Two-dimensional Hamiltonian Research Articles

Quantitative Hardy inequalities for magnetic Hamiltonians

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.

Exploring Exact-Factorization-Based Trajectories for Low-Energy Dynamics near a Conical Intersection.

We study low-energy dynamics generated by a two-dimensional two-state Jahn-Teller Hamiltonian in the vicinity of a conical intersection using quantum wave packet and trajectory dynamics. Recently, these dynamics were studied by comparing the adiabatic representation and the exact factorization, with the purpose to highlight the different nature of topological-phase and geometric-phase effects arising in the two theoretical representations of the same problem. Here, we employ the exact factorization to understand how to accurately model low-energy dynamics in the vicinity of a conical intersection using an approximate description of the nuclear motion that uses trajectories. We find that since nonadiabatic effects are weak but non-negligible, the trajectory-based description that invokes the classical approximation struggles to capture the correct behavior.

Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature.

Programmable integrated photonics for topological Hamiltonians

A variety of topological Hamiltonians have been demonstrated in photonic platforms, leading to fundamental discoveries and enhanced robustness in applications such as lasing, sensing, and quantum technologies. To date, each topological photonic platform implements a specific type of Hamiltonian with inexistent or limited reconfigurability. Here, we propose and demonstrate different topological models by using the same reprogrammable integrated photonics platform, consisting of a hexagonal mesh of silicon Mach-Zehnder interferometers with phase shifters. We specifically demonstrate a one-dimensional Su-Schrieffer-Heeger Hamiltonian supporting a localized topological edge mode and a higher-order topological insulator based on a two-dimensional breathing Kagome Hamiltonian with three corner states. These results highlight a nearly universal platform for topological models that may fast-track research progress toward applications of topological photonics and other coupled systems.

A general expression for vibrational Hamiltonians expressed in oblique coordinates

We examine the properties of oblique coordinates. The coordinates, introduced by Zúñiga et al. [J. Phys. B: At., Mol. Opt. Phys. 52, 055101, (2019)], reduce vibrational mode-mixing and enhance the quality of vibrational assignments in quantum mechanical investigations of two-dimensional model Hamiltonians. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block-diagonal matrix where the blocks are distinguished by the total quanta of vibrational excitation. Using techniques for the polar decomposition of matrices, we present a scheme for finding these coordinates for systems of arbitrary dimensions. Several molecular examples are presented that highlight the advantages of these coordinates.

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

The construction of kinetic equilibrium states is important for studying stability and wave propagation in collisionless plasmas. Thus, many studies over the past decades have been focused on calculating Vlasov–Maxwell equilibria using analytical and numerical methods. However, the problem of kinetic equilibrium of hybrid models is less studied, and self-consistent treatments often adopt restrictive assumptions ruling out cases with irregular and chaotic behaviour, although such behaviour is observed in spacecraft observations of space plasmas. In this paper, we develop a one-dimensional (1-D), quasineutral, hybrid Vlasov–Maxwell equilibrium model with kinetic ions and massless fluid electrons and derive associated solutions. The model allows for an electrostatic potential that is expressed in terms of the vector potential components through the quasineutrality condition. The equilibrium states are calculated upon solving an inhomogeneous Beltrami equation that determines the magnetic field, where the inhomogeneous term is the current density of the kinetic ions and the homogeneous term represents the electron current density. We show that the corresponding 1-D system is Hamiltonian, with position playing the role of time, and its trajectories have a regular, periodic behaviour for ion distribution functions that are symmetric in the two conserved particle canonical momenta. For asymmetric distribution functions, the system is nonintegrable, resulting in irregular and chaotic behaviour of the fields. The electron current density can modify the magnetic field phase space structure, inducing orbit trapping and the organization of orbits into large islands of stability. Thus, the electron contribution can be responsible for the emergence of localized electric field structures that induce ion trapping. We also provide a paradigm for the analytical construction of hybrid equilibria using a rotating two-dimensional harmonic oscillator Hamiltonian, enabling the calculation of analytic magnetic fields and the construction of the corresponding distribution functions in terms of Hermite polynomials.

