Direct observation of the proliferation of ferroelectric loop domains and vortex-antivortex pairs

Atomic-Scale Tracking Topological Phase Transition Dynamics of Polar Vortex-Antivortex Pairs

Non-trivial topological structures, such as vortex-antivortex (V-AV) pairs, have garnered significant attention in the field of condensed matter physics. However, the detailed topological phase transition dynamics of V-AV pairs, encompassing behaviors like self-annihilation, motion, and dissociation, have remained elusive in real space. Here, polar V-AV pairs are employed as a model system, and their transition pathways are tracked with atomic-scale resolution, facilitated by in situ (scanning) transmission electron microscopy and phase field simulations. This investigation reveals that polar vortices and antivortices can stably coexist as bound pairs at room temperature, and their polarization decreases with heating. No dissociation behavior is observed between the V-AV phase at room temperature and the paraelectric phase at high temperature. However, the application of electric fields can promote the approach of vortex and antivortex cores, ultimately leading to their annihilation near the interface. Revealing the transition process mediated by polar V-AV pairs at the atomic scale, particularly the role of polar antivortex, provides new insights into understanding the topological phases of matter and their topological phase transitions. Moreover, the detailed exploration of the dynamics of polar V-AV pairs under thermal and electrical fields lays a solid foundation for their potential applications in electronic devices.

Melting of Unidirectional Charge Density Waves across Twin Domain Boundaries in GdTe3

Solids undergoing a transition from order to disorder experience a proliferation of topological defects. The melting process generates transient quantum states. However, their dynamic nature with a femtosecond lifetime hinders exploration with atomic precision. Here, we suggest an alternative approach to the dynamic melting process by focusing on the interface created by competing degenerate quantum states. We use a scanning tunneling microscope (STM) to visualize the unidirectional charge density wave (CDW) and its spatial progression ("static melting") across a twin domain boundary (TDB) in the layered material GdTe3. Combining the STM with a spatial lock-in technique, we reveal that the order parameter amplitude attenuates with the formation of dislocations and thus two different unidirectional CDWs coexist near the TDB, reducing the CDW anisotropy. Notably, we discovered a correlation between this anisotropy and the CDW gap. Our study provides valuable insight into the behavior of topological defects and transient quantum states.

Quantitative Mapping of Chemical Defects at Charged Grain Boundaries in a Ferroelectric Oxide

Polar discontinuities, as well as compositional and structural changes at oxide interfaces can give rise to a large variety of electronic and ionic phenomena. In contrast to earlier work focused on domain walls and epitaxial systems, this work investigates the relation between polar discontinuities and the local chemistry at grain boundaries in polycrystalline ferroelectric ErMnO3 . Using orientation mapping and scanning probe microscopy (SPM) techniques, the polycrystalline material is demonstrated to develop charged grain boundaries with enhanced electronic conductance. By performing atom probe tomography (APT) measurements, an enrichment of erbium and a depletion of oxygen at all grain boundaries are found. The observed compositional changes translate into a charge that exceeds possible polarization-driven effects, demonstrating that structural phenomena rather than electrostatics determine the local chemical composition and related changes in the electronic transport behavior. The study shows that the charged grain boundaries behave distinctly different from charged domain walls, giving additional opportunities for property engineering at polar oxide interfaces.

Pressure Control of Nonferroelastic Ferroelectric Domains in ErMnO3

Mechanical pressure controls the structural, electric, and magnetic order in solid-state systems, allowing tailoring of their physical properties. A well-established example is ferroelastic ferroelectrics, where the coupling between pressure and the primary symmetry-breaking order parameter enables hysteretic switching of the strain state and ferroelectric domain engineering. Here, we study the pressure-driven response in a nonferroelastic ferroelectric, ErMnO3, where the classical stress-strain coupling is absent and the domain formation is governed by creation-annihilation processes of topological defects. By annealing ErMnO3 polycrystals under variable pressures in the MPa regime, we transform nonferroelastic vortex-like domains into stripe-like domains. The width of the stripe-like domains is determined by the applied pressure as we confirm by three-dimensional phase field simulations, showing that pressure leads to oriented layer-like periodic domains. Our work demonstrates the possibility to utilize mechanical pressure for domain engineering in nonferroelastic ferroelectrics, providing a lever to control their dielectric and piezoelectric responses.

Coherent Many-Body Oscillations Induced by a Superposition of Broken Symmetry States in the Wake of a Quantum Phase Transition

It is now widely accepted that quenches through the critical region of quantum phase transitions result in post-transition states populated with topological defects-analogs of the classical topological defects. However, consequences of the very nonclassical fact that the state after a quench is a superposition of distinct, broken-symmetry vacua with different numbers and locations of defects have remained largely unexplored. We identify coherent quantum oscillations induced by such superpositions in observables complementary to the one involved in symmetry breaking. These oscillations satisfy Kibble-Zurek dynamical scaling laws with the quench rate, with an instantaneous oscillation frequency set primarily by the gap of the system. In addition to the obvious fundamental significance of a superposition of different broken symmetry states, quantum coherent oscillations can be used to verify unitarity and test for imperfections of the experimental implementations of quantum simulators.

Quantum phase transition dynamics in the two-dimensional transverse-field Ising model

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.

