Non-trivial Point Research Articles

Pressure-induced magnetic and topological transitions in non-centrosymmetric MnIn2Te4

In recent years, the study of magnetic topological materials, with their variety of exotic physics, has significantly flourished. In this work, we predict the interplay of magnetism and topology in the non-centrosymmetric ternary manganese compound MnIn2Te4under external hydrostatic pressure, using first-principles calculations and symmetry analyses. At ambient pressure, the ground state of the system is an antiferromagnetic insulator. With the application of a small hydrostatic pressure (∼0.50 GPa), it undergoes a magnetic transition, and the ferromagnetic state becomes energetically favorable. At ∼2.92 GPa, the ferromagnetic system undergoes a transition into a Weyl semimetallic phase, which hosts multiple Weyl points in the bulk. The presence of non-trivial Weyl points have been verified by Wilson bands computations and the presence of characteristic surface Fermi arcs. Remarkably, we discover that the number of Weyl points in this system can be controlled by pressure and that these manifest in an anomalous Hall conductivity (AHC). In addition to proposing a new candidate magnetic topological material, our work demonstrates that pressure can be an effective way to induce and control topological phases, as well as AHC, in magnetic materials. These properties may allow our proposed material to be used as a novel pressure-controlled Hall switch.

Theory of angular momentum transfer from light to molecules

We present a theory describing the interaction of structured light, such as light carrying orbital angular momentum, with molecules. The light-matter interaction Hamiltonian we derive is expressed through couplings between spherical gradients of the electric field and the (transition) electric multipole moments of a particle of any nontrivial rotation point group. Our model can therefore accommodate an arbitrary complexity of the molecular and electric field structure, and it can be straightforwardly extended to atoms or nanostructures. Applying this framework to rovibrational spectroscopy of molecules, we uncover the general mechanism of angular momentum exchange between the spin and orbital angular momenta of light, molecular rotation, and its center-of-mass motion. We show that the nonzero vorticity of Laguerre-Gaussian beams can strongly enhance certain rovibrational transitions that are considered forbidden in the case of nonhelical light. We discuss the experimental requirements for the observation of these forbidden transitions in state-of-the-art spatially resolved spectroscopy measurements. Published by the American Physical Society 2024

Anomalous non-Hermitian skin effect: topological inequivalence of skin modes versus point gap

It has long been believed that skin modes are equivalent to the nontrivial point gap. However, we find that this concomitance can be broken, in that skin modes can be absent or present when the point gap is nontrivial or trivial, respectively, named anomalous non-Hermitian skin effect. This anomalous phenomenon arises whenever unidirectional hopping amplitudes emerge among subsystems, where sub-chains have decoupling-like behaviors and contribute only to the energy levels without particle occupation. The occurrence of anomalous non-Hermitian skin effect is accompanied by changes in open boundary eigenvalues, whose structure exhibits multifold exceptional points and can not be recovered by continuum bands. Moreover, an experimental setup is proposed to simulate this effect. Our results reveal the topologically inequivalence of skin modes and point gap. This effect not only provides a deeper understanding of non-Bloch theory and critical phenomena, but may inspire applications, such as in sensor field.

Semiclassical geometry in double-scaled SYK

We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a coupling λ in which all observables are governed by a non-trivial saddle point. The Liouville description of double-scaled SYK suggests that the correlation functions define a geometry in a two-dimensional bulk, with the 2-point function describing the metric. For small coupling, the fluctuations are highly suppressed, and the bulk describes a rigid (A)dS spacetime. As the coupling increases, the fluctuations become stronger. We study the correction to the curvature of the bulk geometry induced by these fluctuations. We find that as we go deeper into the bulk the curvature increases and that the theory eventually becomes strongly coupled. In general, the curvature is related to energy fluctuations in light operators. We also compute the entanglement entropy of partially entangled thermal states in the semiclassical limit.

Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points

Exceptional points and skin effect, as the two distinct hallmark features unique to the non-Hermitian physics, have each attracted enormous interests. Recent theoretical works reveal that the topologically nontrivial exceptional points can guarantee the non-Hermitian skin effect, which is geometry-dependent, relating these two unique phenomena. However, such novel relation remains to be confirmed by experiments. Here, we realize a non-Hermitian phononic crystal with exceptional points, which exhibits the geometry-dependent skin effect. The exceptional points connected by the bulk Fermi arcs, and the skin effects with the geometry dependence, are evidenced in simulations and experiments. Our work, building an experimental bridge between the exceptional points and skin effect and uncovering the unconventional geometry-dependent skin effect, expands a horizon in non-Hermitian physics.

