Transition pathways connecting crystals and quasicrystals

Transition Pathways Connecting Crystals and Quasicrystals Jianyuan Yin1 , Kai Jiang2 , An-Chang Shi3 ,∗ Pingwen Zhang1 ,† and Lei Zhang4‡ 1 arXiv:2007.15866v1 [cond-mat.mtrl-sci] 31 Jul 2020 School of Mathematical Sciences, Laboratory of Mathematics and Applied Mathematics, Peking University, Beijing 100871, China. 2 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China. 3 Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M1. 4 Beijing International Center for Mathematical Research, Center for Quantitative Biology, Peking University, Beijing 100871, China. (Dated: December 1, 2021) Transition pathways connecting crystalline and quasicrystalline phases are studied using an efficient numerical approach applied to a Landau free-energy functional. Specifically, minimum energy paths connecting different local minima of the Lifshitz–Petrich model are obtained using the highindex saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state. Since the discovery of quasicrystals characterized by quasiperiodic positional order with non-classical rotational symmetries [1], tremendous progresses have been made on the understanding of these fascinating materials [2, 3]. Various quasicrystals have been reported [1, 4–7]. Besides examples from metallic alloys, quasicrystalline order has been observed in different systems including Faraday waves and soft matter [8–15]. While the structure of quasicrystals are now well understood [16], the study of the thermodynamic stability of quasicrystals and phase transitions between crystals and quasicrystals, which requires the examination of the free-energy landscape of the system, remains a challenge. In particular, the emergence of quasicrystalline order from crystalline states, or the nucleation of quasicrystals, represents a long-standing unsolved problem. In general, nucleation of a stable phase from a metastable state could be examined using three approaches, i.e. classical nucleation theory, atomistic theory, and density-functional theory [17, 18]. Within the framework of the density-functional theory, the freeenergy landscape of the system is described by a freeenergy functional determined by the density of the molecular species. Stable and metastable phases of the system correspond to local minima of the free-energy landscape, whereas the minimum energy paths (MEPs) on the free-energy landscape represent most probable transition pathways between different phases. Transition states (i.e. index-1 saddle points) on the pathways can be identified as critical nuclei, representing critical states along the transition pathways. This theoretical framework has been applied successfully to various problems undergoing phase transitions, including the nucleation and growth of crystalline structures [19–21]. In this Letter, we examine the transition pathways connecting quasicrystals and crystals within the framework of density-functional theory. Specifically, we develop an efficient numerical method using the high-index saddle dynamics (HiSD) and apply it to a Landau freeenergy functional, i.e. the Lifshitz–Petrich (LP) model [22], with local minima corresponding to two-dimensional (2D) crystalline and quasicrystalline phases. MEPs connecting various local minima of the model are computed and critical nuclei of the 6-fold crystalline and 12-fold quasicrystalline states are identified. In particular, two MEPs connecting two ordered phases are obtained, revealing that the phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to either a one-stage phase transition or a two-stage phase transition involving a metastable intermediate quasicrystalline state. Although our methodology applies to any free-energy functional, we will focus on the LP model for simplicity. The LP model is a Landau-theory designed to explore quasicrystalline structures with two characteristic length scales [22]. Despite its deceivingly simple form, the LP model possesses a rich phase behaviour containing a number of equilibrium ordered phases with 2-, 6-, and 12-fold symmetries [22, 23]. As such, this simple Landau free-energy provides an ideal model system for the study of transition pathways connecting crystals and quasicrystals. The LP model assumes a scalar order parameter φ(r) corresponding to the density profile of the molecules in a volume V . The free-energy functional of the model is given by [22, 23],  Z

