Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Removal of the singularity in the moment-expansion formalism

Nazakat Ullah

Published Mar 1, 1995• 17 citations
Publisher

Abstract

The expression for the exact ground-state energy as given by Horn and Weinstein [Phys. Rev. D 30, 1256 (1984)] is approximated using a probability distribution function having a \ensuremath{\delta} function and a uniform distribution. The expression obtained up to second order does not have the singularity that one encounters in the connected-moment expansion. It also gives the same kind of expression for the ground-state energy that recently has been given by Mancini, Zhou, and Meier [Int. J. Quant. Chem. 50, 101 (1994)] when the third moment becomes negative.

DOI
10.1103/physreva.51.1808

Related papers

Moment Formalisms applied to a solvable Model with a Quantum Phase Transition. I. Exponential Moment Methods

N. S. Witte, High Energy Physics, R. Shankar

We examine the Ising chain in a transverse field at zero temperature from the point of view of a family of moment formalisms based upon the cumulant generating function, where we find exact solutions for the generating functions and cumulants at arbitrary couplings and hence for both the ordered and disordered phases of the model. In a t-expansion analysis, the exact Horn-Weinstein function E(t) has cuts along an infinite set of curves in the complex Jt-plane which are confined to the left-hand half-plane ℑJt < −1/4 for the phase containing the trial state (disordered), but are not so for the other phase (ordered). For finite couplings the expansion has a finite radius of convergence. Asymptotic forms for this function exhibit a crossover at the critical point, giving the excited state gap in the ground state sector for the disordered phase, and the first excited state gap in the ordered phase. Convergence of the t-expansion with respect to truncation order is found in the disordered phase right up to the critical point, for both the ground state energy and the excited state gap. However convergence is found in only one of the Connected Moments Expansions (CMX), the CMX-LT, and the ground state energy shows convergence right to the critical point again, although to a limited accuracy.

•Jan 1, 2008•11 citations

On the Ground State Energy of the Delta-function Fermi Gas II: Further Asymptotics

Craig A. Tracy, Harold Widom

Trends in mathematics•Jan 1, 2018•11 citations

Exact expression of the ground-state energy for the symmetric anderson model

Norio Kawakami, Ayao Okiji

Physics Letters A•Dec 1, 1981•120 citations

Exact Solution to the Moment Problem for the XY Chain

Nicholas S. Witte

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Néel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn–Weinstein Theorems, but the analytic structure of the generating function <e-tH> in the complex t-plane is quite different from that assumed by the t-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E0~ At-2. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s~ B|s+1|-3/4 exp (C|s+1|1/2). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological, and cannot be applied, while numerical evidence for two of the t-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.

arXiv:cond-mat/9701188•International Journal of Modern Physics B•May 20, 1997•7 citations

Exact expression of the energy gap at first-order quantum phase transitions of a non-stoquastic Hamiltonian

秀稔 西森, Hidetoshi Nishimori

•Oct 1, 2017•0 citations

Connected moments expansion: A new tool for quantum many-body theory

Jerzy Ciosłowski

The connected-moments expansion (CMX) for the ground-state energy is derived from the Horn-Weinstein theorem. This new technique possesses several striking advantages over traditional approaches to calculation of the ground-state energy. Like perturbational theories, the CMX series is conceptually simple but is easier to derive and program and is usually convergent even for very ``crude'' trial kets. Numerical examples reveal potential applicability of CMX for calculation of the correlation energy when the trial ket has the form of multideterminant wave function.

Physical Review Letters•Jan 12, 1987•129 citations
The expression for the exact ground-state energy as given by Horn and Weinstein [Phys. Rev. D 30, 1256 (1984)] is approximated using a probability distribution function having a \ensuremath{\delta} function and a uniform distribution. The expression obtained up to second order does not have the singularity that one encounters in the connected-moment expansion. It also gives the same kind of expression for the ground-state energy that recently has been given by Mancini, Zhou, and Meier [Int. J. Quant. Chem. 50, 101 (1994)] when the third moment becomes negative.
We examine the Ising chain in a transverse field at zero temperature from the point of view of a family of moment formalisms based upon the cumulant generating function, where we find exact solutions for the generating functions and cumulants at arbitrary couplings and hence for both the ordered and disordered phases of the model. In a t-expansion analysis, the exact Horn-Weinstein function E(t) has cuts along an infinite set of curves in the complex Jt-plane which are confined to the left-hand half-plane ℑJt < −1/4 for the phase containing the trial state (disordered), but are not so for the other phase (ordered). For finite couplings the expansion has a finite radius of convergence. Asymptotic forms for this function exhibit a crossover at the critical point, giving the excited state gap in the ground state sector for the disordered phase, and the first excited state gap in the ordered phase. Convergence of the t-expansion with respect to truncation order is found in the disordered phase right up to the critical point, for both the ground state energy and the excited state gap. However convergence is found in only one of the Connected Moments Expansions (CMX), the CMX-LT, and the ground state energy shows convergence right to the critical point again, although to a limited accuracy.
We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Néel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn–Weinstein Theorems, but the analytic structure of the generating function <e-tH> in the complex t-plane is quite different from that assumed by the t-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E0~ At-2. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s~ B|s+1|-3/4 exp (C|s+1|1/2). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological, and cannot be applied, while numerical evidence for two of the t-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.
The connected-moments expansion (CMX) for the ground-state energy is derived from the Horn-Weinstein theorem. This new technique possesses several striking advantages over traditional approaches to calculation of the ground-state energy. Like perturbational theories, the CMX series is conceptually simple but is easier to derive and program and is usually convergent even for very ``crude'' trial kets. Numerical examples reveal potential applicability of CMX for calculation of the correlation energy when the trial ket has the form of multideterminant wave function.