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independent configurations.
simulations, e.g. FIG. 7, such local CDW order persists
even at late stage of the phase-ordering where large do-
mains of orbital order have been established. It is likely
that the local CDW order is stabilized as an intermediate
structure in a multi-domain state during relaxation.
Next we discuss the coarsening of the orbital domains.
As discussed in Sec. I, the C-type orbital order is de-
scribed by an Ising order parameter. And since the or- | [
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] |
scribed by an Ising order parameter. And since the or-
bital order and JT distortion are not subject to a con-
servation law, the coarsening of the orbital/JT order
is expected to be described by the universality class of
the phase ordering of a non-conserved Ising order. It is
worth noting that the coarsening dynamics of Ising or-
der has been thoroughly characterized and classified into
several super-universal classes which depend on whether
the Ising order is conserved and the presence of quenched | [
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] |
the Ising order is conserved and the presence of quenched
disorder.
For non-conserved Ising order, which is also
the case for the C-type orbital order here, the coarsening
dynamics is described by a curvature-driven mechanism
summarized in the Allen-Cahn equation [45]. This results
in a specific power-law growth L ∼ t1/2 of Ising domains
applicable to two and higher dimensions [46–48]. A natu-
ral question is to see whether the coarsening of the orbital | [
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] |
ral question is to see whether the coarsening of the orbital
domains falls into this Allen-Cahn universality class.
To quantify the coarsening dynamics, we consider the
time-dependent correlation length of the orbital/JT or-
der. To this end, we first compute the time-dependent
structural factor of the staggered distortion defined as
S(k, t) = |Qx(k, t)|2, where Qx(k, t) is the Fourier trans-
form the JT Qx configuration
Qx(k, t) = 1
N
�
i
Qx
i eik·ri.
(22) | [
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] |
form the JT Qx configuration
Qx(k, t) = 1
N
�
i
Qx
i eik·ri.
(22)
In FIG. 9 the structure factor is plotted at various time
steps after a sudden quench to a low temperature T =
0.01. The emergence of the C-type order corresponds to
a structure-factor peak at the wave vector K = (π, π).
However, instead of a delta peak which is characteristic
of a long-range order, a diffusive peak was observed even
at late times of the phase ordering.
The finite width | [
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] |
at late times of the phase ordering.
The finite width
of the diffusive peak is due to the presence of multiple
orbital domains of opposite Ising orders. As the system
progresses towards equilibrium, the coarsening of these
ordered domains results in a stronger and sharper peak at
K, as can be seen in FIG. 9. The inverse of the width can
thus provide a quantitative estimate for the characteristic
length scale of ordered domains
L−1(t) = ∆k =
�
k
S(k, t)|k − K|
��
k
S(k, t),
(23) | [
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0.05054250732064247,
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0.20278465747833252,
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0.6012009382247925,
0.037745341658592224,
1.0564981698989868,
0.6838711500167847,
0.228634774684906,
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0.04853538051247597,
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0.4102093577384949,
0.1748892217874527,
0.20904502272605896,
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0.132811039686203,
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] |
length scale of ordered domains
L−1(t) = ∆k =
�
k
S(k, t)|k − K|
��
k
S(k, t),
(23)
This characteristic size of orbital domains also provides
a measure of the orbital correlation, i.e.
Cxx(r, t) ∼ e−|r|/ξ(t)eiK·r,
(24)
where the correlation length is proportional to the char-
acteristic domain size, ξ ∼ L.
FIG. 10 shows the characteristic length as a function
of time for two filling fractions, f = 0.5 (one electron
per site) and f = 0.49, both obtained by averaging over | [
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] |
per site) and f = 0.49, both obtained by averaging over
40 independent ML-Langevin simulations on a 100 × 100
system. The L(t) curves of the f = 0.49 case are also
shown for two different temperatures. For all cases, the
coarsening of the orbital domains shows a clear two-stage
behavior: there is an initial rapid domain growth, which
is followed by a much slower coarsening dynamics. This is
also consistent with the snapshots shown in both FIG. 6
and 7. The initial fast stage, as represented by the two | [
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] |
and 7. The initial fast stage, as represented by the two
nsteps = 200 configurations, is characterized by emerg-
ing orbital domains accompanied by a featureless random
distribution of QA1 and electron density. However, once
the orbital orders are well developed in each domain at
nsteps ≳ 400, or for times t ≳ 20τ0, the coarsening rate is
significantly decreased.
By fitting the initial relatively fast growth with the
Allen-Cahn law, as shown as dashed lines in FIG. 10(a), | [
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] |
Allen-Cahn law, as shown as dashed lines in FIG. 10(a),
one can see that the L(t) from simulations starts to devi-
ate from the
√
t behavior for t ≳ 15. On the other hand,
the snapshots shown in FIG. 6 and 7 clearly show that
there is still coarsening going on even at late times of the
relaxation. If we attempt to fit this late-stage coarsening
using the equation L(t) = L0 +a(t−t0)1/2, the extracted
expansion coefficient a ∼ 10−4 is extremely small. This | [
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] |
expansion coefficient a ∼ 10−4 is extremely small. This
corresponds to a relative domain growth of the order of
∆L/L0 ∼ 10−3 over a time span of 30 oscillation cycles
t0 of the JT modes. This indicates a freezing behavior at
the late stage of the phase ordering regardless of whether
the phase-ordering can be described by the Allen-Cahn
domain growth.
