Pairing Hamiltonian Research Articles

Matrix‐Product State Approach to the Generalized Nuclear Pairing Hamiltonian

AbstractIt is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network (matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used Bardeen‐Cooper‐Schrieffer (BCS) approach, and does not suffer from any mean‐field artifacts. This framework is applied to compute the even‐odd mass differences of all known lead isotopes from to in a very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. Pairing with non‐zero angular momentum is also considered and the lowest excited states in the full configuration space of one major shell is determined, which is demonstrated for the , isotones. To demonstrate the capabilities of the method beyond low‐lying excitations, the first 100 excited states of with singlet pairing and the two‐neutron removal spectral function of are calculated, which relate to a two‐neutron pickup experiment.

Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer

A quantum simulation experiment pairing Hamiltonians of nearest-neighbor interacting superconducting qubits was performed with a complete set of algorithms on an IBM Quantum Computer-IBMq Lima. The experiment revealed that the fidelity is a function of iteration using Suzuki–Trotter decomposition for four different types of nearest-neighbor Heisenberg, XY, transverse, and longitudinal Ising superconducting qubit couplings of Hamiltonians. The experiment displayed the models of how the experimental density matrices shift from the theoretical density matrices and how their behavior changes with different numbers of iterations. It also demonstrated the reconstruction of quantum states and how the states change as a function of iteration with the IBM Quantum Computer-IBMq Lima. The time evolutions of the states for different models were also shown to predict the dominance of each state.

Rotation and Vibration in Tetraquarks

A novel approach is introduced for obtaining precise solutions of the pairing Hamiltonian for tetraquarks, which utilizes an algebraic technique in infinite dimensions. The parameters involved in the transition phase are calibrated based on potential tetraquark candidates derived from phenomenology. Our investigation shows that the rotation and vibration transitional theory delivers a reasonable agreement with other works for heavy tetraquarks compared to other methods. To illustrate the concept, we compute the spectra of several tetraquarks, namely charm, bottom, bottom–charm and open charm and bottom systems, and contrast them with those of other particles.

Supercurrent-enabled Andreev reflection in a chiral quantum Hall edge state

A chiral quantum Hall (QH) edge state placed in proximity to an s-wave superconductor experiences induced superconducting correlations. Recent experiments have observed the effect of proximity-coupling in QH edge states through signatures of the mediating process of Andreev reflection. We present the microscopic theory behind this effect by modeling the system with a many-body Hamiltonian, consisting of an s-wave superconductor, subject to spin-orbit coupling and a magnetic field, which is coupled by electron tunneling to a QH edge state. By integrating out the superconductor we obtain an effective pairing Hamiltonian in the QH edge state. We clarify the qualitative appearance of nonlocal superconducting correlations in a chiral edge state and analytically predict the suppression of electron-hole conversion at low energies (Pauli blocking) and negative resistance as experimental signatures of Andreev reflection in this setup. In particular, we show how two surface phenomena of the superconductor, namely Rashba spin-orbit coupling and a supercurrent due to the Meissner effect, are essential for the Andreev reflection. Our work provides a promising pathway to the realization of Majorana zero-modes and their parafermionic generalizations.

Proton-neutron pairing, α-like quarteting and ground/excited states of N=Z nuclei

Various studies have shown that the proton-neutron (pn) pairing correlations can be accurately described not by a condensate of Cooper pairs, as considered in the majority of mean-field calculations, but by a condensate of α-like quartets. After a short review of the quartet condensate model (QCM), it is discussed the effect of the pn pairing on the ground states of N = Z nuclei, analysed recently in the framework of Skyrme-HF+QCM calculations. An interesting aspect pointed out by these calculations is the strong interdependence between all types of pairing correlations. In particular, when the isoscalar pn pairing channel is switched on, the pairing correlations are redistributed among all the pairing channels without changing significantly the total pairing energy. Due to this reason, for the majority of N = Z nuclei, the binding energy is not affected much when the isoscalar pairing channel is switched on. Yet, in all calculations which include both the isovector and the isoscalar pairing forces, the isoscalar pairing correlations contribute significantly to the binding energies and coexist always with the isovector pn pairing. Finally it is presented a recent extension of the QCM approach to the excited stated of pn pairing Hamiltonians. It will be shown that the low-lying excited states of these Hamiltonians can be described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet, an approach which is analogous to the one-broken-pair approximation which is commonly used for like-particle systems.

