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] |
P. Avouris, B. Bhushan, D. Bimberg, K. von Klitzing,
6
H. Sakaki, R. Wiesendanger, and D. Bimberg (Springer
Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 165–187.
[33] See Supplementary Information, for details of implemen-
tation of the rigorous method of Trotter’s decomposition
with cumulant expansion to the system with two QD
qubits coupled directly and via an optical cavity, linear
optical polarization, exciton-phonon coupling matrix el-
ements and phonon spectral denisty, Hamiltonian trans- | [
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] |
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] |
dimensional exciton–phonon coupling for tunable and ef-
ficient single-photon generation with a carbon nanotube,
Nano Letters 17, 4184 (2017). | [
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] |
Kinetics of orbital ordering in cooperative Jahn-Teller models:
Machine-learning enabled large-scale simulations
Supriyo Ghosh,1 Sheng Zhang,1 Chen Cheng,1 and Gia-Wei Chern1
1Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
(Dated: May 24, 2024)
We present a scalable machine learning (ML) force-field model for the adiabatic dynamics of
cooperative Jahn-Teller (JT) systems. Large scale dynamical simulations of the JT model also shed | [
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] |
cooperative Jahn-Teller (JT) systems. Large scale dynamical simulations of the JT model also shed
light on the orbital ordering dynamics in colossal magnetoresistance manganites. The JT effect
in these materials describes the distortion of local oxygen octahedra driven by a coupling to the
orbital degrees of freedom of eg electrons. An effective electron-mediated interaction between the
local JT modes leads to a structural transition and the emergence of long-range orbital order at low | [
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temperatures. Assuming the principle of locality, a deep-learning neural-network model is developed
to accurately and efficiently predict the electron-induced forces that drive the dynamical evolution
of JT phonons. A group-theoretical method is utilized to develop a descriptor that incorporates
the combined orbital and lattice symmetry into the ML model.
Large-scale Langevin dynamics
simulations, enabled by the ML force-field models, are performed to investigate the coarsening | [
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] |
simulations, enabled by the ML force-field models, are performed to investigate the coarsening
dynamics of the composite JT distortion and orbital order after a thermal quench. The late-stage
coarsening of orbital domains exhibits pronounced freezing behaviors which are likely related to
the unusual morphology of the domain structures.
Our work highlights a promising avenue for
multi-scale dynamical modeling of correlated electron systems.
I.
INTRODUCTION
Colossal magnetoresistance (CMR) refers to an ex- | [
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] |
I.
INTRODUCTION
Colossal magnetoresistance (CMR) refers to an ex-
traordinary enhancement of the electrical conductivity
induced by applying a moderate magnetic field. Mangan-
ites of the formula (R1−xAx)MnO3 where R denotes rare
earth ions and A is a divalent alkaline ion are one of the
canonical systems that exhibit CMR effect [1–4]. In these
compounds several degrees of freedom, including itiner-
ant electrons, localized spins, orbitals, and lattice vibra-
tions, are simultaneously important [5]. | [
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] |
tions, are simultaneously important [5].
The complex
interplay between these degrees of freedom leads to var-
ious symmetry-breaking phases with similar characteris-
tic energy scales. Notable among them are ferromagnetic
metallic phases and several antiferromagnetic insulating
states accompanied by charge and/or orbital orders [6, 7].
In particular, it has been proposed that nanoscale phase
separation resulting from the various competing phases
plays a crucial role in the emergence of CMR [3–5, 8–10]. | [
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] |
plays a crucial role in the emergence of CMR [3–5, 8–10].
The metallic behaviors in manganites are mainly sta-
bilized by the double-exchange (DE) mechanism [11–13],
which originates from a strong Hund’s coupling between
spins of itinerant carriers at the eg orbital and S = 3/2
core spins from the localized t2g electrons. This charge-
spin coupling in manganites and related compounds is de-
scribed by a ferromagnetic Kondo-lattice model [14–16].
Since parallel alignment between core spins facilitates the | [
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] |
Since parallel alignment between core spins facilitates the
delocalization of eg electrons, DE mechanism tends to
enhance ferromagnetism. Importantly, upon small hole
doping, the Kondo-lattice model is shown to exhibit a
mixed-phase regime where phase separation occurs be-
tween antiferromagnetic state, stabilized at half-filling,
and hole-rich ferromagnetic regions [14].
Yet, early theoretical studies already showed that the
DE mechanism alone is not sufficient to describe the | [
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] |
DE mechanism alone is not sufficient to describe the
complex physics of CMR materials.
Indeed, the high-
temperature insulating phases in manganites cannot be
produced by DE models with realistic electron densities.
Strong electron correlation due to the on-site Coulomb
repulsion generally could lead to electron localizations
and the emergence of insulating phases in mangan-
ites [17–20]. Several works have also shown that electron-
phonon couplings via the Jahn-Teller (JT) mechanism | [
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] |
phonon couplings via the Jahn-Teller (JT) mechanism
are crucial to a complete description of the phase dia-
grams and transport behaviors of manganites [21–27].