It is worth noting that the Allen-Cahn domain growth
law is intimately related to the domain morphologies of
12
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law is intimately related to the domain morphologies of
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] |
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ZzAH8gfP5A86MkhM=</latexit>pt − t0
FIG. 10. The characteristic length L(t) of orbital domains
as a function of time, averaging over 40 independent ML-
Langevin simulations. Panel (a) shows L(t) curves for the case
of half-filling and a filling fraction f = 0.49. Panel (b) shows
the L(t) curves at two different temperatures for the case of | [
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] |
the L(t) curves at two different temperatures for the case of
f = 0.49 electron filling. The two dashed lines in (a) and the
two solid lines in (b) are fit using the Allen-Cahn growth law
with a shifted time t0 and an initial length L0. The fitted
expansion coefficients are a = 8.1 × 10−5 and a = 9.1 × 10−5
for temperature T = 0.001 and T = 0.005, respectively.
the inhomogeneous states.
Fundamentally, the Allen-
Cahn equation for the domain-wall motion describes a
curvature-driven coarsening process. | [
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] |
Cahn equation for the domain-wall motion describes a
curvature-driven coarsening process.
Mathematically,
the domain-wall velocity is given by v = −rκ, where κ is
the curvature, and r is a proportional constant. Indeed,
this equation implies a power-law domain growth. For
inhomogeneous states characterized by a characteristic
length L, the domain-wall velocity can be approximated
by v ∼ dL/dt, while the curvature is on average given
by κ ∼ 1/L. The Allen-Cahn equation indicates a dif- | [
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] |
by κ ∼ 1/L. The Allen-Cahn equation indicates a dif-
ferential equation dL/dt ∼ −1/L, which can be readily
integrated to give the growth behavior L ∼ t1/2.
Within the framework of curvature-driven domain
growth, the late-stage freezing observed in our simula-
tions could be attributed to the rather straight interfaces
that separate orbital Ising domains of opposite signs; see
FIG. 6 and 7. Moreover, the nearly straight interfaces
tend to run parallel along the x or y-directions. As dis- | [
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] |
tend to run parallel along the x or y-directions. As dis-
cussed in Sec. I, since the effective orbital interactions
originate from the electrons through a mechanism simi-
lar to RKKY interactions, this interfacial anisotropy can
be attributed to the highly directional hopping of eg elec-
trons. The nearly zero curvature of such straight domain
boundaries indicate a vanishing velocity for the domain-
wall motion.
Consequently, the coarsening is mostly | [
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] |
wall motion.
Consequently, the coarsening is mostly
driven by the corner regions of orbital domains, where a
finite interfacial curvature remains. Let κc be a charac-
teristic curvature of the corner regions, the correspond-
ing length scales of curved interfaces is ℓ ∼ 1/κc.
It
should be noted that this curvature κc is determined by
the competition of the electron and elastic energies of the
JT model. When typical domain size L is greater than
this length scale, curvature driven domain-wall motion is | [
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] |
this length scale, curvature driven domain-wall motion is
suppressed by the interfacial stiffness. This indicates a
threshold length scale Lth ∼ ℓ ∼ 1/κc such that orbital
domains with a linear size L ≳ Lth start to show freezing
behavior. The L(t) curves shown in FIG. 10 seem to be
consistent with this threshold scenario.
V.
CONCLUSION AND OUTLOOK
To summarize, we have presented a comprehensive
study on the coarsening dynamics of orbital order in a
cooperative JT model using a scalable ML force-field ap- | [
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] |
cooperative JT model using a scalable ML force-field ap-
proach. Within the adiabatic approximation, the dynam-
ical evolution of the orbital order is governed by the dy-
namics of the MO6 octahedra with a driving force com-
puted from a quasi-equilibrium electron liquid. We have
generalized the Behler-Parrinello ML force-field method,
which was originally proposed for ML-based quantum
MD simulations, to model the vibronic dynamics of coop-
erative JT systems. Assuming the locality principle for | [
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] |
erative JT systems. Assuming the locality principle for
the electronic forces, a deep-learning neural network was
developed to accurately encode the complex dependence
of the effective forces on local distortion configurations.
In order to preserve the symmetry of the original Hamil-
tonian, a lattice descriptor based on the group-theoretical
bispectrum method was developed to incorporate of the
original Hamiltonian.
The ML models are trained by datasets from exact | [
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] |
original Hamiltonian.
The ML models are trained by datasets from exact
diagonalization of 30 × 30 systems. By integrating the
ML force-field model with the Langevin dynamics simu-
lation, our dynamical benchmarks showed that the ML
model not only accurately predicts the local forces that
drive the lattice dynamics, but also faithfully captures
the dynamical evolution of the JT system. Importantly,
compared with the exact diagonalization calculation for | [
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] |
compared with the exact diagonalization calculation for
electronic structures, the ML algorithms significantly im-
prove the time complexity for electronic force calculation
which in turn reduces the time complexity for dynamical
simulations by orders of magnitudes.
While numerous studies have devoted to the study of
the rich low-temperature phases and the complex phase
separation phenomena in CMR manganites, dynamical
processes such as phase ordering, domain growth, and | [
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] |
processes such as phase ordering, domain growth, and
kinetics of phase separation, are much less explored the-
oretically. This is partly due to the technical difficulty of
large-scale dynamical simulations for such complex sys-
tems.
Our work on the dynamics of orbital ordering
makes an important step towards the multi-scale dynami-
13
cal modeling of manganites and other correlated electron
systems. Our large-scale Langevin simulations enabled
by the ML models uncovered an intriguing freezing phe- | [
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] |
by the ML models uncovered an intriguing freezing phe-
nomenon at the late stage of the phase ordering. The
freezing dynamics is likely caused by the unique geometry
of orbital domains which exhibit rather straight domain
boundaries along the x or y-directions. Within the frame-
work of curvature-driven domain growth, the vanishing
curvature of such straight interfaces implies a stagnated
domain-wall motion.
The successful demonstration of the ML force-field ap- | [
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] |
domain-wall motion.
