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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Modified effective-field approach to low-dimensional spin-1/2 systems

G. Kamieniarz, Raf Dekeyser, Grzegorz Musiał, L. Dębski, M. Bieliński

Published Jul 1, 1997• 7 citations
Publisher

Abstract

Two different generalization schemes for the molecular field approximation are described. Both contain the possibility to incorporate generalized effective fields that induce correlations between the degrees of freedom at the boundary of a chosen cluster. Applications are presented for low-dimensional spin-1/2 systems. For the Ising models the critical couplings and critical exponents as well as the spontaneous magnetization curve are determined. An application of the method to the quantum $S$=$\frac{1}{2}$ one-dimensional Heisenberg model is presented and reliable low-temperature estimates of the specific heat are evaluated. The data are combined with the quantum transfer matrix predictions for large system sizes and this leads to the following prediction for the low-temperature specific heat: ${C=AT}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}=0.35\ifmmode\pm\else\textpm\fi{}0.07$.

DOI
10.1103/physreve.56.144

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Two different generalization schemes for the molecular field approximation are described. Both contain the possibility to incorporate generalized effective fields that induce correlations between the degrees of freedom at the boundary of a chosen cluster. Applications are presented for low-dimensional spin-1/2 systems. For the Ising models the critical couplings and critical exponents as well as the spontaneous magnetization curve are determined. An application of the method to the quantum SSS=12\frac{1}{2}21​ one-dimensional Heisenberg model is presented and reliable low-temperature estimates of the specific heat are evaluated. The data are combined with the quantum transfer matrix predictions for large system sizes and this leads to the following prediction for the low-temperature specific heat: C=AT\ensuremathα{C=AT}^{\ensuremath{\alpha}}C=AT\ensuremathα with \ensuremathα=0.35\ifmmode±\else\textpm\fi0.07\ensuremath{\alpha}=0.35\ifmmode\pm\else\textpm\fi{}0.07\ensuremathα=0.35\ifmmode±\else\textpm\fi0.07.
A formulation is given to map the one-dimensional S = 1/2 quantum spin systems onto the two-dimensional Ising model with many-spin interactions on the basis of the n-spin cluster decomposition (n CLD). The cluster transfer matrix is defined which characterizes the thermodynamical properties of the equivalent system. A prescription is presented to construct the cluster transfar matrix and to evaluate its maximum eigenvalue numerically. The method has been applied to the calculation of the energy and specific heat of the XY model for various values of the cluster size n and th Trotter size m. The convergence properties of the energy as a function of n and m have been studied in detail. The main results obtained are: (1) The low-temperature behavior is improved much more by considering larger spin clusters; (2) the energy depends on the spin cluster through the effective cluster size ne defined by ne =(n+1)/2; and (3) the energy converges linearly with 1/n2em2 except at very low temperatures.
We have extended the range of application of the decimation renormalization-group (RG) method, using a combination of analytic and numerical techniques in calculating thermal and magnetic properties of a wide variety of one-dimensional quantum spin-(1/2) systems. Efforts to improve the accuracy of the approach include increasing the spatial rescaling factor and investigating the effect of free versus periodic boundary conditions for each renormalization cluster. We have used the RG method to calculate the specific heat and susceptibility for chains with uniaxial exchange anisotropy (ranging from the Ising to the XY limits) in both longitudinal and transverse magnetic fields, as well as for the Heisenberg antiferromagnet with alternating couplings, and have compared our results, when possible, with either exact solutions or with numerical extrapolations from finite-chain calculations. Decimation is a high-temperature approximation, but its range of reliability is found in some cases (e.g., transverse susceptibility for the Ising chain, or susceptibility for the Heisenberg ferromagnet in a field) to extend well into the strong-coupling regime. We note that the decimation results for magnetic properties are, in general, more accurate than those for the thermal response.
Low dimensional spin systems are a class of interacting many body systems which often can suitably be described by physical models with possible experimental realizations. A series of inorganic, organometallic and organic systems representing different spin ½ models have been synthesized and their thermodynamics have been investigated. In most of the cases, antiferromagnetic coupling is favored and the system can be well described by Heisenberg model where the unpaired spins interact isotropically to the nearest neighbor with an intrachain coupling constant (J). Isothermal heat capacity plots as a function of field at low temperature captures the Quantum Critical Point (QCP) experimentally, owing to change in symmetry of the ground state. This change of phase is associated with the occurrence of first order quantum phase transition in NH4CuPO4·H2O which has been established by studying field dependent behavior of magnetization and entanglement. On one hand entanglement being a nonlocal property of the spins takes its maximum value when the applied magnetic field is zero. On the other hand magnetization describing the local properties of the individual spins takes its maximum value upon reaching the saturation magnetic field at the expense of entanglement. It signifies a physical picture of quantum information sharing and gives an to opportunity capture the critical magnetic field. We have investigated this scenario in NH4CuPO4·H2O and determined the critical field which shows an excellent match with the value obtained from the field dependent specific heat data.
Quantum spin-1/2 antiferromagnetic Heisenberg trimerized chain with strong intradimer and weak monomer-dimer coupling constants is studied using the many-body perturbation expansion developed about the exact solution of the spin-1/2 Ising-Heisenberg diamond chain. The Heisenberg trimerized chain shows the intermediate one-third plateau, which was also observed in the magnetization curve of the polymeric compound Cu3_33​(P2_22​O6_66​OH)2_22​ affording its experimental realization. Within the strong-coupling method we have obtained the effective Hamiltonians for the magnetic-field range from zero to one-third plateau, and from one-third plateau to the saturation magnetization. The second-order perturbation theory shows reliable results even for the moderate ratio between weaker dimer-monomer and stronger intradimer coupling constants. We have also examined thermodynamic properties and recovered the low-temperature peak in the specific heat. The accuracy of the developed method is tested through a comparison with numerical density-matrix renormalization group and quantum Monte Carlo simulations. Using the results for the effective Hamiltonian we suggest straightforward procedure for finding the microscopic parameters of one-dimensional trimerized magnetic compounds with strong intradimer and weak monomer-dimer couplings. We found the refined values for the coupling constants of Cu3_33​(P2_22​O6_66​OH)2_22​ by matching the theoretical results with the available experimental data for the magnetization and magnetic susceptibility in a wide range of temperatures and magnetic fields.