The JT effect also highlights the importance of collec-
tive orbital behaviors and orbital ordering in mangan-
ites. While long-range orbital order can be induced by
super-exchange interactions as described by the Kugel-
Khomskii Hamiltonians [28–32], the cooperative JT ef-
fect is another mechanism for collective orbital behav- | [
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] |
fect is another mechanism for collective orbital behav-
iors and orbital ordering. Although the JT coupling de-
scribes a localized interaction between the orbital degrees
of freedom of an electron and the vibronic normal modes
of the surrounding MO6 octahedron, an effective long-
range interaction between local vibronic modes is gen-
erated by itinerant electrons. This mechanism is simi-
lar to the electron-mediated effective Ruderman-Kittel-
Kasuya-Yosida (RKKY) interaction between localized | [
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] |
Kasuya-Yosida (RKKY) interaction between localized
magnetic moments in spin glasses [33–35].
The long-
range structural distortion induced by this effective inter-
action, in turn, could localize the electrons and establish
a long-range orbital order [36, 37].
In particular, the orbital order in LaMnO3, which is
the parent compound of many CMR manganites, can be
stabilized by a predominant cooperative JT effect, even
in the absence of on-site Coulomb interaction [37].
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] |
in the absence of on-site Coulomb interaction [37].
In
real materials, it is most likely that both mechanisms
contribute to the stabilization of the observed orbital or-
der and structural distortions. An A-type antiferromag-
netic order with the wave vector q = (0, 0, π) is found to
accompany the observed orbital order. In this magnetic
arXiv:2405.14776v1 [cond-mat.str-el] 23 May 2024
2
structure, the localized t2g spins are ferromagnetically or-
dered within a 2D plane by the DE mechanism, while the | [
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] |
dered within a 2D plane by the DE mechanism, while the
spin-polarized planes are antiferromagnetically aligned
in the c-direction by a moderate super-exchange inter-
action. The orbital order, of the C-type, is characterized
by an in-plane checkerboard arrangement of d3x2−r2 and
d3y2−r2 orbitals and a concomitant K = (π, π) JT dis-
tortions.
Despite extensive studies on the orbital and JT tran-
sitions in manganites, the phase-ordering dynamics of
this composite order has yet to be investigated. As the | [
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] |
this composite order has yet to be investigated. As the
in-plane JT distortions of the C-type order break the
Z2 sublattice symmetry of the square lattice, the or-
bital/structural transition is expected to belong to the
Ising universality class.
While several universal coars-
ening behaviors of Ising-type order have been well es-
tablished, it is unclear whether they can be applied to
describe the kinetics of the C-type orbital order. First-
principles simulations of orbital ordering in the cooper- | [
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] |
principles simulations of orbital ordering in the cooper-
ative JT model are difficult since an electronic structure
problem has to be solved at every time-step in order to
obtain the electronic forces that drive the JT distortions.
For JT models without electron-electron interactions, the
electronic structure can be solved by exact diagonaliza-
tion, without resorting to more sophisticated many-body
techniques.
Yet, even repeated diagonalizations could | [
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] |
techniques.
Yet, even repeated diagonalizations could
be overwhelmingly expensive for large-scale simulations,
which partly explains the lack of progress on the study
of the dynamical aspects of JT phase transition.
In this paper, we develop a machine learning (ML)
force-field model to enable large-scale dynamical simu-
lations of orbital ordering in a representative coopera-
tive JT model on a square lattice. The ML force-field
framework, originally developed in the pioneering works | [
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] |
framework, originally developed in the pioneering works
of Behler and Parrinello [38], and Bart´ok et al. [39], has
now been widely used for large-scale ab inito molecular
dynamics with desired quantum accuracy. Here we gener-
alized this approach for the JT lattice models, in particu-
lar by developing proper descriptors to account for lattice
symmetries. Central to our approach is a deep-learning
neural network that could efficiently and accurately pre- | [
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] |
neural network that could efficiently and accurately pre-
dict the electronic forces that drive the dynamics of JT
modes. Moreover, the ML force-field approach is linearly
scalable, which means the NN model trained by ED solu-
tions from small systems can be directly applied to much
larger lattices without rebuilding or retraining.
We apply the ML force-field models to simulate ther-
mal quenches of a cooperative JT system at half-filling
and a slightly doped regime. The dynamical evolution of | [
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] |
and a slightly doped regime. The dynamical evolution of
the local JT modes is described by an effective Langevin
equation with driving forces obtained from ML mod-
els. In addition to the doublet JT distortions that break
the cubic symmetry of a MO6 octahedron, the dynamics
of the A1 breathing mode is also included. The relax-
ation process after a thermal quench is dominated by
the formation and subsequent coarsening of the C-type
JT/orbital order. Our simulations also find long-lasting | [
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] |
JT/orbital order. Our simulations also find long-lasting
localized meta-stable checkerboard density modulations.
