Class Of Hamiltonians Research Articles

Rosen–Zener transition in a nonlinear nonreciprocal two-level system

The dynamics of the Rosen–Zener transition (RZT) in interacting two-level systems within nonreciprocal systems is examined, where nonreciprocity stems from asymmetric couplings between energy levels. For the linear case, an analytical expression for the RZT probability is derived. In the nonlinear scenario, numerical experiments are utilized to investigate the impact of various initial states and scanning periods on transition probabilities under adiabatic conditions. Additionally, an equivalent classical Hamiltonian is formulated to elucidate the origin of transition asymmetry under different initial state conditions. The influence of energy level detuning on the RZT is explored, leading to the suggestion of new applications for studying parameters of two-level systems related to nonlinearity and nonreciprocity. This opens avenues for potential Rosen–Zener–Stückelberg interferometry implementations.

Caballero–Engel meet Lasry–Lions: A uniqueness result

In a Mean Field Game (MFG) each decision maker cares about the cross sectional distribution of the state and the dynamics of the distribution is generated by the agents’ optimal decisions. We prove the uniqueness of the equilibrium in a class of MFG where the decision maker controls the state at optimally chosen times. This setup accommodates several problems featuring non-convex adjustment costs, and complements the well known drift-control case studied by Lasry–Lions. Examples of such problems are described by Caballero and Engel in several papers, which introduce the concept of the generalized hazard function of adjustment. We extend the analysis to a general “impulse control problem” by introducing the concept of the “Impulse Hamiltonian”. Under the monotonicity assumption (a form of strategic substitutability), we establish the uniqueness of equilibrium. In this context, the Impulse Hamiltonian and its derivative play a similar role to the classical Hamiltonian that arises in the drift-control case.

Energy Barriers for the Spontaneous Magnetization Reversal of the Atomic Co Chains on Pt(664) Surface in the Model with Dzyaloshinskii–Moriya Interaction

The analytical approach has been developed in the framework of the continuous XY-model. This approach allows calculating the spontaneous magnetization reversal time of finite-length atomic chains on the metallic surface. The interaction of the magnetic moments of atoms is described by the classical Hamiltonian, which includes the Heisenberg exchange interaction, the Dzyaloshinskii–Moriya interaction, and the magnetic anisotropy energy. Using the Co/Pt(664) system as an example, it has been shown that the proposed method is in a good agreement with the results of the numerical simulation in the limit of short and long atomic chains. And for atomic chains of intermediate length, it can be used to estimate an upper bound on the spontaneous magnetization reversal time. We obtained the dependences of the spontaneous magnetization reversal time of finite-length Co chains the value of the exchange integral, parameters of the magnetic anisotropy, and also on the value of the projection of the Dzyaloshinskii vector onto the axis perpendicular to the plane containing the magnetic moments of the atoms. It is shown that the proposed method has a wide range of applicability both in terms of temperature and the values of the physical parameters characterizing the magnetic properties of the atomic chains.

Classically Computing Performance Bounds on Depolarized Quantum Circuits

Quantum computers and simulators can potentially outperform classical computers in finding ground states of classical and quantum Hamiltonians. However, if this advantage can persist in the presence of noise without error correction remains unclear. In this paper, by exploiting the principle of Lagrangian duality, we develop a numerical method to classically compute a certifiable lower bound on the minimum energy attainable by the output state of a quantum circuit in the presence of depolarizing noise. We provide theoretical and numerical evidence that this approach can provide circuit-architecture-dependent bounds on the performance of noisy quantum circuits. Published by the American Physical Society 2024

A Quantum Formalism for Abstract Dynamical Systems

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract” because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case.

On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems

Port-Hamiltonian systems were recently extended to include implicitly defined energy and energy ports thanks to a (Stokes-)Lagrange subspace. Here, we study the equivalent port-Hamiltonian representations of two systems with damping, written using either a classical Hamiltonian or a Stokes-Lagrange subspace. Then, we study the Timoshenko beam and Euler-Bernoulli models, the latter being the flow-constrained version of the former, and show how they can be written using either a Stokes-Dirac or Stokes-Lagrange subspace related by a transformation operator. Finally, it is proven that these transformations commute with the flow-constraint projection operator.

