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Selected Publications

Selected publications by Florian Gebhard with abtracts and links

Correlation functions for interacting fermions in the Gutzwiller ansatz

Florian Gebhard and Dieter Vollhardt, Phys. Rev. B 38, 6911 (1988).

We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.
Gutzwiller-correlated wave functions in finite dimensions d: A systematic expansion in 1/d

Florian Gebhard, Phys. Rev. B 41, 9452 (1990).

We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=∞). The limit of d→∞ has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=∞ are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=∞ for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=∞ is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=∞ only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (‘‘spinons’’).
Exact Results for a Hubbard Chain with Long-Range Hopping

Florian Gebhard and Andrei E. Ruckenstein, Phys. Rev. Lett. 68, 244 (1992).

We give the exact spectrum and thermodynamics for a long-range hopping Hubbard chain with linear dispersion. This model exhibits a Mott-Hubbard metal-insulator transition at half filling when the interaction strength U equals the bandwidth W. The solution for U≫W also covers the corresponding t-J model, which reduces to the spin model of Haldane and Shastry at half filling. We mention possible extensions of the model in one and higher dimensions.
Asymptotic Bethe Ansatz Results for a Hubbard Chain with 1/sinh-Hopping

Pierre-Antoine Bares and Florian Gebhard, Europhys. Lett. 29, 573 (1995).

We investigate spin-1/2 electrons with local Hubbard interaction and variable-range hopping amplitudes which decay like sinh (κ)/sinh (κr). Assuming integrability, the asymptotic Bethe-Ansatz approach allows us to derive the generalized Lieb-Wu integral equations from the two-particle phase shift. Due to the nesting property there is a metal-to-insulator transition at U_c (κ > 0) = 0⁺. The charge gap in the singular limit κ = 0 opens when the interaction strength equals the bandwidth, U_c (κ = 0) = W > 0.
Multi-band Gutzwiller wave functions for general on-site interactions

Jörg Bünemann, Werner Weber, and Florian Gebhard, Phys. Rev. B 57, 6896 (1998).

We introduce Gutzwiller wave functions for multiband models with general on-site Coulomb interactions. As these wave functions employ correlators for the exact atomic eigenstates, they are exact both in the noninteracting and atomic limits. We evaluate them in infinite lattice dimensions for all interaction strengths without any restrictions on the structure of the Hamiltonian or the symmetry of the ground state. The results for the ground-state energy allow us to derive an effective one-electron Hamiltonian for Landau quasiparticles, applicable for finite temperatures and frequencies within the Fermi-liquid regime. As applications for a two-band model we study the Brinkman-Rice metal-to-insulator transition at half-band-filling, and the transition to itinerant ferromagnetism for two specific fillings, at and close to a peak in the density of states of the noninteracting system. Our results significantly differ from those for earlier Gutzwiller wave functions where only density-type interactions were included. When the correct spin symmetries for the two-electron states are taken into account, the importance of the Hund’s-rule exchange interaction is even more pronounced, and leads to paramagnetic metallic ground states with large local magnetic moments. Ferromagnetism requires fairly large interaction strengths, and the resulting ferromagnetic state is a strongly correlated metal.
Mott-Hubbard transition in infinite dimensions

Reinhard M. Noack and Florian Gebhard, Phys. Rev. Lett. 82, 1915 (1999).

We calculate the zero-temperature gap and quasiparticle weight of the half-filled Hubbard model with a random dispersion relation. After extrapolation to the thermodynamic limit, we obtain reliable bounds on these quantities for the Hubbard model in infinite dimensions. Our data indicate that the Mott-Hubbard transition is continuous; i.e., the quasiparticle weight becomes zero at the same critical interaction strength at which the gap opens.
Excitons in one-dimensional Mott insulators

Fabian H.L. Essler, Florian Gebhard, and Eric Jeckelmann, Phys. Rev. B 64, 125119 (2001).

