# Emergence of Floquet edge states in the coupled Su-Schrieffer-Heeger model

Carla Borja  School of Physical Sciences and Nanotechnology, Yachay Tech
University, Urcuquí 100119, Ecuador  Esther Gutiérrez  Escuela Superior
Politécnica del Litoral, ESPOL, Departamento de Física, Facultad de Ciencias
Naturales y Matemáticas, Campus Gustavo Galindo Km. 30.5 Vía Perimetral, P. O.
Box 09-01-5863, Guayaquil, Ecuador  Alexander López<EMAIL_ADDRESS>Escuela Superior Politécnica del Litoral, ESPOL, Departamento de Física,
Facultad de Ciencias Naturales y Matemáticas, Campus Gustavo Galindo Km. 30.5
Vía Perimetral, P. O. Box 09-01-5863, Guayaquil, Ecuador

###### Abstract

The emergence of non equilibrium topological phases in low dimensional systems
offers an interesting route for material properties engineering. In
particular, we analyze the dynamical modulation of two coupled one-dimensional
chains, described by the Su-Schrieffer-Heeger model. We find that the
interplay of driving interactions and interchain coupling leads to the
emergence of non-equilibrium flat bands at zero and finite values of the quasi
energy spectrum. Using an effective Hamiltonian approach, we quantify the
emergent topological phases via the winding number and show that oscillations
in the mean pseudospin polarization arise as a consequence of the periodic
modulation. The patterns of these pseudospin oscillations are different for
the static trivial and topological phases offering a dynamical means to
distinguish both physical configurations. The system also exhibits non integer
values of the winding number, which have been recently reported experimentally
in connection to spin textures.

##  I Introduction

Since the pioneering work of Von KlitzingVonKlitzing1980  on the quantum Hall
effect (QHE), there have been a wealth of proposals for the observation of
physical states of matter with so called topological propertiesHaldane1988 ;
Kane2005 ; Bernevig2006 ; Koenig2007 ; Hasan2010 . Essentially, these
topological materials can support a bulk bandgap but have protected gapless
edge states at their boundaries. Typically, the protection is due to either
time reversal symmetry and/or spatial inversion symmetries although it has
been shown that spinless systems might possess protected edge states in
absence of time-reversal symmetryBernevig2014 . Within their proposal, the
authors show that the edge states are protected via the symmetries of a given
subset of the crystallographic point groups. Recently, many theoretical works
have addressed the role of different interactions in the generation of these
boundary protected edge states. From the experimental point of view, the first
observation of such topological properties in $HgTe/(Hg,Cd)Te$ quantum
wellsKoenig2007  paved the road for a plethora of other experimental and
theoretical works showing the potential of these materials for technological
implementations of nanodevices. The experimental realization and the
observation of topological phases in two Koenig2007  and three3DTI
dimensional materials has revealed the technological challenges in observing
dissipationless features in the transport regime of these systems. This in
turn has motivated a quest for lower dimensional systems with emergent
topological nontrivial phases. An experimentally realized one-dimensional
system, where nontrivial edge states can occur is the Su-Schrieffer-Heeger
(SSH) model, originally proposed to discuss the solitons in the poly-acetylene
moleculessh-original . Within this model, non-interacting spinless fermions
hop among lattice sites, subject to staggered potentials associated originally
to single and double bonding of the poly-acetylene molecule. More
interestingly, in the energy spectrum, these edge states appear in the form of
flatbands, meaning that they are localized at the boundaries of the sample.
Some experimental realizations of this one dimensional chain have been put
forward in referencesPlatero2013 ; Nature2013_Kitawaga ; Meier2016 ; acoustic-
ssh ; Nature2019Xie .

Further studies have considered the role of interactions in the SSH model. For
instance, upon the application of an external static electric field to the
one-dimensional SSH model, it is shown in reference Zak2020  that an anomalous
phase dielectric response can be found. In addition, recent
worksdal2015floquet ; StJean  explore the role of periodically driving
interactions in a single chain system and found via the associated non-
equilibrium Zak phaseZak1989  that nontrivial topological phases arise which
enrich the associated phase diagram among trivial and nontrivial sectors. This
Zak phase is directly related to the winding number which is a well
established topological invariant, and intuitively can be understood to
characterize the number of times a closed curve encircles a given point in
parameter space. In physical systems, the nontrivial nature of the winding
number is directly related to the emergence of states at the boundaries of the
sample. Therefore, these emergent states are also termed edge states.

Thus, by analyzing the emergence of boundary edge states induced by
periodically driving protocolsGrifoni1998  offers a means to study nontrivial
responses of low dimensional systems in the nonequilibrium realm.
Interestingly, the role of periodic driving has been extensively considered in
the literature. In pioneering works,Oka09 ; Dora2009 ; NP2011Lindner ;
PRB2014PerezPiskunow2 ; PRB2017Refael ; PRB2018Lindner2 ; PRL2018Refael  it is
shown that nontrivial topology can be induced in static systems by driving
fields creating the so-called Floquet Topological Insulators (FTI) which show
boundary edge states not achievable within the static scenario. Generally
speaking, the topological response of a physical system can be assessed via
several topological invariants, such as the Chern number, spin-Chern number
$\mathcal{Z}_{2}$, and higher order extensions. There is indeed a
classification for these topological systems which allows one to determine to
which class belongs a given systemtopoclass0 ; topoclass1 ; topoclass2 ;
topoclass3 ; topoclass4 . In driven one-dimensional systems, the Zak
phasedal2015floquet  has been proposed as a well defined topological invariant
that serves for characterization of the phase diagram relating trivial and
nontrivial phases in the relevant parameter regimes of the model considered.
These studies rely on the generation of the boundary edge states which encode
the nontrivial topological properties of the system under consideration. This
Zak phase can be related to the winding number and we focus on the last one
for discussing our results.

Recently, Li and co-workers li2017topological  have theoretically explored the
static coupling of two SSH chains and show that the associated phase diagram
derived from the winding number calculation, leads to further interesting
physical regimes. Actually, the interplay of different interactions is
expected to lead to the generation of trivial-nontrivial topological phase
transitions as, for instance, it was originally described in the Kane and
MeleKane2005  proposal for a two-dimensional topological insulator which was
based on the interplay of two competing spin-orbit interactions in graphene.
Their work was in turn an extension of the parity anomaly proposal previously
put forward by HaldaneHaldane1988  who showed that a quantum Hall effect (QHE)
could be achieved without the Landau levels required in its original
experimental realizationVonKlitzing1980 .

The role of electromagnetic radiation in a two chain coupled system, described
via the SSH model has been put forward in referencecoupledssh2020 , where the
authors show that in addition to the static topologically protected flat bands
arising within the gap between conduction and valence, a high frequency
radiation field induces finite-energy curved bands inside the gap of subbands.

In this work, we study two coupled one-dimensional SSH chains, which are
subject to periodic driving, focusing on the interplay among periodic driving
and interchain coupling to generate nontrivial topological configurations,
assessed via the winding number. The motivation for using a driven gate
voltage instead of the radiation field discussed in referencecoupledssh2020
is to offer experimentalists an alternative route to dynamically manipulate
the low dimensional topological properties of the system. Thus, by exploring a
finite-size system, we obtain the quasi energy spectrum and by analyzing the
zero energy edge states we show the emergence of flat bands in the quasienergy
spectrum, which are absent in the static counterpart configuration. In order
to quantify the emergent topological phases, we derive an effective bulk
Hamiltonian within the high frequency regime and give the corresponding quasi
energy spectra in momentum space. Then, it is given a characterization of the
effective high frequency dynamical topological effects by means of the winding
number. It is thus shown that the associated phase diagram allows for
additional trivial-nontrivial transitions, as compared to the static scenario.
This in turn, might lead to feasible proposals for on demand control of the
topological phases via periodic driving protocols on one dimensional systems
described with the SSH model. Moreover, we find that along one of the chains,
the pseudospin polarization, which is related to the occupation differences in
the bipartite system, shows driven oscillations. These driven oscillations can
be used to weight the bulk quasi energy spectrum for suitable initially
prepared states, providing an indirect quantification of interplay among
bandgap opening and difference in sublattice occupation due to the periodic
driving protocol. It allows us to distinguish among trivial and nontrivial
topological configurations.

