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arXiv:cond-mat/0604405 · Physical Review B

Gap exponent of theXXZmodel in a transverse field

A. Langari, S. Mahdavifar

Published Feb 6, 2006• 22 citations
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Abstract

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 $\ensuremath{-}1<\ensuremath{\Delta}<1$.

DOI
10.1103/physrevb.73.054410

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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→\ensuremath∞)(N\ensuremath{\rightarrow}\ensuremath{\infty})(N\ensuremath→\ensuremath∞). 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 (\ensuremathΔ)(\ensuremath{\Delta})(\ensuremathΔ). Our numerical results are in good agreement with the field theoretical approach in the whole range of the anisotropy parameter \ensuremath−1<\ensuremathΔ<1\ensuremath{-}1<\ensuremath{\Delta}<1\ensuremath−1<\ensuremathΔ<1.
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)).
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.
The scaling theory of finite size effects in the limiting bulk behaviour is extended to treat crossover phenomena. The method is used to study quantum critical phenomena in the XY chain in a transverse field at zero temperature, through the scaling of the longitudinal susceptibilities and of the energy gaps between the ground and two first excited states. As expected, no abrupt change in critical exponents is observed for small anisotropy. Because of the finiteness of the system, but the limiting isotropic and anisotropic regions display quite distinct critical behaviour, in good agreement with known results. Another interesting result obtained is that the two energy gaps examined vanish at the critical line with the same critical exponent.
The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents zzz and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.