Degenerate Perturbation Theory Research Articles

Degenerate perturbation theory to quantum search

Degenerate perturbation theory to quantum search

Unveiling the quantum secrets of triel metal triangles: a tale of stability, aromaticity, and relativistic effects.

Low lying electronic states of Al3-, Ga3-, In3-, and Tl3- have been characterized using high level multiconfigurational quasi degenerate perturbation theory on the multiconfigurational self-consistent field. Among these species, the singlet states emerge as the predominant energy minima, displaying remarkable stability. However, within the Tl3- series, our investigation leads to the identification of the high-spin , as the most stable spin state, a result corroborated by previous experimental detection via photoelectron spectroscopy. Similarly, we have also identified the singlet state of In3- as the signal detected previously experimentally. By applying Mandado's rules and an array of aromaticity indicators, it is conclusively demonstrated that both the singlet and quintet states exhibit multiple-fold aromaticity, while the triplets exhibit conflicting aromaticity. Furthermore, this investigation highlights the significant impact of relativistic effects, as they enhance the stability of the state relative to its singlet counterpart. These findings shed new light on the electronic structures and properties of these ions, offering valuable insights into their chemical behavior and potential applications.

Combining three sources of optical anisotropy in a tunable open-access microcavity: From theory to experiment

Photonic spin–orbit (SO) coupling is an important physical mechanism leading to numerous interesting phenomena in the systems of microcavity photons and exciton-polaritons. We report the effect of SO coupling in a tunable open-access microcavity embedded with anisotropic active media. The SO coupling associated with the TE–TM splitting results in an emergent anisotropy, which further leads to fine energy splittings allowing clear observation of the full set of eigenstates, in sharp contrast with the isotropic situation which leads to the isotropic eigenstates of spin vortices. We show that the photonic potential can be engineered by playing with the relation between the emergent anisotropy and the cavity ellipticity. All the experimental results are well reproduced by the degenerate perturbation theory. Our results constitute a significant extension to the research field of microcavity spinoptronics, with potential applications in polarization control and optical property measurement of photonic devices and materials.

Degenerate Perturbation Theory for Models of Quantum Field Theory with Symmetries

We consider Hamiltonians of models describing non-relativistic quantum mechanical matter coupled to a relativistic field of bosons. If the free Hamiltonian has an eigenvalue, we show that this eigenvalue persists also for nonzero coupling. The eigenvalue of the free Hamiltonian may be degenerate provided there exists a symmetry group acting irreducibly on the eigenspace. Furthermore, if the Hamiltonian depends analytically on external parameters then so does the eigenvalue and eigenvector. Our result applies to the ground state as well as resonance states. For our results, we assume a mild infrared condition. The proof is based on operator theoretic renormalization. It generalizes the method used in Griesemer and Hasler (Ann Henri Poincaré 10(3):577–621, 2009) to non-degenerate situations, where the degeneracy is protected by a symmetry group, and utilizes Schur’s lemma from representation theory.

Coupling properties of subjects in a three-body interaction system with an intermediary entity and symmetrical participants on both sides

This paper studies the phenomenon of light-excited entanglement between nucleon spins, which is realized by using a transient light beam to excite the spin of electrons to produce entanglement between the two nucleon spins. This standard three-body interaction system is composed of an intermedium equivalent to a spin-1 quasiparticle and two spin-1/2 nucleons. We describe the entire system, its corresponding effective Hamiltonian, and the eigenvalues and eigenstates. The Hamiltonian is analyzed under the symmetric assumption, which can significantly simplify the effective Hamiltonian and make obtaining the corresponding dynamic characteristics easier. In the calculation process, degenerate perturbation theory calculates the eigenequation, whose degeneracy is not eliminated to a first approximation. We calculate the eigenequation to the second order to obtain an accurate zero-order wave function.

Full characterization of vector eigenstates in symmetrically confined systems with photonic spin-orbit coupling.

The photonic spin-orbit (SO) coupling is a widely investigated effect in optical microcavities leading to various interesting physical phenomena and potential applications. We report the full sets of eigenenergies and eigenstates in a symmetrically confined potential under the effect of SO coupling induced by the transverse-electric transverse-magnetic (TE-TM) splitting, which are derived analytically via the degenerate perturbation theory. We obtained the eigenenergies and the eigenstates from the 1st to the 6th orders of excited manifold, and demonstrate unambiguously that universal rules governing the mode formation exist in such complicated photonic systems, making the modes exhibiting the features of solid and hollow skyrmions as well as spin vortices. We show that these eigenstates can be described by the SO coupled hyperspheres that can be decomposed into a series of higher-order Poincare spheres. Our results significantly extend the area of microcavity spin-optronics to the general theory of eigenvalues in confined systems, and provide an efficient theoretical frame for the information processing using microcavity-based high-dimensional vector states.

Elimination of Chirality in Three-Dimensionally Confined Open-Access Microcavities.

