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. B, Condensed matter

Rigorous bounds for ground-state properties of correlated Fermi systems

Roser Valent�, Claudius Gros, P. J. Hirschfeld, W. Stephan

Published Dec 15, 1991• 14 citations
Publisher

Abstract

We show that upper and lower bounds on the ground-state energy of models describing correlated Fermi systems may be combined to produce bounds on the ground-state magnetization and chemical potential. Such bounds are obtainable through standard variational techniques and through recently developed methods involving exact diagonalization of finite-size clusters. For the Hubbard model on the square lattice, we give rigorous bounds for the magnetization at nonzero magnetic field B and for the chemical potential at nonzero hole density 1-n. The quality of these bounds degrades as B\ensuremath{\rightarrow}0 and n\ensuremath{\rightarrow}1, precluding rigorous statements about the stability of the ferromagnetic state or the existence of a Mott-Hubbard gap. Nevertheless, the tendency towards large-U ferromagnetism and localization is evident. We discuss ways of improving these bounds, including the use of kinetic frustration, nonuniform clusters, and averaging over boundary conditions.

DOI
10.1103/physrevb.44.13203

Related papers

Rigorous lower bounds on the ground-state energy of correlated Fermi systems

Roser Valent�, P. J. Hirschfeld, J C Anglès d'Auriac

We show that the thermodynamic potential E-\ensuremath{\mu}N of an itinerant interacting Fermi system is bounded below by the sum of the thermodynamic potentials of small subsystems into which the infinite system is partitioned. Exact lower bounds for the energy in the thermodynamic limit are therefore easily obtainable by diagonalization of small clusters. In one dimension, these bounds are remarkably close to Bethe-ansatz results for the Hubbard model. We present lower bounds for the two-dimensional Hubbard and t-J Hamiltonians. Such bounds may serve as rigorous tests for approximate treatments of this class of problems.

Physical review. B, Condensed matter•Aug 15, 1991•9 citations

Bounds for the ground-state energy in the Emery model determined by cluster calculations

Thomas Wittmann, Joachim Stolze

We present simple upper and lower bounds for the ground-state energy of the Emery model. These bounds are derived by partitioning the full Hamiltonian into a sum of sub-Hamiltonians that may be treated with the Lanczos algorithm. The energy bounds are used to derive bounds on the magnetization as a function of magnetic field and the hole density as a function of chemical potential. We improve upon the recent variational calculation of Carstensen and Dichtel [Phys. Rev. B 46, 11 040 (1992)], which suggested the existence of phase separation. Our results contain no evidence for phase separation.

Physical review. B, Condensed matter•Aug 1, 1993•7 citations

Phase transitions and exact ground-state properties of the one-dimensional Hubbard model in a magnetic field

Chia‐Wei Yang, ARMEN N. KOCHARIAN, Y. L. Chiang

The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.

Journal of Physics Condensed Matter•Aug 2, 2000•24 citations

Ground-state properties of theU=∞ Hubbard model on a 4×4 cluster

G. Chiappe, Enrique Louis, J. Gala´n +2 more

We analyze the ground-state properties of the U=\ensuremath{\infty} Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster using a Lanczos technique. The quantum numbers of the ground-state, its momentum distribution, the magnetic structure factor, some 〈n${\mathrm{\^{}}}_{\mathbf{k}}$n${\mathrm{\^{}}}_{\mathbf{k}\ensuremath{'}}$〉 dynamic correlation functions, and the overlap of the ground state with the noninteracting one (both before and after a Gutzwiller projection) are given for all fillings. As the number of electrons approaches half-filling the system undergoes a transition from a conventional ground state that overlaps appreciably with the noninteracting one to a new ground state that shows local ferromagnetic correlations and negligible overlap with the corresponding U=0 ground state. A detailed analysis of the wave function corresponding to the ground state of 14 electrons (two holes) proves the existence of a finite shift of the Fermi surface. Our results support the existence of a stable magnetic phase at small but finite hole doping. This new magnetic phase is locally ferromagnetic although the total spin is not maximum.

Physical review. B, Condensed matter•Dec 1, 1993•13 citations

Exact-diagonalization study of the Hubbard model with nearest-neighbor repulsion

Y. Ohta, Kenji Tsutsui, Wataru Koshibae +1 more

An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.

Physical review. B, Condensed matter•Nov 1, 1994•40 citations

Quantum Many-body Bootstrap

Xizhi Han

A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.

arXiv:2006.06002•Jun 10, 2020•6 citations
We show that upper and lower bounds on the ground-state energy of models describing correlated Fermi systems may be combined to produce bounds on the ground-state magnetization and chemical potential. Such bounds are obtainable through standard variational techniques and through recently developed methods involving exact diagonalization of finite-size clusters. For the Hubbard model on the square lattice, we give rigorous bounds for the magnetization at nonzero magnetic field B and for the chemical potential at nonzero hole density 1-n. The quality of these bounds degrades as B\ensuremath{\rightarrow}0 and n\ensuremath{\rightarrow}1, precluding rigorous statements about the stability of the ferromagnetic state or the existence of a Mott-Hubbard gap. Nevertheless, the tendency towards large-U ferromagnetism and localization is evident. We discuss ways of improving these bounds, including the use of kinetic frustration, nonuniform clusters, and averaging over boundary conditions.
We show that the thermodynamic potential E-\ensuremath{\mu}N of an itinerant interacting Fermi system is bounded below by the sum of the thermodynamic potentials of small subsystems into which the infinite system is partitioned. Exact lower bounds for the energy in the thermodynamic limit are therefore easily obtainable by diagonalization of small clusters. In one dimension, these bounds are remarkably close to Bethe-ansatz results for the Hubbard model. We present lower bounds for the two-dimensional Hubbard and t-J Hamiltonians. Such bounds may serve as rigorous tests for approximate treatments of this class of problems.
We present simple upper and lower bounds for the ground-state energy of the Emery model. These bounds are derived by partitioning the full Hamiltonian into a sum of sub-Hamiltonians that may be treated with the Lanczos algorithm. The energy bounds are used to derive bounds on the magnetization as a function of magnetic field and the hole density as a function of chemical potential. We improve upon the recent variational calculation of Carstensen and Dichtel [Phys. Rev. B 46, 11 040 (1992)], which suggested the existence of phase separation. Our results contain no evidence for phase separation.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.
We analyze the ground-state properties of the U=\ensuremath{\infty} Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster using a Lanczos technique. The quantum numbers of the ground-state, its momentum distribution, the magnetic structure factor, some 〈nˆk{\mathrm{\^{}}}_{\mathbf{k}}ˆk​n${\mathrm{\^{}}}_{\mathbf{k}\ensuremath{'}}$〉 dynamic correlation functions, and the overlap of the ground state with the noninteracting one (both before and after a Gutzwiller projection) are given for all fillings. As the number of electrons approaches half-filling the system undergoes a transition from a conventional ground state that overlaps appreciably with the noninteracting one to a new ground state that shows local ferromagnetic correlations and negligible overlap with the corresponding U=0 ground state. A detailed analysis of the wave function corresponding to the ground state of 14 electrons (two holes) proves the existence of a finite shift of the Fermi surface. Our results support the existence of a stable magnetic phase at small but finite hole doping. This new magnetic phase is locally ferromagnetic although the total spin is not maximum.
An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.
A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.