The spin-1/2 Hamiltonian for two coupled isosceles Heisenberg triangles, which is well suited for describing the V$_6$-type magnetic molecules, is studied by exact diagonalization. The quantum phase transition diagram, at zero temperature, is obtained as a function of the theoretical parameters. The zero temperature magnetization is also obtained as a function of the external magnetic field. The thermodynamic behavior of the magnetization, entropy, susceptibility, and specific heat, as a function of temperature, are also computed and the corresponding magnetocaloric effect analyzed for various values of the Hamiltonian parameters.
At arbitrary temperature $T$, we solve for the dynamics of single molecule magnets composed of three classical Heisenberg spins either on a chain with two equal exchange constants $J_1$, or on an isosceles triangle with a third, different exchange constant $J_2$. As $T\rightrarrow\infty$, the Fourier transforms and long-time asymptotic behaviors of the two-spin time correlation functions are evaluated exactly. The lack of translational symmetry on a chain or an isosceles triangle yields time correlation functions that differ strikingly from those on an equilateral trinagle with $J_1=J_2$. At l…
Ground state and thermodynamics of geometrically frustrated spin-1/2 Ising-Heisenberg model on two different but topologically related triangles-in-triangles lattices is investigated in particular. A rigorous mapping based on generalized star-triangle transformations enables one to derive exact analytical results for the free and internal energy, spontaneous magnetization, entropy and specific heat from the known exact results for the corresponding spin-1/2 Ising model on a triangular lattice. It is demonstrated that several unconventional quantum ordered and disordered phases may become the g…
The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three inter-connected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperatur…
The geometric frustration of the spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice is investigated within the framework of an exact analytical method based on the generalized star-triangle mapping transformation. Ground-state and finite-temperature phase diagrams are obtained along with other exact results for the partition function, Helmholtz free energy, internal energy, entropy, and specific heat, by establishing a precise mapping relationship to the corresponding spin-1/2 Ising model on the Kagome lattice. It is shown that the residual entropy of t…
Ground-state and finite-temperature phase diagrams of a geometrically frustrated spin-1/2 Ising-Heisenberg model on a triangle-hexagon lattice are investigated within the generalized star-triangle mapping transformation. It is shown that the ground state is constituted by two different spontaneously long-range ordered phases and one disordered spin-liquid phase.
The spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice is exactly solved by establishing a precise mapping correspondence to the simple spin-1/2 Ising model on Kagome lattice. It is shown that the disordered spin liquid state, which otherwise occurs in the ground state of this frustrated spin system on assumption that there is a sufficiently strong antiferromagnetic intra-trimer interaction, is eliminated from the ground state by arbitrary but non-zero Dzyaloshinskii-Moriya anisotropy.
The ground state and zero-temperature magnetization process of the spin-1/2 Ising-Heisenberg model on two-dimensional triangles-in-triangles lattices is exactly calculated using eigenstates of the smallest commuting spin clusters. Our ground-state analysis of the investigated classical--quantum spin model reveals three unconventional dimerized or trimerized quantum ground states besides two classical ground states. It is demonstrated that the spin frustration is responsible for a variety of magnetization scenarios with up to three or four intermediate magnetization plateaus of either quantum o…
The spin-1/2 Ising-Heisenberg three-leg tube composed of the Heisenberg spin triangles mutually coupled through the Ising inter-triangle interaction is exactly solved in a zero magnetic field. By making use of the local conservation for the total spin on each Heisenberg spin triangle the model can be rigorously mapped onto a classical composite spin-chain model, which is subsequently exactly treated through the transfer-matrix method. The ground-state phase diagram, correlation functions, concurrence, Bell function, entropy and specific heat are examined in detail. It is shown that the spin fr…