2 1 ∇ 2 + 12 ∇ 2 + q 2 φ F(φ) = dr 2  ε 2 α 3 1 4 − φ − φ + φ , (1) 2 3 4 where 1 and q are two characteristic wavelength scales. The thermodynamic behaviour of this model is controlled by two parameters, ε and α, where ε is a temperaturelike parameter and α is a parameter characterizing the asymmetry of the order parameter. Possible equilibrium phases of the model correspond to local minima of the 2 free-energy functional with the mass conservation, which are solutions of the Euler-Lagrange equation of the system, δF δφ = 0. The Euler-Lagrange equation has multiple solutions, correspond to stable/metastable phases, transition states, and high-index saddle points of the model system. The phase diagram of the LP model has been examined by a number of researchers [22, 23]. The first step of the study is to find accurate solutions, corresponding to crystals and quasicrystals, of the EulerLagrange equation for the free-energy functional Eq. (1). Because quasicrystals do not have periodic order, special numerical methods are needed to describe their structures accurately. In general, discretization methods for quasiperiodic structures include the crystalline approximant method (CAM) [24] and the projection method [25]. In this Letter, we adopt the CAM to approximate quasiperiodic structures in the whole space with periodic structures in a large domain with proper sizes. For a given set of d base vectors {e∗1 , · · · , e∗d }, a reciprocal lattice vector k of d-dimensional quasicrystals can be expressed as, k = κ1 e∗1 + · · · + κd e∗d , κj ∈ R. (2) It is important to note that some of the coefficients κj might be irrational numbers. A quasiperiodic function φ(r) can be expanded as, X φ̂(k) exp(ik · r). (3) φ(r) = k Since some reciprocal lattice vectors cannot be represented as linear combinations of e∗i with integer coefficients, proper rational numbers L are chosen such that Lκj of all the concerned reciprocal lattice vectors k could be approximated as integers. As a result, a quasiperiodic function could be approximated by a periodic function with a period 2πL,  X r , (4) φ̂(k) exp ik · φ(r) = L k where the reciprocal lattice vectors k are linear combinations of e∗j with integer coefficients. Within this approximation, the computational domain becomes [0, 2πL)d with periodic boundary conditions for φ(r). For the 2D π . For (d = 2) 12-fold quasicrystals, we have q = 2 cos 12 ∗ ∗ e1 = (1, 0) and√e2 = (0, 1), the coefficients to be approxπ imated are 1, 23 , 21 , q cos 12 , q cos π4 , q cos 5π 12 . According to the simultaneous Diophantine approximation [26], the proper values of L are 30, 82, 112, 306, etc. We have tested the accuracy of varying L and found that L > 112 gave results within the required accuracy. Therefore, we set L = 112 or larger in our numerical calculations, and use the spectral methods for Eq. (4) with N points in each dimension to discretize the order parameter. The nonlinear terms in the Euler-Lagrange equation are treated by using the pseudospectral method [24]. Multiple solutions of the Euler–Lagrange equation = 0 are obtained. The simplest solution is φ(r) = 0, corresponding to the homogeneous state (H). Beside this trivial solution, a number of spatially inhomogeneous solutions, including the 6-fold crystalline state (C6) and the 12-fold quasicrystalline state (QC), have been found. Interestingly, a new lamellar quasicrystalline state (LQ) that is periodic in one dimension and quasiperiodic in the other dimension is identified. It is noted that a similar structure was reported in [27]. The structures of these three ordered phases are shown in real and reciprocal spaces in Fig. 1. It is important to note that the Hessian D2 F(φ) of these ordered states has different multiplicities of zero eigenvalues, corresponding to the numbers of Goldstone modes of these states [28, 29]. While a stable state, corresponding to a local minimum of the free-energy functional, could be calculated by gradient descent algorithms with proper initial configurations, finding a transition state is much more difficult because it does not correspond to a local