The structure of the paper is as follows. In section II we present the model
and discuss the numerical results for the quasi energy spectra showing the
emergence of additional zero energy edge states for a finite size sample. We
also present the bulk properties, within the intermediate and high frequency
regimes for the quasi energy spectrum, winding number and pseudospin
polarization. Finally, we discuss our findings and give concluding remarks in
section III, arguing how a feasible experimental detection scheme of these
findings can be realized within currently available techniques. Finally, the
explicit calculations leading to the main results are given in the appendix
IV.

##  II model

We consider a periodically driven coupled system of two one-dimensional chains
with hopping non-interacting spinless electrons. As shown in panel (a) of
Figure1, which following li2017topological , the static Hamiltonian for the
bipartite system is given as

$H_{0}=\sum_{j=1}^{N}wa^{\dagger}_{j+1}b_{j}+va^{\dagger}_{j}b_{j+1}+ca^{\dagger}_{j}b_{j}+H.c.$
(1)

where $a^{\dagger}_{j}$ and $b^{\dagger}_{j}$ are the creation operators at
sublattice A and B of the $jth$ unite cell, whereas $v$ and $w$ denote the
intra and inter cell coupling strengths, respectively and $c$ measures the
interchain coupling parameter. For $c=0$, onerecovers the original SSH model,
that is extensively studied in the literature as a one dimensional system
supporting localized edge states at its boundaries. Interestingly, the SSH
Hamiltonian was originally proposed as a theoretical model to describe
solitary states in the poly-acetylene molecule ssh-original . Upon coupling
two of these SSH chains, it has been shown in reference li2017topological
that, a richer topological structure emerges where the system can be driven
between trivial and nontrivial topological phases meaning essentially that,
those previously mentioned edge states are present or absent at the system
boundaries.

Figure 1: Schematic representation of the two coupled one dimensional SSH
chains with hopping non-interacting spinless electrons with interchain
coupling strength $c$. (a) The shaded box represents the unit cell with static
intracell (intercell) parameter $v$ ($w$). (b) The system subject to periodic
driving with effective time-dependent parameters $v(t)$ and $w(t)$.

When the two-chain system is subject to a periodic driving protocol, with
effective time-dependent parameters $v(t)=v+G(t)$ and $w(t)=w+G(t)$, as shown
in panel (b) of Figure1, one can effectively describe the system with time-
dependent coupling strengths associated to the periodic contribution
$G(t)=G(t+T)$, with $t$ and $T$ being the time variable and period of the
perturbation, respectively. For an actual realization of the driving
interaction term, we consider the protocol $G(t)=2g\cos\Omega t$, proposed in
reference dal2015floquet  to study non-equilibrium properties of the SSH
system, i.e.

$V(t)=2g\cos\Omega
t\sum_{j=1}^{N}(a^{\dagger}_{j+1}b_{j}+a^{\dagger}_{j}b_{j+1}+h.c.)$ (2)

where $\Omega=2\pi/T$ and $g$ are the frequency and effective driving strength
parameters. Thus, one gets an effective time-dependent Hamiltonian periodic in
time. We now solve the quasi energy spectrum via an exact numerical
diagonalization of the driven problem. We first solve the problem for a finite
sized chain by numerically finding the quasi energy spectrum of the system and
discuss the resulting photoinduced band spectrum.

Although the Floquet theory has appeared more often in the literature, mostly
thanks to the theoretical proposals and experimental realizations of FTI, we
briefly give an account of the main ingredients necessary for dealing with the
quasi energy spectrum of the driven system. Hence, we consider a generic time-
dependent periodic Hamiltonian $H(t+T)=H(t)$, where $T$ is the time period of
the Hamiltonian. From the time-dependent Schrödinger equation, this
Hamiltonian generates the time evolution $i\partial_{t}\Psi(t)=H(t)\Psi(t)$.
The solution can be writing as $\Psi(t)=e^{i\epsilon t}\Phi(t)$, with
$\Phi(t+T)=\Phi(t)$ being the periodic part of the solution, which is termed
the Floquet state and $\epsilon$ is the quasi energyGrifoni1998 . This quasi
energy spectrum stems from the fact that the Hamiltonian is time-dependent,
implying that the energy is not a conserved quantity and it is similar to the
quasi momentum in spatially periodic systems. In addition, given the periodic
nature of the dynamical generator, the quasi energies are also periodic since
they are defined modulo $\hbar\Omega$.

Upon introduction of the ansatz $\Psi(t)=e^{i\epsilon t}\Phi(t)$ into the
time-dependent Schrödinger equation, one gets a time independent evolution
equation of the Floquet states: $\mathcal{H}\Phi(t)=\epsilon\Phi(t)$, with the
Floquet Hamiltonian $\mathcal{H}=i\partial_{t}-H(t)$. Then, one performs a
Fourier expansion and obtains an infinite dimensional eigenvalue equation
whose solutions give the aforementioned quasi energy spectrum of the problem.
Interestingly, the main physical features of a periodically driven system are
obtained from the Floquet spectrum and Floquet states.

(a)  (b)

(c)  (d)

Figure 2: Emergence of non equilibrium flat bands in the quasi energy spectrum
for driving frequency $\hbar\Omega=4c/3$ fixing $c=1$. In panels a-b (c-d) we
show the undriven (driven) regime $g=0$ ($g=0.25c$), as a function of the
intercell parameter $v$, with the intracell parameter satisfying $w/c=v/c$,
and $w/c+1=v/c$ for (a),(c) and (b),(d) respectively.

Thus, we now discuss the solutions of the quasi energy spectrum for the driven
coupled SSH model, first discussing the finite size sample with $N=10$ sites,
corresponding to $20$ dimers, which is a long enough sample to get the main
results reported here. Following referencedal2015floquet , we have restricted
the solution to the first harmonic corrections to the static Hamiltonian.
These harmonics correspond to the Fourier expansion of the periodic
Hamiltonian in the frequency realm and are also term side bands since, as it
was described in the description of the Floquet theory, they are connected to
the periodic nature of the quasi energy spectrum. In order to qualitatively
differentiate the coupled chains from the uncoupled situation, we have set the
interchain coupling term to $c=0.1w$ ($c=w$) to model weakly (strongly)
coupled chains. In the following results, the amplitude of the driving is set
to $g=0.25c$ and the frequency value to $\hbar\Omega=4c/3$.

Hence, we consider Figure2, where we show the results for the strongly coupled
chains. Panel (a) and (b) show the quasi energy spectrum without flat bands of
the undriven Hamiltonian described in referenceli2017topological , for the
parameter configurations $v=w$ and $w=v+c$ respectively. For the undriven
coupled chains, Li et alli2017topological  derived the effective static bulk
Hamiltonian (see next section), and establish the conditions for topologically
trivial and nontrivial static phases. For this purpose, they define

$\displaystyle x$ $\displaystyle=$ $\displaystyle(v+w)\cos k+c$ (3)
$\displaystyle y$ $\displaystyle=$ $\displaystyle(v-w)\sin k$

from which the winding number is obtained as

$\mathcal{N}=\frac{1}{2\pi}\int_{c}\frac{xdy-ydx}{r^{2}}$ (4)

Upon integration, the winding number reads

$\mathcal{N}=\begin{cases}0,&|v+w|<c\\\ {\rm
sign}{(v^{2}-w^{2})},&\text{otherwise}\end{cases}$ (5)

Within this context, the interpretation of the winding number is given in
terms of the loop in momentum space associated to the reciprocal space
representation of the bulk Hamiltonian of the system. As it is described in
the following section, it is calculated to fully characterize any emergent
topological phases.

Once the driving protocol is turned on, the Hamiltonian evolve to a
configuration showing zero quasi energy flat bands in a different $v$-range,
as depicted in the lower panels (c) and (d) of Figure2. In the next section we
show that the configuration $v=w=-0.3,c=1$ that presents edge-localized
states, as shown in Figure2-d, coincides with avoiding crossings in the bulk
spectrum. We also observe that, as expected, not only zero energy flat bands
appear in the quasi energy spectrum but additional side flat bands emerge
which are associated to the periodic nature of the quasi energy spectrum.
Therefore, as we turn on the perturbation, edge-localized states developed in
different energy levels because of the coupling between the Floquet replicas.
Moreover, the driven regime exhibits the ubiquitous bandgaps induced by the
time dependent perturbation. In order to get physical insight into the nature
of the emergence of these high frequency modulation effects, we now analyze
the bulk Hamiltonian and discuss the pseudospin polarization associated to the
differences in the occupation of the bipartite system.