The Third Dimension of Ferroelectric Domain Walls

Ferroelectric domain walls are quasi-2D systems that show great promise for the development of nonvolatile memory, memristor technology, and electronic components with ultrasmall feature size. Electric fields, for example, can change the domain wall orientation relative to the spontaneous polarization and switch between resistive and conductive states, controlling the electrical current. Being embedded in a 3D material, however, the domain walls are not perfectly flat and can form networks, which leads to complex physical structures. In this work, the importance of the nanoscale structure for the emergent transport properties is demonstrated, studying electronic conduction in the 3D network of neutral and charged domain walls in ErMnO₃ . By combining tomographic microscopy techniques and finite element modeling, the contribution of domain walls within the bulk is clarified and the significance of curvature effects for the local conduction is shown down to the nanoscale. The findings provide insights into the propagation of electrical currents in domain wall networks, reveal additional degrees of freedom for their control, and provide quantitative guidelines for the design of domain-wall-based technology.

The observed topological vortex domains and the rotating magnetocaloric effect in the hexagonal RMnO3 (R = Ho, Er, and Yb) crystals

Hexagonal RMnO₃ (R = Er, Ho and Yb) single crystals were grown and their unique vortex domain structures, magnetic properties and magnetocaloric effect (MCE) were comprehensively investigated. The topological vortex domains/structures were clearly illustrated by polarized optical microscope and piezo response force microscopy, confirming a high quality of the crystals. The magnetic transitions related to R ³⁺-Mn³⁺ interactions and anisotropic properties were observed in the RMnO₃ crystals. The broad peaks of magnetic entropy change -ΔS _M appeared around [Formula: see text] revealed that the order of R ³⁺ moments is crucial to the large MCE. A giant rotating MCE (RMCE: ∼10.57 J kg^-1 K^-1) was obtained with magnetic field changing from 0 to 50 kOe in ErMnO₃, accompanied with a large refrigerant capacity (RC: ∼159 J kg^-1). These significant RMCE and RC behaviors are found to be closely related to the R ³⁺-R ³⁺, and R ³⁺-Mn³⁺ interactions in these RMnO₃. These results may open up a possibility for designing low-temperature magnetic cooling devices by tailoring the R-4f and Mn-3d orbit interactions.

Topological Transport of Deconfined Hedgehogs in Magnets

We theoretically investigate the dynamics of magnetic hedgehogs, which are three-dimensional topological spin textures that exist in common magnets, focusing on their transport properties and connections to spintronics. We show that fictitious magnetic monopoles carried by hedgehog textures obey a topological conservation law, based on which a hydrodynamic theory is developed. We propose a nonlocal transport measurement in the disordered phase, where the conservation of the hedgehog flow results in a nonlocal signal decaying inversely proportional to the distance. The bulk-edge correspondence between the hedgehog number and skyrmion number, the fictitious electric charges arising from magnetic dynamics, and the analogy between bound states of hedgehogs in ordered phase and the quark confinement in quantum chromodynamics are also discussed. Our study points to a practical potential in utilizing hedgehog flows for long-range neutral signal propagation or manipulation of skyrmion textures in three-dimensional magnetic materials.

Kibble-Zurek Behavior in Disordered Chern Insulators

Even though no local order parameter in the sense of the Landau theory exists for topological quantum phase transitions in Chern insulators, the highly nonlocal Berry curvature exhibits critical behavior near a quantum critical point. We investigate the critical properties of its real space analog, the local Chern marker, in weakly disordered Chern insulators. Because of disorder, inhomogeneities appear in the spatial distribution of the local Chern marker. Their size exhibits power-law scaling with the critical exponent matching the one extracted from the Berry curvature of a clean system. We drive the system slowly through such a quantum phase transition. The characteristic size of inhomogeneities in the nonequilibrium postquench state obeys the Kibble-Zurek scaling. In this setting, the local Chern marker thus does behave in a similar way as a local order parameter for a symmetry breaking second order phase transition. The Kibble-Zurek scaling also holds for the inhomogeneities in the spatial distribution of excitations and of the orbital polarization.

Conductivity control via minimally invasive anti-Frenkel defects in a functional oxide

Utilizing quantum effects in complex oxides, such as magnetism, multiferroicity and superconductivity, requires atomic-level control of the material's structure and composition. In contrast, the continuous conductivity changes that enable artificial oxide-based synapses and multiconfigurational devices are driven by redox reactions and domain reconfigurations, which entail long-range ionic migration and changes in stoichiometry or structure. Although both concepts hold great technological potential, combined applications seem difficult due to the mutually exclusive requirements. Here we demonstrate a route to overcome this limitation by controlling the conductivity in the functional oxide hexagonal Er(Mn,Ti)O₃ by using conductive atomic force microscopy to generate electric-field induced anti-Frenkel defects, that is, charge-neutral interstitial-vacancy pairs. These defects are generated with nanoscale spatial precision to locally enhance the electronic hopping conductivity by orders of magnitude without disturbing the ferroelectric order. We explain the non-volatile effects using density functional theory and discuss its universality, suggesting an alternative dimension to functional oxides and the development of multifunctional devices for next-generation nanotechnology.

Atomic-Scale Observation of Topological Vortices in the Incommensurate Charge Density Wave of 2H-TaSe2

It has been only recently realized that topological vortices associated with structural distortions or ordered spins are rather common in numerous materials where long-range interactions are not dominant. Incommensurate modulations that frequently occur in charge density wave (CDW) materials are often understood in terms of discommensurations with a periodic phase shift. The accumulation of a one-dimensional (1D) phase shift can result in, for example, CDW dislocations in 2H-TaSe₂ with incommensurate CDW (I-CDW). Since any atomic-scale experimental investigation of CDW dislocations in 2H-TaSe₂ has been lacking, we have performed the atomic-scale observation of 2H-TaSe₂ with I-CDW, stabilized with Pd intercalation or strain, with scanning probe microscopy, and unveiled the existence of topological Z₆ or Z₄ vortices with topologically protected 2D winding movements of atomic displacement vectors. The discovery opens the ubiquitous nature of topological vortex domains and a new avenue to explore new facets of various incommensurate modulations or discommensurations.