Accessing ground-state and excited-state energies in a many-body system after symmetry restoration using quantum computers

We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional postprocessing. The final goal is to develop configuration interaction techniques based on many-body trial states preoptimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and the quantum Krylov approaches for the postprocessing. The latter method combined with the quantum variation after projection leads to very fast convergence toward the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing Hamiltonian.

An iterative approach for the exact solution of the pairing Hamiltonian

A new iterative algorithm is established for the exact solution of the standard pairing problem, based on the Richardson-Gaudin method using the polynomial approach. It provides efficient and robust solutions for both spherical and deformed systems at a large scale. The key to its success is that the initial guess for the solutions of such a large set of the non-linear equations is provided in a physically meaningful and controllable manner. Moreover, one reduces the large-dimensional problem to a one-dimensional Monte Carlo sampling procedure, which improves the algorithm's efficiency and avoids the non-solutions and numerical instabilities that persist in most existing approaches. We calculated the ground state and low-lying excited states of equally spaced systems at different pairing strengths G. We then applied the model to study the quantum phase transitional Sm isotopes and the actinide nuclei Pu isotopes, where an excellent agreement with experimental data is obtained. Program summaryProgram Title: IterV1.mCPC Library link to program files:https://doi.org/10.17632/rjnbhgk2p6.1Licensing provisions: GPLv3Programming language: MathematicaNature of problem: The program calculates exact pairing energies based on a new iterative algorithm. The key is the procedure of determining the initial guesses for the large-set non-linear equations involved in a controllable and physically motivated manner. It provides an efficient and robust solver for both spherical and deformed systems in super large model spaces.Solution method: The new iterative algorithm approach starts with simple systems with k nucleon pairs and n=k levels, which can be solved iteratively by including one pair and one level at each step using the Newton-Raphson algorithm with a Monte Carlo sampling procedure. Then it takes the solutions of those systems as initial values and obtain the converged results for the full space by gradually adding the remaining levels. In this way, one reduces the k-dimensional Monte Carlo sampling procedure to a one-dimensional sampling, which improves the efficiency of the algorithm and avoids the non-solutions and numerical instabilities.

Np-Pair Correlations in the Isovector Pairing Model

A diagonalization scheme for the shell model mean-field plus isovector pairing Hamiltonian in the O(5) tensor product basis of the quasi-spin SUΛ(2) ⊗ SUI(2) chain is proposed. The advantage of the diagonalization scheme lies in the fact that not only can the isospin-conserved, charge-independent isovector pairing interaction be analyzed, but also the isospin symmetry breaking cases. More importantly, the number operator of the np-pairs can be realized in this neutron and proton quasi-spin basis, with which the np-pair occupation number and its fluctuation at the J = 0+ ground state of the model can be evaluated. As examples of the application, binding energies and low-lying J = 0+ excited states of the even–even and odd–odd N∼Z ds-shell nuclei are fit in the model with the charge-independent approximation, from which the neutron–proton pairing contribution to the binding energy in the ds-shell nuclei is estimated. It is observed that the decrease in the double binding-energy difference for the odd–odd nuclei is mainly due to the symmetry energy and Wigner energy contribution to the binding energy that alter the pairing staggering patten. The np-pair amplitudes in the np-pair stripping or picking-up process of these N = Z nuclei are also calculated.

α-Like quartetting in the excited states of proton-neutron pairing Hamiltonians

Previous studies have shown that the ground state of systems of nucleons composed by an equal number of protons and neutrons interacting via proton-neutron pairing forces can be described accurately by a condensate of α-like quartets. Here we extend these studies to the low-lowing excited states of these systems and show that these states can be accurately described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet. This approach, which is analogous to the one-broken-pair approximation employed for like-particle pairing, is analyzed for various isovector and isovector-isoscalar pairing Hamiltonians which can be solved exactly by diagonalization.