###  II.1 Bulk Hamiltonian

Once we have numerically explored the quasi energy spectrum dynamics for the
finite-size system and we have obtained the interesting result that zero
energy edge states emerge in the driven scenario, we explore the dynamical
consequences for the bulk properties, as given via an effective bulk
Hamiltonian. For this purpose, we begin by remembering that, for one chain we
have the effective $2\times 2$ Hamiltonian in momentum space:

$H_{1}=\left(\begin{array}[]{cc}0&ve^{-ika}+we^{ika}\\\
ve^{ika}+we^{-ika}&0\end{array}\right)$ (6)

As it is schematically shown in the panel (a) of Figure1, $v$ ($w$) describes
the intra cell (inter cell) energy coupling within the single chain whereas
$a$ is the lattice spacing, which from now on is set as energy scale (i.e.
$a=1$), such that momenta are measured in units of $a^{-1}$. As shown in
Figure1, the layers are coupled via an effective term of strength $c$, which
leads to the bulk Hamiltonian li2017topological :

$H_{0}=\left(\begin{array}[]{cc}0&ve^{-ik}+we^{ik}+c\\\
ve^{ik}+we^{-ik}+c&0\end{array}\right)$ (7)

(a)  (b)

Figure 3: Momentum dependence of the bulk quasi energy spectra for the
undriven $g=0$ (panel a) and driven $g=0.25c$ (panel b) regimes. We have set
the parameters as $v=w=0.3c$, $c=1$. The ubiquitous photoinduced bandgaps are
present in panel b, and the color code represents the value of the mean
pseudospin polarization.

The bulk energy spectrum of the static Hamiltonian reads

$\varepsilon_{0,s}=s\sqrt{[(v+w)\cos k+c]^{2}+[(v-w)\sin k]^{2}}$ (8)

whereas its eigenstates can be written as

$\ket{\Psi_{s}(k)}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
se^{i\beta_{0}}\end{pmatrix},$ (9)

with

$\tan\beta_{0}=\frac{(v-w)\sin k}{(v+w)\cos k+c}.$ (10)

From the eigenstates, the winding number can be calculated as

$Z_{0}=\frac{i}{2\pi}\int^{\pi}_{-\pi}\bra{\Psi_{s}(k)}\partial_{k}\ket{\Psi_{s}(k)}dk$
(11)

which is equivalent to

$Z_{0}=-\frac{1}{2\pi}\int^{\pi}_{-\pi}\frac{\partial\beta_{0}}{\partial
k}dk,$ (12)

which reproduces the results of the phase diagram given in reference
li2017topological . The driving interaction in the bulk Hamiltonian takes the
form dal2015floquet

$V(t)=2g\cos(\Omega t)\left(\begin{array}[]{cc}0&-i\sin k\\\ i\sin
k&0\end{array}\right)=2g\cos(\Omega t)\sigma_{y}\sin(k)$ (13)

where $\sigma_{y}$ is the corresponding Pauli matrix in the canonical
representation where $\sigma_{z}$ is diagonal. In this manner, we get the
periodic time-dependent Hamiltonian

$H(t)=H_{0}+V(t).$ (14)

The full numerical solution of the intermediate frequency regime leads to the
results shown in figure Figure3 for either trivial and nontrivial topological
configurations. As expected for periodically driven systems, the bulk quasi
energy spectrum shows bandgaps which are due to the coupling among the
different harmonics or sidebands. The color code represents the weight of the
bulk pseudospin contribution with darker (lighter) zones corresponding to the
maximum (minimum) value of the average pseudospin polarization. It is apparent
that, for each configuration one can achieve maximum polarization at either
the center or the boundary of the first Brillouin zone, since there is a
largest weight of the polarization which can be used in order to select
polarization configurations suitable for different momenta.
In order to get more physical insight, we make further analytical progress by
constructing an effective Floquet Hamiltonian to leading order in the driving
strength, as it is explicitly shown below at different frequency regimes.

###  II.2 High frequency limit: Analytical Solution

The high frequency regime allows for an exact analytical treatment of the
driven problem since the Floquet sidebands get decoupled and the effects of
the driving interaction can be effectively captured via an energy
renormalization of the static energy spectrum. After some lengthy calculations
(presented in the appendix), and setting $\xi=2g\sin k/\hbar\Omega$, we get
the effective high frequency Floquet Hamiltonian:

$H_{F}=(v-w)\sin k\sigma_{y}+[(w+v)\cos k+c]J_{0}(\xi)\sigma_{x},$ (15)

where $J_{0}(\xi)$ is the zeroth order Bessel function of the first kind. The
approximate quasi energy spectrum reads then

$\varepsilon_{s}=\sqrt{[(v+w)\cos k+c]^{2}J^{2}_{0}(\xi)+(v-w)^{2}\sin^{2}k}$
(16)

for which the Floquet states are given by

$\ket{\phi_{s}(k)}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
se^{i\beta}\end{pmatrix},$ (17)

with

Figure 4: Phase diagram for the winding number, within the effective high
frequency Floquet Hamiltonian. The parameters are chosen as $c=1$, $g=c$ (left
panel) and $g=2c$ (right panel). Alternating trivial and nontrivial phases are
achieved for large values of the driving strength.

$\tan\beta=\frac{\tan\beta_{0}}{J_{0}(\xi)}=\frac{(v-w)\sin
k}{J_{0}(\xi)[(v+w)\cos k+c]}$ (18)

We can then assess the topological effects of the perturbation by numerically
evaluating the winding number

$Z=\frac{1}{2\pi}\int^{\pi}_{-\pi}dk\frac{d\beta}{dk},$ (19)

which is shown in Figure4, in the regime where the two chains are strongly
coupled $c=w=1$, for effective parameter $g=c$ (left panel) and $g=2c$ (right
panel). First of all, we observe in the right panel that a phase inversion is
induced at large values of the effective coupling, which exemplifies the
feasibility of the driving protocol to induce phase transitions which has been
shown to be attained by monochromatic laser irradiation. Strikingly, upon
analyzing the behaviour of the winding number along the line $c=w+v$ it is
interestingly found that additional $Z=\pm\frac{1}{2}$ arise (see right panel
on Figure4). This apparently paradoxical result emerges by explicit evaluation
of the parameters prior to integration and it is already present in the static
domain ($\xi\rightarrow 0$)

$Z_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}\frac{d}{dk}\left[\arctan{\left(\frac{(v-w)\sin
k}{c+(w+v)\cos k}\right)}\right]dk$ (20)

setting $c=v+w$, for $w=0$ one gets

$\frac{1}{2\pi}\int^{\pi}_{-\pi}\frac{d}{dk}\left[\arctan{\tan k/2}\right]dk$
(21)

which equals $1/2$.

Figure 5: Winding number for the undriven and driven scenarios. At low
interchain coupling values (left panel) $Z=\pm 1/2$ non integer value emerges,
whereas the strongly coupled chains configuration shows and isolated $Z=+1/2$
phase.

Although some theoretical works have put forward the idea of non integer
winding number in non hermitian systems and Ising systems with long range
correlations, and taking into account that these peculiar values happen at the
transition between two integer and well defined phases, their understanding
might require an experimental realization that could establish the physical
implications of the actual emergence of this apparently paradoxical result.

