Exact Perturbation Research Articles

Nonthermal eigenstates and slow relaxation in quantum Fredkin spin chains

We study the dynamics and thermalization of the Fredkin spin chain, a system with local three-body interactions, particle conservation, and explicit kinetic constraints. We consider deformations away from its stochastic point in order to tune between regimes where kinetic energy dominates and those where potential energy does. By means of exact diagonalization, perturbation theory, and variational matrix product states, we show there is a sudden change of behavior in the dynamics that occurs, from one of fast thermalization to one of slow metastable (prethermal) dynamics near the stochastic point. This change in relaxation is connected to the emergence of additional kinetic constraints, which lead to the fragmentation of Hilbert space in the limit of a large potential energy. We also show that this change can lead to thermalization being evaded for special initial conditions because of nonthermal eigenstates (akin to quantum many-body scars). We provide clear evidence for the existence of these nonthermal states for large system sizes even when far from the large-potential-energy limit, and explain their connection to the emergent kinetic constraints. Published by the American Physical Society 2024

Exact perturbation approximations for the conditional moments of a multifactor CIR term structure model with a weak mean-reversion influence

In this paper, we derive exact series expressions for the conditional moments of a multivariate Cox-Ingersoll-Ross term structure model with a weak mean-reversion effect assumption. First, we construct the perturbation solution for the system of Riccati equations, which illustrates the conditional characteristic function of this term structure model. Second, we derive the power series expression for the perturbation solution. Via the power series solutions, we establish the exact series expressions for conditional moments. Furthermore, we demonstrate the error bounds for these perturbation approximations of the conditional moments. Numerical analysis indicates that our method is more efficient, accurate and superior to the Monte Carlo approach with the Euler scheme for computing the conditional moments.

Development of an Adjoint Flux Calculation Technique for Exact Perturbation Theory in Monte Carlo Code MCS

A calculation technique computing the adjoint flux of perturbed system is developed for the exact perturbation theory in Monte Carlo transport. By using a correlated sampling and iterated fission probability methods together, the adjoint flux of perturbed system is calculated during a forward Mote Carlo simulation of an unperturbed system. In the method, no additional particle simulation is required to compute the adjoint flux of perturbed system. The exact perturbation method is implemented in the Monte Carlo code MCS. The results of perturbation method for k-eigenvalue change are compared against differential operator sampling, adjoint-weighted perturbation, and direct perturbation methods for the Godiva benchmark and PLUS7 fuel assembly.

LAST – a Library And Stochastic Transport tool

A new stochastic particle transport tool named LAST has been developed to allow seamless integral data assimilation on nuclear reaction model parameters during the evaluation process of nuclear data. The present work demonstrates its capability to perform actuate once-through information propagation from pre-processed application library to integral multiplication factor. Exact perturbation on reactivity is the starting point of the integral feedback on nuclear reaction model parameters. Its implementation in LAST is also shown.

An anharmonic alliance: exact WKB meets EPT

Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in ħ, were shown to admit ħ0-deformations with Borel resummable expansions which reproduce the original model at ħ0 = ħ. Such expansions were dubbed Exact Perturbation Theory (EPT). We investigate how the above results can be obtained within the framework of the exact WKB method by studying the spectrum of polynomial quantum mechanical systems. Within exact WKB, energy eigenvalues are determined by exact quantization conditions defined in terms of Voros symbols aγi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {a}_{\\gamma_i} $$\\end{document}, γi being their associated cycles, and generally give rise to transseries in ħ. After reviewing how the Borel summability of energy eigenvalues in the quartic anharmonic potential emerges in exact WKB, we extend it to higher order anharmonic potentials with quantum corrections. We then show that any polynomial potential can be ħ0-deformed to a model where the exact quantization condition reads simply aγ = −1 and leads to the EPT Borel resummable series for all energy eigenvalues.

Quantum fluctuation of ferroelectric order in polar metals

The polar metallic phase is an unusual metallic phase of matter containing long-range ferroelectric (FE) order in the electronic and atomic structure. Distinct from the typical FE insulating phase, this phase spontaneously breaks the inversion symmetry without global polarization. Unexpectedly, the FE order is found to be dramatically suppressed and destroyed at moderate ~ 10% carrier density. Here, we propose a general mechanism based on carrier-induced quantum fluctuations to explain this puzzling phenomenon. The quantum kinetic effect would drive the formation of polaronic quasi-particles made of the carriers and their surrounding dipoles. The disruption in dipolar directions can therefore weaken or even destroy the FE order. We demonstrate such polaron formation and the associated FE suppression via a concise model using exact diagonalization, perturbation, and quantum Monte Carlo approaches. This quantum mechanism also provides an intuitive picture for many puzzling experimental findings, thereby facilitating new designs of multifunctional FE electronic devices augmented with quantum effects.

