Hamiltonian Space Research Articles

Fock Space Coupled-Cluster Method for the Ground and Excited States of the NaMg+ Molecular Cation.

This study presents a theoretical investigation of the potential energy curves (PECs) and spectroscopic constants of the NaMg+ molecular cation using the intermediate Hamiltonian Fock space coupled-cluster method with singles and doubles within the (2,0) sector [IH-FS-CCSD(2,0)]. The NaMg+ cation has gained scientific interest due to its potential applications in ultracold chemistry, yet it remains experimentally unexplored. We computed 20 lowest-lying PECs and assigned them to 10 dissociation limits. Our results demonstrated high accuracy, and computed curves are smooth and continuous over the entire range of interatomic distances. The study validates the effectiveness of the IH-FS-CCSD(2,0) method in describing PECs of diatomic molecular cations composed of alkali and alkaline earth metals and provides a solid theoretical foundation for further studies of NaMg+ and similar systems.

The R-mAtrIx Net

We provide a novel neural network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other restrictions, iii) explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to. The neural network training is done by minimizing loss functions encoding Yang–Baxter equation, regularity and other model-specific restrictions such as hermiticity. Holomorphy is implemented via the choice of activation functions. We demonstrate the work of our neural network on the spin chains of difference form with two-dimensional local space. In particular, we reconstruct the R-matrices for all 14 classes. We also demonstrate its utility as an Explorer, scanning a certain subspace of Hamiltonians and identifying integrable classes after clusterisation. The last strategy can be used in future to carve out the map of integrable spin chains with higher dimensional local space and in more general settings where no analytical methods are available.

An Intruder-Free Fock Space Coupled-Cluster Study of the Potential Energy Curves of LiMg+ within the (2,0) Sector.

The potential energy curves (PECs) and spectroscopic constants of the ground and excited states of a LiMg+ molecular cation were investigated. We obtained accurate results for the fifteen lowest-lying states of the LiMg+ cation using the Intermediate Hamiltonian Fock Space Multireference Coupled Cluster (IH-FS-CC) method applied to the (2,0) sector. Relativistic corrections were accounted for using the third-order Douglas-Kroll method. In each instance, smooth PECs were successfully computed across the entire range of interatomic distances from equilibrium to the dissociation limit. The results are in good accordance with previous studies of this molecular cation. Notably, this study marks the first application of IH-FS-CC in investigating a mixed alkali and alkaline earth molecular cation, proving its usability in determining accurate PECs of such diatomics and their spectroscopic constants.

Second order topology in a band engineered Chern insulator

Haldane model is a celebrated tight binding toy model of a Chern insulator in a 2D honeycomb lattice that exhibits quantized Hall conductance in the absence of an external magnetic field. In our work, we deform the bands of the Haldane model smoothly by varying one of its three nearest neighbour hopping amplitudes (t1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_1$$\\end{document}), while keeping the other two (t) fixed. This breaks the C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} symmetry of the Hamiltonian, while the Mx∗T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_x*T$$\\end{document} symmetry is preserved. The symmetry breaking causes the Dirac cones to shift from the K and the K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$'$$\\end{document} points in the Brillouin zone (BZ) to an intermediate M point. This is evident from the Berry curvature plots which show a similar shift in the corresponding values as a function of the deformation parameter, namely t1t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{t_1}{t}$$\\end{document}. We observe two different topological phases of which, one is a topological insulator (TI) and the other is a second order topological insulator (SOTI). The Chern number (C) remains perfectly quantized at a value of C=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=1$$\\end{document} for the TI phase and it goes to zero in the SOTI phase. Furthermore, the evolution of the Wannier charge center (WCC) as the band is smoothly deformed shows a jump in the TI phase indicating the presence of conducting edge modes. We also study the SOTI phase and diagonalize the real space Hamiltonian on a rhombic supercell which shows the presence of in-gap zero energy corner modes. The polarization of the system, namely px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document}, are evaluated, along the x and the y directions, respectively. We see that both px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document} are quantized in the SOTI phase owing to the presence of the inversion symmetry of the system. Finally we establish the SOTI phase as an example of a topological phase with zero Berry curvature and provide an analogy with the two dimensional Su–Schrieffer–Heeger model.

