XXZand Ising spins on the triangular kagome lattice

Abstract

The recently fabricated two-dimensional magnetic materials Cu 9 X 2 ͑cpa͒ 6 · xH 2 O ͑cpa= 2-carboxypentonic acid and X =F,Cl,Br͒ have copper sites which form a triangular kagome lattice ͑TKL͒, formed by introducing small triangles ͑"a-trimers"͒ inside of each kagome triangle ͑"b-trimer"͒. We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite-temperature phase diagram for this XXZ-Ising model, including the residual zero-temperature entropies of the seven ground-state phases. Whereas the disordered ͑spin liquid͒ ground state of the pure Ising TKL model has macroscopic residual entropy ln 72= 4.2767. . . per unit cell, the introduction of transverse ͑quantum͒ couplings between neighboring a-spins reduces this entropy to 2.5258. . . per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions and derive the ground-state phase diagram. A small ͑or even infinitesimal͒ field leads to a new phase that corresponds to a nonintersecting loop gas on the kagome lattice, with entropy 1.4053. . . per unit cell and a mean magnetization for the b-spins of 0.12͑1͒ per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase that maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.

Presence or absence of order by disorder in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices

“…It is worth mentioning that the overall ground-state eigenvector is given by a tensor product over the lowestenergy eigenstates of the cluster Hamiltonians (3), which means that the ground-state phase diagram and individual ground states of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices are completely identical with that reported on previously for the analogous model on triangulated planar lattices composed of the same structural unit [29][30][31]. In this regard, let us merely quote individual ground-state eigenvectors of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices without their detailed description (the interested reader is referred to Ref.…”

“…Recently, the spin-1/2 Ising-Heisenberg model on triangulated (triangles-in-triangles) planar lattices have attracted a great deal of attention [29][30][31][32], because it closely resembles a magnetic structure of real polymeric coordination compounds Cu 9 X 2 (cpa) 6 ·nH 2 O (X=F,Cl,Br) with the geometry of triangulated kagomé lattice [26][27][28]. Besides a clear experimental motivation, the spin-1/2 Ising-Heisenberg model on triangulated planar lattices has afforded a valuable playground for an investigation of the role of local quantum fluctuations within a novel class of exactly solvable classical-quantum spin systems.…”

Numerical exact diagonalization study of triangulated kagome Heisenberg spin system

“…The investigated quantum Heisenberg model has not suggested the qualitative difference in the temperature dependence of susceptibility and the magnetic field dependence of magnetization from the classical Ising [4,5] and the Ising-Heisenberg mixed models [6,7]. The contradiction both in its sign and magnitude of J AB for the interpretation of two physical quantities discussed above is not still resolved and remains as a future problem.…”

“…On the latter, the magnetization measured under magnetic field up to 38T shows the tendency of saturation to one third of full polarization value [2,3]. On the other side, theoretically, Ising model [4,5] and Ising-Heisenberg mixed model [6,7], where Heisenberg spin is resided on A-sublattice and Ising spin on B-sublattice, are treated exactly through a mapping to the ferromagnetic Ising model on the kagome lattice. In these studies, two types of magnetic interaction are taken into account, one for A-A interactions among small triangle J AA and the other for nearest neighbor(nn) A-B interactions J AB .…”

A Consistent Description of Magnetic Properties of the Triangulated-Kagome System Cu9X2(cpa)6·nH2O

“…14, the ''minimum'' in T as observed experimentally has been reproduced, but the obtained magnetization reveals only the 5/9-plateau in the straightforward calculation, without taking into account the non-collinear spin arrangement of trimeric units resulting the 1/3-plateau. The thermodynamics of the Ising model, 16,17) Ising-Heisenberg mixed model, 15,[18][19][20] and Heisenberg model 21) have also been investigated; however, no consensus has not been arrived at even in the sign of the intertrimer exchange constant J AB , 22) because a consistent description of the increment in the gradient of À1 at low temperatures, of the existence of a ''minimum'' in T , and of the appearance of 1/3-plateau has not been successfully given.…”