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physica status solidi (b)

Ladder Ising spin configurations II. magnetic properties

Rexhep Mejdani, A. Gashi, Orion Ciftja, A. P. Lambros

Published Sep 1, 1996• 6 citations
Publisher

Abstract

Abstract In this paper we consider ladders with two, three, and four coupled Ising spin chains, characterized by interchain and intrachain couplings, to study in detail their magnetic behaviour for different ratios of the interaction constants and different values of magnetic field, by using a transfer matrix method and a complex algorithm, realizing the diagonalization and the differentiation. We have proposed a combined algorithm to realize also the generation of the elements of the transfer matrix for more complicated ladder configurations. The problem of Ising ladders is straightforward with a crossover between a 2D high‐temperature region and a 1D low‐temperature region when the 2D correlation length equals the width. To emphasize the fact that there is no finite magnetization in absence of the magnetic field, we study also analytically this case for the ladder with two coupled Ising chains. In presence of the magnetic field, for very weak magnetic fields (of the order 10−8, in reduced units) the magnetization goes from the value one to the value zero in a narrow interval of nonzero temperatures, where also the susceptibility exhibits a very high peak. This narrow interval, increasing the number of chains, is displaced towards higher temperatures. Also, in the case of interchain antiferromagnetic couplings, the formation of interchain spin pairs becomes clear.

DOI
10.1002/pssb.2221970122

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Abstract In this paper we consider ladders with two, three, and four coupled Ising spin chains, characterized by interchain and intrachain couplings, to study in detail their magnetic behaviour for different ratios of the interaction constants and different values of magnetic field, by using a transfer matrix method and a complex algorithm, realizing the diagonalization and the differentiation. We have proposed a combined algorithm to realize also the generation of the elements of the transfer matrix for more complicated ladder configurations. The problem of Ising ladders is straightforward with a crossover between a 2D high‐temperature region and a 1D low‐temperature region when the 2D correlation length equals the width. To emphasize the fact that there is no finite magnetization in absence of the magnetic field, we study also analytically this case for the ladder with two coupled Ising chains. In presence of the magnetic field, for very weak magnetic fields (of the order 10−8, in reduced units) the magnetization goes from the value one to the value zero in a narrow interval of nonzero temperatures, where also the susceptibility exhibits a very high peak. This narrow interval, increasing the number of chains, is displaced towards higher temperatures. Also, in the case of interchain antiferromagnetic couplings, the formation of interchain spin pairs becomes clear.
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 have carried out extensive series studies, at T=0 and at high temperatures, of two-chain and three-chain spin-half ladder systems with antiferromagnetic intrachain and both antiferromagnetic and ferromagnetic interchain couplings. Our results confirm the existence of a gap in the two-chain Heisenberg ladders for all nonzero values of the interchain couplings. Complete dispersion relations for the spin-wave excitations are computed. For three-chain systems, our results are consistent with a gapless spectrum. We also calculate the uniform magnetic susceptibility and specific heat as a function of temperature. We find that as T\ensuremath{\rightarrow}0, for the two-chain system the uniform susceptibility goes rapidly to zero, whereas for the three-chain system it approaches a finite value. These results are compared in detail with previous studies of finite systems. \textcopyright{} 1996 The American Physical Society.
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.
We investigate the thermodynamic behavior of the Kitaev ladder model under varying interchain coupling [Formula: see text] and external magnetic fields using the finite temperature Lanczos method and machine-learning approach. At zero magnetic field, we find that the specific heat exhibits a double-peak structure for finite [Formula: see text]. The characteristic temperature [Formula: see text] of the low-temperature peak increases significantly with increasing [Formula: see text], whereas the high-temperature peak remains insensitive to [Formula: see text]. We argue that stronger interchain coupling elevates the characteristic temperature required for flux excitations. The further analysis of the specific-heat phase diagram under a magnetic field for different [Formula: see text] values indicates that when the external magnetic field continuously increases, the specific heat of the system undergoes a transition from well-defined two peaks to one peak, and the system finally enters a fully polarized phase. Concurrently, the low-temperature peak of the specific heat vanishes, which is corroborated in the thermodynamic limit predicted by the machine-learning approach. As [Formula: see text] decreases, weaker interchain coupling makes the system more susceptible to magnetization. These results demonstrate that the Kitaev ladder with finite [Formula: see text] exhibits qualitatively similar character to the two-dimensional honeycomb Kitaev model under a magnetic field, highlighting the crucial role of interchain coupling in regulating flux excitation properties and stabilizing the spin liquid phase.
The effect of antiferromagnetic interchain coupling in alternating spin-(1,1/2) chains is studied by means of spin-wave theory and density-matrix renormalization group (DMRG). In particular, two limiting cases are investigated, the two-leg ladder and its two-dimensional (2D) generalization. Results of the ground-state properties like energy, spin gap, magnetizations, and correlation functions are reported for the whole range of the interchain coupling ${J}_{\ensuremath{\perp}}.$ For the 2D case the spin-wave results predict a smooth dimensional crossover from 1D to 2D keeping the ground state always ordered. For the ladder system, the DMRG results show that any ${J}_{\ensuremath{\perp}}>0$ drives the system to a gapped ground state. Furthermore the behavior of the correlation functions closely resemble the uniform spin-1/2 ladder. For ${J}_{\ensuremath{\perp}}$ lower than 0.3, however, the gap behaves quadratically as $\ensuremath{\Delta}\ensuremath{\sim}{0.6J}_{\ensuremath{\perp}}^{2}.$ Finally, it is argued that the behavior of the spin gap for an arbitrary number of mixed coupled spin chains is analogous to that of the uniform spin-1/2 chains.
The properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.