Stability of thin-shell-wormhole in Robertson-Walker closed universe I: static universe

We construct a generic thin-shell wormhole (TSW) in the Robertson -Walker (RW) cosmological model of the Universe. The general formalism is presented and here in this paper—the first part of the research- we investigate the stability of the TSW constructed in the static RW universe. To do so, we perform two different stability analyses namely a mechanical linear radial perturbation and a small speed exact (not linearized) perturbation. It is shown that the corresponding TSW is stable against a linear radial perturbation provided that the variable equation of state of the surface energy-momentum tensor is finely tuned. On the other hand, the exact stability analysis in the context of small-speed perturbation reveals that the throat of the TSW for the closed static universe oscillates between the center of the Universe and the outer boundary.

Muscle preflex response to perturbations in locomotion: In vitro experiments and simulations with realistic boundary conditions.

Neuromuscular control loops feature substantial communication delays, but mammals run robustly even in the most adverse conditions. In vivo experiments and computer simulation results suggest that muscles' preflex-an immediate mechanical response to a perturbation-could be the critical contributor. Muscle preflexes act within a few milliseconds, an order of magnitude faster than neural reflexes. Their short-lasting action makes mechanical preflexes hard to quantify in vivo. Muscle models, on the other hand, require further improvement of their prediction accuracy during the non-standard conditions of perturbed locomotion. Our study aims to quantify the mechanical work done by muscles during the preflex phase (preflex work) and test their mechanical force modulation. We performed in vitro experiments with biological muscle fibers under physiological boundary conditions, which we determined in computer simulations of perturbed hopping. Our findings show that muscles initially resist impacts with a stereotypical stiffness response-identified as short-range stiffness-regardless of the exact perturbation condition. We then observe a velocity adaptation to the force related to the amount of perturbation similar to a damping response. The main contributor to the preflex work modulation is not the change in force due to a change in fiber stretch velocity (fiber damping characteristics) but the change in magnitude of the stretch due to the leg dynamics in the perturbed conditions. Our results confirm previous findings that muscle stiffness is activity-dependent and show that also damping characteristics are activity-dependent. These results indicate that neural control could tune the preflex properties of muscles in expectation of ground conditions leading to previously inexplicable neuromuscular adaptation speeds.

Exact formulations of relativistic electrodynamics and magnetohydrodynamics with helically coupled scalar field

We present the general relativistic electrodynamics and magnetohydrodynamics with a helically coupled scalar field. We consider three component system with the fluid, scalar field and electromagnetic fields with the helical coupling. We derive three exact formulations: the covariant formulation, the ADM formulation, and the fully nonlinear and exact perturbation formulation. We also derive the weak-gravity limit with fully relativistic fluid and fields. The latter two formulations are presented in the cosmological context.

First-order perturbation theory of eigenmodes for systems with interfaces

We present an exact first-order perturbation theory for eigenmodes in systems with interfaces causing material discontinuities. We show that when interfaces deform, higher-order terms of the perturbation series can contribute to the eigenmode frequencies in first order in the deformation depth. In such cases, the first-order approximation is different from the usual diagonal approximation and its single-mode result. Extracting additional first-order corrections from all higher-order terms enables us to recover the diagonal formalism in a modified form. A general formula for the single-mode first-order correction to electromagnetic eigenmodes in systems with interfaces is derived, capable of treating dispersive, magnetic, and chiral materials of arbitrary shape.

On the derivation of chiral symmetry breaking in QCD-like theories and S-confining theories

Recent works argue that the pattern of chiral symmetry breaking in QCD-like theories can be derived from supersymmetric (SUSY) QCD with perturbation of anomaly-mediated SUSY breaking (AMSB). Nevertheless, despite the fact that AMSB needs to be a small (but still exact) perturbation, two other major problems remain unsolved: first, in order to derive the chiral symmetry breaking pattern, one needs to minimize the potential along a certain specific direction, identifying this direction fully as an outcome is nontrivial given the moduli space of degenerate vacua in the SUSY limit; second, when SUSY is broken, non-holomorphic states might emerge and be relevant for determining the vacuum structure. In this work, we try to resolve these problems and discuss their physical implica- tions. For this purpose, we focus on SUSY QCD with Nf ≤ Nc + 1 and perturb the theories using AMSB. Without minimizing the potential along a certain specific direction in the moduli space, we successfully derive the expected chiral symmetry breaking pattern when Nf< Nc. However, when Nf = Nc and Nf = Nc + 1, we show that tree-level AMSB would induce runaway directions, along which baryon number is spontaneously broken, and the vacua with broken baryon number can be deeper while the field values are not far from the origin. This implies that phase transitions and/or non-holomorphic physics are necessary. In order to derive the expected chiral symmetry breaking pattern of non-SUSY QCD starting from the SUSY limit and AMSB, baryon number conservation is needed as an input rather than obtained as an output. Moreover, we perform explicit consistency checks on “ultraviolet insensitivity” for different Nf by adding the holomorphic mass term for the last flavor, we find that the “jump” of AMSB potential indeed matches the contribution from the holomorphic mass term. We also show in general that, when tree-level AMSB is not vanishing, the origin of the moduli space in s-confining theories does not persist as a minimum.