Intriguingly, the late-stage growth of the orbital-ordered
domains does not follow the Allen-Cahn law expected for
a non-conserved Ising type order. Instead, a freezing be-
havior is observed for the coarsening of orbital and JT
domains.
The rest of the paper is organized as follows. We dis-
cuss the prototype cooperative JT model on a square | [
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] |
cuss the prototype cooperative JT model on a square
lattice that is relevant for the orbital ordering in man-
ganites in Sec. II. An effective Langevin equation is pre-
sented to describe the adiabatic dynamics of the JT dis-
tortions. In Sec. III we present a scalable ML force field
model for cooperative JT systems.
The ML model is
benchmarked against the exact diagonalization for the
force prediction and time-dependent distortion correla-
tion functions. Large-scale thermal quench simulations | [
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] |
tion functions. Large-scale thermal quench simulations
of the JT model and detailed characterizations of the
phase ordering dynamics and the growth law of orbital
domains are presented in Sec. IV. Finally, a summary
and outlook is presented in Sec. V.
II.
ADIABATIC DYNAMICS OF
COOPERATIVE JAHN-TELLER SYSTEMS
We consider a model of itinerant eg electrons inter-
acting with the vibrational modes of MO6 octahedra on
a square lattice. Since our main interest here is the or- | [
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] |
a square lattice. Since our main interest here is the or-
bital ordering, we will neglect the electron spin degrees of
freedom and the t2g core spins. In LaMnO3, the orbital
ordering (accompanied by a JT transition) takes place at
TOO = 750 K, which is much higher than the magnetic
transition at TM = 120 K, below which an A-type anti-
ferromagnetic order develops. In the paramagnetic phase
above TM, the spin degrees of freedom form a homoge-
neous, fluctuating background and we only consider the | [
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] |
neous, fluctuating background and we only consider the
orbital degrees of freedom in this work. Although orbital
orders in manganites are three-dimensional, as a first step
towards large-scale simulations, in this work we focus on
a two-dimensional version of the cooperative JT systems,
which exhibits the same K = (π, π) C-type order/JT or-
der in the ground state.
The Hamiltonian of the cooperative JT model on a
square lattice is given by
ˆH = ˆHK + ˆHJT + EL.
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] |
square lattice is given by
ˆH = ˆHK + ˆHJT + EL.
(1)
The three terms correspond to the electron kinetic en-
ergy, the electron-phonon coupling, and the lattice elas-
tic energy, respectively. The Hamiltonian ˆHK describes
the nearest-neighbor hopping of eg electrons
ˆHK =
�
γ=x,y
�
⟨ij⟩∥γ
�
µν=a,b
�
tγ
µνˆc†
iµcjν + h.c.
�
,
(2)
where ˆc†
i,µ and ˆci,µ represent the creation and annihila-
tion operators, respectively, of an electron with orbital
flavor µ at the i-th site, a and b denote the two ba- | [
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] |
flavor µ at the i-th site, a and b denote the two ba-
sis dx2−y2 and d3z2−r2 of eg orbitals, tγ
µν denotes the
3
orbital-dependent anisotropic hopping between nearest-
neighbor pairs ⟨ij⟩ parallel to the direction γ = x, y on
the square lattice. The following hopping coefficients are
used in our model calculations [14, 37]: taa
ij = −
√
3tab
ij =
−
√
3tba
ij = 3tbb
ij = tnn for hopping along the x-direction,
and taa
ij
=
√
3tab
ij =
√
3tba
ij = 3tbb
ij = tnn along the y-
direction. | [
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0.10578373819589615
] |
and taa
ij
=
√
3tab
ij =
√
3tba
ij = 3tbb
ij = tnn along the y-
direction.
The second term in ˆH describes the JT coupling:
ˆHJT = −λ
�
i
�
QA1
i
ˆni + QE
i · ˆτ
�
,
(3)
where QA1
i
= Q1
i denotes the breathing mode of the MO6
octahedron at site-i, which couples to the electron num-
ber operator
ˆni = ˆc†
iaˆcia + ˆc†
ibˆcib.
(4)
The doublet QE
i
= (Qx
i , Qz
i ) = (Q2
i , Q3
i ) describes the
JT modes that break the cubic symmetry of the octahe-
dron. Schematic diagrams of these three lattice modes | [
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] |
dron. Schematic diagrams of these three lattice modes
are shown in Fig. 1.
The JT doublet couples to the
pseudo-spin operator ˆτi = (ˆτ x
i , ˆτ z
i ) representing the elec-
tron orbital degrees of freedom:
ˆτ x
i = ˆc†
iaˆcib + ˆc†
ibˆcia,
ˆτ z
i = ˆc†
iaˆcia − ˆc†
ibˆcib.
(5)
The third term in Eq. (1) is the classical elastic energy
of the lattice distortions
EL = K
2
�
i
�
β
�
QA1
i
�2 +
��QE
i
��2�
,
(6)
where K denotes an effective elastic coefficient. The pa- | [
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] |
2
�
i
�
β
�
QA1
i
�2 +
��QE
i
��2�
,
(6)
where K denotes an effective elastic coefficient. The pa-
rameter β is defined as β = (ωA1/ωE)2, where ωA1 and
ωE are the vibration energies for the breathing and dou-
blet JT modes, respectively, assuming that the reduced
masses for these two modes are the same. Following pre-
vious works [37, 40], this parameter is set to β = 2 in the
following calculations.
