It remains a great challenge in condensed matter physics to develop a method to treat strongly correlated many-body systems with balanced accuracy and efficiency. We introduce an extended Gutzwiller (EG) method incorporating a manifold technique, which builds an effective manifold of the many-body Hilbert space, to describe the ground-state (GS) and excited-state (ES) properties of strongly correlated electrons. We systematically apply an EG projector onto the GS and ES of a non-interacting system. Diagonalization of the true Hamiltonian within the manifold formed by the resulting EG wavefunctions gives the approximate GS and ES of the correlated system. To validate this technique, we implement it on even-numbered fermionic Hubbard rings at half-filling with periodic boundary conditions, and compare the results with the exact diagonalization (ED) method. The EG method is capable of generating high-quality GS and low-lying ES wavefunctions, as evidenced by the high overlaps of wavefunctions between the EG and ED methods. Favorable comparisons are also achieved for other quantities including the total energy, the double occupancy, the total spin and the staggered magnetization. With the capability of accessing the ESs, the EG method can capture the essential features of the one-electron removal spectral function that contains contributions from states deep in the excited spectrum. Finally, we provide an outlook on the application of this method on large extended systems.
It remains a great challenge in condensed matter physics to develop a method to treat strongly correlated many-body systems with balanced accuracy and efficiency. We introduce an extended Gutzwiller (EG) method incorporating a manifold technique, which builds an effective manifold of the many-body Hilbert space, to describe the ground-state (GS) and excited-state (ES) properties of strongly correlated electrons. We systematically apply an EG projector onto the GS and ES of a non-interacting system. Diagonalization of the true Hamiltonian within the manifold formed by the resulting EG wavefunctions gives the approximate GS and ES of the correlated system. To validate this technique, we implement it on even-numbered fermionic Hubbard rings at half-filling with periodic boundary conditions, and compare the results with the exact diagonalization (ED) method. The EG method is capable of generating high-quality GS and low-lying ES wavefunctions, as evidenced by the high overlaps of wavefunctions between the EG and ED methods. Favorable comparisons are also achieved for other quantities including the total energy, the double occupancy, the total spin and the staggered magnetization. With the capability of accessing the ESs, the EG method can capture the essential features of the one-electron removal spectral function that contains contributions from states deep in the excited spectrum. Finally, we provide an outlook on the application of this method on large extended systems.
We investigate an extended version of the periodic Anderson model (the so-called periodic Anderson-Hubbard model) with the aim to understand the role of interaction between conduction electrons in the formation of the heavy-fermion and mixed-valence states. Two methods are used: (i) variational calculation with the Gutzwiller wave function optimizing numerically the ground-state energy and (ii) exact diagonalization of the Hamiltonian for short chains. The f-level occupancy and the renormalization factor of the quasiparticles are calculated as a function of the energy of the f orbital for a wide range of the interaction parameters. The results obtained by the two methods are in reasonably good agreement for the periodic Anderson model. The agreement is maintained even when the interaction between band electrons, Ud, is taken into account, except for the half-filled case. This discrepancy can be explained by the difference between the physics of the one- and higher-dimensional models. We find that this interaction shifts and widens the energy range of the bare f level, where heavy-fermion behavior can be observed. For large-enough Ud this range may lie even above the bare conduction band. The Gutzwiller method indicates a robust transition from Kondo insulator to Mott insulator in the half-filled model, while Ud enhances the quasiparticle mass when the filling is close to half filling.
Numerical exact diagonalisation (ED) of the Hubbard model is traditionally constrained by the exponential growth of the Hilbert space. In this study, we implemented a symmetry-based ED framework to investigate the one-dimensional Hubbard model at half-filling for system sizes up to L=16. By exploiting particle number conservation, spin projection, and translation invariance, we significantly reduced the Hamiltonian dimension, enabling the resolution of ground state energies and dispersion relations without additional approximations. Our results confirmed the transition from a metallic to a correlated insulating regime with increasing on-site repulsion U. Furthermore, extending calculations from ten to sixteen sites revealed scaling trends that asymptotically approached the thermodynamic limit predicted by the Bethe Ansatz (𝐿= ∞). These results served as precise, reproducible benchmarks for validating approximate many-body techniques such as the Density Matrix Renormalisation Group (DMRG).
