Search Results (81 total)

We investigate eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices. For two-body random ensemble, we find a strong linear correlation between eigenvalues and diagonal matrix elements if both of them are sorted from the smaller values to larger ones. By using this linear correlation we are able to predict reasonably all eigenvalues of a given Hamiltonian matrix without complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangent function improves the accuracy of predicted eigenvalues near the minimum and maximum.

In this article we study the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues is applicable to many different systems. We improve the accuracy of the formula by considering the third central moment. We show that these formulas are applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.

We study number of spin I states for bosons in this article. We extend Talmi's recursion formulas for number of states with given spin I to boson systems, and we prove empirical formulas for five bosons by using these recursions. We obtain number of states with given spin I and F spin for three and four bosons by using sum rules of six-j and nine-j symbols. We also present empirical formulas of states for d bosons with given spin I and F=F_max-1 and F_max-2.

In this article we study the behavior of collectivity under random two-body interactions in the framework of the fermion dynamical symmetry model (FDSM). We found that a Hamiltonian with the SO(8) symmetry of the FDSM does not give vibrational and rotational modes under random interactions while a Hamiltonian with the SP(6) symmetry does. It is suggested that collective motions such as vibration and rotation are closely related not only to the quadruple-quadruple correlation in the Hamiltonian but also to the dynamical symmetries of the Hamiltonian.

The results and conclusions by Ponomarev and Vdovin [Phys. Rev. C 72, 034309 (2005)] are inadequate to judge the applicability of the modified BCS because they were obtained either in the temperature region, where the use of zero-temperature single-particle spectra is no longer justified, or in too limited configuration spaces.

In this report we study the origin of spin-zero ground-state dominance for even-even nuclei in the presence of random two-body interactions. We evaluate the ground-state energy in terms of the energy centroid and the width of the random Hamiltonian. For both fermions and bosons in a single orbital, we obtain excellent agreement between the spin-I ground state probabilities predicted by using our formula and those obtained by diagonalizing the random Hamiltonian.

In this paper we obtain number of states with a given spin I and a given isospin T for systems with three and four nucleons in a single-j orbit, by using sum rules of six-j and nine-j symbols obtained in earlier works.

In this paper we study energy centroids such as those with fixed spin and isospin and those with fixed irreducible representations for both bosons and fermions, in the presence of random two-body and/or three-body interactions. Our results show that regularities of energy centroids of fixed-spin states reported in earlier works are very robust in these more complicated cases. We suggest that these behaviors might be intrinsic features of quantum many-body systems interacting by random forces.

We study the J-pairing Hamiltonian and find that the sum of eigenvalues of spin-I states equals the sum of the norm matrix elements within the pair basis for four identical particles such as four fermions in a single-j shell or four bosons with spin l. We relate the number of states to sum rules of nine-j coefficients. We obtained sum rules for nine-j coefficients 〈(jj)J,(jj)K:I|(jj)J,(jj)K:I〉 and 〈(ll)J,(ll)K:I|(ll)J,(ll)K:I〉 summing over (1) even J and even K, (2) even J and odd K, (3) odd J and odd K, and (4) both even and odd values for J and K, where j is a half integer and l is an integer.

In this article we study the enumeration of number (denoted as D_I) of spin I states for fermions in a single-j shell and bosons with spin l. We show that D_I can be enumerated by the reduction from SU(n+1) to SO(3). New regularities of D_I are discerned. As an example of our new algorithm, we obtained analytical expressions of D_I for four particles.

In this paper we study the behavior of energy centroids (denoted as E_I̅ ) of spin I states in the presence of random two-body interactions, for systems ranging from very simple systems (e.g., single-j shell for very small j) to very complicated systems (e.g., many-j shells with different parities and with isospin degree of freedom). Regularities of E_I̅ 's discussed in terms of the so-called geometric chaoticity (or quasi-randomness of two-body coefficients of fractional parentage) in earlier works are found to hold even for very simple systems in which one cannot assume geometric chaoticity. It is shown that the inclusion of isospin and parity does not “break” the regularities of E_I̅ 's.