Vibration‐induced symmetry breaking in hybrid light‐matter dimer states

AbstractWe investigate the influence of vibronic coupling on a molecular dimer strongly coupled to a single cavity mode. In the framework of the Holstein‐Tavis‐Cummings model, the energy structure of the molecular dimer is analyzed by numerical exact diagonalization and perturbation theory. Under numerical exact diagonalization, we find that the degeneracy of lower polaritons vanishes in the presence of vibronic coupling. Under the second‐order degenerate perturbation theory, the degeneracy breaking of lower polaritons can be associated with asymmetric indirect interactions mediated by the upper polaritons and the dark states. The consistency of the two approaches confirms the robustness of our simulations, indicating that the vibration‐induced symmetry breaking should be experimentally observed.

Exact and many-body perturbation solutions of the Hubbard model applied to linear chains

This study reports on the accuracy of the GW approximation for the treatment of the Hubbard model compared to exact diagonalization (ED) results. We consider not only global quantities, such as the total energy and the density of states, but also the spatial and spin symmetry of wavefunctions via the analysis of the local density of states. GW is part of the more general Green’s function approach used to develop many-body approximations. We show that, for small linear chains, the GW approximation corrects the mean-field (MF) approach by reducing the total energy and the magnetization obtained from the MF approximation. The GW energy gap is in better agreement with ED, especially in systems of an even number of atoms where, in contrast to the MF approximation, no plateau is observed below the artificial predicted phase transition. In terms of density of states, the GW approximation induces quasi-particles and side satellite peaks via a splitting process of MF peaks. At the same time, GW slightly modifies the localization (e.g., edges or center) of states. We also use the GW approximation results in the context of Löwdin’s symmetry dilemma and show that GW predicts an artificial paramagnetic–antiferromagnetic phase transition at a higher Hubbard parameter than MF does.

On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value for a general semilinear singularly perturbed Neumann boundary value problem, in this paper, the exact critical perturbation value $\varepsilon_c$ which determines the existence or nonexistence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.

Transient dynamics of a magnetic impurity coupled to superconducting electrodes: Exact numerics versus perturbation theory

Impurities coupled to superconductors offer a controlled platform to understand the interplay between superconductivity, many-body interactions, and nonequilibrium physics. In the equilibrium situation, local interactions at the impurity induce a transition from the spin-singlet to the spin-doublet ground state, resulting in a supercurrent sign reversal ($0\text{\ensuremath{-}}\ensuremath{\pi}$ transition). In this work, we apply the exact time-dependent density matrix renormalization group method to simulate the transient dynamics of such superconducting systems. We also use a perturbative approximation to analyze their properties at longer times. These two methods agree for a wide range of parameters. In a phase-biased situation, the system gets trapped in a metastable state characterized by a lower supercurrent compared to the equilibrium case. We show that local Coulomb interactions do not provide an effective relaxation mechanism for the initially trapped quasiparticles. In contrast, other relaxation mechanisms, such as coupling to a third normal lead, make the impurity spin relax for parameter values corresponding to the equilibrium 0 phase. For parameters corresponding to the equilibrium $\ensuremath{\pi}$ phase the impurity converges to a spin-polarized stationary state. Similar qualitative behavior is found for a voltage-biased junction, which provides an effective relaxation mechanism for the trapped quasiparticles in the junction.

The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality

The one-dimensional Fibonacci chain is a toy model central to theoretical studies of the physics of electronic states in quasiperiodic structures. This review surveys the state of the art of methods to study the energy spectra and states of Fibonacci chain models, including exact solutions, renormalization group, perturbation theory, and numerical analysis. Results highlight distinctive properties of Fibonacci chain systems, including projection from higher dimensions, nontrivial topological properties, multifractal states, and hyperuniformity. Questions of current interest include the effects of disorder and interactions and experimental realizations in electronic, cold atom, phononic, and photonic systems.

