Yang-Baxter Equation for the<i>R</i>-Matrix of the One-Dimensional Hubbard Model

We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu equations, paying particular attention to so-called 'string solutions'. Two kinds of string solutions occur: Λ strings, related to spin degrees of freedom and k-Λ strings, describing spinless bound states of electrons. Whereas Λ strings were thoroughly studied in the literature, less is known about k-Λ strings. We carry out a thorough analytical and numerical analysis of k-Λ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach and the quantum transfer matrix approach, respectively. The Yang-Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero temperature limit of the Yang-Yang approach.

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“…(38) and (40) follow from (38) and (39) and vice versa. Let us take the logarithm of (40) and solve (38) for sin k j − Λ.…”

Section: Lieb-wu Equations For a Single Down Spin (I) -Graphicalmentioning

confidence: 98%

Thermodynamics and excitations of the one-dimensional Hubbard model

et al. 2000

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“…It only involves the algebraic properties of the operators appearing in the various matrices, not the specific n-dependent representation. The equations (8) and (9) are the only equations of this type needed for the proof.…”

Section: The R-matrixmentioning

confidence: 99%

“…where λ and h are related through (5). Using the explicit expression (14), it should be possible and it is instructive to try to check that the above equation is satisfied for any value of n. The factors I drop out and half of the terms on both sides of the YBE compensate each other because relations (8) and (9) hold. The eight remaining terms involve highly non-trivial relations.…”

Section: The R-matrixmentioning

confidence: 99%

Exact Integrability of the su(n) Hubbard Model

Maassarani

1998

Mod. Phys. Lett. B

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“…We also have a relation 19) which is nothing but the infinitesimal transformation of (5.7). Using (5.19), one can confirm the commutativity 20) which shows the Lie algebra so(4) symmetry of the conserved currents I (n) .…”

Section: §5 So(4) Symmetry Of the Higher Conserved Currentsmentioning

confidence: 99%

(4) symmetry of the transfer matrix for the one-dimensional Hubbard model

Shiroishi¹,

Ujino²,

Wadati³

1998

J. Phys. A: Math. Gen.

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The SO(4) invariance of the transfer matrix for the one-dimensional Hubbard model is clarified from the QISM (quantum inverse scattering method) point of view. We demonstrate the SO(4) symmetry by means of the fermionic L-operator and the fermionic R-matrix, which satisfy the graded Yang-Baxter relation. The transformation law of the fermionic L-operator under the SO(4) rotation is identified with a kind of gauge transformation, which determines the corresponding transformation of the fermionic creation and annihilation operators under the SO(4) rotation. The transfer matrix is confirmed to be invariant under the SO(4) rotation, which ensures the SO(4) invariance of the conserved currents including the Hamiltonian. Furthermore, we show that the representation of the higher conserved currents in terms of the Clifford algebra gives manifestly SO(4) invariant forms. * JSPS fellow †

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Yang-Baxter Equation for theR-Matrix of the One-Dimensional Hubbard Model

Cited by 57 publications

References 11 publications

Thermodynamics and excitations of the one-dimensional Hubbard model

Thermodynamics and excitations of the one-dimensional Hubbard model

Exact Integrability of the su(n) Hubbard Model

(4) symmetry of the transfer matrix for the one-dimensional Hubbard model

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