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Journal of Physics Condensed Matter

Scaling behavior of the energy gap of the XXZ model in a transverse magnetic field

S. Mahdavifar

Published Sep 21, 2007• 4 citations
Publisher

Abstract

The energy gap of the one-dimensional (1D) spin-[Formula: see text] AF (antiferromagnetic) XXZ model in a transverse magnetic field (h) is studied by using the exact diagonalization technique. We have computed the energy gap in the region of anisotropy parameter -1<Δ<1 and in the vicinity of the critical line Δ = -1. By introducing a practical finite-size scaling approach, we have computed the correct critical exponent of the energy gap in good agreement with the field theoretical and spin-wave approaches. To check this approach, we have also studied the critical exponent of the energy gap of the 1D AF-Heisenberg chain in a staggered magnetic field (h(s)).

DOI
10.1088/0953-8984/19/40/406222

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The energy gap of the one-dimensional (1D) spin-[Formula: see text] AF (antiferromagnetic) XXZ model in a transverse magnetic field (h) is studied by using the exact diagonalization technique. We have computed the energy gap in the region of anisotropy parameter -1<Δ<1 and in the vicinity of the critical line Δ = -1. By introducing a practical finite-size scaling approach, we have computed the correct critical exponent of the energy gap in good agreement with the field theoretical and spin-wave approaches. To check this approach, we have also studied the critical exponent of the energy gap of the 1D AF-Heisenberg chain in a staggered magnetic field (h(s)).
We have calculated numerically the gap exponent of the anisotropic Heisenberg model in the presence of a transverse magnetic field. We have implemented the modified Lanczos method to obtain the excited states of our model with the same accuracy as the ground state. The coefficient of the leading term in the perturbation expansion diverges in the thermodynamic limit $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. We have obtained the relation between this divergence and the scaling behavior of the energy gap. We have found that the opening of the gap in the presence of a transverse field scales with a critical exponent which depends on the anisotropy parameter $(\ensuremath{\Delta})$. Our numerical results are in good agreement with the field theoretical approach in the whole range of the anisotropy parameter $\ensuremath{-}1<\ensuremath{\Delta}<1$.
The one-dimensional spin-1/2 XXZ model in a transverse magnetic field is studied. It is shown that the field induces a gap in the spectrum of the model with the easy-plane anisotropy. Using conformal invariance, the field dependence of the gap is found at small fields. The ground state phase diagram is obtained. It contains four phases with the long-range order of different types and a disordered phase. These phases are separated by critical lines, where the gap and the long-range order vanish. Using scaling estimates, the mean-field approach, and numerical calculations in the vicinity of all critical lines, we find the critical exponents of the gap and the long-range order. It is shown that the transition line between the ordered and disordered phases belongs to the universality class of the transverse Ising model.
Scaling behavior of spin gap of a bond alternating spin-1/2 anisotropic Heisenberg chain has been studied both in ferromagnetic (FM) and antiferromagnetic (AFM) cases. Spin gap has been estimated by using exact diagonalization technique. All those quantities have been obtained for a region of anisotropic parameter Δ defined by 0≤Δ≤1. Spin gap is found to develop as soon as the non-uniformity in the alternating bond strength is introduced in the AFM regime which furthermore sustains in the FM regime as well. Scaling behavior of the spin gap has been studied by introducing scaling exponent. The variation of scaling exponents with Δ is fitted with a regular function.
We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field h3h_{3}h3​ and a small periodic field of strength h1h_{1}h1​. For the case of a transverse staggered field a gap opens, which scales with h1ε1h_{1}^{ε_{1}}h1ε1​​, where ε1=ε1(h3)ε_{1}=ε_{1}(h_{3})ε1​=ε1​(h3​) is given by the critical exponent η1(M(h3))η_{1}(M(h_{3}))η1​(M(h3​)) defined through the transverse structure factor of the model at h1=0h_{1}=0h1​=0. For the case of a longitudinal periodic field with wave vector q=π/2q=π/2q=π/2 and strength hqh_{q}hq​ a plateau is found in the magnetization curve at M=1/4M=1/4M=1/4. The difference of the upper- and lower magnetic field scales with h3u−h3l∼hqε3h_{3}^{u}-h_{3}^{l}\sim h_{q}^{ε_{3}}h3u​−h3l​∼hqε3​​, where ε3=ε3(h3)ε_{3}=ε_{3}(h_{3})ε3​=ε3​(h3​) is given by the critical exponent η3(M(h3))η_{3}(M(h_{3}))η3​(M(h3​)) defined through the longitudinal structure factor of the model at hq=0h_{q}=0hq​=0.
We apply a finite-size scaling approach to the one-dimensional spin-1/21/21/2 Ising model with nearest- and next-nearest-neighbor interactions in the presence of a transverse magnetic field. By using the scaling behavior of the energy gap we are able not only to determine the ferromagnetic-paramagnetic transition line at zero absolute temperature but also to compute the corresponding critical exponent. A comparison with other approaches is made, and aside for small discrepancies likely to occur at the thermodynamic limit, we believe that the present results are quite close to the unknown exact ones.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.