{(0,+)}$ can be
described analogously except that it would be described by new coefficients
$\overline{a}$ and $\overline{b}$ that are related to $a$ and $b$ by reversing
the respective weak phases. The propagator of the $\Sigma^{\prime}(1385)$
baryon has the form [21]

$\displaystyle\Pi^{\mu\nu}(k)$
$\displaystyle=-\frac{(\not{k}+m)}{(k^{2}-m^{2}+im\Gamma)}\Big{(}g^{\mu\nu}-\frac{2}{3}\frac{k^{\mu}k^{\nu}}{m^{2}}$
$\displaystyle\qquad-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}+\frac{1}{3m}(k^{\nu}\gamma^{\mu}-k^{\mu}\gamma^{\nu})\Big{)},$
(7)

corresponding to that of a spin-$\nicefrac{{3}}{{2}}$ fermion with the four-
momentum $k$, mass $m$ and width $\Gamma$.

The matrix element for the two-step decay
$\Xi_{b}^{-}(q)\to\Sigma^{\prime-}[\to\Lambda^{0}(q_{1})\pi^{-}(q_{2})]\pi^{0}(q_{3})$
is given by

${\cal M}_{u}={\cal
M}\Big{(}\Xi_{b}^{-}(q)\to\Sigma^{\prime-}\big{[}\to\Lambda^{0}(q_{1})\pi^{-}(q_{2})\big{]}\pi^{0}(q_{3})\Big{)}=\\\
g_{\scriptscriptstyle\Sigma^{\prime}\Lambda\pi}\bar{u}(q_{1})(a^{\scriptscriptstyle-}+b^{\scriptscriptstyle-}\gamma_{5})\Pi^{\mu\nu}(q_{12})u(q)\;q_{3}^{\mu}q_{2}^{\nu},$
(8)

where $q_{ij}=q_{i}+q_{j}$ and $m$ is the mass of $\Sigma^{\prime}(1385)$
resonance. Similarly, the two-step decay $\Xi_{b}^{-}(q)\to\Sigma^{\prime
0}[\to\Lambda^{0}(q_{1})\pi^{0}(q_{3})]\pi^{-}(q_{2})$ is given by

${\cal M}_{t}={\cal M}\Big{(}\Xi_{b}^{-}(q)\to\Sigma^{\prime
0}\big{[}\to\Lambda^{0}(q_{1})\pi^{0}(q_{3})\big{]}\pi^{-}(q_{2})\Big{)}=\\\
g_{\scriptscriptstyle\Sigma^{\prime}\Lambda\pi}\bar{u}(q_{1})(a^{\scriptscriptstyle
0}+b^{\scriptscriptstyle
0}\gamma_{5})\Pi^{\mu\nu}(q_{13})u(q)\;q_{2}^{\mu}q_{3}^{\nu}.$ (9)

The matrix elements ${\cal M}_{u}$ and ${\cal M}_{t}$ are related by isospin.
The sum of the matrix element of the two contributing modes must be completely
Bose symmetric under the exchange of the two pions and may be written in an
explicitly symmetric form as

${\cal
M}\Big{(}\Xi_{b}^{-}\to\Sigma^{\prime}\big{[}\Lambda\pi\big{]}\pi\Big{)}=g_{\scriptscriptstyle\Sigma^{\prime}\Lambda\pi}\bar{u}(q_{1})\Big{[}(A_{e}+B_{e}\gamma_{5})\\\
\qquad\qquad\big{(}\Pi^{\nu\mu}(q_{12})+\Pi^{\mu\nu}(q_{13})\big{)}+(A_{o}+B_{o}\gamma_{5})\\\
\big{(}\Pi^{\nu\mu}(q_{12})-\Pi^{\mu\nu}(q_{13})\big{)}\Big{]}u(q)q_{2}^{\mu}q_{3}^{\nu}$
(10)

where $A_{e}$, $B_{e}$ and $A_{o}$, $B_{o}$ are the even and odd parts of the
amplitude under the exchange of the two pions and are given by

$\displaystyle A_{e,o}=$ $\displaystyle(a^{\scriptscriptstyle-}\pm
a^{\scriptscriptstyle 0})/2$ $\displaystyle B_{e,o}=$
$\displaystyle(b^{\scriptscriptstyle-}\pm b^{\scriptscriptstyle 0})/2.$ (11)

Figure 4:  Sample correlation-plots for the $p$-wave contributions for the
specified choice of parameters. Plotted are the logarithmic value of the
$p$-wave rates with an arbitrary scale that depends on $C$. Explicit
dependence of the some of the parameters on $p$-wave is suppressed for
simplicity. The contributing intermediate $\Sigma^{\prime}$ resonances are
depicted on the plots. The differences between the two plots arise only due
$C\\!P$ violation effects. Similar plots would arise from the $d$-wave
contributions.

The coefficients $a^{\scriptscriptstyle 0}$, $a^{\scriptscriptstyle-}$,
$b^{\scriptscriptstyle 0}$ and $b^{\scriptscriptstyle-}$ are related (see Fig.
2) to the topological amplitudes [19, 20], which after a phase rotation of
$e^{i\gamma}$ may be written as follows

$\displaystyle a^{\scriptscriptstyle 0}=$
$\displaystyle-\tfrac{1}{2\sqrt{3}}\big{(}C_{p}-P_{p}e^{-i\alpha}\big{)},$
$\displaystyle a^{\scriptscriptstyle-}=$
$\displaystyle-\tfrac{1}{2\sqrt{3}}P_{p}e^{-i\alpha}$ $\displaystyle
b^{\scriptscriptstyle 0}=$
$\displaystyle-\tfrac{1}{2\sqrt{3}}\big{(}C_{d}-P_{d}e^{-i\alpha}\big{)},$
$\displaystyle b^{\scriptscriptstyle-}=$
$\displaystyle-\tfrac{1}{2\sqrt{3}}P_{d}e^{-i\alpha},$ (12)

where $C_{p}$, $P_{p}$ are related to the topological amplitudes contributing
to the ${p}$-wave with analogous definitions for the ${d}$-wave. Here the
$p$-wave and $d$-wave refer to waves in the intermediate $\Sigma^{\prime}\pi$
state. Note that the phase rotation of $e^{i\gamma}$ does not alter any
physical observable. Hence,

$\displaystyle A_{e}=$ $\displaystyle-\tfrac{1}{4\sqrt{3}}C_{p}\equiv x_{p}$
$\displaystyle A_{o}=$
$\displaystyle\tfrac{1}{4\sqrt{3}}C_{p}-\tfrac{1}{2\sqrt{3}}P_{p}e^{-i\alpha}\equiv-
x_{\scriptscriptstyle p}+z_{p}e^{-i\alpha}e^{i\delta_{p}}$ $\displaystyle
B_{e}=$ $\displaystyle-\tfrac{1}{4\sqrt{3}}C_{d}\equiv x_{d}$ $\displaystyle
B_{o}=$
$\displaystyle\tfrac{1}{4\sqrt{3}}C_{d}-\tfrac{1}{2\sqrt{3}}P_{d}e^{-i\alpha}\equiv-
x_{d}+z_{d}e^{-i\alpha}e^{i\delta_{d}},$ (13)

where, $\delta_{(p,d)}$, is the strong phase difference between the penguin
and the tree contribution for the ${p}$\- and ${d}$-waves respectively.

