Evolution of topological defects at two sequential phase transitions of Nd2SrFe2O7

How topological defects, unavoidable at symmetry-breaking phase transitions in a wide range of systems, evolve through consecutive phase transitions with different broken symmetries remains unexplored. Nd2SrFe2O7, a bilayer ferrite, exhibits two intriguing structural phase transitions and dense networks of the so-called type-II Z8 structural vortices at room temperature, so it is an ideal system to explore the topological defect evolution. From our extensive experimental investigation, we demonstrate that the cooling rate at the second-order transition (1290oC) plays a decisive role in determining the vortex density at room temperature, following the universal Kibble-Zurek mechanism. In addition, we discovered a transformation between topologically-distinct vortices (Z8 to Z4 vortices) at the first-order transition (550oC), which conserves the number of vortex cores. Remarkably, the Z4 vortices consist of two phases with an identical symmetry but two distinct magnitudes of an order parameter. Furthermore, when lattice distortion is enhanced by chemical doping, a new type of topological defects emerges: loop domain walls with orthorhombic distortions in the tetragonal background, resulting in unique pseudo-orthorhombic twins. Our findings open a new avenue to explore the evolution of topological defects through multiple phase transitions.

Understanding topological defects at phase transition is particularly important in light of their crucial role as fingerprints of order parameter topologies [7,15] and phase fluctuations at the critical temperature through the Kibble-Zurek mechanism [13,14]. The Kibble-Zurek mechanism describes the scaling of the defect density and how information spreads at a finite speed in a system that is driven through a continuous phase transition. It has been effectively tested on a variety of condensed matter systems such as liquid crystals [16,17], superfluid 3 He [2] and multiferroics [7,18]. However, little is known about how topological defects evolve upon crossing multiple phase transitions.
The prototypical model used to describe spontaneous symmetry-breaking is associated with a Mexican hat-type potential-energy landscape [19][20][21], where the distance from the peak and the angle around the axis represent the magnitude and phase of an order parameter, respectively.
A system characterized by this model energy landscape may have a continuous U(1) symmetry peak near the critical temperature (Tc), which plays a key role in the initial topological vortex-like defect formation at Tc before the system falls to the brim with discrete global minima at low temperatures, leading to discrete vortices. Multiferroic hexagonal manganite is an example of a phase transition described by such an energy landscape [19,21]. Interestingly, the restoration of a continuous U(1) symmetry has been predicted and confirmed at vortex cores even at low temperatures [20,[22][23][24], leading to topological protection from low-energy perturbations [25]. Dark and light colors correspond to the ground state and a metastable state, respectively. In the left path, the brown vortex core acts as a nucleation center, and DWs are associated with local minima, separating the global minima in an energy landscape as shown in Fig. 1c. The vortex structures are preserved when the system switches between global minima and local minima. Thus, for example, metastable-state vortices may evolve from ground-state vortices far below the critical temperature. The first requirement to realize such vortex transformation is to find a system having DWs with metastability that can be controlled.
From our study on (Nd,Tb,Sr)3Fe2O7, n = 2 Ruddlesden-Popper (RP) ferrite, we have discovered a new vortex-to-vortex transformation, which is unexpected, but consistent with topological protection. Ferrite is an intriguing system that exhibits one high-temperature secondorder structural transition and a following first-order structural transition above room temperature in addition to two magnetically ordered states at low temperatures [26][27][28][29], so it is an ideal system to explore topological phase transitions systematically within one system. Figures 1a-1b illustrate the details of various relevant structures. First of all, we have discovered the so-called type-II Z8 structural vortices structural vortices at room-temperature and examined the dynamics of Z8 T' structural vortices utilizing in-situ heating dark-field transmission electron microscopy (DF-TEM).
We have explored the effects of thermally-induced kinetic processes, long-range spontaneous distortion, and chemical doping on the resulting topological defect patterns and attempted to explain the origin of domain configurations by phase-field simulations. Our results show that, despite multiple phase transitions, the Kibble-Zurek mechanism is validated if the length scale is set at a second-order critical temperature. We have also unveiled an unexpected vortex-to-loop transformation due to the enlarged elastic interaction with chemical doping and thermodynamic and strain/pressure controls of topological defect patterns.