Having focused on the winding number and topological phases, we can also
evaluate another physical quantities of interest as the pseudospin
polarization, measuring the population imbalance among the two the bipartite
sublattices. For this purpose, we consider the system to be prepared in an
initial state $\ket{\psi_{k}}$ at $t=0$. Thus, the associated average lattice
polarization can be deduced from the definition

$\displaystyle\braket{\sigma_{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\bra{\psi_{k}}U^{\dagger}(t)\hat{\sigma_{z}}U(t)\ket{\psi_{k}}dt.$

Choosing the system to be initially prepared in an eigenstate of the static
Hamiltonian, given in equation (7), one finds the result (the details of the
calculation are deferred for the appendix)

$\langle\sigma_{z}\rangle=-s\sin(\beta-\beta_{0})\sin(2\pi\Sigma_{k}){\rm
sinc}(2\pi\Sigma_{k}),$ (22)

where $s=\pm 1$ and $\Sigma_{k}=\varepsilon_{k}/\hbar\Omega$. One then finds
an oscillating pattern in the average polarization which is modulated via the
phase difference $\beta-\beta_{0}$ of the dressed and static eigenstates,
respectively. In this manner, the role of the driving interaction is to
generate a non-vanishing average value for the pseudospin polarization. This
can explicitly be seen in Figure5 where we plot equation (22), for both
$w/v<1$ and $w/v>1$ configurations. Remarkably, from equation 22 in either
case, the polarization vanishes in the absence of the driving protocol.
Moreover, this non-vanishing of the $z$ component of the pseudospin degree of
freedom leading to driven pseudospin oscillations has the physical importance
of being connected to the chiral symmetry breaking which is absent in the
static regime. Thus, although from this polarization we can not directly infer
the topological phase of the system we can signal the violation of the chiral
symmetry needed for the system to remain in such topological state.

Figure 6: Average value of the pseudospin polarization as given in equation
(22), where we have chosen the $s=-1$ eigenstate of the static Hamiltonian.
The left (right) figure shows the polarization for the $w/v<1$ ($w/v>1$)
configurations.

##  III Discussion and conclusions

We have explored the emergent properties of two coupled one-dimensional SSH
chains subject to a periodically driving perturbation that drives the system
out of equilibrium. The exact numerical results in a finite size chain show
that emergent zero energy states appear in the quasi energy spectrum for
parameter values that, within the static regime, are absent. The reason for
the appearance of these additional edge states is the contribution arising
from the sidebands which modify the topology of the static energy bands for
the static Hamiltonian. In order to give an effective description of the
underlying physical processes that take place in the nonequilibrium regime, we
have studied the bulk properties of the system. As expected, we have found
that leading order bandgap contributions arise from the lower energy sidebands
which, in the high frequency regime, can renormalize the static energy band
spectrum or lead to band mixing that induces bandgaps responsible for the
topologically non-trivial response of the system in the intermediate frequency
regime. As it was reported in reference Platero2013 , we also find that
alternating phases can be achieved by tuning the effective strength of the
driving protocol. We would expect that these results could be experimentally
probed in the realm of ultracold atoms in optical lattices by a combination of
Bloch oscillations and Ramsey interferometryNature2013_Kitawaga  or by using
phase-sensitive injection, and adiabatic preparation schemes as reported in
reference Meier2016 . Strikingly, we find a configuration leading to a non
integer value for the winding number, which happens at one of the trivial-
nontrivial transition. Interestingly, the possibility for non integer values
of the winding number has been described recently in referencenoninteger . In
their work, the authors propose a generalization of the definition of this
topological invariant to make sense of the anomalous situations for which the
winding number is not well defined. Moreover, a recent experimental
worknonintegerexp , explores the physical feasibility of realizing these
potentially interesting non integer winding numbers in the skyrmion texture
present in $Cu_{2}OSeO_{3}$. The authors determine that although these
noninteger winding number values might be experimentally detected, they appear
to be energetically unstable and thus difficult to measure via their circular
dichroism technique. In addition, these non integer winding numbers have also
been recently predicted in static non hermitian materialsnonhermitian . In
their work, the authors introduced a complex Berry phase, which is used as a
quantized invariant to predict the existence of gapless edge modes in these
static non hermitian systems. The extension of driving protocols to these non
dissipative Hamiltonians, suggesting the generation of new types of Floquet
topological materials has been recently put forwarddrivennh . The authors
discuss and compare the skin effects in non Bloch band theory and their
emergent non Floquet system, which might find application in designing
dissipative photonic topological materials. Therefore, we might expect further
interesting developments dealing with the interplay of coupled chains,
described by non hermitian models, that could be explored elsewhere.

##  IV appendix

Here we present the details of the calculations for the relevant results
discussed in the main text.

###  IV.1 Derivation of the high frequency effective Floquet Hamiltonian

Starting from the periodic time-dependent Hamiltonian

$H(t)=H_{0}+V(t)$ (23)

with $H_{0}$ and $V(t)$ given by equations 7 and 13, respectively we
explicitly show how to derive the effective high frequency Hamiltonian given
in equation 15. In order to do so, we first perform a unitary transformation

$\tilde{H}(t)=P^{\dagger}(t)H(t)P(t)-i\hbar P^{\dagger}(t)\partial_{t}P(t)$
(24)

with the definition $P(t)=e^{\frac{i}{\hbar}\int V(t)dt}$, which in turn leads
to

$\tilde{H}(t)=e^{iM(t)\sigma_{y}}H(t)e^{-iM(t)\sigma_{y}}-V(t)$ (25)

where we have defined

$M(t)=\frac{1}{\hbar}\int f(t)dt\sin k=2g\sin k\frac{\sin(\Omega
t)}{\hbar\Omega}.$ (26)

Therefore, we get

$\displaystyle\tilde{H}(t)$ $\displaystyle=$
$\displaystyle[\tilde{H}(t)-V(t)]+e^{iM(t)\sigma_{y}}[H(t),e^{-iM(t)\sigma_{y}}]$
(27) $\displaystyle=$ $\displaystyle
H_{0}+e^{iM(t)\sigma_{y}}[H_{0},-i\sigma_{y}]\sin M(t)$ $\displaystyle=$
$\displaystyle H_{0}+(\cos M(t)+i\sin M(t)\sigma_{y})[((w+v)\cos
k+c)\sigma_{x},-i\sigma_{y}]\sin M(t)$ $\displaystyle=$ $\displaystyle
H_{0}+[(w+v)\cos k+c]e^{iM(t)\sigma_{y}}\sigma_{z}2\sin M(t)$

The last term can be worked out as follow:

$\displaystyle 2e^{iH(t)\sigma_{y}}\sigma_{z}\sin M(t)$ $\displaystyle=$
$\displaystyle 2(\cos M(t)\mathbb{1}+i\sin M(t)\sigma_{y})\sigma_{z}\sin M(t)$
(28) $\displaystyle=$ $\displaystyle\sin 2M(t)\sigma_{z}-(1+\cos
2M(t))\sigma_{x}$ $\displaystyle=$ $\displaystyle-\sigma_{x}+\sin
2M(t)\sigma_{z}+\cos 2M(t)\sigma_{x}$

which $\rightarrow 0$ as $M(t)$ (i.e. $g$) $\rightarrow 0$. Therefore,

$\tilde{H}(t)=H_{0}+[(\Omega+v)\cos(ka)+c]\sigma_{x}+[\sin
2M(t)\sigma_{z}+\cos 2M(t)\sigma_{x}][(v+w)\cos k+c]$ (29)

At high frequencies we find to leading order $\sin 2M(t)\approx 0$ and $\cos
2M(t)\approx J_{0}(\xi)$, with $\xi=\frac{2g\sin ka}{\hbar\Omega}$ and
$J_{0}(x)$ being the zeroth order Bessel function. Thus, we get the effective
Floquet Hamiltonian

$H_{F}=(v-w)\sin(ka)\sigma_{y}-[(\Omega+v)\cos(ka)+t]J_{0}(\xi)\sigma_{x}.$
(30)

###  IV.2 Derivation of the pseudspin polarization

Here, we start from the definition of the average pseudospin polarization

$\displaystyle\braket{\sigma_{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\bra{\psi_{k}}U^{\dagger}(t)\hat{\sigma_{z}}U(t)\ket{\psi_{k}}dt$
(31) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\bra{\psi_{k}}U^{\dagger}(\theta)\hat{\sigma_{z}}U(\theta)\ket{\psi_{k}}d\theta;\qquad\theta=\Omega
t$

where for simplicity we can choose

$\ket{\psi}=\left(\begin{array}[]{cc}0\\\ 1\end{array}\right)$ (32)