The decay rate for the 3-body final state $\Lambda^{0}\pi^{-}\pi^{0}$ will
depend on the Mandelstam variables $t$ and $u$. This can easily be cast in
terms of $s$ and $\cos\theta$ using Eqs. (1), (2), (3) and (4). It is obvious
to conclude from Fig. 3 that under the exchange of the two pions,
$\theta\leftrightarrow\pi-\theta$. The odd (even) part of the amplitude under
the exchange of two pions must therefore be proportional to odd (even) powers
of $\cos\theta$. Since, Bose correlations are clearly depicted in plots
involving $\hat{s}=s/M^{2}$ and $\theta$, we refer to such plots as
correlation-plots. In Fig 4, we show sample correlation-plots for
$P_{p}/C_{p}=0.6$, $\delta_{p}=\pi/3$ and $\alpha=\pi/2$. Plotted are the
logarithmic values of the $p$-wave rates with an arbitrary scale that depends
on the choice of $P_{p}/C_{p}$ and the $p$-wave rate for the decay
$\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$. In our calculations we have used
$M=5797$ $\mathrm{Me\kern-1.00006ptV}$, $m=1385$
$\mathrm{Me\kern-1.00006ptV}$, $\Gamma=38$ $\mathrm{Me\kern-1.00006ptV}$,
$m_{\Lambda}=1115$ $\mathrm{Me\kern-1.00006ptV}$and $m_{\pi}=135$
$\mathrm{Me\kern-1.00006ptV}$. The effects of $C\\!P$ violation can be
observed as differences between the two plots corresponding to mode and
conjugate mode. The effects of Bose correlation are also obvious. The decay
$\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$ arises due to pure penguin (See Eqns.
(8) and (Measuring $\mathbf{CP}$ violating phase in beauty baryon decays)). If
there are no Bose correlation effects contributing to the decays, the decays
of $\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$ and
$\overline{\Xi}_{b}^{+}\to\overline{\Sigma}^{\prime+}\pi^{0}$ seen on the left
of the two plots in Fig. 4 would look identical. However, the Dalitz plots
show a marked difference between $\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$ and
$\overline{\Xi}_{b}^{+}\to\overline{\Sigma}^{\prime+}\pi^{0}$ on their
respective Dalitz plot, providing a smoking gun evidence of large Bose
correlation effects. This observation is fundamental to our new approach to
measure $C\\!P$ violation. We will show next that the observed distributions
along these narrow resonances provide enough information to extract all
theoretical parameters from experimental data, including the ${p}$\- and
${d}$-wave contributions respectively.

The numerator of the decay rate $N_{\Gamma}$ for both the mode and conjugate
mode are worked out to have the complicated form:

$N_{\Gamma}=\sum_{n=0}^{4}c_{n}(\hat{s})\cos
2n\theta+\sum_{n=0}^{3}d_{n}(\hat{s})\cos(2n+1)\theta,$ (14)

where, all masses and momenta are normalized to $M$, the $\Xi_{b}$ mass for
simplicity, and

$\displaystyle
c_{n}(s)=\big{(}f_{n}^{\scriptscriptstyle(1)}(\hat{s})|A_{e}|^{2}+f_{n}^{\scriptscriptstyle(2)}(\hat{s})|B_{e}|^{2}$
$\displaystyle\qquad\quad+f_{n}^{\scriptscriptstyle(3)}(\hat{s})|A_{o}|^{2}+f_{n}^{\scriptscriptstyle(4)}(\hat{s})|B_{o}|^{2}\big{)}$
(15) $\displaystyle
d_{n}(\hat{s})=\big{(}g_{n}^{\scriptscriptstyle(1)}(\hat{s})\text{Re}(A_{e}A_{o}^{*})+g_{n}^{\scriptscriptstyle(2)}(\hat{s})\text{Re}(B_{e}B_{o}^{*})$
$\displaystyle\qquad\quad+g_{n}^{\scriptscriptstyle(3)}(\hat{s})\text{Im}(A_{e}A_{o}^{*})+g_{n}^{\scriptscriptstyle(4)}(\hat{s})\text{Im}(B_{e}B_{o}^{*})\big{)}.$
(16)

The coefficients $f_{n}^{(i)}$ and $g_{n}^{(i)}$ are functions of $\hat{s}$
and are easily computed in terms of kinematic factors. For a given choice of
$\hat{s}$, $f_{n}^{(i)}$ and $g_{n}^{(i)}$ are just numbers. We henceforth
drop explicit dependence on $\hat{s}$, since our solutions are valid for all
$\hat{s}$. The numerator of the decay rate given in Eq. (14) can be fitted as
a function of $\theta$ to obtain coefficients $c_{n}$ and $d_{n}$. Having
obtained $c_{0}$, $c_{1}$, $c_{2}$ and $c_{3}$ it is obvious that
$|A_{e}|^{2}$, $|A_{o}|^{2}$, $|B_{e}|^{2}$ and $|B_{o}|^{2}$ can easily be
obtained. Similarly, $\text{Re}(A_{e}A_{o}^{*})$, $\text{Re}(B_{e}B_{o}^{*})$,
$\text{Im}(A_{e}A_{o}^{*})$ and $\text{Im}(B_{e}B_{o}^{*})$ can be solved
using $d_{0}$, $d_{1}$, $d_{2}$ and $d_{3}$. In order to solve for the
amplitudes and their interference, one must have a minimum of 8-bins in
$\theta$ and $\hat{s}$ involving both the resonances contributing to the
process. Experimental procedure used in an actual analysis can be more
refined. An identical analysis of the decay rate for the conjugate process
would enable us to solve for $|\overline{A}_{e}|^{2}$,
$|\overline{A}_{o}|^{2}$, $|\overline{B}_{e}|^{2}$, $|\overline{B}_{o}|^{2}$
and $\text{Re}(\overline{A}_{e}\overline{A}_{o}^{*})$,
$\text{Re}(\overline{B}_{e}\overline{B}_{o}^{*})$,
$\text{Im}(\overline{A}_{e}\overline{A}_{o}^{*})$ and
$\text{Im}(\overline{B}_{e}\overline{B}_{o}^{*})$. Having solved the values of
these amplitudes and their interference, our aim is to solve for the weak
phase $\alpha$ and the amplitudes $x_{{p,d}}$ and $z_{{p,d}}$ defined in Eq.
(Measuring $\mathbf{CP}$ violating phase in beauty baryon decays) and the
strong phases $\delta_{{p,d}}$. In order to obtain the solutions for the
$p$-wave parameters we define new intermediate observables $r_{i}$:

$\displaystyle r_{0}=$
$\displaystyle|A_{e}|^{2}=|\overline{A}_{e}|^{2}=x_{p}^{2}$ $\displaystyle
r_{1}=$
$\displaystyle|A_{o}|^{2}+|\overline{A}_{o}|^{2}=2x_{p}^{2}+2z_{p}^{2}-4x_{p}z_{p}\cos\delta_{p}\cos\alpha$
$\displaystyle r_{2}=$
$\displaystyle|A_{o}|^{2}-|\overline{A}_{o}|^{2}=-4\,x_{p}z_{p}\sin\delta_{p}\sin\alpha$
$\displaystyle r_{3}=$
$\displaystyle\text{Re}(A_{e}A_{o}^{*}-\overline{A}_{e}\overline{A}_{o}^{*})=2x_{p}z_{p}\sin\delta_{p}\sin\alpha$
$\displaystyle r_{4}=$
$\displaystyle\text{Im}(A_{e}A_{o}^{*}-\overline{A}_{e}\overline{A}_{o}^{*})=2x_{p}z_{p}\cos\delta_{p}\sin\alpha$
$\displaystyle r_{5}=$
$\displaystyle\text{Re}(A_{e}A_{o}^{*}+\overline{A}_{e}\overline{A}_{o}^{*})=-2x_{p}^{2}+2x_{p}z_{p}\cos\delta_{p}\cos\alpha$
$\displaystyle r_{6}=$
$\displaystyle\text{Im}(A_{e}A_{o}^{*}+\overline{A}_{e}\overline{A}_{o}^{*})=-2x_{p}z_{p}\sin\delta_{p}\cos\alpha.$
(17)

We find the solutions in terms of $r_{i}$ to be,

$\displaystyle x_{p}^{2}$ $\displaystyle=r_{0}$ $\displaystyle\tan\delta_{p}$
$\displaystyle=r_{3}/r_{4}$ $\displaystyle\tan\alpha$
$\displaystyle=-r_{3}/r_{6}=r_{2}/(2r_{6})$ $\displaystyle z_{p}^{2}$
$\displaystyle=\frac{(r_{3}^{2}+r_{4}^{2})(r_{3}^{2}+r_{6}^{2})}{4r_{0}r_{3}^{2}}$
(18)

Since, all the $r_{i}$ are not independent, we find the relations:

$\displaystyle r_{3}^{2}+r_{4}^{2}+r_{5}^{2}+r_{6}^{2}$
$\displaystyle=2r_{0}r_{1}$ $\displaystyle r_{3}r_{5}+r_{4}r_{6}$
$\displaystyle=r_{0}r_{2}$ (19) $\displaystyle r_{2}=-2r_{3}$ (20)

The solutions for the $d$-wave parameters can be obtained similarly. It is
noted that both the $p$\- and $d$\- wave contributions must result in the
measurement of the same weak phase $\alpha$. We hence have two independent
measurements of $\alpha$ corresponding to the two partial waves.

The observed Dalitz plot must have a more complicated structure with several
resonances, and one may wonder if our approach to measure the weak phase would
be possible. The contribution from heavier $\Sigma^{\prime}$ states have a
similar decay dynamics with the same weak phase $\alpha$, but the relevant
amplitudes and strong phases would differ. While these resonances are not a
cause for concern suitable binning cuts would easily remove their
contributions without losing relevant signal sample. Only the interference
with the decay mode
$\Xi_{b}^{-}\to\Lambda^{0}\rho^{-}\to\Lambda^{0}\pi^{-}\pi^{0}$, and modes
involving heavier $\rho$-like resonances, instead of $\rho$, need a closer
look. It is interesting to note that $\Xi_{b}^{-}\to\Lambda^{0}\rho^{-}$ has a
different $\hat{s}$ dependence in the overlap region and contributes only to
the odd part of the amplitude. Once data is available in several more
$\hat{s}$ bins, such interference effects can easily be isolated using the
Dalitz distribution [22]. One does not need to impose cuts to remove the
$\Xi_{b}^{-}\to\Lambda^{0}\rho^{-}$ contributions.

We have demonstrated that Bose correlations arise from two intermediate decays
$\Xi_{b}^{-}\to\Sigma^{\prime 0}\pi^{-}$ and
$\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$ contributing to the final state
$\Xi_{b}^{-}\to\Lambda^{0}\pi^{-}\pi^{0}$. Similar Bose correlations arise in
the decay to conjugate final state
$\overline{\Xi}_{b}^{+}\to\overline{\Lambda}^{0}\pi^{+}\pi^{0}$. We expliticly
show that the weak phase $\alpha$ can be measured using both even and odd
contributions to the amplitudes under pion exchange and comparing the mode and
conjugate-mode correlation-plots. Naively, one may be tempted to incorrectly
assume that Bose correlation effects are too small to be observed. However, we
have shown numerically and explicitly that such correlations are sizable
providing an unequivocal approach to measuring $C\\!P$ violation in baryon
decays. The significant difference between the pure penguin decays of
$\Xi_{b}^{-}\to\Sigma^{\prime-}\pi^{0}$ and
$\overline{\Xi}_{b}^{+}\to\overline{\Sigma}^{\prime+}\pi^{0}$ on their
respective Dalitz plot is a smoking gun evidence of large Bose correlation
effects. Our new approach to measure $C\\!P$ violation critically depends on
such Bose correlations. Interestingly, inexplicably large $C\\!P$ violation
effects are observed in the Dalitz plot of three body $B$-meson decays; we
conjecture that these arise due to Bose correlations on the Dalitz plot.

We thank Sheldon Stone and Marcin Chrząszcz for reading the manuscript and
providing valuable suggestions. The work of SR is supported by the grant
CRG/2019/000362, Science and Engineering Research Board, Government of India.

We dedicate this letter to the memory of Sheldon Stone. He went through our
draft even while he was in hospital and provided useful comments; such was his
love for physics. His forthright attitude and recognition of good physics won
support from many physicists. His presence will forever be missed.

## References

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  * [17] Consider the amplitude $A=a+be^{i\phi}$ which can be re-expressed as $A=a+a_{0}+b_{0}e^{i\phi_{0}}$ (where $a$, $b$, $a_{0}$ and $b_{0}$ are complex amplitudes on account of the strong phase). Note that the amplitudes for the mode and conjugate mode remain invariant even when the weak phase has been reparametrized $\phi\to\phi_{0}$ by also modifying $a\to a+a_{0}$ and $b\to b_{0}$. Since $a$, $b$ and $\phi$ are all unknown the phase $\phi$ cannot be uniquely defined. The amplitude for $\Xi^{-}\to\Sigma^{\prime-}\pi^{0}$ does not allow reparametrization in the defination of the weak phase (see Eq. (Measuring $\mathbf{CP}$ violating phase in beauty baryon decays)) enabling a solution for the weak phase.
  * [18] A. K. Giri, R. Mohanta and M. P. Khanna, Phys. Rev. D 65, 073029 (2002). Even though this method is a straight forward extension with less efficiecy than the approach used to measure $\gamma$ in meson decays, we suggest that $\gamma$ ought to be measured using baryons as well.
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  * [20] S. Roy, R. Sinha and N. G. Deshpande, Phys. Rev. D 102, 053007 (2020). It can be seen that the amplitude $P$ can be absorbed by suitably redefining $E_{2}$ and $E_{3}$ for all the $\Delta S=0,$ $\Lambda_{b}$ and $\Xi_{b}^{0}$ decays. Hence, re-parametrization invariance ensures that weak phases cannot be measured using $\Lambda_{b}$ or $\Xi_{b}^{0}$ decays.
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  * [22] It is well-known that contributions from two interfering resonances can be separated using the distribution of events on the Dalitz plot. A detailed discussion of such an analysis will be presented elsewhere.


# Bulk-boundary correspondence in disordered higher-order topological
insulators

Yu-Song Hu  School of Physical Science and Technology, Soochow University,
Suzhou, 215006, China  Yue-Ran Ding  Institute for Advanced Study, Soochow
University, Suzhou 215006, China  Jie Zhang  School of Physical Science and
Technology, Soochow University, Suzhou, 215006, China  Zhi-Qiang Zhang
<EMAIL_ADDRESS>School of Physical Science and Technology, Soochow
University, Suzhou, 215006, China  Chui-Zhen Chen<EMAIL_ADDRESS>Institute for Advanced Study, Soochow University, Suzhou 215006, China  School
of Physical Science and Technology, Soochow University, Suzhou, 215006, China

###### Abstract

In this work, we study the disorder effects on the bulk-boundary
correspondence of two-dimensional higher-order topological insulators (HOTIs).
We concentrate on two cases: (i) bulk-corner correspondence, (ii) edge-corner
correspondence. For the bulk-corner correspondence case, we demonstrate the
existence of the mobility gaps and clarify the related topological invariant
that characterizes the mobility gap. Furthermore, we find that, while the
system preserves the bulk-corner correspondence in the presence of disorder,
the corner states are protected by the mobility gap instead of the bulk gap.
For the edge-corner correspondence case, we show that the bulk mobility gap
and edge band gaps of HOTIs are no longer closed simultaneously. Therefore, a
rich phase diagram is obtained, including various disorder-induced phase
transition processes. Notably, a disorder-induced transition from the non-
trivial to trivial phase is realized, distinguishing the HOTIs from the other
topological states. Our results deepen the understanding of bulk-boundary
correspondence and enrich the topological phase transitions of disordered
HOTIs.

##  I Introduction

The bulk-boundary correspondence is one of the most important features of the
topological phase of matter. For conventional TIs, the bulk-boundary
correspondence is often referred to as bulk-edge correspondence indicating
that $d$ dimensions systems have gapless states on the ($d-1$)-dimensional
boundary. Recently, the higher-order topological insulators (HOTIs) are
introduced as a novel topological state HOTI1 ; HOTI2 ; HOTI3 ; HOTI4 ; HOTI5
; HOTI6 ; HOTI7 ; HOTI8 ; HOTI9 ; HOTI10 ; HOTI11 ; HOTI12 ; HOTI13 ; HOTI14 ;
HOTI15 ; HOTI16 ; HOTI17 ; HOTI18 ; HOTI19 ; HOTI20 ; HOTI21 ; HOTI22 ; HOTI23
; HOTI24 ; HOTI25 ; HOTI26 ; HOTI27 ; HOTI28 ; HOTI29 ; HOTI30 ; BHZmodel ;
HOTI31 ; HOTI32 ; HOTI33 ; HOTI34 ; HOTI35 ; HOTI36 ; HOTI37 ; HOTI38 ; HOTI39
; HOTI40 ; HOTI41 ; HOTI42 ; HOTI43 ; HOTI44 ; HOTI45 ; HOTI46 ; HOTI47 ;
HOTI48 ; HOTI49 ; HOTI50 ; HOTI51 ; HOTI52 ; HOTI53 ; HOTI54 ; HOTI55 .
Generally, the HOTIs require the existence of both bulk energy gap and the
non-trivial gaps of edge or surface statesHOTI10 ; HOTI11  to ensure the
existence of corner or hinge states. Accordingly, the concept of the bulk-
boundary correspondence for HOTIs is generalized to bulk-corner correspondence
and edge-corner correspondence TI1 ; TI2 ; HOTIbe1 ; HOTIbe2 . For example,
for a two-dimensional HOTIHOTI10  constructed by the quantum spin Hall
modelTI1 ; TI2 , the helical edge states are gapped out with mass domains
sitting at the edges of the sample, which guarantee the existence of the
corner states and the non-trivial quadrupole moment HOTI1 ; HOTI2 ; HOTIqxy1 ;
HOTIqxy2 ; HOTIqxy3 . If the bulk energy gap and edge band gaps of HOTIs are
closed simultaneously, the HOTIs are considered to fall into the bulk-corner
correspondence case TI1 ; TI2 . On the other hand, when the edge band gap of
HOTI is closed without bulk gap closing HOTIbe1 ; HOTIbe2 , the system belongs
to the edge-corner correspondence instead of bulk-corner correspondence cases.

