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arXiv:cond-mat/0310391 · Physical Review B

Correlation amplitudes for the spin- 12 XXZ chain in a magnetic field

Takashi Hikihara, Akira Furusaki

Published Feb 24, 2004• 102 citations
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Abstract

We present accurate numerical estimates for the correlation amplitudes of leading and main subleading terms of the two- and four-spin correlation functions in the one-dimensional spin-$1/2$ $\mathrm{XXZ}$ model under a magnetic field. These data are obtained by fitting the correlation functions, computed numerically with the density-matrix renormalization-group method, to the corresponding correlation functions in the low-energy effective theory. For this purpose we have developed the Abelian bosonization approach to the spin chain under the open-boundary conditions. We use the numerical data of the correlation amplitudes to quantitatively estimate spin gaps induced by a transverse staggered field and by exchange anisotropy.

DOI
10.1103/physrevb.69.064427

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The density-matrix renormalization-group technique is used to calculate the spin-correlation functions 〈SjxSkx〉〈{S}_{j}^{x}{S}_{k}^{x}〉〈Sjx​Skx​〉 and 〈SjzSkz〉〈{S}_{j}^{z}{S}_{k}^{z}〉〈Sjz​Skz​〉 of the one-dimensional S=12S=\frac{1}{2}S=21​ XXZ\mathrm{XXZ}XXZ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerically.
Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-12\frac{1}{2}21​ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field through the combined use of bosonization and density-matrix renormalization-group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.
Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-12\frac{1}{2}21​ representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy UUU. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-12\frac{1}{2}21​ XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of UUU and for a number LLL of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to 1010%10 between the output of analytical and numerical computations, even for relatively small values of UUU. Our analysis implies that, also at finite UUU, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.
By exploiting density-matrix renormalization group techniques, we investigate the spin dynamics of a spin-1/2 one-dimensional J 1 - J 2 XXZ model with competing ferromagnetic J 1 and antiferromagnetic J 2 exchange couplings under applied magnetic fields. Numerical results of spin excitation spectra show that in the field-induced spin quadrupole regime, the longitudinal component has a gapless mode and the transverse component has a gapped mode irrespective of the exchange anisotropy. The excitation gap of the transverse spin excitation increases as the exchange anisotropy increases over the XY-like and Ising-like regions, demonstrating that two-magnon bound states are stabilized due to the easy-axis anisotropy.
We study the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain in both uniform and (perpendicular) staggered magnetic fields using the density-matrix renormalization-group method. This model has been shown earlier to describe the physics of the copper benzoate materials in magnetic field. In the present work, we extend the study to more general case for a systematic investigation of the field-induced gap and related properties of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain. In particular, we explore the high magnetic field regime where interesting behaviors in the field-induced gap, magnetization, and spin correlation functions are found. Careful examination of the low energy properties and magnetization reveals interesting competing effects of the staggered and uniform fields. The incommensurate behavior in the spin correlation functions is demonstrated and discussed in detail. The present work reproduces earlier results in good agreement with experimental data on copper benzoate and predicts new interesting field-induced features at very high magnetic field.
Using the density-matrix-renormalization-group technique, we calculate numerically the low-energy excitation spectrum and magnetization curve of the spin-1 antiferromagnetic chain in a staggered magnetic field, which is expected to describe the physics of the ${R}_{2}{\mathrm{BaNiO}}_{5} (R\ensuremath{\ne}Y)$ family below the N\'eel temperature of the magnetic rare-earth (R) sublattice. These results are valid in the entire range of the staggered field, and agree with those given by the nonlinear $\ensuremath{\sigma}$ model study for small fields, but differ from the latter for large fields. They are consistent with the available experimental data. The correlation functions for this model are also calculated. The transverse correlations display the anticipated exponential decay with shorter correlation length, while the longitudinal correlations show explicitly the induced staggered magnetization.