The physics of strongly correlated materials poses one of the most challenging problems in condensed-matter sciences. Standard approximations applicable to wide classes of materials such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons. Instead, we resort to non-perturbative many-body methods. The calculations are, however, only feasible for rather small model systems. The full Hamiltonian of a real material is approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons is included in an average way by renormalizing the parameters. Realistic calculations of strongly correlated materials need to include sufficiently many of these electronic degrees of freedom. The new generation of massively parallel supercomputers allows for these realistic calculations. However, exploiting their computational power requires newly devised algorithms. As a solver we use the Lanczos method, which needs the full many-body state of the correlated system. It is thus limited by the available main memory. The foremost problem for a distributed-memory implementation is that the multiplication of the Hamiltonian to the many-body state leads to highly non-local memory access patterns. A simple yet important observation leads to an efficient solution: in the kinetic term of the Hamiltonian the electron-spin is conserved. Thus, writing the many-body vector as a matrix v(iup,idn), where the indices label spin-configurations, we find that the hopping term only connects vector elements that differ in one index. Hence, storing entire slices v(iup,:) on one node, the kinetic term for the spin-down electrons is local to that process. After transposing v, the same is true for the hopping of the spin-up electrons. With an efficient matrix transpose, implemented via MPI collectives, we thus obtain a version of the Lanczos method that efficiently well on JUBL, Jülich's Blue Gene/L. Using the code we treated two-orbital Hubbard Models which show interesting Mott-band insulator transitions. Kohn's criterium which determines the phase of a system by ground-state properties alone, namely the response of the ground-state energies to a vector potential, was employed. This vector potential can be equivalently regarded as a change in boundary conditions, requiring complex wave vectors. We found that half-filled systems become metallic in the region between a band and Mott insulating phase. To extrapolate to infinite system sizes and therefore getting access to an arbitrarily high resolution in reciprocal space, we use cluster perturbation theory (CPT). It works by diagonalizing a finite cluster exactly and then treating the hopping between identical clusters perturbatively in strong coupling perturbation theory. This leads to an effectively infinite chain. Due to our efficient Lanczos implementation it is possible to calculate relatively large clusters, minimizing finite-size effects. The combination of our Lanczos code and CPT allowed for a treatment of the the organic metal TTF-TCNQ. We were able to solve a long-standing problem in the interpretation of its experimental spectra. Being an interesting one-dimensional metal TTF-TCNQ has been studied thoroughly for more than 30 years. Theoretical estimates as well as experimental measurements suggest values for the parameters of the Hubbard model t and U. With these values Hubbard model calculations show qualitatively the same features as APRES experiments, namely the signatures of spin-charge separation at the Gamma-point. But in order to fit the experimental data to the numerical calculations t has to be doubled, which translates into a broadening of the spectra. This ad-hoc adjustment is not only unsatisfactory but also the temperature dependence of the spectral function is incorrectly reproduced. We resolved this problem by properly including nearest-neighbor Coulomb interactions. Recent DFT calculations show that indeed the commonly used values of t and U are correct. But Coulomb interactions between electrons on neighboring sites are also significant and must not be neglected. Thus, we studied the effect of the next-neighbor interaction parameter V being at about U/2. Including V in our Lanczos calculations for obtaining the angular-integrated spectral function shows that its effect is to broaden the spectra, comparable to increasing the value of t. Moreover, we found that doubling t indeed mimics the effect of Vapprox U/2. Surprisingly, the broadening of the spectrum as a result of V can be understood in first-order Rayleigh-Schrödinger perturbation theory.
We present a symmetry-projected configuration mixing scheme to describe ground and excited states, with well defined quantum numbers, of the two-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Results for the half-filled $2\ifmmode\times\else\texttimes\fi{}4$, $4\ifmmode\times\else\texttimes\fi{}4$, and $6\ifmmode\times\else\texttimes\fi{}6$ lattices as well as doped $4\ifmmode\times\else\texttimes\fi{}4$ systems, compare well with available results, both exact and from other state-of-the-art approximations. We report spectral functions and density of states obtained from a well-controlled ansatz for the $({N}_{e}\ifmmode\pm\else\textpm\fi{}1)$-electron system. Symmetry projected methods have been widely used for the many-body nuclear physics problem but have received little attention in the solid state community. Given their relatively low (mean-field) computational cost and the high quality of results here reported, we believe that they deserve further scrutiny.
Two of the most popular quantum-mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a one-dimensional (1D) double-well potential, a model of atoms in a quasi-1D optical lattice, or electrons of a hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density-functional theory (DFT) functional and two-site Hubbard models. A 1D adiabatic local-density approximation kernel is constructed for use in time-dependent density-functional theory (TDDFT) and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably to the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with nonadiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.
Extended t-t'-t"-J-U models in which the second-nearest-neighbor hopping (t') and third-nearest-neighbor hopping (t") are included are studied using renormalized mean field theory. The models are meant to be low-energy effective models for the Hubbard models, and hence the Heisenberg exchange integral J and Hubbard repulsion U are related by J = 4t(2)/U. The trial wavefunctions for the ground states are partially Gutzwiller projected Hartree-Fock states. The Gutzwiller projection is implemented by means of a Gutzwiller approximation, and the site double occupancy d is taken as a variational parameter. It is found that a large |t'/t| narrows the band filling range that sustains antiferromagnetism (AFM) in the ground state, enhances the d-wave singlet superconductivity (dSC) in hole overdoped systems, but suppresses the dSC in electron overdoped systems. For a system that has large |t'/t| and |t"/t'|, the superconductivity (SC) at the onset of AFM in hole doped band filling is strongly suppressed. On the excitation occurring, when an electron doped system simultaneously contains SC and AFM, the system is found to have a nodeless gap at the Fermi level. Finally, the result of this study is related to experiments on the superconducting cuprates.