The successful demonstration of the ML force-field ap-
proach for the adiabatic dynamics of orbital/JT systems
also laid the groundwork for applying similar ML meth-
ods to more complicated electronic models of CMR man-
ganites that include spin degrees of freedom. It is worth
noting that a BP-type ML model has been developed for
the adiabatic spin dynamics of the single-band ferromag-
netic Kondo-lattice model [64, 65]. As discussed in Sec. I, | [
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] |
netic Kondo-lattice model [64, 65]. As discussed in Sec. I,
the DE mechanism in this model gives rise to a phase sep-
aration regime upon hole doping, which is an important
ingredient for CMR effect. It is envisioned a BP type neu-
ral network which predicts a local energy based on both
magnetic and JT distortion configurations in an immedi-
ate neighborhood could provide a unified ML force-field
model for such CMR systems.
It is worth noting that other linear-scaling numerical | [
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] |
model for such CMR systems.
It is worth noting that other linear-scaling numerical
techniques, notably the kernel polynomial methods [93–
95], can also be to solve quadratic fermion Hamiltoni-
ans, which is the case of the electron Hamiltonian in the
cooperative JT model considered here. However, KPM
and other similar O(N) methods cannot be directly gen-
eralized to fermionic models with electron-electron in-
teractions such as the on-site Coulomb repulsion which | [
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] |
teractions such as the on-site Coulomb repulsion which
has been shown to also play an important role in CMR
manganites [17–20]. On the other hand, the ML frame-
work presented in this work provides a general linear-
scaling approach to achieve linear scalability even for
JT systems that include on-site Hubbard repulsion or
other electron-electron interactions [21–27]. This is be-
cause, as discussed in Sec. III, linear-scaling electronic
structure methods are fundamentally based on Kohn’s | [
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] |
structure methods are fundamentally based on Kohn’s
locality principle. The divide-and-conquer approach in-
trinsic to the ML force-field models naturally take ad-
vantage of the locality property to achieve linear scala-
bility. On the other hand, the ML models for systems
with electron-electron interactions have to be trained by
datasets from more sophisticated many-body techniques
ranging from Hartree-Fock, Gutzwiller mean-field type
methods to quantum Monte Carlo or dynamical mean-
field theory. | [
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] |
methods to quantum Monte Carlo or dynamical mean-
field theory.
Appendix A: Descriptor for JT systems
Here we present details of the calculations of IR coeffi-
cients. As mentioned in the main text, each block of the
representation matrices corresponds to a group of neigh-
bors at the same distance from the center site-i.
For
the square lattice, these invariant blocks can be classified
into three types illustrated in FIG. 11. We first consider
type-I neighbors labeled as A, B, C, and D in panel (a). | [
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] |
type-I neighbors labeled as A, B, C, and D in panel (a).
Although the octahedral modes Qm at these four sites
form a 12-dimensional representation of the D4 group,
it is automatically decomposed into three 4-dimensional
blocks since the three components of Q transform inde-
pendent of each other. As discussed in Sec. III B, both
QA1 and Qz behave as the singlet A1 representation of
D4, while the Qx mode transforms according to the B1
symmetry. Both are decomposed as 4 = A1+B1+E, and | [
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] |
symmetry. Both are decomposed as 4 = A1+B1+E, and
the corresponding IR coefficients are listed in Eqs. (15)
and (16).
The type-II blocks correspond to neighbors sitting on
the two 45◦ diagonals, as shown in FIG. 11(b). The de-
composition of the breathing and Qz modes, both of A1
symmetry, is given by 4 = A1 + B2 + E. Take the Qz as
an example, the IR coefficients are
f A1 = Qz
A + Qz
B + Qz
C + Qz
D
f B2 = Qz
A − Qz
B + Qz
C − Qz
D
f E
x = Qz
A − Qz
B − Qz
C + Qz
D,
f E
y = Qz
A + Qz
B − Qz
C − Qz
D. | [
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] |
A − Qz
B + Qz
C − Qz
D
f E
x = Qz
A − Qz
B − Qz
C + Qz
D,
f E
y = Qz
A + Qz
B − Qz
C − Qz
D.
On the other hand, the four Qx modes are decomposed as
4 = A2 + B1 + E, with the following linear combinations
f A2 = Qx
A − Qx
B + Qx
C − Qx
D,
f B1 = Qx
A + Qx
B + Qx
C + Qx
D,
f E
x = Qx
A − Qx
B − Qx
C + Qx
D,
f E
y = −Qx
A − Qx
B + Qx
C + Qx
D.
Finally, the type-III blocks correspond to a group of
8 neighbors shown in FIG. 11(c).
Again, the 24-
dimensional representation of the 8 Q vectors is automat- | [
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] |
Again, the 24-
dimensional representation of the 8 Q vectors is automat-
ically decomposed into three 8-dimensional block. Both
the QA1 and Qz blocks are decomposed into six IRs:
8 = A1 + A2 + B1 + B2 + 2E, with the following co-
efficients
f A1 = Qz
A + Qz
B + Qz
C + Qz
D + Qz
E + Qz
F + Qz
G + Qz
H
f A2 = Qz
A − Qz
B + Qz
C − Qz
D + Qz
E − Qz
F + Qz
G − Qz
H
f B1 = Qz
A − Qz
B − Qz
C + Qz
D + Qz
E − Qz
F − Qz
G + Qz
H
f B2 = Qz
A + Qz
B − Qz
C − Qz
D + Qz
E + Qz
F − Qz
G − Qz
H
f E = (Qz
A − Qz
D − Qz | [
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0.4219417870044708,
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0.3306872546672821,
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0.10007510334253311,
0.4901961386203766,
0.3344798684120178,
0.4687223732471466,
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0.4660906493663788,
-0.11347289383411407,
0.9664777517318726,
0.06308416277170181,
0.14677029848098755,
0.12062286585569382,
0.6072744131088257,
-0.13095010817050934,
0.004338559694588184,
0.3470081388950348,
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0.5105790495872498,
0.3186609447002411,
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0.37778350710868835,
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] |
G + Qz
H
f B2 = Qz
A + Qz
B − Qz
C − Qz
D + Qz
E + Qz
F − Qz
G − Qz
H
f E = (Qz
A − Qz
D − Qz
E + Qz
H, Qz
B + Qz
C − Qz
F − Qz
G)
f E′ = (Qz
B − Qz
C − Qz
F + Qz
G, Qz
A + Qz
D − Qz
E − Qz
H)
(A1)
The same decomposition also applies to the Qx blocks,
14
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] |
QH
FIG. 11. Schematic diagrams of the three types of neighbor blocks: (a) type-I with four neighbor sites on the x or y axes, (b)
type-II with four neighbors on the y = ±x diagonals, and (c) type-III with eight neighbors. The arrows represent the doublet
vectors QE = (Qx, Qz).
