Unitary Correlation Operator Method Research Articles

Appearance of the Hoyle state and its breathing mode in C12 despite strong short-range repulsion of the nucleon-nucleon potential

In the $^{12}\mathrm{C}$ nucleus, the Hoyle state $({0}_{2}^{+})$ is considered to be an $\ensuremath{\alpha}$-condensed state, and the ${0}_{3}^{+}$ state is considered to be its breathing mode. We investigated whether the $\ensuremath{\alpha}$ condensation in $^{12}\mathrm{C}$ is realized using a microscopic $3\ensuremath{\alpha}$ cluster model with short-range correlation induced by nucleon-nucleon interactions, where we used the Argonne ${v}_{4}^{\ensuremath{'}}$ potential having a short-range repulsion constructed from the realistic Argonne ${v}_{18}$ potential. Short-range correlation was treated using the unitary correlation operator method, and the Bloch--Brink wave function was adopted as a variational wave function where the $\ensuremath{\alpha}$ cluster motions were treated by generator coordinates. We obtained four ${0}^{+}$ states, including short-range correlation, and analyzed them in terms of the Tohsaki--Horiuchi--Schuck--R\"opke (THSR) wave function. In addition to the ordinary THSR wave function, we defined a second-order TSHR wave function to describe the $2\ensuremath{\hbar}\ensuremath{\omega}$ excitation of the $\ensuremath{\alpha}$-condensed state. The ${0}_{2}^{+}$ state (the Hoyle state) is an $\ensuremath{\alpha}$-condensed state mainly comprising the $^{8}\mathrm{Be}({0}^{+})+\ensuremath{\alpha}(0S$-wave) configuration, and the ${0}_{3}^{+}$ state mainly comprises the $^{8}\mathrm{Be}({0}^{+})+\ensuremath{\alpha}(2S$-wave) configuration, which is regarded as the breathing mode of the $\ensuremath{\alpha}$ condensation excited from the Hoyle state.

Comparative analysis of formalisms and performances of three different beyond-mean-field approaches

We investigate the differences and analogies between the equation of motion phonon method (EMPM) and second Tamm-Dancoff and random-phase approximations (STDA and SRPA) paying special attention to the problem of spurious center-of-mass (c.m.) admixtures. In order to compare them on an equal footing, we perform self-consistent calculations of the multipole strength distributions in selected doubly magic nuclei within a space including up to two-particle--two-hole (2p-2h) basis states using the unitary correlation operator method two-body intrinsic Hamiltonian and we explore the tools each approach supplies for removing the spurious c.m. admixtures. We find that the EMPM and STDA yield exactly the same results when the same intrinsic Hamiltonian is used and the coupling of the Hartree-Fock state with the 2p-2h space is neglected, but, unlike STDA and SRPA, the EMPM offers the possibility to completely remove c.m. admixtures.

Finite particle-number description of symmetric nuclear matter with spin excitations of high-momentum pairs induced by the tensor force

We study symmetric nuclear matter using bare nucleon-nucleon ($NN$) interactions with the finite particle-number approach within finite cubic boxes. Due to the $NN$ correlations originating from the bare $NN$ interaction, two nucleons can be excited to the high-momentum region, leading to the increase of the kinetic energy in nuclear matter. We further consider the spin excitations in the nucleon pairs, where the spins of the two nucleons are changed, and this excitation is important for the tensor correlation. The unitary correlation operator method (UCOM) is used to treat the short-range correlation. The tail correction coming from the neighboring boxes is also included. We demonstrate the contributions of various excitations of nucleon pairs as well as the tail correction to the total energy at normal density. We also discuss the effects of UCOM and correlated nucleon pairs on the density dependence of the total energy. We calculate the equations of state of symmetric nuclear matter using two kinds of the Argonne potentials and the results agree with those from other many-body theories. The density dependences of the Hamiltonian components are also shown.