Theoretical molecular spectroscopy of actinide compounds: the ThO molecule

The tiny-core generalised (Gatchina) relativistic pseudopotential (GRPP) model provides an accurate approximation for many-electron Hamiltonians of molecules containing heavy atoms, ensuring a proper description of the effects of non-Coulombian electron-electron interactions, electronic self-energy and vacuum polarisation. Combining this model with electron correlation treatment in the frames of the intermediate Hamiltonian Fock space coupled cluster theory employing incomplete main model spaces, one obtains a reliable and economical tool for excited state modelling. The performance of this method is assessed in applications to ab initio modelling of excited electronic states of the thorium monoxide molecule with term energies below 20,000 cm. Radiative lifetimes of excited states are estimated using truncated expansions of effective and metric operators in powers of cluster amplitudes.

Multiple silicon dangling-bond charge qubits for quantum computing: a Hilbert-space analysis of the Hamiltonian

Silicon-based dangling-bond charge qubit is one of the auspicious models for universal fault-tolerant solid-state quantum computing. In universal quantum computing, it is crucial to evaluate and characterize the computational Hilbert space and reduce the complexity and size of the computational space. Here, we recognize this problem to understand the complexity and characteristics of the Hilbert space in our dangling-bond qubit model. The size of the desired Hilbert space can prominently be reduced by considering assumptions regarding the qubit loss. Moreover, the dimension of the desired subsets in the space shrinks by a factor of two due to the spin preservation property. Finally, the required classical memory for storage of the qubit information, Hamiltonian and Hilbert space is analysed when the number of qubits grows.

Effects of site dilution on Compensation in Ising Spin-1/2 trilayered triangular Ferrimagnets with non-equivalent planes

Using Monte Carlo simulations with the Metropolis algorithm, the magnetic and thermodynamic behaviours of a spin-1/2, trilayered ferrimagnetic system on triangular monolayers with quenched nonmagnetic impurities are studied. Two different theoretical atoms, A and B, make up the ABA and AAB types of distinct configurations. Like atoms (A-A and B-B) interact ferromagnetically, while unlike atoms (A-B) interact antiferromagnetically. Only the A-layers are randomly site-diluted with dilution percentages ranging from 5% to 45%. Such diluted magnetic thin systems exhibit magnetic compensation which depends sensitively on the concentration of impurities. The phase diagram in the Hamiltonian parameter space related to the occurrence of magnetic compensation phenomenon and the effect of site dilution is discussed in detail. Special attention is given to the mathematical dependencies of compensation temperature on the concentration of nonmagnetic impurities. Depending upon the concentration of nonmagnetic impurities, the compensation and critical points shift with the equilibrium magnetic behaviours changing between distinct ferrimagnetic behaviours. For each combination of the coupling strengths, with values of the impurity concentration above a threshold, compensation appears where previously was absent. Suggested mathematical formulae show how threshold impurity concentration relies on Hamiltonian parameters.

Detecting topological phases in the square–octagon lattice with statistical methods

Electronic systems living on Archimedean lattices such as kagome and square–octagon networks are presently being intensively discussed for the possible realization of topological insulating phases. Coining the most interesting electronic topological states in an unbiased way is however not straightforward due to the large parameter space of possible Hamiltonians. A possible approach to tackle this problem is provided by a recently developed statistical learning method (Mertz and Valentí in Phys Rev Res 3:013132, 2021. https://doi.org/10.1103/PhysRevResearch.3.013132), based on the analysis of a large data sets of randomized tight-binding Hamiltonians labeled with a topological index. In this work, we complement this technique by introducing a feature engineering approach which helps identifying polynomial combinations of Hamiltonian parameters that are associated with non-trivial topological states. As a showcase, we employ this method to investigate the possible topological phases that can manifest on the square–octagon lattice, focusing on the case in which the Fermi level of the system lies at a high-order van Hove singularity, in analogy to recent studies of topological phases on the kagome lattice at the van Hove filling.

