Pauli Hamiltonian Research Articles

Cluster State as a Noninvertible Symmetry-Protected Topological Phase.

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

Pauli Hamiltonians with an Aharonov–Bohm flux

We study a two-dimensional Pauli operator describing a charged quantum particle with spin 1/2 moving on a plane in presence of an orthogonal Aharonov–Bohm magnetic flux. We classify all the admissible self-adjoint realizations and give a complete picture of their spectral and scattering properties. Symmetries of the resulting Hamiltonians are also discussed, as well as their connection with the Dirac operator perturbed by an Aharonov–Bohm singularity.

Atomic structure, electron-impact excitation and collisional-radiative modelling for Ar II

The spectra from singly ionized argon Ar II has significant diagnostic capability in the characterization and modelling of both magnetically-confined fusion and astrophysical plasmas. The literature has several pre-existing data sets for Ar+ but this paper presents the results from 3 new atomic structure and electron-impact scattering models in order to better constrain the differences in atomic data and how they impact well-known plasma diagnostics. Several independent atomic structure methodologies are employed to calculate the energy levels and transition probabilities for each model. The first approach employs a relativistic Dirac–Coulomb Hamiltonian model, the second approach uses a semi-relativistic Breit–Pauli Hamiltonian with the mass-velocity, Darwin and spin–orbit corrections, and in a third case an ICFT approach. Three atomic structure models provide a foundation for Dirac R-matrix, a semi-relativistic ICFT (Intermediate Coupling Frame Transformation) and a Breit–Pauli R-Matrix with Pseudostates (BPRMPS) calculation. Synthetic spectra utilizing these three data sets are compared against measurements taken at the Compact Toroidal Hybrid (CTH) stellarator, and the total radiative power loss is also benchmarked against previous calculations.

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras.

Exactly solvable Hamiltonians that can be diagonalized by using relatively simple unitary transformations are of great use in quantum computing. They can be employed for the decomposition of interacting Hamiltonians either in Trotter-Suzuki approximations of the evolution operator for the quantum phase estimation algorithm or in the quantum measurement problem for the variational quantum eigensolver. One of the typical forms of exactly solvable Hamiltonians is a linear combination of operators forming a modestly sized Lie algebra. Very frequently, such linear combinations represent noninteracting Hamiltonians and thus are of limited interest for describing interacting cases. Here, we propose an extension in which the coefficients in these combinations are substituted by polynomials of the Lie algebra symmetries. This substitution results in a more general class of solvable Hamiltonians, and for qubit algebras, it is related to the recently proposed noncontextual Pauli Hamiltonians. In fermionic problems, this substitution leads to Hamiltonians with eigenstates that are single Slater determinants but with different sets of single-particle states for different eigenstates. The new class of solvable Hamiltonians can be measured efficiently using quantum circuits with gates that depend on the result of a midcircuit measurement of the symmetries.

Geometric post-Newtonian description of massive spin-half particles in curved spacetime

We consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldline γ representing a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood of γ and expand the Dirac equation up to, and including, the second order in the dimensionless parameter given by the ratio of the geodesic distance to the radii defined by spacetime curvature, linear acceleration of γ, and angular velocity of rotation of the employed spatial reference frame along γ. With respect to the time measured by the clock γ, we compute the Dirac Hamiltonian to that order. On top of this ‘weak-gravity’ expansion we then perform a post-Newtonian expansion up to, and including, the second order of , corresponding to a ‘slow-velocity’ expansion with respect to γ. As a result of these combined expansions we give the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field. This extends and partially corrects recent results from the literature, which we discuss and compare in some detail.

Variational vs perturbative relativistic energies for small and light atomic and molecular systems.

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) equation to all orders of the α fine-structure constant. The analysis of the polynomial α dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit-Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The α dependence of the no-pair DCB energy shows that the higher-order (α4Eh) non-radiative QED correction is 5% of the leading-order (α3Eh) non-radiative QED correction for Z = 2 (He), but it is 40% already for Z = 4 (Be2+), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

Ab initio composite strategies and multireference approaches for lanthanide sulfides and selenides.

The f-block ab initio correlation consistent composite approach was used to predict the dissociation energies of lanthanide sulfides and selenides. Geometry optimizations were carried out using density functional theory and coupled cluster singles, doubles, and perturbative triples with one- and two-component Hamiltonians. For the two-component calculations, relativistic effects were accounted for by utilizing a third-order Douglas-Kroll-Hess Hamiltonian. Spin-orbit coupling was addressed with the Breit-Pauli Hamiltonian within a multireference configuration interaction approach. The state averaged complete active space self-consistent field wavefunctions obtained for the spin-orbit coupling energies were used to assign the ground states of diatomics, and several diagnostics were used to ascertain the multireference character of the molecules.

Fine and hyperfine structures of pionic helium atoms

The fine and hyperfine structures of pionic helium metastable states is calculated within the formalism of the Breit–Pauli Hamiltonian by using the variationally generated wave functions in Hylleraas coordinates. Our results not only verify the existing values of Hori et al. [Phys. Rev. A 89, 042515 (2014)] for the fine structure of π 4He+, but also determine the hyperfine structure of π 3He+.

