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Physical Review Letters

Interacting electrons model exactly solvable in any dimension

Arianna Montorsi, Mario Rasetti

Published Mar 18, 1991• 28 citations
Publisher

Abstract

When processes not conserving the double occupation number are inhibited, one obtains a model like the Kivelson-Schrieffer-Su-Heeger extension of the Hubbard model. We solve this model exactly in any number of dimensions. The system free energy is given for any fixed filling. The ground-state energy and double-occupancy order parameter are discussed for d=2.

DOI
10.1103/physrevlett.66.1383

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When processes not conserving the double occupation number are inhibited, one obtains a model like the Kivelson-Schrieffer-Su-Heeger extension of the Hubbard model. We solve this model exactly in any number of dimensions. The system free energy is given for any fixed filling. The ground-state energy and double-occupancy order parameter are discussed for d=2.
The Hubbard and Su-Schrieffer-Heeger (SSH) Hamiltonians are iconic models for understanding the qualitative effects of electron-electron and electron-phonon interactions, respectively. In the two-dimensional square lattice Hubbard model at half filling, the on-site Coulomb repulsion UUU between up and down electrons induces antiferromagnetic (AFM) order and a Mott insulating phase. On the other hand, for the SSH model, there is an AFM phase when the electron-phonon coupling $\ensuremath{\lambda}$ is less than a critical value ${\ensuremath{\lambda}}_{c}$ and a bond order wave when $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$. In this Letter, we perform numerical studies on the square lattice optical Su-Schrieffer-Heeger-Hubbard Hamiltonian, which combines both interactions. We use the determinant quantum Monte Carlo method which does not suffer from the fermionic sign problem at half filling. We map out the phase diagram and find that it exhibits a direct first-order transition between an antiferromagnetic phase and a bond-ordered wave as $\ensuremath{\lambda}$ increases. The AFM phase is characterized by two different regions. At smaller $\ensuremath{\lambda}$ the behavior is similar to that of the pure Hubbard model; the other region, while maintaining long-range AFM order, exhibits larger kinetic energies and double occupancy, i.e., larger quantum fluctuations, similar to the AFM phase found in the pure SSH model.
The ground-state properties of the half-filled Hubbard model are investigated in the framework of lattice density functional theory. The single-particle density matrix ${\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}$ is regarded as the central variable of the many-body problem, where iii and jjj refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.
Despite extensive studies on the one-dimensional Su-Schrieffer-Heeger-Hubbard (SSHH) model, the variant incorporating second-nearest neighbor hopping remains largely unexplored. Here, the topological classification of the extended SSH model is analyzed within the framework of the BDI symmetry class, using the winding number as the corresponding topological invariant. We investigate the ground-state properties of this extended SSHH model using the constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) method. We show that this model exhibits rich topological phases, characterized by robust edge states against interaction. We quantify the properties of these edge states by analyzing spin correlation and second-order Rényi entanglement entropy. The system exhibits long-range spin correlation and near-zero Rényi entropy at half-filling. Besides, there is an anti-ferromagnetic order at quarter-filling. Interestingly, an external magnetic field disrupts this anti-ferromagnetic order, restoring long-range spin correlation and near-zero Rényi entropy. Furthermore, our work provides a paradigm for studying topological properties in large interacting systems via the CP-AFQMC algorithm.