Ferroelectric materials can host a variety of unusually stable and well-defined ultrafine polarization patterns. These patterns, referred to as polar topologies, could be useful for future electronic devices. However, a full understanding of how they form, how they can be controlled and how their properties can be applied in practical designs is still lacking. Here we discuss how these polar topological textures can be classified, created and manipulated. We focus on oxide heterostructures, where structural and electronic interactions lead to different polarization behaviours, including changes in polarization patterns, switching processes, phase transitions and predictable motions of polar features. By examining how these features relate to materials design, we highlight recent progress that may guide future device development. Our aim is to clarify the key physical principles that determine the generation and tunability of these polar topological textures and discuss practical opportunities for their use in next-generation materials and devices. Ferroelectric materials can host robust and diverse polarization textures. This Review examines their formation mechanisms, responses to external stimuli and potential applications in next-generation electronic devices.

Volkov-Pankratov (VP) states are a family of sub-gap states that appear at the smooth interface/domain wall between topologically distinct gaped states. We carry out quantum transport simulations on one- and two-dimensional lattice models to demonstrate the emergence of such states in the edge spectrum of a quantum spin Hall system subjected to a smoothly varying exchange field that switches its sign at a given spatial point. We show the VP states possess non-trivial spin textures that can be characterized by a winding number in real space. It is further demonstrated that the application of an electric field along the edge provides control of this spin texture without altering the winding number. Finally, we illuminate how these spin textures can be read off via the local tunnel magnetoresistance (TMR) response of spin-polarized tunnel probes attached to the edge and the TMR can be controlled by purely electrical means akin to a Datta-Das type spin transistor.

In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons, and fermions defined by local rules. Here we apply it to find connections between bosonic and fermionic lattice models in the realm of condensed-matter physics and uncover a novel fivefold way topology it demands in these systems. At the single-particle level, our connections pair a bosonic and fermionic lattice model, either describing the hopping of number-conserving particles or local couplings between fermion parityconserving particles. The pair are isospectral except for zero modes, such as flat bands, quadratic band touchings, and nexus points, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences. Notably, in this framework, the supercharge operator that generates SUSY is Hermitian and can itself be interpreted as a hopping Hamiltonian on a bipartite lattice, a feature that enables the discovery of materials or model lattices hosting the SUSY partners. To illustrate the power of SUSY, we present 16 use cases of SUSY, that span topics including frustrated magnets, Kitaev spin liquids, and topological superconductors, the majority of which turn out to provide insights into the discovery and design of flat bands and topological materials.

In recent years, several proposals that leverage principles from condensed matter and high-energy physics for engineering laser arrays have been put forward. The most important among these concepts are topology, which enables the construction of robust zero-mode laser devices, and supersymmetry (SUSY), which holds the potential for achieving phase locking in laser arrays. In this work, we show that the relation between supersymmetric coupled bosonic and fermionic oscillators on one side, and bipartite networks (and hence chiral symmetry) on another side can be exploited together with non-Hermitian engineering for building one- and two-dimensional laser arrays with in-phase synchronization. To demonstrate our strategy, we present a concrete design starting from the celebrated Su-Schrieffer-Heeger (SSH) model to arrive at a SUSY laser structure that enjoys two key advantages over those reported in previous works. Firstly, the design presented here features a near-uniform geometry for both the laser array and supersymmetric reservoir (i.e., the widths and distances between the cavity arrays are almost the same). Secondly, the uniform field distribution in the presented structure leads to a far-field intensity that scales as N2 where N is the number of lasing elements. Taken together, these two features can enable the implementation of higher-power laser arrays that are easy to fabricate, and hence provide a roadmap for pushing the frontier of SUSY laser arrays beyond the proof-of-concept phase.

The concept of dynamical freezing is a phenomenon where a suitable set of local observables freezes under a strong periodic drive in a quantum many-body system. This happens because of the emergence of approximate but perpetual conservation laws when the drive is strong enough. In this work, we probe the resilience of dynamical freezing to random perturbations added to the relative phases between the interfering states (elements of a natural basis) in the time-evolving wave function after each drive cycle. We study this in an integrable Ising chain in a time-periodic transverse field. Our key finding is, that the imprinted phase noise melts the dynamically frozen state, but the decay is “slow”: a stretched-exponential decay rather than an exponential one. Stretched-exponential decays (also known as Kohlrausch relaxation) are usually expected in complex systems with time-scale hierarchies due to strong disorders or other inhomogeneities resulting in jamming, glassiness, or localization. Here we observe this in a simple translationally invariant system dynamically frozen under a periodic drive. Moreover, the melting here does not obliterate the entire memory of the initial state but leaves behind a steady remnant that depends on the initial conditions. This underscores the stability of dynamically frozen states.