We present our results on properties of ground states for nucleonic systems in the presence of random two-body interactions. In particular, we calculate probability distributions for parity, seniority, spectroscopic (i.e., in the laboratory frame) quadrupole moments, and discuss α clustering in the ground states. We find that the probability distribution for the parity of the ground states obtained by a two-body random ensemble simulates that of realistic nuclei with A≥70: positive parity is dominant in the ground states of even-even nuclei, while for odd-odd nuclei and odd-mass nuclei we obtain with almost equal probability ground states with positive and negative parity. In addition, assuming pure random interactions, we find that, for the ground states, low seniority is not favored, no dominance of positive values of spectroscopic quadrupole deformation is observed, and there is no sign of α-clustering correlation, all in sharp contrast to realistic nuclei. Considering a mixture of a random and a realistic interaction, we observe a second-order phase transition for the α-clustering correlation probability.

In this paper we show that a system of three fermions is exactly solvable for the case of a single-j in the presence of an angular momentum-J pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-j symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin J=J_max=2j−1, the “non-integer” eigenvalues of three fermions with angular momentum I around the maximum appear as “non-integer” eigenvalues of four fermions if I is around (or larger than) J_max. This pattern is also found in five and six fermion systems. A boson system with spin l exhibits a similar pattern.

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum J. We show that pairs with spin J are reasonable building blocks for the low-lying states. For systems with only a J=J_max pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta I≤2j. For I=0 eigenstates of four fermions in a single-j shell we show that there is only one nonzero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.

In this paper we study the number of angular momentum I states for both fermions and bosons. Suppose that n is the particle number, M=I_max−I, and that J refers to the angular momentum of a single-particle orbit for fermions or the spin L carried by bosons. We prove that the number of states is independent of J if M≤(2J−n+1) for fermions and M≤2J for bosons, and that the number of I states is independent of both n and J if M≤min(n,2J+1−n) for fermions and M≤min(n,2J) for bosons. We also present in this paper empirical formulas for the number of I states of three and four identical fermions or bosons.

The width of the giant dipole resonance (GDR) at finite temperature T in ¹²⁰Sn is calculated within the phonon damping model including the neutron thermal pairing gap determined from the modified BCS theory. It is shown that the effect of thermal pairing causes a smaller GDR width at T≲2  MeV as compared to the one obtained by neglecting pairing. This improves significantly the agreement between theory and experiment, including the most recent data point at T=1  MeV.

In this paper, we report our systematic calculations of angular momentum I ground state probabilities P(I) of boson systems with spin l in the presence of random two-body interactions. It is found that the P(0) dominance is usually not true for a system with an odd number of bosons, while it is valid for an even number of bosons, which indicates that the P(0) dominance is partly connected to the even number of identical particles. It is also noticed that the P(I_max)’s of bosons with spin l do not follow the 1/N (N=l+1, referring to the number of independent two-body matrix elements) relation. The properties of the P(I)’s obtained in boson systems with spin l are discussed.

The modified Hartree-Fock-Bogoliubov (MHFB) theory at finite temperature is derived, which conserves the unitarity relation of the particle-density matrix. This is achieved by constructing a modified-quasiparticle-density matrix, where the fluctuation of the quasiparticle number is microscopically built in. This matrix can be directly obtained from the usual quasiparticle-density matrix by applying the secondary Bogoliubov transformation, which includes the quasiparticle-occupation number. It is shown that, in the limit of constant pairing parameter, the MHFB theory yields the previously obtained modified BCS (MBCS) equations. It is also proved that the modified quasiparticle-random-phase approximation, which is based on the MBCS quasiparticle excitations, conserves the Ikeda sum rule. The numerical calculations of the pairing gap, heat capacity, level density, and level-density parameter within the MBCS theory are carried out for ¹²⁰Sn. The results show that the superfluid-normal phase transition is completely washed out. The applicability of the MBCS up to a temperature as high as T∼5 MeV is analyzed in detail.

In this Brief Report we present the systematics of excitation energies for even-even nuclei in two regions: the 50<Z<~66, 82<N<~104 region, and the 66<Z<82, 82<N<~104 region. Using the N_pN_n scheme, we obtain compact trajectories for the ground band as well as quasi-β and quasi-γ bands. This suggests that the N_pN_n scheme is useful even if one extends it to nonyrast levels, and thus can serve as a general tool to disclose new types of structural evolution for higher excitations, besides the yrast states which have been investigated extensively. It is highlighted that deformations in nonyrast quasibands of nuclei with Z∼80 and N∼104 are often very different from those in the ground bands.