Bounds for the Diameters of Orbital Graphs of Affine Groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer H ≤GL(V ), where the vector space V has dimension d over the prime field, can be bounded in terms of d and log |V|/log |H| only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

Stability Analysis of Output Feedback Type Linear Reset Systems with Non-detectable Jump Dynamics

An output feedback type reset system consists of the output feedback connection of two hybrid dynamics (i.e. plant and controller) and the output based reset condition. In the existing results of Lyapunov based stability analysis for reset systems, Lyapunov-like functions which decrease after jump were usually considered. However, if the plant is a continuous-time system, the closed-loop jump dynamics always has nontrivial stationary points. This implies the existing Lyapunov based analysis is not applicable to a certain class of the reset systems which are practically important. To overcome this difficulty, the author proposes a new type of Lyapunov-like function which has a different property for the component of a state in the null space of an output matrix of the system and its complementary subspace. The existence conditions of the proposed Lyapunov-like function and a nonempty interval of reset forbidden period to guarantee the stability of the temporally regularized system are also discussed.

Rethinking WMMSE: Can Its Complexity Scale Linearly With the Number of BS Antennas?

Precoding design for maximizing weighted sum-rate (WSR) is a fundamental problem for downlink of massive multi-user multiple-input multiple-output (MU-MIMO) systems. It is well-known that this problem is generally NP-hard due to the presence of multi-user interference. The weighted minimum mean-square error (WMMSE) algorithm is a popular approach for WSR maximization. However, its computational complexity is cubic in the number of base station (BS) antennas, which is unaffordable when the BS is equipped with a large antenna array. In this paper, we consider the WSR maximization problem with either a sum-power constraint (SPC) or per-antenna power constraints (PAPCs). For the former, we prove that any nontrivial stationary point must have a low-dimensional subspace structure, and then propose a reduced-WMMSE (R-WMMSE) with linear complexity by exploiting the solution structure. For the latter, we propose a linear-complexity WMMSE approach, named PAPC-WMMSE, by using a novel recursive design of the algorithm. Both R-WMMSE and PAPC-WMMSE have simple closed-form updates and guaranteed convergence to stationary points. Simulation results verify the efficacy of the proposed designs, especially the much lower complexity as compared to the state-of-the-art approaches for massive MU-MIMO systems.

Sextic tensor model in rank 3 at next-to-leading order

We compute the four-loop beta functions of short and long-range multi-scalar models with general sextic interactions and complex fields. We then specialize the beta functions to a U(N)3 symmetry and study the renormalization group at next-to-leading order in N and small ϵ. In the short-range case, ϵ is the deviation from the critical dimension while it is the deviation from the critical scaling of the free propagator in the long-range case. This allows us to find the 1/N corrections to the rank-3 sextic tensor model of [1]. In the short-range case, we still find a non-trivial real IR stable fixed point, with a diagonalizable stability matrix. All couplings, except for the so-called wheel coupling, have terms of order ϵ0 at leading and next-to-leading order, which makes this fixed point different from the other melonic fixed points found in quartic models. In the long-range case, the corrections to the fixed point are instead not perturbative in ϵ and hence unreliable; we thus find no precursor of the large-N fixed point.

Representations attached to elliptic curves with a non‐trivial odd torsion point

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non-split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic representations in terms of the parameters.

Relative Hofer–Zehnder capacity and positive symplectic homology

We study the relationship between a homological capacity $$c_{\mathrm {SH}^+}(W)$$ for Liouville domains W defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on W: if the positive symplectic homology of W is non-zero, then the capacity yields a finite upper bound to the $$\pi _1$$ -sensitive Hofer–Zehnder capacity of W relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of W has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of W in terms of the homological capacity $$c_{\mathrm {SH}}(W)$$ defined using the full symplectic homology. Applications of these statements to cotangent bundles are discussed and use a result by Abbondandolo and Mazzucchelli in the appendix, where the monotonicity of systoles of convex Riemannian two-spheres in $${\mathbb {R}}^3$$ is proved.

Enhancing photon entanglement in a three-mode optomechanical system via imperfect phonon measurements

By considering a 3-mode optomechanical system formed by two cavities interacting with a common mechanical mode, we demonstrate that phonon-counting measurements lead to a significant enhancement of entanglement in the output of the two cavities. This conclusion still holds for an inefficient detector, but the dependence on system parameters changes qualitatively from the ideal limit of perfect projective measurements. We find non-trivial optimal points for the entanglement as functions of detector efficiency, measurement outcome, and optical drive strengths. We characterize both the highest achievable entanglement as well as a ‘typical’ value, obtained at the most likely measurement outcome. Numerical results are well understood within an approximate analytical approach based on perturbation theory around the ideal detector limit.

