Simple Perturbation Theory Research Articles

Cavity-renormalized quantum criticality in a honeycomb bilayer antiferromagnet

Strong light-matter interactions as realized in an optical cavity provide a tantalizing opportunity to control the properties of condensed matter systems. Inspired by experimental advances in cavity quantum electrodynamics and the fabrication and control of two-dimensional magnets, we investigate the fate of a quantum critical antiferromagnet coupled to an optical cavity field. Using unbiased quantum Monte Carlo simulations, we compute the scaling behavior of the magnetic structure factor and other observables. While the position and universality class are not changed by a single cavity mode, the critical fluctuations themselves obtain a sizable enhancement, scaling with a fractional exponent that defies expectations based on simple perturbation theory. The scaling exponent can be understood using a generic scaling argument, based on which we predict that the effect may be even stronger in other universality classes. Our microscopic model is based on realistic parameters for two-dimensional magnetic quantum materials and the effect may be within the range of experimental detection.

Stability of PPT in equilibrium states

We use simple spectral perturbation theory to show that the positive partial transpose property is stable under bounded perturbations of the Hamiltonian, for equilibrium states in infinite dimensions. The result holds provided the temperature is high enough, or equivalently, provided the perturbation is small enough.

Broadband microwave spectroscopy of cyclopentylsilane and 1,1,1-trifluorocyclopentylsilane

The broadband microwave spectra of two related organosilicon molecules, cyclopentylsilane and 1,1,1-trifluorocyclopentylsilane, have been measured and assigned. Both molecules exhibit a Cs-symmetric “envelope” structure, which is confirmed by a determination of the heavy atom backbone rs substitution structures using isotopic substitution in natural abundance, as well as the presence of strong a-type transitions, weaker c-type, and a non-detection of b-type transitions, indicative of a mirror plane perpendicular to the prolate symmetry axis of the molecule. Both molecules appear to have significant zero-point vibrational averaging of their ring twisting and puckering coordinates, which lead to deviations between the ground state experimental structure and the predicted equilibrium structures at the B3LYP-D3/def2-TZVP level of theory. Additionally, we observe an excited vibrational state associated with this large amplitude motion and we use a simple perturbation theory argument to confirm the identity of these excited state carriers.

A simple second order thermodynamic perturbation theory for associating fluids

An approximation within Wertheim's second order perturbation theory is proposed which allows for the development of a general solution for pure component fluids with an arbitrary number and functionality of association sites. The solution is closed, concise and general for all second order effects such as ring formation, steric hindrance and hydrogen bond cooperativity. The approach is validated by comparison to hydrogen bond structure data for liquid water.

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field ε with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the “soft mode” ε, turning them into a simple diagrammatic perturbation theory at large central charge.Our second (equivalent) approach concerns a theory of reparametrization modes, generalizing previous studies in the context of the Schwarzian theory and two-dimensional CFTs. Due to the conformal anomaly in even dimensions, gauge modes of the conformal group acquire an action and are shown to exhibit the same dynamics as the soft mode ε that encodes the physics of the stress tensor shadow. We discuss the calculation of the conformal partial waves or the conformal blocks using our effective field theory. The separation of conformal blocks from shadow blocks is related to gauging of certain symmetries in our effective field theory of the soft mode.These connections explain and generalize various relations between conformal blocks, shadow operators, kinematic space, and reparametrization modes. As an application we study thermal physics in higher dimensions and argue that the theory of reparametrization modes captures the physics of quantum chaos in Rindler space. This is also supported by the observation of the pole skipping phenomenon in the conformal energy-energy two-point function on Rindler space.

Topological Hall Effect from Strong to Weak Coupling

The topological Hall effect (THE) of electrons coupled to a noncoplanar spin texture has been studied so far for the strong- and weak-coupling regimes separately; the former in terms of the Berry phase and the latter by perturbation theory. In this letter, we present a unified treatment in terms of spin gauge field by considering not only the adiabatic (Berry phase) component of the gauge field but also the nonadiabatic component. While only the adiabatic contribution is important in the strong-coupling regime, it is completely canceled by a part of the nonadiabatic contribution in the weak-coupling regime, where the THE is governed by the remaining nonadiabatic terms. We found a new weak-coupling region that cannot be accessed by a simple perturbation theory, where the Hall conductivity is proportional to M, with 2M being the exchange splitting of the electron spectrum.

