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

arXiv:cond-mat/9910461 · Physical review. B, Condensed matter

Convergent method for calculating the properties of many interacting electrons

Roger Haydock

Published Mar 15, 2000• 10 citations
arXivHTMLPDF

Abstract

A method is presented for calculating binding energies and other properties of extended interacting system using the projected density of transitions (PDOT) which is the probability distribution for transitions of different energies induced by a given localized operator, the operator on which the transitions are projected. It is shown that the transition contributing to the PDOT at each energy is the one which disturbs the system least, and so, by projecting on appropriate operators, the binding energies of equilibrium electronic states and the energies of their elementary excitations can be calculated. The PDOT may be expanded as a continued fraction by the recursion method, and as in other cases the continued fraction converges exponentially with the number of arithmetic operations, independent of the size of the system, in contrast to other numerical methods for which the number of operations increases with system size to maintain a given accuracy. These properties are illustrated with a calculation of the binding energies and zone-boundary spin-wave energies for an infinite spin-$\frac{1}{2}$ Heisenberg chain, which is compared with analytic results for this system and extrapolations from finite rings of spins.

DOI
10.1103/physrevb.61.7953

Related papers

A recursive solution of Heisenberg's equation and its interpretation

James F. Annett, W Matthew, W. M. C. Foulkes +1 more

We present the generalization of the recursion method of Haydock and co-workers to systems of many interacting particles. This new method has close similarities to the memory function or Mori formalism, but with some important differences. Heisenberg's equation for the time evolution of a microscopic operator is recursively transformed into a tridiagonal matrix equation. This equation resolves the operator into components corresponding to transitions of different energies. The projected spectrum of transitions has a continued fraction expansion given by the elements of the tridiagonal matrix. We show that for an appropriate choice of inner product this density of transitions obeys a generalization of the black body theorem of electromagnetism, in that it is exponentially insensitive to distant parts of the system. This implies that the projected density of transitions is computationally stable and can be calculated even in macroscopic many-body systems. We argue that the physical content of the density of transitions is determined by the nature of its singular points, such as discrete transitions, continuous spectrum, band edges and van Hove singularities.

Journal of Physics Condensed Matter•Aug 8, 1994•13 citations

Many-electron systems with fractional electron number and spin: Exact properties above and below the equilibrium total spin value

Yuli Goshen, Eli Kraisler

The description of many-electron systems with a fractional electron number, Ntot, and fractional z-projection of the spin, Mtot, is of great importance in physical chemistry, solid-state physics, and materials science. In this study, we analyze the fundamental question of what the ensemble ground state of a general, finite, many-electron system at zero temperature is, with a given Ntot and Mtot, distinguishing between low- and high-spin cases (separated by the boundary spin MB). For the low-spin case, the general form of the ensemble ground state has been rigorously derived in Goshen and Kraisler [J. Phys. Chem. Lett. 15, 2337 (2024)], generalizing the piecewise linearity and the flat-plane conditions for many-electron systems. Here, we provide an alternative proof for this case, discuss the ambiguity in the description of the ground state, and show that this ambiguity can be removed via maximization of the system's entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. We prove three general properties that characterize the ground state at high spins and narrow down the list of pure states it may consist of. We illustrate the aforementioned properties of high-spin cases by examining the ensemble ground state when Mtot approaches MB from above during the addition of (a fraction of) an up- and down-electron to a given system. Furthermore, we relate the frontier orbital energies of Kohn-Sham (KS) density functional theory (DFT) to total energy differences at high spin values, particularly the ionization potential (IP), the fundamental gap, and the spin flip energies. Analyzing the frontier energies on both sides of each boundary in the total energy profile, where the energy slope changes abruptly, we derive expressions for new derivative discontinuities, which are predicted to appear as jumps in the corresponding KS potentials. In this way, we generalize the well-known IP theorem of DFT to cases with fractional electron number and to cases with high spin. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this study are instrumental for the development of advanced approximations in DFT and other many-electron methods.

The Journal of Chemical Physics•Mar 11, 2026•1 citation

Many body density of states of a system of non interacting spinless fermions

R. Lefèvre, Krissia Zawadzki, Grégoire Ithier

Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.

arXiv:2208.02236•New Journal of Physics•May 25, 2023•3 citations

Density-matrix renormalization-group algorithm for quantum lattice systems with a large number of states per site

Robert J. Bursill

A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain and a dimerized $\mathrm{XY}$ spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.

arXiv:cond-mat/9812349•Physical review. B, Condensed matter•Jul 15, 1999•31 citations

Linked-cluster expansion for Jastrow-type wave functions and its application to the electron-gas problem

James D. Talman

A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like $1+\frac{A}{{r}^{2}}$. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like $1+\frac{A}{{r}^{2}}$ for large $r$. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.

