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.

Heinz-Juergen Schmidt Marshall Luban

We provide exact analytical expressions for the magnetic susceptibility function in the high temperature expansion for finite Heisenberg spin systems with an arbitrary coupling matrix, arbitrary single-spin quantum number, and arbitrary number of spins. The results can be used to determine unknown exchange parameters from zero-field magnetic susceptibility measurements without diagonalizing the system Hamiltonian. We demonstrate the possibility of reconstructing the exchange parameters from simulated data for two specific model systems. We examine the accuracy and stability of the proposed method.

H. -J. Schmidt J. Schnack Marshall Luban

O. Waldmann has shown that some spin systems, which fulfill the condition of a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal energies are rather poorly approximated by a quadratic dependence on the total spin quantum number. We comment on this observation and provide the new argument that, under certain conditions, the approximating parabolas appear as natural bounds of the spectrum generated by spin coherent states.

H. -J. Schmidt J. Schnack Marshall Luban

It is shown that that violation of causality in two-dimensional light-front field theories quantized in a finite ``volume'' $L$ with periodic or antiperiodic boundary conditions is marginal and vanishes smoothly in the continuum limit. For this purpose, we derive an exact integral representation for the complete infinite series expansion of the two-point functions of a free massive scalar and fermi field for an arbitrary finite value of $L$ and show that in the $L \to \infty$ limit we retrieve the correct continuum results.

Lubomir Martinovic Marshall Luban

We consider the dynamics and thermodynamics of a pair of magnetic dipoles interacting via their magnetic fields. We consider only the "spin" degrees of freedom; the dipoles are fixed in space. With this restriction it is possible to provide the general solution of the equations of motion in analytical form. Thermodynamic quantities, such as the specific heat and the zero field susceptibility are calculated by combining low temperature asymptotic series and a complete high temperature expansion. The thermal expectation value of the autocorrelation function is determined for the low temperature regime including terms linear in $T$. Furthermore, we compare our analytical results with numerical calculations based on Monte Carlo simulations.

Heinz-Jürgen Schmidt Christian Schröder Eva Hägele Marshall Luban

This paper presents the design, development and testing of GiAnt, an affordable hexapod which is inspired by the efficient motions of ants. The decision to model GiAnt after ants rather than other insects is rooted in ants' natural adaptability to a variety of terrains. This bio-inspired approach gives it a significant advantage in outdoor applications, offering terrain flexibility along with efficient energy use. It features a lightweight 3D-printed and laser cut structure weighing 1.75 kg with dimensions of 310 mm x 200 mm x 120 mm. Its legs have been designed with a simple Single Degree of Freedom (DOF) using a link and crank mechanism. It is great for conquering challenging terrains such as grass, rocks, and steep surfaces. Unlike traditional robots using four wheels for motion, its legged design gives superior adaptability to uneven and rough surfaces. GiAnt's control system is built on Arduino, allowing manual operation. An effective way of controlling the legs of GiAnt was achieved by gait analysis. It can move up to 8 cm of height easily with its advanced leg positioning system. Furthermore, equipped with machine learning and image processing technology, it can identify 81 different objects in a live monitoring system. It represents a significant step towards creating accessible hexapod robots for research, exploration, and surveying, offering unique advantages in adaptability and control simplicity.

Aasfee Mosharraf Bhuiyan Md Luban Mehda Md. Thawhid Hasan Puspo Jubayer Amin Pritom

We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelation function (ACF) is more subtle. Unlike the classical Heisenberg trimer where the ACF approaches a unique non-zero limit for long times, for the quantum trimer the ACF is periodic in time. We present exact values of the time average over one period of the quantum trimer for s less or equal 7 and for infinite temperature. These averages differ from the long-time limit, (9/40)\ln3+(7/30), of the corresponding classical trimer by terms of order 1/(s*s). However, upon applying the Levin u-sequence acceleration method to our quantum results we can reproduce the classical value to six significant figures.

D. Mentrup H. -J. Schmidt J. Schnack Marshall Luban

In an effort to understand the low temperature behavior of recently synthesized molecular magnets we present numerical evidence for the existence of a rotational band in systems of quantum spins interacting with nearest-neighbor antiferromagnetic Heisenberg exchange. While this result has previously been noted for ring arrays with an even number of spin sites, we find that it also applies for rings with an odd number of sites as well as for all of the polytope configurations we have investigated (tetrahedron, cube, octahedron, icosahedron, triangular prism, and axially truncated icosahedron). It is demonstrated how the rotational band levels can in many cases be accurately predicted using the underlying sublattice structure of the spin array. We illustrate how the characteristics of the rotational band can provide valuable estimates for the low temperature magnetic susceptibility.

J. Schnack Marshall Luban

We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w.r.t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N times N coupling matrix which are usually easily evaluated. In addition we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron.

H. -J. Schmidt J. Schnack Marshall Luban

A model relevant for the study of certain molecular magnets is the ring of N=4 classical spins with equal near-neighbor isotropic Heisenberg exchange interactions. Assuming classical Heisenberg spin dynamics, we solve explicitly for the time evolution of each of the spins. Exact triple integral representations are derived for the auto, near-neighbor, and next-nearest-neighbor time correlation functions for any temperature. At infinite temperature, the correlation functions are reduced to quadrature. We then evaluate the Fourier transforms of these functions in closed form, which are double integrals. At low temperatures, the Fourier transform functions explicitly demonstrate the presence of magnons. Our exact results for the infinite temperature correlation functions in the long-time asymptotic limit differ qualitatively from those obtained assuming diffusive spin dynamics. Whether such explicitly non-hydrodynamic behavior would be maintained for large-N rings is discussed.

Richard A. Klemm Marshall Luban