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AIP conference proceedings

Functional renormalization group study of a dipolar spin chain in a transverse magnetic field

Aldo Isidori, Andreas Kreisel

Published Jan 1, 2012• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.4755828

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•Jan 1, 1997•8 citations
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 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 ${[{\ensuremath{\Delta}}^{2}+{v}_{k}^{2}{k}^{2}]}^{1/2}$, 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 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 have studied the antiferromagnetic Ising chain in a transverse magnetic field hx{h}_{x}hx​ and uniform longitudinal field hz.{h}_{z}.hz​. Using the density-matrix renormalization group calculation combined with a finite-size scaling the ground-state phase diagram in (hx,hz){(h}_{x}{,h}_{z})(hx​,hz​) plane is determined. It is shown that there is an order-disordered transition line in this plane and the critical properties belong to the universality class of the two-dimensional Ising model. Based on the perturbation theory in hz{h}_{z}hz​ the scaling behavior of the mass gap in the vicinity of the critical point (hx=1/2,hz=0){(h}_{x}{=1/2,h}_{z}=0)(hx​=1/2,hz​=0) is established. It is found that the form of the transition line near the classical multicritical point (hx=0,hz=1){(h}_{x}{=0,h}_{z}=1)(hx​=0,hz​=1) is linear. The connection of the considered quantum model with the quasi-one-dimensional classical Ising model in the magnetic field is discussed.
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.
Fisher's [Phys. Rev. B 51,51,51, 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian: