Ising-type Hamiltonian Research Articles

Quantum annealing with trigger Hamiltonians: Application to 2-satisfiability and nonstoquastic problems

We study the performance of quantum annealing for two sets of problems, namely, 2-satisfiability (2-SAT) problems represented by Ising-type Hamiltonians and nonstoquastic problems which are obtained by adding extra couplings to the 2-SAT problem Hamiltonians. In addition, we add to the transverse Ising-type Hamiltonian used for quantum annealing a third term, the trigger Hamiltonian with ferromagnetic or antiferromagnetic couplings, which vanishes at the beginning and end of the annealing process. We also analyze some problem instances using the energy spectrum, average energy or overlap of the state during the evolution with the instantaneous low-lying eigenstates of the Hamiltonian, and identify some nonadiabatic mechanisms which can enhance the performance of quantum annealing.

A graph-theoretic approach for multiscale modeling and prediction of crack propagation in polycrystalline materials

A multiscale graph theory-based approach is introduced here to predict the microscale crack path in polycrystalline materials. The crack path is represented as the boundary of the partition of a geometric graph. The partitioning is carried out by optimizing an Ising-type hamiltonian. The hamiltonian parameters are chosen such that each partition cost is the same as the energy of the corresponding crack. The interplay of the loading conditions on the specimen and the microstructure of the material near the crack tip determines the crack growth angle in polycrystalline materials. Two different length scales of macro and micro is incorporated for the crack path by defining the crack total energy as the summation of macroscopic energy release and microscopic surface energy. The former term represents the macroscopically favorable crack growth angle by directing the crack to propagate from the crack tip along the direction of the maximum energy release. The latter term guides the microscopic crack path along the macroscopically preferred direction. At this scale, the crack path naturally accommodates both the intergranular and transgranular fractures. In the case of intergranular fracture, the crack propagates along the grain boundaries, while in the case of transgranular fracture, the crack propagates along the crystallographic cleavage planes. The mixed-mode fracture in a thin foil specimen is studied, and the effect of the dihedral angle of the 2D crack is included in defining the effective surface energy. The model is validated using the analytical results for mixed-mode fracture in an isotropic medium and mode-I fracture in a medium with a preferred crack direction. The proposed method can be used to design materials microstructure with optimal fracture resistance.

Acetylene Adsorption on Pd–Ag Alloys: Evidence for Limited Island Formation and Strong Reverse Segregation from Monte Carlo Simulations

Restructuring of alloy surfaces induced by strongly bound adsorbates is a well-established phenomenon occurring in catalysis and membrane science. In catalytic processes, this restructuring can have profound effects because it alters the ensemble distribution between the as-prepared state of the catalyst and the catalytic surface under operando conditions. This work assesses the restructuring of Pd–Ag alloys induced by adsorption of acetylene in the framework of the ensemble formalism. A detailed Ising-type model Hamiltonian of the (111) surface plane is fitted to extensive density functional theory computations. The equilibrium distributions under a realistic environment are then evaluated by a Monte Carlo approach as a function of temperature and alloy composition. Acetylene induces a strong reverse segregation within the relevant range of temperature. Therefore, the surface of Pd–Ag catalysts is almost entirely covered by Pd for bulk ratios <0.8 Ag–Pd, which is, in general, detrimental to the selectivity of Pd–Ag catalysts. Despite the very strong vertical segregation, acetylene only induces marginal in-plane ordering, that is, the surface triangular ensembles follow random distributions as a function of the surface layer Ag fraction quite closely. This can be explained by two factors: first, triangular sites are not sufficient to fully capture the diversity of acetylene binding energies on Pd–Ag alloy surfaces. Rather, an extended environment including the first coordination sphere is necessary and leads to an overlap in terms of binding energy between weakly binding Pd3 ensembles and strongly binding Pd2Ag ensembles. The second critical aspect is related to lateral interactions, which preclude adsorption of acetylene molecules on nearest neighbor triangular sites. Therefore, in a Pd3 island, roughly two thirds of Pd3 sites would be lost. Our study suggests that the equilibrium structure of these alloy catalysts under operando conditions is far from the state targeted by the catalyst design, revealing a nearly unavoidable reason for loss of selectivity of the catalyst with time of operation.

Observation of Entangled States of a Fully Controlled 20-Qubit System

We generate and characterise entangled states of a register of 20 individually controlled qubits, where each qubit is encoded into the electronic state of a trapped atomic ion. Entanglement is generated amongst the qubits during the out-of-equilibrium dynamics of an Ising-type Hamiltonian, engineered via laser fields. Since the qubit-qubit interactions decay with distance, entanglement is generated at early times predominantly between neighbouring groups of qubits. We characterise entanglement between these groups by designing and applying witnesses for genuine multipartite entanglement. Our results show that, during the dynamical evolution, all neighbouring qubit pairs, triplets, most quadruplets, and some quintuplets simultaneously develop genuine multipartite entanglement. Witnessing genuine multipartite entanglement in larger groups of qubits in our system remains an open challenge.

