# Coulomb instabilities of 3D higher-order topological insulators

Peng-Lu Zhao  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology
(SUSTech), Shenzhen 518055, China  Shenzhen Key Laboratory of Quantum Science
and Engineering, Shenzhen 518055, China  Xiao-Bin Qiang  Shenzhen Institute
for Quantum Science and Engineering and Department of Physics, Southern
University of Science and Technology (SUSTech), Shenzhen 518055, China
Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055,
China  Hai-Zhou Lu  Corresponding author<EMAIL_ADDRESS>Shenzhen
Institute for Quantum Science and Engineering and Department of Physics,
Southern University of Science and Technology (SUSTech), Shenzhen 518055,
China  Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen
518055, China  X. C. Xie  International Center for Quantum Materials, School
of Physics, Peking University, Beijing 100871, China  CAS Center for
Excellence in Topological Quantum Computation, University of Chinese Academy
of Sciences, Beijing 100190, China  Beijing Academy of Quantum Information
Sciences, West Building 3, No. 10, Xibeiwang East Road, Haidian District,
Beijing 100193, China

###### Abstract

Topological insulator (TI) is an exciting discovery because of its robustness
against disorder and interactions. Recently, higher-order TIs have been
attracting increasing attention, because they host 1D topologically-protected
hinge states in 3D or 0D corner states in 2D. A significantly critical issue
is whether the higher-order TIs also survive interactions, but it is still
unexplored. We study the effects of weak Coulomb interaction on a 3D second-
order TI, with the help of a renormalization group calculation. We find that
the 3D higher-order TIs are always unstable, suffering from two types of
topological phase transitions. One is from higher-order TI to TI, the other is
to normal insulator (NI). The first type is accompanied by emergent time-
reversal and inversion symmetries and has a dynamical critical exponent
$\kappa=1$. The second type does not have the emergent symmetries and has non-
universal dynamical critical exponents $\kappa<1$. Our results may inspire
more inspections on the stability of higher-order topological states of matter
and related novel quantum criticalities.

_Introduction_.–As generalizations of the topological insulator (TI) Fu  _et
al._ (2007); Moore and Balents (2007); Murakami (2007); Roy (2009); Fu and
Kane (2007); Hsieh  _et al._ (2008); Zhang  _et al._ (2009); Xia  _et al._
(2009); Shen (2017); Hasan and Kane (2010); Qi and Zhang (2011); Hasan and
Moore (2011), higher-order TIs have been attracting considerable interest
recently Benalcazar  _et al._ (2017a, b); Song  _et al._ (2017); Wang  _et
al._ (2017); Langbehn  _et al._ (2017); Schindler  _et al._ (2018a, b); Ezawa
(2018); Li  _et al._ (2019); Liu  _et al._ (2019); Luo and Zhang (2019); Kudo
_et al._ (2019); Chen  _et al._ (2020); Agarwala  _et al._ (2020); Fu  _et
al._ (2021). A simplest 3D higher-order TI hosts 3D gapped bulk states inside
but topologically protected gapless 1D hinge states and gapped 2D surface
states (Fig. 1). There have been experimental evidences for the higher-order
topology in bosonic systems, including circuitry Peterson  _et al._ (2018);
Imhof  _et al._ (2018); Serra-Garcia  _et al._ (2019), phononics Serra-Garcia
_et al._ (2018), acoustics Xue  _et al._ (2019); Ni  _et al._ (2019); Xue  _et
al._ (2020), and photonics Mittal  _et al._ (2019); El Hassan  _et al._
(2019); Xie  _et al._ (2019); Li  _et al._ (2020a). Despite the theoretical
predictions on material candidates Schindler  _et al._ (2018a, b); Yue  _et
al._ (2019); Wang  _et al._ (2019); Xu  _et al._ (2019); Zhang  _et al._
(2020), there are few observation of higher-order TIs in electronic systems
Schindler  _et al._ (2018b). This raises the concerns on the stability of
higher-order TIs against, _e.g._ disorder Su  _et al._ (2019); Wang and Wang
(2020); Li  _et al._ (2020b); Szabó and Roy (2020). More importantly, it is
still unknown whether higher-order TIs can survive a more intrinsic presence
in electronic systems, the Coulomb interaction González  _et al._ (1999);
Goswami and Chakravarty (2011); Hosur  _et al._ (2012); Moon  _et al._ (2013);
Yang  _et al._ (2014); Hofmann  _et al._ (2014); Lai (2015); Jian and Yao
(2015); Huh  _et al._ (2016); Isobe  _et al._ (2016); Cho and Moon (2016);
Isobe and Nagaosa (2016); Szabó and Roy (2020).

Figure 1: Schematic of a typical 3D second-order topological insulator (TI)
(left) which hosts 3D gapped bulk states in the interior but topologically
protected 2D massive (gapped) Dirac cones on the surfaces and gapless 1D
chiral hinge states. It may turn to a TI (right top) or a normal insulator
(NI) (right bottom), in the presence of the Coulomb interaction.

In this Letter, we study the stability of 3D higher-order TIs in the present
of the Coulomb interaction. We find that the higher-order TIs are always
unstable. Two types of topological phase transition could happen (Fig. 1). In
the first type, the Coulomb interaction could drive a topological phase
transition from higher-order TI to TI. This transition is accompanied by the
recovery of time-reversal and inversion symmetries in the low-energy limit,
and hence is protected by these emergent symmetries. The quantum criticality
for this phase transition is described by a dynamical critical exponent
$\kappa=1$ and a correlation length exponent $\nu=1$. In the second type, the
Coulomb interaction could induce another topological phase transition from
higher-order TI to a normal insulator (NI). There are no emergent symmetry
when this phase transition happens, and its criticality is characterized by
the appearance of non-universal dynamical critical exponents $\kappa<1$. Our
results will be insightful for the ongoing experimental search for higher-
order TIs in electronic systems.

_Model for 3D second-order TIs_.–We start with a four-band Hamiltonian for 3D
second-order TIs Schindler  _et al._ (2018a),

$\displaystyle\mathcal{H}_{\mathrm{0}}({\bf k})$ $\displaystyle=$
$\displaystyle\big{[}M+\sum_{i}t_{i}\cos(ak_{i})\big{]}\tau_{z}\sigma_{0}+\sum_{i}\Delta_{i}\sin\left(ak_{i}\right)$
(1)
$\displaystyle\times\tau_{x}\sigma_{i}+\Delta_{2}\left[\cos(ak_{x})-\cos(ak_{y})\right]\,\tau_{y}\sigma_{0},$

where $i=x,y,z$, $a$ is the lattice constant and $\sigma_{i}$ and $\tau_{i}$
are the Pauli matrices. $M,t_{i},\Delta_{i},\Delta_{2}$ are the hopping
parameters, and we take $t_{x}=t_{y}=t_{\perp}$, and
$\Delta_{x}=\Delta_{y}=\Delta_{\perp}$. This model has no time-reversal
symmetry if $\Delta_{2}\neq 0$ (SI of Sup ). Also, a four-fold rotation
symmetry $R_{4z}\equiv\tau_{0}e^{-\mathrm{i}\frac{\pi}{4}\sigma_{z}}$ is
broken by the nonzero $\Delta_{2}$ term. The combination $R_{4z}\mathcal{T}$
($\mathcal{T}$ is the time-reversal operator) is a symmetry that protects the
3D second-order TI. Another important symmetry is the combination of time
reversal and inversion $\mathcal{IT}$ (SII of Sup ), where
$\mathcal{I}=\tau_{z}{\sigma}_{0}$. With which, the $\mathcal{Z}_{2}$
invariants

$\displaystyle(-1)^{\vartheta}=\prod_{i}\prod_{n=1}^{N/2}\xi_{n}\left(\Gamma_{i}\right),$
(2)

where $\xi_{n}\left(\Gamma_{i}\right)=\pm 1$ is the eigenvalue of
$\mathcal{I}$ for the $n$th occupied energy band at momenta $\Gamma_{i}$, and
$\Gamma_{i}\in\\{(0,0,0),(\pi,\pi,0),(0,0,\pi),(\pi,\pi,\pi)\\}$ representing
all the $R_{4z}\mathcal{T}$-invariant ${\bf k}$ points. As a result, for
$\left|2t_{\perp}-t_{z}\right|<\left|M\right|<\left|2t_{\perp}+t_{z}\right|$,
$(-1)^{\vartheta}=-1$ (SII of Sup ), which represents the second-order TI and
for $\left|M\right|>\left|2t_{\perp}+t_{z}\right|$ or
$\left|M\right|<\left|2t_{\perp}-t_{z}\right|$, $(-1)^{\vartheta}=1$, which
stands for a NI. This difference establishes only when $\Delta_{i}\neq
0\neq\Delta_{2}$. Once $\Delta_{i}=0$, there exist gapless points which break
the insulating nature. Once $\Delta_{2}=0$, time-reversal symmetry recovers
and the phase is a TI. Later, we will show that, even if starting with
$\Delta_{2}\neq 0$ and
$\left|2t_{\perp}-t_{z}\right|<\left|M\right|<\left|2t_{\perp}+t_{z}\right|$,
the Coulomb interaction makes $\Delta_{2}$ flow to zero in the low-energy
limit, leading to a transition from higher-order TI to TI, or causes
$\left|M\right|>\left|2t_{\perp}+t_{z}\right|$, which induces a transition
from higher-order TI to NI.

