Class Of Hamiltonians Research Articles

Spectroscopic response theory with classical mapping Hamiltonians.

Exact quantum dynamics with a time-independent Hamiltonian in a discrete state space can be computed using classical mechanics through the classical Meyer-Miller-Stock-Thoss mapping Hamiltonian. In order to compute quantum response functions from classical dynamics, we extend this mapping to a quantum Hamiltonian with time-dependence arising from a classical field. This generalization requires attention to time-ordering in quantum and classical propagators. Quantum response theory with the original quantum Hamiltonian is equivalent to classical response theory with the classical mapping Hamiltonian. We elucidate the structure of classical response theory with the mapping Hamiltonian, thereby generating classical versions of the two-sided quantum density operator diagrams conventionally used to describe spectroscopic processes. This formal development can provide a foundation for new semiclassical approximations to spectroscopic observables for models in which classical nuclear degrees of freedom are introduced into a mapping Hamiltonian describing electronic states. Calculations of the temperature-dependence of two-dimensional electronic spectra for an exciton dimer using two semiclassical approaches are compared with benchmark calculations using the hierarchical equations of motion method.

Magnetic skyrmions in cylindrical ferromagnetic nanostructures with chiral interactions

We study the geometrical conditions for stabilizing magnetic skyrmions in cylindrical nanostrips and nanotubes of ferromagnetic materials with chiral interactions. We obtain the low-temperature equilibrium state of the system implementing a simulation annealing technique for a classical spin Hamiltonian with competing isotropic exchange and chiral interactions, radial anisotropy and an external field. We address the impact of surface curvature on the formation, the shape and the size of magnetic skyrmions. We demonstrate that the evolution of the skyrmion phase with the curvature is controlled by the competition between two characteristic lengths, namely the curvature radius, $R$ (geometrical length) and the skyrmion radius, $R_{sk}$ (physical length). In narrow nanotubes ($R<R_{sk}$) the skyrmion phase evolves to a stripe phase, while in wide nanotubes ($R>R_{sk}$) a mixed skyrmion-stripe phase emerges. Most interestingly, the mixed phase is characterized by spatially separated skyrmions from stripes owing to the direction of the applied field relative to the surface normal. In the stability regime ($R \gtrsim R_{sk}$) skyrmions remain circular and preserve their size as a consequence of their topological protection. Zero-field skyrmions are shown to be stable on curved nanoelements with free boundaries within the same stability region ($R\gtrsim R_{sk}$). The experimental and technological perspectives from the stability of skyrmions on cylindrical surfaces are discussed.

Orbital angular momentum dynamics of Bose-Einstein condensates trapped in two stacked rings

We investigate the stability and dynamics of the orbital angular momentum modes of a repulsive Bose-Einstein condensate trapped in two tunnel-coupled rings in a stack configuration. Within mean-field theory, we derive a two-state model for the system in the case in which we populate equally both rings with a single orbital angular momentum mode and include small perturbations in other modes. Analyzing the fixed point solutions of the model and the associated classical Hamiltonian, we characterize the destabilization of the stationary states and the subsequent dynamics. By populating a single orbital angular momentum mode with an arbitrary population imbalance between the rings, we derive analytically the boundary between the regimes of Josephson oscillations and macroscopic quantum self-trapping and study numerically the stability of these solutions.

Horizon induces instability locally and creates quantum thermality

The classical Hamiltonian for a chargeless and massless particle in a very near-horizon region is shown to be of the form $H\ensuremath{\sim}xp$ as long as radial motion is concerned. This is demonstrated explicitly for static spherically symmetric black hole and also found to be applicable for specific choice of radial trajectories in the Kerr case. Such feature of horizon leads to unavoidable ``local instability'' in the particle's radial motion as long as near-horizon regime is concerned. We show that at the quantum level this provides thermality in the system. The temperature is found to be given by the Hawking expression. Finally, we conjecture that the automatic instability created by the horizon is responsible for its own temperature and consequently can be a possible physical mechanism for horizon temperature.

Deterministic Quantum Mechanics: The Mathematical Equations

Without wasting time and effort on philosophical justifications and implications, we write down the conditions for the Hamiltonian of a quantum system for rendering it mathematically equivalent to a deterministic system. These are the equations to be considered. Special attention is given to the notion of `locality'. Various examples are worked out, followed by a systematic procedure to generate classical evolution laws and quantum Hamiltonians that are exactly equivalent. What is new here is that we consider interactions, keeping them as general as we can. The quantum systems found, form a dense set if we limit ourselves to sufficiently low energy states. The class is discrete, just because the set of deterministic models containing a finite number of classical states, is discrete. In contrast with earlier suspicions, the gravitational force turns out not to be needed for this; it suffices that the classical system act at a time scale much smaller than the inverse of the maximum scattering energies considered.

Semiclassical work and quantum work identities in Weyl representation

We derive a semiclassical nonequilibrium work identity by applying the Wigner–Weyl quantization scheme to the Jarzynski identity for a classical Hamiltonian. This allows us to extend the concept of work to the leading order in ℏ. We propose a geometric interpretation of this semiclassical Jarzynski relation in terms of trajectories in a complex phase space and illustrate it with the exactly solvable case of the quantum harmonic oscillator.