(d) An example of partitioning the neighborhood into eight symmetry-related regions.
Average
distortion vectors QK obtained from these regions are used to compute the reference IR coefficients f Γ
ref for the descriptor. | [
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-0.29524096846580505,
-0.06227967515587807,
0.13050587475299835,
-0.44543835520744324,
0.20420823991298676,
-0.05828739330172539,
-0.21287164092063904,
-0.7943962216377258,
0.27981966733932495,
0.4829567074775696,
-0.344626784324646,
-0.5328209400177002,
0.8653210401535034,
0.010791373439133167,
0.07856165617704391,
0.27697813510894775,
0.31188976764678955,
-0.2350451648235321,
0.04062129184603691,
0.47213050723075867,
-0.7552061676979065,
-0.1793624609708786,
-0.0476827397942543,
0.3725072741508484,
0.5266191959381104,
-0.1823258250951767,
-0.6749724745750427,
-0.7061187624931335,
-0.5224462747573853,
0.4093211591243744,
-0.1697862148284912,
0.5930964946746826,
-0.2761000096797943,
-0.25090742111206055,
-0.511932373046875,
0.25146088004112244,
0.10423769801855087,
-0.09055030345916748,
-0.3374904692173004,
-0.1181948333978653,
0.03299706429243088,
-0.1165483295917511,
-0.05897821485996246,
-0.43786171078681946,
0.13698413968086243,
-0.13378961384296417
] |
ref for the descriptor.
but with different linear combinations
f A1 = Qx
A − Qx
B − Qx
C + Qx
D + Qx
E − Qx
F − Qx
G + Qx
H
f A2 = Qx
A + Qx
B − Qx
C − Qx
D + Qx
E + Qx
F − Qx
G − Qx
H
f B1 = Qx
A + Qx
B + Qx
C + Qx
D + Qx
E + Qx
F + Qx
G + Qx
H
f B2 = Qx
A − Qx
B + Qx
C − Qx
D + Qx
E − Qx
F + Qx
G − Qx
H
f E = (Qx
A − Qx
4 − Qx
E + Qx
H, −Qx
B − Qx
C + Qx
F + Qx
G)
f E′ = (−Qx
B + Qx
C + Qx
F − Qx
G, Qx
A + Qx
D − Qx
E − Qx
H)
(A2)
Using these formulas, one can systematically decompose | [
0.0412873700261116,
0.4155539572238922,
0.06985471397638321,
-0.35408318042755127,
-0.46580639481544495,
0.4915100634098053,
0.3964039981365204,
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-0.5034120082855225,
0.15491367876529694,
0.6206782460212708,
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0.27433377504348755,
0.2591034770011902,
0.315341591835022,
0.3477976620197296,
0.15764614939689636,
0.06552966684103012,
-0.8678402900695801,
0.23583541810512543,
0.07055625319480896,
0.06940897554159164,
-0.14411215484142303,
0.056339144706726074,
0.14747506380081177,
0.2586490213871002,
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0.6052650809288025,
0.27750465273857117,
0.19232061505317688,
0.10257744789123535,
-0.15710937976837158,
-0.004983043763786554,
0.05641312152147293,
0.2334420531988144,
0.073227658867836,
0.6705182194709778,
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0.15092833340168,
-0.40146583318710327,
0.159426748752594,
0.2471233457326889,
-0.17031195759773254,
-0.0197344571352005,
-0.9832373261451721,
0.1496371477842331,
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0.0006198465125635266,
0.16545933485031128,
0.8965421319007874,
0.03459502011537552,
0.10552700608968735,
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0.05865562707185745,
0.8405597805976868,
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0.4627171456813812,
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0.3758997619152069,
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0.6174445748329163,
0.18309032917022705,
0.25134459137916565,
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0.4410272240638733,
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0.294554740190506,
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0.04210331290960312,
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0.30188697576522827,
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0.22891943156719208,
0.07677985727787018,
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0.7730197310447693,
0.13814441859722137,
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0.25431397557258606,
0.1639404296875,
0.014868694357573986,
0.42226648330688477,
0.10881127417087555,
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0.5077272653579712,
0.12979702651500702,
0.13193605840206146,
0.031141316518187523,
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-0.8286433219909668,
-0.2709943354129791,
-0.8018438816070557,
0.5233969688415527,
0.13564208149909973,
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0.3035755455493927,
-0.3467593491077423,
-0.14550331234931946,
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0.013628577813506126,
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0.058389388024806976,
1.0470221042633057,
0.45659950375556946,
0.9769431948661804,
0.4228277802467346,
0.7051892876625061,
0.41476669907569885,
-0.3928161561489105,
0.5276763439178467,
1.014573335647583,
-0.7294706106185913,
-0.19717583060264587,
-0.027250448241829872,
1.1790374517440796,
-0.10290634632110596,
0.36457040905952454,
0.2638201117515564,
-0.7327320575714111,
0.05243746563792229,
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-0.3629816472530365,
-0.20955245196819305,
-0.2949143648147583,
0.20505677163600922,
-0.09187376499176025,
0.23014220595359802,
-0.36123231053352356,
-0.09916187822818756,
0.09994538873434067,
0.10145606100559235,
0.6971845030784607,
0.02066997066140175,
0.10314089059829712,
-0.3120776116847992,
-0.07117663323879242,
-0.4789848327636719,
-0.40326258540153503,
0.09756249189376831,
-0.17720866203308105,
0.22257544100284576,
-0.4613221287727356,
0.23946523666381836,
0.5699223875999451,
0.1185319572687149,
-0.10662726312875748,
-0.16094592213630676,
0.22100107371807098,
-0.28803420066833496,
0.04665248095989227,
0.15915125608444214,
-0.16791845858097076,
-0.3125913441181183,
-0.5361380577087402,
0.03698785975575447,
0.4755052328109741,
-0.5210824608802795,
-0.4627319574356079,
0.6198992729187012,
0.45440271496772766,
0.3022153079509735,
0.14980819821357727,
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0.4970177710056305,
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0.1857689768075943,
0.21744680404663086,
0.4319024980068207,
0.19452612102031708,
0.5491697788238525,
0.35269296169281006,
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0.4698079526424408,
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1.247304081916809,
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0.2635892629623413,
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0.6276945471763611,