_Coulomb interaction and renormalization group equations._ –The effective
action in Euclidean spacetime for the second-order TI in the presence of the
Coulomb interaction takes the form

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int d\tau d^{3}{\bf
r}\big{\\{}\bar{\psi}\big{[}\left(\partial_{\tau}+ig\phi\right)\gamma_{0}+v_{i}\gamma_{i}\partial_{i}+m+B_{i}\partial_{i}^{2}$
(3) $\displaystyle-
iD\left(\partial_{x}^{2}-\partial_{y}^{2}\right)\gamma_{5}\big{]}\psi+\frac{1}{2}\eta_{i}\left(\partial_{i}\phi\right)^{2}\big{\\}},$

where $\psi$ describes a four-component fermion field and
$\bar{\psi}=\psi^{\dagger}\gamma_{0}$. The $\gamma$ matrices satisfy the
anticommuting algebra
$\left\\{\gamma_{\mu},\gamma_{\nu}\right\\}=2\delta_{\mu,\nu}$. The repeated
index $i$ sums for $i=x,y,z$, and $v_{i}=\Delta_{i}a$, $m=M+2t_{\perp}+t_{z}$,
$B_{i}=t_{i}a^{2}/2$, $D=\Delta_{2}a^{2}/2$, which are obtained by expanding
Eq. (1) around the $\Gamma$ point. The parameter $m$ is the Dirac mass, and
the $B_{i}$ and $D$ terms represent the quadratic corrections to the Dirac
Hamiltonian. We introduce an auxiliary scale field $\phi$ through the Hubbard-
Stratonovich transformation Coleman (2015) to decouple the density-density
Coulomb interaction.
$\left(\eta_{x},\eta_{y},\eta_{z}\right)=\left(1,1,\eta\right)$ is used to
characterize the spatial anisotropy of $\phi$. $g=e/\sqrt{\epsilon}$
representing the coupling between electrons and the scalar field, where $-e$
is the electron charge and $\epsilon$ is the dielectric constant. The Coulomb
interaction does not break the $R_{4z}\mathcal{T}$ and $\mathcal{IT}$
symmetries of Eq. (1) (SIIIA of Sup ), and the topology is still distinguished
by Eq. (2). According to Eq. (1), $m$ solely controls the gap closing and
reopening Murakami (2007); Shen  _et al._ (2011); Shen (2017), and hence its
sign change identifies the phase transition between higher-order TI and NI.

To explore how the Coulomb interaction renormalizes the parameters and
consequently leads to the phase transitions, we perform a Wilsonian momentum-
shell renomalization group analysis Shankar (1994) for Eq. (3). After
redefining the original parameters $B_{i},D,m,v_{i}$, and Coulomb interaction
strength $g$ into dimensionless

$\displaystyle B_{i}\Lambda v^{-1}\eta_{i}^{-1}\rightarrow B_{i},\,\,D\Lambda
v^{-1}\rightarrow D,$ $\displaystyle mv^{-1}\Lambda^{-1}\rightarrow
m,\,\,g^{2}/(4\pi^{2}v\sqrt{\eta})\rightarrow\alpha,$ (4)

where $\Lambda$ is the cutoff, the renormalization group flow equations for
them are found as (SIIIC of Sup )

$\displaystyle dv/d\ell$ $\displaystyle=$
$\displaystyle\left(\kappa-1+\mathcal{F}_{0}^{\perp}\alpha\right)v,$ (5)
$\displaystyle d\gamma^{2}/d\ell$ $\displaystyle=$
$\displaystyle\gamma^{2}\alpha\left[2\left(2\mathcal{F}_{0}^{z}-\mathcal{F}_{0}^{\perp}\right)-\mathcal{F}_{2}^{z}+\mathcal{F}_{2}^{\perp}\right],$
(6) $\displaystyle dm/d\ell$ $\displaystyle=$ $\displaystyle
m+\alpha\left[\left(m-B_{z}\right)\mathcal{F}_{0}^{z}-B_{\perp}\mathcal{F}_{0}^{\perp}\right],$
(7) $\displaystyle dB_{\perp}/d\ell$ $\displaystyle=$ $\displaystyle-
B_{\perp}+\alpha\Big{[}B_{\perp}\left(\mathcal{F}_{1}^{\perp}-\mathcal{F}_{0}^{\perp}\right)$
(8)
$\displaystyle+\left(B_{z}-m\right)\mathcal{F}_{1}^{z}-m\mathcal{F}_{1}^{\perp}\Big{]},$
$\displaystyle dB_{z}/d\ell$ $\displaystyle=$ $\displaystyle-
B_{z}+\alpha\Big{[}B_{z}\left(\mathcal{F}_{2}^{\perp}-\mathcal{F}_{2}^{z}-\mathcal{F}_{1}^{\perp}-3\mathcal{F}_{1}^{z}\right)$
(9)
$\displaystyle+B_{\perp}\left(\mathcal{F}_{0}^{\perp}-2\mathcal{F}_{1}^{\perp}\right)-m\left(3\mathcal{F}_{0}^{z}-\mathcal{F}_{0}^{\perp}\right)\Big{]},$
$\displaystyle dD/d\ell$ $\displaystyle=$
$\displaystyle\left[-1+\alpha\left(\mathcal{F}_{1}^{D}-\mathcal{F}_{0}^{\perp}\right)\right]D,$
(10) $\displaystyle d\alpha/d\ell$ $\displaystyle=$
$\displaystyle-\alpha^{2}\left[\mathcal{F}_{0}^{\perp}+\left(\mathcal{F}_{2}^{z}+\mathcal{F}_{2}^{\perp}\right)/2\right],$
(11)

where $v=v_{x,y}$, $\kappa$ is the dynamic exponent whose value is obtained by
fixing $v$, $\gamma^{2}=v_{z}/(v\eta)$ describing the anisotropy, and $\ell$
is the running scale parameter whose value increase lowers the energy scale.
$\mathcal{F}_{0-2}^{\perp,z},\mathcal{F}_{1}^{D}$ are dimensionless functions
of $m$, $B_{i}$, $D$, and $\gamma^{2}$ (Their expressions in IIIB of Sup ). We
numerically solve the renormalization group equations and obtain the running
of $m,B_{i},D,\alpha,\gamma^{2}$ and $\kappa$ with $\ell$. Despite that the
running of these parameters highly depends on their initial values, _i.e._
their values at the cutoff $\Lambda$, their behaviors can be classified into
two categories, corresponding to two types of phase transitions.

Figure 2: [(a)–(c)] The renormalized $D$, $m$, and $\alpha$ as functions of
the running scale parameter $\ell$. $D$ and $m$ protect the second-order and
first-order topological properties, respectively. The vanishing $D$ and
increasing $m$ means a topological phase transition from second-order TI to
TI. (d) The scale dependence of $\kappa$. The solutions are obtained by fixing
the initial values $m_{0}=B_{\perp}^{0}=1$, $B_{z}^{0}=0.5$,
$\alpha_{0}=0.1=\gamma^{2}_{0}=0.1$ while varying $D_{0}$. The same legends in
(a)–(d). In (b), the $m\left(\ell\right)$ curve of $D_{0}=5,10$, and $20$ are
shifted vertically by $1$ for clarity. Figure 3: [(a), (b), and (d)] The
renormalized $m$, $D$, and $\alpha$ as functions of the running scale
parameter $\ell$. $D$ and $m$ protect the second-order and first-order
topological properties, respectively. The finite $D$ at the sign change of $m$
means a topological phase transition from 3D second-order TI to NI. (c) The
scale dependence of $\kappa$. The solutions are obtained by varying the
initial value of $D$ while fixing the initial values of other parameters as
$m_{0}=0.2$, $B_{\perp}^{0}=2B_{z}^{0}=2$, $\alpha_{0}=0.5$, and
$\gamma^{2}_{0}=0.1$. The same legends in (a)–(d). The orange dashed lines in
(a)–(c) label the values of $\ell$ at the phase transition points. In (b) and
(c), the crossing points of the orange dashed lines with the red, black, blue
lines indicate the values of $D$ and $\kappa$ at the topological phase
transition points. From left to right, $D=0.43,0.72,0.74$ in (b) and
$\kappa=0.79,0.81,0.83$ in (c).

_From second-order TI to TI._ –This phase transition is characterized by a
vanishing $D$ and always positive $m$ at large $\ell$ (the low-energy limit),
as shown in Fig. 2. Figure 2(a) shows that, in a large range of $D_{0}$, $D$
flows to zero rapidly with increasing $\ell$. This behavior reflects that the
renormalization group equation for $D$ [Eq. (10)] only has one stable fixed
point at $D_{\ast}=0$. As $D$ flows to zero, $m$ increases and remains
positive [Fig. 2(b)]. The unrestricted growth of $m$ ceases the rapid decay of
$\alpha$ [Fig. 2(c)] because $\mathcal{F}_{0}^{\perp}\sim
1/|m|,\mathcal{F}_{2}^{\perp,z}\sim 1/|m|^{3}$ for extremely large $m$, which
give rises to $d\alpha/d\ell\sim 0$. Although the effective Coulomb
interaction is marginally irrelevant according to Eq. (11), its values in the
low-energy limit is a small constant instead of zero [see Fig. 2(c)]. With no
sign change of $m$, there is no gap closing and reopening near the $\Gamma$
point when approaching the low-energy limit, and the topological invariant
does not change. However, when $D$ flows to zero, the free part of the
effective model Eq. (3) reduces to the modified Dirac Hamiltonian that
describes TIs Zhang  _et al._ (2009); Shen (2017). According to the previous
results Qi  _et al._ (2008); Wang  _et al._ (2010); Goswami and Chakravarty
(2011); Wang  _et al._ (2012), TIs are immune to weak Coulomb interactions.
Therefore, the low-energy state is a TI with a finite but weak Coulomb
interaction, which means that the second-order TI is unstable due to the
Coulomb interaction. Time-reversal symmetry and inversion symmetry emerge
along with the phase transition from second-order TI to TI. The previous works
have shown the possibilities of emergent Lorentz Nielsen and Ninomiya (1978);
Chadha and Nielsen (1983); Sitte  _et al._ (2009); Anber and Donoghue (2011);
Bednik  _et al._ ; Sibiryakov (2014); Kharuk and Sibiryakov (2016); Roy  _et
al._ , chiral Balachandran and Vaidya (1997); Candido  _et al._ (2018); Szabó
and Roy , and super- Balents  _et al._ (1998); Fendley  _et al._ (2003); Lee
(2007); Yu and Yang (2008, 2010); Bauer  _et al._ (2013); Grover  _et al._
(2014); Ponte and Lee (2014); Jian  _et al._ (2015, 2017); Feldmann  _et al._
(2018); Zhao and Liu (2019) symmetries. Our concrete example above shows the
emergent discrete time-reversal and inversion symmetries, enriching the family
of emergent symmetries Volovik (2008). This phase transition does not need a
large critical value of $\alpha$, and exists at least for $\alpha\sim
10^{-3}$, corresponding to an extremely weak Coulomb interaction (IV of Sup ).
We find that the dynamical critical exponent $\kappa=1$ for this phase
transition, as shown in Fig. 2(d). We also obtain a correlation length
exponent $\nu=1$ by assuming that the spatial correlation length $\xi$
diverges as $\delta\equiv D-D_{*}\rightarrow 0$ in the manner
$\xi\sim|\delta|^{-\nu}$. This definition is similar to the conventional
definition of the correlation length exponent in a symmetry-broken quantum
phase transition Sondhi  _et al._ (1997); Imada  _et al._ (1998).

_From second-order TI to NI._ –This phase transition is characterized by a
sign change of $m$ and a finite $D$ as $m$ changes sign, as shown in Fig. 3.
Here, the finite $D$ guarantees that the topological phase transition to NI
happens before the transition to TI. Figure 3(a) shows that $m$ changes sign
when $D_{0}$ is below a critical value. For comparison, the green line shows a
case in which $m$ does not change sign. Figure 3(b) shows that $D$ does not
vanish when $m$ changes sign. Therefore, for the parameters represented by the
red, black, and blue lines in Fig. 3(a), the transitions from second-order TI
to NI happen and for the case depicted by the green line the transition from
second-order TI to TI happens. Due to the finite $D$, there is no emergent
time-reversal and inversion symmetries at the phase transition point. After
the transition, the emergent time-reversal and inversion symmetries appear in
the low-energy limit of the NI. Interestingly, this transition has no
universal dynamical critical exponent. Figure 3(c) shows that the dynamical
critical exponent has different values for the three cases, all below $1$. A
non-universal $\kappa<1$ is a direct result of the transition happening at a
finite energy scale $\ell$. According to Eq. (5),

$\displaystyle\kappa\left(\ell\right)=1-\alpha\left(\ell\right)\mathcal{F}_{0}^{\perp}\left(\ell\right).$
(12)

The values of $\alpha\left(\ell\right)$ are always positive constant [see Fig.
3(d)], and the non-negative $\mathcal{F}_{0}^{\perp}\left(\ell\right)$
vanishes only as $\ell\rightarrow\infty$. Considering that the sign change of
$m$ always happens at a finite $\ell$, the value of $\kappa$ is smaller than
$1$ and its particular value depends on $\alpha$ and
$\mathcal{F}_{0}^{\perp}\left(\ell\right)$, and hence is non-universal.
However, as shown in Fig. 3(c), the asymptotic value of $\kappa$ as
$\ell\rightarrow\infty$ is still $1$, which describes the low-energy dynamics
of the NI.

We have shown our main results by varying the initial value of $D$ while
fixing those of the other parameters. We could perform similar analyses for
any cases by changing the initial value of one parameter while fixing others,
but our conclusion still holds.

Figure 4: [(a) and (b)] The renormalized $\Delta_{M}$ and $\Delta_{C}$. The
different curves are obtained by fixing the initial values
$m_{0}=B_{\perp}^{0}=B_{z}^{0}=\gamma^{2}_{0}=0.1$, $\alpha_{0}=0.5$ while
varying the initial values of $\Delta_{M}$ in (a) and $\Delta_{C}$ in (b).
[(c) and (d)] Phase diagrams in the $m$–$\alpha$ and $\Delta_{C}$–$\alpha$
planes. In (c), the black, red, and blue lines represent the boundary for the
cases without disorder, with random mass $\Delta_{M}^{0}=0.3$, and with random
chemical potential $\Delta_{C}^{0}=0.3$, respectively. In (d), CDM stands for
the compressible diffusive metal.
$D_{0}=B_{\perp}^{0}=B_{z}^{0}=1,\gamma^{2}_{0}=0.1$ for the both diagrams and
$m_{0}=0.1$ for (d).

_Screened Coulomb interaction._ –The screening of Coulomb interaction cannot
change our main conclusion. Due to the existence of gap $m$, once the Fermi
energy is placed in the gap, the density of state vanishes. The screening
effect is extremely weak, compared to those in metals and semimetals. Using
the random phase approximation, the renormalized boson propagator accounting
for the screened Coulomb potential is

$\displaystyle k_{\perp}^{2}+\eta k_{z}^{2}-\Pi\left(k\right)$
$\displaystyle\sim$ $\displaystyle
k_{\perp}^{2}\left(1+\alpha\mathcal{F}_{2}^{\perp}\ell\right)$ (13)
$\displaystyle+$ $\displaystyle\eta
k_{z}^{2}\left(1+\alpha\mathcal{F}_{2}^{z}\ell\right).$

Considering $\mathcal{F}_{2}^{\perp,z}\sim 1/|m|^{3}\sim 0$ at low energies,
the screening effect therefore can be ignored.

_Effect of coexisting disorder._ – We introduce disorder described by $\delta
H=U_{i}(x)\bar{\psi}\Gamma_{i}\psi$, where $\Gamma_{i}$ is a $4\times 4$
Hermitian matrix and $U_{i}(x)$ is the impurity potential of a Gaussian white-
noise distribution as $\left\langle U_{i}\right\rangle=0$ and $\left\langle
U_{i}({\bf x})U_{j}\left({\bf
x}^{\prime}\right)\right\rangle=\Delta_{i}\delta_{ij}\delta({\bf x}-{\bf
x}^{\prime})$. The types of disorder which respect $R_{4z}\mathcal{T}$ and
$\mathcal{IT}$ symmetries are denoted by $\Gamma_{i}=\mathcal{I}_{4\times 4}$
and $\Gamma_{i}=\gamma_{0}$ (SVI of Sup ), and are referred to as the random
mass and random chemical potential, respectively. The coupling strength
$\Delta_{M}$ for the random mass is irrelevant [Fig. 4(a)] and hence its
existence cannot prevent the phase transitions in the clean system. As shown
in Fig. 4(b), the coupling strength $\Delta_{C}$ has a critical value
$\Delta_{C}^{c}$ for the random chemical potential. Once the initial value
$\Delta_{C}^{0}$ is smaller than $\Delta_{C}^{c}$, the random chemical
potential is also irrelevant and cannot change our conclusion (SVI of Sup ).
Due to the disorder-induced renormalization to the gap $m$, the boundary
between the two kinds of phase transitions are shifted. As shown in Fig. 4(c),
the boundary between the phase transition to TI and NI for the clean system,
random mass, and random chemical potential are different. If
$\Delta_{C}^{0}>\Delta_{C}^{c}$, a disorder-induced phase transition happens
and the system flows to a disorder-dominated phase, dubbed the compressible
diffusive metal Wang and Wang (2020); Sup .