Localized solutions of Lugiato-Lefever equations with focused pump

Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D) accurately describe the dynamics of optical fields in pumped lossy cavities with the intrinsic Kerr nonlinearity. The external pump is usually assumed to be uniform, but it can be made tightly focused too–in particular, for building small pixels. We obtain solutions of the LL equations, with both the focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes supported by the localized pump. In the 1D setting, we first develop a simple perturbation theory, based in the sech ansatz, in the case of weak pump and loss. Then, a family of exact analytical solutions for spatially confined modes is produced for the pump focused in the form of a delta-function, with a nonlinear loss (two-photon absorption) added to the LL model. Numerical findings demonstrate that these exact solutions are stable, both dynamically and structurally (the latter means that stable numerical solutions close to the exact ones are found when a specific condition, necessary for the existence of the analytical solution, does not hold). In 2D, vast families of stable confined modes are produced by means of a variational approximation and full numerical simulations.

Non-Abelian Higgs models: Paving the way for asymptotic freedom

Asymptotically free renormalization group trajectories can be constructed in nonabelian Higgs models with the aid of generalized boundary conditions imposed on the renormalized action. We detail this construction within the languages of simple low-order perturbation theory, effective field theory, as well as modern functional renormalization group equations. We construct a family of explicit scaling solutions using a controlled weak-coupling expansion in the ultraviolet, and obtain a standard Wilsonian RG relevance classification of perturbations about scaling solutions. We obtain global information about the quasi-fixed function for the scalar potential by means of analytic asymptotic expansions and numerical shooting methods. Further analytical evidence for such asymptotically free theories is provided in the large-N limit. We estimate the long-range properties of these theories, and identify initial/boundary conditions giving rise to a conventional Higgs phase.

Nonadiabatic exchange dynamics during adiabatic frequency sweeps

A Bloch equation analysis that includes relaxation and exchange effects during an adiabatic frequency swept pulse is presented. For a large class of sweeps, relaxation can be incorporated using simple first order perturbation theory. For anisochronous exchange, new expressions are derived for exchange augmented rotating frame relaxation. For isochronous exchange between sites with distinct relaxation rate constants outside the extreme narrowing limit, simple criteria for adiabatic exchange are derived and demonstrate that frequency sweeps commonly in use may not be adiabatic with regard to exchange unless the exchange rates are much larger than the relaxation rates. Otherwise, accurate assessment of the sensitivity to exchange dynamics will require numerical integration of the rate equations. Examples of this situation are given for experimentally relevant parameters believed to hold for in-vivo tissue. These results are of significance in the study of exchange induced contrast in magnetic resonance imaging.

Boundary conditions as dynamical fields

The possibility of treating boundary conditions in terms of a bilocal dynamical field is formalized in terms of a boundary action. This allows for a simple path-integral perturbation theory approach to physical effects such as radiation from a time-dependent boundary. The nature of the action which governs the dynamics of the bilocal field is investigated for a limited case (which includes the Robin boundary conditions).

Prediction of Spin Orientations in Terms of HOMO-LUMO Interactions Using Spin-Orbit Coupling as Perturbation.

For most chemists and physicists, electron spin is merely a means needed to satisfy the Pauli principle in electronic structure description. However, the absolute orientations of spins in coordinate space can be crucial in understanding the magnetic properties of materials with unpaired electrons. At low temperature, the spins of a magnetic solid may undergo long-range magnetic ordering, which allows one to determine the directions and magnitudes of spin moments by neutron diffraction refinements. The preferred spin orientation of a magnetic ion can be predicted on the basis of density functional theory (DFT) calculations including electron correlation and spin-orbit coupling (SOC). However, most chemists and physicists are unaware of how the observed and/or calculated spin orientations are related to the local electronic structures of the magnetic ions. This is true even for most crystallographers who determine the directions and magnitudes of spin moments because, for them, they are merely the parameters needed for the diffraction refinements. The objective of this article is to provide a conceptual framework of thinking about and predicting the preferred spin orientation of a magnetic ion by examining the relationship between the spin orientation and the local electronic structure of the ion. In general, a magnetic ion M (i.e., an ion possessing unpaired spins) in a solid or a molecule is surrounded with main-group ligand atoms L to form an MLn polyhedron, where n is typically 4-6, and the d states of MLn are split because the antibonding interactions of the metal d orbitals with the p orbitals of the surrounding ligands L depend on the symmetries of the orbitals involved.1 The magnetic ion M of MLn has a certain preferred spin direction because its split d states interact among themselves under SOC.2,3 The preferred spin direction can be readily predicted on the basis of perturbation theory in which the SOC is taken as perturbation and the split d states as unperturbed states by inspecting the magnetic quantum numbers of its d orbitals present in the HOMO and LUMO of the MLn polyhedron. This is quite analogous to how chemists predict whether a chemical reaction is symmetry-allowed or symmetry-forbidden in terms of the HOMO-LUMO interactions by simply inspecting the symmetries of the frontier orbitals.4,5 Experimentally, the determination of the preferred spin orientations of magnetic ions requires a sophisticated level of experiments, for example, neutron diffraction measurements for magnetic solids with an ordered spin state at a very low temperature. Theoretically, it requires an elaborate level of electronic structure calculations, namely, DFT calculations including electron correlation and SOC. We show that the outcomes of such intricate experimental measurements and theoretical calculations can be predicted by a simple perturbation theory analysis.