Data envelopment analysis model with decision makers’ preferences: a robust credibility approach

Data envelopment analysis (DEA) is one of the widely used methods to measure the efficiency scores of decision making units (DMUs). Conventional DEA is unable to consider both uncertainty in data and decision makers’ (DMs) judgments in the evaluations. This study, to address the shortcomings of the conventional DEA, proposes a new best worst method (BWM)- robust credibility DEA (BWM-RCDEA) model to estimate the efficiency scores of DMUs considering DMs’ preferences and uncertain data, simultaneously. First, to handle uncertainty in input and output variables, fuzzy credibility model has been applied. Additionally, uncertainty in constructing fuzzy sets is modeled using robust optimization with fuzzy perturbation degree. In this paper, two new types of RCDEA models are proposed: RCDEA model with exact perturbation in fuzzy inputs and outputs and RCDEA model with fuzzy perturbation in fuzzy inputs and outputs. In addition, to deal with flexibility of weights and incorporating DMs’ judgement into the RCDEA model, a bi-objective BWM-RCDEA model is introduced. Finally, the proposed bi-objective model is solved using min–max approach. To illustrate the usefulness and capability of the proposed model, efficiency scores of 39 distribution companies in Iran is investigated and results are analyzed and discussed. Finally, based on the results, recommendations have been made for policy makers.

Vibration analysis of irregular-shaped plates on simple supports

In this work, we propose a general perturbative approach for modal analysis of irregular-shaped plates of uniform thickness with uniform boundary conditions. Given a plate of irregular boundary, first, a uniform circular plate of identical thickness and area, centred at the centroid, is determined. The irregular boundary is then treated as a perturbation with a suitable smallness parameter, and is expressed as a generalized Fourier series. The frequency parameter, shape function and boundary conditions are then perturbed in terms of the smallness parameter. The homogeneous zeroth-order equation corresponds to the circular plate, which is exactly solvable. We show that the inhomogeneous equations in the higher orders can also be solved exactly using a particular solution structure. We can then construct the exact perturbative solution up to any order. The proposed method is demonstrated through the modal analysis of simply supported super-circular plates. The results are validated using the numerical results obtained from ANSYS ® , which are an excellent match. Interestingly, the supposedly degenerate modes with an even number of nodal diameters of super-circular plates are found to split naturally.

Emergent symmetries and slow quantum dynamics in a Rydberg-atom chain with confinement

Rydberg atoms in optical tweezer arrays provide a playground for nonequilibrium quantum many-body physics. The PXP model describes the dynamics of such systems in the strongly interacting Rydberg blockade regime and notably exhibits weakly nonergodic dynamics due to quantum many-body scars. Here, we study the PXP model in a strong staggered external field, which has been proposed to manifest quasiparticle confinement in light of a mapping to a lattice gauge theory. We characterize this confining regime using both numerical exact diagonalization and perturbation theory around the strong-field limit. In addition to the expected emergent symmetry generated by the staggered field, we find a second emergent symmetry that is special to the PXP model. The interplay between these emergent symmetries and the Rydberg blockade constraint dramatically slows down the system's dynamics beyond naive expectations. We devise a nested Schrieffer-Wolff perturbation theory to properly account for the new emergent symmetry, and we show that this treatment is essential to understand the numerically observed relaxation time scales. We also discuss connections to Hilbert space fragmentation, and we trace the origin of the new emergent symmetry to a ``nearly-$\text{SU}(2)$'' algebra discovered in the context of many-body scarring.

Robustness verification of $\boldsymbol~K$-NN classifiers via constraint relaxation and randomized smoothing

We study the robustness verification problem for $K$-NN classifiers. The objective of formal robustness verification is to find the exact minimal adversarial perturbation or a guaranteed lower bound of the perturbation. We find that the robustness verification of $K$-NN classifiers could be formalized as a series of quadratic programming problems. Solving these quadratic programming problems is not possible in general because the number of problems grows exponentially with respect to $K$. The constraint relaxation method is proposed to compute the lower bound of the minimal adversarial perturbation in polynomial time. However, we find that the resulting lower bound tends to be extremely loose when $K$ is large; hence, $K$-NN with a large $K$ being less robust is counterintuitive. To tackle this issue, we propose to employ the randomized smoothing method to verify the robustness of $K$-NN classifiers. By exploiting the resistance of $K$-NN to random Gaussian noise, the randomized smoothing method achieves high performance in verification. Our experiments on benchmark datasets show that the smoothed $K$-NN classifier is more verifiably robust than state-of-the-art robust neural networks.