_Phase diagrams._ –To have a global view of the various phases, Fig. 4 shows
two phase diagrams. According to Fig. 4(c), we find that $m$ plays a key role
to determine the type of phase transition. Once $m$ is large enough, the
transition to NI cannot happen. Due to the dominant role of $m$, our
conclusion also apply to other 3D higher-order TIs (e.g., the helical second-
order TI Schindler  _et al._ (2018a)), in particular when the topology depends
on the quadratic or higher-order corrections to the Dirac Hamiltonian (e.g.,
the $D$-term in our model). According to our calculation, these terms are all
irrelevant in the low-energy limit, which causes the transition to TI. The
transition from higher-order TI to NI originates from the interplay between
the anisotropic quadratic term ($B_{i}$) and Dirac mass term ($m$) generated
by the Coulomb interaction, which does not depend on the terms protecting the
higher-order TIs and hence will still exist for other 3D higher-order TIs.

_Discussion_.–Our results show that the Coulomb interaction is critical in the
experiments searching for higher-order TIs. Previously, the transition between
TI and NI has been observed in
$\text{BiTl}\left(\text{S}_{1-\delta}\text{Se}_{\delta}\right)_{2}$ Xu  _et
al._ (2011) by varying $\delta$. Our theory shares a similar spirit, by fixing
the values of the parameters at the cutoff and analyzing their behaviors at a
particular low-energy scale. As a result, the change of initial values of the
parameters is equivalent to the change of the parameters at the particular
energy scale relevant to the experiment. Therefore, the topological phase
transitions from higher-order TI to NI or TI are possible in experiments by
doping higher-order TIs, such as the recently-proposed candidate materials
bismuth Schindler  _et al._ (2018b), Eu$\text{In}_{2}\text{As}_{2}$ Xu  _et
al._ (2019), and Mn$\text{Bi}_{2}\text{Te}_{4}$ Zhang  _et al._ (2020).
Correspondingly, the 1D gapless hinge state will transform into a gapped state
or 2D Dirac cone, which may be observed.

###### Acknowledgements.

We thank helpful discussions with Jing-Rong Wang and Daoyuan Li. This work was
supported by the National Natural Science Foundation of China (12047531,
11534001, 11925402), the Strategic Priority Research Program of Chinese
Academy of Sciences (XDB28000000), the National Key R & D Program
(2016YFA0301700), Guangdong province (2016ZT06D348, 2020KCXTD001), Shenzhen
High-level Special Fund (G02206304, G02206404), and the Science, Technology
and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217,
JCYJ20170412152620376). The numerical calculations were supported by Center
for Computational Science and Engineering of SUSTech.

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# Log-convexity and the overpartition function

Gargi Mukherjee  Institute for Algebra, Johannes Kepler University,
Altenberger Straße 69, A-4040 Linz, Austria.  gargi.mukherjee@dk-
compmath.jku.at

###### Abstract.

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we
obtain an inequality for the sequence $\Delta^{2}\log\
\sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ which states that

$\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}\biggr{)}<\Delta^{2}\log\
\sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}\
\ \text{for}\ n\geq N(\alpha),$

where $\alpha$ is a non-negative real number, $N(\alpha)$ is a positive
integer depending on $\alpha$ and $\Delta$ is the difference operator with
respect to $n$. This inequality consequently implies $\log$-convexity of
$\bigl{\\{}\sqrt[n]{\overline{p}(n)/n}\bigr{\\}}_{n\geq 19}$ and
$\bigl{\\{}\sqrt[n]{\overline{p}(n)}\bigr{\\}}_{n\geq 4}$. Moreover, it also
establishes the asymptotic growth of $\Delta^{2}\log\
\sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ by showing
$\underset{n\rightarrow\infty}{\lim}\Delta^{2}\log\
\sqrt[n]{\overline{p}(n)/n^{\alpha}}=\dfrac{3\pi}{4n^{5/2}}.$

Mathematics Subject Classifications. Primary 05A20; 11N37.
Keywords. Log-convexity; Overpartitions.

##  1\. Introduction

An overpartition of $n$ is a nonincreasing sequence of natural numbers whose
sum is $n$ in which the first occurrence of a number may be overlined and
$\overline{p}(n)$ denotes the number of overpartitions of $n$. For
convenience, define $\overline{p}(0)=1$. For example, there are $8$
overpartitions of $3$ enumerated by
$3,\overline{3},2+1,\overline{2}+1,2+\overline{1},\overline{2}+\overline{1},1+1+1,\overline{1}+1+1$.
Systematic study of overpartition began with the work of Corteel and Lovejoy
[4], although it has been studied under different nomenclature that dates back
to MacMahon. Analogous to Hardy-Ramanujan-Rademacher formula for partition
function (cf. [7],[10]), Zuckerman [13] gave a formula for $\overline{p}(n)$
that reads

$\overline{p}(n)=\frac{1}{2\pi}\underset{2\nmid
k}{\sum_{k=1}^{\infty}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^{2}}{\omega(2h,k)}e^{-\frac{2\pi
inh}{k}}\dfrac{d}{dn}\biggl{(}\dfrac{\sinh\frac{\pi\sqrt{n}}{k}}{\sqrt{n}}\biggr{)},$
(1.1)

where

$\omega(h,k)=\text{exp}\Biggl{(}\pi
i\sum_{r=1}^{k-1}\dfrac{r}{k}\biggl{(}\dfrac{hr}{k}-\biggl{\lfloor}\dfrac{hr}{k}\biggr{\rfloor}-\dfrac{1}{2}\biggr{)}\Biggr{)}$

for positive integers $h$ and $k$. In somewhat a similar spirit as Lehmer [8]
obtained an error bound for the partition function, Engel [6] provided an
error term for $\overline{p}(n)$

$\overline{p}(n)=\frac{1}{2\pi}\underset{2\nmid
k}{\sum_{k=1}^{N}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^{2}}{\omega(2h,k)}e^{-\frac{2\pi
inh}{k}}\dfrac{d}{dn}\biggl{(}\dfrac{\sinh\frac{\pi\sqrt{n}}{k}}{\sqrt{n}}\biggr{)}+R_{2}(n,N),$
(1.2)

where

$\bigl{|}R_{2}(n,N)\bigr{|}<\dfrac{N^{5/2}}{\pi
n^{3/2}}\sinh\biggl{(}\dfrac{\pi\sqrt{n}}{N}\biggr{)}.$ (1.3)

A positive sequence $\\{a_{n}\\}_{n\geq 0}$ is called $\log$-convex if for
$n\geq 1$,

$a^{2}_{n}-a_{n-1}a_{n+1}\leq 0,$

and it is called $\log$-concave if for $n\geq 1$,

$a^{2}_{n}-a_{n-1}a_{n+1}\geq 0.$

Engel [6] proved that $\\{\overline{p}(n)\\}_{n\geq 2}$ is $\log$-concave by
using the asymptotic formula (1.2) with $N=2$ followed by (1.3). Prior to
Engel’s work on overpartitions, $\log$-concavity of partition function $p(n)$
and its associated inequalities has been studied in a broad spectrum, for
example see [1], [2], and [5]. Following the same line of studies, Liu and
Zhang [9] proved a list of inequalitites for overpartition function.
Sun [11] initiated the study on $\log$-convexity problems associated with
$p(n)$, later settled by Chen and Zheng [3, Theorem 1.1-1.2]. In a more
general setting, Chen and Zheng studied $\log$-convexity of
$\\{\sqrt[n]{p(n)/n^{\alpha}}\\}_{n\geq n(\alpha)}$ (cf. [3, Theorem 1.3]).
Moreover, they discovered the asymptotic growth of the sequence
$\Delta^{2}\log\sqrt[n]{p(n)}$ (cf. [3, Theorem 1.4]).
The main objective of this paper is to prove all the theorems [3, Theorem
1.1-1.4] but in context of overpartitions. Our goal is to obtain a much more
general inequality, given in Theorem 1.1, which at once implies [3, Theorem
1.1-1.4] for $\overline{p}(n)$, presented in Corollary 1.2-1.5. More
explicitly, in Theorem 1.1, we get a somewhat symmetric upper and lower bound
of $\sqrt[n]{\overline{p}(n)/n^{\alpha}}$, as shown in (1.4). We note that the
lower bound presented in (1.4) depicts a finer inequality than merely stating
$\Delta^{2}\log\sqrt[n]{\overline{p}(n)/n^{\alpha}}>0$ which implies
$\log$-convexity. In another direction, we note that (1.4) readily suggests
that $\dfrac{3\pi}{4}$ is the best possible constant so as to understand the
asymptotic growth of $\Delta^{2}\log\sqrt[n]{\overline{p}(n)/n^{\alpha}}$,
given in Corollary 1.5.
For $\alpha\in\mathbb{R}_{\geq 0}$, define
$r_{\alpha}(n):=\sqrt[n]{\overline{p}(n)/n^{\alpha}}$.

