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arXiv:cond-mat/0301162 · Physical review. B, Condensed matter

Phase diagram of a coupled tetrahedral Heisenberg model

Onofre Rojas, Francisco Castilho Alcaraz

Published May 5, 2003• 23 citations
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Abstract

The phase diagram of a coupled tetrahedral Heisenberg model is obtained. The quantum chain has a local gauge symmetry and its eigenspectrum is obtained by the composition of the eigenspectra of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ XXZ chains with arbitrary distribution of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{3}{2}$ impurities. The phase diagram is quite rich with an infinite number of phases with ferromagnetic, antiferromagnetic, or ferrimagnetic order. In some cases the ground state and the low-lying eigenlevels of the model can be exactly calculated since they coincide with the eigenlevels of the exactly integrable XXZ chain. The thermodynamical properties of the model at low temperatures are also studied through finite-size analysis.

DOI
10.1103/physrevb.67.174401

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The phase diagram of a coupled tetrahedral Heisenberg model is obtained. The quantum chain has a local gauge symmetry and its eigenspectrum is obtained by the composition of the eigenspectra of spin\ensuremath−12\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ XXZ chains with arbitrary distribution of spin\ensuremath−32\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{3}{2}spin\ensuremath−23​ impurities. The phase diagram is quite rich with an infinite number of phases with ferromagnetic, antiferromagnetic, or ferrimagnetic order. In some cases the ground state and the low-lying eigenlevels of the model can be exactly calculated since they coincide with the eigenlevels of the exactly integrable XXZ chain. The thermodynamical properties of the model at low temperatures are also studied through finite-size analysis.
The asymmetric tetrahedron is composed of all edges of a tetrahedron represented by Ising interactions except for one, which has a Heisenberg-type interaction. This asymmetric tetrahedron is arranged by connecting a vertex whose edges are only Ising-type interactions to another vertex with the same structure for another tetrahedron. The process is replicated and this kind of lattice we call the asymmetric Ising–Heisenberg chain. We have studied the ground-state phase diagram for this kind of model. In particular, we consider two situations in the Heisenberg-type interaction: (i) the asymmetric tetrahedral spin-(1/2, 1/2) Ising-XYZ chain and (ii) the asymmetric tetrahedral spin-(1/2, 1) Ising-XXZ chain, where we have found a rich phase diagram and a number of multicritical points. Additionally we have also studied their thermodynamical properties and the correlation function, using the decorated transformation. We have mapped the asymmetric tetrahedral Ising–Heisenberg chain in an effective Ising chain, and we have also concluded that it is possible to evaluate the partition function including a longitudinal external magnetic field.
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1
The location of the XY phase in the ground-state phase diagram of the spin-1/2 anisotropic ferromagnetic-antiferromagnetic alternating Heisenberg chain is determined. This model reduces to the spin-1 XXZ chain if the ferromagnetic bond is infinitely strong. The phase boundary between the XY phase and the Haldane phase is determined from the analysis of the numerically obtained energy spectrum of finite-sized systems with the open boundary condition. Our results are consistent with the series-expansion results by Yamanaka et al . within their error bars. The boundary between the ferromagnetic phase and the XY phase is also determined by comparing the energy eigenvalues of these two phases.