Hexagonal rare-earth manganites and ferrites: a review of improper ferroelectricity, magnetoelectric coupling, and unusual domain walls

Hexagonal rare-earth manganites and ferrites are well-known improper ferroelectrics with low-temperature antiferromagnetism/weak ferromagnetism. In recent decades, new multi-functional device concepts and applications have provoked the exploration for multiferroics which simultaneously possess ferroelectric and magnetic orders. As a promising platform for multiferroicity, hexagonal manganites and ferrites are attracting great research interest among the fundamental scientific and technological communities. Moreover, the novel type of vortex-like ferroelectric domain walls are locked to the antiphase structural domain walls, providing an extra degree of freedom to tune the magnetoelectric coupling and other properties such as conductance. Here, we summarize the main experimental achievements and up-to-date theoretical understanding of the ferroelectric, magnetic, and magnetoelectric properties, as well as the intriguing domain patterns in hexagonal rare-earth manganites and ferrites. Recent work on non-stoichiometric compounds will also be briefly introduced.

Topological thermalization via vortex formation in ultrafast quenches

We investigate the thermalization of a two-component scalar field across a second-order phase transition under extremely fast quenches. We find that vortices start developing once the thermal bath sets the control parameter to its final value in the nonsymmetric phase. Specifically, we find that vortices emerge as the fluctuating field relaxes and departs macroscopically from its symmetric configuration. The density of primordial vortices at the relaxation time is a decreasing function of the final temperature of the quench. Subsequently, vortices and antivortices annihilate at a rate that eventually determines the total thermalization time. This rate decreases if the theory contains a discrete anisotropy term, which otherwise leaves the primordial vortex density unaffected. Our results thus establish a link between the topological processes involved in the vortex dynamics and the delay in the thermalization of the system.

Hexagonal manganites: Strong coupling of ferroelectricity and magnetic orders

Hexagonal manganites belong to an exciting class of materials exhibiting strong interactions between a highly frustrated magnetic system, the ferroelectric polarization, and the lattice. The existence and mutual interaction of different magnetic ions (Mn and rare earth) results in complex magnetic phase diagrams and novel physical phenomena. A summary and discussion of the various properties, underlying physical mechanisms, the role of the rare earth ions, and the complex interactions in multiferroic hexagonal manganites are presented in this review.

Symmetry Breaking Bias and the Dynamics of a Quantum Phase Transition

The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like τ_{Q}^{-βδ/(1+zν)}, where β, δ, z, ν are the critical exponents and τ_{Q} is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation of quantum phase transition dynamics in the Ising chain with ultracold Rydberg atoms.

Piezoresponse force microscopy and nanoferroic phenomena

Since its inception more than 25 years ago, Piezoresponse Force Microscopy (PFM) has become one of the mainstream techniques in the field of nanoferroic materials. This review describes the evolution of PFM from an imaging technique to a set of advanced methods, which have played a critical role in launching new areas of ferroic research, such as multiferroic devices and domain wall nanoelectronics. The paper reviews the impact of advanced PFM modes concerning the discovery and scientific understanding of novel nanoferroic phenomena and discusses challenges associated with the correct interpretation of PFM data. In conclusion, it offers an outlook for future trends and developments in PFM.

Observation of Uncompensated Bound Charges at Improper Ferroelectric Domain Walls

Low-temperature electrostatic force microscopy (EFM) is used to probe unconventional domain walls in the improper ferroelectric semiconductor Er_0.99Ca_0.01MnO₃ down to cryogenic temperatures. The low-temperature EFM maps reveal pronounced electric far fields generated by partially uncompensated domain-wall bound charges. Positively and negatively charged walls display qualitatively different fields as a function of temperature, which we explain based on different screening mechanisms and the corresponding relaxation time of the mobile carriers. Our results demonstrate domain walls in improper ferroelectrics as a unique example of natural interfaces that are stable against the emergence of electrically uncompensated bound charges. The outstanding robustness of improper ferroelectric domain walls in conjunction with their electronic versatility brings us an important step closer to the development of durable and ultrasmall electronic components for next-generation nanotechnology.

Dynamical Critical Behavior of an Attractive Bose-Einstein Condensate Phase Transition

When matter undergoes a continuous phase transition on a finite timescale, the Kibble-Zurek mechanism predicts universal scaling behavior with respect to structure formation. The scaling is dependent on the universality class and is irrelevant to the details of the system. Here, we examine this phenomenon by controlling the timescale of the phase transition to a Bose-Einstein condensate using sympathetic cooling of an ultracold Bose thermal cloud with tunable interactions in an elongated trap. The phase transition results in a diverse number of bright solitons and gray solitons in the condensate that undergo attractive and repulsive interactions, respectively. The power law dependence of the average soliton number on the timescale of the phase transition is measured for each interaction and compared. The results support the Kibble-Zurek mechanism, in that the scaling behavior is determined by universality and does not rely on the interaction properties.

