Nontrivial Order Research Articles

Transport of Spin Magnetic Multipole Moments Carried by Bloch Quasiparticles.

Magnetic ordering beyond the standard dipolar order has attracted significant attention in recent years, but it remains an open question how to effectively manipulate such nontrivial order parameters using external perturbations such as electric currents or fields. In particular, it is desirable to have a conceptual tool similar to nonequilibrium spin currents in spintronics to describe the creation and transport of multipole moments. In this context, we present a theory for Cartesian spin magnetic multipole moments of Bloch quasiparticles and their transport based on a general gauge-invariant formula obtained using the wave packet approach. As a concrete example, we point out that the low-energy Hamiltonian of phosphorene subject to a perpendicular electric field has a valley structure that hosts magnetic octupole moments. The magnetic octupole moments can be exhibited by an in-plane electric current and lead to accumulation of staggered spin densities at the corners of a rectangular sample. Our Letter paves the way for systematically seeking and utilizing quasiparticles with higher-order magnetic multipole moments in crystal materials towards the emergence of multipoletronics.

On discrete properties of monotone mappings

Let [Formula: see text] and [Formula: see text] be a partial order on [Formula: see text]. We deal with properties of oriented graphs which correspond to the algebra [Formula: see text] in the case that [Formula: see text] is monotone with respect to [Formula: see text]. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that [Formula: see text] is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that [Formula: see text] is monotone with respect to this order will be described; if the number of components of [Formula: see text] is infinite, then the number of such orders is equal to the cardinality of the power set of [Formula: see text].

Line defect RG flows in the ε expansion

A general analysis of line defect renormalisation group (RG) flows in the ε expansion below d = 4 dimensions is undertaken. The defect beta function for general scalar-fermion bulk theories is computed to next-to-leading order in the bulk couplings. Scalar models as well as scalar-fermion models with various global symmetries in the bulk are considered at leading non-trivial order. Different types of potential infrared (IR) defect conformal field theories (dCFTs) and their RG stability are discussed. The possibility of multiple IR stable dCFTs is realised in specific examples with hypertetrahedral symmetry in the bulk. The one-point function coefficient of the order parameter in the stable IR dCFT of the cubic model is computed at next-to-leading order and compared with that in the IR dCFT of the Heisenberg model.

Theory of critical phenomena with long-range temporal interaction

We systematically develop theories of critical phenomena with a prior formed memory of a power-law decaying long-range temporal interaction parameterized by a constant θ > 0 for a space dimension d both below and above an upper critical dimension d c = 6 − 2/θ. We first provide more evidences to confirm the previous theory that a dimensional constant is demanded to rectify a hyperscaling law, to produce correct unique mean-field critical exponents via an effective spatial dimension originating from temporal dimension, and to transform the time and change the dynamic critical exponent. Next, for d < d c , we develop a renormalization-group theory by employing the momentum-shell technique to the leading nontrivial order explicitly and to higher orders formally in ϵ = d c − d but to zero order in ε = 1 − θ and find that more scaling laws besides the hyperscaling law are broken due to the breaking of the fluctuation-dissipation theorem. Moreover, because dynamics and statics are intimately interwoven, even the static critical exponents involve contributions from the dynamics and hence do not restore the short-range exponents even for θ = 1 and the crossover between the short-range and long-range fixed points is discontinuous contrary to the case of long-range spatial interaction. In addition, a new scaling law relating the dynamic critical exponent with the static ones emerges, indicating that the dynamic critical exponent is not independent. However, once is displaced by a series of ϵ and ε such that most values of the critical exponents are changed, all scaling laws are saved again, even though the fluctuation-dissipation theorem keeps violating. Then, for d ≥ d c , we develop an effective-dimension theory by carefully discriminating the corrections of both temporal and spatial dimensions and find three different regions. For d ≥ d c0 = 4, the upper critical dimension of the usual short-range theory, the usual Landau mean-field theory with fluctuations confined to the effective-dimension equal to d c0 correctly describe the critical phenomena with memory, while for d c < d ≤ 4, there exist new universality classes whose critical exponents depend only on the space dimension but not at all on θ. Yet another region consists of d = d c only and the previous theory is retrieved. All these results show that the dimensional constant is the fundamental ingredient of the theories for critical phenomena with memory. However, its value continuously varies with the space dimension and vanishes exactly at d = 4, reflecting the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Moreover, special finite-size scaling appears ubiquitously except for d = d c0.