Bound central orbits

The orbits in any central potential are described analytically and new expressions are derived for their periods and precession rates. A simple perturbation theory allows their delineation to any accuracy. Kepler's equation is generalized to such orbits and angle and action variables are given for them. A new property of orbits under inverse fifth power forces is found.

Damping of mechanical vibrations by free electrons in metallic nanoresonators

We investigate the effect of free electrons on the quality factor ($Q$) of a metallic nanomechanical resonator in the form of a thin elastic beam. The flexural and longitudinal modes of the beam are modeled using thin beam elasticity theory, and simple perturbation theory is used to calculate the rate at which an externally excited vibration mode decays due to its interaction with free electrons. We find that electron-phonon interaction significantly affects the $Q$ of longitudinal modes, and may also be of significance to the damping of flexural modes in otherwise high-$Q$ beams. The finite geometry of the beam is manifested in two important ways. Its finite length breaks translation invariance along the beam and introduces an imperfect momentum conservation law in place of the exact law. Its finite width imposes a quantization of the electronic states that introduces a temperature scale for which there exists a crossover from a high-temperature macroscopic regime, where electron-phonon damping behaves as if the electrons were in the bulk, to a low-temperature mesoscopic regime, where damping is dominated by just a few dissipation channels and exhibits sharp nonmonotonic changes as parameters are varied. This suggests a scheme for probing the electronic spectrum of a nanoscale device by measuring the $Q$ of its mechanical vibrations.

Towards understanding the empty liquid of colloidal platelets: vapour–liquid phase coexistence of square-well oblate ellipsoids

We study the phase behaviour of hard oblate ellipsoids with superimposed short-ranged oblate-shaped square-well attractions using the replica exchange Monte Carlo method and a simple van der Waals type perturbation theory. By fixing the square-well range (λ = 0.25σ‖), we examine the effect of varying the oblates aspect ratio, k = σ⊥/σ‖, on the density profiles, vapour–liquid and isotropic–nematic phase transitions. We observe that the formation of a liquid phase with vanishing density, an empty liquid, is possible with increasing aspect ratio. Surprisingly, the increasing shape anisotropy does not shift the stability region of the nematic phase to low densities, while the oblate-shaped square-well attraction favours disordered states. For k > 1.5, the critical temperature and packing fraction monotonically decrease with k, while the isotropic–nematic phase boundary moves towards higher packing fractions with decreasing temperature for all k. Simulations show that oblates preferably orientate their faces parallel to the vapour–liquid interface when located at the liquid side of the interface region and perpendicular when located at the vapour side.

A strain space framework for numerical hyperplasticity

Numerical simulations of high strain rate plastic flow have historically been built in a hypoelastic framework and use radial return (Wilkins’ method) as the solution algorithm. We show how each of these choices can lead to inaccurate and possibly nonconvergent results. We describe an alternative solution procedure based on a simple multiple time scale perturbation theory that is stable, accurate, computationally efficient and simple to implement. Further extension of these results then leads to a strain space formulation that has additional computational advantages. We illustrate our development with numerical experiments.This paper is dedicated to my friend and colleague Christo Christov on the occasion of his 60th birthday, in recognition of his many important and creative contributions to the formulation of continuum mechanics.

On the existence of stable seasonally varying Arctic sea ice in simple models

Within the framework of lower order thermodynamic theories for the climatic evolution of Arctic sea ice we isolate the conditions required for the existence of stable seasonally‐varying solutions, in which ice forms each winter and melts away each summer. This is done by constructing a two‐season model from the continuously evolving theory of Eisenman and Wettlaufer (2009) and showing that seasonally‐varying states are unstable under constant annual average short‐wave radiative forcing. However, dividing the summer season into two intervals (ice covered and ice free) provides sufficient freedom to stabilize seasonal ice. Simple perturbation theory shows that the condition for stability is determined by the timing of when the ice vanishes in summer and hence the relative magnitudes of the summer heat flux over the ocean versus over the ice. This scenario is examined within the context of greenhouse gas warming, as a function of which stability conditions are discerned.