Simple approach to characterizing band topology in bosonic pairing Hamiltonians

We revisit the problem of characterizing band topology in dynamically-stable quadratic bosonic Hamiltonians that do not conserve particle number. We show this problem can be rigorously addressed by a smooth and local adiabatic mapping procedure to a particle number conserving Hamiltonian. In contrast to a generic fermionic pairing Hamiltonian, such a mapping can always be constructed for bosons. Our approach shows that particle non-conserving bosonic Hamiltonians can be classified using known approaches for fermionic models. It also provides a simple means for identifying and calculating appropriate topological invariants. We also explicitly study dynamically stable but non-positive definite Hamiltonians (as arise frequently in driven photonic systems). We show that in this case, each band gap is characterized by two distinct invariants.

PairDiagSph: Generalization of the exact pairing diagonalization program for spherical systems

We present an efficient program for the exact diagonalization solution of the pairing Hamiltonian in spherical systems with rotational invariance based on the SU(2) quasi-spin algebra. The basis vectors with quasi-spin symmetry considered are generated by using an iterative algorithm. Then the Hamiltonian matrix constructed on this basis is diagonalized with the Lanczos algorithm. All non-zero matrix elements of the Hamiltonian matrix are evaluated “on the fly” by the scattering operator and hash search acting on the basis vectors. The OpenMP parallel program thus developed, PairDiagSph, can efficiently calculate the ground-state eigenvalue and eigenvector of general spherical pairing Hamiltonians. Systems with dimension up to 108 can be calculated in few hours on standard desktop computers. Program summaryProgram Title: PairDiagSph.CPC Library link to program files:https://doi.org/10.17632/t5txsg3s5g.1Code Ocean capsule:https://codeocean.com/capsule/9159628Licensing provisions: CC by NC 3.0.Programming language: Fortran 95.Nature of problem: The exact diagonalization of spherical pairing Hamiltonian can be achieved in the quasi-spin space.Solution method: The program generates the basis vectors via the adjacency excitation algorithm, and diagonalizes the spherical pairing Hamiltonian by the Lanczos + QR algorithm.Additional comments including restrictions and unusual features: The total number of spherical must be less than 64; The maximal dimension that can be handled is restricted by the local RAM capacity.

Exploring non-linear correlators on AGP.

Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential Ansätze in this paper. We investigate two approaches in particular. Similar to Wahlen-Strothman et al. [Phys. Rev. B 91, 041114(R) (2015)], we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting the Schrödinger equation. The second approach is based on approximating the unitary pair-hopper Ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.

Symmetry-Assisted Preparation of Entangled Many-Body States on a Quantum Computer.

Starting from the quantum-phase-estimate (QPE) algorithm, a method is proposed to construct entangled states that describe correlated many-body systems on quantum computers. Using operators for which the discrete set of eigenvalues is known, the QPE approach is followed by measurements that serve as projectors on the entangled states. These states can then be used as inputs for further quantum or hybrid quantum-classical processing. When the operator is associated with a symmetry of the Hamiltonian, the approach can be seen as a quantum-computer formulation of symmetry breaking followed by symmetry restoration. The method, called discrete spectra assisted, is applied to superfluid systems. By using the blocking technique adapted to qubits, the full spectra of a pairing Hamiltonian is obtained.

Role of pair-vibrational correlations in forming the odd-even mass difference

In the random phase approximation (RPA)-amended Nilsson-Strutinskij method of calculating nuclear binding energies [1, 2], the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers N and Z of neutrons and protons and even mass number A = N + Z. Here it is applied to odd-A nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with Z = N – 1 and 25 ⩽ A ⩽ 99. (ii) The odd-A isotopes of In, Sn, and Sb with 46 ⩽ N ⩽ 92. (iii) The odd-A isotopes of Sr, Y, Zr, Nb, and Mo with 60 ⩽ N ⩽ 64. The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper sd shell this correction accounts for almost the entire odd-even mass difference for odd Z and about half of it for odd N. The size and sign of the RPA contribution varies, which is explained qualitatively in terms of a closed expression for a smooth RPA counter term.

