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obF4q49Svc+TdO2y09cCFwzn3cu89QcyZ0o7zbS0tr6yurec28ptb2zu79t5+Q0WJLROIh7JVoAV5UzQuma01YsKQ4DTpvB8GbiNx+oVCwS93oUy/EfcF6jGBtJN8+LAWnqCj8tCNDpDSN1RhdoYrjFH274JSdKdAicTNSgAw13/7qdCOShFRowrFSbdeJtZdiqRnhdJzvJIrGmAxn7YNFTikykunL4zRiVG6qBdJU0 Kjqfp7IsWhUqMwMJ0h1gM1703E/7x2onuXspEnGgqyGxRL+FIR2iSB+oySYnmI0MwkczcisgAS0y0S1vQnDnX14kjUrZPS+f3VUK1esjhwcwTGUwIULqMIt1KAOB7hGV7hzXqyXqx362PWumRlMwfwB9bnD9UYlSY=</latexit>(b) nsteps = 200
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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
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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 <latexit sha1_base64="mOR Kj+qPGo0yiSztaOIscmha2s=">AB/HicbVDLSsNAFL3xWes r2qWbwSK4KknxtRGKblxWsA9oQ5hMJ+3QySTMTIQ6q+4caGI Wz/EnX/jtM1CWw9cOJxzL/feEyScKe0439bK6tr6xmZpq7y9s
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Ox7S14OQzu+APnM8fo5CUDA=</latexit>nsteps = 3000
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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 <latexit sha1_base64="mOR Kj+qPGo0yiSztaOIscmha2s=">AB/HicbVDLSsNAFL3xWes r2qWbwSK4KknxtRGKblxWsA9oQ5hMJ+3QySTMTIQ6q+4caGI
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Ox7S14OQzu+APnM8fo5CUDA=</latexit>nsteps = 3000
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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 <latexit sha1_base64="VDE2OyMgUEgUMKfAuAgjRVmZx0Q=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8cEzQOSJcxOepMhs7PLzKwQlnyCF w+KePWLvPk3TpI9aLSgoajqprsrSATXxnW/nMLK6tr6RnGztLW9s7tX3j9o6ThVDJsFrHqBFSj4BKbhuBnUQhjQKB7WB8O/Pbj6g0j+WDmSToR3QoecgZNVa6b/TdfrniVt05yF/i5aQCOer98mdvELM0QmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JilQEJY2VLGjJXf05kNJ6EgW2M6JmpJe9mfif101
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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 <latexit sha1_base64="4L2mL2JfiTA5AgAlR3No1wib1Q=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr5MEvHiMaB6QLGF
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-0.4061679542064667, -0.2357233315706253, 0.041941504925489426, -0.07171718031167984, 0.026828637346625328, 0.16418400406837463, 0.1901458203792572, 0.38233640789985657, -0.36005303263664246, -0.20153343677520752, -0.25245481729507446, -0.1296939253807068, -0.0647064745426178 ]
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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