The spin quantization problem

Alvin Meckler

The Majorana formula for the transition probability of a general spin is here rederived and brought into a more compact form. The method is based on the use of the projection operators which characterize the spin vector as definitely quantized along a specified unit vector. If the Hamiltonian contains only terms linear in the spin, it is possible to define a moving instantaneous axis along which the moving spin vector maintains its quantization. The Majorana problem is reduced to the calculation of the joint probability of quantization along two different axes.The usual expression of the Majorana formula contains a factor which is the transition probability of a spin-\textonehalf{} vector. It is shown here how this factor is related to the angle through which the general spin is turned.

George Sterman

Maurice A. de Gosson

Zekun Zhuang, John Bradley Marston

Quantum spin orbital liquids (QSOLs) are a novel phase of matter, similar to quantum spin liquids, with quantum fluctuations in both spin and orbital degrees of freedom. We use nonequilibrium Green's function theory to study out-of-equilibrium spin transport in an exactly solvable QSOL model put forward by Yao and Lee. We find that the spin transport problem can be mapped to that of a free fermion problem with effective fermionic baths that have a rapidly varying density of states. In the gapless phase, the spin current ${I}_{s}\text{\ensuremath{-}}{V}_{s}$ relation is thus highly nonlinear, while in the chiral gapped phase, the spin current conductance is quantized to be $1/2\ensuremath{\pi}$ provided that the contacts are sufficiently wide. The quantized conductance is a signature of the topological nature of the chiral gapped QSOL.

Tingting Zhang, Jie Han

Developed from the Ising model, Ising machines are promising to be efficient domain-specific accelerators for solving combinatorial optimization problems (COPs). A simulated bifurcation (SB) algorithm enables an Ising machine to achieve massive parallelism in the simulation of Hamiltonian dynamics. Although SB significantly accelerates the search, more resources are required due to the use of continuous variables for the position of oscillators to obtain discrete spin states, compared to conventional simulated annealing. This article proposes ternary and multiple-value quantized SB (qSB) algorithms by discretizing the position variables used for the hardware-consuming multiply-accumulate (MAC) operations in SB. These quantization schemes do not only reduce the computational complexity, but also improve the solution quality for COPs. Specifically, the ternary qSB with dynamic threshold settings converts the MAC into addition and subtraction. To improve the precision in number representation when solving large-scale COPs, a uniform quantization scheme is applied to provide multiple-valued quantization. Alternatively, a logarithmic qSB leverages the evolution characteristics of position variables and implements multiplication by using simple shift operations. We demonstrate that using the proposed qSB improves the solution quality in a long search and accelerates energy convergence in a short search for solving COPs tackled by up to 2000 fully connected spins.