# Manganites Thesis Statements

### Degeneracy space

In order to investigate the topological properties of defects, the media are characterized by order parameter fields valued in a space of degeneracy^{1,2,3,4,5}. In *R*MnO_{3}, the amplitude (*Q*) that reflects the relative strength of the *K*_{3} mode and the azimuthal tilt angle (φ) of the bipyramid is chosen as two components of the primary order parameter, and the amplitude of the mode (*P*) is the secondary order parameter^{24,26}. The degeneracy space can be determined by minimization of the temperature-dependent bulk free energy density. Figure 1(a–c) show the temperature dependent variation of potential-energy surface. At low temperatures, the surface adopts the form of a “Mexican hat” with six wells, and the energetic barriers between different trimerized states become negligible with increasing *T*. Next, the minimization of bulk free energy decreases *Q* remarkably and give rise to continuous variation of φ (from 0 to 2π), and then the degeneracy space of φ transforms from **R**_{1} (composed of six discrete points) with *Z*_{6} symmetry to **R**_{2} (with the form of a unit circle *S*^{1}) with *U*(1) symmetry^{33}. When *T* > *T*_{s}, the minimum is at *Q* = 0, suggesting that the system transforms to a centro-symmetric phase without distortion, and the degeneracy space **R**_{3} shrinks into a single point, as shown in supplementary section S1.

### Homotopy theory for *R*MnO_{3}

In homotopy theory, topological defects are classified by the elements of homotopy groups associated with the symmetry of the order parameter space. For *R*MnO_{3}, space **R**_{1} is composed of six discrete points, so the zeroth homotopy group π_{0}(**R**_{1}, *x*_{0}) (*x*_{0}: the base point in order parameter space) should be nontrivial and has a one-to-one correspondence with the set of connected components of **R**_{1}. Therefore the elements of π_{0}(**R**_{1}, *x*_{0}) classify domain walls. Meanwhile, the fundamental homotopy group π_{1}(**R**_{1}, *x*_{0}) and any higher order homotopy groups are all trivial because of the discrete *Z*_{6} topology of **R**_{1}. Assumption the degeneracy space of a system remains at **R**_{1}, even when temperature varies, domain walls are the only possible topological defects (see supplementary section S2, Fig. S2(a–c), and Movie I). When temperature approaches *T*_{s}, the degeneracy space expands to **R**_{2}, which is isomorphic to a unit circle *S*^{1}. In such a condition, only the fundamental homotopy group is non-trivial, and it is isomorphic to the additive group of integers, i.e. π_{1}(**R**_{2}, *x*_{0}) = *Z* (*Z* = {0, ±1, ±2, …}). Similarly, this suggests that only one-dimensional vortex strings are the only preferred type of topological defect if the degeneracy space is unchanged. Above the phase transition temperature *T*_{s}, the degeneracy space shrinks, resulting in trivial homotopy groups, and no topological defects exist.

However, the vortex structure observed in *R*MnO_{3} at room temperature is not constructed of domain walls alone, nor of strings alone, but of a combination of walls and strings. This implies that, with decreasing temperature, the topological defects must transform, accompanied by alternation of the degeneracy space. An exact homotopy sequence can describe this process^{14}. Because **R**_{1} is a subset of **R**_{2}, the exact sequence is:

where π_{1}(**R**_{1}, *x*_{0}) = 0 (the trivial group), π_{1}(**R**_{2}, *x*_{0}) = *Z*, π_{0}(**R**_{1}, *x*_{0}) = *Z*_{6}, and π_{0}(**R**_{2}, *x*_{0}) = 0. Homomorphism *i* maps π_{1}(**R**_{1}, *x*_{0}) to the identity element of π_{1}(**R**_{2}, *x*_{0}), so none of the non-trivial elements in π_{1}(**R**_{1}, *x*_{0}) are images of *i*, nor are the kernels of *j*; instead, they are mapped onto the non-trivial elements in π_{1}(**R**_{2}, **R**_{1}, *x*_{0}). According to the theorem, the kernel ker(*k*) of homomorphism *k* is a normal subgroup of π_{1}(**R**_{2}, **R**_{1}, *x*_{0}), and the quotient group π_{1}(**R**_{2}, **R**_{1}, *x*_{0})/ker (*k*) is isomorphic to the image im(*k*) of *k*. Because ker (*k*) = im (*j*), and im (*k*) = ker (*l*), we have

