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arXiv:cond-mat/9706096 · Physical review. B, Condensed matter

Random-mass Dirac fermions in doped spin-Peierls and spin-ladder systems: One-particle properties and boundary effects

M. Steiner, Michele Fabrizio, Alexander O. Gogolin

Published Apr 1, 1998• 45 citations
arXivHTMLPDF

Abstract

Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as $1/\sqrt{x}$ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.

DOI
10.1103/physrevb.57.8290

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Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, the single-particle Green function decays in real space like G(x,\ensuremathϵ)\ensuremath∝(1/x)3/2G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}G(x,\ensuremathϵ)\ensuremath∝(1/x)3/2. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances x\ensuremath∼L\ensuremathϵ\ensuremath∝ln2(1/\ensuremathϵ)x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})x\ensuremath∼L\ensuremathϵ​\ensuremath∝ln2(1/\ensuremathϵ), the Green function crosses over to a standard exponential decay. The scale L\ensuremathϵ{L}_{\ensuremath{\epsilon}}L\ensuremathϵ​ is different from the Thouless localization length \ensuremathλ\ensuremathϵ\ensuremath∝ln(1/\ensuremathϵ){\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})\ensuremathλ\ensuremathϵ​\ensuremath∝ln(1/\ensuremathϵ). Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as 1/x1/\sqrt{x}1/x​ saturating at x\ensuremath∼L\ensuremathϵ.x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.x\ensuremath∼L\ensuremathϵ​. We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.
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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.
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We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.
We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.