Electronic Spectrum Features under the Transition from Axion Insulator Phase to Quantum Anomalous Hall Effect Phase in an Intrinsic Antiferromagnetic Topological Insulator Thin Film

In this paper, we investigate the electron topological states in a thin film of intrinsic antiferromagnetic topological insulator, focusing on their relationship with the magnetic texture. We consider a model for the film with an even number of septuple-layer blocks, which is subject to transition from the phase of an axion insulator to the phase of quantized Hall conductivity under an external magnetic field. In the continuum approach, we model an effective two-dimensional Hamiltonian of the thin film of a topological insulator with non-collinear magnetization, on the basis of which we obtain the energy spectrum and the Berry curvature. The analysis of topological indices makes it possible to construct a topological phase diagram depending on the parameters of the system and the degree of non-collinearity. For topologically different regions of the diagram, we describe the edge electronic states on the side face of the film. In addition, we investigate the spectrum of one-dimensional states on the domain wall separating domains with the opposite canting angle. We also discuss the results obtained and the experimental situation in thin films of the MnBi2Te4 compound.

Algebraic structure of Dirac Hamiltonians in non-commutative phase space

In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations of the graded Lie algebra . As application of our results, we consider the Landau model and a fermion in a finite cylindrical well.

Experimental realization of an extended Fermi-Hubbard model using a 2D lattice of dopant-based quantum dots

The Hubbard model is an essential tool for understanding many-body physics in condensed matter systems. Artificial lattices of dopants in silicon are a promising method for the analog quantum simulation of extended Fermi-Hubbard Hamiltonians in the strong interaction regime. However, complex atom-based device fabrication requirements have meant emulating a tunable two-dimensional Fermi-Hubbard Hamiltonian in silicon has not been achieved. Here, we fabricate 3 × 3 arrays of single/few-dopant quantum dots with finite disorder and demonstrate tuning of the electron ensemble using gates and probe the many-body states using quantum transport measurements. By controlling the lattice constants, we tune the hopping amplitude and long-range interactions and observe the finite-size analogue of a transition from metallic to Mott insulating behavior. We simulate thermally activated hopping and Hubbard band formation using increased temperatures. As atomically precise fabrication continues to improve, these results enable a new class of engineered artificial lattices to simulate interactive fermionic models.

Stochastic multi-configuration time-dependent Hartree for dissipative quantum dynamics with strong intramolecular coupling.

In this article, we explore the dissipation dynamics of a strongly coupled multidimensional system in contact with a Markovian bath, following a system-bath approach. We use in this endeavor the recently developed stochastic multi-configuration time-dependent Hartree approach within the Monte Carlo wave packet formalism [S. Mandal et al., J. Chem. Phys. 156, 094109 (2022)]. The method proved to yield thermalized ensembles of wave packets when intramolecular coupling is weak. To treat strongly coupled systems, new Lindblad dissipative operators are constructed as linear combinations of the system coordinates and associated momenta. These are obtained by a unitary transformation to a normal mode representation, which reduces intermode coupling up to second order. Additionally, we use combinations of generalized raising/lowering operators to enforce the Boltzmann distribution in the dissipation operators, which yield perfect thermalization in the harmonic limit. The two ansatz are tested using a model two-dimensional Hamiltonian, parameterized to disentangle the effects of intramolecular potential coupling, of strong mode mixing observed in Fermi resonances, and of anharmonicity.

Renormalization-group-inspired neural networks for computing topological invariants

We show that artificial neural networks (ANNs) can, to high accuracy, determine the topological invariant of a disordered system given its two-dimensional real-space Hamiltonian. Furthermore, we describe a ``renormalization-group'' (RG) network, an ANN which converts a Hamiltonian on a large lattice to another on a small lattice while preserving the invariant. By iteratively applying the RG network to a ``base'' network that computes the Chern number of a small lattice of set size, we are able to process larger lattices without retraining the system. We therefore show that it is possible to compute real-space topological invariants for systems larger than those on which the network was trained. This opens the door for computation times significantly faster and more scalable than previous methods.

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.

Beyond-mean-field dynamical correlations for nuclear mass table in deformed relativistic Hartree-Bogoliubov theory in continuum * *Supported by the National Natural Science Foundation of China (11875225, 12005175, 11935003, 11875075, 11975031, 12070131001, 11790325), the Fundamental Research Funds for the Central Universities, the National Key R&D Program of China (2017YFE0116700, 2018YFA0404400), the State Key Laboratory of Nuclear Physics and Technology, Peking University (NPT2020ZZ01)

We extend the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) to go beyond-mean-field framework by performing a two-dimensional collective Hamiltonian. The influences of dynamical correlations on the ground-state properties are examined in different mass regions, picking Se, Nd, and Th isotopic chains as representatives. It is found that the dynamical correlation energies (DCEs) and the rotational correction energies in the cranking approximation have an almost equivalent effect on the description of binding energies for most deformed nuclei, and the DCEs can provide a significant improvement for the (near) spherical nuclei close to the neutron shells and thus reduce the rms deviations of by 17%. Furthermore, it is found that the DCEs are quite sensitive to the pairing correlations; taking Nd as an example, a 10% enhancement of pairing strength can raise the DCE by 37%.