Influence of the Magnitude of Ferroelectric Domain Polarization on the Photochemical Reactivity of BaTiO3

The spontaneous polarization of domains in ferroelectric materials has been used to spatially separate photogenerated electrons and holes, reducing recombination and thereby improving the efficiency of photochemical reactions. Here, the influence of the magnitude of the polarization on photochemical reactivity is investigated. The magnitude of the out-of-plane component of the polarization was characterized by scanning Kelvin probe force microscopy (KFM). By examining crystals with orientations that deviate by only a few degrees from (001), two types of domains were identified: those with polarization vectors nearly perpendicular to the surface and those with polarization vectors nearly parallel to the surface. The photochemical reactivity was measured using topographic atomic force microscopy to determine the amount of Ag⁺ (Pb²⁺) that was photochemically reduced (oxidized) to Ag (PbO₂) on the surface. For the reduction reaction, the reactivities of domains with polarizations nearly perpendicular to the surface were only about 3 times greater than the reactivities of the domains with polarizations nearly parallel to the surface, indicating that, for this reaction, the magnitude of the out-of-plane polarization is less important than its sign. For the oxidation of lead, only the domains with polarizations nearly perpendicular to the surface were reactive, indicating that for this reaction, both the sign and magnitude of the polarization are important.

Electrical half-wave rectification at ferroelectric domain walls

Domain walls in ferroelectric semiconductors show promise as multifunctional two-dimensional elements for next-generation nanotechnology. Electric fields, for example, can control the direct-current resistance and reversibly switch between insulating and conductive domain-wall states, enabling elementary electronic devices such as gates and transistors. To facilitate electrical signal processing and transformation at the domain-wall level, however, an expansion into the realm of alternating-current technology is required. Here, we demonstrate diode-like alternating-to-direct current conversion based on neutral ferroelectric domain walls in ErMnO₃. By combining scanning probe and dielectric spectroscopy, we show that the rectification occurs at the tip-wall contact for frequencies at which the walls are effectively pinned. Using density functional theory, we attribute the responsible transport behaviour at the neutral walls to an accumulation of oxygen defects. The practical frequency regime and magnitude of the direct current output are controlled by the bulk conductivity, establishing electrode-wall junctions as versatile atomic-scale diodes.

Observation of magnetic vortex pairs at room temperature in a planar α-Fe2O3/Co heterostructure

Vortices, occurring whenever a flow field 'whirls' around a one-dimensional core, are among the simplest topological structures, ubiquitous to many branches of physics. In the crystalline state, vortex formation is rare, since it is generally hampered by long-range interactions: in ferroic materials (ferromagnetic and ferroelectric), vortices are observed only when the effects of the dipole-dipole interaction are modified by confinement at the nanoscale^1-3, or when the parameter associated with the vorticity does not couple directly with strain ⁴ . Here, we observe an unprecedented form of vortices in antiferromagnetic haematite (α-Fe₂O₃) epitaxial films, in which the primary whirling parameter is the staggered magnetization. Remarkably, ferromagnetic topological objects with the same vorticity and winding number as the α-Fe₂O₃ vortices are imprinted onto an ultra-thin Co ferromagnetic over-layer by interfacial exchange. Our data suggest that the ferromagnetic vortices may be merons (half-skyrmions, carrying an out-of plane core magnetization), and indicate that the vortex/meron pairs can be manipulated by the application of an in-plane magnetic field, giving rise to large-scale vortex-antivortex annihilation.

Polarization Control with Plasmonic Antenna Tips: A Universal Approach to Optical Nanocrystallography and Vector-Field Imaging

Controlling the propagation and polarization vectors in linear and nonlinear optical spectroscopy enables us to probe the anisotropy of optical responses providing structural symmetry selective contrast in optical imaging. Here, we present a novel tilted antenna-tip approach to control the optical vector-field by breaking the axial symmetry of the nanoprobe in tip-enhanced near-field microscopy. This gives rise to a localized plasmonic antenna effect with significantly enhanced optical field vectors with control of both in-plane and out-of-plane components. We use the resulting vector-field specificity in the symmetry selective nonlinear optical response of second-harmonic generation (SHG) for a generalized approach to optical nanocrystallography and imaging. In tip-enhanced SHG imaging of monolayer MoS₂ films and single-crystalline ferroelectric YMnO₃, we reveal nanocrystallographic details of domain boundaries and domain topology with enhanced sensitivity and nanoscale spatial resolution. The approach is applicable to any anisotropic linear and nonlinear optical response and enables the optical nanocrystallographic imaging of molecular or quantum materials.

Studies on the high-temperature ferroelectric transition of multiferroic hexagonal manganite RMnO3

Hexagonal manganites are multiferroic materials with two highly-dissimilar phase transitions: a ferroelectric transition (from P6₃/mmc to P6₃cm) at a temperature higher than 1000 K and an antiferromagnetic transition at T _N  =  65-130 K. Despite its critical relevance to the intriguing ferroelectric domain physics, the details of the ferroelectric transition are not well known to date primarily because of the ultra-high transition temperature. Using high-temperature x-ray diffraction experiments, we show that the ferroelectric transition is a single transition of abrupt order and R-O_p displacement is the primary order parameter. This structural transition is then simultaneously accompanied by MnO₅ tilting and the subsequent development of electric polarization.