Role of the unitary correlation operator on high-momentum antisymmetrized molecular dynamics using the bare NN interaction for 3H and 4He

Abstract We extend high-momentum antisymmetrized molecular dynamics (HMAMD) by incorporating the short-range part of the unitary correlation operator method (UCOM) as the variational method of finite nuclei. In this HMAMD+UCOM calculation of light nuclei, HMAMD is mainly in charge of the tensor correlation with up to four-body correlation, while the short-range correlation is further improved by using UCOM. The binding energies of the $^3$H and $^4$He nuclei are calculated with HMAMD+UCOM using the AV8$'$ bare nucleon–nucleon ($NN$) interaction. The different roles of the short-range and tensor correlations from HMAMD and UCOM are analyzed in the numerical results. Compared with previous calculations based on different variational methods, this newly extended HMAMD+UCOM method can almost provide consistent results with ab initio results.

Finite particle number description of neutron matter using the unitary correlation operator and high-momentum pair methods * *Supported by the National Natural Science Foundation of China (11822503, 11575082, 11947220), by the Fundamental Research Funds for the Central Universities (Nanjing University), by JSPS KAKENHI (JP18K03660, JP16K05351), and by a Project funded by China Postdoctoral Science Foundation (2019M661785). The author N. W.

Using bare Argonne V4' (AV4'), V6' (AV6'), and V8' (AV8') nucleon–nucleon (NN) interactions, the nuclear equations of state (EOSs) for neutron matter are calculated with the unitary correlation operator and high-momentum pair methods. Neutron matter is described using a finite particle number approach with magic number under a periodic boundary condition. The central short-range correlation originating from the short-range repulsion in the interaction is treated by the unitary correlation operator method (UCOM), and the tensor correlation and spin-orbit effects are described by the two-particle two-hole (2p2h) excitations of nucleon pairs, where the two nucleons with a large relative momentum are regarded as a high-momentum (HM) pair. With increasing 2p2h configurations, the total energy per particle of the neutron matter is well-converged under this UCOM+HM framework. Comparing the results calculated with AV4', AV6', and AV8' interactions, we demonstrate the effects of the short-range correlation, tensor correlation, and spin-orbit coupling on the density dependence of the total energy per particle of neutron matter. Moreover, the contribution of each Hamiltonian component to the total energy per particle is discussed. The EOSs of neutron matter calculated within the present UCOM+HM framework agree with the calculations of six microscopic many-body theories, especially the auxiliary field-diffusion Monte Carlo calculations.

Nuclear response using correlated realistic interactions: first-order random phase approximation and beyond

A correlated realistic interaction derived within the Unitary Correlation Operator Method (UCOM) based on the Argonne V18 nucleon-nucleon potential is used in calculations of nuclear response for closed-shell nuclei. Giant resonances are examined in the framework of the random-phase approximation (RPA). The effects of explicit ground-state correla- tions and of higher than first-order configurations are discussed.

Towards ab initio structure and response calculations across the nuclear chart

Starting from the Argonne V18 nucleon-nucleon interaction and using the Unitary Correlation Operator Method, a correlated interaction vucOM has been constructed, which is suitable for calculations within restricted Hilbert spaces. In this work we employ the vucOM in Hartree-Fock, perturbation-theory and RPA calculations. First results are reported on the ground-state properties of various closed-shell nuclei, as well as some excited states. Our results provide valuable information on the properties of the VUCOM and guidance to further optimization. The above scheme offers the prospect of ab initio calculations in nuclei throughout the nuclear chart. It can be used in conjunction with other realistic NN interactions as well, both local and non-local. Various many-body methods can be employed, such as Second RPA, QRPA, Shell Model, etc.

