Exact solution of mean-field plus an extended T = 1 nuclear pairing Hamiltonian in the seniority-zero symmetric subspace
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An efficient quantum circuit for block encoding a pairing Hamiltonian
We present an efficient quantum circuit for block encoding a pairing Hamiltonian often studied in nuclear physics. Our block encoding scheme does not require mapping the creation and annihilation operators to the Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode the Hamiltonian directly using controlled swap operations. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with the quantum singular value transformation to construct an efficient quantum circuit for approximating the density of states of a pairing Hamiltonian. The techniques presented can be extended to encode more general second-quantized Hamiltonians.
97 MATHEMATICS AND COMPUTING↗
Magnetic properties of the second Mott lobe in pairing Hamiltonians
We explore the Mott insulating state of single-band bosonic pairing Hamiltonians using analytical approaches and large-scale density matrix renormalization group calculations. We focus on the second Mott lobe which exhibits a magnetic quantum phase transition in the Ising universality class. We use this feature to discuss the behavior of a range of physical observables within the framework of the one-dimensional quantum Ising model and the strongly anisotropic Heisenberg model. This includes the properties of local expectation values and correlation functions both at and away from criticality. Depending on the microscopic interactions it is possible to achieve either antiferromagnetic or ferromagnetic exchange interactions and we highlight the possibility of observing the E{sub 8} mass spectrum for the critical Ising model in a longitudinal magnetic field.
36 MATERIALS SCIENCE↗
Hartree-Fock-Bogoliubov solution of the pairing Hamiltonian in finite nuclei
We present an overview of the Hartree-Fock-Bogoliubov (HFB) theory of nucleonic superfluidity for finite nuclei. After introducing basic concepts related to pairing correlations, we show how the correlated pairs are incorporated into the HFB wave function. Thereafter, we present derivation and structure of the HFB equations within the superfluid nuclear density functional formalism and discuss several aspects of the theory, including the unitarity of the Bogoliubov transformation in truncated single-particle and quasiparticle spaces, form of the pairing functional, structure of the HFB continuum, regularization and renormalization of pairing fields, and treatment of pairing in systems with odd particle numbers.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Scaling properties of the pairing problem in the strong coupling limit
We study the excited states of the pairing Hamiltonian providing an expansion for their energy in the strong coupling limit. To assess the role of the pairing interaction we apply the formalism to the case of a heavy atomic nucleus. We show that only a few statistical moments of the level distribution are sufficient to yield an accurate estimate of the energy for not too small values of the coupling G and we give the analytic expressions of the first four terms of the series. Further, we discuss the convergence radius G{sub sing} of the expansion showing that it strongly depends upon the details of the level distribution. Furthermore G{sub sing} is not related to the critical values of the coupling G{sub crit}, which characterize the physics of the pairing Hamiltonian, since it can exist even in the absence of these critical points. -- Highlights: •We study the excitation spectrum of the pairing Hamiltonian. •We provide an analytic expansion around the strong coupling limit. •We discuss the convergence radius of the expansion. •We connect the radius with the critical points of H.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Neutron-proton pairing correction in the extended isovector and isoscalar pairing model
An extended O ST (8) model with multi j-orbits is constructed based on the angular momentum decomposition with “pseudo”-spin S for valence nucleons in a j-orbit. It is shown that the isovector S = 0 and T=1 pairs are exactly the J=0 and T=1 pairs in a given j-orbit, while the isoscalar S = 1 pairs are linear combinations of J=odd pairs, with which the pairing Hamiltonian can be used to estimate isovector and isoscalar pairing interactions. As an example of the model application, some low-lying J = 0 + level energies of even-even and odd-odd A=18 – 28 nuclei up to the half-filling in the ds-shell above the 16 O core are fit by the model and compared with the fitting results of the same Hamiltonian in the O LST (8) form. It has been verified from the fitting of both the models that the isoscalar pairing interaction can be neglected in the lower energy part of the spectra of these ds-shell nuclei as far as binding energies and a few J=0 + excited levels of these nuclei are concerned. With the mean-field plus isovector pairing interaction only, neutron-neutron, proton-proton, and neutron-proton pairing contributions at the ground or the lowest J=0 + state of these nuclei are estimated. It is shown that the isovector np-pairing contribution to the binding in the odd-odd N=Z nuclei is systematically larger than that in the even-even nuclei. Furthermore, the isoscalar np-pair content at the lowest J=0+ state of these nuclei is also estimated. In both the O ST (8) and O LST (8) models, it is clearly shown that the isoscalar-pair content in the lowest J=0 + state of the N=Z and N=Z±2 nuclei increases with increasing of the valence nucleons, especially in those even-even nuclei, which indicates the isoscalar pairing correlation to be of importance at low-lying states of N=Z and N=Z±2 nuclei, especially in those even-even nuclei with more valence nucleons up to the half-filling, even though the isoscalar pairing interaction is negligible.