Electronic Structure Calculations and the Ising Hamiltonian.

Obtaining exact solutions to the Schrödinger equation for atoms, molecules, and extended systems continues to be a "Holy Grail" problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, "Can these machines be used to perform electronic structure calculations?" Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H2 , He2 , HeH+, and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. This is an early step in developing generalized methods on such devices for chemical physics.

Relationship between Torsion and Anisotropic Exchange Coupling in a Tb(III)-Radical-Based Single-Molecule Magnet.

The incorporation of paramagnetic ligands within rare-earth ion clusters exhibiting large magnetic anisotropy has provided significant advancement in the design of single-molecule magnets (SMMs) with large blocking temperatures. However, the exchange interaction in such systems is complex and difficult to probe by conventional magnetometry techniques, and little is known about the structural relationships. Inelastic neutron scattering and terahertz electron paramagnetic resonance measurements are used complimentarily to investigate the large exchange interaction between a rare earth-radical pair in a Tb(III)-based SMM complex. The origin of the exchange interaction is investigated for two molecular species in the crystallographic unit cell that exhibit different bonding structures between Tb(III) and a 2pyNO radical. A correlation between the Tb-O-N-C torsion angles and the magnitudes of exchange couplings is found. Interestingly, a large nondegeneracy within the ground-state doublet is present for the larger torsion angle species. It is essential to consider the balance of two channels of exchange coupling, 2p-4f hybridization and 2p-5d charge transfer, to explain this characteristic behavior. The former channel gives the antiferromagnetic interaction, and the latter gives the ferromagnetic one. When an effective Ĵ = (1)/2 Ising-type Hamiltonian is applied, the exchange couplings are evaluated to be antiferromagnetic J(z) = 9.89 meV (79.8 cm(-1)) for the low torsion angle (3.8°) species and J(z) = 7.39 meV (59.6 cm(-1)) for the larger torsion angle (15.8°) species. It is also found that a small percentage of the transverse exchange component must be included for the larger torsion angle to account for the observed nondegenerate ground state. The symmetry of the exchange couplings is discussed by considering the characters of d and f orbitals.

Growth of graph states in quantum networks

We propose a scheme to distribute graph states over quantum networks in the presence of noise in the channels and in the operations. The protocol can be implemented efficiently for large graph sates of arbitrary (complex) topology. We benchmark our scheme with two protocols where each connected component is prepared in a node belonging to the component and subsequently distributed via quantum repeaters to the remaining connected nodes. We show that the fidelity of the generated graphs can be written as the partition function of a classical Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear cluster and results for its decay rate in random graphs with arbitrary (uncorrelated) degree distributions.

Analysis of long-range interaction effects on phase transitions in two-step spin-crossover chains by using Ising-type systems and Monte Carlo entropic sampling technique

The analysis of two-step spin crossover phenomena in one-dimensional systems is performed in the framework of Ising-type model by using Monte Carlo entropic sampling technique. Both short-range and long-range interactions are considered in the Hamiltonian which also takes into account different degeneracies between the molecular states. The density of states associated to each macrostate is computed by using entropic sampling, via a biased Monte Carlo sampling technique, and the corresponding Ising-type Hamiltonian can thus be solved by using a self-consistent approach. By using this semi-analytical method, the effects of long-range interactions on phase transition driven by temperature variation in spin crossover chains have been investigated. Various types of spin crossover phenomena have been identified from gradual two-step transition to the one accompanied by multiple hysteresis loops, which was also recently observed experimentally. A metastable “plateau” has been observed at a high-spin fraction approximately equal to ½ for intermediate temperature region, but this plateau is not part of the low-to-high spin transition in strong cooperative systems and might be reached only by using special reversal curves or other external factors, such as light or pressure. In addition, a standard Monte-Carlo algorithm with Arrhenius dynamics is applied to this system in order to better understand its molecular behavior during the intermediate phase.

Configurational thermodynamics of the Fe-Crσphase

Configurational thermodynamics of the Fe-Cr sigma phase is investigated on the basis of an Ising-type configurational Hamiltonian and a single-site mean-field model for the free energy. The parameters of the statistical models are obtained from efficient first-principles calculations using different computational techniques. We demonstrate that the effective pair and multisite interactions in the sigma phase are relatively small, which allows using a simplified model for distributing Fe and Cr atoms among sublattices. We also show that this system exhibits a nontrivial magnetic behavior at high temperatures, which affects the site occupation by Fe and Cr atoms. The structural variation (volume and c/a) that might be present due to neutron irradiation and thermal expansion can lead to an additional atomic redistribution.

