Strongly correlated fermions

A Hubbard model with a N_d-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to N_d. It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to N_d, it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.

A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single band Hubbard models.

S.C. Zhang has put forward the idea that high-temperature superconductors can be described in the framework of an SO(5)-symmetric theory in which the three components of the antiferromagnetic order-parameter and the two components of the two-particle condensate form a five-component order-parameter with SO(5) symmetry. Interactions small in comparison to this strong interaction introduce anisotropies into the SO(5)-space and determine whether it is favorable for the system to be superconducting or antiferromagnetic. Here the view is expressed that Zhang's derivation of the effective interaction V_eff based on his Hamiltonian H_a is not correct. However, the orthogonality constraints introduced several pages after this 'derivation' give the key to an effective interaction very similar to that given by Zhang. It is shown that the orthogonality constraints are not rigorous constraints, but they maximize the entropy at finite temperature. If the interaction drives the ground-state to the largest possible eigenvalues of the operators under consideration (antiferromagnetic ordering, superconducting condensate, etc.), then the orthogonality constraints are obeyed by the ground-state, too.

Applying the method of continuous unitary transformations to a class of Hubbard models, the derivation of the t/U-expansion for the strong-coupling case is re-examined. The flow equations for the coupling parameters of the higher-order effective interactions can be solved exactly, resulting in a systematic expansion of the Hamiltonian in powers of t/U, valid for any lattice in arbitrary dimension and for general band-filling. The expansion ensures a correct treatment of the operator products generated by the transformation, and only involves the explicit recursive calculation of numerical coefficients. This scheme provides a unifying framework to study the strong-coupling expansion for the Hubbard model, which clarifies and circumvents several difficulties inherent to earlier approaches. Our results are compared with those of other methods, and it is shown that the freedom in the choice of the unitary transformation that eliminates interactions between different Hubbard bands can affect the effective Hamiltonian only at order t^3/U^2 or higher.

The effective field theory that describes the interaction of fluctuations of the attractive Hubbard model in the strong-coupling limit is formulated and applied to obtain the two-loop results for the order parameters of the model at low temperatures and in the leading singularities near two dimensions. Starting from a Grassmann functional integral representation of the strong-coupling model on the restricted fermionic Hilbert space, which corresponds to the anisotropic spin-1/2 Heisenberg quantum antiferromagnet, we derive the effective field theory up to fifth order in the bosonic fluctuation fields. The loop-corrections of the order parameters for superconductivity and charge-ordering and of the chemical potential are evaluated including both the singular planar phase fluctuations and the out-of-plane fluctuations, which become soft at half band-filling. The results are in agreement with the scaling behaviour expected from the corresponding XY and Heisenberg limits of the O(n) nonlinear sigma-model.

The low-temperature result for the superfluid density of the attractive (negative-U) Hubbard model in two dimensions and in the strong-coupling limit is obtained, applying a field-theoretic representation of the model and utilizing a hydrodynamic relation between the order parameter field, the correlation function for soft planar fluctuations, and the superfluid density. This microscopic derivation does not involve self-consistency arguments and allows to include the influence of the out-of-plane pseudo-spin fluctuations which become singular at half band-filling and suppress a Kosterlitz-Thouless transition at finite temperature. The result is in good agreement with quantum Monte Carlo data.

A Grassmann functional integration technique is applied to the anisotropic pseudospin Hamiltonian derived from the Holstein-Hubbard model with strong electron-phonon coupling. It is found that in those regions of the low-temperature phase diagram where superconductivity and the charge-density wave coexist already the static and homogeneous saddle-point solution of the resulting field theory yields a ground-state energy which is lower than the classical result corresponding to the usual mean-field solution. This is due to the presence of quantum fluctuations and leads to new ground-state phase boundaries quantitatively different from those obtained in previous work by conventional operator decoupling schemes.