minimum. Moreover, multiple zero eigenvalues of its Hessian D2 F(φ) could lead to the degeneracy of transition states. The problem is further complicated by the fact that there is no a priori knowledge of the transition states. Most of the existing methods for solving nonlinear equations, such as homotopy methods [30–32] and deflation techniques [33, 34], are inefficient for this degenerate problem because of superabundant solutions from arbitrary translation. The string method [35, 36] requires a good initial path connecting the initial and final states. Because there is no obvious epitaxial relation between the crystals and quasicrystals, a direct interpolation between the initial and final states cannot lead to the MEP. Similarly, surface-walking methods for searching index-1 saddle points, such as gentlest ascent dynamics [37] and dimer-type methods [38], are also not capable of computing the transition states of QC from metastable states because the eigenvectors with zero eigenvalues for the metastable states would be mistaken as the ascent direction, leading to a failure of climbing out of the basin. In order to overcome these difficulties, we apply a numerical method based on the HiSD to compute the highindex saddle points corresponding to the transition states of quasicrystals. The HiSD for index-k saddles (k-HiSD) is governed by the following dynamic equation [39],   k X δF (φ), (5) φ̇ = − I − 2 vj vj⊤  δφ j=1 δF δφ where v1 , · · · , vk represent the ascent directions, which are taken as the eigenvectors of the smallest k eigenvalues of the Hessian D2 F(φ). Specifically, for a metastable state φ∗ whose Hessian D2 F(φ∗ ) has m zero eigenvalues, we use the locally optimal block preconditioned conjugate gradient (LOBPCG) method [40] to calculate {u∗1 , · · · , u∗m } as an orthonormal basis of the nullspace of 3 C6 LQ QC FIG. 1. Three stable ordered states in the √ LP model: C6 with f = F/V = −6.8 × 10−4 , LQ with f = −7.3 × 10−4 , and QC −4 −6 with f = −7.5 × 10 at ε = 5 × 10 , α = 2/2. The computational domain is [0, 2πL]2 with L = 112 and a 20π × 20π square is zoomed in for better illustration. Scale bars represent 10π. Each lower right inset is the diffraction pattern in the reciprocal space and the arrows specify primitive reciprocal vectors (in Z). the Hessian D2 F(φ∗ ) and u∗m+1 as a normalized eigenvector of the smallest positive eigenvalue. Next, we apply the (m + 1)-HiSD by choosing φ(0) = φ∗ + ǫu∗m+1 as the initial search position and vi = u∗i (i = 1, · · · , m + 1) as the initial ascent directions for searching an index(m + 1) saddle. The small positive constant ǫ is used to push the system away from the minimum, which could be regarded as an upward search on a pathway map [41]. By relaxing the (m + 1)-HiSD for time-dependent φ with updated ascent directions vi (i = 1, · · · , m+1) by LOBPCG, a stationary solution φnew can be found, corresponding to a degenerate transition state with only one negative eigenvalue and m zero eigenvalues in most cases. If φnew turns out to be a high-index saddle point, for instance, an index-k saddle (k 6 m), we then implement (k − 1)-HiSD to apply a downward search on a pathway map [41] to search lower-index saddles until the degenerate transition state is located. The MEP is then obtained by following the gradient flow dynamics along positive and negative unstable directions of the transition state. FIG. 2. Transition pathway from H to QC with ε = −0.01, α = 1. The QC critical nucleus shows a circular shape with a small amplitude. Scale bars represent 10π. First, we present the MEP connecting a homogeneous phase to a quasicrystal. By choosing ε = −0.01 and α = 1, QC has a lower free-energy density of f = F/V = −2.7 × 10−3 than H with f = 0. The critical nucleus of QC from H is shown in Fig. 2.It is composed of a clear QC pattern at the centre of the nucleus surrounded by damping density waves. The density wave at the centre of the nucleus has a much smaller amplitude than that of the corresponding QC state, indicating that the assumption of classical nucleation theory would be invalid in this case. Along the transition pathway and beyond the critical state, the nucleus grows isotropically with an increasing amplitude at the centre, eventually reaching a full QC phase (see