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arXiv:cond-mat/9701094 · Physical review. B, Condensed matter

Complete pseudohole and heavy-pseudoparticle operator representation for the Hubbard chain

J. M. P. Carmelo, Nuno M. R. Peres

Published Aug 15, 1997• 16 citations
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Abstract

We introduce the pseudohole and heavy-pseudoparticle operator algebra that generates all Hubbard-chain eigenstates from a single reference vacuum. In addition to the pseudoholes already introduced for the description of the low-energy physics, this involves the heavy pseudoparticles associated with Hamiltonian eigenstates whose energy spectrum has a gap relative to the many-electron ground state. We introduce a generalized pseudoparticle perturbation theory that describes the relevant finite-energy ground-state transitions. In the present basis these excitations refer to a small density of excited pseudoparticles. Our operator basis goes beyond the Bethe-ansatz solution and it is the suitable and correct starting point for the study of the finite-frequency properties, which are of great relevance for the understanding of the unusual spectral properties detected in low-dimensional novel materials.

DOI
10.1103/physrevb.56.3717

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We introduce the pseudohole and heavy-pseudoparticle operator algebra that generates all Hubbard-chain eigenstates from a single reference vacuum. In addition to the pseudoholes already introduced for the description of the low-energy physics, this involves the heavy pseudoparticles associated with Hamiltonian eigenstates whose energy spectrum has a gap relative to the many-electron ground state. We introduce a generalized pseudoparticle perturbation theory that describes the relevant finite-energy ground-state transitions. In the present basis these excitations refer to a small density of excited pseudoparticles. Our operator basis goes beyond the Bethe-ansatz solution and it is the suitable and correct starting point for the study of the finite-frequency properties, which are of great relevance for the understanding of the unusual spectral properties detected in low-dimensional novel materials.
We introduce an operator algebra for the description of the low-energy physics of one-dimensional, integrable, multicomponent quantum liquids. Considering the particular case of the Hubbard chain in a magnetic field and chemical potential, we show that at low energy its Bethe-ansatz solution can be interpreted in terms of a pseudoparticle-operator algebra. Our algebraic approach provides a concise interpretation of, and justification for, several recent studies of low-energy excitations and trasnport which have been based on detailed analyses of specific Bethe-ansatz eigenfunctions and eigenenergies. A central point is that the exact ground state of the interacting many-electron problem is the noninteracting pseudoparticle ground state. Furthermore, in the pseudoparticle basis, the quantum problem becomes perturbative, i.e., the two-pseudoparticle forward-scattering vertices and amplitudes do not diverge, and one can define a many-pseudoparticle perturbation theory. We write the general quantum-liquid Hamiltonian in the pseudoparticle basis and show that the pseudoparticle-perturbation theory leads, in a natural way, to the generalized Landau-liquid approach.
We consider the Hubbard chain in a magnetic field and chemical potential. We introduce a pseudohole basis where all states are generated from a single reference vacuum. This allows the evaluation for all sectors of Hamiltonian symmetry of the model of the expression of the σ electron and hole operators at Fermi momentum ±kFσ and vanishing excitation energy in terms of pseudohole operators. In all sectors and to leading order in the excitation energy the electron and hole are constituted by one c pseudohole, one s pseudohole, and one topological momenton. These three quantum objects are confined in the electron or hole and cannot be separated. We find that the set of different pseudohole types which in pairs constitute the two electrons and two holes associated with the transitions from the (N↑,N↓) ground state to the (N↑+1,N↓), (N↑,N ↓ +1) and (N ↑ −1,N↓), (N↑,N ↓ −1) ground states, respectively, transform in the representation of the symmetry group of the Hamiltonian in the initial-ground-state sector of parameter space. We also find the pseudohole generators for the half-filling holon and zero-magnetic-field spinon. The pseudohole basis introduced in this paper is the only suitable for the extension of the present type of operator description to the whole Hilbert space. 1 I.
Based on a recently introduced operator algebra for the description of a class of integrable quantum liquids we define the ground states for all canonical ensembles of these systems. We consider the particular case of the Hubbard chain in a magnetic field and chemical potential. The ground states of all canonical ensembles of the model can be generated by acting on the electron vacuum (densities n1) or hole vacuum (densities n>1), with suitable pseudoparticle creation operators. We also evaluate the energy gaps of the non-lowest-weight states and non-highest-weight states of the \ensuremath{\eta} spin and spin algebras relative to the corresponding ground states. For all sectors of parameter space and symmetries the exact ground state of the many-electron problem is in the pseudoparticle basis the noninteracting pseudoparticle ground state. This plays a central role in the pseudoparticle perturbation theory.
We write the Hamiltonian of the Bose gas with two-body repulsive \ensuremath{\delta}-function potential in a pseudoparticle-operator basis which diagonalizes the problem via the Bethe ansatz. In this operator basis the original bosonic interactions are represented by zero-momentum forward-scattering interactions between Landau-liquid pseudoparticles. We find that this pseudoparticle-operator algebra is complete: all the Hamiltonian eigenstates are generated by acting pseudoparticle operators on the system vacuum. It is shown that one boson of vanishing momentum and energy is a composite of a one-pseudoparticle excitation and a collective pseudoparticle excitation. These excitations have finite opposite momenta and cannot be decomposed. Our formalism enables us to calculate the various quantities that characterize the static and dynamic behavior of the system at low energies.
We show that the recently developed pseudoparticle-operator algebra which generates the low-energy Hamiltonian eigenstates of multicomponent integrable systems with contact interactions also provides a natural operator representation for the Virasoro algebras associated with the conformal-invariant character of the low-energy spectrum of these models. Studying explicitly the Hubbard chain in a nonzero chemical potential and external magnetic field, we establish that the pseudoparticle-perturbation theory provides a correct starting point for the construction of a suitable critical-point Hamiltonian. We derive explicit expressions in terms of pseudoparticle operators for the generators of the Virasoro algebras and the energy-momentum tensor, describe the conformal-invariant character of the critical point from the point of view of the response to curvature of the two-dimensional space time, and discuss the relation to Kac-Moody algebras and dynamical separation.
In this paper we consider the one-dimensional Hubbard model and study the deviations from the ground-state values of double occupation which result from creation or annihilation of holons, spinons, and pseudoparticles. These quantum objects are such that all energy eigenstates are described by their occupancy configurations. The band-momentum dependence of the obtained double-occupation spectra provides important information on the degree of localization/delocalization of the real-space lattice electron site distribution configurations associated with the pseudoparticles. We also study the band momentum, on-site electronic repulsion, and electronic density dependence of the pseudoparticle energy bands. The shape of these bands plays an important role in the finite-energy spectral properties of the model. Such a shape defines the form of the lines in the momentum-energy/frequency plane where the peaks and edges of the one-electron and two-electron spectral weight of physical operators are located. Our findings are useful for the study of the one-electron and two-electron spectral-weight distribution of physical operators.