20 ≤ A ≤ 38↗
Proton-neutron pairing and alpha-type condensation in nuclei
We summarize a recent work (N. Sandulescu et al, arXiv:1507.04144) on isoscalar and isovector proton-neutron pairing treated in a formalism which conserves exactly the particle number and the isospin. The formalism is designed for self-conjugate (N=Z) systems of nucleons moving in an axially deformed mean field and interacting through the most general isovector and isoscalar pairing interactions. The ground state of these systems is described by a superposition of two types of condensates, i.e., condensates of isovector quartets, built by two isovector pairs coupled to the total isospin T=0, and condensates of isoscalar proton-neutron pairs. The comparison with the exact solutions of realistic isovector-isoscalar pairing Hamiltonians shows that this formalism is able to describe accurately the pairing correlations energies. It is also shown that, contrary to the majority of HFB calculations, in the present formalism the isovector and isoscalar pairing correlations coexist together for any pairing interactions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Coupled bipolarons and optical phonons as a model for high-T(sub c) superconductors
The coherence length of the new high-temperature superconductors reaches very small value which is comparable to the dimensions of unit cell of these compounds. This means that a pair consists of two holes occupying the same site or two adjacent sites. Such a situation seems to be described by a model of the local-pairs (bipolarons). The origin of local-pairs may come not only from strong enough electron or hole-phonon interaction but also from other interactions. Independent of the specific nature of such local-pairs, they can undergo a Bose-like condensation to the superconducting state at a critical temperature which is usually much lower than the temperature of the pair formation. An interplay of ferroelectric and superconducting properties is considered within the model of hole-like local-pairs interacting with optical phonons. Therefore, researchers extend the usual local-pair Hamiltonian by including a direct interaction between the local-pairs and the optical phonons. These optical phonons are known to play an important role in the ferroelectric transition, if any, and they transform into an additional pseudo-acoustic branch at the ferroelectric critical temperature. (This is associated with nonzero electric polarization due to the existence of two separate lattices composed of negative and positive ions, respectively.)
Kasperczyk, J.↗
Coupled bipolarons and optical phonons as a model for high-Tc superconductors
The coherence length of the new high-temperature superconductors reaches a small value which is comparable to the dimensions of the unit cell of the compound. This means that a pair consists of two holes occupying the same site or two adjacent sites. Such a situation is described by a model of the local-pairs (bipolarons). The origin of local-pairs may come not only from strong enough electron or hole-phonon interaction but also from other interactions. Independent of the specific nature of such local-pairs, they can undergo a Bose-like condensation to the superconducting state at a critical temperature which is usually much lower than the temperature of the pair formation. An interplay of ferroelectric and superconducting properties is considered within the model of hole-like local-pairs interacting with optical phonons. Therefore, researchers extend the usual local-pair Hamiltonian by including a direct interaction between the local-pairs and the optical phonons. These optical phonons are known to play an important role in the ferroelectric transition and they transform into an additional pseudo-acoustic branch at the ferroelectric critical temperature. (This is associated with nonzero electric polarization due to the existence of two separate lattices composed of negative and positive ions, respectively.)