Notice that, within our effective Floquet Hamiltonian,

$U(t)=e^{-iH_{k}t/\hbar}=e^{-i\Sigma_{k}\sigma_{k}\theta}=U(\theta)$ (33)

where

$\begin{split}\Sigma_{k}\sigma_{k}&=\frac{1}{\hbar\Omega}\left[(v-w)\sin(ka)\sigma_{y}-J_{0}(\xi)\left[(v+w)\cos(ka)+c\right]\sigma_{x}\right]\\\
&=\Sigma_{k}\begin{pmatrix}0&e^{-i\beta}\\\
e^{i\beta}&0\end{pmatrix}\end{split}$ (34)

with

$\tan\beta=-\frac{(v-w)\sin(ka)}{J_{0}(\xi)\left[(v+w)\cos(ka)+c\right]}$ (35)

and

$\Sigma_{k}=\frac{1}{\hbar\Omega}\sqrt{(v+w)^{2}\sin^{2}(ka)+J_{0}^{2}(\xi)\left[(v+w)\cos(ka)+c\right]}=\frac{\varepsilon_{k}}{\hbar\Omega}$
(36)

Therefore

$U(\theta)=\cos(\Sigma_{k}\theta)\mathbb{1}-i\sin(\Sigma_{k}\theta)\sigma_{k}$
(37)

which leads to

$\begin{split}U^{\dagger}(\theta)\sigma_{z}U(\theta)&=\begin{pmatrix}\cos\theta&i\sin\theta
e^{-i\beta}\\\ i\sin\theta
e^{i\beta}&\cos\theta\end{pmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\cos\theta&-i\sin\theta e^{-i\beta}\\\
-i\sin\theta e^{i\beta}&\cos\theta\end{pmatrix}\\\
&=\begin{pmatrix}\cos\theta&-i\sin\theta e^{-i\beta}\\\ i\sin\theta
e^{i\beta}&-\cos\theta\end{pmatrix}\begin{pmatrix}\cos\theta&-i\sin\theta
e^{-i\beta}\\\ -i\sin\theta e^{i\beta}&\cos\theta\end{pmatrix}\\\
&=\begin{pmatrix}\cos 2\theta&i\sin 2\theta e^{-i\beta}\\\ i\sin 2\theta
e^{i\beta}&-\cos 2\theta\end{pmatrix}\\\
&=\begin{pmatrix}\cos(2\Sigma_{k}\theta)&-i\sin(2\Sigma_{k}\theta)e^{-i\beta}\\\
i\sin(2\Sigma_{k}\theta)e^{-i\beta}&-\cos(2\Sigma_{k}\theta)\end{pmatrix}\end{split}$
(38)

Upon substitution on the relation
$\left\langle\psi_{k}|U^{\dagger}(\theta)\sigma_{z}U(\theta)|\psi_{k}\right\rangle$
we get

$\begin{pmatrix}0&1\end{pmatrix}\begin{pmatrix}-i\sin(2\Sigma_{k}\theta)e^{-i\beta}\\\
-\cos(2\Sigma_{k}\theta)\end{pmatrix}=-\cos(2\Sigma_{k}\theta)$ (39)

Thus, we only had to evaluate

$\begin{split}\left\langle\sigma_{z}(k)\right\rangle&=\frac{-1}{2\pi}\int_{0}^{2\pi}\cos\left(2\frac{\varepsilon_{k}}{\hbar\omega}\theta\right)d\theta=\frac{\sin\left(2\frac{\varepsilon_{k}}{\hbar\Omega}\theta\right)}{2\pi\left(2\frac{\varepsilon_{k}}{\hbar\Omega}\theta\right)}\Big{|}_{0}^{2\pi}\\\
&={\rm sinc}\left[\frac{4\varepsilon_{k}\pi}{\hbar\Omega}\right]\end{split}$
(40)

Another interesting scenario arises when the system is initially prepared in
an eigenstate of the static Hamiltonian

$H_{0}=\varepsilon_{0k}\begin{pmatrix}0&e^{-i\beta_{0}}\\\
e^{i\beta_{0}}&0\end{pmatrix}$ (41)

where

$\begin{split}\varepsilon_{0k}&=\sqrt{[(v+w)\cos(ka)+c]^{2}+(v-w)^{2}\sin^{2}(ka)}\\\
\end{split}$ (42)

and

$tan\beta_{0}=\frac{(v-w)\sin(ka)}{(v+w)\cos(ka)+c}$ (43)

The corresponding eigenstates read

$|\psi_{\nu}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ \nu
e^{i\beta_{0}}\end{pmatrix}$ (44)

with $\nu=\pm 1$ such that
$H_{0}|\psi_{\nu}\rangle=\nu\epsilon_{0k}|\psi_{\nu}\rangle$. Thus, within
this approximate scenario we get for the $\nu=-1$ state

$\begin{split}\langle\psi_{-1}|U^{\dagger}(\theta)\sigma_{z}U(\theta)|\psi_{-1}\rangle&=\frac{1}{2}\begin{pmatrix}1&-e^{-i\beta_{0}}\end{pmatrix}\begin{pmatrix}\cos(2\Sigma_{k}\theta)&-i\sin(2\Sigma_{k}\theta)e^{-i\beta}\\\
i\sin(2\Sigma_{k}\theta)e^{-i\beta}&-\cos(2\Sigma_{k}\theta)\end{pmatrix}\begin{pmatrix}1\\\
-e^{i\beta_{0}}\end{pmatrix}\\\
&=\frac{1}{2}\begin{pmatrix}1&-e^{-i\beta_{0}}\end{pmatrix}\begin{pmatrix}\cos(2\Sigma_{k}\theta)+i\sin(2\Sigma_{k}\theta)e^{-i(\beta-\beta_{0})}\\\
i\sin(2\Sigma_{k}\theta)e^{i\beta}+\cos(2\Sigma_{k}\theta)e^{i\beta_{0}}\end{pmatrix}\\\
&=-\sin(2\Sigma_{k}\theta)\sin(\beta-\beta_{0})\end{split}$ (45)

Thus, the result for the average polarization reads now

$\begin{split}\langle\sigma_{z}\rangle&=-\frac{1}{2\pi}\int_{0}^{2\pi}\sin(2\Sigma_{k}\theta)\sin(\beta-\beta_{0})d\theta\\\
&=\sin(\beta-\beta_{0})\left[\frac{\cos(2\Sigma_{k}2\pi)-1}{2\pi
2\Sigma_{k}}\right]\\\ &=-\sin(\beta-\beta_{0})\sin(2\pi\Sigma_{k}){\rm
sinc}(2\pi\Sigma_{k})\end{split}$ (46)

###### Acknowledgements.

The authors thank R. Molina for useful discussions. This work is part of the
project: Estudio de la fractalidad en sistemas electrónicos at ESPOL.

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${\dagger}$${\dagger}$footnotetext: These authors contributed equally to this
work.$\ast$$\ast$footnotetext: Email<EMAIL_ADDRESS>(S.W.);
<EMAIL_ADDRESS>(Y.S.)

# Prediction of ambient pressure superconductivity in cubic ternary hydrides
with MH6 octahedra

Feng Zheng† School of Science, Jimei University, Xiamen 361021, China
Department of Physics, Xiamen University, Xiamen 361005, China  Zhen Zhang†
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011,
USA  Shunqing Wu∗ Department of Physics, Xiamen University, Xiamen 361005,
China  Qiubao Lin  School of Science, Jimei University, Xiamen 361021, China
Renhai Wang  School of Physics and Optoelectronic Engineering, Guangdong
University of Technology, Guangzhou 510006, China  Yimei Fang  Department of
Physics, Xiamen University, Xiamen 361005, China  Cai-Zhuang Wang  Department
of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA  Ames
National Laboratory, U.S. Department of Energy, Ames, IA 50011, USA  Vladimir
Antropov  Department of Physics and Astronomy, Iowa State University, Ames, IA
50011, USA  Ames National Laboratory, U.S. Department of Energy, Ames, IA
50011, USA  Yang Sun∗ Department of Physics, Xiamen University, Xiamen 361005,
China  Kai-Ming Ho  Department of Physics and Astronomy, Iowa State
University, Ames, IA 50011, USA

(Jan. 28, 2024)

###### Abstract

Exploring high-temperature superconducting (high-$T_{c}$) material at ambient
pressure holds immense significance for physics, chemistry, and materials
science. In this study, we perform a high-throughput screening of strong
electron-phonon interactions in X2MH6 compounds (X = Li, Na, Mg, Al, K, Ca,
Ga, Rb, Sr, and In; M are $3d$, $4d$, and $5d$ transition metals). These
compounds have a cubic structure featuring an MH6 octahedron motif. Our
screening calculations suggest that 26 compounds exhibit dynamic stability and
strong electron-phonon coupling. Among these 26 compounds, Mg2RhH6, Mg2IrH6,
Al2MnH6, and Li2CuH6 show promising energetic stability and $T_{c}$ of more
than 50 K at ambient pressure. This study underscores promising high-$T_{c}$
compounds at ambient pressure with distinctive MH6 motifs.