Meanwhile, the study on the disorder-induced phase transitions of higher-order
topological phases also attracted great interest very recentlyHOTIq1 ; HOTIq2
; HOTIq3 ; HOTIq4 ; HOTId1 ; HOTId4 ; HOTId5 ; HOTId6 ; HOTId7 ; HOTId8 . The
disorder-induced phase transitions in HOTIsHOTIq1 ; HOTIq2  show distinct
features such as higher-order topological Anderson insulators (HOTAIs),
comparing to those in conventional topological phasesTAI0 ; TAI1 ; TAI2 ; TAI3
; TAI4 ; TAI5 ; TAI6 ; TAI7 ; TAI8 ; TAI9 . The numerical calculations
indicate that the energy gap closes and reopens by the disorder, and the
topological phase transitions are observedHOTIq1 ; HOTIq2 . In the study of
Yang et al.HOTIq2 , the existence of extended bulk states is discussed.
Furthermore, the presence of mobility edge is also clarified. In Li et al.’s
study, however, they pay more attention to the edge band gaps. The edge band
gaps are considered to be closed and reopenedHOTIq1 . They also notice that
the localization length along a specific direction diverges for samples under
open boundary conditions, similar to those in one-dimensional Su-Schrieffer-
Heeger modelsTAI0 .

However, previous studies mainly concentrated on the evolution of the bulk
energy gap and edge energy gap. Although the mobility gap for HOTIs has been
discussed, the topological invariant protected by the mobility gap is still
unclearTAI6 ; TAI7 ; TAI8 ; TAI9 . Furthermore, to demonstrate the influence
of disorder on the bulk-corner correspondence, it is of great value to study
the evolutions of mobility gaps and the correlated topological invariant. More
importantly, due to the breakdown of bulk-corner correspondence, the disorder-
induced gap closing and the related topological phase transitions of HOTI with
edge-corner correspondence should be abundant. It is essential to study the
disorder-induced gap closing of HOTIs, which will deepen the understanding of
the topological features of the disordered HOTIs.

In this paper, we study the disorder effects on the evolution of both edge and
bulk gaps, as well as the bulk-boundary correspondence in HOTIs. We
investigate both the bulk-corner correspondence and edge-corner correspondence
cases. For the bulk-corner correspondence case, we first demonstrate the
existence of mobility gaps. Further, the topological invariant named as spin
Chern numberCS1 ; CS2 ; CS3  $\mathcal{C}_{s}$ is revealed to characterize the
bulk states. By comparing the spin Chern number with the quadrupole moment, we
find bulk-corner correspondence of the HOTIs remains intact in the presence of
disorder. A phase diagram is given to elucidate the evolution of the mobility
gap by varying the disorder strength. For the edge-corner correspondence case,
the disorder-induced edge energy gap closing is achieved. Importantly, unlike
the previously discovered HOTAI, a phase transition from topological non-
trivial phases to trivial ones is identified due to the competition between
two renormalized parameters. In addition, a variety of gap-closing processes
with disorder strength $W\leq 5t$ (where the band renormalization dominates)
are obtained, which enriches the disorder-induced phase transitions of HOTIs.

The rest of this paper is organized as follows: In Sec. II, we present the
details of the model and the methods. In Sec. III, we concentrate on the gap
closing and phase transitions of the bulk-corner correspondence cases. In Sec.
IV, The edge-corner correspondence cases are considered. Finally, a brief
discussion and summary are presented in Sec. V.

##  II model and methods

Following previous studies, the Hamiltonian can be written as HOTI30 ; HOTIbe1
:

$\displaystyle\begin{split}\mathcal{H}(\textbf{k})&=[-m+t\cos k_{x}+t\cos
k_{y}]\tau_{z}\sigma_{z}\\\ &+\lambda\sin k_{x}\tau_{x}\sigma_{z}-\lambda\sin
k_{y}\tau_{y}\sigma_{z}\\\
&+\Delta(k_{x},k_{y})\tau_{0}\sigma_{x},\end{split}$ (1)

in the basis $(A_{\uparrow},A_{\downarrow},B_{\uparrow},B_{\downarrow})$ with
$A,B$ ($\uparrow,\downarrow$) representing the orbit (spin) degrees of
freedom. The Pauli matrices (identity matrix) $\tau_{x/y/z}$ ($\tau_{0}$) and
$\sigma_{x/y/z}$ ($\sigma_{0}$) act on orbit and spin space, respectively.
$m$, $\lambda$ and $t$ are model parameters and we set $\lambda=t$ throughout
the paper. Here $\Delta(k_{x},k_{y})=\Delta_{0}+\Delta_{x}\cos
k_{x}+\Delta_{y}\cos k_{y}$ with $\Delta_{x}=-\Delta_{y}$. If
$\Delta(k_{x},k_{y})=0$, the Hamiltonian is exactly the Bernevig-Hughes-Zhang
(BHZ) modelBHZmodel  and can describe a quantum spin-Hall effect with a pair
of helical edge modes. Generally, when $\Delta(k_{x},k_{y})\neq 0$ , the edge
helical states with opposite group velocities will couple and gapped out [see
Fig. 1(a)]. To be specific, the $(\Delta_{x}\cos k_{x}-\Delta_{x}\cos k_{y})$
can create the mass domain walls along the edge of the sample and lead to a
HOTI with corner states. Importantly, the bulk-corner correspondence exists
only when $\Delta_{0}=0$ and the Hamiltonian has the edge-corner
correspondence instead for $\Delta_{0}\neq 0$ HOTIbe1  [see appendix for more
details].

Figure 1: (Color online). Three different gaps for higher-order topological
insulators, which are edge band gap, bulk energy gap, and mobility gap. (b)
The schematic diagram of sample in real space with $N\times N$ primitive
cells. For clean samples, the edge band gap along $x$ and $y$ directions are
$f(m,\Delta_{0},\Delta_{x})$ and $f(m,\Delta_{0},\Delta_{y})$, respectively.
$f(m,\Delta_{0},\Delta_{x/y})$ is a function of $m$, $\Delta_{0}$ and
$\Delta_{x/y}$. When $f(m,\Delta_{0},\Delta_{x})>0$ and
$f(m,\Delta_{0},\Delta_{y})<0$, the edge band gaps construct the mass domain
walls. The red and blue arrows are the helical edge states. We concentrate on
the edge band gap along $y$ direction.

Before starting calculation, we define the quadrupole momentHOTIq1 ; HOTIq2 :

$Q_{xy}=\frac{1}{2\pi}Im\\{log[det(U^{\dagger}\widehat{q}U)\sqrt{det(\widehat{q}^{\dagger})}]\\},$
(2)

where $\widehat{q}\equiv exp[i2\pi\widehat{x}\widehat{y}/(N^{2})]$. $N$ is the
sample size and $U$ is constructed by the eigenvalues of occupied states with
the projection operator $P_{r}=UU^{\dagger}$. $\widehat{x}$ ($\widehat{y}$) is
the coordinate operator along $x$ ($y$) direction. Note that $Q_{xy}$ is
quantized since the Hamiltonian $\mathcal{H}$ has the chiral symmetry
$\mathcal{S}=\tau_{0}\sigma_{y}$, i.e.
$\mathcal{S}\mathcal{H}(\textbf{k})\mathcal{S}^{-1}=-\mathcal{H}(\textbf{k})$.