Kasperczyk, J.↗
Exact methods for self interacting neutrinos
The effective many-body Hamiltonian which describes vacuum oscillations and self interactions of neutrinos in a two flavor mixing scheme under the single angle approximation has the same dynamical symmetries as the well known BCS pairing Hamiltonian. These dynamical symmetries manifest themselves in terms of a set of constants of motion and can be useful in formulating the collective oscillation modes in an intuitive way. In particular, we show that a neutrino spectral split can be simply viewed as an avoided level crossing between the eigenstates of a mean field Hamiltonian which includes a Lagrange multiplier in order to fix the value of an exact many-body constant of motion. We show that the same dynamical symmetries also exist in the three neutrino mixing scheme by explicitly writing down the corresponding constants of motion.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Correlating AGP on a quantum computer
For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose and implement a unitary correlator on AGP and benchmark it on the ground state of the pairing Hamiltonian. Furthermore, the results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.
97 MATHEMATICS AND COMPUTING↗
Metropolis-style random sampling of quantum gates for the estimation of low-energy observables
In this work, we propose a quantum algorithm to compute low-energy expectation values of a quantum Hamiltonian by sampling a partition function associated with the average energy of that Hamiltonian. For any given quantum circuit-Hamiltonian pair, there is an associated average energy. The sampling is done through an accept/reject Metropolis-style algorithm on the quantum gates of the circuit itself. Observables calculated under the canonical ensemble from these samples of circuits are extrapolated from higher energies to the ground state.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Excitation spectrum for a Bose gas with repulsive core and attractive well.
He II excitation spectrum for interparticle potential using pair Hamiltonian with repulsive core and attractive well
Brown, G. V.↗
Determinant quantum Monte Carlo study of d -wave pairing in the plaquette Hubbard hamiltonian
We used the determinant Quantum Monte Carlo (DQMC) to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters - a two-dimensional square lattice consisting of elemental 2x2 plaquettes with hopping t and on-site repulsion U coupled by an interplaquette hopping t' ≤ t. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half-filling by the inhomogeneous hopping or not. For U/t = 4, DQMC indicates an optimal t'/t ≈ 0.4 at which the pairing vertex is most attractive. We also found that optimal t'/t increases with U/t. We then contrast our results for this plaquette model with a Hamiltonian which instead involves a regular pattern of site energies whose large site energy limit is the three band CuO 2 model; we show that there the inhomogeneity rapidly, and monotonically, suppresses pairing.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Entropy Pair Functional Theory: Direct Entropy Evaluation Spanning Phase Transitions
We prove that, within the class of pair potential Hamiltonians, the excess entropy is a universal, temperature-independent functional of the density and pair correlation function. This result extends Henderson’s theorem, which states that the free energy is a temperature dependent functional of the density and pair correlation. The stationarity and concavity of the excess entropy functional are discussed and related to the Gibbs–Bugoliubov inequality and to the free energy. We apply the Kirkwood approximation, which is commonly used for fluids, to both fluids and solids. Approximate excess entropy functionals are developed and compared to results from thermodynamic integration. The pair functional approach gives the absolute entropy and free energy based on simulation output at a single temperature without thermodynamic integration. We argue that a functional of the type, which is strictly applicable to pair potentials, is also suitable for first principles calculation of free energies from Born–Oppenheimer molecular dynamics performed at a single temperature. In conclusion, this advancement has the potential to reduce the evaluation the free energy to a simple modification to any procedure that evaluates the energy and the pair correlation function.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Generalization of Lieb's variational principle to Bogoliubov–Hartree–Fock theory
In its original formulation, Lieb's variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization we prove for both fermion and boson systems with semi-bounded Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over pure quasifree states. In particular, the Hamiltonian is not assumed to preserve the number of particles. To shed light on the relation between our result and the usual formulation of Lieb's variational principle in terms of one-particle density matrices, we also include a characterization of pure quasifree states by means of their generalized one-particle density matrices.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Branched Hamiltonians and supersymmetry
Some examples of branched Hamiltonians are explored both classically and in the context of quantum mechanics, as recently advocated by Shapere and Wilczek. These are in fact cases of switchback potentials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian chaotic dynamical systems in a classical setting, and as encountered in analogous renormalization group flows for quantum theories which exhibit RG cycles. In conclusion, a basic two-worlds model, with a pair of Hamiltonian branches related by supersymmetry, is considered in detail.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