Similarly,

By combining these two equations, we obtain π_{1}(**R**_{2}, **R**_{1}, *x*_{0}) = *Z* × *Z*_{6}, which implies that, although a representative loop in real space around a string may start and end in **R**_{1}, the loop necessarily traverses a region (or regions) of **R**_{2} as it encircles the string. Normally, this region will be narrow for energetic reasons and form walls terminating on the vortex string. The elements (*m*, *n*) in π_{1}(**R**_{2}, **R**_{1}, *x*_{0}) suggest that pure domain walls and “string-wall-bounded” vortex structures are permitted forms of topological defects. The vortices can be classified by the elements (*m*, *n*) with *m *≠ 0, and elements (0, *n*) with *n *≠ 0 can classify the isolated domain walls that are not bounded with strings, i.e. the stripe, circle or annular domains that can coexist with vortices^{34,35} (see supplementary section S2). Normally, vortices with a low winding number *m *= ±1 (i.e. the vortex and anti-vortex) are the most common structures observed in *R*MnO_{3}. On the other hand, when *T* > *T*_{s} and the degeneracy space shrinks into **R**_{3}, all vortex cores become nonsingular in association the annihilation of strings. This transition corresponds to a process in which all elements in π_{1}(**R**_{2}, *x*_{0}) are mapped onto the identity (π_{1}(**R**_{3}, *x*_{0}) = 0) by the following homomorphism *m*:

This fact suggests that the bounded structure of strings and walls does not appear immediately after the structural phase transition. Instead, at first, only vortex strings emerge from the topological defect-free, high temperature phase. With declining temperature, domain walls appear progressively in string-attached or isolated forms, whereupon **R**_{1} becomes dominant. Since the intrinsic topology of strings is not broken by alternation of the degeneracy space in this system, it is protected by the formation of “string-wall-bounded” structures and high temperature residual features within vortex cores and domain walls. Thus the cores and walls appear in distinct regions that are composed mainly of the order parameter values belonging to **R**_{2}.

### Numerical simulation

Evolution of order parameter field with changing temperature, described in the previous section, is depicted in Fig. 2(a–c), which were obtained from numerical simulation (see supplementary section S3). These three images correspond to three typical states at *T* > *T*_{s}, *T*_{s} > *T*≫ 0 and *T *= 0, respectively. Figure 2(a) shows a defect-free homogenous state with all *Q* values in close proximity to 0. Just below the structure phase transition temperature *T*_{s}, *Q* increases slightly to minimize bulk free energy, which remains insensitive to the value of φ, so there is no preference for φ at this stage. However, the spatially configuration of order parameter field has noticeable impact on the gradient energy, so smooth variation of φ among adjacent sites is energetically preferred. The system in this state is analogous to the quasi-liquid phase of the *x*-*y* model in spin system because their degeneracy spaces both adopt *U*(1) symmetry, accompanied by the topological excitations (bounded vortex-antivortex pairs without domain walls) that minimize the total energy of the system^{36}, as shown in Fig. 2(b). A further decrease of temperature results in the increase of *Q*, which drives the system into six-fold degeneracy; then domains/domain walls emerge, lowering both the local bulk energy and gradient energy, and each core is surrounded by six domain walls, as shown in Fig. 2(c). It is also notable that the positions of vortex cores change little from Fig. 2(b,c) and that no nucleation or annihilate of cores is observed. This implies that these strings are stable across a large temperature region. The density of vortex cores is controlled mainly by the rate at which the temperature decreases across *T*_{s}. During the transformation from Fig. 2(b,c), the intrinsic topology of strings is not affected by symmetry-breaking of the degeneracy space, and it is protected by the formation of “string-wall-bounded” structures. So order parameter values that belong to **R**_{2} can be preserved within vortex cores and walls, as shown in the two enlarged panels in Fig. 2. With such microstructure features, the high stability of vortex cores under external electric field is also demonstrated by our simulations, which coincide with our experimental results (see supplementary Figs S3 and S4).