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0.6576179265975952,
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0.4441632628440857,
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0.3757033348083496,
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0.2434137463569641,
0.34832513332366943,
0.09587736427783966,
-0.13544023036956787,
-0.2657524347305298,
0.2762443721294403,
0.45490050315856934,
0.1100364699959755,
-0.6775202751159668,
-0.3283117115497589,
-0.2955562472343445,
0.45751991868019104,
0.03643710911273956,
0.10361891239881516,
0.9047238826751709,
-0.4340081512928009,
0.4320426285266876,
0.5547490119934082,
-0.0798725038766861,
0.4689515233039856,
-0.20028038322925568,
0.02026865817606449,
0.2019239068031311,
-0.015845973044633865,
0.10394848883152008,
-0.1349673867225647,
0.04797804728150368,
0.2774391174316406,
0.08344562351703644,
0.3977839946746826,
-0.1727587878704071,
0.1334921270608902,
-0.17307879030704498,
0.3504083752632141,
-0.1091064065694809,
-0.08725380897521973,
0.5692239999771118,
-0.25266462564468384,
0.6289584636688232,
-0.43475547432899475,
-0.17866915464401245,
0.04609750956296921,
0.3163618743419647,
0.3648549020290375,
-0.4801037609577179,
-0.06852826476097107,
-0.10443601757287979,
0.22089844942092896,
-0.07016295194625854,
-0.1713682860136032,
0.1395263522863388,
-0.23346003890037537,
-0.4573734402656555,
0.006953165866434574,
-0.40699848532676697,
-0.13211888074874878,
-0.4233018457889557,
-0.8173709511756897,
0.10658196359872818,
1.084815502166748,
0.15122529864311218,
-0.32905539870262146,
0.1844719797372818,
-1.035971999168396,
0.3978869616985321,
0.032277971506118774,
-0.16744627058506012,
0.6429897546768188,
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-0.5113480091094971,
-0.0957358255982399,
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0.23973268270492554,
0.3644576668739319,
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0.39185962080955505,
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-0.06223205849528313,
-0.11998976767063141,
0.8551149368286133,
-0.011519603431224823,
-1.2054576873779297,
0.34434327483177185,
0.09166774153709412,
-0.3090246617794037,
-0.12256809324026108,
0.5283841490745544,
0.3218652307987213,
0.2847359776496887,
0.19868336617946625,
0.0466102659702301,
0.08228529989719391,
-0.25412502884864807,
-0.23475711047649384,
-0.6991656422615051,
0.5107693076133728,
-5.794312953948975,
0.00659484788775444,
-0.773542582988739,
0.17475871741771698,
-0.9391632080078125,
-0.40299198031425476,
0.29572269320487976,
-0.20831353962421417,
-0.36909815669059753,
-0.5535692572593689,
0.09221617132425308,
0.24522659182548523,
0.8154791593551636,
0.672758162021637,
-0.014454254880547523,
0.12322968989610672,
0.795103132724762,
0.17058254778385162,
0.2665267288684845,
0.44268736243247986,
-0.11913327872753143,
0.08073446154594421,
0.047162655740976334,
-0.08497108519077301,
0.25111496448516846,
0.19929753243923187,
-1.0601651668548584,
0.4090823233127594,
-0.16923736035823822,
-0.5566099882125854,
-0.1505136787891388,
0.20087386667728424,
0.35063436627388,
-0.3993041217327118,
0.33880364894866943,
0.3918582797050476,
0.3810991048812866,
0.030930697917938232,
0.032743994146585464,
-0.518124520778656,
0.5265310406684875,
0.03037395142018795,
-0.02976829931139946,
-0.03291100263595581,
0.06672085076570511,
0.27551591396331787,
-0.30125775933265686,
0.28144657611846924,
-0.23943543434143066,
0.6512970328330994,
0.07834666967391968,
-0.2269829362630844,
0.13865022361278534,
0.14621911942958832,
0.11544553935527802,
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0.1640176624059677,
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0.16723018884658813,
0.02513457089662552,
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0.9301626086235046,
0.5316951274871826,
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0.4255810081958771,
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0.31714779138565063,
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0.044637281447649,
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0.37941113114356995,
0.06987746804952621,
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0.14368221163749695,
0.08696687966585159,
0.14821875095367432,
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0.11363080143928528,
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0.2750416100025177,
0.6294407248497009,
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0.06115502119064331,
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0.09782402962446213,
0.2051912546157837,
0.608593225479126,
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0.6533006429672241,
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1.1384979486465454,
0.2340753972530365,
0.24689190089702606,
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0.6175485849380493,
0.45737072825431824,
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0.5029007196426392,
0.07160801440477371,
0.05721065774559975,
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0.023976266384124756,
-0.08895619958639145,
0.3047812879085541,
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-0.9529805779457092,
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0.19726873934268951,
0.09595265239477158,
0.490433931350708,
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0.4075835943222046,
0.427432656288147,
0.05290595069527626,
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0.6653907895088196,
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0.6368650197982788,
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0.38957834243774414,
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0.24470239877700806,
1.0973163843154907,
0.14173875749111176,
0.3441872298717499,
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0.07245611399412155,
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0.2153739631175995,
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0.08453720808029175,
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0.0546041838824749,
0.06967233121395111,
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0.6751075387001038,
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] |
F − Qx
G, Qx
A + Qx
D − Qx
E − Qx
H)
(A2)
Using these formulas, one can systematically decompose
the octahedral distortion modes Qj within the neighbor-
hood into IRs.