##  I Introduction

The idea of metallic hydrogen processing a significant high-pressure
superconductivity was suggested by Abrikosov [1] and Ashcroft [2] a long time
ago. However, the problem of making metallic hydrogen remains unsolved [3] and
the search for corresponding superconductivity was shifted to the area of
systems with partial hydrogen content, assuming that in such compounds, one
can reduce the physical pressure of hydrogen metallization due to chemical
pressure. In this way, the first superconducting hydride Th4H15 with $T_{c}$
$\sim$ 8.05-8.35 K was reported in 1970 [4]. After that, the superconductivity
of the Pd-H system was discovered with a $T_{c}$ of 8-9 K [5]. The Pd-(Cu, Ag,
and Au)-H system was also deeply studied with $T_{c}$ of 16.6, 15.6, and 13.6
K [6]. It appears that chemical pressure was not strong enough, and hydride
superconductivity research in recent years has shifted to using both physical
high-pressure and chemical pressure ideas with many successful discoveries,
including theoretical ones. Many H-rich compounds [7, 8, 9, 10, 11, 12, 13,
14, 15, 16] have been predicted to be promising conventional superconductors
with high $T_{c}$ under high pressure, using state-of-the-art crystal
structure search methods [17, 18, 19, 20, 21]. Among them, H3S [22], LaH10
[23, 24], CaH6 [25, 26], YH9 [27, 28, 29] and YH6 [27] have been
experimentally confirmed to have a $T_{c}$ greater than 200 K under high
pressures. These remarkable discoveries represent a significant milestone,
suggesting that conventional mechanisms through electron-phonon interaction
can achieve superconductivity close to room temperature [30, 31]. Although
hydrides can trap H atoms within their host lattice to achieve
superconductivity, the extremely high-pressure conditions required to
stabilize the structures is still challenging. It has only been attained in a
few experiments and hinders their practical applications. Therefore, the
research on superconductivity in hydrides necessitates a shift toward
exploring novel H-rich compounds that can be superconducting at ambient
pressure.

Extensive theoretical calculations have been conducted to shed light on
ambient-pressure superconducting hydrides. Vocaturo et al. predicted that
PdCuH2 possesses superconducting properties with $T_{c}$ $\sim$ 34 K at
ambient pressure [32]. Al4H ($Pm\bar{3}m$ space group) [33] and (Be4)2H (by
combining $hcp$ H and Be) [34] were also predicted to have $T_{c}$ of 54 and
72-84 K, respectively. However, these two structures are both metastable with
energy above convex hull ($E_{d}$) of 136 and 202 meV/atom [35]. Using
machine-learning accelerated high-throughput techniques, Tiago et al.
identified several hydride and hydrogen-containing materials with high-$T_{c}$
at ambient pressure, such as Li2PdH2 (43 K), PdH (37 K), ZrH3 (24.9 K), and
KCdH3 (23.4 K) [36]. Very recently, by employing ab initio random structure
search (AIRSS) across the periodic table, Dolui et al. predicted cubic Mg2IrH6
to be a superconductor at ambient pressure, with the $T_{c}$ up to 160 K [37].
The Mg2IrH6 is isostructural with the Mg2FeH6 family [38], which was
previously synthesized. Sanna et al. also investigated the superconducting
properties of the Mg2MH6 family (M = Rh, Ir, Pd, and Pt) and suggested their
superconducting $T_{c}$ are in the range of 45-80 K [35]. The crystal
structure of Mg2MH6 is a cubic phase with an MH6 octahedron structural motif.
A natural question remains whether other ternary superconducting hydrides can
be stabilized in this structural framework at ambient pressure.

In this paper, we employ our recently developed fast screening method [39] to
investigate the possible superconductivity in cubic X2MH6 compounds (X = Li,
Na, Mg, Al, K, Ca, Ga, Rb, Sr and In; M is $3d$, $4d$, and $5d$ transition
metal) at ambient pressure. The screening is based on the electron-phonon
coupling strength (EPC) at the $\Gamma$ point ($\lambda_{\Gamma}$), which was
found to be an efficient descriptor to identify conventional superconducting
materials in hydrides and borides [14, 39, 40, 41, 42, 43, 44]. We will
explore their superconducting behavior, energetic and dynamic stabilities
under ambient conditions.

##  II Computational methods

The structure optimizations were performed by using the projector-augmented
wave (PAW) [45] representations with density functional theory implemented in
the Vienna ab initio simulation package (VASP) [46, 47]. The exchange and
correlation energy is treated within the spin-polarized generalized gradient
approximation (GGA) and parameterized by Perdew-Burke-Ernzerhof (PBE) formula
[48]. A plane-wave basis was used with a kinetic energy cutoff of 520 eV.
Brillouin zone integrations were approximated by using special k-point
sampling of the Monkhorst-Pack scheme [49] with $2\pi\times 0.02$
$\rm{\AA}^{-1}$. Lattice vectors and atomic coordinates were fully relaxed
until the force on each atom was less than 0.01 eV$\cdot\rm{\AA}^{-1}$. The
fast screening of EPC constant $\lambda_{\Gamma}$ at the Brillouin zone center
was carried out based on the frozen-phonon method [39]. Here, the
$\lambda_{\Gamma}$ can be defined by

$\lambda_{\Gamma}=\sum_{\nu}\lambda_{\Gamma\nu},$ (1)

where $\sum_{\nu}$ indicates the summation of all modes at zone-center
$\Gamma$. $\lambda_{\Gamma\nu}$ is defined by

$\lambda_{\Gamma\nu}=\frac{\widetilde{\omega}_{\Gamma\nu}^{2}-\omega_{\Gamma\nu}^{2}}{4\omega_{\Gamma\nu}^{2}}$
(2)

where the $\omega_{\Gamma\nu}$ and $\widetilde{\omega}_{\Gamma\nu}$ are
screened and unscreened phonon frequencies of mode $\nu$ at zone-center,
respectively. The zone-center phonon was computed by the PHONOPY software [50,
51], with a k-point sampling grid of $2\pi\times 0.02$ $\rm{\AA}^{-1}$ spacing
and a criterion of self-consistent calculation 10-8 eV.

The full Brillouin-zone EPC calculation was performed with the Quantum
ESPRESSO (QE) code [52, 53] based on the density-functional perturbation
theory (DFPT) [54]. The pseudopotentials (stringent norm-conserving set) from
the PSEUDODOJO project [55] for PBE functional were used. The kinetic energy
cutoffs were 120 Ry for wave functions and 600 Ry for potentials. Monkhorst-
Pack’s sampling scheme [49] was adopted for Brillouin-zone sampling with a
k-point grid of $2\pi\times 0.02$ $\rm{\AA}^{-1}$. The DFPT calculations were
performed with the k-mesh of exactly the sampling scheme and set the q-mesh to
half of the k-mesh. The convergence threshold for self-consistency was
$1\times 10^{-14}$ Ry.