We employ dynamical density-matrix renormalization-group (DDMRG) and field-theory methods to determine the frequency-dependent optical conductivity in one-dimensional extended, half-filled Hubbard models. The field-theory approach is applicable to the regime of “small” Mott gaps which is the most difficult to access by DDMRG. For very large Mott gaps the DDMRG recovers analytical results obtained previously by means of strong-coupling techniques. We focus on exciton formation at energies below the onset of the absorption continuum. As a consequence of spin-charge separation, these Mott-Hubbard excitons are bound states of spinless, charged excitations (“holon-antiholon” pairs). We also determine exciton binding energies and sizes. In contrast to simple band insulators, we observe that excitons exist in the Mott-insulating phase only for a sufficiently strong intersite Coulomb repulsion. Furthermore, our results show that the exciton binding energy and size are not related in a simple way to the strength of the Coulomb interaction.
Atomic correlations in itinerant ferromagnets: Quasi-particle bands of nickel

J. Bünemann, F. Gebhard, T. Ohm, R. Umstätter, S. Weiser, W. Weber, R. Claessen, D. Ehm, A. Harasawa, A. Kakizaki, A. Kimura, G. Nicolay, S. Shin, and V.N. Strocov, Europhys. Lett. 61, 667 (2003).

The Gutzwiller theory is demonstrated to resolve most of the long-standing discrepancies between experiment and theory on the quasi-particle bands of ferromagnetic nickel. This is confirmed by new angle-resolved photoelectron spectroscopy data along various high-symmetry lines of the bulk Brillouin zone obtained under full control of the three-dimensional momentum. Our findings support the view of itinerant ferromagnetism as a consequence of atomic correlations.
Tomonaga-Luttinger parameters and spin excitations in the dimerized extended Hubbard model

Satoshi Ejima, Florian Gebhard, and Satoshi Nishimoto, Phys. Rev. B 74, 245110 (2006).

We study the one-dimensional extended Hubbard model with alternating size of the hopping integrals using the density-matrix renormalization group method. We calculate the spin gap, the Tomonaga-Luttinger parameter, and the charge-density-wave order parameter for various dimerizations, interaction strengths, and band fillings. At half band-filling the spin and charge excitations are gapped but these gaps disappear for infinitesimal hole doping. At quarter filling, the umklapp scattering in the half-filled lower Peierls band generates a gap for the charge excitations but the gapless spin excitations can be described in terms of an effective antiferromagnetic Heisenberg model. Beyond a critical strength for the nearest-neighbor interaction, the dimerized extended Hubbard model at quarter filling develops a charge-density-wave ground state. The dimerization and the nearest-neighbor Coulomb interaction strongly reduce the Tomonaga-Luttinger parameter from its value for the bare Hubbard model. We discuss the relevance of our findings for the Bechgaard salts.
Equivalence of Gutzwiller and slave-boson mean-field theories for multiband Hubbard models

Jörg Bünemann and Florian Gebhard, Phys. Rev. B 76, 193104 (2007).

We demonstrate that a recently introduced slave-boson mean-field theory is equivalent to our Gutzwiller theory for multiband Hubbard models with general onsite interactions. We relate the different objects that appear in both approaches at zero temperature and discuss the limitations of both methods.
Excited states in polydiacetylene chains: A density matrix renormalization group study

Gergely Barcza, William Barford, Florian Gebhard, and Örs Legeza, Phys. Rev. B 87, 245116 (2013).

