New solvable many-body problems in the plane

Abstract

We revisit an integrable (indeed, superintegrable and solvable) many-body model introduced almost two decades ago by Gibbons and Hermsen and by Wojciechowski, and we modify it so that its generic solutions are all isochronous (namely, completely periodic with fixed period). We then show how this model (or rather the more basic dynamical system that underlies its solvable character, and other avatars of it) can be conveniently reinterpreted as (rotation-invariant) models in the plane; and we thereby present several new (solvable, isochronous and rotation-invariant) many-body problems in the plane.     

Pairing mechanism in the doped Hubbard antiferromagnet: the 4×4 model as a test case

We introduce a local formalism, in terms of eigenstates of number operators, having well defined point symmetry, to solve the Hubbard model at weak coupling on a N × N square lattice (for even N). The key concept is that of W = 0 states, that are the many-body eigenstates of the kinetic energy with vanishing Hubbard repulsion. At half filling, the wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. Further, we state a general theorem which allows to find all the W = 0 pairs (two-body W = 0 singlet states). We show that, in special cases, this assigns the ground state symmetries at least in the weak coupling regime. The N = 4 case is discussed in detail. To study the doped half filled system, we enhance the group theory analysis of the 4×4 Hubbard model introducing an Optimal Group which explains all the degeneracies in the one-body and many-body spectra. We use the Optimal Group to predict the possible ground state symmetries of the 4×4 doped antiferromagnet by means of our general theorem and the results are in agreement with exact diagonalization data. Then we create W = 0 electron pairs over the antiferromagnetic state. We show analitycally that the effective interaction between the electrons of the pairs is attractive and forms bound states. Computing the corresponding binding energy we are able to definitely predict the exact ground state symmetry. Received 24 October 2000     

Properties of then-body correlation functions near the liquid-gas critical point. Correlation inequalities

We investigate the behavior of the many-body correlation functions in the vicinity of the gas-liquid critical point. We use the framework of the liquid state theory and, accordingly, no reference to an effective Landau-Ginzburg Hamiltonian is made. The critical condition is introduced by means of the equation of state. From the Baxter equation relating the many-body correlation functionsh(n) andh(n+1), we find that the integrals of all theh(n) diverge at the critical point. Then we present strong arguments and this leads to GKS-like inequalities, under some limiting conditions: the interparticle distances must be large and the thermodynamic state of the system must be close to the critical point. In order to get these inequalities, an upper bound forh(n) is obtained. Particular attention must be paid to the fact that the usual asymptotic approximations of the liquid state theory are no longer valid.     

Nucleation of Instability of the Meissner State of 3-Dimensional Superconductors

This paper concerns a nonlinear partial differential system in a 3-dimensional domain involving the operator curl², which is a simplified model used to examine nucleation of instability of the Meissner state of a superconductor as the applied magnetic field reaches the superheating field. We derive a priori C ^2+α estimates for a weak solution H, the curl of the magnetic potential, and determine the location of the maximal points of |curlH| which correspond to the nucleation of instability of the Meissner state. We show that, if the penetration length is small, the solution exhibits a boundary layer. If the applied magnetic field is homogeneous, |curlH| is maximal around the points on the boundary where the applied field is tangential to the surface.     

Efficient electronic structure theory via hierarchical scale-adaptive coupled-cluster formalism: I. Theory and computational complexity analysis

ABSTRACT

A novel reduced-scaling, general-order coupled-cluster approach is formulated by exploiting hierarchical representations of many-body tensors, combined with the recently suggested formalism of scale-adaptive tensor algebra. Inspired by the hierarchical techniques from the renormalisation group approach, H/H²-matrix algebra and fast multipole method, the computational scaling reduction in our formalism is achieved via coarsening of quantum many-body interactions at larger interaction scales, thus imposing a hierarchical structure on many-body tensors of coupled-cluster theory. In our approach, the interaction scale can be defined on any appropriate Euclidean domain (spatial domain, momentum-space domain, energy domain, etc.). We show that the hierarchically resolved many-body tensors can reduce the storage requirements to O(N), where N is the number of simulated quantum particles. Subsequently, we prove that any connected many-body diagram consisting of a finite number of arbitrary-order tensors, e.g. an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN) floating-point operations. On top of that, we suggest an additional approximation to further reduce the computational complexity of higher order coupled-cluster equations, i.e. equations involving higher than double excitations, which otherwise would introduce a large prefactor into formal O(NlogN) scaling.     