In addition to the quadrupole moment $Q_{xy}$, we will use the spin Chern
number $\mathcal{C}_{s}=|\mathcal{C}_{+}-\mathcal{C}_{-}|/2$ CS1 ; CS2 ; CS3 .
Here the Chern number $\mathcal{C}_{\pm}$ for different spin is given byHOTId4
; HOTId7 ; CS1 :

$\mathcal{C}_{\pm}=-\frac{2\pi i}{N_{x}N_{y}}\sum_{n,\alpha}\langle
n,\alpha|P_{\pm}[-i[\widehat{x},P_{\pm}],-i[\widehat{y},P_{\pm}]]|n,\alpha\rangle,$
(3)

$|n,\alpha\rangle$ is the eigenvector for site $n$ and orbit $\alpha$.
$P_{\pm}=U_{\pm}U_{\pm}^{\dagger}$ are the projection operators for different
spin components, and $U_{\pm}$ are determined from the eigenvectors of
$P_{r}(\tau_{0}\sigma_{z})P_{r}$. Although the spin up ($\uparrow$) and spin
down ($\downarrow$) parts of the Hamiltonian are coupled by
$\Delta(k_{x},k_{y})$, the spin Chern number is still well defined when the
eigenvalue of $P_{r}(\tau_{0}\sigma_{z})P_{r}$ has a gapCS1 . $C_{s}=1$
suggests the existence of edge states and the non-trivial features of bulk
states.

At last, the Anderson disorder is introduced as HOTId7
$H_{n}^{w}=\operatorname{diag}\\{\varepsilon_{n}^{1},\varepsilon_{n}^{2}\\}\otimes\sigma_{z}$,
to preserve the symmetry $\mathcal{S}H^{w}_{n}\mathcal{S}^{-1}=-H_{n}^{w}$.
The random onsite potential $\varepsilon_{n}^{1}$, $\varepsilon_{n}^{2}$ for
site $n$ satisfy the uniform distribution
$\varepsilon_{n}^{1},\varepsilon_{n}^{2}\in[-W/2,W/2]$ with the disorder
strength $W$. For a weak disorder strength, the renormalization of model
parameters can be evaluated via the self-consistent Born approximation (SCBA)
of self-energy HOTId7 ; HOTIq1 ; HOTIq2

$\Sigma^{r}=\sum_{i=1}^{2}\frac{W^{2}}{48\pi^{2}}\iint_{BZ}d^{2}\textbf{k}\gamma_{i}[E_{F}+i0^{+}-\mathcal{H}(\textbf{k})-\Sigma^{r}]^{-1}\gamma_{i}.$
(4)

with $\gamma_{1,2}=(\tau_{0}\pm\tau_{z})\otimes\sigma_{z}$/2. The integral is
on the first Brillouin zone and $E_{F}$ is the Fermi energy.

Figure 2: (Color online). (a), (b), and (c) are the quadrupole moment
$Q_{xy}$, the eigenvalue $E_{2N^{2}+1}$ and DOS versus disorder strength $W$,
respectively. The parameters are set as $m=t$, $\Delta_{x}=-\Delta_{y}=0.5$,
$\Delta_{0}=0$. In our numerical calculations, a square sample with size $N$
is used. It has $4N^{2}$ eigenvalues in total. The bulk gap equals to
$2E_{2N^{2}+1}$. The average DOS and the geometric average DOS are
$\rho_{ave}$ and $\rho_{type}$, respectively. The color of the curves are the
same as the labels for (c) and (f). The blue (red) line for inset of (b) shows
$E_{2N^{2}+1}$ versus $W$ for open (periodic) boundary condition, which shows
the evolution of bulk (edge) gap. OB and PB corresponds to the open and
periodic boundary condition, respectively. (d)-(f) are the same with (a)-(c),
except that $m=2.05t$. The dashed lines roughly shows the bulk energy gap
closing points. The sample size is $N=40$.

##  III HOTI with bulk-corner correspondence

Different from the traditional two-dimensional topological insulators, the
HOTIs could have the bulk-corner correspondence or edge-corner
correspondenceHOTIbe1 . In this section, we consider $\Delta_{0}=0$ in Eq. (1)
and the Hamiltonian of the HOTI has the bulk-corner correspondence.

###  III.1  mobility gap and the related topological invariant

For the clean HOTIs with bulk-corner correspondence, the evolutions of the
bulk energy gap are correlated with the edge band gaps and the topological
phase transitions. To be specific, the bulk energy gap closing determines the
topological phase transition of the HOTIs in the same way as in the
traditional topological insulators. However, when the disorder is considered,
the bulk gap will be replaced by the mobility gap for both the 2D traditional
topological insulatorsTAI6 ; TAI7 ; TAI8 ; TAI9  and HOTIsHOTIq1 .

To demonstrate the existence of the mobility gap of HOTI, we compare the
$Q_{xy}$ with both the bulk energy gap and the edge energy gap and plot them
as a function of the disorder strength $W$ in Fig. 2. The $Q_{xy}$ and the
bulk energy gap are evaluated utilizing a square sample with size $N$ [see
Fig. 1(b)] under periodic boundary conditions in both $x$ and $y$ directions.
On the other hand, the edge energy gap is calculated with open and periodic
boundary conditions in $x$ and $y$ directions, respectively. Because the whole
spectrum has $4N^{2}$ eigenvalues in total as well as symmetry $\mathcal{S}$,
The energy gap is determined by the $(2N^{2}+1)$-th eigenvalue $E_{2N^{2}+1}$
and the energy gap closes when $E_{2N^{2}+1}=0$.

In Figs. 2(a) and (d), the system starts from a HOTI phase, and a normal
insulator exists in the clean limit. For a weak disorder, one can see a
disorder-induced topological phase transition from a normal insulator to a
HOTI phase in Fig. 2 (d) and the bulk energy gap closes in Fig. 2 (e), which
correspond to the creation of the HOTAI. When the disorder is strong enough,
bulk energy gaps are closed for both HOTIs and HOTAIs in Figs .2(b) and (e).
Notably, the $Q_{xy}$ is still quantized when the bulk energy gap closes at
$W\approx 4t$. This means that the topological phase transition from HOTI to
normal insulator is independent of the bulk energy gap when disorder $W\geq
4t$.

In order to clarify the localization properties of bulk states, we calculate
the average density of states (DOS) and the geometric average DOSTAI6 ; TAI7 ;
TAI8  given by

$\displaystyle\begin{split}\rho_{ave}(E)=\sum_{i=1}^{L}\rho_{i}/L,~{}~{}\rho_{type}(E)=\exp[\sum_{i=1}^{L}\ln\rho_{i}/L].\end{split}$
(5)

where $L$ is the number of the considered sites.
$\rho_{i}(E)=-\frac{1}{\pi}Im[G^{r}_{i,i}(E)]$ with
$G^{r}(E)=[E+i0^{+}-H]^{-1}$, and $H$ is the real space Hamiltonian. In Figs.
2(c) and (f), $\rho_{ave}$, $\rho_{type}$ and the ratio
$\rho_{type}/\rho_{ave}$ are also plotted as a function of $W$. Generally,
$\rho_{type}$ is smaller than $\rho_{ave}$, $\rho_{type}/\rho_{ave}$ tends to
be finiteTAI6 ; TAI7 ; TAI8  for extended states and $\rho_{type}/\rho_{ave}$
will approach zero when the states are much more localized. In Fig. 2(c) when
$W>4t$, the density of state $\rho_{ave}\neq 0$ meaning that the bulk energy
gap disappears. In this circumstance, the quantized quadrupole moment
$Q_{xy}=1/2$ is protected by the mobility gap instead. By further increasing
disorder strength to $W\approx 6t$, $\rho_{type}/\rho_{ave}$ shows a peak and
$Q_{xy}$ decreases to about $1/4$ [half of its quantized value], meaning that
the mobility gap is closed with the emergence of delocalized states at $E=0$.

Figure 3: (Color online). The spin Chern number $C_{s}$ and quadrupole moment
$Q_{xy}$ versus disorder strength $W$. (a) $m=t$, (b) $m=2.05t$. The sample
size is $N=40$. (c) and (d) are the scaling of $Q_{xy}$ and $C_{s}$,
respectively. Disorder strength is fixed at $W=4t$ with $m=t$. Other
parameters are the same as those in Fig. 2.  Figure 4: (Color online). The
normalized localization length $\lambda_{y}/N$ versus energy $E$ for different
sample sizes $N$. The disorder strength is: (a) $W=2t$, (b) $W=3.5t$, (c)
$W=4t$, (d) $W=5t$, (e) $W=5.5t$, (f) $W=6t$, (g) $W=7t$, (h) $W=10t$,
respectively. Parameters are set as $m=t$, $\Delta_{0}=0$, and
$\Delta_{x}=0.5t$. The periodic boundary condition is adopted for (a)-(h).