II. POSSIBLE SCENARIOS OF VORTEX TRANSFORMATION
Intriguing structural type-II Z8 T' structural vortices (eight-state vortex-antivortex pairs) were recently found in Ca2SrTi2O7, n = 2 RP bi-layered perovskite [30]. They exhibit a unique real-space topology in which domains and DWs are intricately intertwined with two sets of octahedral tilts marked by red and black arrows (the bottom cartoon of Fig. 1b). Figure 2a illustrates those sets of arrows, rotating every 45 degrees, are associated with two distinct 4 symmetries: an orthorhombic O' phase (Amam) and tetragonal T' phase (P42/mnm), respectively.
The smooth 90°-rotating, for example, from an up-tilt to a right-tilt state ( Fig. 1b yellow to red domains), occurs via a diagonal-tilt state (light-red DW in Fig. 1b), resulting in a sequence of all consecutive eight states around an un-tilted core in brown (Fig. 2a). Note that four states of the O' phase appear narrow as DWs and the other four states of the T' phase remain broad as domains.
This type of domain configurations with large disparity in size is called type-II domains; otherwise, type-I. Emphasize that the symmetry in the order parameter space is 4-fold, but 4 distinct domain states and 4 distinct domain wall states are present around a vortex core, and an oxygen displacement vector relevant to the order parameter rotates 8 times around vortex core, so we call these vortices as type-II Z8 vortices. Experimentally, however, Ca2SrTi2O7 reveal a direct tetragonal T to T' structural transition, indicating a significant instability of the desired orthorhombic O' phase [30].
On the other hand, Nd2SrFe2O7 ferrite, the structural analogy of titanates, was reported to  meV, which is about two times of that in hexagonal YMnO3 [19]. Full computational details [33][34][35][36][37] are given in Supplemental Material section I [31]. In addition to two sequential structural transitions, the system is in-plane antiferromagnetic below ~550 K [27][28][29] and spins are aligned along the c axis below 15 K (the details are discussed in Supplemental Material section II and Fig.   S1 [31]).

IV. IN-SITU HEATING DOMAIN OBSERVATION
To investigate the possible vortex-to-vortex transformation that we discussed earlier, we further performed in-situ TEM heating experiments above Tc2 on Nd2SrFe2O7 single crystals. Two specimens with initial low (Figs. 5b-5c) and high (Figs. 5e-5f) type-II Z8 T' vortex densities, resulting from different cooling rate across Tc1, were examined. Upon heating, the S2 = ½(200)T = (110)T' spots begin to fade (Fig. 5a) while the S1/S3 spots become stronger over a wide temperature range, indicating the hysteretic phase transformation from the T' to O'Tc2 phases. In real space, four-level colors exist; rectangular domains with dark-and light-grey contrasts appear above Tc2

V. SPONTANEOUS DISTORTION EFFECT ON DOMAIN PATTERNS.
Since type-II Z8 T' topological defects are associated with octahedral distortions, we turn our attention to the effects of spontaneous distortion on the vortex formation. It is well known that the amplitude of local oxygen octahedral distortions in RP compounds can be manipulated by varying the ionic size mismatch (i.e. the so-called tolerance factor [38]). We have prepared high quality polycrystalline samples of the solid solution Nd2CaxSr1-xFe2O7 in the range of 0 ≤ x ≤ 0.6, with the aim of increasing the tilt amplitude by the partial chemical substitutions of small Ca 2+ for large Sr 2+ . We establish the phase diagram (Fig. 7a) and spontaneous distortion (Fig. 7b)  A similar pattern was found in Tb2Ca0.65Sr0.35Fe2O7 (Fig. 7g) We then discuss the underlying mechanisms behind the formation of vortices and T'/O' loops based on phase-field simulations ( Fig. 8a-8b). When the temperature is between Tc1 and Tc2, the stable domain structure is orthorhombic twins with the possible coexistence of Z4 O'Tc2 vortices.
For a slow cooling across Tc1 and large spontaneous distortion, the Z4 O'Tc2 vortex density is very low. For simplicity, the initial domain configuration in our simulation is an O'Tc2 twin as shown in Fig. 8a, with the two domains correspond to φ = 1π⁄4 (light red) and 7π⁄4 (light yellow), respectively. When the temperature is cooled below Tc2, T' domains will replace O'Tc2 domains.
According to the energy landscape shown in Fig. 1c, the yellow T' domain (φ = 8π⁄4) naturally becomes the majority matrix since the corresponding order parameter is close to the two initial O'Tc2 domains. The red (φ = 2π⁄4) and green (φ = 6π⁄4) T' domains grow as inclusions within the matrix, and the boundaries of the inclusion are O' nano domains (Fig. 8b), consistent with the experimental observation ( Fig. 7e and Supplemental Material Fig. S9 [31]). The full evolution process is demonstrated in supplemental Movie [31]. The elimination of blue T' domain (φ = 4π⁄4) can be understood as a long switching path or a high energy barrier across the center peak (Fig.   1c)

VI. Conclusions
In summary, we discovered and explored the formation of topological defects of type-II Z8 The expected vortex-to-vortex transformation outlined in the left path of Fig. 1b was not observed in our system, but may require a system with two continuous (i.e., second-order) phase transitions.
In addition, unprecedented type-II Z8 T' (anti)vortex to T'/O' loop evolution, occurring with enhanced spontaneous distortion, represents a finely balanced set of tilt order parameters, which may enable external thermodynamic and strain/pressure control. It should be further investigated how these real-space crystallographic order parameter topologies presented here influence the configurations of antiferromagnetic domains and DWs through spin-lattice coupling in this bilayer ferrite. Furthermore, there are numerous materials undergoing multiple structural and/or magnetic phase transitions, our findings is a stepping stone to study the evolution of topological defects in any of those materials.