As discussed in Sec. III B, the various
coefficients of an IR in the decomposition is represented
by a vector
f Γ
r = (f (Γ,r)
1
, f (Γ,r)
2
, · · · , f (Γ,r)
nΓ
)
where Γ = A1, A2, B1, B2, or E labels the IR of D4 group,
nΓ is the dimension of the IR, and r enumerates the mul-
tiple occurrences of Γ in the decomposition of Ci. A set | [
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0.3171045482158661,
0.5281860828399658,
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0.7875874042510986,
0.3251195251941681,
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0.47140341997146606,
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0.08602408319711685,
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0.1745751053094864,
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0.4362627863883972,
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0.6564932465553284,
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0.08944536000490189,
0.047488048672676086,
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0.46228593587875366,
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0.20561344921588898,
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0.12301105260848999,
0.02902340516448021,
0.15969406068325043,
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0.0468691885471344,
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-0.025337323546409607,
-0.22295811772346497,
1.1264331340789795,
0.6297791600227356,
-0.303560346364975,
-0.4477812647819519,
0.006532033905386925,
-0.6660318374633789,
-0.2576434910297394,
0.42586761713027954,
-0.24577857553958893,
0.06335737556219101,
-0.15877538919448853,
0.3120184540748596,
-0.006698453798890114,
0.38898611068725586,
0.5893357992172241,
-0.33098018169403076,
-0.353366494178772,
0.2425461709499359,
0.3810679316520691,
-0.23317189514636993,
0.5808898210525513,
0.07785557210445404,
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0.022798284888267517,
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0.1184018924832344,
-0.8032763004302979,
-0.5782690048217773,
-0.0032013009767979383,
0.05477137863636017,
0.2943044602870941,
0.0670362189412117,
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-0.3020223081111908,
-0.19350488483905792,
0.4788389801979065,
-0.5727623105049133,
0.7261873483657837,
0.45171359181404114,
-0.2987215518951416,
0.4270443320274353,
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-0.6038573980331421,
0.2355400174856186,
0.08587948977947235,
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0.5104224681854248,
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0.1820685714483261,
-0.25191015005111694,
-0.6838281154632568,
-0.8935258984565735,
0.47450461983680725,
-0.7910496592521667,
0.0302394088357687,
-0.149996817111969,
0.16420814394950867,
0.23568198084831238,
-0.21145859360694885,
-0.2762340307235718,
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0.2075732946395874,
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0.23750752210617065,
0.0982532873749733,
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0.20709091424942017,
-0.08089735358953476,
0.8710758090019226,
0.265300989151001,
0.0313447006046772,
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0.7073909044265747,
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0.19674047827720642,
0.7245876789093018,
0.5817791223526001,
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0.3541457951068878,
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0.4513437747955322,
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-0.7446686029434204,
0.01831778697669506,
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0.17579713463783264,
0.11952820420265198,
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0.09332432597875595,
0.4166412651538849,
0.38311198353767395,
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0.009641272015869617,
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0.1640188843011856,
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0.28250446915626526,
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0.2061929702758789,
0.609341561794281,
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0.33981338143348694,
0.40916264057159424,
0.14770089089870453,
0.6760714650154114,
0.41908499598503113,
-0.19233182072639465,
-0.9094822406768799,
0.1793373078107834,
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0.14789599180221558,
0.34267640113830566,
0.27068156003952026,
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0.6132733225822449,
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0.1021924763917923,
0.11256915330886841,
0.4964549243450165,
0.9852176308631897,
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0.062332600355148315,
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0.39884281158447266,
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0.590420126914978,
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0.03805672749876976,
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0.46732115745544434,
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0.9449909925460815,
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0.2724260091781616,
0.3544776737689972,
0.07498100399971008,
0.4423457384109497,
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0.11610881239175797,
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0.24656087160110474,
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0.40400880575180054,
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0.18985679745674133,
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0.3630429208278656,
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0.2582145631313324,
0.3825068175792694,
0.1155817061662674,
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0.26239079236984253,
0.19311557710170746,
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0.3812914192676544,
0.21382734179496765,
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0.698337197303772,
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0.20775717496871948,
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0.6514601111412048,
0.06325849890708923,
0.008394535630941391,
-0.31277453899383545,
-0.08800584077835083,
-0.040318526327610016,
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-0.4296075105667114,
-0.06954330950975418,
-0.4467126131057739
] |
tiple occurrences of Γ in the decomposition of Ci. A set
of invariants called the power spectrum can be readily
obtained
pΓ
r =
��f Γ
r
��2 .