Quaternionic quantum mechanics for N\u2009=\u20091, 2, 4 supersymmetry

BackgroundQuaternions have emerged as powerful tools in higher-dimensional quantum mechanics as they provide homogeneous four-dimensional structure in quantum field theories, offer compact representations, and incorporate spin naturally. Quantum field theories then lead to the unification of fundamental interactions so the use of quaternion becomes necessary when we are dealing with higher-dimensional theories. On the other hand, supersymmetry is the theory of bosons and fermions and is an essential constituent of grand unified theories. The use of quaternion in supersymmetric field theories provides an excellent framework for higher-dimensional unification theories.ResultA complete theory for supersymmetric quaternionic quantum mechanics has been constructed for N = 1, 2, 4 supersymmetry in terms of one, two, and four supercharges and Hamiltonians, respectively. It has been shown that N = 4 SUSY is the quaternionic extension of the N = 2 complex SUSY and N = 1 real SUSY; also spin is the natural outcome of using quaternion units. Pauli and Dirac Hamiltonian and their relationship have also been obtained in quaternion space. It has been shown that quaternionic quantum mechanics are superior to ordinary and complex quantum mechanics because in the quaternion framework we do not need three different theories for N = 1,2,4 SQM but a single theory only.ConclusionsIt has been concluded that N = 1 real SUSY is equal to N = 2 complex SUSY which in turn is equal to N = 4 quaternion SUSY so one can arrive at higher-dimensional quantum field theories starting from lower-dimensional quantum theories. Higher-dimensional quaternion field theories are suitable for nonphotonic light cone particles which are not allowed in complex QFT, also noncommutative nature of quaternion gives an extra degree of freedom and may provide the possibility of some new particle, dark matter, or new phenomenon. Though quaternions provide an excellent framework in higher-dimensional field theories, there are certain challenges due to their noncommutativity as calculations become tedious where large terms are involved. Keeping in view the noble features of quaternion, we expect some development to get a better understanding of N = 8 supergravity, maximal supergravity (D = 11 − n), and maximal supersymmetry theories (N = 10) in terms of quaternion operators.

On the Breit interaction in an explicitly correlated variational Dirac-Coulomb framework.

The Breit interaction is implemented in the no-pair variational Dirac-Coulomb (DC) framework using an explicitly correlated Gaussian basis reported in the previous paper [P. Jeszenszki, D. Ferenc, and E. Mátyus, J. Chem. Phys. 156, 084111 (2022)]. Both a perturbative and a fully variational inclusion of the Breit term are considered. The no-pair DC plus perturbative Breit and the no-pair DC-Breit energies are compared with perturbation theory results including the Breit-Pauli Hamiltonian and leading-order non-radiative quantum electrodynamics corrections for low Z values. Possible reasons for the observed deviations are discussed.

Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D

We study locality preserving automorphisms of operator algebras on D-dimensional uniform lattices of prime p-dimensional qudits quantum cellular automata (QCAs), specializing in those that are translation invariant (TI), and map every prime p-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of the unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Furthermore, we prove that for any D, the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for D = 3 and any odd prime p and show that the Witt group of the finite field Fp is a subgroup of the group C(D=3,p) of all TI Clifford QCA modulo trivial ones. That is, C(D=3,p≡1mod4)⊇Z2×Z2 and C(D=3,p≡3mod4)⊇Z4. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian, which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of this paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.

Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics

Since variational symplectic integrators for the guiding center was proposed [1,2], structure-preserving geometric algorithms have become an active research field in plasma physics. We found that the slow manifolds of the classical Pauli particle enable a family of structure-preserving geometric algorithms for guiding center dynamics with long-term stability and accuracy. This discovery overcomes the difficulty associated with the unstable parasitic modes for variational symplectic integrators when applied to the degenerate guiding center Lagrangian. It is a pleasant surprise that Pauli's Hamiltonian for electrons, which predated the Dirac equation and marks the beginning of particle physics, reappears in classical physics as an effective algorithm for solving an important plasma physics problem. This technique is applicable to other degenerate Lagrangians reduced from regular Lagrangians.

Classical simulation of noncontextual Pauli Hamiltonians

Noncontextual Pauli Hamiltonians decompose into sets of Pauli terms to which joint values may be assigned without contradiction. We construct a quasi-quantized model for noncontextual Pauli Hamiltonians. Using this model, we give an algorithm to classically simulate noncontextual VQE. We also use the model to show that the noncontextual Hamiltonian problem is NP-complete. Finally, we explore the applicability of our quasi-quantized model as an approximate simulation tool for contextual Hamiltonians. These results support the notion of noncontextuality as classicality in near-term quantum algorithms.

