Gauge Invariance and Mass.

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrodinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Moller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pade and quadratic approximants) that can improve the overall accuracy of the Moller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

https://iopscience.iop.org/article/10.1088/1361-648X/abe795/pdf

Perturbation theory in the complex plane: exceptional points and where to find them.

We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration.

https://www.mdpi.com/2075-1680/9/4/126/pdf

Nonlinear Approximations to Critical and Relaxation Processes

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

From Asymptotic Series to Self-Similar Approximants

The generalized alternating method of Schwarz can be presented as an infinite sequence of all the mutual interactions between inclusions in the boundary value problem stated for a composite. Schwarz's method for overlapping domains has been assigned to decomposition methods widely used in pure numerical computations. In the same time, the generalized alternating method for non-overlapping inclusions is convenient in symbolic-numerical computations described in this paper. Implementation of the explicit and implicit schemes yields new approximate analytical formulas for the effective properties of dispersed composites. Moreover, the exactly derived precision of their validity in concentration and in contrast parameter is established. The first order iteration corresponds to the famous Maxwell's self-consistent approach. 

Effective properties of two-dimensional dispersed composites. Part I. Schwarz's alternating method

We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Pade-techniques and their various post-Pade extensions by means of root and factor approximants. Factor approximants are applied to finding critical points. Roots are employed within the context of finding critical index. Additive self-similar approximants are discussed and DLog additive recursive approximants are introduced as their generalization. They are applied to the problem of interpolation. Several examples of interpolation are considered.

Padé and Post-Padé Approximations for Critical Phenomena

Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor approximants are applied together with various natural optimization conditions of non-perturbative nature. The role of control parameter is played by the critical index by itself. The minimal derivative condition imposed on critical amplitude appears to bring the most reasonable, uniquely defined results. The minimal difference condition also imposed on amplitudes produces upper and lower bound on the critical index. While one of the bounds is close to the result from the minimal difference condition, the second bound is determined by the non-optimized factor approximant. One would expect that for the minimal derivative condition to work well, the bounds determined by the minimal difference condition should be not too wide. In this sense the technique of optimization presented above is self-consistent, since it automatically supplies the solution and the bounds. In the case of effective viscosity of passive suspensions the bounds could be found that are too wide to make any sense from either of the solutions. Other optimization conditions imposed on the factor approximants, lead to better estimates for the critical index for the effective viscosity. The optimization is based on equating two explicit expressions following from two different definitions of the critical index, while optimization parameter is introduced as the trial third-order coefficient in the expansion.

Optimized Factor Approximants and Critical Index

We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to the indeterminate (divergent) problem my means of power transformation. Most challenging indeterminate for the continued roots problems of calculating critical amplitudes, can be successfully attacked by performing proper power transformation to be found from the optimization imposed on the parameters of power transform. The self-similar continued roots were derived by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the fundamental symmetry principle of functional self-similarity, realized constructively in the space of approximations. Our approach to the solution of the indeterminate problem is to replace it with the determinate problem, but with some unknown control parameter b in place of the known critical index β. From optimization conditions b is found in the way making the problem determinate and convergent. The index β is hidden under the carpet and replaced by b. The idea is applied to various, mostly quantum-mechanical problems. In particular, the method allows us to solve the problem of Bose-Einstein condensation temperature with good accuracy.

Simon Gluzman

Papers

Padé and Post-Padé Approximations for Critical Phenomena

Optimized Factor Approximants and Critical Index

Continued Roots, Power Transform and Critical Properties