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

Hamiltonian system-based analytical solutions for the free vibration of edge-cracked thick rectangular plates

Analytical modeling for cracked plates’ free vibration is of great significance as a crack alters the properties of plate structures and thus greatly affects the mechanical behaviors of such key engineering components. This paper presents novel analytical solutions for the free vibration of thick rectangular plates with a crack normal to an edge, a kind of typical edge-cracked plates. To resolve the jump discontinuity caused by the edge crack, an original plate is decomposed into four elementary plates, whose free vibration problems are solved analytically via an unusual symplectic superposition method (SSM). The SSM is implemented in the Hamiltonian system-based symplectic space, where several mathematical treatments such as the variable separation and the symplectic eigenvector expansion are valid, rendering rigorous step-by-step derivations without pre-defining solution forms. By incorporating the solutions of the elementary plates, the eventual solutions of the edge-cracked thick plates are obtained. Such a symplectic framework provides a new route to solving various complicated plate problems that cannot be analytically handled previously. Comprehensive free vibration results such as natural frequencies and vibration modes of edge-cracked thick plates are presented. High accuracy and fast convergence of the framework are confirmed. The present analytical solutions are also utilized to study the effects of both the crack length and crack location on free vibration behaviors of edge-cracked thick plates.

New symplectic analytic solutions for buckling of CNT reinforced composite rectangular plates

This paper presents new analytic solutions for buckling of non-Lévy-type carbon nanotube (CNT) reinforced composite rectangular plates, including cantilever, free, and clamped ones. The buckling problems of such types of plates cannot be handled by some conventional analytic methods through expressing mechanical quantities in pre-defined solution forms, which, however, is excluded by the present analytic symplectic superposition method. The present method not only provides a more rigorous solution procedure but also enables more types of plates to be analytically solved. The governing partial differential equation is first expressed in the Hamiltonian system-based symplectic space, and then the variable separation as well as the symplectic eigen expansion are utilized to analytically solve two elementary buckling problems. By a skillful superposition of the elementary solutions, the eventual buckling solutions are obtained. Comprehensive buckling load/mode results are tabulated as new benchmarks. With the new analytic solutions, the parametric studies on CNT distributions, CNT volume fractions, aspect ratios, and boundary conditions are conducted to reveal the effects on the plate buckling performance.

Quantum phase transitions in a model Hamiltonian exhibiting entangled simultaneous fermion-pair and exciton condensations

Quantum states of a novel Bose-Einstein condensate, in which both fermion-pair and exciton condensations are simultaneously present, have recently been realized theoretically in a model Hamiltonian system. Here we identify quantum phase transitions in that model between fermion-pair and exciton condensations based on a geometric analysis of the convex set of ground-state two-particle reduced density matrices (2-RDMs). The 2-RDM set provides a finite representation of the infinite parameter space of Hamiltonians that readily reveals a fermion-pair condensate phase and two distinct exciton condensate phases, as well as the emergence of first- and second-order phase transitions as the particle number of the system is increased. The set, furthermore, shows that the fermion-exciton condensate (FEC) lies along the second-order phase transition between the exciton and fermion-pair condensate phases. The detailed information about the exciton and fermion-pair phases, the forces behind these phases, as well as their associated transitions provides additional insight into the formation of the FEC condensate, which we anticipate will prove useful in its experimental realization.

Riemann-Encircling Exceptional Points for Efficient Asymmetric Polarization-Locked Devices.

Dynamically encircling exceptional points (EPs) have unveiled intriguing chiral dynamics in photonics. However, the traditional approach based on an open manifold of Hamiltonian parameter space fails to explore trajectories that pass through an infinite boundary. Here, by mapping the full parameter space onto a closed manifold of the Riemann sphere, we introduce a framework to describe encircling-EP loops. We demonstrate that an encircling trajectory crossing the north vertex can realize near-unity asymmetric transmission. An efficient gain-free, broadband asymmetric polarization-locked device is realized by mapping the encircling path onto L-shaped silicon waveguides.

Unraveling the topology of dissipative quantum systems

We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. The latter emerge in the unraveling of Markovian quantum master equations and/or in continuous quantum measurements. Ensemble-averaging quantum trajectories at the occurrence of quantum jumps, i.e., the jump times, gives rise to a discrete, deterministic evolution which is highly sensitive to the presence of dark states [Gneiting et al., Phys. Rev. A 104, 062212 (2021)]. We show for a broad family of translation-invariant collapse models that the set of dark-state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians, which is also reflected by the corresponding jump-time dynamics. The topological character of the latter can then be observed, for instance, in the transport behavior, exposing an infinite hierarchy of topological phase transitions. We develop our theory for one- and two-dimensional two-band Hamiltonians and show that the topological behavior is directly manifest for chiral, $\mathcal{PT}$, or time-reversal-symmetric Hamiltonians.