Exceptional Point of the Neutral Heavy Higgs System H2–H3

We study the singularity of the surface that represents the masses of the isolated doublet of heavy, neutral Higgs bosons, H ₂–H ₃, in a toy model based on the MSSM with CP violation, in parameter space. These two heavy, neutral Higgs bosons are coherent and, for large values of the masses, nearly degenerate. In this scenario, mixing between the mass eigenstates of the H ₂–H ₃ system could be very large and exact degeneracy is possible. As function of the Lagrangian parameters, the physical mass of the doublet has an algebraic branch point of rank one at the exceptional point where the two masses are equal. The real and imaginary parts of the masses in the doublet have branch cuts that start at the same branch point but extend in opposite directions in parameter space. Associated with this branch point, the propagator of the mixing doublet of neutral heavy Higgs bosons has a double pole in the complex s-plane of the energy squared. We computed the mass surface of the isolated doublet of H ₂–H ₃ bosons as function of the Lagrangian parameters in the neighbourhood of the exceptional point in a toy model of the system H ₂–H ₃. We also computed the trajectories of the poles of the transition matrix for values of the Lagrangian parameters close to the exceptional point and explained the characteristic change of identity seen in these trajectories in the s-plane as a manifestation of the topology of the two-sheeted mass surfaces in the space of Lagrangian parameters.     

Multiply connected Fermi sphere and fermion condensation

We examine the structure of the ground state of a homogeneous Fermi liquid beyond the instability point of the Fermi-like quasiparticle momentum distribution in the effective-functional method with a strong repulsive effective interaction. A numerical study of the initial stage of rearrangement of the ground state, based on a simple effective functional, showed that there exists a temperature T ₀, above which the behavior of the system is the same as in the theory of fermion condensation, and for T<T ₀ the scenario of rearrangement of the ground state is different. At low temperatures an intermediate structure arises, with a multiply connected quasiparticle momentum distribution. The transition of this structure with growth of the coupling constant to a state with a fermion condensate is discussed. Zh. éksp. Teor. Fiz. 114, 2078–2088 (December 1998)     

ISOCHRONOUS DYNAMICAL SYSTEM AND DIOPHANTINE RELATIONS I

We identify a solvable dynamical system — interpretable to some extent as a many-body problem — and point out that — for an appropriate assignment of its parameters — it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data). We then identify its equilibrium configurations and investigate its behavior in their neighborhood. We thereby identify certain matrices — of arbitrary order — whose eigenvalues are all rational numbers: a Diophantine finding.     

A path integral approach to effective non-linear medium

In this article, we propose a new method to compute the effective properties of non-linear disordered media. We use the fact that the effective constants can be defined through the minimum of an energy functional. We express this minimum in terms of a path integral allowing us to use many-body techniques. We obtain the perturbation expansion of the effective constants to second order in disorder, for any kind of non-linearity. We apply our method to the case of strong non-linearities (i.e. , where is fluctuating from point to point), and to the case of weak non-linearity (i.e. where and fluctuate from point to point). Our results are in agreement with previous ones, and could be easily extended to other types of non-linear problems in disordered systems. Received: 13 May 1998 / Accepted: 27 July 1998     

How to test for diagonalizability: the discretized PT-invariant square-well potential

Given a non-Hermitian matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note explains how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upon discretizing the configuration space, the system is described by a matrix of dimension three which turns out not to be diagonalizable for a critical strength of the interaction. The systems develops a three-fold degenerate eigenvalue, and two of the three eigenfunctions disappear at this exceptional point, giving a difference between the algebraic and geometric multiplicity of the eigenvalue equal to two. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

The Scattering Problem in PT‐Symmetric Periodic Structures of 1D Two‐Material Waveguide Networks

A PT‐symmetric periodic structure with two‐material waveguide networks is constructed. In this study, how changing the number of cells affects the transmission properties is investigated. The results show that the PT‐unbroken (broken) region of the system is only determined by the cell structure, regardless of the number of unit cells. This means that any system has the same exceptional points (EPs), regardless of the number of cells and as long as the cell structure is consistent. In addition, it is confirmed that the coherent perfect absorbers and lasers (CPA lasers) occur in our model. The transfer matrix method is used to derive a sufficient condition for achieving the CPA laser point. A simple, effective formula for predicting the CPA laser state in an N unit cell system is derived.     

Asymptotically isochronous systems

Abstract

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear (“harmonic oscillator”) force.     

Algebraic connectivity analysis in molecular electronic structure theory I: coulomb potential,tensor connectivity, ε-approximation

In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.     

Instability of layered quantum liquids: 1. Local-field correction in superlattices

For a superlattice with periodd the Singwi et al. (Phys. Rev.176, 589 (1968)) approach for the local-field correction and the static structure factor is formulated. With two approximations we reduce the resulting three-dimensional integral-equation into a one-dimensional integral-equation. For the local-field correction we present analytical results for small wave numbers and large wave numbers. An expression of Hubbard-typ is derived for the local-field correction. Explicit results for boson superlattices and electron superlattice are given. A charge-density-wave instability in a layered Bose condensate withr _s>r_scd^3/4 is discovered.r _s is the small parameter of the random-phase-approximation. The charge-density-wave instability is due to a many-body anomaly (short-range correlations) in layered structures and is a general property of layered quantum liquids. We find the charge-density-wave instability in a layered electron gas forr _s>r_scd. Double-quantum-well structures are also considered. The effects of a finite well width is calculated. The general implications of the charge-density-wave instability for microscopically layered quantum liquids are pointed out.     