In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons and fermions. Here we show how this fundamental concept can be applied to connect bosonic and fermionic lattice models in the realm of condensed matter physics, e.g., to identify a variety of (bosonic) phonon and magnon lattice models which admit topologically nontrivial free fermion models as superpartners. At the single-particle level, the bosonic and the fermionic models that are generated by the SUSY are isospectral except for zero modes, such as flat bands, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences and discuss further ramifications such as the definition of supersymmetric topological invariants for generic bosonic systems. Notably, a Hermitian form of the supercharge operator, the generator of the SUSY, can itself be interpreted as a hopping Hamiltonian on a bipartite lattice. This allows us to identify a wide class of interconnected lattices whose tight-binding Hamiltonians are superpartners of one another or can be derived via squaring or square-rooting their energy spectra all the while preserving band topology features. We introduce a five-fold way symmetry classification scheme of these SUSY lattice correspondences, including cases with a non-zero Witten index, based on a topological classification of the underlying Hermitian supercharge operator. These concepts are illustrated for various explicit examples including frustrated magnets, Kitaev spin liquids, and topological superconductors.

Theoretical models of a spin-polarized voltage probe (SPVP) tunnel-coupled to the helical edge states (HES) of a quantum spin Hall system (QSHS) are studied. Our first model of the SPVP comprises NP spin-polarized modes (subprobes), each of which is locally tunnel-coupled to the HES, while the SPVP, as a whole, is subjected to a self-consistency condition ensuring zero average current on the probe. We carry out a numerical analysis which shows that the optimal situation for reading off spin-resolved voltage from the HES depends on the interplay of the probe-edge tunnel-coupling and the number of modes in the probe (NP ). We further investigate the stability of our findings by introducing Gaussian fluctuations in (i) the tunnel-coupling between the subprobes and the HES about a chosen average value and (ii) spin-polarization of the subprobes about a chosen direction of the net polarization of SPVP. We also perform a numerical analysis corresponding to the situation where four such SPVPs are implemented in a self-consistent fashion across a ferromagnetic barrier on the HES and demonstrate that this model facilitates the measurements of spin-resolved four-probe voltage drops across the ferromagnetic barrier. As a second model, we employ the edge state of a quantum anomalous Hall state (QAHS) as the SPVP which is tunnel-coupled over an extended region with the HES. A two-dimensional lattice simulation for the quantum transport of the proposed device setup comprising a junction of QSHS and QAHS is considered and a feasibility study of using the edge of the QAHS as an efficient spin-polarized voltage probe is carried out in presence of an optimal strength of the disorder.

Volkov-Pankratov (VP) states are a family of sub-gap states which appear at the smooth interface/domain wall between topologically distinct gapped states. We study the emergence of such states in the edge spectrum of a quantum spin Hall system subjected to a smoothly varying mass term (Zeeman field) that switches sign at a given spatial point. Both the VP states at non-zero energy and the zero energy Jackiw-Rebbi mode stay localized at the interfacial region, however, the former feature several distinctive signatures compared to the latter such as non-trivial spin textures that can be characterized by a winding number in real space. On applying an electric field, the texture deforms leaving its winding number unaltered. Moreover, the VP states exhibit an intriguing interplay between the electric and the magnetic field with a collapse of the spectrum onto the zero mode when they are equal in magnitude. Quantum transport simulations on a 2D lattice model are performed to undergird our theoretical prediction.

Weyl semimetals exhibit exotic transport responses, among which, recently Goos-Haenchen (GH) and Imbert-Fedorov (IF) effects have received a revived attention, which are, otherwise, well-studied phenomena in optical systems and certain electronic systems. Besides the usual parametric dependence of the shifts inherited from the underlying Hamiltonian to describe the Weyl system and/or that induced by external controls, the IF shift further carries a topological identity -- it depends on the chirality of the Weyl cones. A realistic system of Weyl semimetal naturally accommodates surface potentials induced by impurities present on its surface that could pose impediments to observe clean transport signatures predicted in theoretical models. Classifying these potentials, we study their effects on GH and IF shifts to provide useful guidance to future experiments that are tuned to the objective of characterizing Weyl semimetals and for a possible realisation of novel devices based on these phenomena. A transfer matrix-based approach is invoked to study the profile of Weyl wavefunctions across the interface which is hosting the impurity potentials, revealing that such potentials can lead to several discerning effects which, in certain cases, extend even to nullifying the IF shift completely and giving rise to phenomenon like valley inversion.