Improved accuracy in degenerate variational integrators for guiding centre and magnetic field line flow

First-order-accurate degenerate variational integration (DVI) was introduced in Ellison et al. (Phys. Plasmas, vol. 25, 2018, 052502) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introduce second-order-accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second-order scheme with a preserved two-form by composing a first-order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second-order-accurate DVI schemes. We test these second-order schemes numerically on two systems having non-canonical variables, namely the magnetic field line and guiding centre systems. Variational integration for Hamiltonian systems with non-uniform time steps, in terms of an extended phase space Hamiltonian, is generalized to non-canonical variables. It is shown that preservation of proper degeneracy leads to single-step (one-step) methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second-order-accurate as well as first-order schemes, and can be applied to adapt the time stepping to an error estimate.

Geometric K-homology and the Freed–Hopkins–Teleman theorem

We construct a map at the level of cycles from the equivariant twisted K-homology of a compact, connected, simply connected Lie group G to the Verlinde ring, which is inverse to the Freed–Hopkins–Teleman isomorphism. As an application, we prove that two of the proposed definitions of the quantization of a Hamiltonian loop group space—one via twisted K-homology of G and the other via index theory on non-compact manifolds—agree with each other

Interpolation results for pathwise Hamilton-Jacobi equations

We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov, Holder, and variation norms, and criteria for the Hamiltonians are presented in terms of both regularity and structure. We also explore various properties of functions that are representable as the difference of convex functions, the largest space of Hamiltonians for which the equation is well-posed for all continuous paths. Finally, we discuss some open problems and conjectures.

Approximate two-body generating Hamiltonian for the particle-hole Pfaffian wave function

We present two 2-body Hamiltonians that approximate the exact PH-Pfaffian wavefunction with their ground states for all the system sizes where this wavefunction has been numerically constructed to date. The approximate wavefunctions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimisation procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a non-interacting model. They belong to a finite region in the variational space of Hamiltonians where each point approximately generates the PH-Pfaffian. We diagonalize the identified Hamiltonians for up to 20 electrons and find that for them the PH-Pfaffian shift appears energetically more favorable. Possibility to interpret the data in terms of composite fermions is discussed.

From 2d droplets to 2d Yang-Mills

We establish a connection between time evolution of free Fermi droplets and partition function of generalised q-deformed Yang-Mills theories on Riemann surfaces. Classical phases of (0+1) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces H+ and H− associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large N limit, equal to the quadratic Casimir of u(N). We establish an exact mapping between states in H± and geometries of droplets. In particular, coherent states in H± correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in H± is equal to the chiral and anti-chiral partition function of 2d Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space H+⊗H− admits a composite basis, we show that correlation between two classical droplet geometries is equal to the full U(N) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in H± and higher point correlators in 2d Yang-Mills on Riemann surface. There are special states in H± whose transition amplitudes are equal to the partition function of 2d q-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the q-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.

Effect of a uniform random external magnetic field with spatiotemporal variation on compensation in Ising spin-1/2 trilayered square ferrimagnets.

Trilayered spin-1/2 Ising ferrimagnets are interesting thin systems for compensation phenomenon. In this work, a Metropolis Monte Carlo study is performed on the magnetic and thermodynamic response of such a system on square Bravais lattice, driven by uniform random external magnetic field with spatiotemporal variations. In two distinct configurations, the surface layers are made up of A and the middle layer is made up of B atoms in an ABA-type stacking while in AAB-type stacking, the top layer and the middle layer is made up of A atoms while the bottom layer is made up of B atoms. The magnetic coupling between the like atoms (A-A and B-B) is ferromagnetic while between the unlike atoms (A-B), it is antiferromagnetic. For the time-dependent external uniform random field, the mean is always set to zero and the standard deviation is varied until spin-field energy is comparable to the dominant cooperative energy of the system. The findings show that the observed compensation and critical points shift and steady-state magnetic behaviors shift among N, L, P, and Q, etc., type of ferrimagnetic behaviors, depending on the strength of external uniform random field. The compensation phenomenon even vanishes after crossing a finite threshold of standard deviation of the magnetic field for particular choices of the other controlling parameters. Thus islands of ferrimagnetic phase without compensation appear within the phase area with compensation of field-free case, in the two-dimensional Hamiltonian parameter space. For both the configurations, the areas of such islands even grow with increasing standard deviation of the external field, σ, obeying a scaling relation of the form: f(σ,A(σ))=σ^{-b}A(σ) with b_{ABA}=1.958±0.122 and b_{AAB}=1.783±0.118.