The Renormalization Group Treatment of the Hubbard Model

The article presents the renormalization group treatment to the Hubbard model. To begin with, the bosonization of Hubbard model Hamiltonian is performed. We have obtained the sine-Gordon Hamiltonian. We have further approximated this Hamiltonian by the Hamiltonian of ⁴-theory. Then we utilized Wilson's results of the renormalization group method and obtained the recursion formula for the Hubbard model. Having solved these formulas we have obtained the critical indices for the Hubbard model.     

Flow equations and the strong-coupling expansion for the Hubbard model

Applying the method of continuous unitary transformations to a class of Hubbard models, we reexamine the derivation of thet/U expansion for the strong-coupling case. 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 oft/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 ordert ³/U² or higher.     

Entanglement in Finite Quantum Systems Under Twisted Boundary Conditions

In a recent publication, we have discussed the effects of boundary conditions in finite quantum systems and their connection with symmetries. Focusing on the one-dimensional Hubbard Hamiltonian under twisted boundary conditions, we have shown that properties, such as the ground-state and gap energies, converge faster to the thermodynamical limit (\(L \rightarrow \infty \)) if a special torsion Θ^? is adjusted to ensure particle-hole symmetry. Complementary to the previous research, the present paper extends our analysis to a key quantity for understanding correlations in many-body systems: the entanglement. Specifically, we investigate the average single-site entanglement 〈S_j〉 as a function of the coupling U/t in Hubbard chains with up to L =?8 sites and further examine the dependence of the per-site ground-state ??₀ on the torsion Θ in different coupling regimes. We discuss the scaling of ??₀ and 〈S_j〉 under Θ^? and analyze their convergence to Bethe Ansatz solution of the infinite Hubbard Hamiltonian. Additionally, we describe the exact diagonalization procedure used in our numerical calculations and show analytical calculations for the case study of a trimer.     

Stability of Ground States of 2d Strongly Asymmetric Correlated-Hopping Hubbard Model

The asymmetric correlated-hopping Hubbard model is analysed perturbatively for large values of the Coulomb interaction U. An effective Hamiltonian is obtained up to terms of the order U ^–3. For d=2 and in the limit of the strong asymmetry, the orderings of the ground states are found (confirming earlier nonrigorous results). Their thermal and quantum stability is proved. These results have been obtained by an application of the quantum Pirogov–Sinai theory in the variant developed by Datta, Fernandez, Fröhlich, and Rey-Bellet.     

Non perturbative calculations for three particles in a linear chain within the generalized Hubbard model

A real-space method has been introduced to study the pairing problem within the generalized Hubbard Hamiltonian. This method includes the bond-charge interaction term as an extension of the previously proposed mapping method [1] for the Hubbard model. The generalization of the method is based on mapping the correlated many-body problem onto an equivalent site- and bond-impurity tight-binding one in a higher dimensional space, where the problem can be solved exactly. In a one-dimensional lattice, we analyzed the three particle correlation by calculating the binding energy at the ground state, using different values of the bond-charge, the on-site (U) and the nearest-neighbor (V) interactions. A pairing asymmetry is found between electrons and holes for the generalized hopping amplitude, where the hole pairing is not always easier than the electron case. For some special values of the hopping parameters and for all kinds of interactions in the Hubbard Hamiltonian, an analytical solution is obtained. Received 21 January 2000 and Received in final form 18 July 2000     

Variational theory for correlated lattice fermions in high dimensions

A diagrammatic approach to the evaluation of correlated variational wave functions for strongly interacting fermions is presented. Diagrammatic rules for the calculation of the one-particle density matrix and the Hubbard interaction are derived which are valid for arbitraryd-dimensional lattices. An exact evaluation of expectation values is performed in the limitd=. The wellknown Gutzwiller approximation is seen to become the exact result for the expectation value of the Hubbard Hamiltonian in terms of the Gutzwiller wave function ind=. An efficient procedure to correct the Gutzwiller approximation in finite dimensions is developed. A detailed discussion of expectation values ind= in terms of explicit antiferromagnetic wave functions is given. Thereby an approximate result for the ground state energy of the Hubbard model, obtained recently within a slave-boson approach, is recovered.     

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.     

The su(n) Hubbard model

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I introduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that makes it the natural su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the Hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard Hamiltonian in any dimension.     