Nanolaser-based emulators of spin Hamiltonians

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

Rapid fair sampling of the XY spin Hamiltonian with a laser simulator

Coupled oscillators such as lasers, OPO's and BEC polaritons can rapidly and efficiently dissipate into a stable phase locked state that can be mapped onto the minimal energy (ground state) of classical spin Hamiltonians. However, for degenerate or near degenerate ground state manifolds, statistical fair sampling is required to obtain a complete knowledge of the minimal energy state, which needs many repetitions of simulations under identical conditions. We show that with dissipatively coupled lasers such fair sampling can be achieved rapidly and accurately by exploiting the many longitudinal modes of each laser to form an ensemble of identical but independent simulators, acting in parallel. We fairly sampled the ground state manifold of square, triangular and Kagome lattices by measuring their coherence function identifying manifolds composed of a single, doubly degenerate, and highly degenerate ground states, respectively.

Exact mapping between a laser network loss rate and the classical XY Hamiltonian by laser loss control

AbstractRecently, there has been growing interest in the utilization of physical systems as heuristic optimizers for classical spin Hamiltonians. A prominent approach employs gain-dissipative optical oscillator networks for this purpose. Unfortunately, these systems inherently suffer from an inexact mapping between the oscillator network loss rate and the spin Hamiltonian due to additional degrees of freedom present in the system such as oscillation amplitude. In this work, we theoretically analyze and experimentally demonstrate a scheme for the alleviation of this difficulty. The scheme involves control over the laser oscillator amplitude through modification of individual laser oscillator loss. We demonstrate this approach in a laser network classical XY model simulator based on a digital degenerate cavity laser. We prove that for each XY model energy minimum there corresponds a unique set of laser loss values that leads to a network state with identical oscillation amplitudes and to phase values that coincide with the XY model minimum. We experimentally demonstrate an eight fold improvement in the deviation from the minimal XY energy by employing our proposed solution scheme.

Toverline{T} deformations, massive gravity and non-critical strings

The Toverline{T} deformation of a 2 dimensional field theory living on a curved space- time is equivalent to coupling the undeformed field theory to 2 dimensional ‘ghost-free’ massive gravity. We derive the equivalence classically, and using a path integral formulation of the random geometries proposal, which mirrors the holographic bulk cutoff picture. We emphasize the role of the massive gravity Stückelberg fields which describe the diffeomorphism between the two metrics. For a general field theory, the dynamics of the Stückelberg fields is non-trivial, however for a CFT it trivializes and becomes equivalent to an additional pair of target space dimensions with associated curved target space geometry and dynamical worldsheet metric. That is, the Toverline{T} deformation of a CFT on curved spacetime is equivalent to a non-critical string theory in Polyakov form, with a non-zero B-field. We give a direct proof of the equivalence classically without relying on gauge fixing, and determine the explicit form for the classical Hamiltonian of the Toverline{T} deformation of an arbitrary CFT on a curved spacetime. When the QFT action is a sum of a CFT plus an operator of fixed scaling dimension, as for example in the sine-Gordon model, the equivalence to a non-critical theory string holds with a modified target space metric and modified B-field. Finally we give a stochastic path integral formulation for the general Toverline{T}+Joverline{T}+Toverline{J} deformation of a general QFT, and show that it reproduces a recent path integral proposal in the literature.

Realizing spin Hamiltonians in nanoscale active photonic lattices.

Spin models arise in the microscopic description of magnetic materials and have been recently used to map certain classes of optimization problems involving large degrees of freedom. In this regard, various optical implementations of such Hamiltonians have been demonstrated to quickly converge to the global minimum in the energy landscape. Yet, so far, an integrated nanophotonic platform capable of emulating complex magnetic materials is still missing. Here, we show that the cooperative interplay among vectorial electromagnetic modes in coupled metallic nanolasers can be utilized to implement certain types of spin Hamiltonians. Depending on the topology/geometry of the arrays, these structures can be governed by a classical XY Hamiltonian that exhibits ferromagnetic and antiferromagnetic couplings, as well as geometrical frustration. Our results pave the way towards a scalable nanophotonic platform to study spin exchange interactions and could address a variety of optimization problems.

Casimir effect in polymer scalar field theory

In this paper, we study the Casimir effect in the classical geometry of two parallel conducting plates, separated by a distance $L$, due to the presence of a minimal length $\lambda$ arising from a background independent (polymer) quantization scheme. To this end, we polymer-quantize the classical Klein-Gordon Hamiltonian for a massive scalar field confined between the plates and obtain the energy spectrum. The minimal length scale of the theory introduces a natural cutoff for the momenta in the plane parallel to the plates and a maximum number of discrete modes between the plates. The zero-point energy is calculated by summing over the modes, and by assuming $\lambda \ll L$, we expressed it as an expansion in powers of $1/N$, being $N=L/ \lambda$ the number of points between the plates. Closed analytical expressions are obtained for the Casimir energy in the cases of small and large scalar mass limits.