It is worth noting that the breathing and JT modes
of each octahedra are independent of each other in this | [
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>6
FIG. 1.
Schematic diagram of the vibronic modes for the
MnO6 octahedron: (a) the symmetry-preserving breathing | [
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] |
MnO6 octahedron: (a) the symmetry-preserving breathing
mode, (b) and (c) the symmetry-breaking JT modes.
In
terms of oxygen displacements, the coordinates of these nor-
mal modes are: QA1 = (−X1 +X2 −X3 +X4 −X5 +X6)/
√
6,
Qx = (−X1 +X2 +X3 −X4)/2, and Qz = (−X1 +X2 −X3 +
X4 + 2X5 − 2X6)/
√
12, where Xi denotes the x coordinates
of the i-th oxygen, and so on.
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] |
FIG. 2. Schematic diagram of (a) C-type orbital order, and
(b) the concomitant antiferro-distortive JT order. Also shown
in panel (b) is the vector representation of the orbital/JT
order. The arrows represent either the doublet vector QE or
the expectation value of the orbital pseudo-spin ⟨ˆτ⟩. These
two vectors are related to each other via Eq. (7) in the ground
state.
elastic model. More realistic approach naturally needs to
include couplings between neighboring octahedra, for ex-
ample: E′
L = �
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] |
include couplings between neighboring octahedra, for ex-
ample: E′
L = �
ij Kmn
ij Qm
i Qn
j , where the indices m, n =
1, x, z. The equilibrium structural distortion is thus de-
termined by both such direct elastic interactions as well
as the effective electron-mediated interaction through
the cooperative JT effect discussed above.
In general,
as same oxygen atoms are shared by two neighboring
octahedra, nearest-neighbor couplings between these Q
modes are antiferromagnetic, i.e. K > 0, which are com- | [
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] |
modes are antiferromagnetic, i.e. K > 0, which are com-
patible with both the C-type orbital order and the meta-
stable CDW order to be discussed later. For simplicity,
in this work we neglect the direct elastic couplings and
focus on the intrinsic cooperative JT mechanism.
The ground state of the cooperative JT model in
Eq. (1) at half-filling exhibits a C-type orbital ordering,
characterized by a wave vector K = (π, π), accompanied
by a structural distortion of the antiferro-distortive order | [
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] |
by a structural distortion of the antiferro-distortive order
with a predominate Qx mode [37, 41–44]. A schematic
diagram of the orbital/JT order is shown in Fig. 2. The
orbital order can be described by the expectation values
of the pseudo-spins. To this end, minimization of the to-
tal energy with respect to the JT distortions yields the
relation
⟨ˆτi⟩ = K
λ QE
i ,
(7)
It is worth noting that the linear JT coupling in Eq. (3)
of a single octahedron is given by the inner product | [
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] |
of a single octahedron is given by the inner product
(ˆτ · QE), which implies that the interaction is invari-
ant under simultaneous rotation of the orbital pseudo-
spin and JT doublet vector. Inclusion of quadratic JT
couplings of the form QQτ reduces this O(2) symmetry
to a 3-fold degeneracy already at the single-octahedron
level. On the other hand, in the cooperative JT scenario,
the determination of the lattice distortions needs to in-
clude the kinetic energy of itinerant electrons on a lattice. | [
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] |
clude the kinetic energy of itinerant electrons on a lattice.
Since the electron hopping in Eq. (2) is anisotropic with
a strong orbital-dependence, the O(2) symmetry of the
4
doublet vector is reduced to a two-fold mirror symmetry
Qx → −Qx for the half-filled cooperative JT system [36].
The resultant minima are found to be along the Qx direc-
tion QE
∗ = ±(Q∗, 0), and the checkerboard arrangement
of these two symmetry-related minima QE
∗ in the C-type
order comes from a dominant electron-mediated nearest- | [
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] |
∗ in the C-type
order comes from a dominant electron-mediated nearest-
neighbor interaction of antiferro-distortive sign [37].
The Z2 mirror symmetry discussed above corresponds
to a tetragonal lattice symmetry. As shown in Fig. 1(b),
changing the sign of Qx sends a tetrahedron elongated in
the x-direction to one along the y-direction. The eigen-
states of ˆτx with eigenvalue +1 and −1 are dominated by
d3x2−r2 and d3y2−r2, respectively. The breaking of the | [
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] |
d3x2−r2 and d3y2−r2, respectively. The breaking of the
global Z2 symmetry of the cooperative JT system leads
to the C-type orbital order which can be described viewed
as a N´eel type order of Ising variable Qx. On symmetry
ground, the orbital JT phase transition is expected to
belong to the Ising universality class. A detailed ther-
modynamic study of the orbital/JT phase transition by,
e.g. Monte Carlo simulations, remain to be done.