In two dimensions, all the states tend to be localized in the presence of
disorder, and the emergence of delocalized states is generally accompanied by
topological invariants MBL1 ; MBL2 . Therefore, in the following, we explore
the topological invariants that characterize the topological phase in the
mobility gap. To start with, we plot the quadrupole moment $Q_{xy}$ as shown
in Fig. 3(c). Unfortunately, we find that the quantization of $Q_{xy}=0.5$ is
destroyed when the Fermi energy slightly deviates from $E=0$. That’s because
the quadrupole moment $Q_{xy}$ is protected by the symmetry
$\mathcal{S}=\tau_{0}\sigma_{y}$, which is preserved only when the system is
half-filled, namely $E=0$.

Next, we consider the spin Chern number $\mathcal{C}_{s}=1$ to distinguish the
HOTI from the trivial ones, because Eq. (1) can be viewed as the quantum spin
Hall state with gapped edge states and the bulk-edge correspondence also holds
(see appendix). Different from the Chern number, $\mathcal{C}_{s}=1$ can
determine the existence of gapped edge states, which are essential for HOTIs.
As illustrated in Figs. 3(a) and (b), the spin Chern number $\mathcal{C}_{s}$
perfectly fits the evolution of $Q_{xy}$ by increasing disorder strength. The
quadrupole moment equals to $0.5$ when $\mathcal{C}_{s}=1$. Besides,
$\mathcal{C}_{s}$ also captures the existence of HOTAI, shown in Fig. 3(b).
Thus, it is appropriate to adopt the spin Chern number to determine HOTIs
within the mobility gap. Regarding
$\mathcal{S}\mathcal{H}\mathcal{S}^{-1}=-\mathcal{H}$, the system is symmetric
about $E=0$, and we only pay attention to $E\geq 0$ in all our calculations.
In Fig. 3(d), we do finite-size scaling of $\mathcal{C}_{s}$ and find
$\mathcal{C}_{s}\rightarrow 1$ for energy roughly within $E\in[0,t]$. This
confirms the quantization of $C_{s}$ within the entire mobility gap of the
disordered HOTIs.

Generally, the $C_{s}=1$ only ensures the existence of gapped edge states
within the mobility gap. Nevertheless, due to the overlaps between $Q_{xy}$
and $\mathcal{C}_{s}$ curves [see Figs. 3(a) and (b)], it is appropriate to
conclude that $\mathcal{C}_{s}=1$ accompanied with $Q_{xy}=0.5$ at $E=0$
should also ensure the existence of corner states in the mobility gap. This
interprets why the HOTIs could have the bulk-corner correspondence.

###  III.2 mobility gap closing and phase transition

In order to study the evolution of the mobility gap versus disorder, we
calculate the normalized localization lengthNL0 ; NL1 ; NL2 ; NL3 ; NL4 ; NL5
$\lambda_{y}/N$ along $y$ direction with $m=t$. The localized and delocalized
states can be distinguished by the finite-size scaling analysis of
$\lambda_{y}/N$. $\lambda_{y}/N$ decreases (increases) for localized
(extended) states with $N$ increasing, while $\lambda_{y}/N$ is independent of
$N$ at critical points. For small disorder strength $W$ in Figs. 4(a)-(c), the
system shows a mobility gap near $E=0$ with $\lambda_{y}/N$ decreasing, and
then it encounters a critical point. Notably, for $W=4t$, the critical point
is at $E\approx t$, which is consistent with the scaling of $\mathcal{C}_{s}$
in Fig. 3(d).

With an increase of $W$, the mobility gap decreases and eventually disappears
when $W\approx 7t$ [see Figs. 4(d)-(h)], where all the states are localized
for the entire energy range. Note that similar results can also be found for
HOTAI with $m=2.05t$, as shown in the appendix.

Figure 5: (Color online). (a) the scaling of $C_{s}$ versus $W$ with $E=0$.
(b) is the same with (a), expect for $Q_{xy}$ instead. (c) The plot of
mobility edge versus disorder strength $W$. The blue squares are obtained with
the help of localization length. The red dashed line and the red square are
determined by (a). $m=t$, $\Delta_{x}=0.5t$, and $\Delta_{0}=0$ for all the
plots.

To further demonstrate that the HOTI becomes a trivial insulator when the
mobility gap closes, we perform the scaling analysis of the spin Chern number
and quadrupole moment under different disorder strengths. As shown in Figs.
5(a) and (b), $\mathcal{C}_{s}\rightarrow 1$ in the thermodynamic limit when
$W<6.8t$, while $Q_{xy}$ increases monotonously with the increase of $N$ in
this disorder region. On the other hand, when $W>6.8t$, both $Q_{xy}$ and
$\mathcal{C}_{s}$ decrease with increasing $N$, and they tend to approach zero
for larger $N$. These results are in accordance with those in Fig. 3(a), where
the $\mathcal{C}_{s}$ and $Q_{xy}$ curves almost overlap.

We summarize the above results in the phase diagram in Fig. 5(c). The blue
squares are obtained by scaling of the normalized localization length in Fig.
4, while the red square is obtained with the help of Fig. 5(a) and (b). One
can see the HOTAIs (with $Q_{xy}=0.5$) and the trivial insulator phase (with
$Q_{xy}=0.0$) are separated by the mobility edge. Therefore, we conclude that
the HOTIs or HOTAIs are protected by the mobility gap in the presence of
disorder.

We close this section by discussing the relation between the edge band gap and
the existence of corner states. The inset of Fig. 2(b) shows the edge band gap
and bulk band gap are closed simultaneously at $W\approx 4t$. However,
$Q_{xy}\approx 0.5$ is still quantized in this case. Therefore, it is
appropriate to deduce that the edge band still has the non-trivial mass domain
walls to host the corner states. Meanwhile, the mass-domain walls induced by
edge gaps are filled with localized states. At present, it is difficult to
clearly demonstrate whether these edge gaps are closed or not due to the
localized states. A careful characterization of edge states under strong
disorder requires further investigation.

##  IV HOTI with edge-corner correspondence

This section focuses on HOTIs with edge-corner correspondence, where the bulk-
corner correspondence breaks down. In particular, we pay attention to the weak
disorder strength cases so that the SCBA can describe the disorder-induced
band renormalization.

###  IV.1 edge band gap closing

Figure 6: (Color online). (a) and (b) $Q_{xy}$ and the bulk energy gap versus
$\Delta_{0}$ and disorder strength $W$. The dashed lines are
$\Delta_{0}=0.4t$. (c) shows the typical plot with $\Delta_{0}=0.4t$.
(c1)-(c3) $C_{s}$, $Q_{xy}$, bulk gap and edge band gap evolution with the
increasing of $W$. The parameters are $m=t$ and $\Delta_{x}=0.5t$. The sample
size is $N=40$. (d) A schematic plot of gap evolution by increasing disorder
strength.

Next, we show that the bulk-corner correspondence breaks down for the
Hamiltonian in Eq. (1) with $\Delta_{0}\neq 0$. We plot the $Q_{xy}$ and the
bulk energy gap as a function of $W$ and $\Delta_{0}$ in Fig. 6(a) and (b),
respectively. The quantized quadrupole moment $Q_{xy}=0.5$ (the HOTIs) region
decreases with $\Delta_{0}$ increasing in Fig. 6(a), while the bulk energy gap
is independent of $\Delta_{0}$ as shown in Fig. 6(b). This is inconsistent
with the fact that the bulk energy gap of clean samples is independent of
$\Delta_{0}$ for such cases. Therefore, the bulk-corner correspondence no
longer holds.