A quantum fluctuation description of charge qubits

We consider a specific instance of a superconducting circuit, the so-called charge-qubit, consisting of a capacitor and a Josephson junction that we describe by means of the BCS microscopic model in terms of two tunnelling superconducting systems in the strong-coupling quasi-spin formulation. Then, by means of collective observables we derive the Hamiltonian governing the quantum behaviour of the circuit in the limit of a large number N of quasi-spins. Our approach relies on suitable quantum fluctuations, i.e. on collective quasi-spin operators, different from mean-field observables, that retain a quantum character in the large-N limit. These collective operators generate the Heisenberg algebra on the circle and we show that their dynamics reproduces the phenomenological one generated by the charge qubit Hamiltonian obtained by quantizing the macroscopic classical Hamiltonian of the circuit. The microscopic derivation of the emergent, large-N behaviour provides a rigorous setting to investigate more in detail both general quantum circuits and quantum macroscopic scenarios; in particular, in the specific case of charge-qubits, it allows to explicitly obtain the temperature dependence of the critical Josephson current in the strong coupling regime, a result not accessible using standard approximation techniques.

Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Transverse Magnetization in Spin-Orbit Coupled Antiferromagnets.

Some antiferromagnets under a magnetic field develop magnetization perpendicular to the field as well as more conventional ones parallel to the field. So far, the transverse magnetization (TM) has been attributed to either the spin canting effect or the presence of cluster magnetic multipolar ordering. However, a general theory of TM based on microscopic understanding is still missing. Here, we construct a general microscopic theory of TM in antiferromagnets with cluster magnetic multipolar ordering by considering classical spin Hamiltonians with spin anisotropy that arises from the spin-orbit coupling. First, from general symmetry analysis, we show that TM can appear only when all crystalline symmetries are broken other than the antiunitary mirror, antiunitary twofold rotation, and inversion symmetries. Moreover, by analyzing spin Hamiltonians, we show that TM always appears when the degenerate ground state manifold of the spin Hamiltonian is discrete, as long as it is not prohibited by symmetry. On the other hand, when the degenerate ground state manifold is continuous, TM generally does not appear except when the magnetic field direction and the spin configuration satisfy specific geometric conditions under single-ion anisotropy. Finally, we show that TM can induce the anomalous planar Hall effect, a unique transport phenomenon that can be used to probe multipolar antiferromagnetic structures. We believe that our theory provides a useful guideline for understanding the anomalous magnetic responses of the antiferromagnets with complex magnetic structures.

Classical spin Hamiltonians are context-sensitive languages

Classical spin Hamiltonians are a powerful tool to model complex systems, characterized by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here, we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognizing spin configurations and their energies. We compare it with the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.

Generalized Langevin equationwith shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian.

We provide a first-principles derivation of the Langevin equation with shear flow and its corresponding fluctuation-dissipation theorems. Shear flow of simple fluids has been widely investigated by numerical simulations. Most studies postulate a Markovian Langevin equationwith a simple shear drag term in the manner of Stokes. However, this choice has never been justified from first principles. We start from a particle-bath system described by a classical Caldeira-Leggett Hamiltonian modified by adding a term proportional to the strain-rate tensor according to Hoover's DOLLS method, and we derive a generalized Langevin equationfor the sheared system. We then compute, analytically, the noise time-correlation functions in different regimes. Based on the intensity of the shear rate, we can distinguish between close-to-equilibrium and far-from-equilibrium states. According to the results presented here, the standard, simple, and Markovian form of the Langevin equationwith shear flow postulated in the literature is valid only in the limit of extremely weak shear rates compared to the effective vibrational temperature of the bath. For even marginally higher shear rates, the (generalized) Langevin equationis strongly non-Markovian, and nontrivial fluctuation-dissipation theorems are derived.

Nonlinear Landau-Zener tunneling under higher-order dispersion

We studied nonlinear Landau-Zener tunneling under the effect of higher-order dispersion, transformed the Gross-Pitaevskii (GP) equation with higher-order dispersion terms into a nonlinear two-energy level form using a two-mode approximation, and comprehensively analyzed the loop structure of the lowest-energy band and the nonlinear Landau-Zener tunneling. The results show that, when third-order dispersion coefficient is not zero, there is a spatial inversion symmetry breaking of the energy band structure and unbalanced Landau-Zener tunneling. By analyzing the fixed point's nature of the classical Hamiltonian, we obtain a law for the variation of the loop structure and adiabatic tunneling probability with the dispersion coefficient, consistent with our numerical analysis results. In addition, we analyzed the real-time evolution of solitons with transverse bias and observed the tunneling phenomenon. Consistent with our analysis, the adjustment of the dispersion term can effectively control optical tunneling, which provides an alternative idea for optical switching.