######  Theorem 1.1.

Let $\alpha\in\mathbb{R}_{\geq 0}$ and

$N(\alpha):=\begin{cases}\text{$\max\Biggl{\\{}\Bigl{[}\dfrac{3490}{\alpha}\Bigr{]}+2,\Bigl{\lceil}\Bigl{(}\dfrac{4(11+5\alpha)}{3\pi}\Bigr{)}^{4}\Bigr{\rceil},5505\Biggr{\\}}$}&\quad\text{if
$\alpha\in\mathbb{R}_{>0}$},\\\ \text{$4522$}&\quad\text{if $\alpha=0$.}\\\
\end{cases}$

Then for $n\geq N(\alpha)$,

$\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}\biggr{)}<\Delta^{2}\log
r_{\alpha}(n-1)<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}.$ (1.4)

######  Corollary 1.2.

The sequence $\bigl{\\{}\sqrt[n]{\overline{p}(n)/n^{\alpha}}\bigr{\\}}_{n\geq
N(\alpha)}$ is $\log$-convex.

  * Proof.

From (1.4), it is immediate that

$\dfrac{r_{\alpha}(n+1)r_{\alpha}(n-1)}{r^{2}_{\alpha}(n)}>1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}\
\ \text{for all}\ \ n\geq N(\alpha).$

We finish the proof by observing that

$1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}>1\ \ \text{for all}\ \
n\geq N(\alpha).$

∎

######  Corollary 1.3.

The sequences $\bigl{\\{}\sqrt[n]{\overline{p}(n)/n}\bigr{\\}}_{n\geq 19}$ and
$\bigl{\\{}\sqrt[n]{\overline{p}(n)}\bigr{\\}}_{n\geq 4}$ are $\log$-convex.

  * Proof.

In order to prove $\bigl{\\{}\sqrt[n]{\overline{p}(n)/n}\bigr{\\}}_{n\geq 19}$
and $\bigl{\\{}\sqrt[n]{\overline{p}(n)}\bigr{\\}}_{n\geq 4}$ are
$\log$-convex, after corollary 1.2, it remains to check numerically for
$19\leq n\leq 5504$ and $4\leq n\leq 4521$, which is done in ‘Mathematica’
interface. ∎

######  Corollary 1.4.

For all $n\geq 2$, we have

$\dfrac{\sqrt[n]{\overline{p}(n)}}{\sqrt[n+1]{\overline{p}(n+1)}}\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}>\dfrac{\sqrt[n-1]{\overline{p}(n-1)}}{\sqrt[n]{\overline{p}(n)}}.$
(1.5)

  * Proof.

It is an immediate implication of (1.4) as it is only left over to verify
(1.5) for $2\leq n\leq 4522$, which we did numerically in ‘Mathematica’. ∎

######  Corollary 1.5.

$\lim_{n\rightarrow\infty}n^{5/2}\Delta^{2}\log r_{\alpha}(n)=\frac{3\pi}{4}.$
(1.6)

  * Proof.

Multiplying both side of (1.4) by $n^{5/2}$ and taking limit as $n$ tends to
infinity, we get (1.6). ∎

##  2\. Proof of Theorem 1.1

In this section, we give a proof of Theorem 1.1. First, we state the Lemma 2.1
[3, Lemma 2.1] of Chen and Zheng which will be useful in the proofs of Lemmas
2.2-2.4. These lemmas further direct to get upper bound and lower bound of
$\Delta^{2}\log r_{\alpha}(n)$ respectively in Lemma 2.5 and 2.6, finally
results (1.4).

######  Lemma 2.1.

[3, Lemma 2.1] Suppose $f(x)$ has a continuous second derivative for
$x\in[n-1,n+1]$. Then there exists $c\in(n-1,n+1)$ such that

$\Delta^{2}f(n-1)=f(n+1)+f(n-1)-2f(n)=f^{\prime\prime}(c).$ (2.1)

If $f(x)$ has an increasing second derivative, then

$f^{\prime\prime}(n-1)<\Delta^{2}f(n-1)<f^{\prime\prime}(n+1).$ (2.2)

Conversely, if $f(x)$ has a decreasing second derivative, then

$f^{\prime\prime}(n+1)<\Delta^{2}f(n-1)<f^{\prime\prime}(n-1).$ (2.3)

We start by laying out a brief outline of Engel’s primary set up [6] for
proving $\log$-concavity of $\\{\overline{p}(n)\\}_{n\geq 2}$. Setting $N=3$
in (1.2), we express $\overline{p}(n)$ as

$\overline{p}(n)=\overline{T}(n)+\overline{R}(n),$ (2.4)

where

$\displaystyle\overline{T}(n)$ $\displaystyle=$
$\displaystyle\dfrac{\overline{c}}{\overline{\mu}(n)^{2}}\biggl{(}1-\dfrac{1}{\overline{\mu}(n)}\biggr{)}e^{\overline{\mu}(n)},$
(2.5) $\displaystyle\overline{R}(n)$ $\displaystyle=$
$\displaystyle\dfrac{1}{8n}\biggl{(}1+\dfrac{1}{\overline{\mu}(n)}\biggr{)}e^{-\overline{\mu}(n)}+R_{2}(n,3)$
(2.6)

with $\overline{c}=\dfrac{\pi^{2}}{8}$ and $\overline{\mu}(n)=\pi\sqrt{n}$. In
order to estimate the upper and lower bound of $\Delta^{2}\log
r_{\alpha}(n-1)$, it is necessary for us to express $\Delta^{2}\log
r_{\alpha}(n-1)$ in the following form

$\displaystyle\Delta^{2}\log r_{\alpha}(n-1)$ $\displaystyle=$
$\displaystyle\Delta^{2}\dfrac{1}{n-1}\log\overline{p}(n-1)-\alpha\
\Delta^{2}\dfrac{1}{n-1}\log(n-1)$ $\displaystyle=$
$\displaystyle\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)+\Delta^{2}\dfrac{1}{n-1}\log\biggl{(}1+\dfrac{\overline{R}(n-1)}{\overline{T}(n-1)}\biggr{)}-\alpha\
\Delta^{2}\dfrac{1}{n-1}\log(n-1).$

Define

$\overline{E}(n-1)=\log\biggl{(}1+\dfrac{\overline{R}(n-1)}{\overline{T}(n-1)}\biggr{)}$
(2.8)

and rewrite (2) as

$\Delta^{2}\log
r_{\alpha}(n-1)=\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)+\Delta^{2}\dfrac{1}{n-1}\overline{E}(n-1)-\alpha\
\Delta^{2}\dfrac{1}{n-1}\log(n-1)$ (2.9)

Therefore, in order to estimate $\Delta^{2}\log r_{\alpha}(n-1)$, it is
sufficient to estimate each of the three factors, appearing on the right hand
side of (2.9).

######  Lemma 2.2.

Let

$\displaystyle\overline{G}_{1}(n)$ $\displaystyle=$
$\displaystyle\dfrac{3\pi}{4(n+1)^{5/2}}-\dfrac{5\log\overline{\mu}(n-1)}{(n-1)^{3}},$
(2.10) $\displaystyle\overline{G}_{2}(n)$ $\displaystyle=$
$\displaystyle\dfrac{3\pi}{4(n-1)^{5/2}}-\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}+\dfrac{4}{(n-1)^{3}}.$
(2.11)

Then for $n\geq 2$, we have

$\overline{G}_{1}(n)<\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)<\overline{G}_{2}(n).$
(2.12)

  * Proof.