Supplemental Movie 1). It is noted that the nucleation and growth of quasicrystals from the homogenous phase has also been simulated using a dy- Next we demonstrate how a quasicrystal would transform to a crystal. We choose ε = 0.05, α = 1 so that QC is a metastable state with f = −5.3 × 10−3 and C6 is a stable state with f = −6.3 × 10−3 . For this case we found two transition pathways connecting QC to C6. In the one-stage transition pathway, a circular critical nucleus of C6 shown in Fig. 3(a) is observed. Interestingly, the growing C6 nucleus after the critical state shows that a transient state along the transition pathway contains a new LQ interphase connecting the C6 and QC states (see Supplemental Movie 2 for QC → C6 transition pathway). This finding indicates that the LQ could serve as an intermediate state connecting the QC and C6 phases. Indeed, a two-stage transition pathway from QC to C6 has been obtained from our calculations. This namic phase-field crystal model in [42]. The structure of the critical nucleus and subsequent growing patterns predicted in the current study are consistent with the simulation results. cri cal nucleus f 1.0e-8 0.1 H 0 QC 0 1.2 -2.7e-3 -0.4 transi on pathway 4 two-stage pathway reveals a first transition from QC → LQ and a second transition from LQ → C6 as shown in Fig. 3(a). Nucleation at the first stage shows an ellipsoidal critical nucleus of LQ with the periodic direction as the major axis. The energy barrier of the LQ nucleus (∆f = 4.5 × 10−6 ) is lower than the energy barrier of the C6 nucleus (∆f = 5.3 × 10−6 ), indicating that the QC → LQ transition pathway is the more probable one. After the QC → LQ transition, the second-stage transition follows the formation of another ellipsoidal critical nucleus of C6 with the quasiperiodic direction of LQ as the major axis and eventually to the C6 phase (see Supplemental Movie 3 for QC → LQ → C6 transition pathway). It is noted that two transition pathways have been observed for the gyroid to lamellar transitions of block copolymers [21]. Furthermore, the appearance of a metastable intermediate state as a precursor of the stable phase is consistent with Ostwald’s step rule [43]. Finally we present results on the emergence of a quasiperiodic structure from a periodic structure. By √ choosing ε = 5 × 10−6 and α = 2/2, C6 becomes a metastable state with f = −6.8 × 10−4 and QC is a stable state with f = −7.5 × 10−4 . Again, two transition pathways are computed in this case (see Fig. 3(b)). It is noted that, along the one-stage transition pathway, a larger computation domain with L = 306 was used to avoid the effect of finite domain size on growth dynamics. Similar to the phase transition from QC to C6, a circular critical nucleus of QC is found on the one-stage transition pathway. After nucleation, the size of the QC nucleus increases with the appearance of the LQ interphase (see Supplemental Movie 4 for C6 → QC transition pathway). On the other hand, a two-stage transition pathway from C6 to QC via a metastable LQ, as shown in Fig. 3(b), has been obtained. The critical nucleus of LQ at the first stage assumes an ellipsoidal shape with the quasiperiodic direction as the major axis. The energy barrier of LQ nucleus (∆f = 1.7 × 10−6 ) is lower than that of C6 nucleus (∆f = 4.6 × 10−6 ), indicating that the C6 → LQ transition would be more likely chosen than the direct C6 → QC transition pathway. Because LQ and QC have similar free energies in this case, a small driving force from LQ to QC transition is expected. Therefore, a much larger critical nucleus of QC is found. A full C6 → LQ → QC transition pathway is shown in Supplemental Movie 5. In summary, we developed an efficient numerical method based on the HiSD to accurately compute critical nuclei and transition pathways between crystals and quasicrystals. The computational challenge of the problem stems from the existence of multiple zero eigenvalues of the Hessian of different ordered phases. We solved this challenging problem by applying the HiSD to search for high-index saddle points, resulting in the solutions corresponding to the degenerate critical nuclei. Application of the numerical method to the Lifshitz–Petrich model reveals an interesting set of transition pathways connecting