Inverse Holstein-Primakoff transformation of bosonic operators —Application to a bilayer model

An inversion of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of an SU(2) algebra. In association with more common quadratic combinations for fermionic operators, that inverse transformation sets a quantum Hamiltonian fully in terms of SU(2) operators. A subsequent application of the prescription by Lieb, to obtain the classical limit for spin operators, then allows one to write effciently a classical Hamiltonian for the system. This process is illustrated for a bilayer model undergoing an (electron-hole)-to-exciton quantum phase transition.

Hamiltonian-Driven Adaptive Dynamic Programming With Approximation Errors

In this article, we consider an iterative adaptive dynamic programming (ADP) algorithm within the Hamiltonian-driven framework to solve the Hamilton-Jacobi-Bellman (HJB) equation for the infinite-horizon optimal control problem in continuous time for nonlinear systems. First, a novel function, "min-Hamiltonian," is defined to capture the fundamental properties of the classical Hamiltonian. It is shown that both the HJB equation and the policy iteration (PI) algorithm can be formulated in terms of the min-Hamiltonian within the Hamiltonian-driven framework. Moreover, we develop an iterative ADP algorithm that takes into consideration the approximation errors during the policy evaluation step. We then derive a sufficient condition on the iterative value gradient to guarantee closed-loop stability of the equilibrium point as well as convergence to the optimal value. A model-free extension based on an off-policy reinforcement learning (RL) technique is also provided. Finally, numerical results illustrate the efficacy of the proposed framework.

Continuum mechanics of nonideal crystals: Microscopic approach based on projection-operator formalism.

We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. The starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the defect concentration. Isothermal and adiabatic elastic constants are obtained from equilibrium correlations, while transport coefficients are given as Green-Kubo formulas, providing the basis for their measurement in atomistic simulations or colloidal experiments. The approach to the linearized continuum mechanics and results are compared to others from the literature.

Two types of metastable hopfions in bulk magnets

Localized magnetic topological solitons with Hopf index of 1 in an unbounded bulk magnet are studied theoretically, starting with the classical micromagnetic Hamiltonian. It is shown analytically that (like Bloch and Néel walls in classical magnetism) there are two possible stable configurations of these nanoobjects, which are equivalent topologically. Their stability regions are plotted on a phase diagram in the magnetic field — uniaxial anisotropy space for different normalized magnetostatic interaction strengths. For some parameters both types of hopfions can be stable.

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Stefankovic, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Stefankovic, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Stefankovic, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

Quantum to classical mapping of the two-dimensional toric code in an external field

Kitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1-dimensional classical representation. In these notes, we provide details of a Suzuki-Trotter expansion of the partition function of the toric code Hamiltonian in the presence of an external magnetic field. By coupling additional degrees of freedom in the form of a matter field that can subsequently be gauged away, we explicitly derive a classical Hamiltonian on a cubic lattice which takes the form of a non-isotropic 3D Ising gauge theory.

Quantum-inspired Search Method for Low-energy States of Classical Ising Hamiltonians

We develop a quantum-inspired numerical procedure for searching low-energy states of a classical Hamiltonian composed of two-body fully-connected random Ising interactions and a random local longitudinal magnetic field. In this method, we introduce infinitesimal quantum interactions that do not commute with the original Ising Hamiltonian, and repeatedly generate and truncate direct product states, inspired by the Krylov subspace method, to obtain the low-energy states of the original classical Ising Hamiltonian. The computational cost is controlled by the form of infinitesimal quantum interactions (e.g., one-body or two-body interactions) and the numbers of infinitesimal interaction terms introduced, different initial states considered, and low-energy states kept during the iteration. For a demonstrate of the method, here we introduce as the infinitesimal quantum interactions pair products of Pauli $X$ operators acting on different sites and on-site Pauli $X$ operators into the random Ising Hamiltonian, in which the numerical cost is $O(N^3)$ per iteration with the system size $N$. We consider 120 instances of the random coupling realizations for the random Ising Hamiltonian with $N$ up to 600 and search the 120 lowest-energy states for each instance. We find that the time-to-solution by the quantum-inspired method proposed here, with parallelization in terms of the different initial states, for searching the ground state of the random Ising Hamiltonian scales approximately as $N^5$ for $N$ up to 600. We also examine the basic physical properties such as the ensemble-averaged ground-state and first-excited energies and the ensemble-averaged number of states in the low-energy region of the random Ising Hamiltonian.

Cyclability in graph classes

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.