In analyzing the evolution of vortex pattern, we find that the emergence of *U*(1) is critical for the formation of vortices, even though the final state is of *Z*_{6} symmetry (see Movie II and III in supplementary materials). Note also that theoretical simulations based on phase-field methods or six-state clock model can yield certain results in good agreement with our conclusion: the initial states set for Monte-Carlo calculations are disordered states that can be excited only at high temperatures, the calculation steps correspond to a quenching or annealing process accompanied by spontaneous symmetry breaking from *U*(1) to *Z*_{6}, and the six-fold vortices observed in the final state are the inevitable product of this evolution process^{37,38,39,40,41}.

### Relationship between order parameter and structural distortion

Because the structural distortions are associated with the condensation of the *K*_{3} and modes, the corrugation state of *R* layers will change in accord with the variation of the order parameters^{24,26,31,42}. Group -theoretical analysis shows that there are three distinct symmetry-allowed isotropy subgroups (*P*6_{3}*cm*, and *P*3*c*1) for the *K*_{3}-irreducible representation of *P*6_{3}/*mmc*, so these three structures can exist in the *R*MnO_{3} system^{42}, and they correspond to φ ∈ Φ, φ ∈ Φ’ and φ ∉ (Φ∪Φ’), respectively (Φ = {0, π/3, 2π/3, π, 4π/3 and 5π/3}, Φ’ = Φ + π/6). Figure 3(a) shows the *P*6_{3}*cm* unit cell along the direction in which displacements of *R* atoms are obviously visible. The relationship between displacements of *R* atoms and φ is summarized in Fig. 3(b). (This relationship is qualitatively verified by our first-principles calculation, see supplementary section S4). In addition, because the value of *Q* can be used to quantify the amplitude of the *K*_{3} mode^{24}, a decrease of *Q* is accompanied by a reduction of atomic displacements at all *R* sites. Our simulation results show that *Q* decreases only slightly and φ deviates little from Φ within domain walls, so it is reasonable that the unit cells in these regions do not show significant difference from the bulk regions. This coincides with the results given in ref. 31: although φ deviates from Φ within several unit cells, the absolute displacements of *R* atom change little at domain walls. However, more significant deviation of the order parameter from **R**_{1} space happens in the core regions: φ varies continuously around the center, and *Q* decreases dramatically from outside in. So, in contrast to what we observed in the bulk region with *P*6_{3}*cm* symmetry, the atomic structure at a vortex core is more likely to adopt the features of the high energy phase and *P*3*c*1, so significant distortions of *R* layers are expected to be found within the cores.

### Atomic image at domain wall

In order to clarify the atomic structural features within vortex cores in *R*MnO_{3}, we employed high-resolution high angle annular dark field (HAADF) STEM. Figure 4(a) shows an atomically resolved HAADF image of a vortex structure in Y_{0.9}In_{0.1}MnO_{3}, taken along the [110] crystal orientation. The sample thickness in this region is estimated to be 70 nm by electron energy loss spectra. Slight doping of indium in YMnO_{3} can increase the vortex core density, helping one to find proper vortex cores in TEM samples^{43}. Six kinds of domains, denoted as α^{+}, β^{−}, γ^{+}, α^{−}, β^{+} and γ^{−}, with alternating polarization and structural phases, can be identified in this region. However, it is commonly noted that the domain boundaries often show a complex variation of atomic contrast within a few (or tens of) unit cells, where the shift of *R* ions differs from that in domain regions. In vortex center, the imaging of Y columns becomes much more complicated (enlarged image is shown in Fig. S6). This phenomenon implies that unfamiliar structural features may exist within core regions.

An experimental image for a typical domain wall in YMnO_{3} is reproduced in Fig. 4(b). It is clear that the shifts of Y atoms yield a polarization change from left to right. In particular, the atomic structure in the middle region (with a span of about 5 unit cells and marked in red) shows irregular image streaking. Careful examination reveals that one in every three Y atoms become ambiguous, the less bright atomic columns are accompanied by atomic streaking along the * c* axis. This structural feature is obvious in the middle region and becomes invisible at both sides of the red-marked region. Experimental observations of a variety of samples show that atomic contrast anomalies always occur on the 4(b) Wyckoff positions in

*P*6

_{3}

*cm*symmetry. The “down-down-up” poling configuration can be converted into “down-up-up” or “up-down-up”, depending on whether the transition starts at the (2/3, 1/3,

