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Scientific Reports

Spin conductivity in two-dimensional Heisenberg model on Lieb lattice under magnetic field and spin-orbit interactions

Farshad Azizi, Hamed Rezania

Published Jul 2, 2025• 12 citations
Publisher

Abstract

We investigate here the impact of an out-of-plane magnetic field and spin-orbit interaction on the spin conductivity of the two-dimensional Heisenberg model on a Lieb lattice. In this study, the Hamiltonian of the spin model has been transformed into a strongly interacting bosonic gas using a hard boson transformation. In this transformation, the occupation of a boson at each site is restricted by adding a hard core repulsion. To determine the excitation spectrum of the mapped model, the Green's function method has been employed. Based on the spectrum of the bosonic gas, the two-particle Green's function related to the spin conductivity of the two-dimensional Heisenberg model has been calculated. Computational results indicate that with an increase in the strength of the Dzyaloshinskii-Moriya interaction, the peak position in the dynamic spin conductivity shifts to higher frequencies under a constant magnetic field. However, the magnetic field does not affect the peak position of the dynamic spin conductivity. On the other hand, the intensity of dynamic spin conductivity increases with the strength of the Dzyaloshinskii-Moriya interaction. Our findings suggest that for a range of values of Dzyaloshinskii-Moriya interaction strength, the static transverse structure factor continuously decreases with the magnetic field. Additionally, for each value of the magnetic field, the temperature dependence of the static spin conductivity of localized electrons on the lattice exhibits a limited temperature peak.

DOI
10.1038/s41598-025-06018-5

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We investigate here the impact of an out-of-plane magnetic field and spin-orbit interaction on the spin conductivity of the two-dimensional Heisenberg model on a Lieb lattice. In this study, the Hamiltonian of the spin model has been transformed into a strongly interacting bosonic gas using a hard boson transformation. In this transformation, the occupation of a boson at each site is restricted by adding a hard core repulsion. To determine the excitation spectrum of the mapped model, the Green's function method has been employed. Based on the spectrum of the bosonic gas, the two-particle Green's function related to the spin conductivity of the two-dimensional Heisenberg model has been calculated. Computational results indicate that with an increase in the strength of the Dzyaloshinskii-Moriya interaction, the peak position in the dynamic spin conductivity shifts to higher frequencies under a constant magnetic field. However, the magnetic field does not affect the peak position of the dynamic spin conductivity. On the other hand, the intensity of dynamic spin conductivity increases with the strength of the Dzyaloshinskii-Moriya interaction. Our findings suggest that for a range of values of Dzyaloshinskii-Moriya interaction strength, the static transverse structure factor continuously decreases with the magnetic field. Additionally, for each value of the magnetic field, the temperature dependence of the static spin conductivity of localized electrons on the lattice exhibits a limited temperature peak.
We address a detailed analysis of the dynamical spin conductivity of an anisotropic spin ladder in the direction of the legs by means of a quantum many-particle approach. Bond operator transformation has been employed in order to map the original spin model Hamiltonian to a bosonic one. This bosonic Hamiltonian should include an infinite hard-core repulsion to preserve the SU(2) algebra of the original model. Using a Green's function approach, the energy spectrum of quasiparticle excitation has been obtained. Linear response theory has been implemented to obtain spin conductivity via calculation of time-ordered energy current correlation. The results show the spin conductivity possessing a nonzero value in a finite frequency interval. Increasing the rung-coupling exchange constant moved this frequency interval to higher frequencies. Also, the spin conductivity presents a small nonzero value at very low frequency for low rung-coupling constants. Furthermore, the effects of spin anisotropies on both leg and rung couplings are investigated. A tiny dependence of the spin transport on the anisotropy in the leg direction is observed in contrast to the strong one on the anisotropy along the rung direction, due to the direct effect of the triplet density. Additionally, the frequency behavior of the spin conductivity of the isotropic case has been studied due to the temperature effects. The temperature dependence of static spin conductivity has been found for various rung-coupling exchange constants and anisotropies in both coupling strengths.
We use the method of recurrence relations to investigate the dynamics of the one-dimensional spin-1/21/21/2 Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction in a magnetic field perpendicular to the DM axis. Our results are valid in the high-temperature limit. We determine exactly the first four recurrants, which produce the time-dependent correlation functions. In order to extend our results to longer times, we introduce a new extrapolation procedure for obtaining higher-order recurrants. Our extrapolation mechanism produces good results when compared to the well-known behavior of the relaxation function in the limit of the XY\mathit{XY}XY and isotropic Heisenberg models. The relaxation function and the spectral density function are then analyzed for several values of the external field BBB for the Heisenberg model with DM interactions in one dimension, in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. We find that the external field produces stronger and faster oscillations in relaxation functions and a suppression of the central peak in the spectral density. That is accompanied by the appearance of a peak centered at a finite frequency, due to enhancement of the collective mode of spins precessing about the external field.
We study the thermodynamics of a spin-1/2 XYZ Heisenberg chain with a Dzyaloshinskii-Moriya interaction. This model describes the low-energy behaviors of a one-dimensional two-component bosonic model with a synthetic spin-orbit coupling in the deep insulating region. In the limit ${U}^{\ensuremath{'}}/U\ensuremath{\rightarrow}\ensuremath{\infty}$, where UUU is the strength of the onsite intracomponent repulsion and ${U}^{\ensuremath{'}}$ is the intercomponent one, we solve our model exactly by Jordan-Wigner transformation, and thus provide a benchmark for our following numerical approach. In other cases, we calculate the entropy and the specific heat numerically by the transfer-matrix renormalization-group method. Their low-temperature behaviors depend crucially on the properties of the zero-temperature phases. A refined ground-state phase diagram is then deduced from their low-temperature behaviors. Our findings offer an alternative way to detect those distinguishable phases experimentally.
We study the effects of charge degrees of freedom on the spin excitation dynamics in quasi-one-dimensional magnetic materials. Using the density matrix renormalization group method, we calculate the dynamical spin structure factor of the Hubbard model at half electronic filling on a chain and on a ladder geometry, and compare the results with those obtained using the Heisenberg model, where charge degrees of freedom are considered frozen. For both chains and two-leg ladders, we find that the Hubbard model spectrum qualitatively resembles the Heisenberg spectrum---with low-energy peaks resembling spinonic excitations---already at intermediate on-site repulsion as small as $U/t\ensuremath{\sim}2--3$, although ratios of peak intensities at different momenta continue evolving with increasing U/tU/tU/t converging only slowly to the Heisenberg limit. We discuss the implications of these results for neutron scattering experiments and we propose criteria to establish the values of U/tU/tU/t of quasi-one-dimensional systems described by one-orbital Hubbard models from experimental information.
We calculate the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$ for the columnar valence bond solid phase of a two-dimensional frustrated antiferromagnet. In particular, we consider the square lattice spin-$1/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model with an additional Dzyaloshinskii-Moriya interaction between the spins and in the presence of an external magnetic field. We concentrate on the intermediate parameter region of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, where a quantum paramagnetic phase is stable, and consider a Dzyaloshinskii-Moriya vector pattern associated with the couplings between the spins in the CuO2{\mathrm{CuO}}_{2}CuO2​ planes of the YBa2Cu3O6{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6}YBa2​Cu3​O6​ compound. We describe the columnar valence bond solid phase within the bond-operator formalism, which allows us to map the Heisenberg model into an effective interacting boson model written in terms of triplet operators. The effective boson model is studied within the harmonic approximation, and the triplon excitation bands of the columnar valence bond solid phase are determined. We then calculate the Berry curvature and the Chern numbers of the triplon excitation bands and, finally, determine the thermal Hall conductivity due to triplons as a function of the temperature. We find that the Dzyaloshinskii-Moriya interaction yields a finite Berry curvature for the triplon bands, but the corresponding Chern numbers vanish. Although the triplon excitations are topologically trivial, the thermal Hall conductivity of the columnar valence bond solid phase in the square lattice antiferromagnet is finite at low temperatures. Our results complement a previous study by Samajdar et al., Phys. Rev. B 99, 165126 (2019) concerning the thermal Hall effect due to spinons of a spin-liquid phase on a square lattice. We also comment on the relations of our results with a no-go condition for a thermal Hall effect previously derived for ordered magnets by Katsura et al., Phys. Rev. Lett. 104, 066403 (2010).
This work is a sequel to our preceding work [Phys. Rev. B 77, 045319 (2008)]. Here we compare pure-spin and pseudospin dynamics using as an example a system of two quantum dots, a pair of localized conduction-band electrons in an nnn-doped GaAs semiconductor. Pure-spin dynamics is obtained by tracing out the orbital degrees of freedom, whereas pseudospin dynamics retains (as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-orbit interaction result in a nonunitary evolution in pure-spin dynamics, whereas these interactions contribute to the effective pseudospin Hamiltonian via terms that are asymmetric in spin permutations, in particular, the Dzyaloshinskii-Moriya (DM) spin-orbit interaction. We numerically investigate the nonunitary effects in the dynamics of the triplet-state population, purity, and Lamb energy shift as a function of interdot distance and magnetic field difference $\ensuremath{\Delta}\stackrel{P\vec}{B}$. The spin-orbit interaction is found to produce effects of roughly 4 orders of magnitude smaller than those due to $\ensuremath{\Delta}\stackrel{P\vec}{B}$ in the pure-spin model. We estimate the spin-orbit interaction magnitude in the DM-interaction term. Our estimate gives a smaller value than that recently obtained by Kavokin [Phys. Rev. B 64, 075305 (2001)], who did not include double-occupancy effects. We show that a necessary and sufficient condition for obtaining a universal set of quantum logic gates, involving only two spins, in both pure-spin and pseudospin models is that the magnetic field inhomogeneity $\ensuremath{\Delta}\stackrel{P\vec}{B}$ and the Heisenberg interaction are both nonvanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recently proposed by Lyon [Phys. Rev. A 74, 052338 (2006)].