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arXiv:cond-mat/0208580 · Physical review. B, Condensed matter

Luttinger liquid behavior in spin chains with a magnetic field

Gábor Fáth

Published Oct 24, 2003• 33 citations
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Abstract

Antiferromagnetic Heisenberg spin chains in a sufficiently strong magnetic field are Luttinger liquids, whose parameters depend on the actual magnetization of the chain. Here we present precise numerical estimates of the Luttinger liquid dressed charge Z, which determines the critical exponents, by calculating the magnetization and quadrupole operator profiles for $S=1/2$ and $S=1$ chains using the density matrix renormalization group method. Critical amplitudes and the scattering length at the chain ends are also determined. Although both systems are Luttinger liquids the characteristic parameters differ considerably.

DOI
10.1103/physrevb.68.134445

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Antiferromagnetic Heisenberg spin chains in a sufficiently strong magnetic field are Luttinger liquids, whose parameters depend on the actual magnetization of the chain. Here we present precise numerical estimates of the Luttinger liquid dressed charge Z, which determines the critical exponents, by calculating the magnetization and quadrupole operator profiles for S=1/2S=1/2S=1/2 and S=1S=1S=1 chains using the density matrix renormalization group method. Critical amplitudes and the scattering length at the chain ends are also determined. Although both systems are Luttinger liquids the characteristic parameters differ considerably.
A recently derived general formula and older numerical results are combined to deduce the behavior of the transverse correlation exponent for the spin-1 Heisenberg antiferromagnetic chain in an applied magnetic field $\ensuremath{\eta}\ensuremath{\approx}1∕2\ensuremath{-}(2.0)m+O({m}^{2})$, where mmm is the magnetization per site. A comparison with the O(3) nonlinear $\ensuremath{\sigma}$ model is also made.
We study the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron-scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighboring chains.
The S =1/2 bond alternating Heisenberg spin chain in a magnetic field is studied by the numerical diagonalization and the finite size scaling based on the conformal field theory. It is found that the magnetic state (0< m <1) is gapless (the central charge c =1) and described by the Luttinger liquid, which corresponds to a spinless Fermion system with repulsive interaction for antiferromagnetic-antiferromagnetic alternating chains, but attractive one for ferromagnetic-antiferromagnetic chains. The critical exponents δ and η are determined at the critical fields. The results suggest that the phase transitions belong to the same universality class as the S =1 antiferromagnetic chain for both alternating cases.
We report a combined analytical and density matrix renormalized group study of the antiferromagnetic XXZXXZXXZ spin-1/21/21/2 Heisenberg chain subject to a uniform Dzyaloshinskii-Moriya (DM) interaction and a transverse magnetic field. The numerically determined phase diagram of this model, which features two ordered Ising phases and a critical Luttinger liquid, one with fully broken spin-rotational symmetry, agrees well with the predictions of Garate and Affleck [I. Garate and I. Affleck, Phys. Rev. B 81, 144419 (2010)]. We also confirm the prevalence of the Nz{N}^{z}Nz N\'eel Ising order in the regime of comparable DM and magnetic field magnitudes.
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We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.