*z*) or (0, 0,

*z*) site. Similar phenomena also happen in other kinds of domain walls (See Fig. S7). Based on theoretical simulations and experimental observations, we suggest that the contrast anomalies observed at domain walls arise from the spatial structure of these defects (see supplementary section S6): in TEM experiments, we cannot expect that domain walls are always parallel to the observation direction (e.g. the [110] zone axis in the present case). Considering a slice sample perpendicular to the [110] direction, as shown in Fig. 4(c), two adjacent domains partially overlap along [110] because of the sloped domain wall. Figure 4(d) is a schematic atomic model illustrating a profile projection of the slice (only

*R*atoms are shown). Along the [110] direction, some columns contain oppositely distorted

*R*atoms (the inset of Fig. 4(d)). These columns are arranged periodically, and the ratio between atoms with opposite distortion varies with the depth of the domain wall. Since this kind of structural feature could affect the channeling effect and dynamic scattering process when electrons pass through a sample

^{44,45}, the contrast of these atomic columns will change significantly in comparison with bulk regions. In order to verify the influence of this “overlapping effect” on HAADF imaging at domain walls, we performed STEM simulations via the QSTEM software package

^{46}. These calculations are based on the “sharp transition model” in which ferroelectric polarization flips immediately at the wall

^{31}. The result is shown in Fig. 4(e). As the proportion of upward shifted Y atoms increases from 10% to 90%, the variation of atomic contrast matches well with our experimental results. When the proportion is 50%, the ambiguous contrast becomes most significant, and its intensity center approaches the middle plane of two Mn layers. So this may lead to the erroneous interpretation that there is an extra atomic column at the high-temperature mirror plane

^{27}. Moreover, “overlapping effect” also occurs in vortex core region. Because of the complex spatial configuration of vortex cores, we can see two thirds of Y columns show ambiguous contrast periodically in the center regions (see supplementary section S7).

### Atomic structure at vortex core

In order to reveal the structural features at the vortex core, we used the two-dimensional (2D) Gaussian function to fit the intensity and position of each *R* atom in Fig. 4(a). After that, the contrast features can be described quantitatively by fitting parameters: the intensity centers are determined by the 2D peak positions and the degree of ambiguous is quantified by the longitudinal standard deviation (see supplementary section S8). Figure 5(a) is a map of displacement of the Y intensity center (determined by the deviation of each Y intensity center from the middle plane of two adjacent Mn layers: the value is negative (positive) when the intensity center is below (above) this plane), and Fig. 5(e) shows a map of the standard deviation, illustrating the elongation of the Y atomic image. These two mappings can be used to directly identify the change in position of atomic intensity centers and regions where “overlapping effect” exists.

Considering the triple superstructure in *R*MnO_{3}, we can periodically label the *R* atoms as *R*_{1}, *R*_{2} and *R*_{3} on the projective plane along the [110] direction (as shown in Fig. 3(a)). Figure 5(b–df–h) are maps of center deviation and longitudinal standard deviation for these three Y sites, respectively. In Fig. 5(b), the intensity center of the Y_{1} site shifts downward in the left area and upward in the right area, and these two areas transition smoothly over several unit cells. This means that the intensity centers shift upward gradually from left to right in this transition region. Similarly, the transition regions can also be found in Fig. 5c,d. Note that the standard deviation at the Y_{1} site falls to a low value in the vortex core region, as shown in Fig. 5(f), i.e. no “overlapping effect” exists here. So, the continuous change of intensity center shown in Fig. 5(b) exactly reflects the continuous atomic distortion at Y_{1} sites in core region. The continuous shift of Y atoms demonstrates that the atomic structure within the intersection of the six domains is differs from the domain regions and hence cannot be described by *P*6_{3}*cm* symmetry. Instead, this structural feature coincides with the and *P*3*c*1 symmetry as analyzed in a previous section. Based on the span of this transitional area, the diameter of the core region can be estimated to be about 50 Å (see supplementary section S9 and Fig. S11). However, this value is much larger than the result given in ref. 32 (about 4 unit cells); this discrepancy may originate from the fact that, in that paper, the “overlapping effect” was not considered and the core region was determined simply by visual inspection instead of quantified measurement. As we discussed above, the real atomic positions may be misjudged due to the “overlapping effect”.

