Boundary effect in random spin systems

We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$ . We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$ . Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$. Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

Physical ReviewDec 10, 1969118 citations

Effect of randomness on spin systems

Yutaka Okabe, Kazuhisa Kaneda, Yusuke Tomita +5 more

Jan 1, 19980 citations

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

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.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.

arXiv:cond-mat/9706096Physical review. B, Condensed matterApr 1, 199845 citations

Boundary-driven Lindblad dynamics of random quantum spin chains: strong disorder approach for the relaxation, the steady state and the current

Cécile Monthus

Abstract The Lindblad dynamics of the XX quantum chain with large random fields h j (the couplings J j can be either uniform or random) is considered for boundary-magnetization-drivings acting on the two end-spins. Since each boundary-reservoir tends to impose its own magnetization, we first study the relaxation spectrum in the presence of a single reservoir as a function of the system size via some boundary-strong-disorder renormalization approach. The non-equilibrium-steady-state in the presence of two reservoirs can be then analyzed from the effective renormalized Linbladians associated to the two reservoirs. The magnetization is found to follow a step profile, as found previously in other localized chains. The strong disorder approach allows to compute explicitly the location of the step of the magnetization profile and the corresponding magnetization-current for each disordered sample in terms of the random fields and couplings.Abstract The Lindblad dynamics of the XX quantum chain with large random fields h j (the couplings J j can be either uniform or random) is considered for boundary-magnetization-drivings acting on the two end-spins. Since each boundary-reservoir tends to impose its own magnetization, we first study the relaxation spectrum in the presence of a single reservoir as a function of the system size via some boundary-strong-disorder renormalization approach. The non-equilibrium-steady-state in the presence of two reservoirs can be then analyzed from the effective renormalized Linbladians associated to the two reservoirs. The magnetization is found to follow a step profile, as found previously in other localized chains. The strong disorder approach allows to compute explicitly the location of the step of the magnetization profile and the corresponding magnetization-current for each disordered sample in terms of the random fields and couplings.

arXiv:1701.02102Journal of Statistical Mechanics Theory and ExperimentApr 1, 201720 citations

\mathfrak{H}

. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about

{T}_{c}

. Furthermore, at

{T}_{c}

the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At

{T}_{c}

this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when

m

the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

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.