Aspects of thermal one-point functions and response functions in AdS black holes

We revisit the problem of analytically computing the one-point functions for scalar fields in planar AdS black holes of arbitrary dimension, which are sourced by the Weyl squared tensor. We analyze the problem in terms of power series expansions around the boundary using the method of Frobenius. We clarify the pole structure of the final answer in terms of operator mixing, as argued previously by Grinberg and Maldacena. We generalize the techniques to also obtain analytic results for slowly modulated spatially varying sources to first nontrivial order in the wave vector for arbitrary dimension. We also study the first-order corrections to the one-point function of the global AdS black hole at large mass, where we perturb in terms that correspond to the curvature of the horizon.

A functional approach to the Van der Waals interaction

We evaluate the quantum interaction energy between two neutral atoms, using a functional integral approach. In our setup, each atom has an electron bound to the nucleus via a harmonic potential. The resulting expression for the vacuum energy becomes the Van der Waals interaction at the first non-trivial order in an expansion in powers of the fine structure constant, encompassing both the long and short distance behaviours. We also explore the opposite, strong-coupling limit, which yields a result for the interaction energy as well as a threshold for the existence of a vacuum decay probability, manifested here as an imaginary part for the effective action. In the weak-coupling limit, we also study the effect of using a general central potential for the internal structure of the atoms.

Ultrasmall Polar Skyrmions and Merons in SrTiO3 Heterostructures by Polaron Engineering.

Topological objects with skyrmionic textures in ferroelectrics, i.e., polar skyrmions, are promising technological paradigms in next-generation electronic devices. While breakthrough discoveries of stable polar skyrmions approximately ten nanometers in size have been very recently witnessed in complex systems, such a nontrivial topological order in ferroelectrics inevitably disappears below the ferroelectric critical size of several nanometers. Herein, we propose a strategy to overcome this limitation and achieve ultrasmall and isolated polar skyrmions by engineering excess-electron polarons in otherwise nonferroelectric SrTiO3 heterostructures. Our first-principle calculations demonstrate that a polaron localized at a SrTiO3 surface induces a Neel-type polar skyrmion as small as 1.8 nanometers attributed to the effect of atomic-scale surface roughness. Furthermore, we show that this polar topological structure is tunable by the choice of heterostructures and by the mechanical approach, which undergoes a phase transition to a meron state in the twisted boundary and to an antiskyrmion state in the surface with external shear strain, respectively. Such ultraminiaturization of skyrmions and their transitions unexpectedly unravels the formula of ultrasmall topological orders originating from the interplay between an electron polaron and structural symmetry breaking, which is completely different from the common mechanism of geometric confinement for larger-scale skyrmions. Our results not only provide a mechanism for the exploration of polar skyrmions and their rich topological transitions but also hold potential for ultrahigh-density memories.

Integrated negative geometries in ABJM

We study, in the context of the three-dimensional mathcal{N} = 6 Chern-Simons-matter (ABJM) theory, the infrared-finite functions that result from performing L − 1 loop integrations over the L-loop integrand of the logarithm of the four-particle scattering amplitude. Our starting point are the integrands obtained from the recently proposed all-loop projected amplituhedron for the ABJM theory. Organizing them in terms of negative geometries ensures that no divergences occur upon integration if at least one loop variable is left unintegrated. We explicitly perform the integrations up to L = 3, finding both parity-even and -odd terms. Moreover, we discuss a prescription to compute the cusp anomalous dimension Γcusp of ABJM in terms of the integrated negative geometries, and we use it to reproduce the first non-trivial order of Γcusp. Finally, we show that the leading singularities that characterize the integrated results are conformally invariant.

Non-Lifshitz invariants corrections to Dzyaloshinskii-Moriya interaction energy

We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling. The resulting free-energy density of the system contains linear terms in the spatial gradient of the magnetic texture, which mark an instability of the system towards the formation of nontrivial magnetic orders such as skyrmions or chiral domain walls. We perform a microscopic analysis of DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants features. In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order in the small exchange coupling. The higher-order terms may even lead to an enhancement of DMI interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of the conduction band. Finally, we also study the DMI free energy generated from Rashba spin-orbit interaction in different symmetry groups.

Chiral edge dynamics and quantum Hall physics in synthetic dimensions with an atomic erbium Bose-Einstein condensate

Quantum Hall physics is at the heart of research on both matter and artificial systems, such as cold atomic gases, with nontrivial topological order. We report on the observation of a chiral edge current by transferring atomic wave packets simultaneously to opposite edges of a synthetic Hall system realized in the two-dimensional state space formed by one spatial and one synthetic dimension encoded in the $J=6$ electronic spin of erbium atoms. To characterize the system, the Hall drift of the employed atomic Bose-Einstein condensate in the lowest Landau-like level is determined. The topological properties are verified by determining the local Chern marker, and upon performing low-lying excitations both cyclotron and skipping orbits are observed in the bulk and edges, respectively. Future prospects include studies of novel topological phases in cold-atom systems.