La_{1/3}Ca_{2/3}MnO_{3} is known to have a unidirectional superstructure aligned along the *a*-axis at low temperatures^{3,4,5}. The fundamental lattice at room-temperature and the superstructure at low temperatures are both orthorhombic with the space group of *Pnma*^{4,5}. From the ELC perspective, the superstructure, with a periodicity 3 lattice spacings, breaks the translational symmetry along one direction with respect to the fundamental lattice. With distinct structures along the *a* and *c* axes compared to the fundamental lattice, the superstructure breaks the point group rotational symmetry, as well (see supplementary material for details on the classification of ELC phases). Therefore, this electronic phase with LR superstructure can be classified as an electronic smectic. The superstructure can be probed by the SLRs either in electron diffraction (ED) patterns obtained from a large volume of the material or in electron nanodiffraction (END) patterns using an electron beam smaller than 2 nm in diameter (Fig. S1). The correlation length, measured from the width of the SLRs in the ED patterns (black symbols in Fig. 1a), decreases from ~70 nm at *T* = 98 K to ~4 nm at *T* = 306 K, suggesting that the superstructure loses its LR coherence upon warming but without a sharp signature of a transition. On the other hand, the SLRs shift from commensurate (*q* = 0.33 in this case) to incommensurate (C-IC transition; see supplementary material for more discussions in the C-IC transition) at *T*_{1} = 210 ± 10 K (Fig. 1b) upon warming, measured from both the ED and the END patterns (see supplementary material for the measurements using electron diffraction and synchrotron x-ray scattering).

To identify the temperature at which the superstructure transforms from LR-SR, we utilized scanning electron nanodiffraction (SEND) imaging^{33} to map the intensity of the SLRs in the END patterns in real space; the spatial distribution of the superstructure order (red) is shown in Fig. 1c. The superstructure order is LR at low *T*~113 K, shown by the homogeneously distributed red color in the scanned area. The intensity fluctuation in minor regions at the top of the SEND map at 113 K is comparable to the measurement uncertainty. The superstructure map starts to break into separated areas above *T*~210 K, and the superstructure regions continue to shrink on further warming. To obtain a measure of this change, the correlation length of the ordered areas in the SEND maps is indicated by the red dots in Fig. 1a, revealing a sharp transition at *T*_{1} = 210 ± 10 K, beyond which the disordered regions start to percolate and the ordered regions become isolated. We note firstly that our real-space characterization yields a LR-SR transition temperature that is the same as that for the C-IC transition, in contrast to the results from the spatially-averaged diffraction measurements in ref. 6. Secondly, because there is no LR superstructure at *T* > *T*_{1}, the translational symmetry of the electronic structure for the bulk is effectively the same as the fundamental lattice and only the rotational symmetry of the electronic superstructure remains broken with respect to the fundamental lattice. Namely the electronic phase as a whole in the bulk has a LR nematic order. At *T* > *T*_{2}~310 K, the SLRs have undetectable intensities and the electronic structure transforms into an isotropic phase, with the same translational and rotational symmetry as the fundamental lattice. Therefore, our observations identify an electronic smectic-nematic transition at *T*_{1}~210 K and an electronic nematic-isotropic transition at *T*_{2}~310 K in La_{1/3}Ca_{2/3}MnO_{3}.

The coincidence of the electronic smectic-nematic transition (the same as the LR-SR transition by the definition) with the C-IC transition stimulated further exploration to understand the relationship. A recent work of Nie and coworkers^{34} indicates that, for a 3D system at finite temperature, a commensurate stripe phase is stable against weak disorder (likely to be charge disorders) and therefore is consistent with smectic order, but an incommensurate stripe phase is not; the resulting “vestigial order” is a nematic, consisting of SR incommensurate stripes and a disordered counterpart^{34}. Our experimental observations present a concrete example of that theoretical proposal. More interestingly, an empirical rule *q* = 1 − *x*, where *q* is the wave number of the superstructure and *x* is the doping level, was found to describe the ground state of LCMO crystals at both commensurate or incommensurate doping levels^{3}. Accordingly the effective doping level *x*_{eff}, derived from *x*_{eff} = 1 − *q*, inside the ordered nano-regions is 0.75 ± 0.02 at 295 K, significantly larger than the nominal doping of 0.67. Note that the variation in *q* with temperature is seen consistently in both the real-space and spatially averaged measurements. Charge neutrality would require that the increase in doping of the ordered regions be compensated by a reduction in the disordered regions. We will return to the point later and present evidence for corresponding charge inhomogeneity.