The emergent optical activity (OA) caused by anisotropic light emitter in microcavities is an important physical mechanism discovered recently, which leads to Rashba-Dresselhaus photonic spin-orbit (SO) coupling. In this study, we report a sharp contrast of the roles of the emergent OA in free and confined cavity photons, by observing the optical chirality in a planar-planar microcavity and its elimination in a concave-planar microcavity, evidenced by polarization-resolved white-light spectroscopy, which agrees well with the theoretical predictions based on the degenerate perturbation theory. Moreover, we theoretically predict that a slight phase gradient in real space can partially restore the effect of the emergent OA in confined cavity photons. The results are significant additions to the field of cavity spinoptronics and provide a novel method for manipulating photonic SO coupling in confined optical systems.

Strong-disorder renormalization group approach to the Anderson model using Rayleigh–Schrödinger perturbation theory

Previous work proposed a strong-disorder renormalization approach for the Anderson model, using it to calculate the density of states and the inverse participation ratio, Johri and Bhatt (2014). This is interesting because of the potential for expansion to higher dimensions and to interacting systems. The original proposal used a non-standard perturbation theory approach which avoided degeneracies. We implemented the same structure but with standard Rayleigh–Schrödinger perturbation theory. Here degeneracies do arise, and we consider two approaches, one in which renormalization is suppressed if degeneracy is present and a second in which the most common form of degeneracy is handled using standard degenerate perturbation theory. The version in which degeneracies are not handled performs similarly to the original proposal, and the addition of degeneracies provides some improvement.

Waves in the Earth’s core. III. A perturbative approach to quasi-free-decay modes

We consider the canonical problem of magnetic field decay in an electrically conducting fluid ball. The problem is closely allied to the problem of the decay modes of a rigid ball, and the spatial form of the eigenmodes survives largely intact. The decaying but oscillatory behaviour of the new fluid eigenmodes first discovered by Schmitt a decade ago (and named quasi-free-decay (QFD) modes) is deduced by application of perturbation methods to the case of rapid rotation and a static applied background magnetic field that is uniform and axial. Some, but not all, of the rigid-case poloidal eigenmodes share decay rates with other toroidal modes, necessitating the use of both degenerate and non-degenerate perturbation theory within this paper. The perturbation theory is developed in terms of the Elsasser number Λ (measuring the competition between Coriolis and Lorentz forces), and the analytic results are in striking accord with numerical calculations even when Λ is of O ( 1 ) . We find linear scaling of the QFD eigenfrequency with Λ and small changes in the decay rate that scale with Λ 2 . Although the modes are overdamped (quality factor Q < 1 ), they are not strongly overdamped when the applied field is strong Λ ∼ 1 .

Propagation of spin channel waves

A mutilated model is constructed to approximate the collision term of spin Boltzmann equation that incorporates newly appearing collisional invariants, i.e., the total angular momentum. With recourse to degenerate perturbation theory, the dispersion relations of hydrodynamic modes are formulated, among which spin modes are responsible for spin equilibration. We find that the nonlocality does not change the sound speed but slows down the propagation of spin channel waves. The damping rates of spin modes are close to those of spinless modes over a reasonable parameter value range. The results reveal that both spin and momentum should be treated simultaneously in a unified transport framework. In the nonrelativistic limit, the short-wavelength behavior for normal modes is also explored and there exists a critical point for every distinct discrete mode over which only quasiparticle modes contribute.

Beyond Hard-Core Bosons in Transmon Arrays

Arrays of transmons have proven to be a viable medium for quantum information science and quantum simulations. Despite their widespread popularity as qubit arrays, there remains yet untapped potential beyond the two-level approximation or, equivalently, the hard-core boson model. With the higher excited levels included, coupled transmons naturally realize the attractive Bose-Hubbard model. The dynamics of the model has been difficult to study due to unfavorable scaling of the dimensionality of the Hilbert space with the system size. In this work, we present a framework for describing the effective unitary dynamics of highly-excited states of coupled transmons based on high-order degenerate perturbation theory. This allows us to describe various collective phenomena -- such as bosons stacked onto a single site behaving as a single particle, edge-localization, and effective longer-range interactions -- in a unified, compact and accurate manner. While our examples deal with one-dimensional chains of transmons for the sake of clarity, the theory can be readily applied to more general geometries.

A wave function correction-based approach to the identification of resonances for vibrational perturbation theory.

An approach for identifying resonances in vibrational perturbation theory calculations is introduced. This approach makes use of the corrections to the wave functions that are obtained from non-degenerate perturbation theory calculations to identify spaces of states that must be treated with degenerate perturbation theory. Pairs of states are considered to be in resonance if the magnitude of expansion coefficients in the corrections to the wave functions in the non-degenerate perturbation theory calculation is greater than a specified threshold, χmax. This approach is applied to calculations of the vibrational spectra of CH4, H2CO, HNO3, and cc-HOONO. The question of how the identified resonances depend on the value of χmax and how the choice of the resonance spaces affects the calculated vibrational spectrum is further explored for H2CO. The approach is also compared to the Martin test [J. M. L. Martin et al., J. Chem. Phys. 103, 2589-2602 (1995)] for calculations of the vibrational spectra of H2CO and cc-HOONO.