Strong-coupling perturbation theory of the Hubbard model

The strong-coupling perturbation theory of the Hubbard model is presented and carried out to order (t/U)⁵ for the one-particle Green function in arbitrary dimension. The spectral weight is expressed as a Jacobi continued fraction and compared with new Monte-Carlo data of the one-dimensional, half-filled Hubbard model. Different regimes (insulator, conductor and short-range antiferromagnet) are identified in the temperature-hopping integral (T,t) plane. This work completes a first paper on the subject (Phys. Rev. Lett. 80, 5389 (1998)) by providing details on diagrammatic rules and higher-order results. In addition, the non half-filled case, infinite resummations of diagrams and the double occupancy are discussed. Various tests of the method are also presented. Received 25 October 1999     

Phase transitions in electron systems with short-range pairing interactions: Ground state

We propose a real-space, tight-binding model of electrons with short-range pairing interactions. The model involves a competition between the ordinary single particle hoppingt and an attractive interactionV between the singlet electronic pairs formed on neighboring lattice sites. The Hamiltonian effectively describes a mechanism for pair formation. We study the ground-state properties of its onedimensional version using numerically exact finite chain calculations for up toN= 10 sites. The ground-state wave functions, the energy spectrum, and various ground-state correlation functions are calculated with the help of an exactly equivalent system of two coupledS=1/2 spin chains. The results indicate the existence of a transition between the band and the localized pairs situation. The transition takes place forV/t= 1.4–0.1 and appears to be of essential singularity type. Comparison with other models used for pairing phenomena, like the negativeU-Hubbard model is made.     

Variational pair theory of the attractive and extended Hubbard model ind-dimensions

We present a variational approach for treating the Hubbard Hamiltonian in one, two and three dimensions. It is based on 2M-fermion wavefunctions which are allowed to form correlated spin-singlet pairs. Expressions for the ground state energy and correlation functions are derived in terms of general pair coefficient functions. The presented approach offers a convenient starting point for improved variational treatments that allow to include different specific types of pair correlations. We present first applications to the attractive and to the extended Hubbard model using a very simple ansatz for the pair coefficient functions. The ground state energy, chemical potential, order parameter, momentum distribution as well as spin-spin and density-density correlation functions follow from a system of coupled nonlinear equations that has to be solved selfconsistently. All quantities are given for arbitrary band-filling in one, two and three dimensions. Our results are compared with those of other approximations and for the one-dimensional case with the exact results of Krivnov and Ovchinnikov.     

On the <Emphasis Type="Italic">SU</Emphasis>(2)×<Emphasis Type="Italic">SU</Emphasis>(2) symmetry in the Hubbard model

We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.     

Spin-wave renormalization by mobile holes in a two-dimensional quantum antiferromagnet

The coupling of antiferromagnetic spin excitations and propagating holes has been studied theoretically on a square lattice in order to investigate the dependence of antiferromagnetic order on hole doping, being of relevance, e.g., for the Cu–3 d⁹ system in antiferromagnetic CuO₂-planes of high-T_c superconductors. An effective Hamiltonian has been used, which results from a 2D Hubbard model (hopping integral t) with holes and with strong on-site Coulomb repulsion U. Bare antiferromagnetic excitations and holes with energies of the same order of magnitude t²/U are interacting via a coupling term being proportional to t and allowing holes to hop by emitting and absorbing spinwaves. In terms of a self-consistent one-loop approximation the renormalization of the spectral function both of holes and antiferromagnetic spin excitations are calculated.     

On the spectrum of the Heisenberg Hamiltonian

The quantum, antiferromagnetic, spin-1/2 Heisenberg Hamiltonian on thed-dimensional cubic lattice ^d is considered for any dimensiond. First the anisotropic case is considered for small transversal coupling and a convergent expansion is given for a family of eigenprojections which is complete in all finite-volume truncations. Then the general case is considered, for which an upper bound to the ground-state energy is given which is optimal for strong enough anisotropy. This bound is expressed through a functional involving the statistical expectation value at finite temperature of a certain correlation function of an Ising model defined on the lattice ^d itself.     

