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Physical review. B, Condensed matter

Magnetic-field effects on the correlation functions in the one-dimensional strongly correlated Hubbard model

Masao Ogata, Tadao Sugiyama, Hiroyuki Shiba

Published Apr 1, 1991• 55 citations
Publisher

Abstract

The momentum distribution function and spin-correlation functions are studied for the one-dimensional large-U Hubbard model with an emphasis on the effects of the magnetic field. A direct calculation of the correlation functions is carried out with the exact wave function, whose explicit form for the large-U limit was derived in a previous paper. Singularities are detected at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$+2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ in the momentum distribution function ${\mathit{n}}_{\mathrm{\ensuremath{\sigma}}}$(k), where ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$ is the Fermi wave number of the noninteracting electrons with a given magnetization. Similarly, the longitudinal spin-correlation function ${\mathit{S}}^{\mathit{z}\mathit{z}}$(k) has singularities at 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$ and 2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$, while the transverse one ${\mathit{S}}^{\mathit{x}\mathit{x}}$(k) is singular at ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$+${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$ and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$-${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$. The nature of the singularities (i.e., critical exponents) is determined from an analysis of the size dependence. As a complementary approach, those critical exponents for the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit are explicitly calculated with conformal field theory. In this limit the exponents depend only on the magnetization and are independent of the electron density. The resulting exponents are consistent with the first results.