Choice of dynamics for spin-crossover systems

We present a generalized version of the Ising-type Hamiltonian describing the spin-crossover (SC) solids, accounting for the tunneling effect, lattice phonons, and spin-phonon interactions. We show that under Lang-Firsov-type transformation, the model becomes isomorphic to an effective Ising-type Hamiltonian in which the ``exchange'' coupling between the spin operators originates from a multiphonon interaction, thus explaining the origin of the elastic interaction between the SC units. In addition, we investigated theoretically the quantum nonequilibrium version of the present model in the frame of the Born-Oppenheimer approach. As a result, we could derive naturally a Glauber-type master equation with Arrhenius-type dynamics that agrees with experimental data. The present theory gives the microscopic origin of the Arrhenius-type dynamics which is usually set phenomenologically in most of the models describing the dynamics of spin-crossover phenomena.

An Efficient Cluster Expansion Method for Binary Solid Solutions: Application to the Halite-Silvite, NaCl-KCl, System

Research Article| January 01, 2010 An Efficient Cluster Expansion Method for Binary Solid Solutions: Application to the Halite-Silvite, NaCl-KCl, System Victor Vinograd; Victor Vinograd Institute of Geosciences, University of Frankfurt, Frankfurt a.M., Germany, v.vinograd@kristall.uni-frankfurt.de, b.winkler@kristall.uni-frankfurt.de Search for other works by this author on: GSW Google Scholar Björn Winkler Björn Winkler Institute of Geosciences, University of Frankfurt, Frankfurt a.M., Germany, v.vinograd@kristall.uni-frankfurt.de, b.winkler@kristall.uni-frankfurt.de Search for other works by this author on: GSW Google Scholar Reviews in Mineralogy and Geochemistry (2010) 71 (1): 413–436. https://doi.org/10.2138/rmg.2010.71.19 Article history first online: 09 Mar 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation Victor Vinograd, Björn Winkler; An Efficient Cluster Expansion Method for Binary Solid Solutions: Application to the Halite-Silvite, NaCl-KCl, System. Reviews in Mineralogy and Geochemistry 2010;; 71 (1): 413–436. doi: https://doi.org/10.2138/rmg.2010.71.19 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyReviews in Mineralogy and Geochemistry Search Advanced Search Abstract The system NaCl-KCl is used as an example to introduce the reader to a variety of concepts, which form the modern solid solution theory. The chapter starts with the discussion of the chemical stress both end members experience in the process of mixing. This stress is treated as a homogeneous deformation of the end members within the concept of the virtual crystal approximation, VCA. It is shown that the VCA predicts the existence of a positive excess enthalpy in the NaCl-KCl solid solution, but drastically overestimates its magnitude. The inaccuracy of the VCA is attributed to its failure to describe local relaxations. Further, the local relaxations are linked to the effects of intracrystalline reactions of NaNa + KK = 2NaK type, and it is shown that the excess enthalpy can be written as a sum of such interactions acting at various distances within the supercell. The decomposition of the enthalpy of mixing into pairwise interactions is then discussed in the relation to a more general cluster expansion technique, which considers effective interactions from clusters of various sizes and shapes including many-body interactions. Then we argue that a wide range of materials interesting for geosciences can be described within the concept of pairwise interactions only and introduce the supercell expansion method dealing with the effective pairwise interactions. It is then shown that the calculation of the effective pairwise interactions can be performed with different methods. The traditional approach consisting in the calculation of the static energies of a large number of randomly varied structures and solving for the pairwise interactions with the least squares method is compared to a deterministic algorithm, which is based on the consideration of the excess energies of the structures with double defects inserted at all possible pairwise distances within the supercell. This Double Defect Method, DDM, is then applied to the system NaCl-KCl. The effective interactions obtained are used to construct an Ising-type Hamiltonian, from which the temperature dependent mixing enthalpy is evaluated with the Monte Carlo method. Then the temperature-dependent enthalpies are thermodynamically integrated to obtain the free energies of mixing and to construct the temperature-composition diagram. A comparison of the calculated diagram with the experimentally known phase relations shows that a certain thermodynamic effect is missed in the simulations. This effect is identified as the excess vibrational entropy and the DDM calculations are repeated within the quasi-harmonic approximation, which evaluates both the static and the vibrational components of the free energy. These calculations are used to derive the temperature-dependent pairwise interactions, which via the Monte Carlo simulations lead to a much better agreement with the experiment. Finally we discuss applications of the DDM approach to solid solutions with coupled substitution and to multicomponent solutions. You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

