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Brazilian Journal of Physics

The effective charge velocity of spin-¹/2 superlattices

Jereson Silva-Valencia, R. Franco, Marcos Sergio Figueira

Published Sep 1, 2006• 0 citations
Publisher

Abstract

We calculate the spin gap of homogeneous and inhomogeneous spin chains, using the White's density matrix renormalization group technique. We found that the spin gap is related to the ration between the spin velocity and the correlation exponent. We consider a spin superlattice, which is composed of a repeated pattern of two spin-¹/2 XXZ chains with different anisotropy parameters. The behavior of the charge velocity as a function of the anisotropy parameter and the relative size of sub-chains was investigated. We found reasonable agreement between the bosonization results and the numerical ones.

DOI
10.1590/s0103-97332006000600035

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We calculate the spin gap of homogeneous and inhomogeneous spin chains, using the White's density matrix renormalization group technique. We found that the spin gap is related to the ration between the spin velocity and the correlation exponent. We consider a spin superlattice, which is composed of a repeated pattern of two spin-¹/2 XXZ chains with different anisotropy parameters. The behavior of the charge velocity as a function of the anisotropy parameter and the relative size of sub-chains was investigated. We found reasonable agreement between the bosonization results and the numerical ones.
Using the density matrix renormalization group technique, we study spin superlattices composed of a repeated pattern of two spin-12XXZ\frac{1}{2}\phantom{\rule{0.3em}{0ex}}XXZ21​XXZ chains with different anisotropy parameters. The magnetization curve can exhibit two plateaus, a nontrivial plateau with the magnetization value given by the relative sizes of the subchains and another trivial plateau with zero magnetization. We find good agreement of the value and the width of the plateaus with the analytical results obtained previously. In the gapless regions away from the plateaus, we compare the finite-size spin gap with the predictions based on bosonization and find reasonable agreement. These results confirm the validity of the Tomonaga-Luttinger liquid superlattice description of these systems.
We study spin superlattices composed of a repeated pattern of two long spin-12\frac{1}{2}21​ XXZ chains with different anisotropy parameters. They can be viewed as the limit of p-merized chains when the number of sites per cluster is very large. Magnetization plateaus are found, with magnetization values that depend on the relative sizes of the subchains. In certain regions of parameter space, the low-energy properties are described in terms of a Luttinger liquid superlattice parametrized by an effective velocity and an effective compactification radius.
We have calculated the charge gap and spin gap for the two-chain Hubbard model as a function of the on-site Coulomb interaction and the interchain hopping amplitude. We used the density matrix renormalization group method and developed a method to calculate separately the gaps numerically for the symmetric and antisymmetric modes with respect to the exchange of the chain indices. We have found very different behaviors for the weak and strong interaction cases. Our calculated phase diagram is compared to the one obtained by Balents and Fisher using the weak coupling renormalization group technique.
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-1/21/21/2 XXZ\mathrm{XXZ}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.
The anisotropic XXZXXZXXZ spin-12\frac{1}{2}21​ Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction sixsjx+siysjy{s}_{i}^{x}{s}_{j}^{x}+{s}_{i}^{y}{s}_{j}^{y}six​sjx​+siy​sjy​ induces an antiferromagnetic correlation in the out-of-plane siz{s}_{i}^{z}siz​ component, at higher temperatures in the antiferromagnetic XXZXXZXXZ chain, dominantly at low temperatures in the ferromagnetic XXZXXZXXZ chain, and, in-between, at all temperatures in the XYXYXY chain. We find that the converse effect also occurs in the antiferromagnetic XXZXXZXXZ chain: an antiferromagnetic sizsjz{s}_{i}^{z}{s}_{j}^{z}siz​sjz​ interaction induces a correlation in the sixy{s}_{i}^{xy}sixy​ component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XYXYXY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.
We investigate Luttinger liquid superlattices, periodic structures composed of two kinds of one-dimensional systems of interacting electrons. We calculate several properties of the low-energy sector: the effective charge and spin velocities, the compressibility, various correlation functions, the Landauer conductance, and the Drude weight. The low-energy properties are subsumed into effective parameters, much like homogeneous one-dimensional systems. A generic result is the weighted average nature of these parameters, in proportion to the spatial extent of the underlying subunits, pointing to the possibility of ``engineered'' structures. As a specific realization, we consider a one-dimensional Hubbard superlattice, which consists of a periodic arrangement of two long Hubbard chains with different coupling constants and different hopping amplitudes. This system exhibits a rich phase diagram with several phases, both metallic and insulating. We have found that gapless insulating phases are present over a wide range of parameters.