We study theoretically polydiacetylene chains diluted in their monomer matrix. We employ the density matrix renormalization group method on finite chains to calculate the ground state and low-lying excitations of the corresponding Peierls–Hubbard-Ohno Hamiltonian which is characterized by the electron transfer amplitude t₀ between nearest neighbors, by the electron-phonon coupling constant α, by the Hubbard interaction U, and by the long-range interaction V. We treat the lattice relaxation in the adiabatic limit, i.e., we calculate the polaronic lattice distortions for each excited state. Using chains with up to 102 lattice sites, we can safely perform the extrapolation to the thermodynamic limit for the ground-state energy and conformation, the single-particle gap, and the energies of the singlet exciton, the triplet ground state, and the optical excitation of the triplet ground state. The corresponding gaps are known with high precision from experiments. We determine a coherent parameter set (t₀=2.4eV,α=3.4eV/Å,U=6eV,V=3eV) from a fit of the experimental gap energies to the theoretical values which we obtain for 81 parameter points in the four-dimensional search space (t₀,α,U,V). We identify dark in-gap states in the singlet and triplet sectors as seen in experiments. Using a fairly stiff spring constant, the length of our unit cell is about 1% larger than its experimental value.
Relaxation of ideal classical particles in a one-dimensional box

Florian Gebhard and Kevin zu Münster, Annalen der Physik (Berlin) 523, 552 (2011).

We study the deterministic dynamics of non‐interacting classical gas particles confined to a one‐dimensional box as a pedagogical toy model for the relaxation of the Boltzmann distribution towards equilibrium. Hard container walls alone induce a uniform distribution of the gas particles at large times. For the relaxation of the velocity distribution we model the dynamical walls by independent scatterers. The Markov property guarantees a stationary but not necessarily thermal velocity distribution for the gas particles at large times. We identify the conditions for physical walls where the stationary velocity distribution is the Maxwell distribution. For our numerical simulation we represent the wall particles by independent harmonic oscillators. The corresponding dynamical map for oscillators with a fixed phase (Fermi–Ulam accelerator) is chaotic for mesoscopic box dimensions.
Gutzwiller Theory of Band Magnetism in LaOFeAs

Tobias Schickling, Florian Gebhard, Jörg Bünemann, Lilia Boeri, Ole K. Andersen, and Werner Weber, Phys. Rev. Lett. 108, 036406 (2012).

We use the Gutzwiller variational theory to calculate the ground-state phase diagram and quasiparticle bands of LaOFeAs. The Fe3d−As4p Wannier-orbital basis obtained from density-functional theory defines the band part of our eight-band Hubbard model. The full atomic interaction between the electrons in the iron orbitals is parametrized by the Hubbard interaction U and an average Hund’s-rule interaction J. We reproduce the experimentally observed small ordered magnetic moment over a large region of (U,J) parameter space. The magnetically ordered phase is a stripe spin-density wave of quasiparticles.
Gutzwiller density functional theory: a formal derivation and application to nickel

Tobias Schickling, Jörg Bünemann, Florian Gebhard, and Werner Weber, New J. Phys. 16, 093034 (2014).

We present a detailed derivation of the Gutzwiller density functional theory (DFT) that covers all conceivable cases of symmetries and Gutzwiller wave functions. The method is used in a study of ferromagnetic nickel where we calculate ground state properties (lattice constant, bulk modulus, spin magnetic moment) and the quasi-particle band structure. Our method resolves most shortcomings of an ordinary density functional calculation on nickel. However, the quality of the results strongly depends on the particular choice of the double-counting correction. This constitutes a serious problem for all methods that attempt to merge DFT with correlated-electron approaches based on Hubbard-type local interactions.
Quasiparticle bands and structural phase transition of iron from Gutzwiller density-functional theory

Tobias Schickling, Jörg Bünemann, Florian Gebhard, and Lilia Boeri, Phys. Rev. B 93, 205151 (2016).

We use the Gutzwiller density-functional theory to calculate ground-state properties and band structures of iron in its body-centered-cubic (bcc) and hexagonal-close-packed (hcp) phases. For a Hubbard interaction U=9eV and Hund's-rule coupling J=0.54eV, we reproduce the lattice parameter, magnetic moment, and bulk modulus of bcc iron. For these parameters, bcc is the ground-state lattice structure at ambient pressure up to a pressure of p_c=41GPa where a transition to the nonmagnetic hcp structure is predicted, in qualitative agreement with experiment (p_exp,c=10,...,15GPa). The calculated band structure for bcc iron is in good agreement with ARPES measurements. The agreement improves when we perturbatively include the spin-orbit coupling.
Gutzwiller variational approach to the two-impurity Anderson model for a metallic host at particle-hole symmetry

Thorben Linneweber, Jörg Bünemann, Zakaria M.M. Mahmoud, and Florian Gebhard, J. Phys.: Condens. Matter 29, 445603 (2017).