Large Scale Two-Dimensional Flux-Closure Domain Arrays in Oxide Multilayers and Their Controlled Growth

Ferroelectric flux-closures are very promising in high-density storage and other nanoscale electronic devices. To make the data bits addressable, the nanoscale flux-closures are required to be periodic via a controlled growth. Although flux-closure quadrant arrays with 180° domain walls perpendicular to the interfaces (V-closure) have been observed in strained ferroelectric PbTiO₃ films, the flux-closure quadrants therein are rather asymmetric. In this work, we report not only a periodic array of the symmetric flux-closure quadrants with 180° domain walls parallel to the interfaces (H-closure) but also a large scale alternative stacking of the V- and H-closure arrays in PbTiO₃/SrTiO₃ multilayers. On the basis of a combination of aberration-corrected scanning transmission electron microscopic imaging and phase field modeling, we establish the phase diagram in the layer-by-layer two-dimensional arrays versus the thickness ratio of adjacent PbTiO₃ films, in which energy competitions play dominant roles. The manipulation of these flux-closures may stimulate the design and development of novel nanoscale ferroelectric devices with exotic properties.

Topological Defects in Hexagonal Manganites: Inner Structure and Emergent Electrostatics

Diverse topological defects arise in hexagonal manganites, such as ferroelectric vortices, as well as neutral and charged domain walls. The topological defects are intriguing because their low symmetry enables unusual couplings between structural, charge, and spin degrees of freedom, holding great potential for novel types of functional 2D and 1D systems. Despite the considerable advances in analyzing the different topological defects in hexagonal manganites, the understanding of their key intrinsic properties is still rather limited and disconnected. In particular, a rapidly increasing number of structural variants is reported without clarifying their relation, leading to a zoo of seemingly unrelated topological textures. Here, we combine picometer-precise scanning-transmission-electron microscopy with Landau theory modeling to clarify the inner structure of topological defects in Er_1-xZr_xMnO₃. By performing a comprehensive parametrization of the inner atomic defect structure, we demonstrate that one primary length scale drives the morphology of both vortices and domain walls. Our findings lead to a unifying general picture of this type of structural topological defects. We further derive novel fundamental and universal properties, such as unusual bound-charge distributions and electrostatics at the ferroelectric vortex cores with emergent U(1) symmetry.

Characteristics and controllability of vortices in ferromagnetics, ferroelectrics, and multiferroics

Topological defects in condensed matter are attracting e significant attention due to their important role in phase transition and their fascinating characteristics. Among the various types of matter, ferroics which possess a switchable physical characteristic and form domain structure are ideal systems to form topological defects. In particular, a special class of topological defects-vortices-have been found to commonly exist in ferroics. They often manifest themselves as singular regions where domains merge in large systems, or stabilize as novel order states instead of forming domain structures in small enough systems. Understanding the characteristics and controllability of vortices in ferroics can provide us with deeper insight into the phase transition of condensed matter and also exciting opportunities in designing novel functional devices such as nano-memories, sensors, and transducers based on topological defects. In this review, we summarize the recent experimental and theoretical progress in ferroic vortices, with emphasis on those spin/dipole vortices formed in nanoscale ferromagnetics and ferroelectrics, and those structural domain vortices formed in multiferroic hexagonal manganites. We begin with an overview of this field. The fundamental concepts of ferroic vortices, followed by the theoretical simulation and experimental methods to explore ferroic vortices, are then introduced. The various characteristics of vortices (e.g. formation mechanisms, static/dynamic features, and electronic properties) and their controllability (e.g. by size, geometry, external thermal, electrical, magnetic, or mechanical fields) in ferromagnetics, ferroelectrics, and multiferroics are discussed in detail in individual sections. Finally, we conclude this review with an outlook on this rapidly developing field.

Functional electronic inversion layers at ferroelectric domain walls

Ferroelectric domain walls hold great promise as functional two-dimensional materials because of their unusual electronic properties. Particularly intriguing are the so-called charged walls where a polarity mismatch causes local, diverging electrostatic potentials requiring charge compensation and hence a change in the electronic structure. These walls can exhibit significantly enhanced conductivity and serve as a circuit path. The development of all-domain-wall devices, however, also requires walls with controllable output to emulate electronic nano-components such as diodes and transistors. Here we demonstrate electric-field control of the electronic transport at ferroelectric domain walls. We reversibly switch from resistive to conductive behaviour at charged walls in semiconducting ErMnO₃. We relate the transition to the formation-and eventual activation-of an inversion layer that acts as the channel for the charge transport. The findings provide new insight into the domain-wall physics in ferroelectrics and foreshadow the possibility to design elementary digital devices for all-domain-wall circuitry.

Kibble-Zurek Scaling during Defect Formation in a Nematic Liquid Crystal

Symmetry-breaking phase transitions are often accompanied by the formation of topological defects, as in cosmological theories of the early universe, superfluids, liquid crystals or solid-state systems. This scenario is described by the Kibble-Zurek mechanism, which predicts corresponding scaling laws for the defect density ρ. One such scaling law suggests a relation ρ≈τ_Q^-1/2 with τ_Q the change of rate of a control parameter. In contrast to the scaling of the defect density during annihilation with ρ≈t^-1 , which is governed by the attraction of defects of the same strength but opposite sign, the defect formation process, which depends on the rate of change of a physical quantity initiating the transition, has only rarely been investigated. Herein, we use nematic liquid crystals as a different system to demonstrate the validity of the predicted scaling relation for defect formation. It is found that the scaling exponent is independent of temperature and material employed, thus universal, as predicted.

Topological phase transformations and intrinsic size effects in ferroelectric nanoparticles

Composite materials comprised of ferroelectric nanoparticles in a dielectric matrix are being actively investigated for a variety of functional properties attractive for a wide range of novel electronic and energy harvesting devices. However, the dependence of these functionalities on shapes, sizes, orientation and mutual arrangement of ferroelectric particles is currently not fully understood. In this study, we utilize a time-dependent Ginzburg-Landau approach combined with coupled-physics finite-element-method based simulations to elucidate the behavior of polarization in isolated spherical PbTiO₃ or BaTiO₃ nanoparticles embedded in a dielectric medium, including air. The equilibrium polarization topology is strongly affected by particle diameter, as well as the choice of inclusion and matrix materials, with monodomain, vortex-like and multidomain patterns emerging for various combinations of size and materials parameters. This leads to radically different polarization vs. electric field responses, resulting in highly tunable size-dependent dielectric properties that should be possible to observe experimentally. Our calculations show that there is a critical particle size below which ferroelectricity vanishes. For the PbTiO₃ particle, this size is 2 and 3.4 nm, respectively, for high- and low-permittivity media. For the BaTiO₃ particle, it is ∼3.6 nm regardless of the medium dielectric strength.

Conductivity Contrast and Tunneling Charge Transport in the Vortexlike Ferroelectric Domain Patterns of Multiferroic Hexagonal YMnO_{3}

We deduce the intrinsic conductivity properties of the ferroelectric domain walls around the topologically protected domain vortex cores in multiferroic YMnO_{3}. This is achieved by performing a careful equivalent-circuit analysis of dielectric spectra measured in single-crystalline samples with different vortex densities. The conductivity contrast between the bulk domains and the less conducting domain boundaries is revealed to reach up to a factor of 500 at room temperature, depending on the sample preparation. Tunneling of localized defect charge carriers is the dominant charge-transport process in the domain walls that are depleted of mobile charge carriers. This work demonstrates that, via equivalent-circuit analysis, dielectric spectroscopy can provide valuable information on the intrinsic charge-transport properties of ferroelectric domain walls, which is of high relevance for the design of new domain-wall-based microelectronic devices.

Dynamic Off-Equilibrium Transition in Systems Slowly Driven across Thermal First-Order Phase Transitions

We study the off-equilibrium behavior of systems with short-range interactions, slowly driven across a thermal first-order transition, where the equilibrium dynamics is exponentially slow. We consider a dynamics that starts in the high-T phase at time t=t_{i}<0 and ends at t=t_{f}>0 in the low-T phase, with a time-dependent temperature T(t)/T_{c}≈1-t/t_{s}, where t_{s} is the protocol time scale. A general off-equilibrium scaling (OS) behavior emerges in the limit of large t_{s}. We check it at the first-order transition of the two-dimensional q-state Potts model with q=20 and 10. The numerical results show evidence of a dynamic transition, where the OS functions show a spinodal-like singularity. Therefore, the general mean-field picture valid for systems with long-range interactions is qualitatively recovered, provided the time dependence is appropriately (logarithmically) rescaled.

Facilitation of Ferroelectric Switching via Mechanical Manipulation of Hierarchical Nanoscale Domain Structures

Heterogeneous ferroelastic transition that produces hierarchical 90° tetragonal nanodomains via mechanical loading and its effect on facilitating ferroelectric domain switching in relaxor-based ferroelectrics were explored. Combining in situ electron microscopy characterization and phase-field modeling, we reveal the nature of the transition process and discover that the transition lowers by 40% the electrical loading threshold needed for ferroelectric domain switching. Our results advance the fundamental understanding of ferroelectric domain switching behavior.

Hall effect in charged conducting ferroelectric domain walls

Enhanced conductivity at specific domain walls in ferroelectrics is now an established phenomenon. Surprisingly, however, little is known about the most fundamental aspects of conduction. Carrier types, densities and mobilities have not been determined and transport mechanisms are still a matter of guesswork. Here we demonstrate that intermittent-contact atomic force microscopy (AFM) can detect the Hall effect in conducting domain walls. Studying YbMnO₃ single crystals, we have confirmed that p-type conduction occurs in tail-to-tail charged domain walls. By calibration of the AFM signal, an upper estimate of ∼1 × 10¹⁶ cm⁻³ is calculated for the mobile carrier density in the wall, around four orders of magnitude below that required for complete screening of the polar discontinuity. A carrier mobility of∼50 cm²V⁻¹s⁻¹ is calculated, about an order of magnitude below equivalent carrier mobilities in p-type silicon, but sufficiently high to preclude carrier-lattice coupling associated with small polarons.

Conduction in ferroelectric domain walls is now an established phenomenon, yet fundamental aspects of transport physics remain elusive. Here, Campbell et al. report the type, density and mobility of carriers in conducting domain walls in ytterbium manganite using nanoscale Hall effect measurements.

Manipulation of Nanoscale Domain Switching Using an Electron Beam with Omnidirectional Electric Field Distribution

Reversible ferroelectric domain (FD) manipulation with a high spatial resolution is critical for memory storage devices based on thin film ferroelectric materials. FD can be manipulated using techniques that apply heat, mechanical stress, or electric bias. However, these techniques have some drawbacks. Here we propose to use an electron beam with an omnidirectional electric field as a tool for erasable stable ferroelectric nanodomain manipulation. Our results suggest that local accumulation of charges contributes to the local electric field that determines domain configurations.