crystalline and quasicrystalline phases. For the transitions between the crystalline C6 and quasicrystalline QC phases, two transition pathways, corresponding to a one-stage direct transition and a two-stage indirect transition, have been obtained. We found that a new, one-dimensional quasicrystalline, LQ phase, which shows periodicity in one direction and quasiperiodicity in the other direction, plays a crucial role to connect C6 with QC. Compared with one-stage transition pathway between C6 and QC, the two-stage transition pathway C6 ↔ LQ ↔ QC is, consistent with the Ostwald’s step rule, more probable because the LQ nucleus has a lower energy barrier. The current study reveals the mechanism of the structural transition during quasicrystal nucleation. The accurate numerical results provide a comprehensive picture of critical nuclei and transition pathways between periodic and quasiperiodic structures. Furthermore, the proposed methodology is applicable to a wide range of physical problems undergoing phase transitions. This work was supported by the National Natural Science Foundation of China (Grants No. 11861130351, No. 21790340, No. 11421101, No. 11771368) and the Natural Science and Engineering Research Council (NSERC) of Canada. J. Y. acknowledges the support from the Elite Program of Computational and Applied Mathematics for Ph.D. Candidates of Peking University. ∗ † ‡ [1] [2] [3] [4] [5] [6] [7] [8] [9] shi@mcmaster.ca pzhang@pku.edu.cn zhangl@math.pku.edu.cn D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951 (1984). C. Janot, Quasicrystals: A Primer (Clarendon Press, Oxford, 1992). J.-B. Suck, M. Schreiber, and P. Häussler, eds., Quasicrystals: An Introduction to Structure, Physical Properties and Applications (Springer, Berlin Heidelberg, 2002). W. Steurer, Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals, Z. Kristallogr. 219, 391 (2004). A. P. Tsai, Icosahedral clusters, icosaheral order and stability of quasicrystals—a view of metallurgy, Sci. Technol. Adv. Mater. 9, 013008 (2008). S. Fischer, A. Exner, K. Zielske, J. Perlich, S. Deloudi, W. Steurer, P. Lindner, and S. Förster, Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry, Proc. Natl. Acad. Sci. U.S.A. 108, 1810 (2011). M. Martinsons, M. Sandbrink, and M. Schmiedeberg, Colloidal trajectories in two-dimensional light-induced quasicrystals with 14-fold symmetry due to phasonic drifts, Acta Physica Polonica A 126, 568 (2014). W. S. Edwards and S. Fauve, Parametrically excited quasicrystalline surface waves, Phys. Rev. E 47, R788 (1993). X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, and J. K. Hobbs, Supramolecular dendritic liquid quasicrys- 5 (a) f one-stage transi on two-stage transi on cri cal nucleus transient state Δf=5.3e-6 1.2 0.6 LQ QC 0 -5.3e-3 LQ Δf=4.5e-6 Δf=3.2e-6 C6 -5.8e-3 -6.3e-3 transi on pathway (b) f transient state 0.8 LQ Δf=4.6e-6 0.4 C6 0 -6.8e-4 Δf=1.7e-6 LQ Δf=9.3e-6 QC -7.3e-4 -7.5e-4 transi on pathway FIG. 3. Transition pathways between QC and C6. (a) from QC to C6; (b) from C6 to QC. One-stage transition pathway (dot dash curve) includes a circular critical nucleus of QC and an inserted transient state after nucleation showing LQ serves as an interface between QC and C6. Two-stage transition pathway (solid curve) includes two ellipsoidal critical nuclei and a metastable LQ. Two critical nuclei are partially zoomed in for better illustration in (b). All critical nuclei are labeled by white rings and all scale bars represent 10π. tals, Nature 428, 157 (2004). [10] K. Hayashida, T. Dotera, A. Takano, and Y. Matsushita, Polymeric quasicrystal: Mesoscopic quasicrystalline tiling in ABC star polymers, Phys. Rev. Lett. 98, 195502 (2007). [11] D. V. Talapin, E. V. Shevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray, Quasicrystalline order in self-assembled binary nanoparticle superlattices, Nature 461, 964 (2009). [12] J. Zhang and F. S. Bates, Dodecagonal quasicrystalline morphology in a poly(styrene-b-isoprene-b-styrene-bethylene oxide) tetrablock terpolymer, J. Am. Chem. Soc. 134, 7636 (2012). [13] M. Huang, K. Yue, J. Wang, C.