Back to sources – the role of losses and coherence in super-resolution imaging revisited

Photon losses are intrinsic for any translationally invariant optical imaging system with a non-trivial Point Spread Function, and the relation between the transmission factor and the coherence properties of an imaged object is universal – we demonstrate the rigorous proof of this statement, based on the principles of quantum mechanics. The fundamental limit on the precision of estimating separation between two partially coherent sources is then derived. The careful study of the role of photon losses allows to resolve conflicting claims present in previous works. We compute the Quantum Fisher Information for the generic model of optical 4f imaging system, and use prior considerations to validate the result for a general, translationally invariant imaging apparatus. We prove that the spatial-mode demultiplexing (SPADE) measurement, optimal for non-coherent sources, remains optimal for an arbitrary degree of coherence. Moreover, we show that some approximations, omnipresent in theoretical works about optical imaging, inevitably lead to unphysical, zero-transmission models, resulting in misleading claims regarding fundamental resolution limits.

On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

We study the local eigenvalue statistics $\xi_{\omega,E}^N$ associated with the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as $N^\alpha$, for $0 < \alpha < \frac{1}{2}$. We consider the limit points associated with the random variables $\xi_{\omega,E}^N [I]$, for $I \subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random variables with $0 \leq \alpha < \frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every $E \in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables $\xi_{\omega,E}^N [I]$ and associated quantities related to the intensities, as $N$ tends towards infinity, and employs known localization bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for $0 < \alpha < \frac{1}{2}$, we prove that any nontrivial limit points of the random variables $\xi_{\omega,E}^N [I]$ are distributed according to Poisson distributions.

Fully Dynamic (Δ +1)-Coloring in O (1) Update Time

The problem of (Δ +1)-vertex coloring a graph of maximum degree Δ has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA’18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized algorithm for maintaining a (Δ +1)-coloring with O (log Δ) expected amortized update time. In this article, we present an improved randomized algorithm for (Δ +1)-coloring that achieves O (1) amortized update time and show that this bound holds not only in expectation but also with high probability. Our starting point is the state-of-the-art randomized algorithm for maintaining a maximal matching (Solomon, FOCS’16). We carefully build on the approach of Solomon, but, due to inherent differences between the maximal matching and (Δ +1)-coloring problems, we need to deviate significantly from it in several crucial and highly nontrivial points. 1

Molecular response for nematic superconducting media in a hollow cylinder: a numerical approach

In the context of the Ginzburg–Landau formalism proposed by Barci et al. in 2016 for nematic superconductivity, and by performing a numerical treatment based on the shooting method, we analyze the behaviour of the radial distribution of the nematic order parameter when the superconducting order parameter reaches the typical non-trivial saddle point. We consider the cases of a hollow cylindrical domain, with a disk or an annular domain as its cross section, whether the order parameter is subjected to Neumann or Dirichlet boundary conditions. We conclude that depending on the corresponding situation a non trivial solution holds if certain relations between the radii are satisfied. Moreover, we observe a saturation effect on each instance that constitutes a purely geometrical consequence on the relation between the typical sizes and shapes of the samples. These conclusions can be useful for further experimental realizations and extensions to the interaction of the nematic (superconducting) order parameters with electromagnetic fields.

Counting rational points on elliptic curves with a rational 2-torsion point

Let E/\mathbb{Q} be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if E has full rational 2-torsion, the number N_E(B) of rational points with Weil height bounded by B is \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) . In this paper we exploit the method of descent via 2-isogeny to extend this result to elliptic curves with just one nontrivial rational 2-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound N_E(B) = \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) for these curves and to remove a deep transcendence theory ingredient from the proof.

Antiferromagnetic topological crystalline insulator and mixed Weyl semimetal in two-dimensional NpAs monolayer

Magnetic topological states have attracted significant attentions due to their intriguing quantum phenomena and potential applications in topological spintronic devices. Here, we propose a two-dimensional material NpAs monolayer as a candidate for multiple topological states accompanied with the changes of magnetic structures. Under the antiferromagnetic configuration, the long-awaited topological crystalline insulator (TCI) emerges with a nonzero mirror Chern number and a giant band gap of 630 meV, and remarkably a pair of gapless edge states can be tailored by rotating the magnetization directions while the TCI phase survives. Moreover, we establish the existence of quantum anomalous Hall effect and nontrivial nodal points under the ferromagnetic configuration, thereby giving rise to the mixed Weyl semimetal after adding the magnetization direction to topological classification. Our findings not only provide an ideal candidate for uncovering exotic topological characters with magnetism but also put forward potential applications in topological spintronics.

Minimal models of rational elliptic curves with non-trivial torsion

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $$E/{\mathbb {Q}}$$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $$E/{\mathbb {Q}}$$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for E. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which E has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterlé pertaining to the primes at which an elliptic curve over a number field K with a non-trivial K-torsion point can have additive reduction.

Universal points in the asymptotic spectrum of tensors

Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiringX\mathcal {X}of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements ofX\mathcal {X}in terms ofthe asymptotic spectrum ofX\mathcal {X}, which is defined as the collection of semiring homomorphisms fromX\mathcal {X}to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information.Universal spectral pointsare elements in the asymptotic spectrum of the semiring ofalltensors. Finding all universal spectral points suffices to find the asymptotic spectrum ofanysub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, calledquantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.