Homotopy-Theoretic Study &Atomic-Scale Observation of Vortex Domains in Hexagonal Manganites

Essential structural properties of the non-trivial "string-wall-bounded" topological defects in hexagonal manganites are studied through homotopy group theory and spherical aberration-corrected scanning transmission electron microscopy. The appearance of a "string-wall-bounded" configuration in RMnO3 is shown to be strongly linked with the transformation of the degeneracy space. The defect core regions (~50 Å) mainly adopt the continuous U(1) symmetry of the high-temperature phase, which is essential for the formation and proliferation of vortices. Direct visualization of vortex strings at atomic scale provides insight into the mechanisms and macro-behavior of topological defects in crystalline materials.

Quasi-Long-Range Order and Vortex Lattice in the Three-State Potts Model

We show that the order-disorder phase transition in the three-state Potts ferromagnet on a square lattice is driven by a coupled proliferation of domain walls and vortices. Raising the vortex core energy above a threshold value decouples the proliferation and splits the transition into two. The phase between the two transitions exhibits an emergent U(1) symmetry and quasi-long-range order. Lowering the core energy below a threshold value also splits the order-disorder transition but the system forms a vortex lattice in the intermediate phase.

Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensional O(N) vector models

We consider the dynamical off-equilibrium behavior of the three-dimensional O(N) vector model in the presence of a slowly varying time-dependent spatially uniform magnetic field H(t)=h(t)e, where e is an N-dimensional constant unit vector, h(t)=t/t(s), and t(s) is a time scale, at fixed temperature T≤T(c), where T(c) corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at h(i)<0, correspondingly t(i)<0, and end at time t(f)>0 with h(t(f))>0, or vice versa. We show that the magnetization displays an off-equilibrium scaling behavior close to the transition line H(t)=0. It arises from the interplay among the time t, the time scale t(s), and the finite size L. The scaling behavior can be parametrized in terms of the scaling variables t(s)(κ)/L and t/t(s)(κ(t)), where κ>0 and κ(t)>0 are appropriate universal exponents, which differ at the critical point and for T<T(c). In the latter case, κ and κ(t) also depend on the shape of the lattice and on the boundary conditions. We present numerical results for the Heisenberg (N=3) model under a purely relaxational dynamics. They confirm the predicted off-equilibrium scaling behaviors at and below T(c). We also discuss hysteresis phenomena in round-trip protocols for the time dependence of the external field. We define a scaling function for the hysteresis loop area of the magnetization that can be used to quantify how far the system is from equilibrium.

Universality in the Dynamics of Second-Order Phase Transitions

When traversing a symmetry-breaking second-order phase transition at a finite rate, topological defects form whose number dependence on the quench rate is given by simple power laws. We propose a general approach for the derivation of such scaling laws that is based on the analytical transformation of the associated equations of motion to a universal form rather than employing plausible physical arguments. We demonstrate the power of this approach by deriving the scaling of the number of topological defects in both homogeneous and nonhomogeneous settings. The general nature and extensions of this approach are discussed.

Hexagonal RMnO3: a model system for two-dimensional triangular lattice antiferromagnets

The hexagonal RMnO3(h-RMnO3) are multiferroic materials, which exhibit the coexistence of a magnetic order and ferroelectricity. Their distinction is in their geometry that both results in an unusual mechanism to break inversion symmetry and also produces a two-dimensional triangular lattice of Mn spins, which is subject to geometrical magnetic frustration due to the antiferromagnetic interactions between nearest-neighbor Mn ions. This unique combination makes the h-RMnO3 a model system to test ideas of spin-lattice coupling, particularly when both the improper ferroelectricity and the Mn trimerization that appears to determine the symmetry of the magnetic structure arise from the same structure distortion. In this review we demonstrate how the use of both neutron and X-ray diffraction and inelastic neutron scattering techniques have been essential to paint this comprehensive and coherent picture of h-RMnO3.

Off-equilibrium scaling behaviors across first-order transitions

We study off-equilibrium behaviors at first-order transitions (FOTs) driven by a time dependence of the temperature across the transition point T(c), such as the linear behavior T(t)/T(c)=1±t/t(s) where t(s) is a time scale. In particular, we investigate the possibility of nontrivial off-equilibrium scaling behaviors in the regime of slow changes, corresponding to large t(s). We consider the two-dimensional Potts models, which provide an ideal theoretical laboratory to investigate issues related to FOTs driven by thermal fluctuations. We put forward general ansatzes for off-equilibrium scaling behaviors around the time t=0 corresponding to T(c). Then we present numerical results for the q=10 and 20 Potts models. We show that off-equilibrium scaling behaviors emerge at FOTs with relaxational dynamics, when appropriate boundary conditions are considered, such as mixed boundary conditions favoring different phases at the opposite sides of the system, which enforce an interface in the system.

Functional domain walls in multiferroics

During the last decade a wide variety of novel and fascinating correlation phenomena has been discovered at domain walls in multiferroic bulk systems, ranging from unusual electronic conductance to inseparably entangled spin and charge degrees of freedom. The domain walls represent quasi-2D functional objects that can be induced, positioned, and erased on demand, bearing considerable technological potential for future nanoelectronics. Most of the challenges that remain to be solved before turning related device paradigms into reality, however, still fall in the field of fundamental condensed matter physics and materials science. In this topical review seminal experimental findings gained on electric and magnetic domain walls in multiferroic bulk materials are addressed. A special focus is put on the physical properties that emerge at so-called charged domain walls and the added functionality that arises from coexisting magnetic order. The research presented in this review highlights that we are just entering a whole new world of intriguing nanoscale physics that is yet to be explored in all its details. The goal is to draw attention to the persistent challenges and identify future key directions for the research on functional domain walls in multiferroics.

