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] |
FIG. 5. Comparison of time-dependent correlation functions
Cxx(r, t) defined in Eq. (19) obtained from Langevin simu-
lations with the ML force-field model and the ED method.
The correlation functions were obtained from 40 independent
thermal quench simulations on a 30 × 30 lattice with a filling
fraction f = 0.49.
point, this is feasible since the breathing mode as well as
its corresponding force F A1 are scalars, They are invari-
ant under the point-group symmetry transformations, | [
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] |
ant under the point-group symmetry transformations,
and the same descriptor can be also used for the corre-
sponding NN model. The fact that no automatic differen-
tiation is required for the breathing modes also enhances
the efficiency. But more importantly, we found that the
prediction accuracy of the F A1 forces is significantly im-
proved. With this hybrid approach, rather accurate force
predictions were achieved for all three vibronic modes,
as shown in FIG. 4. The histogram of prediction errors | [
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] |
as shown in FIG. 4. The histogram of prediction errors
(δ = F ML − F ED), shown on the right panels of FIG. 4,
are characterized by a small MSE of σF ∼ 0.005 for the
force predictions.
It is worth noting that both the electron density and
the local orbital configuration can be directly obtained
from the ML model, thanks to the local and linear cou-
pling between electrons and octahedral modes. As shown
in Eq. (9), for example, the expectation value of orbital
pseudo-spin can be obtained as: ⟨ˆτi⟩ = ( ˆF E | [
0.05554338917136192,
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] |
pseudo-spin can be obtained as: ⟨ˆτi⟩ = ( ˆF E
i + KQE
i )/λ,
where ˆF E
i
= h(Ci) is the total doublet force predicted
by the ML model. Similarly, the on-site electron number
can also be obtained from the predicted forces ˆF A1
i
for
the breathing mode: ⟨ˆni⟩ = ( ˆF A1
i
+ βKQA1
i )/λ.
Next we integrated the trained ML models into
the Langevin dynamics method and conducted thermal
quench simulations of the JT model. We initiated the
simulations with a random initial state, which was sud- | [
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] |
simulations with a random initial state, which was sud-
denly quenched to a temperature of T = 0.01 at t = 0.
The results from the ML-Langevin simulations were then
compared with those from ED simulations.
As the
ground-state is characterized by a checkerboard pattern
of the Qx JT distortions, we computed the correlation
9
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] |
FIG. 6. Snapshots of local breathing mode QA1 and orbital Ising order parameter ϕi at various time steps after a thermal
quench of a half-filled f = 0.5 JT model. An initially random configuration is suddenly quenched to a temperature T = 0.001
at time t = 0 (nstep = 0). The ML-Langevin dynamics was used to simulate the relaxation of the system toward equilibrium.
The red and blue regions correspond to orbital domains with order parameters ϕ = +1 and −1, respectively. A time step | [
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] |
∆t = 0.05τ0 is used in the simulations.
function
Cxx(rij) = ⟨Qx
i Qx
j ⟩
(19)
between the Qx modes at two octahedra separated by
a distance rij along either the x- or the y-direction.
FIG. 5 displays the time-dependent correlation functions
obtained from both ML and ED Langevin simulations,
demonstrating excellent agreement. The relaxation pro-
cess is characterized by a gradual built-up of staggered
correlations of the Qx distortion, characteristic of the C- | [
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] |
correlations of the Qx distortion, characteristic of the C-
type orbital/JT oder. This dynamical benchmark shows
that the ML model not only accurately predicts the
forces, but also captures the dynamical evolution of the
JT system.
IV.
COARSENING OF ORBITAL ORDER IN
COOPERATIVE JAHN-TELLER MODEL
The ML force-field model is applied to study the
large-scale coarsening dynamics of orbital order in the
JT model.
In addition to the linear scalability of ML | [
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] |
JT model.
In addition to the linear scalability of ML
method, the efficiency is further enhanced by running the
simulations on GPU machines. We performed the ther-
mal quench simulations on a 100 × 100 lattice where an
initial state with random local distortions was suddenly
cooled to a low temperature T = 0.01 at time t = 0. As
discussed above, the low-temperature C-type orbital/JT
order exhibits a broken Z2 symmetry, which is physi-
cally related to both the sublattice and mirror transfor- | [
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] |
cally related to both the sublattice and mirror transfor-
mation about the diagonals. For the case of half-filling,
this ground state is also characterized by a uniform elec-
tron density of ⟨ˆni⟩ = 1. The lattice distortions in the
ground state can be described by two parameters:
QA1
i
= η,
QE
i = (δ, 0) exp(iK · ri).
(20)
where K = (π, π) is the ordering wave vector, and the
phase factor exp(iK · ri) = ±1 for lattice sites at the two
sublattices. The parameter η = λ/βK is the amplitude of | [
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] |
sublattices. The parameter η = λ/βK is the amplitude of
the uniform expansion of octahedra, while δ = λ⟨ˆτ x⟩/K
can be viewed as a global Z2 order parameter for the
staggered JT distortions. To characterize the inhomoge-
neous states with multiple orbital domains after a ther-
mal quench, we define a scalar order parameter that mea-
sures the local staggered JT distortion
ϕi =
�
Qx
i − ⟨Qx
nn⟩i
�
exp(iK · ri).