We present exact integral representations of the time-dependent spin-spin correlation functions for the classical Heisenberg N-spin `squashed' equivalent neighbor model, in which one spin is coupled via the Heisenberg exchange interaction with strength $J_1$ to the other N-1 spins, each of which is coupled via the Heisenberg exchange coupling with strength $J_2$ to the remaining N-2 spins. At low temperature T we find that the N spins oscillate in four modes, one of which is a central peak for a semi-infinite range of the values of the exchange coupling ratio. For the N=4 case of four spins on…
Here we consider the Ising-Heisenberg model in the expanded Kagomé lattice, also known as triangle-dodecagon (3-12) or star lattice. This model can still be understood as a decorated honeycomb lattice. Assuming that the Heisenberg spins are at the vertices of the triangle while other spins are of the Ising type. Thus, this model is equivalent to an effective Kagomé Ising lattice, through the decoration transformation technique. Thus this means that the model is exactly solvable so we can study the most relevant properties of this model. Like the phase diagram at zero temperature, exhibiting a …
The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate thermodynamic quantities as density of states, specific heat, susceptibility and spin autocorrelation functions. These calculations are performed (semi-)analytically and shown to agree with corresponding Monte Carlo simulations. For the long-time autocorrelation function, we find, for certain values of the coupling constants, a decay to constant values in the form of an $1/t$ damped harmonic oscillation and propose a t…
We investigate classes of quantum Heisenberg spin systems which have different coupling constants but the same energy spectrum and hence the same thermodynamical properties. To this end we define various types of isospectrality and establish conditions for their occurence. The triangle and the tetrahedron whose vertices are occupied by spins 1/2 are investigated in some detail. The problem is also of practical interest since isospectrality presents an obstacle to the experimental determination of the coupling constants of small interacting spin systems such as magnetic molecules.
An efficient algorithm is developed for quantum spin tubes in the context of the tensor network representations. It allows to efficiently compute the ground-state fidelity per lattice site, which in turn enables us to identify quantum critical points, at which quantum spin tubes undergo quantum phase transitions. As an illustration, we investigate the isosceles spin 1/2 antiferromagnetic three-leg Heisenberg tube. Our simulation results suggest that two Kosterlitz-Thouless transitions occur as the degree of asymmetry of the rung interaction is tuned, thus offering an alternative route towards …
Classical Heisenberg and XY models on zigzag ladder lattices with nearest-neighbor bilinear-biquadratic exchange: Exact solution for the ground-state problem
We use computer simulations and scaling arguments to investigate statistical and structural properties of a semiflexible ribbon composed of isosceles triangles. We study two different models, one where the bending energy is calculated from the angles between the normal vectors of adjacent triangles, the second where the edges are viewed as semiflexible polymers so that the bending energy is related to the angles between the tangent vectors of next-nearest neighbor triangles. The first model can be solved exactly whereas the second is more involved. It was recently introduced by Liverpool and G…
A mixed Ising-Heisenberg spin system consisting of triangular XXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration--iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground state phase diagrams, thermodynamic and magnetic properties of the system are examined.
Recent developments of theoretical studies on spin nanotubes are reviewed, especially focusing on the $S=1/2$ three-leg spin tube. In contrast to the three-leg spin ladder, the tube has a spin gap in case of the regular-triangle unit cellwhen the rung interaction is sufficiently large. The effective theory based on the Hubbard Hamiltonian indicates a quantum phase transition to the gapless spin liquid due to the lattice distortion to an isosceles triangle. This is also supported by the numerical diagonalization and the density matrix renormalization group analyses. Furthermore, combining analy…
Exact and rigorous solutions of the ground-state problem for the classical Heisenberg model with nearest-neighbor interactions on two- and three-dimensional lattices composed of zigzag (triangular) ladders are obtained in a very simple way, with the use of a cluster method. It is shown how the geometrical frustration due to the presence of triangles as structural units leads to the emergence of incommensurate spiral orderings and their collinear limits. Interestingly, these orderings are determined by a single angle (along with the signs of the interactions between neighboring spins); therefor…