Evolution of the statistical distribution in a topological defect network

The complex networks of numerous topological defects in hexagonal manganites are highly relevant to vastly different phenomena from the birth of our cosmos to superfluidity transition. The topological defects in hexagonal manganites form two types of domain networks: type-I without and type-II with electric self-poling. A combined phase-field simulations and experimental study shows that the frequencies of domains with N-sides, i.e. of N-gons, in a type-I network are fitted by a lognormal distribution, whereas those in type-II display a scale-free power-law distribution with exponent ∼2. A preferential attachment process that N-gons with a larger N have higher probability of coalescence is responsible for the emergence of the scale-free networks. Since the domain networks can be observed, analyzed, and manipulated at room temperature, hexagonal manganites provide a unique opportunity to explore how the statistical distribution of a topological defect network evolves with an external electric field.

Kibble-Zurek mechanism in colloidal monolayers

The Kibble-Zurek mechanism describes the evolution of topological defect structures like domain walls, strings, and monopoles when a system is driven through a second-order phase transition. The model is used on very different scales like the Higgs field in the early universe or quantum fluids in condensed matter systems. A defect structure naturally arises during cooling if separated regions are too far apart to communicate (e.g., about their orientation or phase) due to finite signal velocity. This lack of causality results in separated domains with different (degenerated) locally broken symmetry. Within this picture, we investigate the nonequilibrium dynamics in a condensed matter analog, a 2D ensemble of colloidal particles. In equilibrium, it obeys the so-called Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) melting scenario with continuous (second order-like) phase transitions. The ensemble is exposed to a set of finite cooling rates covering roughly three orders of magnitude. Along this process, we analyze the defect and domain structure quantitatively via video microscopy and determine the scaling of the corresponding length scales as a function of the cooling rate. We indeed observe the scaling predicted by the Kibble-Zurek mechanism for the KTHNY universality class.

Configuration of ripple domains and their topological defects formed under local mechanical stress on hexagonal monolayer graphene

Ripples in graphene are extensively investigated because they ensure the mechanical stability of two-dimensional graphene and affect its electronic properties. They arise from spontaneous symmetry breaking and are usually manifested in the form of domains with long-range order. It is expected that topological defects accompany a material exhibiting long-range order, whose functionality depends on characteristics of domains and topological defects. However, there remains a lack of understanding regarding ripple domains and their topological defects formed on monolayer graphene. Here we explore configuration of ripple domains and their topological defects in exfoliated monolayer graphenes on SiO2/Si substrates using transverse shear microscope. We observe three-color domains with three different ripple directions, which meet at a core. Furthermore, the closed domain is surrounded by an even number of cores connected together by domain boundaries, similar to topological vortex and anti-vortex pairs. In addition, we have found that axisymmetric three-color domains can be induced around nanoparticles underneath the graphene. This fascinating configuration of ripple domains may result from the intrinsic hexagonal symmetry of two-dimensional graphene, which is supported by theoretical simulation using molecular dynamics. Our findings are expected to play a key role in understanding of ripple physics in graphene and other two-dimensional materials.

Duality of topological defects in hexagonal manganites

We show that the spontaneous symmetry breaking in multiferroic hexagonal manganites can be chemically manipulated to yield two complementary ground states: the well-known ferroelectric P6(3)cm and an antipolar P3c phase. Both symmetry breakings yield topologically protected vortex defects, with the antipolar vortices dual to those of the ferroelectric. This duality stems from the existence of 12 possible angles of MnO5 tilting, and broad strain-free walls with low energy spontaneously emerge through an intermediate P3c1 state, providing a complete unified symmetry description.

Quench in the 1D Bose-Hubbard model: topological defects and excitations from the Kosterlitz-Thouless phase transition dynamics

Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain predictions can lead to results that are inconsistent with a more careful KZM analysis based on causality - on the comparison of the relaxation time of the order parameter with the "time distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.

Direct observation of multiferroic vortex domains in YMnO3

Topological vortices with swirling ferroelectric, magnetic and structural anti-phase relationship in hexagonal RMnO₃ (R = Ho to Lu, Y, and Sc) have attracted much attention because of their intriguing behaviors. Herein, we report the structure of multiferroic vortex domains in YMnO₃ at atomic scale using state-of-the-art aberration-corrected scanning transmission electron microscopy (STEM). Two types of displacements were identified among six domain walls (DWs); six translation-ferroelectric domains denoted by α+, γ−, β+, α−, γ+ and β−, respectively, were recognized, demonstrating the interlocking nature of the anti-vortex domain. We found that the anti-vortex core is about four unit cells wide. In addition, we reconstructed the vortex model with three swirling pairs of DWs along the [001] direction. These results are very critical for the understanding of topological behaviors and unusual properties of the multiferroic vortex.

Unfolding of vortices into topological stripes in a multiferroic material

Multiferroic hexagonal RMnO(3) (R=rare earths) crystals exhibit dense networks of vortex lines at which six domain walls merge. While the domain walls can be readily moved with an applied electric field, the vortex cores so far have been impossible to control. Our experiments demonstrate that shear strain induces a Magnus-type force pulling vortices and antivortices in opposite directions and unfolding them into a topological stripe domain state. We discuss the analogy between this effect and the current-driven dynamics of vortices in superconductors and superfluids.