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arXiv:1108.6064 · Physical Review B

Quantum criticality of dipolar spin chains

Aldo Isidori, Annika Ruppel, Andreas Kreisel, Peter Kopietz, Alexander Mai, R. M. Noack

Published Nov 14, 2011• 4 citations
arXivHTMLPDF

Abstract

We show that a chain of Heisenberg spins interacting with long-range dipolar forces in a magnetic field $h$ perpendicular to the chain exhibits a quantum critical point belonging to the two-dimensional Ising universality class. Within linear spin-wave theory, the magnon dispersion for small momenta $k$ is ${[{\ensuremath{\Delta}}^{2}+{v}_{k}^{2}{k}^{2}]}^{1/2}$, where ${\ensuremath{\Delta}}^{2}\ensuremath{\propto}|h\ensuremath{-}{h}_{c}|$ and ${v}_{k}^{2}\ensuremath{\propto}|\phantom{\rule{-0.16em}{0ex}}\mathrm{ln}k|$. For fields close to ${h}_{c}$ linear spin-wave theory breaks down, and we investigate the system using density-matrix and functional renormalization group methods. The Ginzburg regime where non-Gaussian fluctuations are important is found to be rather narrow on the ordered side of the transition and very broad on the disordered side.

DOI
10.1103/physrevb.84.184417

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We show that a chain of Heisenberg spins interacting with long-range dipolar forces in a magnetic field hhh perpendicular to the chain exhibits a quantum critical point belonging to the two-dimensional Ising universality class. Within linear spin-wave theory, the magnon dispersion for small momenta kkk is [\ensuremathΔ2+vk2k2]1/2{[{\ensuremath{\Delta}}^{2}+{v}_{k}^{2}{k}^{2}]}^{1/2}[\ensuremathΔ2+vk2​k2]1/2, where \ensuremathΔ2\ensuremath∝∣h\ensuremath−hc∣{\ensuremath{\Delta}}^{2}\ensuremath{\propto}|h\ensuremath{-}{h}_{c}|\ensuremathΔ2\ensuremath∝∣h\ensuremath−hc​∣ and vk2\ensuremath∝∣lnk∣{v}_{k}^{2}\ensuremath{\propto}|\phantom{\rule{-0.16em}{0ex}}\mathrm{ln}k|vk2​\ensuremath∝∣lnk∣. For fields close to hc{h}_{c}hc​ linear spin-wave theory breaks down, and we investigate the system using density-matrix and functional renormalization group methods. The Ginzburg regime where non-Gaussian fluctuations are important is found to be rather narrow on the ordered side of the transition and very broad on the disordered side.
We study a one-dimensional spin-chain interacting with long-range dipolar forces in the presence of a transverse magnetic field. The system exhibits a quantum critical point belonging to the two-dimensional Ising universality class, whose critical behavior is investigated using the nonperturbative density-matrix and functional renormalization group methods. Interestingly, we find that linear spin-wave theory predicts a magnon dispersion characterized by a logarithmic singularity in the magnon velocity for vanishing momenta, due to the peculiar behavior of dipole-dipole interactions in one dimension. However, in the vicinity of the critical point this logarithmically divergent term is renormalized to zero by the effects of quantum fluctuations, signaling the reemergence of scale invariance, in accordance with the Ising critical scenario.
We study the role of long-range dipolar interactions on antiferromagnetic spin chains, from the classical $S\ensuremath{\rightarrow}\ensuremath{\infty}$ limit to the deep quantum case S=1/2S=1/2S=1/2, including a transverse magnetic field. To this end, we combine different techniques such as classical energy minima, classical Monte Carlo, linear spin waves, bosonization, and density matrix renormalization group (DMRG). We find a phase transition from the already reported dipolar ferromagnetic region to an antiferromagnetic region for high enough antiferromagnetic exchange. Thermal and quantum fluctuations destabilize the classical order before reaching magnetic saturation in both phases, and also close to zero field in the antiferromagnetic phase. In the extreme quantum limit S=1/2S=1/2S=1/2, extensive DMRG computations show that the main phases remain present with transition lines to saturation significatively shifted to lower fields, in agreement with the bosonization analysis. The overall picture maintains a close analogy with the phase diagram of the anisotropic XXZ spin chain in a transverse field.
The density-matrix renormalization-group technique is employed to show explicitly that S=2S=2S=2 quantum spin chains belong to the same universality class as S=1S=1S=1 spin chains. The dispersion of the single-magnon band for periodic chains is shown to be $\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{q}^{2}}$, where $\ensuremath{\Delta}$ is the Haldane gap and ${\ensuremath{\omega}}_{q}=v|\mathrm{sin}(\ensuremath{\pi}\ensuremath{-}q)|$ with $v=\ensuremath{\Delta}\ensuremath{\xi}$, $\ensuremath{\xi}$ as the correlation length, $\ensuremath{\pi}\ensuremath{-}q=2\ensuremath{\pi}n/L$, LLL as the chain length, and nnn as an integer. Moreover, the existence of spin-1/21/21/2 topological excitations due to the breaking of one singlet bond is demonstrated by finite-size scaling of correlation functions decaying as $\mathrm{exp}(\ensuremath{-}r/\ensuremath{\xi})/\sqrt{L}$. These stringent numerical tests, as well as the repulsive interactions between magnons, provide strong evidence in favor of the existence of a universality class for integer quantum spin chains conjectured by Haldane.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
Antiferromagnetic Heisenberg integer-spin chains are characterized by a spin-liquid ground state with no long-range order, due to the relevance of quantum fluctuations. Spin anisotropy, however, freezes quantum fluctuations, and the system is magnetized in the presence of a sufficiently large easy-axis anisotropy. We numerically investigate the case S=1, by means of the density-matrix renormalization group, and find that the freezing of the spin liquid into a Néel spin solid, with increasing easy-axis anisotropy, is a continuous quantum phase transition. Numerical evidence indicates that the transition is not in the two-dimensional Ising universality class.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.