Pauli Hamiltonian for a spin one-half particle carrying a non-Abelian charge in the presence of non-Abelian external fields

We study the solution of non-relativistic approximation of Dirac equation in the presence of non-Abelian external fields. The dynamics of the system is described by the Pauli Hamiltonian for a spin one-half particle carrying a non-Abelian charge. This non-Abelian charge, at the quantum level, is proportional to the generators of the U(2) gauge group and reveals the effect of a spin-orbit coupling across the non-Abelian gauge field. By choosing a suitable non-Abelian gauge field, the total angular momentum is the conserved quantity of the system as well as the corresponding supercharge. The Hamiltonian of the system is then invariant under the transformations generated by the supercharge. This allows us to apply the algebraic method. The Landau levels obtained are deformed by the non-Abelian contribution of the gauge potential. Each of these energy levels are doubly degenerate except the lowest Landau level whose eigenvalue is zero. In the Abelian limit, the Landau level energies are recovered with a fourfold degeneracy. These results correspond to the supersymmetric extension of the ordinary Landau levels.

Measurement reduction in variational quantum algorithms

Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We approach this problem from the perspective of contextuality, and use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. We investigate this technique for lattice Hamiltonians, random Pauli Hamiltonians, and electronic structure Hamiltonians. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10--30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.

Low-lying electronic terms of diatomic molecules AB (A = Sc–Ni, B = Cu/Ag/Au)

Low-lying electronic terms of 24 heteronuclear diatomic molecules AB (A = Sc–Ni, B = Cu/Ag/Au) have been systematically studied. Scalar relativistic effects were included by use of the spin-free Douglas–Kroll–Hess (DKH) Hamiltonian. The complete active space self-consistent field (CASSCF) method, followed by multi-reference configuration interaction (MRCI), was used to construct full potential energy curves (PECs). Spectroscopic constants as well as dipole moments are also reported. Spin-orbit coupling was subsequently calculated perturbatively via the spin-orbit terms of the Breit–Pauli Hamiltonian based on the CASSCF wavefunctions. Full spin-orbit coupled PECs were thus constructed for the low-lying terms of NiCu and NiAg. In addition, Kramers-restricted configuration interaction (KRCI) with the exact 2-component (X2C) Hamiltonian was utilised for the atoms and for the Cu-containing diatomic molecules. The influence of relativistic effects, spin-orbit coupling, core correlation effects and basis set incompleteness was probed for some selected cases. It is shown that for 23 out of the 24 diatomic molecules considered here a direct correspondence exists between the ground state terms (GSTs) of the monocations and those of the diatomic molecules with and .

Time Global Finite-Energy Weak Solutions to the Many-Body Maxwell–Pauli Equations

We study the quantum mechanical many-body problem of $N$ nonrelativistic electrons interacting with their self-generated classical electromagnetic field and $K$ static nuclei. The system of coupled equations governing the dynamics of the electrons and their self-generated electromagnetic field is referred to as the many-body Maxwell-Pauli equations. Here we construct time global, finite-energy, weak solutions to the many-body Maxwell-Pauli equations under the assumption that the fine structure constant $\alpha$ and the atomic numbers are not too large. The particular assumptions on the size of $\alpha$ and the atomic numbers ensure that we have energetic stability of the many-body Pauli Hamiltonian, i.e., the ground state energy is finite and uniformly bounded below with lower bound independent of the magnetic field and the positions of the nuclei. This work serves as an initial step towards understanding the connection between the energetic stability of matter and the wellposedness of the corresponding dynamical equations.

Středa formula for charge and spin currents

We consider a 2-dimensional Bloch–Landau–Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the [Formula: see text]-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect of the well-known formula by Středa, which is formulated instead for charge transport.

Spin current generation and control in carbon nanotubes by combining rotation and magnetic field

We study the quantum dynamics of ballistic electrons in rotating carbon nanotubes in the presence of a uniform magnetic field. When the field is parallel to the nanotube axis, the rotation-induced electric field brings about the spin–orbit interaction which, together with the kinetic, inertial, and Zeeman terms, compose the Schrödinger–Pauli Hamiltonian of the system. Full diagonalization of this Hamiltonian yields the eigenstates and eigenenergies leading to the calculation of the charge and spin currents. Our main result is the demonstration that, by suitably combining the applied magnetic field intensity and rotation speed, one can tune one of the currents to zero while keeping the other one finite, giving rise to a spin current generator.

Covariant version of the Pauli Hamiltonian, spin-induced noncommutativity, Thomas precession, and the precession of spin

We show that there is a manifestly covariant version of the Pauli Hamiltonian with equations of motion quadratic on spin and field strength. Relativistic covariance inevitably leads to noncommutative positions: classical brackets of the position variables are proportional to the spin. It is the spin-induced noncommutativity that is responsible for transforming the covariant Hamiltonian into the Pauli Hamiltonian, without any appeal to the Thomas precession formula. The Pauli theory can be thought to be $1/c^2$ approximation of the covariant theory written in special variables. These observations clarify the long standing question on the discrepancy between the covariant and Pauli Hamiltonians. We also discuss the transformational properties of spin axis in the passage from laboratory to comoving and instantaneous frames, and reveal the role of Thomas spin-vector in the covariant scheme.