The properties of neutron star in the framework of relativistic Hartree–Fock model with unitary correlation operator method

The properties of neutron star are studied in the framework of relativistic Hartree–Fock (RHF) model with realistic nucleon–nucleon (NN) interactions, i.e., Bonn potentials. The strong repulsion of NN interaction at short range is properly removed by the unitary correlation operator method (UCOM). Meanwhile, the tensor correlation is neglected due to the very rich neutron environment in neutron star, where the total isospin of two nucleons can be approximately regarded as [Formula: see text]. The equations of state of neutron star matter are calculated in [Formula: see text] equilibrium and charge neutrality conditions. The properties of neutron star, such as mass, radius and tidal deformability, are obtained by solving the Tolman–Oppenheimer–Volkoff equation and tidal equation. The maximum masses of neutron from Bonn A, B, C potentials are around [Formula: see text]. The radius are [Formula: see text][Formula: see text]km at [Formula: see text], respectively. The corresponding tidal deformabilities are [Formula: see text]. All of these properties are satisfied with the recent observables from the astronomical and gravitational wave devices and are consistent with the results from the relativistic Brueckner–Hartree–Fock model.

Variational calculation of nuclear matter in a finite particle number approach using the unitary correlation operator and high-momentum pair methods

We propose a new variational method for describing nuclear matter from nucleon-nucleon interaction. We use the unitary correlation operator method (UCOM) for central correlation to treat the short-range repulsion and further include the two-particle two-hole (2p2h) excitations of nucleon pair involving a large relative momentum, which is called 'high-momentum pair'(HM). We describe nuclear matter in finite size with finite particle number on periodic boundary condition and increase the 2p2h configurations until we get the convergence of the total energy per particle. We demonstrate the validity of this 'UCOM+HM' framework by applying it to the symmetric nuclear and neutron matters with the Argonne V4$^\prime$ potential having short-range repulsion. The nuclear equations of state obtained in UCOM+HM are fairly consistent to those of other calculations such as Brueckner-Hartree-Fock and auxiliary field diffusion Monte Carlo in the overall density region.

Origin of fine structure of the giant dipole resonance in sd -shell nuclei

A set of high resolution zero-degree inelastic proton scattering data on 24Mg, 28Si, 32S, and 40Ca provides new insight into the long-standing puzzle of the origin of fragmentation of the Giant Dipole Resonance (GDR) in sd-shell nuclei. Understanding is provided by state-of-the-art theoretical Random Phase Approximation (RPA) calculatios for deformed nuclei using for the first time a realistic nucleon-nucleon interaction derived from the Argonne V18 potential with the unitary correlation operator method and supplemented by a phenomenological three-nucleon contact interaction. A wavelet analysis allows to extract significant scales both in the data and calculations characterizing the fine structure of the GDR. The fair agreement supports that the fine structure arises from ground-state deformation driven by alpha clustering.

Successive variational method of the tensor-optimized antisymmetrized molecular dynamics for central interaction in finite nuclei

Tensor-optimized antisymmetrized molecular dynamics (TOAMD) is the basis of the successive variational method for nuclear many-body problem. We apply TOAMD to finite nuclei to be described by the central interaction with strong short-range repulsion, and compare the results with the unitary correlation operator method (UCOM). In TOAMD, the pair-type correlation functions and their multiple products are operated to the AMD wave function. We show the results of TOAMD using the Malfliet-Tjon central potential containing the strong short-range repulsion. Adding the double products of the correlation functions in TOAMD, the binding energies are converged quickly to the exact values of the few-body calculations for s-shell nuclei. This indicates the high efficiency of TOAMD for treating the short-range repulsion in nuclei. We also employ the s-wave configurations of nuclei with the central part of UCOM, which reduces the short-range relative amplitudes of nucleon pair in nuclei to avoid the short-range repulsion. In UCOM, we further perform the superposition of the s-wave configurations with various size parameters, which provides a satisfactory solution of energies close to the exact and TOAMD values.