The calculations of superconducting $T_{c}$ are based on the Eliashberg
spectral equation $\alpha^{2}F(\omega)$ defined commonly now as

$\alpha^{2}F(\omega)=\frac{1}{2\pi
N\left(E_{f}\right)}\sum_{q\nu}\frac{\gamma_{q\nu}}{\hbar\omega_{q\nu}}\delta\left(\omega-\omega_{q\nu}\right),$
(3)

where $N(E_{f})$ is the density of states at the Fermi level, $\omega_{q\nu}$
denotes the phonon frequency of the mode $\nu$ with wave vector q.
$\gamma_{q\nu}$ is the phonon linewidth defined as

$\gamma_{q\nu}=\frac{2\pi\omega_{q\nu}}{\Omega_{BZ}}\sum_{ij}\int
d^{3}k\left|g_{k,q\nu}^{ij}\right|^{2}\delta\left(\epsilon_{q,i}-E_{f}\right)\delta\left(\epsilon_{k+q,j}-E_{f}\right),$
(4)

where $\epsilon_{q,i}$ and $\epsilon_{k+q,j}$ are eigenvalues of Kohn-Sham
orbitals at given bands and vectors. q and k are wave vectors, and $i$ and $j$
denote indices of energy bands. $g_{k,q\nu}^{ij}$ is the EPC matrix element.
The EPC constant $\lambda$ can be determined through summation over the first
Brillouin zone or integration of the spectral function in frequency space,

$\lambda=\sum_{q\nu}\lambda_{q\nu}=2\int\frac{\alpha^{2}F(\omega)}{\omega}d\omega,$
(5)

where the EPC constant $\lambda_{q\nu}$ for mode $\nu$ at wave vector q using
Eq. (5) can be written as

$\lambda_{q\nu}=\frac{\gamma_{q\nu}}{\pi\hbar
N\left(E_{f}\right)\omega_{q\nu}^{2}}.$ (6)

The superconducting $T_{c}$ is determined with the analytical McMillan
equation modified by the Allen-Dynes (A-D) equation [56, 57],

$T_{c}=\frac{\omega_{\log}}{1.2}\exp\left[\frac{-1.04(1+\lambda)}{\lambda\left(1-0.62\mu^{*}\right)-\mu^{*}}\right],$
(7)

where $\omega_{\log}$ is the logarithmic average frequency

$\omega_{\log}=\exp\left[\frac{2}{\lambda}\int\frac{\mathrm{d}\omega}{\omega}\alpha^{2}(\omega)F(\omega)\log\omega\right]$
(8)

and the effective screened Coulomb repulsion constant $\mu^{*}$ was 0.1 in our
calculations.

##  III Results and Discussion

###  III.1 High throughput screening of superconductivity in X2MH6

Figure 1: (a) The prototype structure of X2MH6. The X, M, and H atoms are
colored in orange, green and white, respectively. (b) Schematic of screening
workflow. (c) The number of nonmagnetic compounds of X2MH6 with X in different
groups. (d) The elements involved in the screening. Orange indicates the
elements of X. Green indicates the elements of M.

Figure 1 summarizes our screening process of superconducting ternary hydrides
at ambient pressure. As shown in Fig. 1(a), the prototype structure X2MH6 is a
cubic phase with MH6 octahedron structural motif, which is isostructural with
Mg2FeH6 [38]. By replacing X with Li, Na, Mg, Al, K, Ca, Ga, Rb, Sr, and In,
and M with $3d$, $4d$, and $5d$ transition metals (seen Fig. 1d), we generate
290 candidates for screening. The screening criteria are shown in Fig. 1 (b).
The X2MH6 compounds are all relaxed by the structural optimization at 0 GPa.
After spin-polarized calculations, we remove magnetic compounds (magnetic
moments are larger than 0.3 $\mu_{B}$/M) since the conventional
superconductivity is mutually exclusive with magnetism. This step reduces the
population of the candidate pool to 199 compounds. We find that the number of
nonmagnetic compounds of X2MH6 can be influenced by the different groups of X
ions, as shown in Fig. 1(c). X2MH6 with X in the IIIA group has the highest
number of nonmagnetic compounds, followed by those in the IIA and IA groups.
These nonmagnetic compounds are screened by the zone-center EPC strength
calculations. By setting a threshold of 0.3 ($\lambda_{\Gamma}>0.3$), 32
materials are selected for further calculations. Among them, 26 compounds are
dynamically stable. To evaluate their thermodynamic stability, the energy
above the convex hull ($E_{d}$) is computed and shown in Table 1. The
reference phases in the convex hulls are obtained from the Material Project
database [58]. The $E_{d}$ represents the energy of a material to decompose
into a set of stable phases. If the $E_{d}$ = 0, it indicates that the phase
is stable. However, a metastable phase with small $E_{d}$ can also be
synthesized, given that many registered structures in the Inorganic Crystal
Structure Database (ICSD) are metastable, according to the recent survey [59].
We employ the $E_{d}$ less than 80 meV/atom as the criterion, corresponding to
a typical thermal energy of $\sim$ 900 K [60, 61]. After these screening, six
low-energy superconductors are ultimately identified, i.e., Mg2CoH6, Mg2RhH6,
Mg2IrH6, Ca2PtH6, Al2MnH6 and Li2CuH6. We then conduct full Brillouin-zone EPC
and $T_{c}$ calculations and identify four compounds with $T_{c}$ greater than
50 K at ambient pressure, i.e., Mg2RhH6, Mg2IrH6, Al2MnH6 and Li2CuH6. The
superconducting behaviors of these compounds are studied in the next section.
Superconductivity calculations for the remaining compounds are listed in
Supporting Information.

Table 1: The zone-center EPC strength ($\lambda_{\Gamma}$) and energy above convex hull ($E_{d}$) for 26 dynamically stable ternary hydrides. Ternary hydrides | $\lambda_{\Gamma}$ |  $E_{d}$ (meV/atom)
---|---|---
Mg2RhH6 | 6.23 | 0
Mg2IrH6 | 3.27 | 0
Ca2PtH6 | 0.75 | 9
Mg2CoH6 | 0.42 | 29
Al2MnH6 | 0.96 | 80
Li2CuH6 | 0.41 | 80
Ca2PdH6 | 0.40 | 97
Sr2CuH6 | 0.31 | 99
Ca2CuH6 | 0.32 | 109
Na2CuH6 | 0.30 | 112
Na2AuH6 | 0.31 | 113
Rb2CuH6 | 0.30 | 113
Al2TcH6 | 1.67 | 127
In2TcH6 | 1.27 | 151
Ga2TcH6 | 0.70 | 172
Mg2PtH6 | 1.56 | 185
In2MnH6 | 0.32 | 210
Li2AuH6 | 0.64 | 217
K2AgH6 | 0.30 | 259
Rb2AgH6 | 0.31 | 266
Na2AgH6 | 0.31 | 286
Rb2CdH6 | 0.35 | 324
Ga2RuH6 | 0.65 | 327
K2CdH6 | 0.65 | 386
Rb2ZnH6 | 1.35 | 422
Ga2MnH6 | 1.14 | 778

###  III.2 Superconductivity of Mg2RhH6, Mg2IrH6, Al2MnH6 and Li2CuH6

####  III.2.1 Mg2RhH6 and Mg2IrH6

Figure 2: The electronic band structure and density of states projected into
atoms and orbitals for (a) Mg2RhH6 and (b) Mg2IrH6.

Figure 2 shows the band structures and density of states projected into atoms
and orbitals for Mg2RhH6 and Mg2IrH6. One can see that they have nearly
identical electronic structures, which are both metallic with a van Hove
singularity at the Fermi level ($E_{f}$). The calculated total DOS at $E_{f}$
are 3.09 and 3.17 eV-1 per f.u. for Mg2RhH6 and Mg2IrH6, respectively. Near
the $E_{f}$, the states are predominantly composed of transition metals
$e_{g}^{*}$ antibonding states mixed with H-1$s$ ($\sim$ 0.30 and $\sim$ 0.28
eV-1 per f.u. for Mg2RhH6 and Mg2IrH6, respectively) and Mg-3$s$ states. In
the valence band of these two compounds, sharp peaks are observed in the
$t_{2g}$ states, corresponding to the flat bands along the high-symmetry path.
The $t_{2g}$ nonbonding bands of Mg2RhH6 and Mg2IrH6 exhibit a significant
energy gap with their $e_{g}^{*}$ antibonding states of $\sim$ 2.6 and $\sim$
3.3 eV, respectively. The $e_{g}$ bonding bands at a lower energy level also
display a sharp peak. We note that the summation of the projected DOS in the
PAW sphere is smaller than the total DOS because of significant charge
accumulations in the interstitial region. The intersititial contribution to
the density of states at the Fermi level is significant in these systems.