As discussed in the main text, a descriptor based only on
power spectrum suffers from spurious symmetry since the
relative phases between different IRs are not accounted
for. A complete descriptor is given by the bispectrum
coefficients [77, 81]
BΓ,Γ1,Γ2
r,r1,r2
=
�
m,n,l
CΓ;Γ1,Γ2
m,n,l
f Γ ∗
r,mf Γ1
r1,nf Γ2
r2,l,
where CΓ;Γ1,Γ2
m,n,l | [
0.3275877833366394,
-0.27405011653900146,
0.8595934510231018,
-0.026788707822561264,
-0.21239523589611053,
0.0498441718518734,
0.15923263132572174,
-0.8191563487052917,
-0.3121393322944641,
-0.136103555560112,
0.028904281556606293,
0.45811450481414795,
-0.22068054974079132,
0.1707175225019455,
-0.1345287561416626,
0.7715086936950684,
0.3605346977710724,
0.5085299611091614,
-0.8936946988105774,
-1.0680902004241943,
0.5701161623001099,
-0.11739509552717209,
0.700472354888916,
-0.23583535850048065,
0.22318536043167114,
0.6627631783485413,
0.11271928250789642,
0.13837596774101257,
0.0951758399605751,
0.55416339635849,
-0.45046886801719666,
0.7298532128334045,
-0.12773144245147705,
0.057364847511053085,
0.2600366175174713,
0.3512062132358551,
0.6421204805374146,
-0.12428083270788193,
0.3208066523075104,
-0.5623764395713806,
0.2776770293712616,
-0.26740962266921997,
0.08172980695962906,
0.3293348252773285,
-0.4160679578781128,
-0.6729432344436646,
-1.2429550886154175,
0.513163149356842,
-0.06273680925369263,
-0.22432594001293182,
-0.46487241983413696,
0.22389107942581177,
-0.1691003441810608,
0.22622555494308472,
0.238875612616539,
0.44230207800865173,
-0.5889192223548889,
0.1270969808101654,
0.21933048963546753,
0.10261031985282898,
0.32897433638572693,
0.16553017497062683,
0.08298460394144058,
0.6205698251724243,
0.11878549307584763,
0.5140514373779297,
-0.1896168291568756,
0.0972668007016182,
-0.029384642839431763,
-0.010571385733783245,
0.5745392441749573,
0.053312357515096664,
0.02712053619325161,
0.27869629859924316,
-0.09442773461341858,
0.9155571460723877,
-0.13910788297653198,
-0.3529600501060486,
0.5446690917015076,
0.1696377694606781,
-0.13086052238941193,
-0.41815078258514404,
0.5559467673301697,
-0.5948123335838318,
0.34497618675231934,
0.03502550348639488,
0.04132887348532677,
0.2556988596916199,
-0.44331106543540955,
0.5352218747138977,
0.11224992573261261,
-0.8784663081169128,
0.2842300832271576,
0.41929692029953003,
0.018082451075315475,
0.3277585804462433,
0.03260204941034317,
-0.2645939290523529,
0.4407184422016144,
-0.2501395642757416,
-0.12859170138835907,
0.2334093302488327,
-0.026725757867097855,
-0.42287564277648926,
-0.5349085927009583,
-0.43713197112083435,
-0.41802865266799927,
-0.5689751505851746,
-0.41725650429725647,
-0.1865214705467224,
-0.8900710940361023,
-0.05740761011838913,
0.14734803140163422,
-0.24938565492630005,
-0.034932855516672134,
0.10785358399152756,
0.32729625701904297,
-0.04485160484910011,
-0.036591652780771255,
0.4007323086261749,
0.03658241033554077,
0.9559838771820068,
0.050769418478012085,
0.25528693199157715,
0.6655558347702026,
0.1341172158718109,
0.5743171572685242,
0.7185155153274536,
-0.26223820447921753,
-0.003906658384948969,
0.07276065647602081,
0.7629176378250122,
0.03456882759928703,
-0.14273639023303986,
-0.08088886737823486,
-0.6410531997680664,
-0.23427347838878632,
-0.03001917153596878,
-0.08237689733505249,
-0.0016753431409597397,
-0.33783480525016785,
-0.00785739254206419,
-0.5868587493896484,
-0.31525561213493347,
0.0634203553199768,
-0.3065415024757385,
-0.573976457118988,
-0.036568209528923035,
0.0668492242693901,
0.14531046152114868,
0.28092142939567566,
-0.02042911760509014,
-0.23904560506343842,
0.135635644197464,
-0.14930109679698944,
-0.1933063417673111,
-0.048622071743011475,
0.23411288857460022,
-0.23933832347393036,
-0.45157915353775024,
0.0614006407558918,
-0.20072676241397858,
0.2151063084602356,
-0.032189060002565384,
0.5999143719673157,
-0.20216158032417297,
0.10760557651519775,
-0.2954627573490143,
-0.019400592893362045,
0.33656537532806396,
-0.24671514332294464,
-0.11915741115808487,
0.09442510455846786,
0.3675335645675659,
-0.15350909531116486,
-0.2100226879119873,
0.23518593609333038,
0.364840567111969,
0.0708780512213707,
-0.09091579169034958,
-0.7099792957305908,
-0.2031555026769638,
-0.016681402921676636,
-0.5498563647270203,
0.19040420651435852,
0.35929813981056213,
0.38916584849357605,
0.25670328736305237,
0.2473675161600113,
-0.2366892695426941,
0.26924461126327515,
-0.20524491369724274,
0.6613656878471375,
-0.9214017391204834,
0.7715467214584351,
0.11188298463821411,
-0.6558821201324463,
0.38884231448173523,
-0.025382723659276962,
0.15846557915210724,
-0.1314866840839386,
0.15396921336650848,
0.650564432144165,
-0.1616620272397995,
-0.5229949355125427,
0.29250359535217285,
-0.28504249453544617,
0.22288860380649567,
0.006661410443484783,
-0.028021497651934624,
0.25233668088912964,
-0.5768012404441833,
-0.19184233248233795,
-0.033469196408987045,
0.06996000558137894,
-0.21970537304878235,
-0.2292969971895218,
-0.10460597276687622,
0.5807087421417236,
0.03270278126001358,
0.05060723051428795,
-0.033312324434518814,
0.1333351731300354,
0.5622450709342957,
0.025611767545342445,
1.0136641263961792,
0.6222382187843323,
-0.09550081938505173,
0.8822176456451416,
-0.4154258072376251,
-0.09919177740812302,
-0.47139444947242737,
0.6359169483184814,
-0.07955653965473175,
-0.2890618145465851,
0.49753522872924805,
0.3062647879123688,
-0.02036186121404171,
0.5831001996994019,
-0.5943610072135925,
0.03930439427495003,
0.4409787952899933,
-1.0462080240249634,
-0.094687320291996,
0.17379462718963623,
