Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.

Jan 1, 20110 citations

Superconductivity within the extended Hubbard model

Rene Jursa, S. Wermbter, G. Czycholl

Czechoslovak Journal of PhysicsFeb 1, 19962 citations

Mott transition in the one-dimensional SU (n) Hubbard model

E. Szirmai, Jenő Sólyom

The one-dimensional SU $(n)$ Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for $n>2$ . It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive $U$ . This behavior is thus different from that at $1∕n$ filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical ${U}_{\mathrm{c}}$ is needed for the Mott transition.The one-dimensional SU$(n)$ Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for $n>2$. It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive $U$. This behavior is thus different from that at $1∕n$ filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical ${U}_{\mathrm{c}}$ is needed for the Mott transition.

Physical Review BMay 16, 200521 citations

Strong coupling Hubbard model and superconductivity

M. Noga

Czechoslovak Journal of PhysicsNov 1, 19890 citations

Non-Linear Canonical Methods in Strongly Correlated Electron Systems

Matteo Bazzanella

In this thesis some new ideas to perform the analysis of Strongly Correlated Electronic Systems (SCES) are developed. In particular the use of non-linear canonical transformations is considered thoroughly. Using such transformations it is possible, in some circumstances, to simplify the quantum problem redefining the fermionic degrees of freedom used to describe the system. To understand and use effectively these non-linear transformations it is convenient to work in the Majorana fermion representation, i.e., to represent the quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom. In a fermionic system, different equivalent sets of (emergent) Majorana fermions can be used to build the fermionic operators that characterize the system. The non-linear transformations can be seen as a way to mix these equivalent sets. Thanks to this insight, it becomes possible to determine the full structure of the group of canonical transformations and to identify an advantageous framework, which allows their use in the study of a generic SCES system. To test these statements the Hubbard and the Kondo lattice models were intensively studied making use of non-linear canonical transformations, obtaining interesting results in both cases. For example, in the Hubbard model a free fermion mean-field description of the paramagnetic Mott insulator was identified, while in the Kondo lattice it was possible to describe already at mean-field level the spin-selective Kondo insulating phase, consistently (from a quantitative and qualitative point of view) with the known numerical results. Moreover the method elaborated for the study of the Hubbard model is suitable for a systematic generalization to other situations and shows great room for improvement. These results prove that, thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean-field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many SCES systems.In this thesis some new ideas to perform the analysis of Strongly Correlated Electronic Systems (SCES) are developed. In particular the use of non-linear canonical transformations is considered thoroughly. Using such transformations it is possible, in some circumstances, to simplify the quantum problem redefining the fermionic degrees of freedom used to describe the system. To understand and use effectively these non-linear transformations it is convenient to work in the Majorana fermion representation, i.e., to represent the quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom. In a fermionic system, different equivalent sets of (emergent) Majorana fermions can be used to build the fermionic operators that characterize the system. The non-linear transformations can be seen as a way to mix these equivalent sets. Thanks to this insight, it becomes possible to determine the full structure of the group of canonical transformations and to identify an advantageous framework, which allows their use in the study of a generic SCES system. To test these statements the Hubbard and the Kondo lattice models were intensively studied making use of non-linear canonical transformations, obtaining interesting results in both cases. For example, in the Hubbard model a free fermion mean-field description of the paramagnetic Mott insulator was identified, while in the Kondo lattice it was possible to describe already at mean-field level the spin-selective Kondo insulating phase, consistently (from a quantitative and qualitative point of view) with the known numerical results. Moreover the method elaborated for the study of the Hubbard model is suitable for a systematic generalization to other situations and shows great room for improvement. These results prove that, thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean-field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many SCES systems.

Nov 14, 20140 citations

Superconductivity in the Hubbard-Holstein model

Hiroki Goto, Takashi Hotta, Yasutami Takada +3 more

Jan 1, 19950 citations

The one-dimensional SU

(n)

Hubbard model is investigated using an analytical, perturbative renormalization-group treatment in the fermionic formulation of the model for

n>2

. It is found that in the half-filled case, the umklapp processes couple the spin and charge degrees of freedom and generate a finite gap for arbitrary finite positive

U

. This behavior is thus different from that at

1∕n

filling, where there is one particle per site, in which case the dynamics of the two degrees of freedom can be separated and a critical

{U}_{\mathrm{c}}

is needed for the Mott transition.