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arXiv:0804.2379

The Hubbard model on a square lattice in the one- and two-electron subspace

J. M. P. Carmelo

Published Apr 15, 2008• 1 citation
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Abstract

In this paper the spin configurations of the ground state and one- and two-electron excited states of the Hubbard model on the square lattice are studied. We profit from a general rotated-electron description, which is consistent with the model global $SO(3)\times SO(3)\times U(1)$ symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion $U>0$. The above states are within that description generated by occupancy configurations of charge $c$ fermions and spin-singlet two-spinon $s1$ bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the $c$ fermions have no internal structure, that of the spin-neutral $s1$ bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La$_2$CuO$_4$. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.

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A description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetry

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In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$, on-site repulsion $U$, and $N_a^2\gg 1$ sites consistent with its exact global $SO(3)\times SO(3)\times U(1)$ 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 $U/4t>0$ 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 $c$ fermions, spin-1/2 spinons, and $η$-spin-1/2 $η$-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the $η$-spinon (and spinons) that are not invariant under that transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$-spin-neutral (and spin-neutral) $2ν$-$η$-spinon (and $2ν$-spinon) composite $ην$ fermions (and $sν$ fermions) where $ν=1,2,...$ is the number of $η$-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 $x=0$ hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations $x>0$.

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In this paper the spin configurations of the ground state and one- and two-electron excited states of the Hubbard model on the square lattice are studied. We profit from a general rotated-electron description, which is consistent with the model global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion U>0U>0U>0. The above states are within that description generated by occupancy configurations of charge ccc fermions and spin-singlet two-spinon s1s1s1 bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the ccc fermions have no internal structure, that of the spin-neutral s1s1s1 bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La2_22​CuO4_44​. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.
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.
In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N2a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La2CuO4 is shortly reviewed.
In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with Na2≫1N_a^2\gg 1Na2​≫1 sites, effective transfer integral ttt, and onsite repulsion UUU has in a suitable one- and two-electron subspace. Such an operator description is that consistent with the model exact global symmetry recently extended to SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1). There is a large consensus that in the thermodynamic limit Na2→∞N_a^2\rightarrow\inftyNa2​→∞ long-range antiferromagnetic order occurs in the spin-density m=0m=0m=0 ground state of the half-filled Hubbard model on the square lattice. Here we find that the corresponding spontaneous symmetry breaking lowers the symmetry of such a state from SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) for Na2≫1N_a^2\gg 1Na2​≫1 large but finite to [U(2)×U(1)]/Z22≡[SO(3)×U(1)×U(1)]/Z2[U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2[U(2)×U(1)]/Z22​≡[SO(3)×U(1)×U(1)]/Z2​ for Na2→∞N_a^2\rightarrow\inftyNa2​→∞. Moreover, we argue that the spin effective lattice being identical to the original lattice is a necessary condition for the occurrence of ground-state long-range antiferromagnetic order in the limit Na2→∞N_a^2\rightarrow\inftyNa2​→∞. Consistently, strong evidence is provided that for very small hole concentration $0
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.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.
I present a non-orthogonal and overcomplete set of states that spans the S_tot=0 Hilbert subspace for a fermionic system with four possible spin configurations per site, on a half-filled lattice with an even number of sites. This set consists of possible pairing of the spins into three bond types: singlet bonds and positive and negative charge bonds. Although I present the specific case for the half-filled Hubbard model in one-dimension, these sets of states can be trivially extended to higher dimensions. The overlap of these generalized valence bond states form closed loops with properties that are related to operator expectation values in this basis. I verify the correctness of this basis by solving the model via exact diagonalization and comparing with known results. There is evidence of structure in the ground state that may be useful to simulations using variational wave functions.