Also of interest is the transition dynamics of the simul- | [
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] |
Also of interest is the transition dynamics of the simul-
taneous JT distortion and orbital ordering. Of particu-
lar interest is the coarsening behaviors of orbital domains
and whether the resultant domain-growth law falls into
well-established universality classes of phase-ordering dy-
namics. To this end, we consider the dynamical evolution
of the JT systems based on the approximation that elec-
tron relaxations are much faster than the dynamics of
JT distortions. This adiabatic approximation is similar | [
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] |
JT distortions. This adiabatic approximation is similar
to the Born-Oppenheimer approximation widely used in
the ab initio molecular dynamics methods [49]. In par-
ticular, this means that the electronic contribution to the
driving forces are computed from an equilibrium Fermi
liquid of the instantaneous lattice distortion.
As both
breathing and JT distortions are vibronic normal modes,
their dynamics are governed by an effective Newton equa-
tion of motion
µA1
d2QA1
i
dt2
+ γA1
dQA1
i
dt
= − ∂EL
∂QA1
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] |
tion of motion
µA1
d2QA1
i
dt2
+ γA1
dQA1
i
dt
= − ∂EL
∂QA1
i
− ∂⟨HJT⟩
∂QA1
i
+ ηA1
i (t),
µE
d2QE
i
dt2
+ γE
dQE
i
dt
= − ∂EL
∂QE
i
− ∂⟨HJT⟩
∂QE
i
+ ηE
i (t). (8)
Here µα is the effective mass of the octahedral normal
modes α = A1 and E, the corresponding damping coef-
ficients and Langevin noises are denoted by γα and ηα,
respectively. For simplicity, we shall assume the same pa-
rameters for the breathing and JT modes. In particular,
as discussed above, the difference in effective mass can | [
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] |
as discussed above, the difference in effective mass can
be accounted for by the ratio β of the elastic constants.
As in standard Langevin method, the thermal forces are
Gaussian random variables with zero mean and variance
consistently related to the damping coefficients γ through
the dissipation-fluctuation theorem.
The deterministic forces have two contributions, cor-
responding to the first two terms on the right-hand side:
the elastic restoring forces and the electronic forces via
JT coupling. | [
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0.953713059425354,
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0.7686728835105896,
0.47879770398139954,
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0.5681383013725281,
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-0.28390899300575256,
1.0695521831512451,
-0.13229982554912567,
0.3025118410587311,
-0.5118477940559387,
0.6365708708763123,
-0.4514167904853821,
0.09685051441192627,
0.5847393870353699,
-0.19363319873809814,
-0.2871902585029602,
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0.1287374198436737,
0.023608606308698654,
0.3635677397251129,
-0.3096323013305664,
-0.11138781905174255,
-0.37674522399902344,
-0.20275576412677765,
-0.23713351786136627,
0.9419028162956238,
0.17971062660217285,
0.33737823367118835,
0.10606224089860916,
0.4073914587497711,
0.2083829790353775,
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0.08803526312112808,
-0.1788972169160843,
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0.8050220608711243,
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0.17497985064983368,
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0.043569326400756836,
0.07533973455429077,
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0.17217972874641418,
-0.14618413150310516,
-0.1589721292257309,
0.5772802233695984,
0.06562282145023346,
0.07653066515922546,
-0.17884604632854462,
-0.08958058804273605,
-0.2737155258655548,
0.4393141269683838,
0.22682490944862366,
-0.4412359893321991,
-0.4707324504852295,
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-0.48634612560272217,
-0.06749044358730316,
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-0.27183645963668823,
-0.017637059092521667,
-0.30069878697395325
] |
the elastic restoring forces and the electronic forces via
JT coupling.
The calculation of forces can be simpli-
fied using the Hellmann-Feynman theorem: ∂⟨ ˆH⟩/∂Q =
⟨∂ ˆH/∂Q⟩. Using Eq. (3) and (6), one obtains the follow-
ing expressions for the driving forces
F A1
i
= −βKQA1
i
+ λ⟨ˆni⟩,
F E
i
= −KQE
i + λ⟨ˆτi⟩.
(9)
The equilibrium condition F E
i
= 0 gives the relation in
Eq. (7).
As discussed above, expectation values ⟨· · · ⟩
are computed based on an equilibrium electron liquid | [
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] |
are computed based on an equilibrium electron liquid
corresponding to the instantaneous lattice configuration
{QA1
i , QE
i }. Explicitly, for example, the expectation val-
ues of orbital pseudo-spins are given by
⟨ˆτi⟩ = 1
Ze
Tr
�
ˆτi e
−β ˆ
He
�
QA1
i
,QE
i
��
(10)
where ˆHe = ˆHK + ˆHJT is the electron Hamiltonian, β =
1/kBT is the inverse temperature, and Ze = Tre−β ˆ
He is
the electron partition function. As the electron Hamilto-
nian He is quadratic in the fermion creation/annihilation | [
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] |
nian He is quadratic in the fermion creation/annihilation
operators, it can be solved by the exact diagonalization
(ED) in real space. Yet, since the electronic forces have
to be computed at every time-step of the Langevin dy-
namics simulation, the O(N 3) time complexity of ED can
be overwhelmingly time-consuming for large-scale simu-
lations.