Kibble-Zurek theory (KZ) stands out as the most robust theory of defect generation in the dynamics of phase transitions. KZ utilizes the structure of equilibrium states away from the transition point to estimate the excitations due to the transition using adiabatic and impulse approximations. Here we show, the actual nonequilibrium dynamics lead to a qualitatively different scenario from KZ, as far correlations between the defects (rather than their densities) are concerned. For a quantum Ising chain, we show, this gives rise to a Gaussian spatial decay in the domain wall (kinks) correlations, while KZ would predict an exponential fall. We propose a simple but general framework on top of KZ, based on the "quantum coarsening" dynamics of local correlators in the supposed impulse regime. We outline how our picture extends to generic interacting situations.

Many advancements have been made in the field of topological mechanics. The majority of the work, however, concerns the topological invariant in a linear theory. In this Letter, we present a generic prescription to define topological indices that accommodates nonlinear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of a topological index in differential geometry known as the Poincaré-Hopf index, which features the topological invariant of nonlinear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected nonlinear ZMs in the future.

Many advancements have been made in the field of topological mechanics. The majority of the works, however, concerns the topological invariant in a linear theory. We, in this work, present a generic prescription of defining topological indices which accommodates non-linear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of the Poincare-Hopf index, that features the topological invariant of non-linear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected non-linear ZMs in the future.

We derive a general relation between the bosonic and fermionic entanglement in the ground states of supersymmetric quadratic Hamiltonians. For this, we construct canonical identifications between bosonic and fermionic subsystems. Our derivation relies on a unified framework to describe both, bosonic and fermionic Gaussian states in terms of so-called linear complex structures $J$. The resulting dualities apply to the full entanglement spectrum between the bosonic and the fermionic systems, such that the von Neumann entropy and arbitrary Renyi entropies can be related. We illustrate our findings in one and two-dimensional systems, including the paradigmatic Kitaev honeycomb model. While typically SUSY preserves features like area law scaling of the entanglement entropies on either side, we find a peculiar phenomenon, namely, an amplified scaling of the entanglement entropy ("super area law") in bosonic subsystems when the dual fermionic subsystems develop almost maximally entangled modes.

Topological protection of edge state in quantum spin Hall systems relies only on time-reversal symmetry. Hence, S-z conservation on the edge can be relaxed which can have an interferometric manifestation in terms of spin Berry phase. Primarily it could lead to the generation of spin Berry phase arising from a closed loop dynamics of electrons. Our work provides a minimal framework to generate and detect these effects by employing both spin-unpolarized and spin-polarized leads. We show that spin-polarized leads could lead to resonances or antiresonances in the two-terminal conductance of the interferometer. We further show that the positions of these antiresonances (as a function of energy of the incident electron) get shifted owing to the presence of spin Berry phase. Finally, we present simulations of a device setup using KWANT package which put our theoretical predictions on a firm footing.

In topological mechanics, the identification of a mechanical system's rigidity matrix with an electronic tight-binding model allows to infer topological properties of the mechanical system, such as the occurrence of `floppy' boundary modes, from the associated electronic band structure. Here we introduce an approach to systematically construct topological mechanical systems by an exact supersymmetry (SUSY) that relates the bosonic (mechanical) and fermionic (e.g. electronic) degrees of freedom. As examples we discuss mechanical analogues of the Kitaev honeycomb model and of a second-order topological insulator with floppy corner modes. Our SUSY construction naturally defines hitherto unexplored topological invariants for bosonic (mechanical) systems, such as bosonic Wilson loop operators that are formulated in terms of a SUSY-related fermionic Berry curvature.

Pancharatnam's experimental findings on amplitude interferometry of polarized light during 1950's was an early example of the Berry phase. But a similar experimental realization of the geometric phase in the context of solid-state electronic systems where the polarization state of the photon is replaced by spin-polarized states of the electron remains unexplored. This is primarily due to the fact that the generation of Pancharatnam's geometric phase involves a discrete number of cyclic projective measurements on the polarized states of light, and an equivalent cyclic operation on electron spin is much harder to implement in a solid-state setting. In the present paper, we show that the edge states of the quantum spin Hall effect in conjunction with tunnel coupled spin-polarized electrodes (SPEs) provide us with a unique opportunity to generate Pancharatnam's type of geometric phase locally in space, which can be detected via electronic current measurements. We show that the controlled manipulation of the polarization directions of the SPEs results in coherent oscillations in the cross-correlated current noise, which can be attributed to a multiparticle version of Pancharatnam's geometric phase, and is directly related to the phenomenon of intensity interferometry. We demonstrate that the interference patterns produced due to the manipulation of the geometric phase in our proposed setup show a remarkable immunity to orbital dephasing owing to their spatially local origin.