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arXiv:cond-mat/0305568

Normal-Ordered Wave-Function Factorization of the 1D Hubbard Model for Finite Values of the On-site Repulsion U

J. M. P. Carmelo

Published May 23, 2003• 0 citations
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Abstract

In this paper we find that in the thermodynamic limit and for the the ground-state normal-ordered 1D Hubbard model the wave function of excited states contained in few-electron excitations factorizes for all values of the on-site Coulombian repulsion. This factorization results from the non-interacting character of the pseudofermions whose occupancy configurations describe these excited states. Our study includes the introduction of the pseudoparticle - pseudofermion unitary transformation and of an operator algebra for both the pseudoparticles and the pseudofermions. The pseudofermions are non interacting and thus have no residual interactions, in contrast to the corresponding pseudoparticles, whose statistics we classify according to the generalized Pauli principle. The physics behind the invariance of the pseudoparticles under the above transformations for specific values of the bare momentum is also studied and discussed.

DOI
10.48550/arxiv.cond-mat/0305568

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In this paper we find that in the thermodynamic limit and for the the ground-state normal-ordered 1D Hubbard model the wave function of excited states contained in few-electron excitations factorizes for all values of the on-site Coulombian repulsion. This factorization results from the non-interacting character of the pseudofermions whose occupancy configurations describe these excited states. Our study includes the introduction of the pseudoparticle - pseudofermion unitary transformation and of an operator algebra for both the pseudoparticles and the pseudofermions. The pseudofermions are non interacting and thus have no residual interactions, in contrast to the corresponding pseudoparticles, whose statistics we classify according to the generalized Pauli principle. The physics behind the invariance of the pseudoparticles under the above transformations for specific values of the bare momentum is also studied and discussed.
In this paper we find that in the thermodynamic limit and for the the ground-state normal-ordered 1D Hubbard model the wave function of the excited energy eigenstates which span the Hilbert subspace where the finite-number-electron excitations are contained factorizes for all values of the on-site Coulombian repulsion U. This factorization results from the absence of residual energy interactions for the pseudofermions whose occupancy configurations describe these states. Our study includes the introduction of the pseudoparticle - pseudofermion unitary transformation and of an operator algebra for both the pseudoparticles and the pseudofermions. As the corresponding pseudoparticles, the cνcνcν pseudofermions (and sνsνsν pseudofermions) are ηηη-spin zero 2ν2ν2ν-holon composite quantum objects (and spin zero 2ν2ν2ν-spinon composite quantum objects) where ν=1,2,...ν=1, 2,...ν=1,2,.... The pseudofermion description is the most suitable for the study of the finite-energy spectral properties.
The momentum, electronic density, spin density, and interaction dependences of the exponents that control the ( k , ω ) -plane singular features of the σ = ↑ , ↓ one-electron spectral functions of the 1D Hubbard model at finite magnetic field are studied. The usual half-filling concepts of one-electron lower Hubbard band and upper Hubbard band are defined in terms of the rotated electrons associated with the model Bethe-ansatz solution for all electronic density and spin density values and the whole finite repulsion range. Such rotated electrons are the link of the non-perturbative relation between the electrons and the pseudofermions. Our results further clarify the microscopic processes through which the pseudofermion dynamical theory accounts for the one-electron matrix elements between the ground state and excited energy eigenstates.
This thesis report deals with the 1D Hubbard model and the quantum objects that diagonalize the normal ordered Hubbard hamiltonian, among those the so called PseudoFermions (PFs). These PFs have no residual energy interactions, are eta-spin and spin zero objects, and are the scatterers and the scattering centers of the theory. The S-matrix of this representation is a simple phase factor, involving the phase shifts of the zero energy forward momentum scattering events. A PF dynamical theory is developed and applied to the one-electron removal and lower Hubbard band addition cases. For any value of the on-site effective Coloumb repulsion and electronic density, and in the limit of zero magnetization, we derive closed form expressions for these spectral functions, showing the emergence of the power-law type behavior of correlation functions of Luttinger liquids. However, our expressions are valid for the entire elementary excitation energy bandwidth. The singular behavior of the theoretical spectral weight leads to a spectral weight distribution detectable by photo emission and photo absorption experiments. As an important contribution to the understanding of quasi 1D materials, we are able to reproduce for the whole energy bandwidth, the experimental spectral distributions recently found for the organic compound TTF-TCNQ. This confirms the validity of the PF dynamical theory, and provides a deeper understanding of low dimensional correlated systems.
In this paper we describe the pseudoparticles, holons, and spinons whose occupancy configurations describe the energy eigenstates of the one-dimensional (1D) Hubbard model in terms of rotated electrons. Rotated electrons are related to electrons by a mere unitary transformation such that rotated electron double occupation is a good quantum number. Moreover, we introduce the concepts of local pseudoparticle and effective pseudoparticle lattice. Our findings provide useful information about the relation of the exotic pseudoparticles, holons, and spinons that diagonalize the non-perturbative many-electron problem to the original electrons. We provide an example showing how the derived local pseudoparticle representation can be used in the evaluation of finite-energy few-electron spectral function expressions.
This pre-print deals with the one dimensional Hubbard model, as described by the Pseudofermion Dynamical Theory (PDT), with the purpose of (1) deriving a novel expression for the one electron spectral function for all values of the on-site repulsion U/tU/tU/t and filling n∈(0,1)n \in (0,1)n∈(0,1), at vanishing magnetisation m→0m \rightarrow 0m→0, and (2) discover how to correctly compare the results originating from two different theoretical frameworks in the U/t→∞U/t \rightarrow \inftyU/t→∞ limit, as a first-test of the novel expressions obtained in this paper. Thus, an exact expression of the spectral function is obtained, which is furthermore successfully compared with previously known results in the U→∞U \rightarrow \inftyU→∞ limit. Following the PDT, the expression for the one electron spectral function factorises into a spin part and a charge part for all values of the on-site repulsion U/tU/tU/t, where the dynamical quantum objects are spin zero and ηηη-spin (charge) zero singlet pairs of so-called rotated electrons, which in turn are obtained from the original electrons by a unitary transformation. The spectral function is exemplified for U/t=400U/t = 400U/t=400, with the purpose of comparing it with the same function obtained by other authors (and other means) in the U→∞U \rightarrow \inftyU→∞ limit. The main pillars of the PDT is presented in a summarised form. For example, we will only be interested in excited energy eigenstates which originate the most significant singular features of the spectral map in the (k,ω)(k,ω)(k,ω) plane, safely ignoring higher order contributions. Even though emphasis is given on step-by-step derivations where necessary, derivations that have been done elsewhere and/or do not notably contribute to the physical understanding, are sometimes avoided. Therefore, references for further study are given throughout the paper.
In the first paper of this series it was found that the ηηη-spin 1/2 holons, spin 1/2 spinons, and ccc pseudoparticles whose occupancy configurations describe the energy eigenstates of the one-dimensional Hubbard model emerge from the electron - rotated-electron unitary transformation. In this second paper we discuss and clarify how the relation of the electrons to the above objects can be used in a program for evaluation of finite-energy few-electron spectral functions. As a first step, here we characterize the dominant holon and spinon microscopic physical processes that originate more than 99% of the few-electron spectral weight. These dominant processes are related to exact selection rules for the values of the number of holons and spinons generated or annihilated by application onto a ground state of rotated-electron operators. We also generalize the concepts of a lower Hubbard band and upper Hubbard bands to all values of on-site Coulombian repulsion.