III.
MACHINE LEARNING FORCE-FIELD
MODELS
Here we present a machine-learning (ML) framework
for computing the electronic forces in cooperative JT sys- | [
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] |
for computing the electronic forces in cooperative JT sys-
tems with a linear-scaling complexity. Fundamentally, as
pointed out by W. Kohn, linear-scaling electronic struc-
ture methods are possible mainly because of the locality
nature or “nearsightedness” principle of many-electron
systems [50, 51].
Importantly, modern ML techniques
provide an explicit and efficient approach to incorpo-
rate the locality principle into the implementation of
O(N) methods. Perhaps the most prominent and suc- | [
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] |
O(N) methods. Perhaps the most prominent and suc-
cessful demonstration of this approach is the ML force-
field methods developed in quantum chemistry to enable
large-scale ab initio molecular dynamics (MD) simula-
tions [38, 39, 52–63]. Similar to the Langevin dynamics
for cooperative JT systems described above, the atomic
forces in ab initio MD are computed by solving, for exam-
ple, the Kohn-Sham equation, which has to be repeated
at every time-step [49]. The central idea behind the lin- | [
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at every time-step [49]. The central idea behind the lin-
ear scalability of ML force-field methods is the divide-
and-conquer approach proposed in the pioneering works
of Behler and Parrinello [38] and Bart´ok et al. [39]. The
ML model is trained to produce a local atomic energy
from a finite neighborhood. The atomic forces are ob-
tained indirectly from the total energy, which is the sum
of all atomic energies.
Similar ML frameworks have recently been developed | [
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of all atomic energies.
Similar ML frameworks have recently been developed
to enable large-scale dynamical simulations in several
5
FIG. 3. Schematic diagram of the ML force-field model for the cooperative JT models. A lattice descriptor transforms the
neighborhood distortion configuration Ci into effective coordinates {Gm} which are then fed into a fully connected neural
network (NN). The output node of the NN corresponds to the local site-energy ϵi. The combination of the descriptor and the | [
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] |
NN provides an approximation for the universal function ε(·). The corresponding total potential energy E is obtained from the
summation of the local energies. Automatic differentiation is employed to compute the derivatives ∂EML/∂Qi for the effective
forces acting on the breathing and JT modes.
condensed-matter lattice systems [64–70].
In particu-
lar, the ML force-field approach was applied to the fer-
romagnetic Kondo-lattice or s-d models [64, 65].
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] |
romagnetic Kondo-lattice or s-d models [64, 65].
The
strong-coupling regime of this model corresponds to the
DE mechanism which is another important component of
CMR physics as discussed in Sec. I. The spin dynamics
in the DE system is driven by itinerant electrons, simi-
lar to the cooperative JT models. Based on a general-
ized Behler-Parrinello scheme, a neural network model
was trained to predict local effective fields induced by
propagating electrons. The ML force-field methods have | [
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] |
propagating electrons. The ML force-field methods have
also been applied to the semiclassical Holstein model
on a square lattice [69], a canonical system for study-
ing the physics of electron-phonon coupling and phonon-
assisted charge-density wave orders. The scalar Einstein
phonons in the standard Holstein model can be viewed as
a simplified model for the breathing mode QA1. Large-
scale dynamical simulations enabled by the ML meth-
ods unveiled intriguing anomalous coarsening behavior | [
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] |
ods unveiled intriguing anomalous coarsening behavior
of charge-density waves in the Holstein model [69].
A.
Behler-Parrinello machine-learning framework
The Langevin dynamics discussed in Sec. II can be
viewed as a MD method for the octahedral normal modes.
This prompts a generalization of the Behler-Parrinello
(BP) approach for the adiabatic dynamics of JT systems.
However, there are two major differences between the two
systems. First, both breathing and JT modes are defined | [
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] |
systems. First, both breathing and JT modes are defined
on a lattice, in contrast to coordinates of atoms in free
space. Second, the symmetry of the JT modes is tied to
the symmetry of the underlying lattice, while MD sys-
tems are characterized by continuous translational and
rotational symmetries.
The BP scheme is modified to
account for these two important issues; see FIG. 3 for a
schematic diagram of the ML force-field model for the JT
systems. First, the total energy of the system in the adi- | [
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] |
systems. First, the total energy of the system in the adi-
abatic approximation is given by the expectation value of
the Hamiltonian in Eq. (1), which is to be approximated
by a ML energy EML. As in the BP scheme, this system
energy is partitioned into local energies, each associated
with a lattice site:
⟨ ˆH⟩ ≈ EML =
�
i
ϵi =
�
i
ε(Ci).
