Class Of Hamiltonians Research Articles

Quantum Approximate Markov Chains are Thermal

We prove that any one-dimensional (1D) quantum state with small quantum conditional mutual information in all certain tripartite splits of the system, which we call a quantum approximate Markov chain, can be well-approximated by a Gibbs state of a short-range quantum Hamiltonian. Conversely, we also derive an upper bound on the (quantum) conditional mutual information of Gibbs states of 1D short-range quantum Hamiltonians. We show that the conditional mutual information between two regions A and C conditioned on the middle region B decays exponentially with the square root of the length of B. These two results constitute a variant of the Hammersley-Clifford theorem (which characterizes Markov networks, i.e. probability distributions which have vanishing conditional mutual information, as Gibbs states of classical short-range Hamiltonians) for 1D quantum systems. The result can be seen as a strengthening - for 1D systems - of the mutual information area law for thermal states. It directly implies an efficient preparation of any 1D Gibbs state at finite temperature by a constant-depth quantum circuit.

Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

The classical and quantum-mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics (i.e., $p_{j} (\vec{x}) \rightarrow p_{j}(\vec{x}) -e A_{j}(\vec{x})$ , where $p_{j}(\vec{x})$ is the j-th component of the classical PDM-canonical-momentum), it admits a necessarily different and vital form in quantum mechanics (i.e., $\hat{p}_{j}(\vec{x})/\sqrt{m(\vec{x})} \rightarrow (\hat{p}_{j}(\vec{x})$ $ -e A_{j}(\vec{x}))/\sqrt{m(\vec{x})}$ , where $ \hat{p}_{j}(\vec{x})$ is the j-th component of the quantum PDM-momentum operator). Under our point transformation settings, only one of the two commonly used vector potentials (i.e., $\vec{A} (\vec{x}) \sim (-x_{2}, x_{1}, 0)$ ) is found eligible and is considered for our Illustrative examples.

Model of ballistic-diffusive thermal transport in HAMR media

A model is presented that considers the grain size and temperature dependence of the thermal conductivity and specific heat of L10 FePt nanoparticles and is applied to study the temperature distribution within the top recording layer in heat assisted magnetic recording. Ballistic effects on the thermal conductivity are included assuming diffuse scattering of the electron heat carriers at interfaces and grain boundaries. The magnetic contribution to the specific heat during laser heating close to the Curie temperature is determined using Monte Carlo simulations based on the effective classical spin Hamiltonian of chemically ordered L10 FePt. Following the application of a nanosecond laser pulse, the perpendicular heat flow through a FePt/MgO/heat-sink stack is calculated using the finite element method. The temperature drop across the thickness of the FePt layer is shown to be significant for grains of small in-plane size. Size induced grain temperature variations are small due to the temperature dependence of the specific heat of FePt, so the resultant contribution to transition jitter is expected to be limited in extent.

Effective Ising model for correlated systems with charge ordering

Collective electronic fluctuations in correlated materials give rise to various important phenomena, such as existence of the charge ordering, superconductivity, Mott insulating and magnetic phases, plasmon and magnon modes, and other interesting features of such systems. Unfortunately, description of these correlation effects requires significant efforts, since they almost entirely rely on strong local and nonlocal electron-electron interactions. Some collective phenomena, such as magnetism, can be sufficiently described by a simple Heisenberg-like models that are formulated in terms of bosonic variables. This fact suggests that other many-body excitations can also be described by simple bosonic models in spirit of the Heisenberg theory. Here we derive an effective bosonic action for charge degrees of freedom for the extended Hubbard model and define a physical regime where the obtained action reduces to a classical Hamiltonian of an effective Ising model.

Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p . We assume that their Poisson commutator vanishes, is a constant, a constant times H, or a constant times K. In the quantum case H and K are operators and their Lie commutator has one of the above properties. We use two copies of such pairs to generate two-dimensional superintegrable systems in the Euclidean space E2, allowing the separation of variables in Cartesian coordinates. Nearly all known separable superintegrable systems in E2 can be obtained in this manner and we obtain new ones for N = 4.

Magnetic properties of nanowires with ferromagnetic core and antiferromagnetic shell

We present a Monte Carlo study of the magnetic properties of thin cylindrical nanowires composed of a ferromagnetic core and an antiferromagnetic shell implementing a classical spin Hamiltonian. We address systematically the impact of interface exchange coupling on the loop characteristics and the magnetization reversal mechanism. We study the effect of shell polycrystallinity on the characteristic fields of the isothermal hysteresis loop (coercivity, exchange-bias). We demonstrate that coupling to a polycrystalline antiferromagnetic shell increases the critical core diameter for transition from transverse to vortex domain walls.