We study Gutzwiller-correlated wave functions as variational ground states for the two-impurity Anderson model (TIAM) at particle-hole symmetry as a function of the impurity separation R. Our variational state is obtained by applying the Gutzwiller many-particle correlator to a single-particle product state. We determine the optimal single-particle product state fully variationally from an effective non-interacting TIAM that contains a direct electron transfer between the impurities as variational degree of freedom. For a large Hubbard interaction U between the electrons on the impurities, the impurity spins experience a Heisenberg coupling proportional to V²/U where V parameterizes the strength of the on-site hybridization. For small Hubbard interactions we observe weakly coupled impurities. In general, for a three-dimensional simple cubic lattice we find discontinuous quantum phase transitions that separate weakly interacting impurities for small interactions from singlet pairs for large interactions.
Symmetric single-impurity Kondo model on a tight-binding chain: Comparison of analytical and numerical ground-state approaches

Gergely Barcza, Kevin Bauerbach, Fabian Eickhoff, Frithjof B. Anders, Florian Gebhard, and Örs Legeza, Phys. Rev. B 101, 075132 (2020). (Editors' Suggestion)

We analyze the ground-state energy, local spin correlation, impurity spin polarization, impurity-induced magnetization, and corresponding zero-field susceptibilities of the symmetric single-impurity Kondo model on a tight-binding chain with bandwidth W=2D where a spin-1/2 impurity at the chain center interacts with coupling strength J_K with the local spin of the bath electrons. We compare perturbative results and variational upper bounds from Yosida, Gutzwiller, and first-order Lanczos wave functions to the numerically exact extrapolations obtained from the density-matrix renormalization group (DMRG) method and from the numerical renormalization group (NRG) method performed with respect to the inverse system size and Wilson parameter, respectively. In contrast to the Lanczos and Yosida wave functions, the Gutzwiller variational approach becomes exact in the strong-coupling limit JK≫W, and reproduces the ground-state properties from DMRG and NRG for large couplings JK≳W, with a high accuracy. For weak coupling, the Gutzwiller wave function describes a symmetry-broken state with an oriented local moment, in contrast to the exact solution. We calculate the impurity spin polarization and its susceptibility in the presence of magnetic fields that are applied globally or only locally to the impurity spin. The Yosida wave function provides qualitatively correct results in the weak-coupling limit. In DMRG, chains with about 10^3 sites are large enough to describe the susceptibilities down to JK/D≈0.6. For smaller Kondo couplings, only the NRG provides reliable results for a general host-electron density of states ρ_0(ε). To compare with results from Bethe ansatz that become exact in the wide-band limit, we study the impurity-induced magnetization and zero-field susceptibility. For small Kondo couplings, the zero-field susceptibilities at zero temperature approach χ_0(JK≪D)/(gμ_B)2≈exp[1/(ρ_0(0)J_K)]/[2CD√πeρ_0(0)J_K], where ln(C) is the regularized first inverse moment of the density of states. Using NRG, we determine the universal subleading corrections up to second order in ρ_0(0)J_K.
Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions
Florian Gebhard, Kevin Bauerbach, and Örs Legeza, Phys. Rev. B 106, 205133 (2022). (Editors' Suggestion)

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band filling with nearest-neighbor electron transfer amplitude t and interaction V. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For V>Vc=2t, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at V(2)c,2≈1.5t. Second-order Hartree Fock theory is almost variational because the density of quasiparticle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites, which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasiparticle density, and the density-density correlation function to locate Vc from the DMRG data. Tracing the breakdown of the Luttinger liquid and the peak in the quasiparticle density at the band edge permits us to reproduce Vc with an accuracy of one percent.