Effective Boseben Hubbard interaction with enhanced nonlinearity in an array of coupled cavities

An array of coupled cavities, each of which contains an N four-level atom, is investigated. When cavity fields dispersively interact with the atoms, an effective Bose—Hubbard model can be achieved. By numerically comparing the full Hamiltonian with the effective one, we find that within the parameters region, the effective Hamiltonian can completely account for the Mott-insulator as well as the phase transition from the similar Mott-insulator to superfluid. Through jointly adjusting the classical Rabi frequency and the detuning, the nonlinearity can be improved.     

PPP Hamiltonian for polar polycyclic aromatic hydrocarbons

Total energies of charged states and configurations of different spin multiplicity of two polar non-alternant polycyclic aromatic hdrocarbons (PAH), namely, pentaheptafulvalene and azulene, calculated by means of a Multi-Configurational (MCSCF) method which includes correlation only amongst π orbitals, have been fitted by exact solutions of the Pariser-Parr-Pople (PPP) and the Hubbard Hamiltonians for π electrons. As both molecules are planar, such an approach is in principle feasible. As found in our previous analysis of PAH, PPP fittings are significantly better than those attained with the Hubbard Hamiltonian. In addition, parameters for the Hubbard Hamiltonian are around twice those derived for the PPP model, indicating that parameters are not model independent. Fitted PPP parameters are close to those derived from a similar study of the PAH 2, 5, 8-trihydrogenated phenalene and those originally proposed by Pariser et al. providing further support to a wide applicability of the fitted parameters. Fittings obtained for a MCSCF method that MM298001also includes σσ and σπ correlations (MCSCF/MP2) are slightly less accurate giving an on-site repulsion 10–15% smaller. The accuracy of the fittings further diminishes when parameters are derived from energies obtained by means of a DFT method (B3LYP) with an additional decrease in U of 5–25%. In the latter two cases, parameters have to be considered as effective, accounting for effects of σ orbitals not explicitly included in the model Hamiltonians. Electron affinities, ionization energies and dipole moments, calculated by means of the model Hamiltonians, are compared to those derived from DFT and ab initio methods and, whenever available, to experimental data.     

The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions

We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g ~ a/r ² is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ~ r ⁻² above the ground state energy. ?2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.     

General spectral function expressions of a 1D correlated model

We introduce a method that allows the evaluation of general expressions for the spectral functions of the one-dimensional Hubbard model for all values of the on-site electronic repulsion U. The spectral weights are expressed in terms of pseudofermion operators such that the spectral functions can be written as a convolution of pseudofermion dynamical correlation functions. Our results are valid for all finite energy and momentum values and are used elsewhere in the study of the unusual finite-energy properties of quasi-one-dimensional compounds and the new quantum systems of ultra-cold fermionic atoms on an optical lattice.     

Low Temperature Phase Diagrams¶of Fermionic Lattice Systems

We consider fermionic lattice systems with Hamiltonian H=H ^{(0)}+λH _Q , where H ^{(0)} is diagonal in the occupation number basis, while H _Q is a suitable “quantum perturbation”. We assume that H ^{(0)} is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitations, while H _Q is a finite range or exponentially decaying Hamiltonian that can be written as a sum of even monomials in the fermionic creation and annihilation operators. Mapping the d dimensional quantum system onto a classical contour system on a d+1 dimensional lattice, we use standard Pirogov–Sinai theory to show that the low temperature phase diagram of the quantum system is a small perturbation of the zero temperature phase diagram of the classical system, provided λ is sufficiently small. Particular attention is paid to the sign problems arising from the fermionic nature of the quantum particles. As a simple application of our methods, we consider the Hubbard model with an additional nearest neighbor repulsion. For this model, we rigorously establish the existence of a paramagnetic phase with commensurate staggered charge order for the narrow band case at sufficiently low temperatures. Received: 23 December 1996/ Accepted: 7 April 1999     

Exact thermodynamics of the Hubbard chain: free energy and correlation lengths

We study the one-dimensional Hubbard model at finite temperatures in the quantum transfer matrix approach. The eigenvalue equations of this matrix are obtained by a nested Bethe ansatz. The largest and next-largest eigenvalues yield the free energy as well as the correlation lengths of the system. An equivalent set of four integral equations is derived from the Bethe ansatz equations. The limit of Trotter-Suzuki number N → ∞ is taken analytically. For half-filling the final equations are studied in the low-temperature limit yielding analytic expressions for the free energy and spin-spin correlation length. Numerical results are presented for intermediate temperatures.