Bifurcation of Relative Equilibria for Vortices and General Homogeneous Potentials

We examine the relative equilibria of the four vortex problem where three vortices have equal strength, that is $$ \varGamma _1 = \varGamma _2 = \varGamma _3 = 1 $$ and $$ \varGamma _4 = m $$, where m is a parameter. We study the problem in the case of the classical logarithmic vortex Hamiltonian and in the case the Hamiltonian is a homogeneous function of degree $$ \alpha $$. First we study the bifurcations emanating from the equilateral triangle configuration with the fourth vortex in the barycenter. Then, in the vortex case, we show that all the convex configurations contain a line of symmetry. We also give a precise count of the number of concave asymmetric configurations. Our techniques involve a combination of analysis and computational algebraic geometry.

Analyticity of the free energy for quantum Airy structures

It is shown that the free energy associated to a finite-dimensional Airy structure is an analytic function at each finite order of the -expansion. Its terms are interpreted as objects living on the zero locus of the classical hamiltonians. The geometry of this variety is studied. The structure of singularities of the free energy is described. To this end topological recursion equations are expressed in a form particularly suitable for semiclassical analysis. It involves a differential operator which is a deformation of the exterior derivative. Its local properties are derived. The developed formalism is applied in several examples. Global properties of the obtained partition functions are investigated. In the case of a divergent -series, a simple resummation is performed.

The Benefits of Affine Quantization

Canonical quantization has served wonderfully for the quantization of a vast number of classical systems. That includes single classical variables, such as p and q, and numerous classical Hamiltonians H(p,q), as well as field theories, such as π(x) and φ(x), and many classical Hamiltonians H(π,φ. However, in all such systems, there are situations for which canonical quantization fails. This includes certain particle and field theory problems. Affine quantization involves a simple recombination of classical variables that lead to a new chapter in the process of quantization, and which is able to solve a vast variety of normally insoluble systems, such as quartic interactions in scalar field theory in spacetime dimensions 4 and higher, as well as the quantization of Einstein’s gravity in 4 spacetime dimensions.

Size and chemical order dependence of magnetic-ordering temperature and spin structure in Fe@Ni and Ni@Fe core-shell nanoparticles.

The effect of particle size and chemical order on the temperature-dependent magnetic properties of Fe@Ni and Ni@Fe core-shell nanoparticles is studied in the framework of a classical spin Hamiltonian and Monte Carlo simulations. We found that the mean temperature-dependent magnetization and magnetic-ordering temperature are strongly affected by both the particle size (in size range of 4 to 16 nm) and core-shell chemical order. As a main result, we report the depression of the magnetic ordering-temperature with decreasing size of the elemental Fe and Ni nanoparticles. More specifically, in the case of Fe and Ni nanoparticles, the magnetic-ordering temperature is lowered by 40 (195 K) to 300 (175 K) compared to the bulk value for nanoparticle diameters ranging from 16 to 4 nm, respectively, consistent with previous theoretical data. We further provide a comprehensive insight into the magnetic properties of Fe@Ni and Ni@Fe nanoparticles, unveiling a rich and distinct magnetic-ordering temperature and spin structure that emphatically depends on the core/shell ratio.

Erratum: “Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities” [J. Math. Phys. 58, 073502 (2017)

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Effect of lattice vibrations on magnetic properties of metals at finite temperatures

We investigate the effect of electron-phonon interaction in the dynamic spin-fluctuation theory of metallic magnetism. Electron-phonon interaction is represented by fluctuations of the crystal lattice potential. We illustrate the theoretical results in the example of Fe and Ni. We show that the effect of phonons on temperature dependence of magnetic characteristics is appreciable in Fe, but not as large as in the static single-site spin-fluctuation theory and spin dynamics using classical Hamiltonians. On the contrary, in Ni the effect of phonons is small in agreement with the results of the spin dynamics.

Effect of Phonons on the Magnetic Characteristics of Metals at Finite Temperatures

We consider the electron-phonon interaction in the framework of dynamic spin-fluctuation theory and obtain an expression for the self-energy part that depends explicitly on the spin fluctuations and lattice vibrations. We illustrate the theoretical results with the example of iron. We show that the effect of phonons on temperature dependence of the magnetic characteristics of metals is appreciable but not as large as in the static single-site approximation of the spin-fluctuation theory and spin dynamics using classical Hamiltonians.

Dynamics of dissipative topological defects in coupled phase oscillators

The dynamics of topological defects in a system of coupled phase oscillators, arranged in one and two-dimensional arrays, was numerically investigated using the Kuramoto model. After a rapid decay of the number of topological defects, a long-time quasi steady state with few topological defects was detected. Two competing time scales governed the dynamics corresponding to the dissipation rate and the coupling quench rate. The density of topological defects scales as a power law function of the coupling quench rate ρ ∼ Cν with ν = 0.25. Reducing the number of topological defects improves the long time coherence and order parameter of the system, enhancing the probability to reach a global minimal loss state that can be mapped to the ground state of a classical XY spin Hamiltonian.