To gain further insight into the edge-corner correspondence of the system, we
compare the spin Chern number $\mathcal{C}_{s}$ with the quadrupole moment
$Q_{xy}$ and plot them against $W$ with fixed $\Delta_{0}=0.4t$ as shown in
Fig. 6(c1). Notably, the spin Chern number $\mathcal{C}_{s}\approx 1$ is more
robust than the quadrupole moment $Q_{xy}=0.5$, and it remains a quantized
value for $W<5t$. These results are distinct from the bulk-corner
correspondence cases where $C_{s}$ and $Q_{xy}$ curves overlap as shown in
Figs. 3(a) and (b). Furthermore, we show the bulk and edge energy gaps in
Figs. 6(c1) and (c2), to compare $\mathcal{C}_{s}$ and $Q_{xy}$ with the two
energy gaps. It is clear that the robustness of $Q_{xy}$ is mainly determined
by the edge band gap closing at $W\approx 3t$, while $\mathcal{C}_{s}$ remains
quantized until the bulk energy gap closes at $W\approx 4.5t$. This justifies
the fact that $Q_{xy}=0.5$ results from the mass domain walls of edge modes
while $\mathcal{C}_{s}$ is protected by the bulk gap.

Now we come to summarize the gap closing process for the edge-corner
correspondence case in Fig. 6(d). In the clean limit, the system is
topological non-trivial, with $\mathcal{C}_{s}=1$ and $Q_{xy}=0.5$ that lead
to helical edge modes in the bulk gap and corn states in the edge gap. By
turning on $W$, the edge gap fills up and closes, destroying the quantized
quadrupole moment, i.e., $Q_{xy}\neq 0.5$. Further increasing $W$, the bulk
energy gap is closed while $\mathcal{C}_{s}$ is still quantized because of the
mobility bulk gap. At last, both $\mathcal{C}_{s}$ and $Q_{xy}$ equal to zero,
and the system is an Anderson insulator when the mobility gap is closed. We
note that, because the bulk and edges gaps close simultaneously in Sec. III
with $\Delta_{0}=0$, $\mathcal{C}_{s}$ and $Q_{xy}$ curves overlap, and the
system recovers the bulk-corner correspondence.

Figure 7: (Color online). (a) shows $Q_{xy}$ versus disorder strength $W$. (b)
and (c) show $\lambda_{y}$ versus $W$ for open boundary and periodic boundary
conditions, respectively. (d) is the typical plot of the averaged wavefunction
$|\psi(i_{x}=1,i_{y})|^{2}$ with $i_{y}\in[1,N]$ for different disorder
strength. The open boundary condition is adopted with $\Delta_{0}=0.4t$,
$\Delta_{x}=0.5t$, $m=t$.

At last, we evaluate the localization length $\lambda_{y}$ along $y$ direction
with periodic and open boundary conditions to accurately determine the edge
and bulk energy gap closing points, respectively. In Fig. 7(b), the
localization length has a peak sitting at $W\approx 3.1t$ under the open
boundary condition. Besides, the position is independent of the sample size.
On the other hand, such a peak disappears in the periodic boundary cases [see
Fig. 7(c)], which implies that the peak shown in Fig. 7(b) should be related
to the features of edge statesHOTIq1 . Next, we compare $\lambda_{y}$ with the
scaling of $Q_{xy}$, as shown in Fig. 7(a). $Q_{xy}$ gradually loses
quantization near the peak at $W\approx 3.1t$, where $Q_{xy}$ almost decreases
to half of its quantized value. Moreover, we plot the disorder-averaged wave
functions of corner states along $y$ direction $|\psi(i_{x}=1,i_{y})|^{2}$
against $W$ with $i_{y}\in[1,N]$ explicitly. Figure 7(c) clearly shows that
the corner states spread into the edge along direction $y$ and disappear when
$W\approx 3.1t$. Thus, we conclude that the sharp peak of $\lambda_{y}$ can be
used to accurately determine the edge band gap closing point where the corner
states disappear due to the edge-corner correspondence.

###  IV.2 gap closing based on SCBA

Figure 8: (Color online). (a) $\ln(\lambda_{y})$ versus $(W,\Delta_{0})$.
$\lambda_{y}$ is the localization length along $y$ direction. The square and
dashed line are obtained by SCBA. The sample size is $N=8$. (b) and (c)
renomalized $m_{r}$ and $\Delta_{0r}$ versus $\Delta_{0}$ for different
disorder strength $W$, respectively. (d) The schematic diagram of the path in
the phase diagram. Parameters are the same as those in Fig. 7.

To elucidate the edge band gap closing under disorder, we evaluate the
disorder-induced band renormalization using the SCBA method. At first glance,
it seems to contradict the previous theory of disorder-induced topological
Anderson insulatorsHOTIq1 ; HOTIq2  since gap closing induces a phase
transition from $Q_{xy}=0.5$ (non-trivial) to $0$ (trivial). In the following,
we will clarify it by showing that the renormalization to both topological
mass $m$ and $\Delta_{0}$ term can give rise to a much richer phase diagram.

Generally, the topological invariants $C_{s}$ and $Q_{xy}$ of the Hamiltonian
$\mathcal{H}({\bf k})$ are determined by $m$, $\Delta_{0}$ and $\Delta_{x,y}$.
In particular, the edge gap in $x$ ($y$) direction is determined by a function
$\bar{\Delta}_{x,y}=f(m,\Delta_{0},\Delta_{x,y})$. When the gap
$\bar{\Delta}_{x}$ and $\bar{\Delta}_{y}$ have opposite sign, there are corner
states near the domain walls, giving rise to $Q_{xy}=0.5$. Therefore, we focus
on the renormalization of $\Delta_{0}$ and $m$ in the following. In the
presence of disorder, the band structure is normalized by the disorder-induced
self-energy $\Sigma^{r}$ according to the SCBA. $\Sigma^{r}$ can be decomposed
as: $\Sigma^{r}=\sum_{ij}\Sigma^{r}_{ij}\tau_{i}\sigma_{j}$ with
$i,j=0,x,y,z$. The mass $m$ and $\Delta_{0}$ are renormalized into
$m_{R}\equiv m+m_{r}=m-\Sigma^{r}_{zz}$ and
$\Delta_{0R}\equiv\Delta_{0}+\Delta_{0r}=\Delta_{0}+\Sigma^{r}_{0x}$,
respectively. In Fig. 8(a), the phase boundary predicted by SCBA is perfectly
fitted to gap-closing points indicated by the maximum localization length.
This suggests that the phase transition from non-trivial to trivial can be
described by the disorder-induced self-energy $\Sigma^{r}$. It is important to
notice that the disorder-induced renormalization suggests $m_{r}<0$ and
$\Delta_{0r}>0$ for various $\Delta_{0}$ when $\Delta_{0}\neq 0$, shown in
Figs. 8(b) and (c).

Now we can explain the disorder-induced phase transition from $Q_{xy}=0.5$ to
$Q_{xy}=0$. Figure 8(d) shows the phase diagram in the clean limit with phase
boundaries determined by the edge band gap closing and spin Chern number [see
appendix for more details]. If we start from the red square (with
$\Delta_{0}=0.4t$ and $m=t$) in Fig. 8(d), $m$ is renormalized to $m_{R}<m$,
i.e., the red square shifts to the white square, and the sample tends to be
more topological. It is consistent with the previous studies of disorder-
induced topological Anderson insulators. However, since $\Delta_{0}\neq 0$,
$\Delta_{0}$ is also renormalized along the vertical direction so that
$\Delta_{0R}$ increases. In addition, $\Delta_{0r}$ is larger than $m_{r}$
when $W\rightarrow 5t$, as shown in Fig. 8(b) and (c). The white square
continuously shifts to the black square by crossing the phase boundary when
the disorder is strong enough. Therefore, a phase transition from $Q_{xy}=0.5$
to $Q_{xy}=0$ happens with $\mathcal{C}_{s}=1$ unchanged. These predictions
are consistent with the results shown in Fig. 6(c). Furthermore, $m_{r}$ is
almost independent of $\Delta_{0}$, while $\Delta_{0r}$ increases with an
increase of $\Delta_{0}$. Thus, the critical disorder strength between
$Q_{xy}=0.5$ and $Q_{xy}=0$ decreases for larger $\Delta_{0}$. This explains
what we observe in Fig. 6(a).

Figure 9: (Color online). (a) The flow under different parameters with
disorder strength $W\in[0,5t]$. The black (red) squares correspond to $W=0$
($W=5t$). The direction of the arrows suggests that disorder strength $W$
increases. All the curves are obtained based on the SCBA calculations. The
arrows with dashed line are the three conditions already known. Four solid
lines are four possible gap closing cases shown in (b)-(e). Parameter is set
as $\Delta_{x}=0.5t$. The phase boundary is $m=2t-2\Delta_{0}$. The sample
size is $N=30$.