The perturbation theory of matrix mechanics based on its canonical transformations

We elucidate that the canonical transformations of matrix mechanics are just the transformations of different representations of the standard quantum mechanics, reproduce the non-degenerate, degenerate and time-dependent perturbation theory in three men’s paper, we solve the mystery of time-dependent perturbation theory of matrix mechanics. The Kramers’ dispersion formula from time-dependent perturbation theory of matrix mechanics is also derived from that of wave mechanics.

Non-equilibrium random walks on multiplex networks

We introduce a non-equilibrium discrete-time random walk model on multiplex networks, in which at each time step the walker first undergoes a random jump between neighboring nodes in the same layer, and then tries to hop from one node to one of its replicas in another layer. We derive the so-called supra-Markov matrix that governs the evolution of the occupation probability of the walker. The occupation probability at stationarity is different from the weighted average over the counterparts on each layer, unless the transition probabilities between layers vanish. However, they are approximately equal when the transition probabilities between layers are very small, which is given by the first-order degenerate perturbation theory. Moreover, we compute the mean first passage time (MFPT) and the graph MFPT (GrMFPT) that is the average of the MFPT over all pairs of distinct nodes. Interestingly, we find that the GrMFPT can be smaller than that of any layer taken in isolation. The result embodies the advantage of global search on multiplex networks.

Tricritical curve of massive chiral Gross-Neveu model with isospin

We reconsider the two-flavor version of the massive, chiral Gross-Neveu model in 1+1 dimensions. Its phase diagram as a function of baryon chemical potential, isospin chemical potential and temperature has previously been explored. We recapitulate the results, adding the missing tricritical curves. They can be determined exactly by extending the standard stability analysis, using fourth order almost degenerate perturbation theory. Results for three different bare masses are presented and discussed.

Quantum phase transitions in the K -layer Ising toric code

We investigate the quantum phase diagram of the $K$-layer Ising toric code corresponding to $K$ layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays ${\mathbb{Z}}_{2}^{K}$ topological order originating from the toric codes in each layer, the system shows ${\mathbb{Z}}_{2}$ topological order in the high-Ising limit. The latter is demonstrated for general $K$ by deriving an effective low-energy model in $K\mathrm{th}$-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases $K=3$ and $K=4$ we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both $K$ separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.

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.

Nonrelativistic Conformal Invariance in Mesoscopic Two-Dimensional Fermi Gases.

Two-dimensional Fermi gases with universal short-range interactions are known to exhibit a quantum anomaly, where a classical scale and conformal invariance is broken by quantum effects at strong coupling. We argue that in a quasi two-dimensional geometry, a conformal window remains at weak interactions. Using degenerate perturbation theory, we verify the conformal symmetry by computing the energy spectrum of mesoscopic particle ensembles in a harmonic trap, which separates into conformal towers formed by so-called primary states and their center-of-mass and breathing-mode excitations, the latter having excitation energies at precisely twice the harmonic oscillator energy. In addition, using Metropolis importance sampling, we compute the hyperradial distribution function of the many-body wave functions, which are predicted by the conformal symmetry in closed analytical form. The weakly interacting Fermi gas constitutes a system where the nonrelativistic conformal symmetry can be revealed using elementary methods, and our results are testable in current experiments on mesoscopic Fermi gases.

Instabilities of spin-1 Kitaev spin liquid phase in presence of single-ion anisotropies

We study the spin-one Kitaev model on the honeycomb lattice in the presence of single-ion anisotropies. We consider two types of single ion anisotropies: A ${D}_{111}$ anisotropy which preserves the symmetry between $X, Y$, and $Z$ bonds but violates flux conservation and a ${D}_{100}$ anisotropy that breaks the symmetry between $X, Y$, and $Z$ bonds but preserves flux conservation. We use series expansion methods, degenerate perturbation theory, and exact diagonalization to study these systems. Large positive ${D}_{111}$ anisotropy leads to a simple product ground state with conventional magnonlike excitations, while large negative ${D}_{111}$ leads to a broken symmetry and degenerate ground states. For both signs there is a phase transition at a small $|{D}_{111}|\ensuremath{\approx}0.12$ separating the more conventional phases from the Kitaev spin liquid phase. With large ${D}_{100}$ anisotropy, the ground state is a simple product state, but the model lacks conventional dispersive excitations due to the large number of conservation laws. Large negative ${D}_{100}$ leads to decoupled one-dimensional systems and many degenerate ground states. No evidence of a phase transition is seen in our numerical studies at any finite ${D}_{100}$. Convergence of the series expansion extrapolations all the way to ${D}_{100}=0$ suggests that the nontrivial Kitaev spin liquid is a singular limit of this type of single-ion anisotropy going to zero, which also restores symmetry between the $X, Y$, and $Z$ bonds.

Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.