Oscillatory behaviour of charge transfer probabilities in ion-atom collisions

The oscillatory behaviour of charge transfer probability as a function of impact parameter in ion–atom collisions often referred to as Stueckelberg oscillations is discussed in terms of simple first-order time-dependent perturbation theory. The system of He+2 + H is used to make the treatment specific. Symmetry consequences for transfer probabilities due to nonadiabatic coupling terms are analysed that apply to general ion–atom systems.

General Perturbative Approach for Spectroscopy, Thermodynamics, and Kinetics: Methodological Background and Benchmark Studies.

A general second-order perturbative approach based on resonance- and threshold-free computations of vibrational properties is introduced and validated. It starts from the evaluation of accurate anharmonic zero-point vibrational energies for semirigid molecular systems, in a way that avoids any singularity. Next, the degeneracy corrected second-order perturbation theory (DCPT2) is extended to a hybrid version (HDCPT2), allowing for reliable computations even in cases where the original formulation faces against severe problems, including also an automatic treatment of internal rotations through the hindered-rotor model. These approaches, in conjunction with the so-called simple perturbation theory (SPT) reformulated to treat consistently both energy minima and transition states, allow one to evaluate degeneracy-corrected partition functions further used to obtain vibrational contributions to properties like enthalpy, entropy, or specific heat. The spectroscopic accuracy of the HDCPT2 model has been also validated by computing anharmonic vibrational frequencies for a number of small-to-medium size, closed- and open-shell, molecular systems, within an accuracy close to that of well established but threshold-dependent perturbative-variational models. The reliability of the B3LYP/aug-N07D model for anharmonic computations is also highlighted, with possible improvements provided by the B2PLYP/aug-cc-pVTZ models or by hybrid schemes. On a general grounds, the overall approach proposed in the present work is able to provide the proper accuracy to support experimental investigations even for large molecular systems of biotechnological interest in a fully automated manner, without any ad hoc scaling procedure. This means a fully ab initio evaluation of thermodynamic and spectroscopic properties with an overall accuracy of about, or better than, 1 kJ mol(-1), 1 J mol(-1) K(-1) and 10 cm(-1) for enthalpies, entropies, and vibrational frequencies, respectively.

Dual fermion dynamical cluster approach for strongly correlated systems

We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size ${L}_{c}$ embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as $\mathcal{O}(1/{L}_{c})$, so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order $\mathcal{O}(1/{L}_{c}^{3})$ with third-order and three-body corrections down by an additional factor of $\mathcal{O}(1/{L}_{c})$.

Simple analytic equations of state for the generalized hard-core Mie(α, β) and Mie(α, β) fluids from perturbation theory

New, simple and analytic perturbation theory equations of state for generalized hard-core Mie HCMie(α, β) and Mie(α, β) fluids are proposed. They are based on the second-order Barker–Henderson perturbation theory in the macroscopic compressibility approximation and the new analytical expression of the radial distribution function of hard spheres, g HS(r), developed by Sun in terms of a polynomial expansion of base functions adapted to the square-well and Sutherland potentials [Can. J. Phys. 83 (2005) 55], the combination of which yields the HCMie(α, β) and Mie(α, β) functions. The compressibility factors, the residual internal energies and the radial distribution function at contact with the hard core are then obtained from this equation of state for the HCLJ(12, 6) potential, which is a particular case of the HCMie(α, β) potentials with α = 12 and β = 6. The results are in good agreement with the existing Monte Carlo (MC) simulation data, and compare favorably with those obtained from five other equations of state, three of which contain numerical coefficients fitted to the Monte Carlo results. For the Mie(α, 6) (α = 8, 10, 12), fluids, the present equation of state is a good representation of recent molecular dynamics (MD) simulations of the pressure and internal energy. It is more accurate than the statistical associating fluid theory of variable range (SAFT-VR Mie(n, 6)) theory for n = 8, and 10, while for n = 12 the SAFT-VR theory is best. For the Mie(14, 7) fluid, which is outside the range of application of the SAFT-VR theory, the results for the pressure are in good agreement with the analytical equation of state obtained from the MC simulation data.