Global Phase Diagrams of Three Point Vortices

In this paper, we investigate the dynamics of three-vortex systems and give a global phase diagram analysis with different types of parameters on the plane. By using the invariants to construct canonical transformations, the Hamiltonian of the three-vortex system is reduced to a desired two-dimensional Hamiltonian. Then, we analyze its phase diagrams by considering all possible situations. In each case, the stabilities of the equilibrium points and singular points are established by using numerical methods, and we obtain the distribution of trajectories in the phase space.

Edge state in AB-stacked bilayer graphene and its correspondence with the Su-Schrieffer-Heeger ladder

We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the Chern number of the corresponding two-dimensional (2D) bulk Hamiltonian is zero. The existence and topological features of edge states when two layers ended with the same or different edge terminations (zigzag, bearded, armchair) are discussed. The edge states (non-dispersive bands near the Fermi level) are states localized at the edge of graphene nanoribbon that only exists in certain range of momentum $k_y$. Their existence near the Fermi level are protected by the chiral symmetry with topology well described by coupled Su-Schrieffer-Heeger (SSH) chains model, i.e., SSH ladder, based on the bulk-edge correspondence of one-dimensional (1D) systems. These zero-energy edge states can exist in the whole $k_y$ region when two layers have zigzag and bearded edges, respectively. Winding number calculation shows a topological phase transition between two distinct non-trivial topological phases when crossing the Dirac points. Interestingly, we find the stacking configuration of BLG ribbon is important since they can lead to unexpected edge states without protection from the chiral symmetry both near the Fermi level in armchair-armchair case and in the gap within bulk bands that are away from Fermi level in the general case. The influence of interlayer next nearest neighbor (NNN) interaction and interlayer bias are also discussed to fit the realistic graphene materials, which suggest the robust topological features of edge states in BLG systems.

Reduction scheme for coupled Dirac systems

We analyze a class of coupled quantum systems whose dynamics can be understood via two uncoupled, lower-dimensional quantum settings with auxiliary interactions. The general reduction scheme, based on algebraic properties of the potential term, is discussed in detail for two-dimensional Dirac Hamiltonian. We discuss its possible application in description of Dirac fermions in graphene or bilayer graphene in presence of distortion scattering or spin–orbit interaction. We illustrate the general results on the explicit examples where the involved interactions are non-uniform in space and time.

The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

In this article, we provide an expansion (up to the fourth order of the coupling constant) of the energy of the ground state of the Hamiltonian of a quantum mechanical particle moving inside a parabolic well in the x-direction and constrained by the presence of a two-dimensional impurity, modelled by an attractive two-dimensional isotropic Gaussian potential. By investigating the associated Birman–Schwinger operator and exploiting the fact that such an integral operator is Hilbert–Schmidt, we use the modified Fredholm determinant in order to compute the energy of the ground state created by the impurity.

Kramers nodal line metals

Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs.

Spin and charge order in doped spin-orbit coupled Mott insulators

We study a two-dimensional single band Hubbard Hamiltonian with antisymmetric spin-orbit coupling. We argue that this is the minimal model to understand the electronic properties of locally non-centrosymmetric transition-metal (TM) oxides such as Sr$_2$IrO$_4$. Based on exact diagonalizations of small clusters and the random phase approximation, we investigate the correlation effects on charge and magnetic order as a function of doping and of the TM-oxygen-TM bond angle $\theta$. For small doping and $\theta$ $\lesssim$ $15^\circ$ we find dominant commensurate in-plane antiferromagnetic fluctuations while ferromagnetic fluctuations dominate for $\theta$ $\gtrsim$ $25^\circ$. Moderately strong nearest-neighbor Hubbard interactions can also stabilize a charge density wave order. Furthermore, we compare the dispersion of magnetic excitations for the hole-doped case to resonant inelastic X-ray scattering data and find good qualitative agreement.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points.

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i)Fermi points and (ii)exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.