0.21262606978416443,
-0.4211235046386719,
0.10821222513914108,
0.08867347985506058,
-0.26230496168136597,
0.18977850675582886,
0.3565809428691864,
-0.12567904591560364,
-1.0910779237747192,
-0.035864390432834625,
0.08429469913244247,
-0.025474200025200844,
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0.14193189144134521,
0.1018642783164978,
-0.5335351228713989,
-0.1023598238825798,
-0.04238886758685112,
-0.3760910928249359,
-0.14622215926647186,
-0.42400020360946655,
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0.9123353958129883,
0.25733843445777893,
0.057340092957019806,
-0.10496201366186142,
-0.6757795810699463,
0.16765658557415009,
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0.05347972735762596,
-0.6761388182640076,
-0.7846391797065735,
0.33880066871643066,
0.37579160928726196,
0.9560304284095764,
0.1380474865436554,
-0.6176233291625977,
-0.03729765862226486,
0.09498312324285507,
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-0.2360459566116333,
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0.29276132583618164,
0.16893205046653748,
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0.2735573351383209,
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0.3427135944366455,
0.17084626853466034,
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0.156306192278862,
-0.49776020646095276,
-0.4448837637901306,
-0.06061553210020065,
0.2034558206796646,
-0.8102242350578308,
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] |
BΓ,Γ1,Γ2
r,r1,r2
=
�
m,n,l
CΓ;Γ1,Γ2
m,n,l
f Γ ∗
r,mf Γ1
r1,nf Γ2
r2,l,
where CΓ;Γ1,Γ2
m,n,l
are the Clebsch-Gordan coefficients of
the point group, which are introduced to account for
the different transformation properties of the three IRs.
Although the bispectrum coefficients provide a faithful
symmetry-invariant representation of the neighborhood
distortion configurations, the rather large number of such
coefficients combined with the fact that many of them | [
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] |
coefficients combined with the fact that many of them
are redundant render this an impractical approach for
descriptors.
A more efficient method for descriptor is based on the
idea of reference IRs, which serve as an intermediate to
store the phase information, as discussed in Sec. III B.
To obtain coefficients of these reference IRs, we first par-
tition the neighborhood (defined as within a radius rc
from the center site) into 8 symmetry-related regions,
as shown in FIG. 11(d). | [
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0.09608837217092514,
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0.25832071900367737,
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0.5558976531028748,
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0.08042766153812408,
0.7355260252952576,
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0.46402615308761597,
0.6892744302749634,
0.06103198975324631,
0.010135536082088947,
0.19743478298187256,
0.3880756199359894,
0.4955938160419464,
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0.5771471858024597,
-0.06351763010025024,
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-0.4975746273994446,
0.16802017390727997,
0.1757264882326126,
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0.023422054946422577,
0.13644647598266602,
-0.08396436274051666,
0.19730572402477264,
-0.44672998785972595,
0.02582540549337864,
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0.38166072964668274,
0.23063960671424866,
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0.6681119799613953,
0.24383404850959778,
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0.23935800790786743,
0.34702304005622864,
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0.0528813935816288,
0.5503647923469543,
-1.0036964416503906,
-0.19539234042167664,
0.014149969443678856,
0.7818498015403748,
0.6752340793609619,
-0.2160361260175705,
-0.5973896384239197,
0.08631103485822678,
-0.4837312400341034,
0.3251648247241974,
0.24092762172222137,
0.7661241888999939,
-0.4745296835899353,
-0.09362304955720901,
-0.7556344270706177,
0.4219878315925598,
-0.24854908883571625,
-0.14425651729106903,
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-0.24574154615402222,
-0.3091862201690674,
0.014119436964392662,
0.3928668200969696
] |
from the center site) into 8 symmetry-related regions,
as shown in FIG. 11(d).
An average distortion vector
QK =
1
M
�
j∈K Qj is obtained for each of the regions,
where M is the number of sites in each region. This con-
struction has the advantage that the resultant reference
IRs are less sensitive to small variations in the neighbor-
hood.
The reference IR coefficients f Γ
ref are computed
using Eqs. (A1) and (A2). The reference bispectrum co-
efficients BΓ,Γ1,Γ2
ref | [
0.02580583468079567,
0.07090682536363602,
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] |
ref are computed
using Eqs. (A1) and (A2). The reference bispectrum co-
efficients BΓ,Γ1,Γ2
ref
which are part of the feature variables
in Eq. (17) are obtained by substituting f Γ
ref into the
above bispectrum equation.
ACKNOWLEDGMENTS
The authors thank Puhan Zhang for useful discussions.
The work was supported by the US Department of En-
ergy Basic Energy Sciences under Contract No.
DE-
SC0020330. The authors also acknowledge the support
of Research Computing at the University of Virginia.
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] |
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