(11)
Here we have invoked the locality assumption and express
the site-energy ϵi as a function of lattice distortions in a | [
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] |
the site-energy ϵi as a function of lattice distortions in a
local neighborhood, denoted as Ci. Explicitly, we define
the local distortion configuration as
Ci =
�
Qj
�� |rj − ri| < rc
�
,
(12)
where rc is a cutoff distance which is determined by
the locality of the forces, and we have grouped the
breathing and JT modes into a three-component vector
Q = (Q1, Q2, Q3) = (QA1, Qx, Qz) for convenience. It
should be noted that the vector notation here does not
imply an underlying O(3) rotation symmetry. The com- | [
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] |
imply an underlying O(3) rotation symmetry. The com-
plex dependence of the site-energy on local environment
is encoded in the function ε(·) which is universal for a
given JT electronic Hamiltonian and electron filling frac-
tion.
Importantly, this universal function is to be ap-
proximated by a ML model in the BP approach.
As shown in FIG. 3, there are two central components
in the BP-type ML model: a descriptor and a learning
model. The former is to transform the neighborhood con- | [
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model. The former is to transform the neighborhood con-
figuration into a proper feature variables, while the latter
is to approximate the universal function ε(·). In this work
a feedforward neural network (NN) is employed as the
learning model which, according to universal approxima-
tion theorem [71, 72], offers the capability of accurately
representing complex functions to the desired accuracy.
The total energy is given by the sum of all site-energies
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] |
The total energy is given by the sum of all site-energies
6
in the system, which are obtained by applying the same
ML model to all lattice sites. The fact that the same
ML model is used for all lattice sites simply reflects the
translational symmetry of the original Hamiltonian. The
effective forces acting on the octahedral normal modes
are given by the derivatives of the total energy
Fi = −∂EML
∂Qi
,
(13)
which can be efficiently computed using automatic dif- | [
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] |
Fi = −∂EML
∂Qi
,
(13)
which can be efficiently computed using automatic dif-
ferentiation [73, 74]. This expression also indicates that
the ML predicted forces are conservative, which is an ap-
pealing feature of the PB-type ML models.
B.
Descriptors for lattice distortions
The BP scheme also allows for a systematic approach
to incorporate symmetry requirements into ML models
through feature engineering.
Since the local energy ϵi
at the output of the NN is a scalar, it is invariant under | [
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] |
Since the local energy ϵi
at the output of the NN is a scalar, it is invariant under
symmetry operations of the system. However, despite the
powerful approximation capability of NNs, such symme-
try constraints can only be learnt statistically, sometimes
with the help of techniques such as data augmentation.
However, the symmetry constraints cannot be exactly
implemented based on deep-learning alone. A descrip-
tor is introduced here to provide a proper representation | [
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] |
tor is introduced here to provide a proper representation
of the neighborhood configuration in such a way that
the representation is itself invariant under transforma-
tions of the relevant symmetry group. The resultant fea-
ture variables are input to the NN model. As a result,
symmetry-related configurations Ci are described by ex-
actly the same feature variables, which in turn produces
exactly the same local energy at the output.
The importance of descriptors in the implementation of | [
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] |
exactly the same local energy at the output.
The importance of descriptors in the implementation of
ML force field models for quantum MD was also empha-
sized in the original works of Behler and Parrinello [38].
A set of feature variables called the atom-centered sym-
metry functions (ACSFs) are introduced to represent
local atomic configurations such that the rotation and
reflection symmetries are exactly incorporated into the
atomic energy function [38, 54]. The building blocks of | [
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] |
atomic energy function [38, 54]. The building blocks of
ACSFs are relative distances and angles of atomic po-
sition vectors, which are manifestly invariant under ro-
tations of the SO(3) symmetry group [54]. The ACSF
descriptor is physically intuitive, yet to some extent ad
hoc, approach to ML force-field models for MD systems.
Since then many atomic descriptors have been proposed
and implemented [39, 52, 53, 75–80].
Since the JT models are defined on a lattice, the rele- | [
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] |
and implemented [39, 52, 53, 75–80].
Since the JT models are defined on a lattice, the rele-
vant symmetry group is reduced from the SO(3) rotation
group of free space to the point group associated with
the underlying lattice. Moreover, the JT modes are also
characterized by well-defined transformation rules of the
same point group. To describe the combined symmetry
transformations, consider the neighborhood Ci centered
at site-i, and a discrete rotation or reflection ˆg of the | [
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] |
at site-i, and a discrete rotation or reflection ˆg of the
point group that sends site-j to k, i.e. Rki = O(ˆg) · Rji,
where O(ˆg) is the orthogonal matrix representation of ˆg.
The transformation of the octahedral distortions is de-
scribed by
˜Q(γ,m)
k
= M (γ)
mn(ˆg) Q(γ,n)
j
.
(14)
Here γ indicates the irreducible representation (IR) of the
vibronic modes, and M (γ) is the orthogonal transforma-
tion matrix of IR-γ, the indices m, n = 1, 2, · · · nγ label | [
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] |
tion matrix of IR-γ, the indices m, n = 1, 2, · · · nγ label
the different components in this IR. For example, for the
double QE, a matrix corresponding to a 90◦-rotation is
M E(Cπ/4) = diag(−1, 1).