(21)
where ⟨Qx
nn⟩i ≡ (Qx
i+ˆx + Qx
i−ˆx + Qx
i+ˆy + Qx
i−ˆy)/4 denotes | [
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] |
nn⟩i
�
exp(iK · ri).
(21)
where ⟨Qx
nn⟩i ≡ (Qx
i+ˆx + Qx
i−ˆx + Qx
i+ˆy + Qx
i−ˆy)/4 denotes
the average JT distortion at the four nearest neighbors
of site-i. A nonzero ϕi thus indicates the presence of a
local difference in JT distortion. Indeed, this Ising order
parameter is ϕi = δ in the long-range staggered JT order
described in Eq. (20).
FIG. 6 show the snapshots of the breathing-mode am-
plitude QA1
i
and the local Ising order parameter ϕi at
different times after a thermal quench of a half-filled | [
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] |
i
and the local Ising order parameter ϕi at
different times after a thermal quench of a half-filled
JT model. The red and blue regions, corresponding to
ϕi = +1 and −1, respectively, are C-type orbital orders
related by the Z2 sublattice symmetry. The two types
of orbital domains are separated by interfaces of vanish-
ing ϕi, corresponding to the white regions. On the other
10
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] |
FIG. 7. Snapshots of local breathing mode QA1 and orbital Ising order parameter ϕi at various time steps after a thermal
quench for a JT model with f = 0.49 electron filling.
hand, a rather uniform ordering of QA1 is quickly de-
veloped throughout the system, except at the interfaces
of the two orbital domains. The nearly constant ampli-
tude QA1 corresponds to the uniform charge distribution
⟨ˆni⟩ ≈ 1 in the ground state as discussed above. At the
interfaces that separate Ising domains of opposite signs, | [
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] |
interfaces that separate Ising domains of opposite signs,
there are segments with an electron density both above
and below the average value of one electron per site.
The snapshots during the relaxation of a JT system
with an electron filling of f = 0.49 are shown in FIG. 7.
Overall, a relaxation behavior similar to that of the half-
filled case was obtained. However, there is a major dif-
ference regarding the breathing mode and the electron
density. In contrast to the milder inhomogeneity for the | [
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] |
density. In contrast to the milder inhomogeneity for the
half-filling case, the amplitude of the QA1 mode as well
as the electron density are significantly reduced at the
interfaces, as shown in the top panels of FIG. 7. As the
C-type orbital/JT order is stabilized in the half-filling
limit, this result is consistent with the phase separation
scenario where nearly all of the doped holes go into the
interfaces between orbital domains.
Another intriguing feature of the QA1 configurations is | [
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] |
interfaces between orbital domains.
Another intriguing feature of the QA1 configurations is
the localized checkerboard modulations of the breathing-
mode amplitudes.
Such localized checkerboard modu-
lations can be seen in both filling fractions, but more
pronounced in the f = 0.49 case. A close-up view of this
localized structure is shown in FIG. 8(a). A region of
checkerboard modulation of the QA1 amplitudes is en-
closed in orbital domains represented by a N´eel-type or- | [
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] |
closed in orbital domains represented by a N´eel-type or-
der of the doublet QE vectors predominantly in the Qx
direction. As the electron number ˆni directly couples to
the breathing mode, this checkerboard structure also in-
dicates a concomitant charge density wave (CDW) char-
acterized by the wave vector K = (π, π).
It is worth
noting that these CDW states are essentially the same as
the CDW state of the Holstein model at half-filling [90–
92]. In terms of the orbital Ising parameter ϕ, shown in | [
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] |
92]. In terms of the orbital Ising parameter ϕ, shown in
FIG. 8(b), the checkerboard pattern is accompanied by a
vanishing orbital order, similar to the interface that sep-
arates different orbital domains. However, the average
electron density of the CDW is found to be exactly one
electron per site, which means that the doped holes are
not accommodated in such local structures.
It is worth noting that these localized CDW structures
are also obtained in the ED-based Langevin simulations | [
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] |
are also obtained in the ED-based Langevin simulations
on smaller systems, indicating that they are not an arti-
fact of ML models.
However, such charge modulation
patterns are mostly meta-stable during the relaxation
process. Our ED simulations on a 30 × 30 lattice found
that a state initialized to a homogeneous CDW order
is unstable and will decay into the C-type order. Yet,
as shown in the snapshots from large-scale ML-Langevin | [
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as shown in the snapshots from large-scale ML-Langevin
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] |
FIG. 8.
Localized checkerboard modulation of breathing
mode QA1 and electron density.
An example of such local
structures is displayed in terms of (a) the amplitude of breath-
ing mode as well as the QE vector and (b) the orbital Ising
order parameter.
11
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] |
PIkVYTj2aIwZVDFcBIF7FBsGIjXSAsqL4V4gESCsdWF6HsPj7/0XzHIqVvmVKhdzuPIgSNwDE6BA1xQA9egDhoAg3vwCJ7Bi/FgPBmvxtusdcmY
zxyAHzLevwAUi5Ra</latexit>nsteps = 3000
FIG. 9. Structure factor S(k, t) of the JT Qx distortion mode
at different time steps after suddenly cooled to a temperature
T = 0.001, starting from a random state. The system size is
100 × 100 with filling fraction f = 0.49, and averaged over 40
independent configurations.
simulations, e.g. FIG. 7, such local CDW order persists | [
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