Estimate of the location of the neutron drip line for calcium isotopes from an exact Hamiltonian with continuum pair correlations

The eastern region of the calcium isotope chain of the nuclei chart is, nowadays, of great activity. The experimental assessment of the limit of stability is of interest to confirm or improve microscopic theoretical models. The goal of this work is to provide the drip line of the calcium isotopes from the exact solution of the pairing Hamiltonian which incorporates explicitly the correlations with the continuum spectrum of energy. The modified Richardson equations, which include correlations with the continuum spectrum of energy modeled by the continuum single particle level density, is used to solve the many-body system. Three models are used, two isospin independent models with core 40Ca and 48Ca, and one isospin dependent model. One and two-neutron separation energies and occupation probabilities for bound and continuum states are calculated from the solution of the Richardson equations. The one particle drip line is found at the nucleus 57Ca, while the two neutron drip line is found at the nucleus 60Ca from the isospin independent model and at 66Ca from the isospin dependent one.

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. The results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.

Neutron-proton pairing correction in the extended isovector and isoscalar pairing model

An extended ${\mathrm{O}}_{\mathcal{S}T}$(8) model with multi-$j$ orbits is constructed based on the angular momentum decomposition with ``pseudo''-spin $\mathcal{S}$ for valence nucleons in a $j$ orbit. It is shown that the isovector $\mathcal{S}=0$ and $T=1$ pairs are exactly the $J=0$ and $T=1$ pairs in a given $j$ orbit, while the isoscalar $\mathcal{S}=1$ pairs are linear combinations of $J=\mathrm{odd}$ 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 $^{16}\mathrm{O}$ core are fit by the model and compared with the fitting results of the same Hamiltonian in the ${\mathrm{O}}_{LST}$(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 ${\mathrm{O}}_{\mathcal{S}T}$(8) and ${\mathrm{O}}_{LST}$(8) models, it is clearly shown that the isoscalar pair content in the lowest $J={0}^{+}$ state of the $N=Z$ and $N=Z\ifmmode\pm\else\textpm\fi{}2$ 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 $N=Z\ifmmode\pm\else\textpm\fi{}2$ nuclei, especially in those even-even nuclei with more valence nucleons up to the half-filling, even though the isoscalar pairing interaction is negligible.

Pairing, quasi-spin and seniority

We present our concise notes for the lectures and tutorials on pairing, quasi-spin and seniority delivered at SERB school on Role of Symmetries in Nuclear Physics, AMITY University, $2019$. Starting with some generic features of residual nucleon-nucleon interactions, we provide detailed derivation of the matrix elements for the $\delta$-interaction which is the basis for the standard pairing Hamiltonian. The eigen values for standard pairing Hamiltonian are then obtained within the quasi-spin formalism. The algebra involving quasi-spin operators is performed explicitly using the annihilation and creation operators for single nucleon together with the application of Wicks theorem. These techniques are expected to be helpful in deriving the mean-field equations for the $Hartree-Fock$, $Bardeen-Cooper-Schrieffer$ and $Hartree-Fock$ Bogoliubov theories.

Quartet condensation induced by the isovector pairing force

The phenomenon of quartet condensation in the ground state of an isovector pairing Hamiltonian for an even–even N = Z system is investigated. For this purpose we follow the evolution of the ground state from an unperturbed regime up to a strongly interacting one in a formalism of collective pairs. These pairs are those resulting from the diagonalization of the pairing Hamiltonian in a space of two particles coupled to isospin T = 1. The ground state is found to rapidly evolve from a product of distinct T = 0 quartets, each one formed by two of the above pairs, to a condensate of identical quartets built only with the pair corresponding to the lowest energy. This finding establishes a link between the complicated structure of the exact ground state and the simple approximation scheme of the quartet condensation model. The mechanism at the basis of this quartet condensation turns out to be the same which is responsible for the development of a pair condensate in the ground state of a like-particle pairing Hamiltonian.

Geminal Replacement Models Based on AGP.

The antisymmetrized geminal power (AGP) wave function has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP-based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J-CI provides excellent accuracy in energies when applied to the pairing Hamiltonian.