Physical Review A•Oct 1, 1974•24 citations

High-order convergent expansions for quantum many particle systems

Martin P. Gelfand, Rajiv R. P. Singh

We review recent advances in the calculations of high-order convergent expansions for quantum many-particle systems. Calculations for ground state properties, including correlation functions and static susceptibilities, for spin models as well as for models of many fermions, such as Hubbard and Kondo models, are discussed. A historical perspective to the subject is provided. Recently important technical advances have been made in perturbative calculations of the excitation spectra of quantum many-particle systems, which enable the calculation of these spectra to high orders. The method, along with its applications, are explained. Fairly comprehensive, though simplified, algorithms for generating lists of relevant clusters, their lattice embeddings and subclusters are presented. The perturbative recursion relations and their computer implementation are also discussed in detail. A compilation is made of various series expansion studies that have been carried out for condensed matter problems. The scope and limitations of these methods are explained, and several open problems are noted.

Advances In Physics•Jan 1, 2000•120 citations
A method is presented for calculating binding energies and other properties of extended interacting system using the projected density of transitions (PDOT) which is the probability distribution for transitions of different energies induced by a given localized operator, the operator on which the transitions are projected. It is shown that the transition contributing to the PDOT at each energy is the one which disturbs the system least, and so, by projecting on appropriate operators, the binding energies of equilibrium electronic states and the energies of their elementary excitations can be calculated. The PDOT may be expanded as a continued fraction by the recursion method, and as in other cases the continued fraction converges exponentially with the number of arithmetic operations, independent of the size of the system, in contrast to other numerical methods for which the number of operations increases with system size to maintain a given accuracy. These properties are illustrated with a calculation of the binding energies and zone-boundary spin-wave energies for an infinite spin-12\frac{1}{2}21​ Heisenberg chain, which is compared with analytic results for this system and extrapolations from finite rings of spins.
We present the generalization of the recursion method of Haydock and co-workers to systems of many interacting particles. This new method has close similarities to the memory function or Mori formalism, but with some important differences. Heisenberg's equation for the time evolution of a microscopic operator is recursively transformed into a tridiagonal matrix equation. This equation resolves the operator into components corresponding to transitions of different energies. The projected spectrum of transitions has a continued fraction expansion given by the elements of the tridiagonal matrix. We show that for an appropriate choice of inner product this density of transitions obeys a generalization of the black body theorem of electromagnetism, in that it is exponentially insensitive to distant parts of the system. This implies that the projected density of transitions is computationally stable and can be calculated even in macroscopic many-body systems. We argue that the physical content of the density of transitions is determined by the nature of its singular points, such as discrete transitions, continuous spectrum, band edges and van Hove singularities.
The description of many-electron systems with a fractional electron number, Ntot, and fractional z-projection of the spin, Mtot, is of great importance in physical chemistry, solid-state physics, and materials science. In this study, we analyze the fundamental question of what the ensemble ground state of a general, finite, many-electron system at zero temperature is, with a given Ntot and Mtot, distinguishing between low- and high-spin cases (separated by the boundary spin MB). For the low-spin case, the general form of the ensemble ground state has been rigorously derived in Goshen and Kraisler [J. Phys. Chem. Lett. 15, 2337 (2024)], generalizing the piecewise linearity and the flat-plane conditions for many-electron systems. Here, we provide an alternative proof for this case, discuss the ambiguity in the description of the ground state, and show that this ambiguity can be removed via maximization of the system's entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. We prove three general properties that characterize the ground state at high spins and narrow down the list of pure states it may consist of. We illustrate the aforementioned properties of high-spin cases by examining the ensemble ground state when Mtot approaches MB from above during the addition of (a fraction of) an up- and down-electron to a given system. Furthermore, we relate the frontier orbital energies of Kohn-Sham (KS) density functional theory (DFT) to total energy differences at high spin values, particularly the ionization potential (IP), the fundamental gap, and the spin flip energies. Analyzing the frontier energies on both sides of each boundary in the total energy profile, where the energy slope changes abruptly, we derive expressions for new derivative discontinuities, which are predicted to appear as jumps in the corresponding KS potentials. In this way, we generalize the well-known IP theorem of DFT to cases with fractional electron number and to cases with high spin. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this study are instrumental for the development of advanced approximations in DFT and other many-electron methods.
Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.
We review recent advances in the calculations of high-order convergent expansions for quantum many-particle systems. Calculations for ground state properties, including correlation functions and static susceptibilities, for spin models as well as for models of many fermions, such as Hubbard and Kondo models, are discussed. A historical perspective to the subject is provided. Recently important technical advances have been made in perturbative calculations of the excitation spectra of quantum many-particle systems, which enable the calculation of these spectra to high orders. The method, along with its applications, are explained. Fairly comprehensive, though simplified, algorithms for generating lists of relevant clusters, their lattice embeddings and subclusters are presented. The perturbative recursion relations and their computer implementation are also discussed in detail. A compilation is made of various series expansion studies that have been carried out for condensed matter problems. The scope and limitations of these methods are explained, and several open problems are noted.