Structures in 9,10Be and 10B studied with the tensor-optimized shell model

We investigate the structures of $^{9,10}$Be and $^{10}$B with the tensor-optimized shell model (TOSM) using the effective interaction based on the bare nucleon-nucleon interaction AV8$^\prime$. The tensor correlation is treated in TOSM with the full optimization of 2p2h configurations including high-momentum components. The short-range correlation is described in the unitary correlation operator method (UCOM). It is found that the level orders of the low-lying states of $^{9,10}$Be and $^{10}$B are entirely reproduced. For $^9$Be, ground band states are located relatively in higher energy than the experiments, which indicates the missing $\alpha$ clustering correlation in these states as seen in the case of $^8$Be with TOSM. In addition, the tensor force gives the larger attraction for $T$=1/2 states than for $T$=3/2 ones for $^9$Be. For $^{10}$Be, the tensor contribution of $0^+_2$ shows the largest value among the $0^+$ states. This can be related to the $\alpha$ clustering correlation in this state. It is also found that the level order of three nuclei depends on the tensor force in comparison with the results obtained with the Minnesota interaction without the tensor force.

Ab initio Monte Carlo shell model calculations for 7Li and 9Li low-lying spectra**Supported by Fundamental Research Funds for the Central Universities (JUSRP1035), National Natural Science Foundation of China (11305077)

The low-lying spectra of 7Li and 9Li are investigated within an ab initio Monte Carlo Shell Model (MCSM) employing a realistic potential obtained via the Unitary Correlation Operator Method (UCOM). The MCSM calculations in a 4-major-shells model space for the binding energy and mass quadrupole moment of 7,9Li show good convergence when the MCSM dimension reaches 20. The excitation energy of the Jπ = 1/2− state for 7Li and the magnetic moments for 7,9Li ground states in the MCSM with a treatment of spurious center-of-mass motion are close to the experimental data. Correct level ordering of Jπ = 3/2− and 1/2− states for 7,9Li can be reproduced due to the inclusion of three-body correlations in the MCSM+UCOM. However, the excitation energy of Jπ = 1/2− state for 9Li is not reproduced in the MCSM mainly due to the lack of larger model space.

No-core Monte Carlo shell model calculations with unitary correlation operator method and similarity renormalization group**Supported by Fundamental Research Funds for the Central Universities (JUSRP1035), National Natural Science Foundation of China (11305077)

The unitary correlation operator method (UCOM) and the similarity renormalization group theory (SRG) are compared and discussed in the framework of the no-core Monte Carlo shell model (MCSM) calculations for 3H and 4He. The treatment of spurious center-of-mass motion by Lawson's prescription is performed in the MCSM calculations. These results with both transformed interactions show good suppression of spurious center-of-mass motion with proper Lawson's prescription parameter βc.m. values. The UCOM potentials obtain faster convergence of total energy for the ground state than that of SRG potentials in the MCSM calculations, which differs from the cases in the no-core shell model calculations (NCSM). These differences are discussed and analyzed in terms of the truncation scheme in the MCSM and NCSM, as well as the properties of the potentials of SRG and UCOM.

Nucleon-nucleon potentials in phase-space representation

A phase-space representation of nuclear interactions, which depends on the distance $\stackrel{P\vec}{r}$ and relative momentum $\stackrel{P\vec}{p}$ of the nucleons, is presented. A method is developed that permits us to extract the interaction $V(\stackrel{P\vec}{r},\stackrel{P\vec}{p})$ from antisymmetrized matrix elements given in a spherical basis with angular momentum quantum numbers, either in momentum- or coordinate-space representation. This representation visualizes in an intuitive way the nonlocal behavior introduced by cutoffs in momentum space or renormalization procedures that are used to adapt the interaction to low-momentum many-body Hilbert spaces, as done in the unitary correlation operator method (UCOM) or with the similarity renormalization group (SRG). It allows us to develop intuition about the various interactions and illustrates how the softened interactions reduce the short-range repulsion in favor of nonlocality or momentum dependence while keeping the scattering phase shifts invariant. It also reveals that these effective interactions can have undesired complicated momentum dependencies at momenta around and above the Fermi momentum. Properties, similarities, and differences of the phase-space representations of the Argonne and the N3LO chiral potential and their UCOM and SRG derivatives are discussed.