Discrete Time Crystal Enabled by Stark Many-Body Localization.

Discrete time crystals (DTCs) have recently attracted increasing attention, but most DTC models and their properties are only revealed after disorder average. In this Letter, we propose a simple disorder-free periodically driven model that exhibits nontrivial DTC order stabilized by Stark many-body localization (MBL). We demonstrate the existence of the DTC phase by analytical analysis from perturbation theory and convincing numerical evidence from observable dynamics. The new DTC model paves a new promising way for further experiments and deepens our understanding of DTCs. Since the DTC order does not require special quantum state preparation and the strong disorder average, it can be naturally realized on the noisy intermediate-scale quantum hardware with much fewer resources and repetitions. Moreover, in addition to the robust subharmonic response, there are other novel robust beating oscillations in the Stark-MBL DTC phase that are absent in random or quasiperiodic MBL DTCs.

Asymptotically safe Hilbert–Palatini gravity in an on-shell reduction scheme

We study the renormalization flow of Hilbert–Palatini gravity to lowest non-trivial order. We find evidence for an asymptotically safe high-energy completion based on the existence of an ultraviolet fixed point similar to the Reuter fixed point of quantum Einstein gravity. In order to manage the quantization of the large number of independent degrees of freedom in terms of the metric as well as the connection, we use an on-shell reduction scheme: for this, we quantize all degrees of freedom beyond Einstein gravity at a given order that remain after using the equations of motion at the preceding order. In this way, we can straightforwardly keep track of the differences emerging from quantizing Hilbert–Palatini gravity in comparison with Einstein gravity. To lowest non-trivial order, the difference is parametrized by fluctuations of an additional abelian gauge field. The critical properties of the ultraviolet fixed point of Hilbert–Palatini gravity are similar to those of the Reuter fixed point, occurring at a smaller Newton coupling and exhibiting more stable higher order exponents.

Deflection in higher dimensional spacetime and asymptotically non-flat spacetimes

Using a perturbative technique, in this work we study the deflection of null and timelike signals in the extended Einstein–Maxwell spacetime, the Born–Infeld gravity and the charged Ellis–Bronnikov (CEB) spacetime in the weak field limit. The deflection angles are found to take a (quasi-)series form of the impact parameter, and automatically takes into account the finite distance effect of the source and observer. The method is also applied to find the deflections in CEB spacetime with arbitrary dimension. It’s shown that to the leading non-trivial order, the deflection in some n-dimensional spacetimes is of the order . We then extended the method to spacetimes that are asymptotically non-flat and studied the deflection in a nonlinear electrodynamical scalar theory. The deflection angle in such asymptotically non-flat spacetimes at the trivial order is found to be not π anymore. In all these cases, the perturbative deflection angles are shown to agree with numerical results extremely well. The effects of some nontrivial spacetime parameters as well as the signal velocity on the deflection angles are analyzed.

Three-point energy correlators and the celestial block expansion

We study the three-point energy correlator (EEEC), defined as a matrix element of a product of three energy detectors at different locations on the celestial sphere. Lorentz symmetry implies that the EEEC can be decomposed into special functions called celestial blocks. We compute three-point celestial blocks in an expansion around the collinear limit, where the three detectors approach each other on the celestial sphere. The leading term is a traditional d – 2-dimensional four-point conformal block, and thus the collinear EEEC behaves like a conformally-invariant four-point function in d – 2 dimensions. We obtain the coefficients of the conformal block decomposition for the collinear EEEC at leading nontrivial order in weakly-coupled \U0001d4a9 = 4 SYM and QCD. These data allow us to make certain all-orders predictions for the collinear EEEC in various kinematic limits, including the OPE limit and the double lightcone limit. We also study Ward identities satisfied by the EEEC and compute contact terms in the EEEC in weakly-coupled \U0001d4a9 = 4 SYM. Finally, we study the celestial block expansion of the EEEC in planar \U0001d4a9 = 4 SYM at strong coupling, determining celestial block coefficients to leading and first subleading order at large λ.

Emergent higher-symmetry protected topological orders in the confined phase of U(1) gauge theory

