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arXiv:cond-mat/0405411

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

J. M. P. Carmelo

Published May 18, 2004• 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 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 $cν$ pseudofermions (and $sν$ pseudofermions) are $η$-spin zero $2ν$-holon composite quantum objects (and spin zero $2ν$-spinon composite quantum objects) where $ν=1, 2,...$. The pseudofermion description is the most suitable for the study of the finite-energy spectral properties.

DOI
10.48550/arxiv.cond-mat/0405411

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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 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.
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 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.
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.
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.
A description of the Hubbard model on a square lattice consistent with its global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral ttt, on-site repulsion UUU, and Na2≫1N_a^2\gg 1Na2​≫1 sites consistent with its exact global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t>0U/4t>0U/4t>0 the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge ccc fermions, spin-1/2 spinons, and ηηη-spin-1/2 ηηη-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the ηηη-spinon (and spinons) that are not invariant under that transformation have ηηη spin 1/2 (and spin 1/2) but are part of ηηη-spin-neutral (and spin-neutral) 2ν2ν2ν-ηηη-spinon (and 2ν2ν2ν-spinon) composite ηνηνην fermions (and sνsνsν fermions) where ν=1,2,...ν=1,2,...ν=1,2,... is the number of ηηη-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x=0x=0x=0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x>0x>0x>0.
Abstract. In this paper we derive general expressions for few-electron spectral functions of the one-dimensional Hubbard model for values of the excitation energy in the vicinity of the M th upper-Hubbard band lower limit. Here M = 1, 2,... is the rotated-electron double occupation, which vanishes for the ground state and is a good quantum number for all values of the on-site Coulomb repulsion U. Our studies rely on a combination of symmetries of the model with a recent finite-energy holon and spinon description of the quantum problem. We apply our general scheme to the one-electron addition spectral function, dynamical structure factor, and spin singlet Cooper pair addition spectral function. Our results provide physically interesting information about the finite-energy spectral properties of the many-electron one-dimensional quantum liquid. PACS numbers: 71.10.Fd, 64.60.Fr, 11.30.-j, 71.10.PmOne and Two-Electron Spectral Function Expressions... 2 1.