To characterize the evolution of defects in the superstructure across the smectic-nematic transition, we performed dark-field imaging as shown in the top row of Fig. 2. At a temperature far below the transition, the superstructure order exhibits uniform stripe-like contrast in the *a-c* plane. On warming to 160 K, two pairs of dislocations can be seen (highlighted by blue and green dashed ellipses), with each pair appearing to break a stripe from the middle. At 200 K, just below the transition, an additional dislocation pair can be seen (yellow dashed ellipse) and the spatial separation of a pair has increased (green ellipse). When the temperature is above *T*_{1}, the proliferation of defect destroys the periodic order, making it impossible to distinguish individual dislocations.

The coherence of the superstructure can be better visualized by mapping the phase function of the superstructure, as done in the bottom row of Fig. 2 (see more details in supplementary Fig. S3). Clearly, the appearance of discrete dislocations has little effect on the long-range phase order below the electronic smectic-nematic transition; it is responsible only for local phase discrepancies from the uniform background. Because the dislocations always appear to form in pairs, each pair can be treated as a local singularity that does not affect the LR coherence of the superstructure. Indeed, the proliferation of the dislocations at 220 K causes the percolation of the disordered region, corresponding to the transition to the nematic phase. These observations provide the first direct confirmation of the key role of dislocations, as proposed by the ELC theory more than a decade ago^{18}.

As mentioned before, there has been a long and lively debate over the nature of the superstructure modulation in doped manganites with *x* ≥ 0.5^{1,4,6,7,8,9,10,11,12,13,14,15,16}. For *x* = 0.67, in particular, neutron^{4} and electron^{3,6,9,10} diffraction studies provide strong evidence for a stripe-like order involving individual rows of Mn^{3+} separated by double rows of Mn^{4+} ions (corresponding to inserting “solitons” of Mn^{4+} into the a so-called CE-type state that occurs for *x* = 0.5)^{7}. The extra electron on each Mn^{3+} site is associated with either a or Wannier orbital (with substantial weight on neighboring O atoms)^{11,12}, resulting in large Mn-O bond length splittings of 0.1 Å^{4}. The modulation wave vector is oriented along the orthorhombic *a* axis, but much of the displacements are transverse to that direction. This ordered state is supported by theoretical calculations^{14,35}; the superstructure modulation is illustrated in Fig. 3b. The controversy is largely associated with the thermally- or doping-induced C-IC transition^{6,7,8,13} and with disputed reports of sliding charge-density waves^{15,16}. Those phenomena have led to proposals that the modulation might involve a relatively weak, uniform variation of charge^{8}, as in a charge-density wave, or the development of discommensurations due to competing order parameters^{6,8}.

Our observation that the commensurate-incommensurate transition occurs via dislocations is compatible with the model of stripe-like order in the commensurate phase. A model for the formation of the dislocation pairs in the superstructure is shown in Fig. 3a, with the defect-free superstructure demonstrated in Fig. 3b. We propose, as illustrated in the middle panel of Fig. 3a, that a thermal excitation can cause a defect in which the electron on one orbital hops to orbital by thermal excitation, along with the elastic distortions of four neighboring MnO_{6} octahedra. This configuration will certainly cost energy due to the elastic strain relative to the orbitally-ordered MnO_{6} octahedra above and below it; however, it should cost relatively little energy to extend this defect along the stripe direction, allowing the pair of dislocations to separate. The entropic free-energy gain from such configurations may compensate for the elastic-energy costs. We want to emphasize here that the dislocations observed in Fig. 2 are defects in the electronic superstructure and seem to be edge-type dislocations. As suggested by our model (Fig. 3a), dislocations in the superstructure do not necessarily indicate defects in the crystal lattice. Indeed, based on the thermal evolution of dislocations in the smectic phase shown in Fig. 2, the dislocations in the superstructure are most likely not related to defects of the average crystal structure. Our observations demonstrate that orbital order can play a dominant role in the LR commensurate superstructures.