Phases of wave functions and level repulsion

Avoided level crossings are associated with exceptional points which are the singularities of the spectrum and eigenfunctions, when considered as functions of a complex coupling parameter. It is shown that the wave function of one state changes sign but not the other, if the exceptional point is encircled in the complex plane. An experimental setup is suggested where this peculiar phase change could be observed. Received: 7 January 1999 / Received in final form: 15 March 1999     

The influence of many-body interactions on the de Haas-van Alphen effect

The de Haas-van Alphen (dHvA) effect, or Landau quantum oscillatory magnetization of metals, has been widely used to explore the single-particle aspects of electrons in metals with the aim of determining their Fermi surfaces. Its role in studying many-body effects in metals is less familiar, even though the influence of such interactions is well known. We present a general field-theoretic approach to this problem which shows that the paradigm for understanding the influence of many-body interactions in the dHvA effect should be shifted from the intuitively reasonable but potentially misleading arguments based on the electron self-energy on the real energy axis to an analysis of the self-energy along the imaginary energy axis. When viewed in this way, the dHvA effect assumes the role of a many-body self-energy filter in which the real part of the self-energy renormalizes the dHvA frequency while the imaginary part renormalizes independently the dHvA amplitude. We obtain a general theory for the dHvA effect in an interacting system which preserves the structure of the original non-interacting theory of Lifshitz and Kosevich. We then apply this extended Lifshitz-Kosevich theory to the analysis of several problems of interest, including electron-electron and electron-phonon interactions, heavy fermions and type II superconductors.     

A Remark on the Notion of Independence of Quantum Integrals of Motion in the Thermodynamic Limit

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, surprisingly enough, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by dwelling upon an important aspect of quantum integrability—the notion of independence of quantum integrals of motion (QIMs). We point out that a widely accepted definition of functional independence of QIMs is flawed, and suggest a new definition. Our study is motivated by the PXP model—a model of N spins 1/2 possessing an extensive number of binary QIMs. The number of QIMs which are independent according to the common definition turns out to be equal to the number of spins, N. A common wisdom would then suggest that the system is completely integrable, which is not the case. We discuss the origin of this conundrum and demonstrate how it is resolved when a new definition of independence of QIMs is employed.

     

Different ways of dealing with Compton scattering and positron annihilation experimental data

Different ways of dealing with one-dimensional (1D) spectra, measured e.g., in the Compton scattering or angular correlation of positron annihilation radiation (ACAR) experiments, are presented. Using the example of divalent hexagonal close packed metals, we show what kind of information on the electronic structure one can get from 1D profiles interpreted in terms of either 2D or 3D momentum densities.2D and 3D densities are reconstructed from merely two and seven 1D profiles, respectively. Applied reconstruction techniques are particular solutions of the Radon transform in terms of orthogonal Gegenabauer polynomials. We propose their modification connected with so-called two-step reconstruction.The analysis is performed both in the extended p and reduced k zone schemes. It is demonstrated that if the positron wave function or many-body effects are strongly momentum dependent, analysis of 2D densities folded into k space may lead to wrong conclusions concerning the Fermi surface. In the case of 2D ACAR data in Mg, we found very strong many-body effects. PACS 71.18.+y; 13.60.Fz; 87.59.Fm     

Monte Carlo calculations of angular momentum projected many-body matrix elements in the Pairing Plus Quadrupole Model

We extend the recently presented formalism for Monte Carlo calculations of the partition function, for both even and odd particle number systems (Phys. Rev. C 59, 2500 (1999)), to the calculation of many-body matrix elements of the type <ψ| e ^{- βℋ}|ψ> where |ψ> is a many-body state with a definite angular momentum, parity, neutron and proton numbers. For large β such matrix elements are dominated by the lowest eigenstate of the many-body Hamiltonian ℋ, corresponding with a given angular momentum parity and particle number. Emphasis is placed on odd-mass nuclei. Negligible sign fluctuations in the Monte Carlo calculation are found provided the neutron and proton chemical potentials are properly adjusted. The formalism is applied to the J ^π = 0⁺ state in ¹⁶⁶ Er and to the J ^π = 9/2^-, 13/2⁺, 5/2^- states in ¹⁶⁵ Er using the pairing-plus-quadrupole model. Received: 28 April 2000 / Accepted: 20 September 2000     

Two-body exceptional points in open dissipative systems

We study two-body non-Hermitian physics in the context of an open dissipative system depicted by the Lindblad master equation.Adopting a minimal lattice model of a handful of interacting fermions with single-particle dissipation,we show that the non-Hermitian effective Hamiltonian of the master equation gives rise to two-body scattering states with state-and interaction-dependent parity-time transition.The resulting two-body exceptional points can be extracted from the trace-preserving density-matrix dynamics of the same dissipative system with three atoms.Our results not only demonstrate the interplay of parity-time symmetry and interaction on the exact few-body level,but also serve as a minimal illustration on how key features of non-Hermitian few-body physics can be probed in an open dissipative many-body system.