Higher-order Spin Interactions Research Articles

Designing Ising machines with higher order spin interactions and their application in solving combinatorial optimization

The Ising model provides a natural mapping for many computationally hard combinatorial optimization problems (COPs). Consequently, dynamical system-inspired computing models and hardware platforms that minimize the Ising Hamiltonian, have recently been proposed as a potential candidate for solving COPs, with the promise of significant performance benefit. However, prior work on designing dynamical systems as Ising machines has primarily considered quadratic interactions among the nodes. Dynamical systems and models considering higher order interactions among the Ising spins remain largely unexplored, particularly for applications in computing. Therefore, in this work, we propose Ising spin-based dynamical systems that consider higher order (> 2) interactions among the Ising spins, which subsequently, enables us to develop computational models to directly solve many COPs that entail such higher order interactions (i.e., COPs on hypergraphs). Specifically, we demonstrate our approach by developing dynamical systems to compute the solution for the Boolean NAE-K-SAT (K ≥ 4) problem as well as solve the Max-K-Cut of a hypergraph. Our work advances the potential of the physics-inspired ‘toolbox’ for solving COPs.

Spin-induced deformations and tests of binary black hole nature using third-generation detectors

In a recent letter [N. V. Krishnendu et al., PRL 119, 091101 (2017)] we explored the possibility of probing the binary black hole nature of coalescing compact binaries, by measuring their spin-induced multipole moments, observed in advanced LIGO detectors. Coefficients characterizing the spin-induced multipole moments of Kerr black holes are predicted by the "no-hair" conjecture and appear in the gravitational waveforms through quadratic and higher order spin interactions and hence can be directly measured from gravitational wave observations. We assess the capabilities of the third-generation gravitational wave interferometers such as Cosmic Explorer and Einstein Telescope in carrying out such measurements and use them to test the binary black hole nature of observed binaries. In this paper, we extend the investigations of our previous work, by proposing to measure (a) spin-induced quadrupole effects, (b) simultaneous measurements of spin-induced quadrupole and octupole effects, in the context of the third-generation detectors. We find that, using third-generation detectors the symmetric combination of coefficients associated with spin-induced quadrupole moment of each binary component may be constrained to a value $\leq 1.1$ while a similar combination of coefficients for spin-induced octupole moment may be constrained to $\leq 2$, where both combinations take the value of 1 for a binary black hole system. Further, we consider two different binary black hole populations, as proxies of the population that will be observed by the third generation detectors, and obtain the resulting distribution of the spin-induced quadrupole coefficient. These estimates suggest that third-generation detectors can accurately constrain the first four multipole moments of the compact objects (mass, spin, quadrupole, and octupole) facilitating a thorough probe of their black hole nature.

Thermal valence-bond-solid transition of quantum spins in two dimensions

We study the $S=1/2$ Heisenberg ($J$) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions ($Q$) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large $Q/J$ and diverging when $Q/J$ approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge $c=1$ and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of $U(1)$ symmetry of the order parameter at $T={T}_{c}$ when ${T}_{c}\ensuremath{\rightarrow}0$.

Conical Spin-Spiral State in an Ultrathin Film Driven by Higher-Order Spin Interactions

We report a transverse conical spin spiral as the magnetic ground state of a double-layer Mn on a W(110) surface. Using spin-polarized scanning tunneling microscopy, we find a long-range modulation along the [001] direction with a periodicity of 2.4 nm coexisting with a local row-wise antiferromagnetic contrast. First-principles calculations reveal a transverse conical spin-spiral ground state of this system which explains the observed magnetic contrast. The canting of the spins is induced by higher-order exchange interactions, while the spiraling along the [001] direction is due to frustrated Heisenberg exchange and Dzyaloshinskii-Moriya interaction.

Itinerant Magnets on a Triangular Cu(111) Lattice

We investigate complex spin structures of frustrated two-dimensional Cr, Mn, and Fe monolayer magnets on a triangular lattice provided by the Cu(111) substrate. First we establish a zero-temperature phase diagram of possible spin structures on the basis of the classical Heisenberg model up to the third-nearest neighbor exchange interaction. Second we carried out first-principles total energy calculations on the basis of the vector-spin density formulation of the density functional theory using the full potential linearized augmented plane wave (FLAPW) method in film geometry for a set of complex non-collinear spin structures. We found, the ground state of Fe is ferromagnetic, Cr exhibits a coplanar, two-dimensional non-collinear 120 Néel state and Mn a three-dimensional non-collinear ground state, the 3Q-state. Incommensurate spin-spiral states are expected for a FeMn alloy on Cu(111). We employ the constrained local moment method to estimate the exchange parameters of the model Hamiltonians. We show that for Mn higher-order spin interactions are the origin of the 3Q-state for Mn. The combination of ab initio calculations and model Hamiltonians provides a powerful tool to investigate the magnetic structures of complex magnetic systems.

