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Physical review. B, Condensed matter

Density-matrix renormalization-group calculation of correlation functions in the one-dimensional Hubbard model

Shaojin Qin, Shoudan Liang, Zhao-Bin Su, Lu Yu

Published Aug 15, 1995• 30 citations
Publisher

Abstract

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=${\mathit{k}}_{\mathit{F}}$ and 3${\mathit{k}}_{\mathit{F}}$, as well as the power-law singularity of the spin-correlation function, are computed for large U (U/t=${10}^{3}$). The momentum-distribution exponent at k=${\mathit{k}}_{\mathit{F}}$ and the spin-correlation exponent are in agreement with the analytic results, while the calculated exponent for momentum distribution at k=3${\mathit{k}}_{\mathit{F}}$ 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.

DOI
10.1103/physrevb.52.r5475

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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 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 Szz{\mathit{S}}^{\mathit{z}\mathit{z}}Szz(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 Sxx{\mathit{S}}^{\mathit{x}\mathit{x}}Sxx(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.
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 calculate 2kF2{k}_{F}2kF​ and 4kF4{k}_{F}4kF​ charge-density wave (CDW) and spin-density wave (SDW) autocorrelation functions for the one-dimensional Hubbard model using a computer renormalization-group technique previously developed by us. We describe the method and expected accuracy of the results. The correlation functions are calculated for Hubbard parameter Ut=2,10,50\frac{U}{t}=2,10,50tU​=2,10,50 and for two different band fillings. We obtain, describe, and compare for the first time the magnitudes and temperature dependences of the correlation functions. We find, inter alia, that the 4kF4{k}_{F}4kF​ instability arises at very large UUU and that the 2kF2{k}_{F}2kF​ SDW and 4kF4{k}_{F}4kF​ CDW dominate at large UUU.
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 large-distance behavior of the spin-spin correlation function of the one-dimensional repulsive Hubbard model is evaluated analytically in the strong-coupling regime at quarter filling. In this case, its power-law decay is characterized by an exponent \ensuremath{\gamma}=3/2. We have found that this behavior is generally valid at any nonzero doping, although our argument is not mathematically rigorous away from quarter filling. These results strongly suggest that the renormalization-group scaling to the Tomonaga-Luttinger model is exact in the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model.