The RKKY Coupling of Two Magnetically Doped Monolayers in Thin Films

The studies of interaction between localized magnetic moments and charge carriers usually assume a contact form of the exchange potential. However, as the localized magnetic moments originate from the inner electronic orbitals (e.g. of d-type in diluted magnetic semiconductors), they manifest a finite spatial extension [1]. Consequently, it seems natural to introduce a diffused exchange potential. This, in turn, modifies the perturbative RKKY-type interaction between the localized spins. Such an effect for the bulk case has been considered so far in Refs. [2,3]. In the bulk media, the relevant lengthscale to be compared with the distribution width is given by an inverse of the Fermi wavevector. However, for the important case of ultrathin film (quantum well) geometry, the film thickness constitutes another vital lengthscale and the spatial extension of the potential is not negligible compared with that, independently on the concentration of the carriers. In our work we apply the Gauss-type exchange coupling potential to the RKKY mechanism of localized spins interaction in ultrathin film modeled by an infinite quantum well [4,5]. We consider a model system based on fcc structure with orientation (001), when two monolayers inside the film contain magnetic ions. Such choice reflects the geometry of experimentally investigated structures of magnetically-doped and undoped semiconductors (such as MnGaAs/GaAs). The model is parametrized by the total thickness of the film (i.e. thickness of the quantum well), the distance between the magnetic layers as well as the concentration of free charge carriers inside the film. We especially study the coupling energy of two magnetic monolayers as a function of their separation, total width of the quantum well and the charge concentration. Our main aim is to explore the range of parameters for which the interaction energy favors ferromagnetic or antiferromagnetic orientation of magnetizations in these layers. In addition we calculate the MFA critical temperatures (Curie/Neel) for the system. We make comparison of our results with the limiting case when the bulk (3D) RKKY interaction coupling is taken into account. Such comparison emphasizes the importance of quantum size effects in the thin films. The considerations are based on the Ising-type Hamiltonian of interaction between the localized spins. To calculate the energy, summation is performed over the lattice sites up to the cut-off distance assuring the satisfactory convergence of the interaction energy per spin, so that we do not have to use the continuous approximation. We emphasize that the diffusion of the contact potential has a noticeable effect on the interaction energy. What is crucial, it also removes the divergence of the RKKY thin-film integral in the direction perpendicular to the film plane.

Density fluctuations and correlations of confined fluids

The density fluctuations about the equilibrium structure of fluids confined by parallel planar walls are analyzed for the cases of identical and symmetrically opposed fields at the walls. We determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) for conditions both above and below the wetting transition temperature T w of the semi-infinite system and corroborate in all cases that the equilibrium configurations are stable. We identify the fluctuations close to the walls and in the middle of the slab and discuss their effect when the wall separation L diverges. For competing walls above T w the localized fluctuation with lowest eigenvalue describes the displacements of the incipient wetting films that become unimpeded interfacial translations for L→∞. Below T w the fluctuations with lowest eigenvalue correspond to stiffer deformations extended across the slab. For identical walls above T w coexisting states display incipient prewetting films and the lowest eigenvalue describes the nature of their growth as L increases. We also calculate the pair correlation function for the inhomogeneous states and, for symmetrically opposed walls, we obtain standard Ornstein–Zernike (OZ) behavior at the walls, but find significant deviations from this law at the interface-like region in the middle of the slab. To model fluids with short-ranged forces we use a ferromagnetic Ising-type Hamiltonian in mean-field approximation.

Ab initio theory of surface segregation: Self-consistent determination of the concentration profile.

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme1–3, or by the generalized perturbation method (GPM)4. The GPM relies on the force theorem5, according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex.

Dilution Effects in a Spin Crossover System, Modelled in Terms of Direct and Indirect Intermolecular Interactions

We have studied by 57 Fe Mossbauer spectroscopy and magnetic measurements the spin transition exhibited by the series: [Fe x Co 1 -x(4,4'-bis-1,2,4-triazole)2(NCS)2].H 2 O, consisting of a spin-crossover iron(II) system diluted in a high-spin cobalt(II) matrix. We performed Monte Carlo simulations of the thermal variation of the high-spin fraction, using a two-level Ising-type Hamiltonian. A good agreement with experiment is obtained by the introduction of an indirect interaction of the spin-crossover units via the non-spin crossover unit, the origin of which is discussed in steric terms.

Hamiltonian approach to kinetic Ising models.

In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr\odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.