-H. Hsu, L. Wang, and S. Z. D. Cheng, Frank–Kasper and related quasicrystal spherical phases in macromolecules, Sci. China Chem. 61, 33 (2018). [14] M. N. Holerca, D. Sahoo, B. E. Partridge, M. Peterca, [15] [16] [17] [18] [19] X. Zeng, G. Ungar, and V. Percec, Dendronized poly(2oxazoline) displays within only five monomer repeat units liquid quasicrystal, a15 and σ Frank–Kasper phases, J. Am. Chem. Soc. 140, 16941 (2018). A. P. Lindsay, R. M. Lewis, B. Lee, A. J. Peterson, T. P. Lodge, and F. S. Bates, A15, σ, and a quasicrystal: Access to complex particle packings via bidisperse diblock copolymer blends, ACS Macro Lett. 9, 197 (2020). W. Steurer and S. Deloudi, Crystallography of Quasicrystals: Concepts, Methods and Structures (Springer, Berlin, Heidelberg, 2009). D. Kashchiev, Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000). J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. iii. nucleation in a two-component incompressible fluid, J. Chem. Phys. 31, 688 (1959). L. Zhang, L.-Q. Chen, and Q. Du, Morphology of critical nuclei in solid-state phase transformations, Phys. Rev. 6 Lett. 98, 265703 (2007). [20] R. Backofen and A. Voigt, A phase-field-crystal approach to critical nuclei, J. Phys. Condens. Matter 22, 364104 (2010). [21] X. Cheng, L. Lin, W. E, P. Zhang, and A.-C. Shi, Nucleation of ordered phases in block copolymers, Phys. Rev. Lett. 104, 148301 (2010). [22] R. Lifshitz and D. M. Petrich, Theoretical model for Faraday waves with multiple-frequency forcing, Phys. Rev. Lett. 79, 1261 (1997). [23] K. Jiang, J. Tong, P. Zhang, and A.-C. Shi, Stability of two-dimensional soft quasicrystals in systems with two length scales, Phys. Rev. E 92, 042159 (2015). [24] P. Zhang and X. Zhang, An efficient numerical method of Landau–Brazovskii model, J. Comput. Phys. 227, 5859 (2008). [25] K. Jiang and P. Zhang, Numerical methods for quasicrystals, J. Comput. Phys. 256, 428 (2014). [26] H. Davenport and K. Mahler, Simultaneous Diophantine approximation, Duke Math. J. 13, 105 (1946). [27] Z. Jiang, S. Quan, N. Xu, L. He, and Y. Ni, Growth modes of quasicrystals involving intermediate phases and a multistep behavior studied by phase field crystal model, Phys. Rev. Materials 4, 023403 (2020). [28] J. Goldstone, A. Salam, and S. Weinberg, Broken symmetries, Phys. Rev. 127, 965 (1962). [29] A.-C. Shi, Nature of anisotropic fluctuation modes in ordered systems, J. Phys. Condens. Matter 11, 10183 (1999). [30] C. Chen and Z. Xie, Search extension method for multiple solutions of a nonlinear problem, Comput. Math. with Appl. 47, 327 (2004). [31] D. Mehta, Finding all the stationary points of a potentialenergy landscape via numerical polynomial-homotopycontinuation method, Phys. Rev. E 84, 025702 (2011). [32] W. Hao, J. D. Hauenstein, B. Hu, and A. J. Sommese, A bootstrapping approach for computing multiple solutions of differential equations, J. Comput. Appl. Math. 258, 181 (2014). [33] K. M. Brow and W. B. Gearhart, Deflation techniques for the calculation of further solutions of a nonlinear system, Numer. Math. 16, 334 (1971). [34] P. E. Farrell, Á. Birkisson, and S. W. Funke, Deflation techniques for finding distinct solutions of nonlinear partial differential equations, SIAM J. Sci. Comput. 37, A2026 (2015). [35] W. E, W. Ren, and E. Vanden-Eijnden, String method for the study of rare events, Phys. Rev. B 66, 052301 (2002). [36] Q. Du and L. Zhang, A constrained string method and its numerical analysis, Commun. Math. Sci. 7, 1039 (2009). [37] W. E and X. Zhou, The gentlest ascent dynamics, Nonlinearity 24, 1831 (2011). [38] L. Zhang, Q. Du, and Z. Zheng, Optimization-based shrinking dimer method for finding transition states, SIAM J. Sci. Comput. 38, A528 (2016). [39] J. Yin, L. Zhang, and P. Zhang, High-index optimizationbased shrinking dimer method for finding high-index saddle points, SIAM J. Sci. Comput. 41, A3576 (2019). [40] A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput. 23, 517 (2001). [41] J. Yin, Y. Wang, J. Z. Y. Chen, P. Zhang, and L. Zhang, Construction of a pathway map on a complicated energy landscape, Phys. Rev. Lett. 124, 090601 (2020). [42] S. Tang, Z. Wang, J. Wang, K. Jiang, C. Liang, Y. Ma, W. Liu, and Y. Du, An atomic scale study of twodimensional quasicrystal nucleation controlled by multiple length scale interactions, Soft Matter 16, 5718 (2020). [43] W. Ostwald, Studies on the formation and change of solid matter, Z. Phys. Chem. 22, 289 (1897).