Using the definition of $\overline{T}(n)$ (2.5), we write

$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)=\sum_{i=1}^{4}\Delta^{2}\
\overline{g}_{i}(n-1),$ (2.13)

where

$\displaystyle\overline{g}_{1}(n)$ $\displaystyle=$
$\displaystyle\dfrac{\overline{\mu}(n)}{n},$
$\displaystyle\overline{g}_{2}(n)$ $\displaystyle=$
$\displaystyle-\dfrac{3\log\ \overline{\mu}(n)}{n},$
$\displaystyle\overline{g}_{3}(n)$ $\displaystyle=$ $\displaystyle\dfrac{\log\
(\overline{\mu}(n)-1)}{n},$ $\displaystyle\text{and}\ \ \overline{g}_{4}(n)$
$\displaystyle=$ $\displaystyle\dfrac{\log\overline{c}}{n}.$

It can be easily checked that for $n\geq 3$,
$\overline{g}^{{}^{\prime\prime\prime}}_{1}(n)<0$,
$\overline{g}^{{}^{\prime\prime\prime}}_{2}(n)>0$,
$\overline{g}^{{}^{\prime\prime\prime}}_{3}(n)<0$, and
$\overline{g}^{{}^{\prime\prime\prime}}_{4}(n)<0$. As a consequence, for
$n\geq 3$,
$\overline{g}^{{}^{\prime\prime}}_{1}(n),\overline{g}^{{}^{\prime\prime}}_{3}(n)$,
and $\overline{g}^{{}^{\prime\prime}}_{4}(n)$ are decreasing, whereas
$\overline{g}^{{}^{\prime\prime}}_{2}(n)$ is increasing. Applying Lemma 2.1,
we get for $i\in\\{1,3,4\\}$,

$\overline{g}^{{}^{\prime\prime}}_{i}(n+1)<\Delta^{2}\
\overline{g}_{i}(n-1)<\overline{g}^{{}^{\prime\prime}}_{i}(n-1)$ (2.14)

and

$\overline{g}^{{}^{\prime\prime}}_{2}(n-1)<\Delta^{2}\
\overline{g}_{2}(n-1)<\overline{g}^{{}^{\prime\prime}}_{2}(n+1).$ (2.15)

From (2.13) and (2.14)-(2.15), we obtain for all $n\geq 3$,

$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)<\overline{g}^{{}^{\prime\prime}}_{1}(n-1)+\overline{g}^{{}^{\prime\prime}}_{2}(n+1)+\overline{g}^{{}^{\prime\prime}}_{3}(n-1)+\overline{g}^{{}^{\prime\prime}}_{4}(n-1)$
(2.16)

and

$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)>\overline{g}^{{}^{\prime\prime}}_{1}(n+1)+\overline{g}^{{}^{\prime\prime}}_{2}(n-1)+\overline{g}^{{}^{\prime\prime}}_{3}(n+1)+\overline{g}^{{}^{\prime\prime}}_{4}(n+1),$
(2.17)

where

$\displaystyle\overline{g}^{{}^{\prime\prime}}_{1}(n)$ $\displaystyle=$
$\displaystyle\dfrac{3\pi}{4n^{5/2}},$ (2.18)
$\displaystyle\overline{g}^{{}^{\prime\prime}}_{2}(n)$ $\displaystyle=$
$\displaystyle\dfrac{9}{2n^{3}}-\dfrac{6\log\overline{\mu}(n)}{n^{3}},$ (2.19)
$\displaystyle\overline{g}^{{}^{\prime\prime}}_{3}(n)$ $\displaystyle=$
$\displaystyle\dfrac{2\log(\overline{\mu}(n)-1)}{n^{3}}-\dfrac{5\pi}{4n^{5/2}(\overline{\mu}(n)-1)}-\dfrac{\pi^{2}}{4n^{2}(\overline{\mu}(n)-1)^{2}},$
(2.20) $\displaystyle\text{and}\ \ \overline{g}^{{}^{\prime\prime}}_{4}(n)$
$\displaystyle=$ $\displaystyle\dfrac{2\log\overline{c}}{n^{3}}.$ (2.21)

We first estimate the upper bound of
$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)$ by (2.16) and (2.18)-(2.21).

$\begin{split}\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)<&\dfrac{3\pi}{4(n-1)^{5/2}}+\dfrac{9}{2(n+1)^{3}}-\dfrac{6\log\overline{\mu}(n+1)}{(n+1)^{3}}\\\
&+\dfrac{2\log(\overline{\mu}(n-1)-1)}{(n-1)^{3}}-\dfrac{5\pi}{4(n-1)^{5/2}(\overline{\mu}(n-1)-1)}-\dfrac{\pi^{2}}{4(n-1)^{2}(\overline{\mu}(n-1)-1)^{2}}\\\
&+\dfrac{2\log\overline{c}}{(n-1)^{3}}\\\
&=\dfrac{3\pi}{4(n-1)^{5/2}}+\overline{U}_{1}(n)+\overline{U}_{2}(n),\end{split}$
(2.22)

where

$\displaystyle\overline{U}_{1}(n)$ $\displaystyle=$
$\displaystyle-\dfrac{6\log\overline{\mu}(n+1)}{(n+1)^{3}}+\dfrac{2\log(\overline{\mu}(n-1)-1)}{(n-1)^{3}}$
(2.23) $\displaystyle\text{and}\ \ \overline{U}_{2}(n)$ $\displaystyle=$
$\displaystyle\dfrac{9}{2(n+1)^{3}}-\dfrac{5\pi}{4(n-1)^{5/2}(\overline{\mu}(n-1)-1)}-\dfrac{\pi^{2}}{4(n-1)^{2}(\overline{\mu}(n-1)-1)^{2}}+\dfrac{2\log\overline{c}}{(n-1)^{3}}.$

It can be easily check that for all $n\geq 2$,

$\overline{U}_{2}(n)<\dfrac{4}{(n-1)^{3}}.$ (2.25)

For an upper bound of $\overline{U}_{1}(n)$, we observe that for all $n\geq
15$,

$\dfrac{2}{(n-1)^{3}}<\dfrac{3}{(n+1)^{3}}\ \ \text{and}\ \
\log(\overline{\mu}(n)-1)<\log\overline{\mu}(n+1),$ (2.26)

that is,

$\dfrac{2\log(\overline{\mu}(n-1)-1)}{(n-1)^{3}}<\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}.$
(2.27)

Consequently for $n\geq 15$ we get,

$\overline{U}_{1}(n)<-\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}$ (2.28)

Invoking (2.25) and (2.28) into (2.22), we have for $n\geq 15$,

$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)<\dfrac{3\pi}{4(n-1)^{5/2}}-\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}+\dfrac{4}{(n-1)^{3}}=\overline{G}_{2}(n).$
(2.29)

For lower bound of $\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)$, using
(2.17) and (2.18)-(2.21) we obtain

$\begin{split}\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)>&\dfrac{3\pi}{4(n+1)^{5/2}}+\dfrac{9}{2(n-1)^{3}}-\dfrac{6\log\overline{\mu}(n-1)}{(n-1)^{3}}\\\
&+\dfrac{2\log(\overline{\mu}(n+1)-1)}{(n+1)^{3}}-\dfrac{5\pi}{4(n+1)^{5/2}(\overline{\mu}(n+1)-1)}-\dfrac{\pi^{2}}{4(n+1)^{2}(\overline{\mu}(n+1)-1)^{2}}\\\
&+\dfrac{2\log\overline{c}}{(n+1)^{3}}\\\
&=\dfrac{3\pi}{4(n+1)^{5/2}}+\overline{L}_{1}(n)+\overline{L}_{2}(n),\end{split}$
(2.30)

where

$\displaystyle\overline{L}_{1}(n)$ $\displaystyle=$
$\displaystyle-\dfrac{6\log\overline{\mu}(n-1)}{(n-1)^{3}}+\dfrac{2\log(\overline{\mu}(n+1)-1)}{(n+1)^{3}}$
(2.31) $\displaystyle\text{and}\ \ \overline{L}_{2}(n)$ $\displaystyle=$
$\displaystyle\dfrac{9}{2(n-1)^{3}}-\dfrac{5\pi}{4(n+1)^{5/2}(\overline{\mu}(n+1)-1)}-\dfrac{\pi^{2}}{4(n+1)^{2}(\overline{\mu}(n+1)-1)^{2}}+\dfrac{2\log\overline{c}}{(n+1)^{3}}.$

Similarly as before, one can check that for $n\geq 9$,

$\overline{L}_{2}(n)>0\ \ \ \text{and}\ \ \
\overline{L}_{1}(n)>-\dfrac{5\log\overline{\mu}(n-1)}{(n-1)^{3}}.$ (2.33)

(2.30) and (2.33) yield for $n\geq 9$,

$\Delta^{2}\dfrac{1}{n-1}\log\overline{T}(n-1)>\dfrac{3\pi}{4(n+1)^{5/2}}-\dfrac{5\log\overline{\mu}(n-1)}{(n-1)^{3}}=\overline{G}_{1}(n).$
(2.34)

(2.29) and (2.34) together imply (2.12) for $n\geq 15$. We finish the proof by
checking (2.12) numerically for $2\leq n\leq 14$. ∎

######  Lemma 2.3.

For $n\geq 38$,

$\bigl{|}\Delta^{2}\
\dfrac{1}{n-1}\overline{E}(n-1)\bigr{|}<\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}.$
(2.35)

  * Proof.