Figure 3: The $\gamma_{q\nu}$-weighted (red circles) phonon spectrum,
projected phonon density of states (PHDOS) and Eliashberg spectral function
$\alpha^{2}F(\omega)$ for (a) Mg2RhH6 and (b) Mg2IrH6. (c) The vibration
pattern for double degenerate $E_{g}$ modes at the $\Gamma$ point. The
displacements are denoted by red arrows.

Figure 3 displays the phonon linewidth ($\lambda_{q\nu}$)-weighted phonon
spectrum, the projected phonon density of states (PHDOS), and Eliashberg
spectral function $\alpha^{2}F(\omega)$ for Mg2RhH6 and Mg2IrH6. The phonon
spectra of these two compounds are divided into three distinct regions. In the
low-frequency region, the spectra are primarily dominated by transition metals
and Mg atoms. The intermediate and high-frequency regions mainly arise from
the vibrations of H atoms. As depicted in Fig. 3(a), for Mg2RhH6, the high-
frequency H modes significantly contribute to the phonon linewidth, especially
the double degenerate phonon modes $E_{g}$ at the $\Gamma$ point. Their
vibrational modes are shown in Fig. 3(c), which involve the breathing
vibrations of H atoms in the MH6 octahedra. The intermediate-frequency region
from $\sim$ 0.075 to $\sim$ 0.105 eV and low-frequency modes at $\sim$ 0.021
eV along $\Gamma$ point corresponding to H vibrations also exhibits phonon
linewidth. The integrated EPC parameter $\lambda$ for Mg2RhH6 is 2.62. The
logarithmic average frequency ($\omega_{\log}$) can be derived from the A-D
equation of 352.9 K. The $T_{c}$ is predicted to be 59 K. Similar to Mg2RhH6,
the phonon linewidth in Mg2IrH6 is primarily contributed by high and
intermediate-frequency H modes, as shown in Fig. 3(b). The high-frequency
phonon modes in Mg2IrH6 show slight hardening compared to Mg2RhH6, where the
frequency of $E_{g}$ modes increases from 0.149 to 0.189 eV. The EPC constant
$\lambda$ for Mg2IrH6 is 2.56 with $\omega_{\log}$ of 391.4 K. The $T_{c}$ is
predicted to be 65 K for Mg2IrH6.

####  III.2.2 Al2MnH6

Figure 4: Electronic structure and electron-phonon calculations for Al2MnH6.
(a) Electronic band structure and density of states projected into atoms and
orbital. (b) The $\gamma_{q\nu}$-weighted (red circles) phonon spectrum,
projected phonon density of states (PHDOS) and Eliashberg spectral function
$\alpha^{2}F(\omega)$.

Figure 4 (a) presents the electronic structure of Al2MnH6. Analogous to
Mg2RhH6 and Mg2IrH6, a van Hove singularity is observed at $E_{f}$,
contributed by flat bands along the high-symmetry path (2/3$\Gamma$-X-1/2W and
1/2$\Gamma$-U). The calculated total DOS is 3.25 eV-1 per f.u, primarily
contributed from transition metal (Mn) antibonding $e_{g}^{*}$ states mixed
with H (0.21 eV-1 per f.u) and Al states. The weaker Mn $t_{2g}$ nonbonding
bands are located near 2.5 eV, exhibiting a sharp peak. The Mn $e_{g}$ bonding
bands are at a lower energy level, showing strong hybridization with H-1$s$
bands over a broad energy range, approximately from -5 to -14 eV.

EPC calculations for Al2MnH6 are shown in Fig. 4(b). The phonon spectrum of
Al2MnH6 is divided into three distinct regions. The phonon modes of Al and Mn
atoms are located in the low-frequency range, from 0 to $\sim$ 0.048 eV. The H
modes are in two regions, from $\sim$ 0.065 to $\sim$ 0.126 eV and $\sim$
0.142 to $\sim$ 0.247 eV. Unlike Mg2RhH6 and Mg2IrH6, the doubly degenerated
phonon modes $E_{g}$ at the $\Gamma$ point do not exhibit a strong phonon
linewidth in Al2MnH6. Only the high-frequency mode 22 of the H atom
contributes significantly to the phonon linewidth along the Brillouin zone
boundary. The H modes at $\sim$ 0.102 eV along $\Gamma$ and L points also
exhibit substantial phonon linewidth. The EPC constant $\lambda$ for Al2MnH6
is 1.38, lower than that of Mg2RhH6 and Mg2IrH6. However, the predicted
$T_{c}$ of Al2MnH6 can reach up to 66 K, due to a high logarithmic average
phonon frequency $\alpha^{2}F(\omega)$ ($\sim$ 631.1 K).

####  III.2.3 Li2CuH6

Figure 5: Electronic structure and electron-phonon calculations for Li2CuH6.
(a) Electronic band structure and density of states projected into atoms and
orbital. (b) The $\gamma_{q\nu}$-weighted (red circles) phonon spectrum,
projected phonon density of states (PHDOS) and Eliashberg spectral function
$\alpha^{2}F(\omega)$.

Figure 5(a) displays the electronic structure of Li2CuH6. It also exhibits a
van Hove singularity at $E_{f}$, with a total DOS of 1. 51 eV-1 per f.u.,
smaller than that of Mg2RhH6, Mg2IrH6 and Al2MnH6. The contribution of
Cu-$e_{g}^{*}$ antibonding states and H-1$s$ states ( 0.40 eV-1 per f.u.) is
almost equal at $E_{f}$. In the valence band, the Cu-$t_{2g}$ nonbonding bands
and $e_{g}$ bonding bands exhibit sharp peaks, resulting in distinct flat
bands along the entire high-symmetry path.

Figure 5(b) presents EPC calculations for Li2CuH6. Similar to Mg2RhH6 and
Mg2IrH6, the doubly degenerated phonon modes $E_{g}$ at the $\Gamma$ point in
Li2CuH6 significantly contribute to the phonon linewidth, with a lower
frequency of 0.143 eV. The integrated EPC parameter $\lambda$ for Li2CuH6 is
2.30. The $\omega_{\log}$ derived from Eq. (8), is 500.1 K, leading to a
predicted $T_{c}$ of 80 K.

##  IV Conclusions

By employing the high-throughput screening method of zone-center electron-
phonon interaction, we thoroughly investigate the superconductivity of cubic
ternary hydrides X2MH6 (X = Li, Na, Mg, Al, K, Ca, Ga, Rb, Sr and In; M is
$3d$, $4d$, and $5d$ transition metal) at ambient pressure. We identify 26
compounds demonstrating dynamic stability with high $\lambda_{\Gamma}$. Among
these, several have been verified to be superconductors through our EPC
calculations. Notably, four compounds (Mg2RhH6, Mg2IrH6, Al2MnH6 and Li2CuH6)
have a $T_{c}$ above 50 K and a low energy above convex hull ($E_{d}<80$
meV/atom), making them promising candidates for experimental synthesis. The
$T_{c}$ of Li2CuH6 is 80 K, which is higher than the liquid-nitrogen
temperature. The calculated electronic structures of these four compounds
reveal that they all possess a sharp van Hove singularity at the Fermi level,
accounting for their high $T_{c}$. Our study indicates a significant number of
candidates displaying superconductivity in cubic ternary hydrides X2MH6 system
at ambient pressure, which also demonstrates an effective strategy to explore
conventional superconductors in ternary hydrides by taking into account
distinctive structural motifs like MH6 octahedron. This work is expected to
reignite interest in investigating the new ternary hydrides with high $T_{c}$
at ambient pressure.

##  V Acknowledgments

The work at Jimei University was supported by the Natural Science Foundation
of Xiamen City (Grant No. 3502Z202372015) and the Research Foundation of Jimei
University (Grant No. ZQ2023013). The work at Xiamen University was supported
by the Natural Science Foundation of Xiamen (Grant No. 3502Z202371007) and the
Fundamental Research Funds for the Central Universities (Grant No.
20720230014). V.A. was supported by the U.S. Department of Energy, Office of
Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames
National Laboratory is operated for the U.S. Department of Energy by Iowa
State University under Contract No. DE-AC02-07CH11358. K.-M.H. acknowledges
support from National Science Foundation Award No. DMR2132666.

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