DOI
10.1103/physrevb.43.8401

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The momentum distribution function and spin-correlation functions are studied for the one-dimensional large-U Hubbard model with an emphasis on the effects of the magnetic field. A direct calculation of the correlation functions is carried out with the exact wave function, whose explicit form for the large-U limit was derived in a previous paper. Singularities are detected at kF\ensuremathσ{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}kF\ensuremathσ​ and kF\ensuremathσ{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}kF\ensuremathσ​+2kF\ensuremath−\ensuremathσ{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}kF\ensuremath−\ensuremathσ​ in the momentum distribution function n\ensuremathσ{\mathit{n}}_{\mathrm{\ensuremath{\sigma}}}n\ensuremathσ​(k), where kF\ensuremathσ{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}kF\ensuremathσ​ is the Fermi wave number of the noninteracting electrons with a given magnetization. Similarly, the longitudinal spin-correlation function Szz{\mathit{S}}^{\mathit{z}\mathit{z}}Szz(k) has singularities at 2kF\ensuremath↑{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}kF\ensuremath↑​ and 2kF\ensuremath↓{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}kF\ensuremath↓​, while the transverse one Sxx{\mathit{S}}^{\mathit{x}\mathit{x}}Sxx(k) is singular at kF\ensuremath↑{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}kF\ensuremath↑​+kF\ensuremath↓{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}kF\ensuremath↓​ and kF\ensuremath↑{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}kF\ensuremath↑​-kF\ensuremath↓{\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}kF\ensuremath↓​. The nature of the singularities (i.e., critical exponents) is determined from an analysis of the size dependence. As a complementary approach, those critical exponents for the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit are explicitly calculated with conformal field theory. In this limit the exponents depend only on the magnetization and are independent of the electron density. The resulting exponents are consistent with the first results.
A recent study of the large-U limit of the one-dimensional (1D) Hubbard model is presented and discussed. It is pointed out first that the wave function has a simple structure in this limit. Namely, it is a product of Slater determinant of noninteracting spinless fermions and the wave function of 1D S=1/2 Heisenberg antiferromagnet. Secondly, by using this property, a direct calculation of momentum distribution and spin correlation function is carried out for the ground state at zero-field and finite-field cases. The results show various singularities exactly at the same position where one expects for the small-U case in both zero-field and finite-field cases. The critical exponents estimated from the size dependence are in reasonable agreement with those predicted by the Tomonaga-Luttinger-liquid and the conformal field theory.
The momentum distribution function n(k) and spin-correlation function S(k) are determined for the one-dimensional large-U Hubbard model with various electron densities. The present work is featured with an application of the Bethe-ansatz wave function, which has a simple form in the large-U limit for any electron density. Namely, its charge degrees of freedom are expressed as a Slater determinant of spinless fermions, while its spin degrees of freedom are equivalent to the one-dimensional S=(1/2 Heisenberg model. The singularity of n(k) at k=kF{k}_{F}kF​ is analyzed from the system size dependence. In addition to the kF{k}_{F}kF​ singularity, n(k) has a weak singularity at k=3kF{k}_{F}kF​; however, no detectable singularity is present at 2kF{k}_{F}kF​, which one might expect from the spinless fermion wave function. The singularity of S(k) at 2kF{k}_{F}kF​ is also examined in detail. It is concluded from the size dependence that, when the system is away from half-filling, the nature of the singularity at 2kF{k}_{F}kF​ is different from that in the Heisenberg model. The results are compared with the behavior in the weak-correlation regime examined with the perturbation calculation, the prediction of g-ology, and recent Monte Carlo calculations.
We present a general method for the calculation of correlation functions in the repulsive one-dimensional Hubbard model at less than half-filling in a magnetic field h. We describe the dependence of the critical exponents that drive their long-distance asymptotics on the Coulomb coupling, the density, and h. This dependence can be described in terms of a set of coupled Bethe-Ansatz integral equations. It simplifies significantly in the strong-coupling limit, where we give explicit formulas for the dependence of the critical exponents on the magnetic field. In particular, we find that at small field the functional dependence of the critical exponents on h can be algebraic or logarithmic---depending on the operators involved. In addition, we evaluate the singularities of the Fourier images of the correlation functions. It turns out that switching on a magnetic field gives rise to singularities in the dynamic field-field correlation functions that are absent at h=0.
We present a general method for the calculation of correlation functions in the repulsive one-dimensional Hubbard model at less than half-filling in a magnetic field h. We describe the dependence of the critical exponents that drive their long-distance asymptotics on the Coulomb coupling, the density, and h. This dependence can be described in terms of a set of coupled Bethe-Ansatz integral equations. It simplifies significantly in the strong-coupling limit, where we give explicit formulas for the dependence of the critical exponents on the magnetic field. In particular, we find that at small field the functional dependence of the critical exponents on h can be algebraic or logarithmic---depending on the operators involved. In addition, we evaluate the singularities of the Fourier images of the correlation functions. It turns out that switching on a magnetic field gives rise to singularities in the dynamic field-field correlation functions that are absent at h=0.
We have studied correlation functions of the one-dimensional Hubbard model using the density-matrix numerical renormalization-group approach. The singularity exponents of the momentum distribution at k=kF{\mathit{k}}_{\mathit{F}}kF​ and 3kF{\mathit{k}}_{\mathit{F}}kF​, as well as the power-law singularity of the spin-correlation function, are computed for large U (U/t=103{10}^{3}103). The momentum-distribution exponent at k=kF{\mathit{k}}_{\mathit{F}}kF​ and the spin-correlation exponent are in agreement with the analytic results, while the calculated exponent for momentum distribution at k=3kF{\mathit{k}}_{\mathit{F}}kF​ is about 3/4, which disagrees with the analytically predicted value 9/8. We also discuss several techniques for an accurate calculation of the correlation functions.
The momentum distribution and the spin-correlation functions are studied for the one-dimensional t-J model, in which the exchange interaction J is anisotropic. The discussion is focused on the nontrivial limit that J is much smaller than t. In this limit the exact ground-state wave function is a product of the spinless-fermion Slater determinant and the spin-wave function of the s=1/2 anisotropic Heisenberg antiferromagnet. With this wave function, correlation functions are calculated explicitly for finite but large systems and the nature of their singularities is studied. It is found that for the XY-like anisotropy the critical behavior of correlation functions can be described by the Tomonaga-Luttinger model, while for the Ising-like anisotropy the model seems to belong to the universality class of the Luther-Emery model of the one-dimensional electron gas.