Role of tensor force in light nuclei with tensor-optimized shell model

We investigate the structures of light nuclei focusing on the role of the tensor force. We describe the tensor and short-range correlations with the tensor optimized shell model (TOSM) and the unitary correlation operator method (UCOM), respectively. In the TOSM, the 2p2h states are fully optimized to describe the large tensor contribution that brings high momentum components explicitly in the wave function. We use a bare nucleon-nucleon interaction AV8' and discuss the structures of the He and Li isotopes, such as the excitation energy spectra and the radii. We also investigate the structures of 8Be from the ground-band states to the highly excited states. These states of 8Be are indicated to have different structures of the 2α clustering states and the shell-model like states.

Giant resonances based on unitarily transformed two-nucleon plus phenomenological three-nucleon interactions

We investigate giant resonances of spherical nuclei on the basis of the Argonne V18 potential after unitary transformation within the similarity renormalization group or the unitary correlation operator method supplemented by a phenomenological three-body contact interaction. Such Hamiltonians can provide a good description of ground-state energies and radii within Hartree–Fock plus low-order many-body perturbation theory. The standard random phase approximation is applied here to calculate the isoscalar monopole, isovector dipole, and isoscalar quadrupole excitation modes of the , , and nuclei. Thanks to the inclusion of the three-nucleon interaction and despite the minimal optimization effort, a reasonable agreement with experimental centroid energies of all three modes has been achieved. The role and scope of the Hartree–Fock reference state in RPA methods are discussed.

From nucleon-nucleon interaction matrix elements in momentum space to an operator representation

Starting from the matrix elements of the nucleon-nucleon interaction in momentum space we present a method to derive an operator representation with a minimal set of operators that is required to provide an optimal description of the partial waves with low angular momentum. As a first application we use this method to obtain an operator representation for the Argonne potential transformed by means of the unitary correlation operator method and discuss the necessity of including momentum-dependent operators. The resulting operator representation leads to the same results as the original momentum space matrix elements when applied to the two-nucleon system and various light nuclei. For applications in fermionic and antisymmetrized molecular dynamics, where an operator representation of a soft but realistic effective interaction is indispensable, a simplified version using a reduced set of operators is given.

Shell and alpha cluster structures in 8Be with tensor-optimized shell model

We study the shell and alpha cluster structures in the ground and excited states of 8Be in terms of the tensor-optimized shell model (TOSM). In TOSM, the tensor correlation is optimized in the full space of 2p2h configurations involving high-momentum components. The short-range correlation is treated with the unitary correlation operator method (UCOM). We use the effective interaction based on the bare nucleon-nucleon interaction AV8$^\prime$. The 8Be states consist of two groups of ground band states and highly excited states with the isospin T=0 and T=1. It is found that the tensor contributions of the ground band states are stronger than the highly excited states and that the kinetic energies and the central contributions of the ground band states are almost twice the 4He values. These features suggest two-alpha clustering for the ground band states in 8Be. We also estimate the correlation energy of the $\alpha$ clustering using the alpha cluster model. In the highly excited states, the calculated spectrum in TOSM reproduces the experimental level order and the relative energies of each level. This agreement suggests that those states can be interpreted as shell-like states. The level order is found to be sensitive to the presence of the tensor force in comparison with the results using the Minnesota effective interaction without the tensor force. It is also found that the tensor contributions in the T=0 states are stronger than the T=1 states, which is consistent with the state dependence of the tensor force.

No-Core MCSM calculation for 10Be and 12Be low-lying spectra

The low-lying excited states of 10Be and 12Be are investigated within a no-core Monte Carlo Shell Model (MCSM) framework employing a realistic potential obtained via the Unitary Correlation Operator Method. The excitation energies of the 2+1 and 2+2 states and the B(E2; 2+1,2 → 0+g.s.) for 10Be in the MCSM with a standard treatment of spurious center-of-mass motion show good agreement with experimental data. Some properties of low-lying states of 10Be are studied in terms of quadrupole moments and E2 transitions. The E2 transition probability of 10C, the mirror nucleus of 10Be, is also presented with a good agreement to experiment. The triaxial deformation of 10Be and 10C is discussed in terms of the B(E2) values.