Green's function theory of quasi-two-dimensional spin-half Heisenberg ferromagnets: stacked square versus stacked kagom\'e lattice

We consider the thermodynamic properties of the quasi-two-dimensional spin-half Heisenberg ferromagnet on the stacked square and the stacked kagom\'e lattices by using the spin-rotation-invariant Green's function method. We calculate the critical temperature $T_C$, the uniform static susceptibility $\chi$, the correlation lengths $\xi_\nu$ and the magnetization $M$ and investigate the short-range order above $T_C$. We find that $T_C$ and $M$ at $T>0$ are smaller for the stacked kagom\'e lattice which we attribute to frustration effects becoming relevant at finite temperatures.Comment: shortened version as published in PR

Quantum J1J_1--J2J_2 antiferromagnet on the stacked square lattice: Influence of the interlayer coupling on the ground-state magnetic ordering

Using the coupled-cluster method (CCM) and the rotation-invariant Green's function method (RGM), we study the influence of the interlayer coupling $J_\perp$ on the magnetic ordering in the ground state of the spin-1/2 $J_1$-$J_2$ frustrated Heisenberg antiferromagnet ($J_1$-$J_2$ model) on the stacked square lattice. In agreement with known results for the $J_1$-$J_2$ model on the strictly two-dimensional square lattice ($J_\perp=0$) we find that the phases with magnetic long-range order at small $J_2< J_{c_1}$ and large $J_2> J_{c_2}$ are separated by a magnetically disordered (quantum paramagnetic) ground-state phase. Increasing the interlayer coupling $J_\perp>0$ the parameter region of this phase decreases, and, finally, the quantum paramagnetic phase disappears for quite small $J_\perp \sim 0.2 ... 0.3 J_1$.Comment: 4 pages, 3 figure

Absence of magnetic order for the spin-half Heisenberg antiferromagnet on the star lattice

We study the ground-state properties of the spin-half Heisenberg antiferromagnet on the two-dimensional star lattice by spin-wave theory, exact diagonalization and a variational mean-field approach. We find evidence that the star lattice is (besides the \kagome lattice) a second candidate among the 11 uniform Archimedean lattices where quantum fluctuations in combination with frustration lead to a quantum paramagnetic ground state. Although the classical ground state of the Heisenberg antiferromagnet on the star exhibits a huge non-trivial degeneracy like on the \kagome lattice, its quantum ground state is most likely dimerized with a gap to all excitations. Finally, we find several candidates for plateaux in the magnetization curve as well as a macroscopic magnetization jump to saturation due to independent localized magnon states.Comment: new extended version (6 pages, 6 figures) as published in Physical Review

Absence of long-range order in a spin-half Heisenberg antiferromagnet on the stacked kagome lattice

We study the ground state of a spin-half Heisenberg antiferromagnet on the stacked kagome lattice by using a spin-rotation-invariant Green's-function method. Since the pure two-dimensional kagome antiferromagnet is most likely a magnetically disordered quantum spin liquid, we investigate the question whether the coupling of kagome layers in a stacked three-dimensional system may lead to a magnetically ordered ground state. We present spin-spin correlation functions and correlation lengths. For comparison we apply also linear spin wave theory. Our results provide strong evidence that the system remains short-range ordered independent of the sign and the strength of the interlayer coupling

Smooth stable and unstable manifolds for stochastic partial differential equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds

Ground state and low-lying excitations of the spin-1/2 XXZ model on the kagome lattice at magnetization 1/3

We study the ground state and low-lying excitations of the S=1/2 XXZ antiferromagnet on the kagome lattice at magnetization one third of the saturation. An exponential number of non-magnetic states is found below a magnetic gap. The non-magnetic excitations also have a gap above the ground state, but it is much smaller than the magnetic gap. This ground state corresponds to an ordered pattern with resonances in one third of the hexagons. The spin-spin correlation function is short ranged, but there is long-range order of valence-bond crystal type.Comment: 2 pages, 1 figure included, to appear in Physica B (proceedings of SCES'04

The Heisenberg antiferromagnet on the square-kagomé lattice

We discuss the ground state, the low-lying excitations as well as high-field thermodynamics of the Heisenberg antiferromagnet on the two-dimensional square-kagom´e lattice. This magnetic system belongs to the class of highly frustrated spin systems with an infinite non-trivial degeneracy of the classical ground state as it is also known for the Heisenberg antiferromagnet on the kagom´e and on the star lattice. The quantum ground state of the spin-half system is a quantum paramagnet with a finite spin gap and with a large number of non-magnetic excitations within this gap. We also discuss the magnetization versus field curve that shows a plateaux as well as a macroscopic magnetization jump to saturation due to independent localized magnon states. These localized states are highly degenerate and lead to interesting features in the low-temperature thermodynamics at high magnetic fields such as an additional low-temperature peak in the specific heat and an enhanced magnetocaloric effect.Ми обговорюємо основний стан, низьколежачi збудження, а також термодинамiку у сильному полi антиферомагнетика Гайзенберга на двовимiрнiй ґратцi квадратне кагоме. Ця магнiтна система належить до класу сильно фрустрованих спiнових систем з безмежним нетривiальним виродженням класичного основного стану, так само, як i антиферомагнетик Гайзенберга на ґратцi кагоме i на ґратцi зiрка. Квантовий основний стан спiн-половина системи є квантовим парамагнетиком iз скiнченою спiновою щiлиною та великим числом немагнiтних збуджень всерединi щiлини. Ми також обговорюємо криву намагнiченiсть-поле, у якiй є плато та макроскопiчний стрибок намагнiченостi до значення насичення через стани, що називаються незалежними локалiзованими магнонами. Цi локалiзованi стани є сильно виродженими та зумовлюють цiкавi риси низькотемпературної термодинамiки у сильному магнiтному полi, такi як додатковий низькотемпературний пiк у теплоємностi i посилений магнiтокалоричний ефект

Linear independence of localized magnon states

At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number $N$ of spins and hence they have a finite entropy also in the thermodynamic limit $N\to \infty$ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagom\'{e}, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.Comment: 23 pages, 6 figure

Quantum kagomé antiferromagnet in a magnetic field: Low-lying nonmagnetic excitations versus valence-bond crystal order

We study the ground state properties of a quantum antiferromagnet on the kagom\'e lattice in the presence of a magnetic field, paying particular attention to the stability of the plateau at magnetization 1∕3 of saturation and the nature of its ground state. We discuss fluctuations around classical ground states and argue that quantum and classical calculations at the harmonic level do not lead to the same result in contrast to the zero-field case. For spin S=1∕2 we find a magnetic gap below which an exponential number of nonmagnetic excitations are present. Moreover, such non-magnetic excitations also have a (much smaller) gap above the threefold degenerate ground state. We provide evidence that the ground state has long-range order of valence-bond crystal type with nine spins in the unit cell.Facultad de Ciencias Exacta