Short-Time Propagators and the Born-Jordan Quantization Rule.

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.

Semiempirical molecular dynamics (SEMD) simulations: Parameterization and validation for biological systems precursors

The design and development of novel drugs could benefit considerably from more reliable atomistic modeling schemes able to deal with the appropriate level of theoretical description on size and time scales relevant for complex life-science systems. This has not been possible, mainly due to limitations of the scaling and performance of quantum chemical algorithms on large biomolecules. Recently, we presented a novel approach that allows us to apply quantum simulations to the kinds of biological domains that are typical for classical Hamiltonians. For this we exploited a novel and efficient large-scale parallel sparse matrix–matrix multiplication algorithm. The scheme is sufficiently general to deal with any type of quantum scheme, and we opted our algorithm to an easily parametrizable semiempirical Neglect of Diatomic Differential Orbitals (NDDO) Hamiltonian. In this paper, we present a novel NDDO parameterization specifically tailored for treating intra- and inter-molecular interactions in proteins. We include a validation of static and dynamic properties for individual amino acids as well as for small polypeptides using both experimental and density functional theory benchmarks for reference. We also demonstrate the preeminence of this approach for describing bond-breaking reactions in terms of time-to-solution compared to traditional quantum simulations.

An Exact Version of the Egorov Theorem for Schrödinger Operators in $$L^{2}({\mathbb {T}})$$ L 2 ( T )

We provide an exact version of the Egorov Theorem for a class of Schrodinger operators in $$L^2({\mathbb {T}})$$ , where $${\mathbb {T}}={\mathbb {R}}/2\pi {\mathbb {Z}}$$ is the one-dimensional torus. We show that the classical Hamiltonian, after the symplectic transformation to action coordinates, can be composed with a toroidal semiclassical $$\psi $$ do in order to recover the Schrodinger operator. This result turns out to be strictly related to the Bohr-Sommerfeld quantization rules as well as to the inverse spectral problem and the periodic homogenization of Hamilton–Jacobi equations.

Classical N-reflection equation and Gaudin models

We introduce the notion of N-reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the N=2 case. The basic theory is established and illustrated with several examples of solutions of the N-reflection equation associated with the rational and trigonometric r-matrices. A central result is the construction of a Poisson algebra associated with a non-skew-symmetric r-matrix whose form is specified by a solution of the N-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of BC_L-type.

Dirac’s Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space

The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.

Wave‐Packet Dynamics in Nonlinear Disordered Chains under the Effect of Acoustic Waves

In this work, the wave‐packet dynamics is considered under the effect of diagonal disorder distribution, electron–lattice interaction, and acoustic‐wave pumping. Electronic transport is taken into account quantum mechanically, whereas lattice vibrations are described through a classical nonlinear Morse Hamiltonian. The electronic hopping term takes the form of an exponential function of the effective distance between nearest‐neighbor lattice elements. The electron wave‐packet is initially localized at the first site of the chain alongside a Gaussian pulse generator. The results suggest that the disorder distribution breaks down the solitonic energy profile that exists within this kind of nonlinear Morse lattice. On the other hand, it is shown that the acoustic pumping sustains the wave packet dynamics within this disordered nonlinear model.

κ-Deformed Photon and Jaynes-Cummings Model

In this paper, we discuss the κ-deformed boson algebra. We present the infinite dimensional representation of this algebra. Assuming that the photon obeys the κ-deformed algebra instead of the ordinary boson algebra, we discuss the κ-deformed quantum optical electromagnetic field whose classical Hamiltonian is written in terms of an infinite set of real harmonic oscillators. We also discuss the second quantization of light when we regard the photon as the quantum particle obeying the κ-deformed boson algebra. Finally we discuss the interaction between a two-level atom coupled to a single κ-photon.

Exchange bias effect in cylindrical nanowires with ferromagnetic core and polycrystalline antiferromagnetic shell

We study numerically the exchange bias effect in thin cylindrical nanowires composed of a ferromagnetic core and an antiferromagnetic shell implementing a classical spin Hamiltonian and Monte Carlo simulations. We address systematically the effect of shell polycrystallinity on the characteristic fields of the isothermal hysteresis loop (coercivity, exchange-bias) and their angular dependence upon the direction of the reversing and cooling fields. We relate the observed trends to modifications of the underlying magnetization reversal mechanism. We fit our simulation results to an extended Stoner-Wohlfarth model with effective off-axis unidirectional anisotropy and demonstrate that shell polycrystallinity leads to maximum exchange bias effect in an off-axis direction. Our results are in qualitative agreement with recent experimental studies of Co/CoO nanowires.