We consider compact ${U}^{\ensuremath{\kappa}}(1)$ gauge theory in $3+1$ dimensions with a general $2\ensuremath{\pi}$-quantized topological term ${\ensuremath{\sum}}_{I,J=1}^{\ensuremath{\kappa}}\frac{{K}_{IJ}}{4\ensuremath{\pi}}{\ensuremath{\int}}_{{M}^{4}}{F}^{I}\ensuremath{\wedge}{F}^{J}$, where $K$ is an integer symmetric matrix with even diagonal elements and ${F}^{I}=d{A}^{I}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, where ${k}_{i}$ are the diagonal elements of the Smith normal form of $K$ and ${\mathbb{Z}}_{0}^{(1)}$ is regarded as a $U(1)$ 1-symmetry. In the ${U}^{\ensuremath{\kappa}}(1)$ confined phase, the boundary can have a phase whose infrared (IR) properties are described by Chern-Simons field theory. Such a phase has a ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this continuum field theory, thus acting as its ultraviolet completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We find that the short-range entangled gapped phase of the lattice model, corresponding to the confined phase of the ${U}^{\ensuremath{\kappa}}(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, whose SPT invariant is $\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I,J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}{B}_{I}\ensuremath{\smile}{B}_{J}+{B}_{I}\underset{1}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I&lt;J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}\phantom{\rule{1.0pt}{0ex}}d{B}_{I}\underset{2}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}$. Here, the background $\mathbb{R}/\mathbb{Z}$-valued 2-cochains ${B}_{I}$ satisfy $\phantom{\rule{1.0pt}{0ex}}d{B}_{I}={\ensuremath{\sum}}_{I}{B}_{I}{K}_{IJ}=0\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}1$ and describe the symmetry twist of the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the nontrivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

Emergent fermionic gauge theory and foliated fracton order in the Chamon model

The Chamon model is an exactly solvable spin Hamiltonian exhibiting nontrivial fracton order. In this paper, we dissect two distinct aspects of the model. First, we show that it exhibits an emergent fractonic gauge theory coupled to a fermionic subsystem symmetry-protected topological state under four stacks of ${\mathbb{Z}}_{2}$ planar symmetries. Second, we show that the Chamon model hosts 4-foliated fracton order by describing an entanglement renormalization group transformation that exfoliates four separate stacks of 2D toric codes from the bulk system.

Pair density wave and loop current promoted by Van Hove singularities in moiré systems

We theoretically show that in the presence of conventional or higher order Van Hove singularities (VHS), the bare finite momentum pairing, also known as the pair density wave (PDW), susceptibility can be promoted to the same order of the most divergent bare BCS susceptibility through a valley-contrasting flux $3\ensuremath{\phi}$ in each triangular plaquette at $\ensuremath{\phi}=\ensuremath{\pi}/3$ and $\ensuremath{\pi}/6$ in moir\'e systems. This makes the PDW order a possible leading instability for an electronic system with repulsive interactions. We confirm that it indeed wins over all other instabilities and becomes the ground state under certain conditions through the renormalization group calculation and a flux insertion argument. Moreover, we also find that a topological nontrivial loop current order becomes the leading instability if the Fermi surface with conventional VHS is perfectly nested at $\ensuremath{\phi}=\ensuremath{\pi}/3$. Similar to the Haldane model, this loop current state has the quantum anomalous Hall effect. If we dope this loop current state or introduce a finite next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, the chiral $d$-wave PDW becomes the dominant instability. Experimentally, the flux can be effectively tuned by an out-of-plane electric field in moir\'e systems based on graphene and transition metal dichalcogenides.

Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control.

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

Abundance of second order topology in C3 symmetric two-dimensional insulators

We have screened 71 two-dimensional (2D) materials with C 3 symmetry for non-trivial second order topological order and find that 28 compounds exhibit an obstructed atomic limit (OAL). In the case of C 3 symmetry, the second order topology can be calculated from bulk symmetry indicator invariants, which predict the value of fractional corner charges in symmetry conserving nanoflakes. The procedure is exemplified by MoS2 in the H-phase, which constitutes a generic example of a 2D OAL material and the predicted fractional corner charges is verified by direct calculations of nanoflakes with armchair edges. We also determine the bulk topological polarization, which always lead to gapless states at zigzag edges and thus deteriorates the concept of fractional corner charges in nanoflakes with zigzag edges that are typically more stable that armchair flakes. We then consider the case of TiCl2, which has vanishing polarization as well as an OAL and we verify that the edge states of nanoflakes with zigzag edges may indeed by passivated such that the edges remain insulating and the corner charges are well defined. For the 28 OAL materials we find that 16 have vanishing polarization and these materials thus constitute a promising starting point for experimental verification of second order topology in a 2D material.

Соотношение унитарности и унитарные ограничения для теории скалярных полей с разными скоростями звука

Motivated by scalar-tensor gravities, we consider a theory which contains massless scalar fields with different sound speeds. We derive unitarity relations for partial wave amplitudes of $2 \to 2$ scattering, with explicit formulas for contributions of two-particle intermediate states. Making use of these relations, we obtain unitarity bounds both in the most general case and in the case considered in literature for unit sound speed. These bounds can be used for estimating the strong coupling scale of a pertinent EFT. We illustrate our unitarity relations by explicit calculation to the first non-trivial order in couplings in a simple model of two scalar fields with different sound speeds.