In the following, let us show that the renormalization to both $m$ and
$\Delta_{0}$ can give rise to much richer phase transitions than the
previously discovered TAI. As shown in Fig .9(a), we plot some possible paths
in ($m$, $\Delta_{0}$) space. Similar to Fig. 8(d), the red square and the
black square correspond to the starting point ($W=0$) and the final point
(W=5t), respectively. The solid and dashed lines connecting the red and black
squares are paths obtained by SCBA. The arrow suggests the renormalization
direction as the disorder strength $W$ increases.

Let us verify the three dashed lines (that starting from $\Delta_{0}=0$ and
$\Delta_{0}=0.4$) in Fig. 9(a), which are the three paths studied in previous
sections. For $\Delta_{0}=0$, two different renormalization paths are given
for $m=t$ and $m=2.05t$. They are correlated with Figs. 3(a) and (b) as
follow. $Q_{xy}=0.5$ and $C_{s}=1$ are unchanged for Fig. 3(a), while the
trivial one ($Q_{xy}=0$ and $C_{s}=0$) is transformed into nontrivial with
$Q_{xy}=0.5$ and $C_{s}=1$ for Fig. 3(a). These two paths are traditional
since $\Delta_{0}=0$, and only $m$ is renormalized. On the other hand, for the
path starting from $\Delta_{0}=0.4t$ and $m=t$ as discussed in Fig. 9(a),
$Q_{xy}=0.5$ decreases to zero with $C_{s}=1$ which corresponds to the case in
Fig. 6(c)

In addition, four new paths marked with the solid lines are shown in Fig.
9(a). First, the yellow path suggests that no phase transition exists, in
agreement with the plot of $C_{s}$ and $Q_{xy}$ in Fig. 9(b). Then, even when
the starting point moves close to the phase boundary [see solid blue line in
Fig. 9(a)] the HOTAI is not available for larger $\Delta_{0}$ as verified in
Fig. 9(c). Nevertheless, figure 9(d) shows that rich phase transitions are
possible as indicated by the magenta line in Fig. 9(a). The spin Chern number
is changed $\mathcal{C}_{s}=0\rightarrow 1$ with the renormalized topological
mass $m_{R}\approx 2t$ and then the edge band gap is closed with
$Q_{xy}\approx 0.5$ [see appendix for more details]. At last, we find only
$\mathcal{C}_{s}=0\rightarrow 1$ for the case shown in Fig. 9(e).

##  V discussion and summary

In section III, we have discussed the existence of mass-domain-walls induced
by edge gaps for bulk-corner correspondence case with $\Delta_{0}=0$, since
$Q_{xy}\rightarrow 0.5$ for the strong disorder strength. However, whether
such edge gaps are closed or not is still unclear because of the existence of
the localized bulk states. Moreover, in section IV, when the bulk-corner
correspondence breaks down by disorder for $\Delta_{0}\neq 0$, the edge band
gap is closed and the mobility gap is still alive. Due to the specific
characteristics of HOTIs without bulk-corner correspondence, the gap closing
of HOTIs under strong disorder could be more complicated. These problems
deserve further investigation.

Figure 10: (Color online). The normalized localization length $\lambda_{y}/N$
versus energy $E$ for different sample sizes $N$. The disorder strength is:
(a) $W=2t$, (b) $W=3t$, (c) $W=4t$, (d) $W=5t$, (e) $W=6t$, (f) $W=6.5t$, (g)
$W=7t$, respectively. (h) shows the summation of the critical points.
Parameters are set as $m=2.05t$, $\Delta_{0}=0$, and $\Delta_{x}=0.5t$. The
periodic boundary condition is adopted.

In summary, we studied the gap closing for HOTIs with or without bulk-corner
correspondence. For the bulk-corner correspondence case, we found that the
bulk-corner correspondence still holds after considering the disorder effect.
The existence of the mobility gap was also revealed. Furthermore, we
demonstrated that the mobility gap closing induces the phase transition from
HOTIs to Anderson insulator, and the quadrupole moment $Q_{xy}$ approaches
zero under strong disorder correspondingly. We concentrated on the weak
disorder strength regions for the edge-corner correspondence case, where the
disorder-induced band renormalization dominates. Importantly, the disorder-
induced phase transition from $Q_{xy}=0.5$ to $Q_{xy}=0$ was obtained. This
transition originates from the disorder-induced renormalization of both $m$
and $\Delta_{0}$, distinct from the HOTAI cases. We found several phase
transitions based on the renormalization, and the corresponding gap closing
was also clarified.

##  VI ACKNOWLEDGEMENT

We are grateful to Qiang Wei, Hongfang Liu, and Hua Jiang for fruitful
discussions. This work was supported by NSFC under Grant N0. 11974256, and the
NSF of Jiangsu Province under Grant N0. BK20190813. Z.Q.Z. was supported by
National Basic Research Program of China (Grant No. 2019YFA0308403), NSFC
under Grant No. 11822407, and a Project Funded by the Priority Academic
Program Development of Jiangsu Higher Education Institutions.

Y.S.H. and Y.R.D. contribute equally to this work.

##  Appendix A the mobility gap for HOTAI

We present the mobility gap evolution of the HOTAI shown in Fig. 3(b) with
$m=2.05t$. The main results are similar to those in Fig. 4. The mobility gap
first increases, shown in Fig. 10(h). Then, the mobility gap decreases due to
the localization effect.

##  Appendix B The bulk-corner, edge-corner, and bulk-edge correspondence

Figure 11: (Color online). (a) and (b) The bulk energy gap and edge band gap
versus $m$ and $\Delta_{0}$, which are calculated under periodic and open
boundary along $x$ direction, respectively. Periodic boundary along $y$
direction is used. (c) The evolution of spin Chern number. The green solid
lines with square marks are the phase boundary. The quadrupole moment with
$Q_{xy}=0$ and $Q_{xy}=0.5$ are separated by $m=2t-2\Delta_{0}$ (d) and (e)
the evolution of bulk energy gap versus disorder strength $W$ for Fig. 9(d)
and (e), respectively. $\Delta_{x}$ is fixed at $0.5t$. Other parameters are
given in the figure. The sample size is $N=20$.

For the model shown in Eq. (1), there are three possible cases for the bulk-
boundary correspondence. (i) When bulk-corner correspondence holds, the bulk
energy (mobility) and edge band gap should be closed simultaneously with
$Q_{xy}$ varying. (ii) The bulk-edge correspondence in this paper means that
$\mathcal{C}_{s}=1$ ensures the existence of edge states with or without edge
band gaps, where $\mathcal{C}_{s}$ changes only when the bulk energy
(mobility) gap is closed. (iii) For edge-corner correspondence, $Q_{xy}$ that
determines the corner states is correlated with the edge band gap instead of
bulk energy gap.

To clarify the bulk-corner, bulk-edge, and edge-corner correspondenceHOTIbe1 ,
we plot the evolution of bulk energy gap and edge band gap in Figs. 11(a) and
(b) for samples without disorder. Fig. 11(c) shows the spin Chern number
$\mathcal{C}_{s}$. For $\Delta_{0}=0$, the bulk gap is closed when $m=2t$.
Furthermore, $\mathcal{C}_{s}$ and $Q_{xy}$ change simultaneously, where the
bulk-edge and bulk-corner correspondence exist. When $\Delta_{0}\neq 0$, the
bulk energy gap is closed only when $(m=0,\Delta_{0}=0)$ and
$(m=0,\Delta_{0}=t)$ [see Fig. 11(a)]. For the rest regions, the bulk gap
always exists. However, $\mathcal{C}_{s}$ changes only when $m=2t$, which is
independent of $\Delta_{0}$ [see Fig. 11(c)]. Thus, the bulk-edge
correspondence breaks down. By comparing the bulk energy gap in Fig. 11(a)
with $Q_{xy}$ shown in Fig. 11(b), it is obvious that the bulk-corner
correspondence breaks. Nevertheless, the edge-corner correspondence exists. We
note that, in Fig. 11(b), the bulk energy gap is used to replace the edge
energy gap for convenience when the edge states are absent. Further, we
compare the bulk energy gap in Figs. 11(d) and (e) and $C_{s}$ shown in Fig.
9(d) and (e) and find that the bulk energy gap is not closed when $C_{s}$
jumps from zero to one or vice versus when the bulk-edge correspondence breaks
down. The $C_{s}$ changes value only when $m=2t$, which is corresponding to
the gap closing condition in BHZ model.

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