Quantum Atomistic Approach for Interacting Spins

The phenomenological Landau theory of the spin precession has been used to reproduce the out-of-equilibrium properties of many magnetic systems. However, such an approach suffers from some serious limitations. The main reason is that the spin and the angular momentum of the atoms are described by the classical theory of the angular momentum. We derive a discrete model that extends the Landau theory to the quantum mechanical framework. Our approach is based on the application of the quantification procedure to the classical Hamiltonian of an array of interacting spins. The connection with the classical dynamics is discussed.

Longitudinal integration measure in classical spin space and its application to first-principle based simulations of ferromagnetic metals

The classical Heisenberg type spin Hamiltonian is widely used for simulations of finite temperature properties of magnetic metals often using parameters derived from first principles calculations. In itinerant electron systems, however, the atomic magnetic moments vary their magnitude with temperature and the spin Hamiltonian should thus be extended to incorporate the effects of longitudinal spin fluctuations (LSF). Although the simple phenomenological spin Hamiltonians describing LSF can be efficiently parameterized in the framework of the constrained Local Spin Density Approximation (LSDA) and its extensions, the fundamental problem concerning the integration in classical spin space remains. It is generally unknown how to integrate over the spin magnitude. Two intuitive choices of integration measure have been used up to date – the Murata-Doniach scalar measure and the simple three dimensional vector measure. Here we derive the integration measure by considering a classical limit of the quantum Heisenberg spin Hamiltonian under conditions leading to the proper classical limit of the commutation relations for all values of the classical spin magnitude and calculate the corresponding ratio of the number of quantum states. We show that the number of quantum states corresponding to the considered classical spin magnitude is proportional to this magnitude and thus a non-trivial integration measure must be used. We apply our results to the first-principles simulation of the Curie temperatures of the two canonical ferromagnets bcc Fe and fcc Ni using a single-site LSF Hamiltonian with parameters calculated in the LSDA framework in the Disordered Local Moment approximation and a fixed spin moment constraint. In the same framework we compare our results with those obtained from the scalar and vector measures.

Exotic states of matter with polariton chains

We consider linear periodic chains of exciton-polariton condensates formed by pumping polaritons non-resonantly into a linear network. To the leading order, such a sequence of condensates establishes relative phases as to minimize a classical one-dimensional XY Hamiltonian with nearest and next to nearest neighbors. We show that the low energy states of polaritonic linear chains demonstrate various classical regimes: ferromagnetic, antiferromagnetic and frustrated spiral phases, where quantum or thermal fluctuations are expected to give rise to a spin liquid state. At the same time nonlinear interactions at higher pumping intensities bring about phase chaos and novel exotic phases.

From physical assumptions to classical and quantum Hamiltonian and Lagrangian particle mechanics

The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible—specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts—will give us a classical Hamiltonian system. Assuming the system is irreducible—specifying the state and the dynamics of the whole system tells us nothing about the state and the dynamics of its substructure—will give us a quantum Hamiltonian system. Assuming kinematic equivalence, that studying trajectories is equivalent to studying state evolution, will give us Lagrangian mechanics and limit the form of the Hamiltonian/Lagrangian to the one with scalar and vector potential forces.

Quantum approximate optimization algorithm for MaxCut: A fermionic view

Farhi et al. recently proposed a class of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), for approximately solving combinatorial optimization problems. A level-p QAOA circuit consists of p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here, we analytically and numerically study parameter setting for QAOA applied to MaxCut. For level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the ring of disagrees, or the 1D antiferromagnetic ring, we provide an analysis for arbitrarily high level. Using a fermionic representation, the evolution of the system under QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of QAOA for any p. It also greatly simplifies numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional sub-manifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.

Three-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability

As a straightforward generalization and extension of our previous paper [A. V. Turbiner et al., “Three-body problem in 3D space: Ground state, (quasi)-exact-solvability,” J. Phys. A: Math. Theor. 50, 215201 (2017)], we study the aspects of the quantum and classical dynamics of a 3-body system with equal masses, each body with d degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. The quantum (and classical) Hamiltonian for which these states are eigenfunctions is derived. It corresponds to a three-dimensional quantum particle moving in a curved space with special d-dimension-independent metric in a certain d-dependent singular potential, while at d = 1, it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is d-independent; it has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h(3) typical for the H3 Calogero model as in the d = 3 case. We find an exactly solvable three-body S3-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly solvable three-body sextic polynomial type potential with singular terms. For both models, an extra first order integral exists. For d = 1, the whole family of 3-body (two-dimensional) Calogero-Moser-Sutherland systems as well as the Tremblay-Turbiner-Winternitz model is reproduced. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to d > 1 leads to two primitive quasi-exactly solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.