Heisenberg-Integrable Spin Systems

Abstract

Abstract. We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P . For these systems the time evolution can be explicitely calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.

Equilibration and thermalization of classical systems

“…As expected, the oscillators trace out a regular pattern in the (x, p) plane. Also, the trajectory of a magnetic moment of a ring of four sites is regular but, in contrast, if we open the ring to make a chain, the behavior changes from regular to chaotic, in agreement with theory [22].…”

“…and p are real constants, and (a, b, c) form a right-handed set of orthogonal unit vectors [30,31]. More generally, equation (12) has simple analytical solutions for N = 2 and 3 [22]. The motion of N = 4 magnetic moments arranged on a ring is regular [22].…”

“…More generally, equation (12) has simple analytical solutions for N = 2 and 3 [22]. The motion of N = 4 magnetic moments arranged on a ring is regular [22]. For N > 4, the system exhibits chaotic motion [22], except for special initial conditions such as the spin-wave and soliton solutions [32].…”

“…Poincaré-type map of the differences X = (x 0 − x 1 )/2 and P = ( p 0 − p 1 )/2 √ 2 for a ring of N = 512 oscillators, an integrable system, (left) and of the differences X = S x 1 − S x 2 and P = S z 1 − S z 2 for a ring of N = 512 magnetic moments, a chaotic system[22] (right). Each graph contains 5000 points.…”

Spin transport in the XXZ model at high temperatures: Classical dynamics vs. quantum S=1/2 autocorrelations

“…with J r±1 = ±J S r ×(S x r±1 , S y r±1 , ∆ S z r±1 ) as the incoming and outgoing spin currents. Since this set of equations is non-integrable in terms of the Liouville-Arnold theorem [34,35], exact analytical solutions can only be given for mostly trivial initial configurations. We thus solve the set of equations numerically using a 4th order Runge-Kutta algorithm with a fixed time step of δt J = 0.01; however, the presented results in the work at hand will not depend on this particular choice of the time step, cf.…”