Using (2.8), we get for $n\geq 2$,

$\Delta^{2}\
\dfrac{1}{n-1}\overline{E}(n-1)=\dfrac{1}{n+1}\log(1+\overline{e}(n+1))-\dfrac{2}{n}\log(1+\overline{e}(n))+\dfrac{1}{n-1}\log(1+\overline{e}(n-1)),$
(2.36)

where

$\overline{e}(n)=\dfrac{\overline{R}(n)}{\overline{T}(n)}.$

Taking absolute value of $\Delta^{2}\ \dfrac{1}{n-1}\overline{E}(n-1)$ in
(2.36), we obtain for all $n\geq 2$,

$\bigl{|}\Delta^{2}\
\dfrac{1}{n-1}\overline{E}(n-1)\bigr{|}\leq\dfrac{1}{n+1}|\log(1+\overline{e}(n+1))|+\dfrac{2}{n}|\log(1+\overline{e}(n))|+\dfrac{1}{n-1}|\log(1+\overline{e}(n-1))|.$
(2.37)

Therefore, it is enough to estimate $|\overline{e}(n)|$. Before proceed to
estimate, let us recall the bound of Engel [6](cf. (1.3)) for $N=3$ that
yields for $n\geq 1$,

$|R_{2}(n,3)|<\dfrac{9\sqrt{3}}{2n\ \overline{\mu}(n)}e^{\overline{\mu}(n)/3}$
(2.38)

by making use of the fact that $\sinh(x)<\dfrac{e^{x}}{2}$ for $x>0$.
Recalling the definitions in (2.5)-(2.6), we obtain

$\displaystyle|\overline{e}(n)|$ $\displaystyle=$
$\displaystyle\Biggl{|}\dfrac{\biggl{(}1+\dfrac{1}{\overline{\mu}(n)}\biggr{)}}{\biggl{(}1-\dfrac{1}{\overline{\mu}(n)}\biggr{)}}e^{-2\overline{\mu}(n)}+\dfrac{R_{2}(n,3)}{\dfrac{1}{8n}\biggl{(}1-\dfrac{1}{\overline{\mu}(n)}\biggr{)}e^{\overline{\mu}(n)}}\Biggr{|}$
(2.39) $\displaystyle\leq$
$\displaystyle\dfrac{\biggl{(}1+\dfrac{1}{\overline{\mu}(n)}\biggr{)}}{\biggl{(}1-\dfrac{1}{\overline{\mu}(n)}\biggr{)}}e^{-2\overline{\mu}(n)}+\dfrac{36\sqrt{3}}{\overline{\mu}(n)\biggl{(}1-\dfrac{1}{\overline{\mu}(n)}\biggr{)}}e^{-2\overline{\mu}(n)/3}\
\ \ (\text{by}\ \eqref{lem3eqn4})$ $\displaystyle=$
$\displaystyle\dfrac{e^{-2\overline{\mu}(n)/3}}{\overline{\mu}(n)-1}\Biggl{[}\bigl{(}\overline{\mu}(n)+1\bigr{)}e^{-4\overline{\mu}(n)/3}+36\sqrt{3}\Biggr{]}$
$\displaystyle=$
$\displaystyle\dfrac{e^{-\overline{\mu}(n)/12}}{\overline{\mu}(n)-1}\Biggl{[}\Bigl{(}\bigl{(}\overline{\mu}(n)+1\bigr{)}e^{-4\overline{\mu}(n)/3}+36\sqrt{3}\Bigr{)}e^{-\overline{\mu}(n)/2}\Biggr{]}e^{-\overline{\mu}(n)/12}.$

It can be easily check that

$\dfrac{e^{-\overline{\mu}(n)/12}}{\overline{\mu}(n)-1}<1\ \ \ \text{for all}\
n\geq 1$ (2.40)

and

$\Bigl{(}\bigl{(}\overline{\mu}(n)+1\bigr{)}e^{-4\overline{\mu}(n)/3}+36\sqrt{3}\Bigr{)}e^{-\overline{\mu}(n)/2}<1\
\ \ \text{for all}\ n\geq 7.$ (2.41)

Invoking (2.40) and (2.41) into (2.39), we obtain for $n\geq 7$

$|\overline{e}(n)|<e^{-\overline{\mu}(n)/12}$ (2.42)

and consequently for $n\geq 38$,

$e^{-\overline{\mu}(n)/12}<\dfrac{1}{5}.$ (2.43)

Putting together (2.42) and (2.43), we get for all $n\geq 38$,

$|\overline{e}(n)|<\dfrac{1}{5}.$ (2.44)

Next we note that for all $n\geq 38$,

$|\log(1+\overline{e}(n))|\leq\dfrac{|\overline{e}(n)|}{1-|\overline{e}(n)|}<\dfrac{5}{4}\
|\overline{e}(n)|$ (2.45)

because of the fact that, for $|x|<1$,

$|\log(1+x)|<\dfrac{|x|}{1-|x|}.$

From (2.37) and (2.45), we obtain for $n\geq 38$,

$\bigl{|}\Delta^{2}\
\dfrac{1}{n-1}\overline{E}(n-1)\bigr{|}<\dfrac{5}{4}\Bigl{(}\dfrac{|\overline{e}(n+1)|}{n+1}+2\dfrac{|\overline{e}(n)|}{n}+\dfrac{|\overline{e}(n-1)|}{n-1}\Bigr{)}.$
(2.46)

Plugging (2.42) into (2.46), we have for $n\geq 38$,

$\displaystyle\bigl{|}\Delta^{2}\ \dfrac{1}{n-1}\overline{E}(n-1)\bigr{|}$
$\displaystyle<$
$\displaystyle\dfrac{5}{4}\Bigl{(}\dfrac{e^{-\overline{\mu}(n+1)/12}}{n+1}+2\dfrac{e^{-\overline{\mu}(n)/12}}{n}+\dfrac{e^{-\overline{\mu}(n-1)/12}}{n-1}\Bigr{)}$
(2.47) $\displaystyle<$
$\displaystyle\dfrac{5}{n-1}e^{-\overline{\mu}(n-1)/12}$

because the sequence $\Bigl{\\{}\dfrac{1}{n}\
e^{-\overline{\mu}(n)/12}\Bigr{\\}}_{n\geq 1}$ is decreasing. ∎

######  Lemma 2.4.

For $\alpha\in\mathbb{R}_{>0}$ and $n\geq 7$,

$-\dfrac{2\alpha\log(n-1)}{(n-1)^{3}}+\dfrac{3\alpha}{(n-1)^{3}}<-\alpha\
\Delta^{2}\dfrac{1}{n-1}\log(n-1)<-\dfrac{2\alpha\log(n+1)}{(n+1)^{3}}+\dfrac{3\alpha}{(n+1)^{3}}.$
(2.48)

  * Proof.

We observe that, for $n\geq 7$,

$\Biggl{(}-\dfrac{\log
n}{n}\Biggr{)}^{{}^{\prime\prime\prime}}=-\dfrac{11}{n^{4}}+\dfrac{6\log
n}{n^{4}}>0.$

Setting $f(n):=-\dfrac{\log n}{n}$ and applying Lemma 2.1, we obtain for
$n\geq 7$,

$-\dfrac{2\log(n-1)}{(n-1)^{3}}+\dfrac{3}{(n-1)^{3}}<-\Delta^{2}\dfrac{1}{n-1}\log(n-1)<-\dfrac{2\log(n+1)}{(n+1)^{3}}+\dfrac{3}{(n+1)^{3}}.$
(2.49)

Since $\alpha$ is a positive real number, from (2.49), we obtain (2.48). ∎

######  Lemma 2.5.

For $\alpha\in\mathbb{R}_{\geq 0}$ and $n\geq 4021$,

$\Delta^{2}\log
r_{\alpha}(n-1)<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}.$ (2.50)

  * Proof.

