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SignificanceSr2RuO4 is distinctive among unconventional superconductors, in that in addition to exhibiting evidence for strong correlations, it is stoichiometric and extremely clean. As a result, its electronic structure is unusually well characterized, rendering it an ideal platform for developing a deep understanding of the mechanism behind the emergence of the superconducting state from a Fermi liquid. Toward that end, an unambiguous determination of the pairing symmetry is an essential step. For more than 2 decades, the preponderance of evidence pointed to a triplet spin pairing state and only recently has this interpretation been challenged. By field-dependent NMR Knight shift measurements, we eliminate from further consideration all candidate purely odd-parity triplet pairing states. Unambiguous identification of the superconducting order parameter symmetry in Sr2RuO4 has remained elusive for more than a quarter century. While a chiral p-wave ground state analogue to superfluid 3He-A was ruled out only very recently, other proposed triplet-pairing scenarios are still viable. Establishing the condensate magnetic susceptibility reveals a sharp distinction between even-parity (singlet) and odd-parity (triplet) pairing since the superconducting condensate is magnetically polarizable only in the latter case. Here field-dependent 17O Knight shift measurements, being sensitive to the spin polarization, are compared to previously reported specific heat measurements for the purpose of distinguishing the condensate contribution from that due to quasiparticles. We conclude that the shift results can be accounted for entirely by the expected field-induced quasiparticle response. An upper bound for the condensate magnetic response of <10% of the normal state susceptibility is sufficient to exclude all purely odd-parity candidates.

The dynamic magnetic and thermodynamic behaviors of the mixed-spin (1/2, 1, 3/2) Ising edge-modified graphene-like nanoparticles in a time-dependent magnetic field are studied by Monte Carlo simulation. The influence of temperature, exchange coupling, crystal field, and time-dependent magnetic field on the dynamic order parameter, magnetic susceptibility, and internal energy are investigated. The simulations reveal that the compensation behavior of the system occurs in a certain parameter range. Strong crystal field and weak exchange coupling promote the compensation behavior in the system, while increasing the amplitude of the oscillating magnetic field reduces the compensation temperature. Furthermore, the system shows the interesting multiple dynamic hysteresis behaviors under the influence of different parameters.

We study the collective relaxation dynamics appearing in weakly coupled Lohe oscillators in a large coupling regime. The Lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the Kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the Lyapunov functional approach and order parameter approach. In this paper, we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid. In particular, we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the Lohe sphere model. We also present new gradient flow formulations for the Lohe matrix models with the same one-body Hamiltonians on some group manifolds. As a direct application of this new formulation, we show that every bounded Lohe flow which originated from any initial configuration converges asymptotically.

A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the antiferromagnetic coupling. We term this behavior ferromagnetic mesoscopic magnetization. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one-spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.

Cells cooperate as groups to achieve structure and function at the tissue level, during which specific material characteristics emerge. Analogous to phase transitions in classical physics, transformations in the material characteristics of multicellular assemblies are essential for a variety of vital processes including morphogenesis, wound healing, and cancer. In this work, we develop configurational fingerprints of particulate and multicellular assemblies and extract volumetric and shear order parameters based on this fingerprint to quantify the system disorder. Theoretically, these two parameters form a complete and unique pair of signatures for the structural disorder of a multicellular system. The evolution of these two order parameters offers a robust and experimentally accessible way to map the phase transitions in expanding cell monolayers and during embryogenesis and invasion of epithelial spheroids.

We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length 2 N with zero chemical potential can be mapped to two independent Kitaev chains of length N . It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su–Schrieffer–Heeger Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen–Cooper–Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.

We propose a family of order parameters to detect the symmetry fractionalization class of anyons in 2D topological phases. This fractionalization class accounts for the projective, as opposed to linear, representations of the symmetry group on the anyons. We focus on quantum double models on a lattice enriched with an internal symmetry in the framework of G-isometric projected entangled pair states. Unlike previous schemes based on reductions to effective 1D systems (dimensional compactification), the order parameters presented here can be probed on genuinely two-dimensional geometries, and are local: they rely on operations on few neighbouring particles in the bulk. The power of these order parameters is illustrated with several combinations of topological content and symmetry. We demonstrate that a strictly finer phase distinction than that provided by dimensional compactification can be obtained. As particular examples, the resolution power of these order parameters is illustrated for a case with non-abelian topological order, and for another with symmetries that involves permutation of anyons.

Recently, Baldwin and Swingle (J Stat Phys 190(7):125, 2023) considered spin glass models with additional conventional order parameters characterizing single-replica properties. These parameters are distinct from the standard order parameter, the overlap, used to measure correlations between replicas. A “min-max” formula for the free energy was prescribed in Baldwin and Swingle (2023). We rigorously verify this prescription in the setting of vector spin glass models featuring additional deterministic spin interactions. Notably, our results can be viewed as a generalization of the Parisi formula for vector spin glass models in Panchenko (Ann Probab 46(2):865–896, 2018), where the order parameter for self-overlap is already present.

Birefringence, based on the crystal orientation of a cellulose nanocrystal (CNC) film, has been investigated for the determination of crystal orientation. The prime focus of our study is to establish a simple and low-cost experimental technique using a standard UV–Vis spectrometer to determine the order parameter (S) for both isotropic and anisotropic configurations. Self-standing CNC films of various order parameters (S: 0–0.95) were prepared using shear flow with varying shear rates. The transmitted light intensity of CNC films between cross polarizers for the bright and dark fields were measured to determine the linear dichroitic ratio, which were used to calculate the order parameters of different crystalline arrangements. Two-dimensional X-ray diffraction results were compared to the birefringence technique to explore the presence of short-range amorphous order, which is significant in higher order parameter (S > 0.60) specimens. Thus, the new birefringence technique was shown to obtain a more accurate measurement of order parameter with an easier method using more common and lower cost equipment.

This research explored the impact of four different electrolytes on the orientation of cellulose nanocrystals (CNCs) in shear-cast films prepared from aqueous CNC gels. Changes in the aqueous CNC gels’ rheological properties with electrolyte addition were correlated to the orientation and optical properties of dried CNC films. Film alignment was qualitatively assessed using cross-polarized optical microscopy and quantified by order parameters computed by UV–Vis transmission spectroscopy. Electrolyte addition resulted in an increased alignment in dried CNC films. For pure CNCs, the film order parameters remained constant at approximately 0.3 for shear rates from 20 s−1 to 100 s−1. However, higher order parameters were achieved in the presence of electrolytes. Notably, an order parameter of 0.88 was achieved at a shear rate of only 20 s−1. In addition, films produced from dispersions containing electrolytes exhibited improved clarity and haze. The results of this work highlight that electrolyte addition can enable higher order parameters at lower shear rates and facilitate the development of aligned CNC films for applications such as polarizers, clear coatings, and piezoelectric materials.