Mixed ising spin system with higher-order spin interaction

For the mixed Ising spin systems with spins S and R (considered cases of S=1 and R= 1 2 , S=1 and R= 3 2 , S=1 and R=2) the Curie temperature T c and the magnetizations 〈 S z 〉 and 〈 R z 〉 have been calculated by making use of four-spin model approximation for various values of the higher-order spin interactions.

Mixed ising spin system with higher-order spin terms

For the mixed Ising spin system with spins S = 1 and R = ½ and that with spins S = 1 and R = 3/2, the Curie temperature T c and the magnetizations 〈S z〉 and 〈R z〉 have been calculated by making use of the four-spin model approximation for various values of the higher-order spin interactions.

Exact ground state phase diagrams of extended ANNNI and BNNNI models

The ground state phase diagram of spinS=3/2 extended Axial Next Nearest Neighbour Ising (ANNNI) model with higher order spin interaction is obtained exactly by using the Fundamental Spin Arrangement (FAS) method, and compared with those ofS=1/2 and 1 models. The results are also applicable to the Biaxial Next Nearest Neighbour Ising (BNNNI) model where there is competition between interactions within two directions.

Magnetic phase diagram of the S=1 axial next-nearest-neighbour Ising (ANNNI) model with higher-order spin interaction

The three-dimensional S=1 axial next-nearest-neighbour Ising (ANNNI) model is discussed. In addition to the ferromagnetic intra- and interlayer interactions J0 and J1 two types of competing antiferromagnetic interactions are considered between next-nearest layers, i.e. ordinary two-site two-spin J2SiSi+2 ((2-2) model) or three-site four-spin J3SiSi+12Si+2 ((3-4) model) interactions. For both models, the magnetic phase diagrams are obtained by means of the site-dependent molecular field approximation. The phase boundaries among paramagnetic, ferromagnetic and various modulated phases are determined by analysing frequency-dependent susceptibility chi (q), or by solving coupled equations of (Si) and (S12) for spins up to 17, iteratively. The constructed 'devil's flower' of the (3-4) model is characteristically different from that of the (2-2) model, since three-site four-spin interaction is effectively temperature dependent. For the (3-4) model, a new behaviour of re-entrant transition is found for the interaction ratio -0.50>J3/J1>-0.53. The role of the entropy term for this re-entrant transition is discussed in detail by analysing the free energies around the multiphase point (T=0, J3/J1=-0.5).

S=1 axial next-nearest neighbour Ising (ANNNI) model with higher order spin interaction

The three-dimensional S=1 axial next-nearest neighbour Ising (ANNNI) model is discussed. In addition to the ferromagnetic intra- and inter-layer interactions J0 and J1, two types of competing antiferromagnetic interactions are considered between next-nearest layers, i.e. the ordinary two-site two-spin interaction, J2SiS1+2, ((2-2) model) or three-site four-spin interaction, J3SiSi+12Si+2, ((3-4) model). For both models, the ground state is rigorously obtained by means of the transfer matrix method, and proved to change from ferromagnetic to antiphase spin structure at J2 (or J3)/J1=-1/2 with the increase of competition. From magnetization, internal energy, specific heat and 'absolute magnetization' calculated by the Monte Carlo simulation with Fourier transformation, phase boundaries among paramagnetic, ferromagnetic and modulated phases are determined with the location of the Lifshitz point. It is confirmed that for the (3-4) model the self-spin correlation, Si+12, included in three-site four-spin interaction weakens the frustration between J1 and J3 at high temperatures. In the vicinity of the paramagnetic phase transition, the correlation along the z-direction becomes considerably weaker than the correlation in xy-plane owing to the frustration along the z-direction. Consequently, for both (2-2) and (3-4) model there exists the temperature region in which the system behaves as quasi two-dimensional system.

Higher Order Spin Interaction and Anisotropy Energy in the Antiferromagnetic Structures of the Pyrite Type Crystal

The effect of the anisotropy energy on the first kind antiferromagnetic structure in the pyrite type crystal is discussed in a phenomenological model, in which higher order spin interactions are taken into consideration. The obtained results are compared with experimental ones.

Impurity Magnons in MnF2:Co

Complete understanding of magnetic impurity modes can be obtained from their energies and wavefunctions. Energies are obtained from absorption, Raman, and fluorescent measurements but wavefunction information from g factors is most useful when the impurity and host g factor are very dissimilar. To this end, impurity modes in MnF2:Co2+ have been studied by far ir absorption and Raman scattering, respectively. The g factor for the s0 mode is measured to be 3.21 and differs substantially from 4.24, the value obtained by EPR in the diamagnetic isomorph MgF2. The s0 energy and g factor can be explained consistently by exchange mixing of the ground state with excited orbital states of Co2+. The g factor for the pair mode at 170.3 cm−1 was found to be 1.61, significantly larger than g (s0) - g (Mn2+) = 1.2. The Ising pair mode energies and g factors have been renormalized by including transverse exchange to second order. The correct energies for the impurity modes are obtained but the g factors of the pair modes do not agree with experiment. Higher-order spin interactions produced by the orbital motion on the cobalt ion do not significantly alter this result.