Using (2.12), (2.35) and (2.48) into (2.9), we obtain for $n\geq 38$,

$\Delta^{2}\log
r_{\alpha}(n-1)<\overline{G}_{2}(n)+\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}-\dfrac{2\alpha\log(n+1)}{(n+1)^{3}}+\dfrac{3\alpha}{(n+1)^{3}}.$
(2.51)

Note that for all $n\geq 4$,

$-\dfrac{2\alpha\log(n+1)}{(n+1)^{3}}+\dfrac{3\alpha}{(n+1)^{3}}\leq 0$ (2.52)

and for $n\geq 4021$,

$\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}<\dfrac{5}{(n-1)^{3}}.$
(2.53)

Therefore from (2.52)-(2.53), for all $n\geq 4021$, it follows that

$\Delta^{2}\log
r_{\alpha}(n-1)<\dfrac{3\pi}{4(n-1)^{5/2}}-\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}+\dfrac{9}{(n-1)^{3}}.$
(2.54)

Apparently, for all $n\geq 93$,

$\dfrac{3\pi}{4(n-1)^{5/2}}-\dfrac{3\log\overline{\mu}(n+1)}{(n+1)^{3}}+\dfrac{9}{(n-1)^{3}}<\dfrac{3\pi}{4n^{5/2}}-\dfrac{9\pi^{2}}{32n^{5}}.$
(2.55)

Using the fact that for $x>0$, $\log(1+x)>x-\dfrac{x^{2}}{2}$, from (2.54) and
(2.55), we finally arrive at

$\Delta^{2}\log r_{\alpha}(n-1)<\log\Bigl{(}1+\dfrac{3\pi}{4n^{5/2}}\Bigr{)}.$
(2.56)

∎

######  Lemma 2.6.

For $\alpha>0$ and
$n\geq\max\Biggl{\\{}\Bigl{[}\dfrac{3490}{\alpha}\Bigr{]}+2,\Bigl{\lceil}\Bigl{(}\dfrac{4(11+5\alpha)}{3\pi}\Bigr{)}^{4}\Bigr{\rceil},5505\Biggr{\\}}$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}\biggr{)}.$
(2.57)

  * Proof.

Using (2.12), (2.35) and (2.48) into (2.9), we obtain for $n\geq 250$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\overline{G}_{1}(n)-\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}-\dfrac{2\alpha\log(n-1)}{(n-1)^{3}}+\dfrac{3\alpha}{(n-1)^{3}}.$
(2.58)

It is easy to check that for
$n\geq\text{max}\Bigl{\\{}\Bigl{[}\dfrac{3490}{\alpha}\Bigr{]}+2,4522\Bigr{\\}}:=N_{1}(\alpha)$,

$-\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}+\dfrac{3\alpha}{(n-1)^{3}}>\dfrac{3\alpha}{(n-1)^{3}}-\dfrac{10470}{(n-1)^{4}}>0.$
(2.59)

Therefore for all $n\geq N_{1}(\alpha)$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\dfrac{3\pi}{4(n+1)^{5/2}}-\dfrac{5\log\overline{\mu}(n-1)}{(n-1)^{3}}-\dfrac{2\alpha\log(n-1)}{(n-1)^{3}}.$
(2.60)

It is immediate that for $n\geq 11$,

$\log\overline{\mu}(n-1)<\log(n-1)$ (2.61)

and for $n\geq 5505$,

$\log(n-1)<(n-1)^{1/4}.$ (2.62)

Putting (2.61) and (2.62) into (2.60), we obtain for
$n\geq\text{max}\\{N_{1}(\alpha),5505\\}$,

$\displaystyle\Delta^{2}\log r_{\alpha}(n-1)$ $\displaystyle>$
$\displaystyle\dfrac{3\pi}{4(n+1)^{5/2}}-\dfrac{5+2\alpha}{(n-1)^{11/4}}$
(2.63)

It remains to show that

$\dfrac{3\pi}{4(n+1)^{5/2}}-\dfrac{5+2\alpha}{(n-1)^{11/4}}>\dfrac{3\pi}{4n^{5/2}}-\dfrac{11+5\alpha}{n^{11/4}}.$
(2.64)

For $n\geq\
\text{max}\Biggl{\\{}\Biggl{\lceil}\Biggr{(}\dfrac{15\pi}{8(\alpha+1)}\Biggr{)}^{4/3}\Biggr{\rceil},5\Biggr{\\}}:=N_{2}(\alpha)$,
it follows that

$\dfrac{11+5\alpha}{n^{11/4}}-\dfrac{5+2\alpha}{(n-1)^{11/4}}\underset{n\geq
5}{>}\dfrac{1+\alpha}{n^{11/4}}>\dfrac{15\pi}{8n^{7/2}}\underset{n\geq
1}{>}\dfrac{3\pi}{4}\Biggl{(}\dfrac{1}{n^{5/2}}-\dfrac{1}{(n+1)^{5/2}}\Biggr{)}.$
(2.65)

From (2.63) and (2.64), we obtain for $n\geq\
\text{max}\Bigl{\\{}N_{1}(\alpha),N_{2}(\alpha)\Bigr{\\}}$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\dfrac{3\pi}{4n^{5/2}}-\dfrac{11+5\alpha}{n^{11/4}}.$ (2.66)

It is easy to check that for
$n\geq\Biggl{\lceil}\Bigl{(}\dfrac{4(11+5\alpha)}{3\pi}\Bigr{)}^{4}\Biggr{\rceil}:=N_{3}(\alpha)$,

$\dfrac{3\pi}{4n^{5/2}}-\dfrac{11+5\alpha}{n^{11/4}}>0$ (2.67)

and using the fact that for $x>0$, $x>\log(1+x)$, we finally get for
$n\geq\text{max}\\{N_{1}(\alpha),N_{3}(\alpha),5505\\}$
$\Bigl{(}\text{since},N_{3}(\alpha)>N_{2}(\alpha)\ \text{for}\
\alpha>0\Bigr{)}$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\log\Bigl{(}1+\dfrac{3\pi}{4n^{5/2}}-\dfrac{11+5\alpha}{n^{11/4}}\Bigr{)}.$
(2.68)

∎

_Proof of Theorem 1.1_: For $\alpha\in\mathbb{R}_{>0}$, from (2.50) and (2.57)
we obtain for all $n\geq\
\text{max}\Biggl{\\{}\Bigl{[}\dfrac{3490}{\alpha}\Bigr{]}+2,\Bigl{\lceil}\Bigl{(}\dfrac{4(11+5\alpha)}{3\pi}\Bigr{)}^{4}\Bigr{\rceil},5505\Biggr{\\}}$,

$\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11+5\alpha}{n^{11/4}}\biggr{)}<\Delta^{2}\log
r_{\alpha}(n-1)<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}.$ (2.69)

For $\alpha=0$, we have already seen that for $n\geq 4021$,

$\Delta^{2}\log
r_{\alpha}(n-1)<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}.$ (2.70)

For $\alpha=0$, from (2.58) we get for $n\geq 250$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\overline{G}_{1}(n)-\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}.$
(2.71)

Following the same approach, it can be checked that for $n\geq 4522$,

$-\dfrac{5}{n-1}e^{-\dfrac{\overline{\mu}(n-1)}{12}}>-\dfrac{10470}{(n-1)^{4}}$
(2.72)

and consequently for $n\geq 476$,

$\overline{G}_{1}(n)-\dfrac{10470}{(n-1)^{4}}>\dfrac{3\pi}{4n^{5/2}}-\dfrac{11}{n^{11/4}}>0.$
(2.73)

So, for $\alpha=0$, by (2.71)-(2.73), we obtain for $n\geq 4522$,

$\Delta^{2}\log
r_{\alpha}(n-1)>\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11}{n^{11/4}}\biggr{)}.$
(2.74)

Putting (2.70) and (2.74), for $n\geq 4522$, it follows that

$\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}-\frac{11}{n^{11/4}}\biggr{)}<\Delta^{2}\dfrac{1}{n-1}\log\overline{p}(n-1)<\log\biggl{(}1+\frac{3\pi}{4n^{5/2}}\biggr{)}.$
(2.75)

This finishes the proof. ∎

##  3\. Conclusion

We conclude this paper by considering the following problem;

######  Problem 3.1.

Let $\alpha$ be a non-negative real number. Then for each $r\geq 1$, does
there exists a positive integer $N(r,\alpha)$ so that for all $n\geq
N(r,\alpha)$, one can obtain both upper bound and lower bound of
$(-1)^{r}\Delta^{r}\log r_{\alpha}(n)$ that finally shows the asymptotic
growth of $(-1)^{r}\Delta^{r}\log r_{\alpha}(n)$ as $n$ tends to infinity?

For $r=2$, we have already seen that one can estimate $(-1)^{r}\Delta^{r}\log
r_{\alpha}(n)$, as given in Theorem 1.4 and its asymptotic growth is reflected
in Corollary 1.5.

Acknowledgements

The author would like to express sincere gratitude to her advisor Prof. Manuel
Kauers for his valuable suggestions on the paper. The research was funded by
the Austrian Science Fund (FWF): W1214-N15, project DK13.

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