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A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of {\it a topological order (TO) with gappable boundary in one higher dimension}; we call this the {\it symTO} of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its {\it maximal} emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a definition of {\it maximal symTO}. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator. 

We study the nonlinear $$\sigma$$-model in $${(d+1)}$$-dimensional spacetime with connected target space $$K$$ and show that, at energy scales below singular field comfigurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $$d$$-representations of a discrete $$d$$-group $$\mathbb{G}^{(d)}$$ (i.e. the emergent symmetry is the dual of the invertible $$d$$-group $$\mathbb{G}^{(d)}$$ symmetry). The $$d$$-group $$\mathbb{G}^{(d)}$$ is determined such that its classifying space $$B\mathbb{G}^{(d)}$$ is given by the $$d$$-th Postnikov stage of $$K$$. In $(2+1)$D and for finite $$\mathbb{G}^{(2)}$$, this symmetry is always holo-equivalent to an invertible $${0}$$-form---ordinary---symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $$\sigma$$-model spontaneously breaks this symmetry, and when $$\mathbb{G}^{(d)}$$ is finite, it is described by the deconfined phase of $$\mathbb{G}^{(d)}$$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $$\sigma$$-model in $${(3+1)}$$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $$\mathbb{C}P^{N-1}$$ nonlinear $$\si$$-models in the large-$$N$$ limit. 

Measurements of rovibrational spectra of clusters provide physical insight only if spectral lines can be assigned to pairs of quantum states, and further insight is obtained if one can deduce the quantitative energy-level pattern. Both steps can be so difficult that some measured spectra remain unassigned, one example isorthoH₂–CO. To extend the scope of spectroscopic insights, we propose to use theoretical information in interpretation of spectra. We first performed high accuracy, full-dimensional calculations of theorthoH₂–CO spectrum, at the highest practically achievable levels of electronic structure theory and quantum nuclear dynamics. Then, an iterative, theory-guided method developed here allowed us to fully interpret the spectrum oforthoH₂–CO, extending the range of van der Waals clusters for which spectroscopy can provide physical insights. 

Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear generically in gapless states of quantum matter constraining the low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) $$S^{\,}_3$$ symmetry and the other with non-invertible $$\mathsf{Rep}(S^{\,}_3)$$ symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by (intrinsic) non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1D symmetry-topological-order (SymTO) of type $$\overline{\mathrm{JK}}^{\,}_4\times \mathrm{JK}^{\,}_4$$. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual $$S^{\,}_3$$-symetric and $$\mathsf{Rep}(S^{\,}_3)$$-symmetric models. 

Emergent inductance has attracted significant interest for its relevance in both interesting fundamental physics and practical applications in magnetic devices that demand miniaturization without compromising inductance. In this Letter, we report the discovery of a stepwise magnetic field-induced emergent magneto-inductance (EML) effect in Permalloy (Py) thin films deposited on polycarbonate (PC) substrates. Remarkably, Py/PC devices exhibit an exceptionally large inductance variation exceeding 1 μH at room temperature, and intriguingly, a sign reversal of inductance occurs around the zero magnetic field. The dependencies of the EML effect on frequency, step magnetic field changes, and film width can be explained from the theory based on the spin motive force driven by transient domain wall motion. This study opens up exciting avenues for advancing our understanding of emergent inductance in fundamental physics and paves the way for practical applications in flexible magnetic devices. 

To enable the practical use of skyrmion-based devices, it is essential to achieve a balance between energy efficiency and thermal stability while also ensuring reliable electrical detection against noise. Understanding how a skyrmion interacts with material disorder and external perturbations is thus essential. Here, we investigate the electronic noise of a single skyrmion under the influence of thermal fluctuations and spin currents in a magnetic thin film. We detect the thermally induced noise with a 1/ f γ signature in the strong pinning regime but a random telegraph noise in the intermediate pinning regime. Both the thermally dominated and current induced telegraph like signals are detected in the weak pinning regime. Our results provide a comprehensive electronic noise picture of a single skyrmion, demonstrating the potential of noise fluctuation as a valuable tool for characterizing the pinning condition of a skyrmion. These insights could also aid in the development of low-noise and reliable skyrmion-based devices.