Importantly, a proper representation of the neighbor-
hood Ci needs to be invariant under these coupled sym-
metry transformations. A systematic approach to derive
invariants of a symmetry group is the group-theoretical
bispectrum method [81].
Atomic descriptors based on | [
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] |
bispectrum method [81].
Atomic descriptors based on
bispectrum coefficients have been used in conjunction
with Gaussian processing learning models for quantum
MD simulations [39, 77]. The group-theoretical method
has also been employed to develop a general theory of
descriptors for electronic lattice models in condensed-
matter systems [66, 82–84]. In particular, a descriptor
based on the idea of reference IRs was developed for
the Holstein model [69], which is essentially an electron- | [
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] |
the Holstein model [69], which is essentially an electron-
phonon model with the breathing modes.
Here we outline the group-theoretical derivation of in-
variant feature variables for the JT models; more details
can be found in Appendix A. First, the octahedral distor-
tions as represented by the set Ci essentially form a high-
dimensional representation of the point group. They can
then be decomposed into fundamental IRs of the point
group. This decomposition can be highly simplified as | [
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] |
group. This decomposition can be highly simplified as
the original representation matrix is automatically block-
diagonalized, with each block corresponding to a fixed
distance from the center-site.
Standard methods can
then be applied to the decomposition of each block [85].
For the square-lattice JT system described by the D4
point-group symmetry, there are three types of blocks
with a dimension of either 4 or 8. The derivation of the
relevant IRs for each block can be further simplified. This | [
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] |
relevant IRs for each block can be further simplified. This
is because while the JT modes QE transform as a doublet
in the octahedral group Oh, they are reduced to the direct
sum of two 1D IRs when restricted to the D4 group. Ex-
plicitly, under symmetry operations of D4, both QA1 and
Qz transform as A1 IR, while Qx transforms according to
IR B1, acquiring a −1 under reflections about the y = ±x
diagonals. Take the vibronic modes {QA, QB, QC, QD}
at the four nearest-neighbor sites as an example, the four | [
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] |
at the four nearest-neighbor sites as an example, the four
Qz modes can be decomposed as 4Qz = 1A1 +1B1 +1E,
with the following IR coefficients:
f A1 = Qz
A + Qz
B + Qz
C + Qz
D,
f B1 = Qz
A − Qz
B + Qz
C − Qz
D,
(15)
f E = (Qz
A − Qz
C, Qz
B − Qz
D).
The decomposition of the breathing QA1 modes is de-
scribed by the same formulas.
On the other hand,
7
while the four JT Qx modes are also decomposed as
4Qx = 1A1 + 1B1 + 1E, the B1 symmetry of Qx gives
rise to different IR coefficients:
f A1 = Qx
A − Qx | [
0.12051991373300552,
0.08042032271623611,
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-0.6291132569313049,
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0.490123450756073,
0.863789439201355,
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0.0994916781783104,
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0.7353513240814209,
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0.17692580819129944,
0.9897878170013428,
0.450489342212677,
0.16432742774486542,
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0.49700814485549927,
0.04148386791348457,
0.5931158065795898,
0.18725241720676422,
-0.030076323077082634,
0.37018999457359314,
0.36125385761260986,
0.09901928901672363,
0.24217644333839417,
0.19262737035751343,
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0.721168041229248,
0.4231107831001282,
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0.5127936601638794,
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0.15094806253910065,
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-0.2851480543613434,
-0.6121021509170532,
-0.12037637829780579,
-0.374929815530777,
0.2649567127227783,
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0.05896729975938797,
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0.792624294757843,
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0.29398801922798157,
0.1508389264345169,
0.3736499547958374,
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0.06861173361539841,
-0.4418257474899292,
-0.6587878465652466,
0.5413744449615479,
0.11581756919622421,
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0.059473175555467606,
-0.5352349877357483,
-0.01873387023806572,
0.3506620526313782,
0.2863560616970062,
0.3209076523780823,
-0.15711604058742523,
-0.3452386260032654,
0.07156793773174286,
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0.32518336176872253,
-0.3071340322494507,
0.5539366006851196,
0.07553538680076599
] |
rise to different IR coefficients:
f A1 = Qx
A − Qx
B + Qx
C − Qx
D,
f B1 = Qx
A + Qx
B + Qx
C + Qx
D,
(16)
f E = (Qx
A − Qx
C, −Qx
B + Qx
D).
The calculation of IR coefficients for other types of blocks
is presented in Appendix A.
For convenience, we arrange the coefficients of the r-
th IR of type Γ in the overall decomposition of Ci into a
vector f Γ
r = (f (Γ,r)
1
, f (Γ,r)
2
, · · · , f (Γ,r)
nΓ
). As the sum of the
squared coefficients of an IR, pΓ
r =
��f Γ
r
��2, is manifestly | [
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