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arXiv:1207.6337 · Physical Review B

Weak localization and magnetoresistance in a two-leg ladder model

Michael P. Schneider, Sam T. Carr, I. V. Gornyi, A. D. Mirlin

Published Oct 23, 2012• 2 citations
arXivHTMLPDF

Abstract

We analyze the weak-localization correction to the conductivity of a spinless two-leg ladder model in the limit of strong dephasing ${\ensuremath{\tau}}_{\ensuremath{\varphi}}\ensuremath{\ll}{\ensuremath{\tau}}_{\mathrm{tr}}$, paying particular attention to the presence of a magnetic field, which leads to an unconventional magnetoresistance behavior. We find that the magnetic field leads to three different effects: (i) negative magnetoresistance due to the regular weak-localization correction, (ii) effective decoupling of the two chains, leading to positive magnetoresistance, and (iii) oscillations in the magnetoresistance originating from the nature of the low-energy collective excitations. All three effects can be observed depending on the parameter range, but it turns out that large magnetic fields always decouple the chains and thus lead to the curious effect of magnetic-field-enhanced localization.

DOI
10.1103/physrevb.86.155141

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We analyze the weak-localization correction to the conductivity of a spinless two-leg ladder model in the limit of strong dephasing \ensuremathτ\ensuremathφ\ensuremath≪\ensuremathτtr{\ensuremath{\tau}}_{\ensuremath{\varphi}}\ensuremath{\ll}{\ensuremath{\tau}}_{\mathrm{tr}}\ensuremathτ\ensuremathφ​\ensuremath≪\ensuremathτtr​, paying particular attention to the presence of a magnetic field, which leads to an unconventional magnetoresistance behavior. We find that the magnetic field leads to three different effects: (i) negative magnetoresistance due to the regular weak-localization correction, (ii) effective decoupling of the two chains, leading to positive magnetoresistance, and (iii) oscillations in the magnetoresistance originating from the nature of the low-energy collective excitations. All three effects can be observed depending on the parameter range, but it turns out that large magnetic fields always decouple the chains and thus lead to the curious effect of magnetic-field-enhanced localization.
This paper considers a finite-sized two-leg ladder of spin-1/2 particles with local inter-chain Ising interaction. We include the effects of an external magnetic field by incorporating a Zeeman-type coupling between the magnetic field and the spins localized on the lattice sites. To analyze the system, we utilize the transfer-matrix formalism to calculate the partition function. We compute the magnetization for different configurations by varying the system's parameters. We observe the presence 3of spontaneous magnetization within the system. As the inter-chain Ising interaction changes its sign, the magnetization value changes from negative to positive values. Furthermore, we identify a phase transition from diamagnetic to ferromagnetic states when examining the temperature dependence of the magnetization at zero magnetic fields. We observe a step-like behavior in the magnetization curve in the low temperature limit.
The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin-1/21/21/2 ladder, and a two-legged spin-1/21/21/2 ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an XXZ\mathrm{XXZ}XXZ spin-1/21/21/2 chain in a magnetic field; the second-order terms give corrections to the XXZ\mathrm{XXZ}XXZ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.
We study the magnetic orbital effect of a doped two-leg ladder in the presence of a magnetic field component perpendicular to the ladder plane. Combining both low-energy approach (bosonization) and numerical simulations (density-matrix renormalization group) on the strong coupling limit ($t\text{\ensuremath{-}}J$ model), a rich phase diagram is established as a function of hole doping and magnetic flux. Above a critical flux, the spin gap is destroyed and a Luttinger liquid phase is stabilized. Above a second critical flux, a reentrance of the spin gap at high magnetic flux is found. Interestingly, the phase transitions are associated with a change of sign of the orbital susceptibility. Focusing on the small magnetic field regime, the spin-gapped superconducting phase is robust, but immediately acquires algebraic transverse (i.e., along rungs) current correlations which are commensurate with the 4kF4{k}_{F}4kF​ density correlations. In addition, we have computed the zero-field orbital susceptibility for a large range of doping and interaction ratio J∕tJ∕tJ∕t: we found strong anomalies at low J∕tJ∕tJ∕t only in the vicinity of the commensurate fillings corresponding to $\ensuremath{\delta}=1∕4$ and 1∕21∕21∕2. Furthermore, the behavior of the orbital susceptibility reveals that the nature of these insulating phases is different: while for $\ensuremath{\delta}=1∕4$ a 4kF4{k}_{F}4kF​ charge density wave is confirmed, the $\ensuremath{\delta}=1∕2$ phase is shown to be a bond order wave.
In this paper we continue and extend a systematic study of plateaux in magnetization curves of antiferromagnetic Heisenberg spin-1/2 ladders. We first review a bosonic field-theoretical formulation of a single XXZ\mathrm{XXZ}XXZ chain in the presence of a magnetic field, which is then used for an Abelian bosonization analysis of NNN weakly coupled chains. Predictions for the universality classes of the phase transitions at the plateaux boundaries are obtained in addition to a quantization condition for the value of the magnetization on a plateau. These results are complemented by and checked against strong-coupling expansions. Finally, we analyze the strong-coupling effective Hamiltonian for an odd number NNN of cylindrically coupled chains numerically. For N=3N=3N=3 we explicitly observe a spin gap with a massive spinon-type fundamental excitation and obtain indications that this gap probably survives the limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$.
We analyze an anisotropic spin-1/2 two-leg ladder in the presence of various types of random perturbations. The generic phase diagram for the pure system, in a way similar to spin one chains, consists of four phases: an antiferromagnet, a Haldane gap or a singlet phase (depending on the sign of interchain coupling), and two XY\mathrm{XY}XY phases designated by XY1\mathrm{XY}1XY1 and XY2\mathrm{XY}2XY2. The effects of disorder depend crucially on whether it preserves XY\mathrm{XY}XY symmetry (random field along zzz and random exchange) or not (random anisotropy or random XY\mathrm{XY}XY fields). In all cases we computed the new phase diagram and the correlation length for the disordered system. The ladder exhibits a remarkable stability to disorder with XY\mathrm{XY}XY symmetry. Not only the singlet phases but also the massless XY1\mathrm{XY}1XY1 phase are totally unaffected by disorder in stark contrast with the single chain case. Even in the presence of an external magnetic field breaking the spin gap most of the phase diagram (including the XY\mathrm{XY}XY point) remains unaffected by the disorder, again in opposition with the single chain case. This stabilization towards disorder is similar, albeit stronger, to the one occurring for fermionic ladders. On the other hand, the XY2\mathrm{XY}2XY2 phase is very strongly suppressed. Disorder breaking XY\mathrm{XY}XY symmetry has much stronger effects and most of the phases at the exception of the singlet one are now destroyed. Interestingly, the disordered XY1\mathrm{XY}1XY1 phase has a much shorter correlation length than the disordered XY2\mathrm{XY}2XY2 phase contrarily with the case of perturbations preserving the rotation symmetry around the zzz axis. The ladder system is thus in fact much more anisotropic than its single chain counterpart with identical exchanges. Finally we examine the case of strong disorder or weak interchain exchange for XY\mathrm{XY}XY symmetric disorder. Close to the isotropic point, when the interchain exchange is increased the system undergoes a transition between two decoupled disordered spin-1/2 chains to a singlet ladder unaffected by disorder. For more XY\mathrm{XY}XY like systems, the transition occurs in two steps with a nondisordered XY1\mathrm{XY}1XY1 phase occurring between the decoupled chains and the singlet phase. Comparison with other disordered gapped systems such as spin one chains is discussed.