In the nematic phase observed at temperatures above *T*_{1}~210 K, dislocations are no longer relevant as the superstructure exists only in isolated areas. These isolated areas continuously shrink in size with warming, and the superstructure in these confined areas appears to be truly incommensurate, without the appearance of any dislocations^{9}. This indicates that the origin of the incommensurability measured in the nematic phase cannot be defects in the superstructure, such as the previously proposed as discommensurations^{6}, but must, instead, be intrinsic in nature. Charge-density-wave (CDW) order was once hypothesized to be the possible origin of the uniformly incommensurate superstructure in La_{1−x}Ca_{x}MnO_{3}^{13,15}. However, as we demonstrated above, the direct imaging of the superstructure here comes from the orbital contribution, and we propose a new model for the incommensurate superstructure, shown in Fig. 3c. The original model for the charge and orbital ordering structure with a commensurate three lattice-spacing period has the or orbital located only at so-called Mn^{3+} O_{3} octahedra, shown in Fig. 3b. These orbitals can be considered as the combination of the and orbitals with the orbital mixing angle θ = 120° or −120° in the formula . In the incommensurate superstructure model shown in Fig. 3c, the orbital mixing angle θ can vary continuously from one column of MnO_{3} octahedra to the next, so that the orientation of the *e*_{g} orbital rotates as one progresses along the *a* axis. The sinusoidal variation of the amplitude of the atomic displacements (transverse) along the *a* axis is only the simplest model that we could propose, and it can be refined with further experimental evidence and theoretical insight (see supplementary discussion section 4).

We emphasize that the proposed model here is very distinct from the conventional concept of the ordered orbitals, in which they must be aligned with the Jahn-Teller distortions in doped manganites. Theoretical calculations have already pointed out that the conventional understanding of the electronic structures by electron-lattice coupling alone, i.e., directly relating orbital ordering with Jahn-Teller distortion in a linear energy term, could be “misleading” or “insufficient to stabilize the orbital ordered state”^{11,36}. Those calculations indicate that a better approach is to include electron-electron coupling and to take self-consistency into account to determine the electronic structure at a more realistic level^{11,36}. As a result of self-consistency, it is possible to expect various angles between the driving Jahn-Teller distortion directions and the orientations of the consequently ordered orbitals. This is consistent with the model we have proposed here and is supported by our direct observations in real-space. A good analogy to such a mechanism may be the canted spin ordering in many antiferromagnetic systems, where the directions of the spins cannot be driven by the magnetic field alone, but also involve the spin-spin interactions, especially when the driving magnetic fields are not strong. The orbital degree of freedom, sometimes described in terms of a pseudo-spin, is demonstrated here to resemble actual spin ordering scenarios, such as a spin-density-wave (SDW).

To further explore the mechanism of the smectic-nematic transition, a mean-field theory has been employed to test a charge-only version of the model in Fig. 3a. We first constructed a Ginzburg-Landau (GL) free energy which prefers unidirectional density-wave order along the *a*-axis (see supplemental material for details). In a region marked by the red box in Fig. 4a, the sign of the anisotropy in the free energy is reversed so that the perpendicular stripe direction is locally preferred. By numerically minimizing the GL free energy, the order within the red box is reduced, similar to the appearance of a pair of dislocations (Fig. 4a). The difference in order parameter between regions inside and outside of the red box should result in variations in local charge density. To visualize this effect, we compute the local charge density, yielding the results in Fig. 4b. Interestingly, we find that the average charge density within the red box is about 10% lower than the surrounding ordered area (for the parameters used in the calculation). When the disordered patches are small, the shift in the average charge density should be small, but when they proliferate, the increased charge density in the regions with the superstructure order should become noticeable, resulting in an increase in *x*_{eff}, qualitatively consistent with the observed change in *q* (Fig. 1b).

This “charge rich” and “charge poor” electronic phase separation scenario was theoretically proposed by previous work^{37}, but is lacking of direct experimental observations to confirm. Spatially-resolved spectroscopic results, obtained by scanning a small electron beam over the La_{1/3}Ca_{2/3}MnO_{3} sample, provide further support for the charge segregation scenario. The energy separation *E*_{s} of the pre-peak and the main-peak at the oxygen K edge in the electron energy-loss spectra (EELS) has been shown to have a direct correlation to the doping level *x* in doped manganites^{38,39}. A linear relationship between the *E*_{s} and *x* in La_{1−x}Ca_{x}MnO_{3} (0 ≤ *x* ≤ 1) (Fig. S5b) is used as a spectroscopic method to quantify the local *x*_{eff}. Most of the spectra collected at *T* = 300 K as a function of position are the same as the black spectrum shown in Fig. 5a, while a small fraction matches the red one. The sized of the electron probe used for the EELS measurements was ~1.5 nm in diameter, several times the average lattice constants. It follows that the *E*_{s} measurements average over a distribution of Mn ions, thus providing an averaged measure of charge density. Using Gaussian curve fitting, *E*_{s} were measured to be 6.85 eV and 7.16 eV for the black set and red set of spectra, respectively, giving *x*_{eff} to be 0.67 ± 0.04 (the nominal doping of the bulk) and 0.77 ± 0.04, respectively. It is worthwhile to highlight that the *x*_{eff} = 0.77 ± 0.04 from a few locations using EELS is quantitatively consistent with the *x*_{eff} = 0.75 ± 0.02 obtained using END results. Moreover, the evolution of the regions with extra charge density upon cooling is shown in the line plot of *E*_{s} in real-space as a function of temperature (Fig. 5b). The material starts in a homogeneous state at high temperature, consistent with the electronic isotropic phase. Upon cooling, areas with extra charge grow in size and the charge deviation (the value above the shaded band) decreases. The line plot of *E*_{s} is back to a homogeneous state at low temperatures, as expected. It should be noted that the value of *E*_{s} in the high-temperature homogeneous state (no superstructure) is the same as that in the low-temperature homogeneous state (with LR superstructure), consistent with the expectation that *E*_{s} probes the locally-averaged charge state while being insensitive to the structural distortion associated with the orbital ordering. We note that the scanning direction for the displayed data is along the *a* axis; however, there is no qualitative difference observed for other scanning directions. It is very interesting that the results from the END and EELS analyses, which measure distinct local properties, are quantitatively consistent concerning the size, temperature dependence and local charge deviation from nominal doping level, revealing a scenario of electronic phase separation at the nanoscale (see supplemental material for more details) during the ELC phase transitions in La_{1/3}Ca_{2/3}MnO_{3}.

We note that LR antiferromagnetic spin order (see dashed lines in Fig. 3b, which correspond to CE-type chains) was reported in this material to appear at *T* < 150 K^{4,5}, much lower than both of the ELC phase transitions. Therefore, we ignore the spin effect in the ELC transitions in La_{1/3}Ca_{2/3}MnO_{3}. Based on the proposed model for the incommensurate orbital ordering and the observation of the charge segregation, the C-IC transition could be a result of competing mechanisms arising from the charge-orbital interplay. Specifically, the softening of the orbital excitations, or orbitons, might cause the C-IC transition, while the local charge fluctuation/segregation gives rise to the electronic phase separation and breaks the LR phase into SR. The incommensurate orbital ordering observed here shares common features with some SDW structures. In the latter case, the spin value at each possible location in a SDW is a combination of two spin eigenstates, and SDW systems have been observed to have the C-IC transition, as well. The symmetry breaking and transition in the SDW phase using ELC classification has received considerable attention^{18,19,20,21,22,23,24}. Therefore, we expect that the ELC theory can provide more guidance for further exploration of orbitally-ordered structures in correlated materials in the future.

Finally, it is worthwhile to highlight that for La_{1/3}Ca_{2/3}MnO_{3}, the crystalline symmetry decreases from cubic to orthorhombic at 1100 K^{40}, indicating that a certain degree of anisotropy between the *a* and *c* directions is already present at high temperature. However, the crystal lattice shows a rapid rise of the anisotropy on cooling below *T*~310 K^{4,5} without further symmetry breaking, i.e., retaining the same orthorhombic space group. This rise is clearly associated with the electronic isotropic-nematic transition, driven by electronic correlations, as the change in structural anisotropy is much too large to result from any mechanism driven only by the lattice. A similar situation, involving the growth of nematic order in a crystal with pre-existing broken rotation symmetry is known to occur in YBa_{2}Cu_{3}O_{6+x}^{21,22,41,42,43,44}.

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