We respond to the Comment by Ponomarev. We reconfirm the correctness of our results and conclusions published in Phys. Rev. C 63, 044302 (2001). In particular, we reiterate that the aim of our previous study is to use for the calculations of E1 resonances in neutron-rich isotopes the same set of parameters whose values are chosen to reproduce the giant dipole resonance in the corresponding double closed-shell nuclei.

The contributions of quasiparticle correlations and of continuum coupling upon the superfluid properties of neutron-rich Ni isotopes are studied within the modified BCS (MBCS) approximation at finite temperature. The effect of quasiparticle correlations is included using a secondary Bogoliubov-type canonical transformation explicitly involving the quasiparticle occupation numbers at temperature T. The effect of continuum coupling is taken into account via the finite widths of the resonant states. It is shown that the combined effect of thermal quasiparticle correlations and of continuum coupling washes out the sharp superfluid-normal phase transition given by the standard finite-temperature BCS calculations. Within the proposed resonant-continuum MBCS approximation the fluctuations of particle number also become more suppressed especially at high temperature for nuclei closer to the drip line. Finally, it is found within the same approximation that the two-neutron separation energy for ⁸⁴Ni drops to zero at T≃0.8 MeV.

In this paper, we discuss the regularities of average energies with a fixed angular momentum I (denoted as E_I’s) in many-body systems interacting via a two-body random ensemble. It is found that E_I’s with I∼I_min (minimum of I) or I∼I_max (maximum of I) have large probabilities [denoted as P(I)] to be the smallest in energy, and P(I) is close to zero elsewhere. A simple argument assuming the randomness of the two-particle coefficients of fractional parentage is given to explain these observations. A compact trajectory of the energy E_I vs I(I+1) is found to be robust. Other regularities, such that there are two or three sizable P(I)’s with I∼I_min but P(I)≪P(I_max)’s with I∼I_max, and that the coefficients C defined by 〈E_I〉_min=CI(I+1) are sensitive to the orbits and not sensitive to particle number, etc., are discovered and studied for the first time.

In this paper we discuss in detail the P(I)’s, angular momentum I probabilities in the ground states, of many-body systems interacting via a two-body random ensemble (TBRE). In particular, we extensively apply an approach introduced in an earlier paper and compare the predicted P(I)’s with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single-j shell and d boson systems, where elegant agreements between the predicted P(I)’s and those obtained by diagonalizing a TBRE Hamiltonian are achieved. We find that d boson systems systematically present counterexamples of angular momentum 0 ground state dominance when the number of d bosons is 6κ±1 with κ a natural number, which suggests that certain fundamental symmetry (say, time reversal invariance) of the Hamiltonian cannot ensure the occurrence of angular momentum 0 ground state dominance. Next, we apply the same approach to more complicated cases, such as even or odd number of fermions in a large single-j shell or a many-j shell, sd-boson or sdg-boson systems, etc. We find that the simple approach proposed in an earlier paper is also well applicable, and thus it is a universal approach. The numerical experiments provide a guideline to tell which interactions are essential to produce a sizable P(I) in a many-body system. This disproves a popular idea that the angular momentum 0 ground state (0 g.s.) dominance may be independent of two-body interactions. Some matrix elements which are useful to understand the observed regularities are given or addressed in detail. In this paper we also report a synchronous staggering between the 0 g.s. probabilities of even numbers of fermions in a single-j shells and j g.s. probabilities of odd numbers of fermions in a single-j shell when j is small. The low seniority chain of 0 g.s. using the same set of two-body interactions is confirmed, but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even numbers of fermions in a single-j shell. Some interesting results taking a displaced two-body random ensemble are presented for the I ground state probabilities.

The low-lying collective states involving many nucleons interacting by the two-body random ensemble (TBRE) interactions are investigated in a collective SD-pair subspace, with the collective pairs defined by the two-nucleon system. It is found that in this truncated pair subspace, collective vibrations arise naturally for a general TBRE Hamiltonian whereas collective rotations do not. A Hamiltonian restricted to include only a few randomly generated separable terms is able to produce collective rotational behavior, as long as it includes a reasonably strong quadrupole-quadrupole component. Similar results arise in the full shell model space. These results suggest that the structure of the Hamiltonian is key to producing generic collective rotation.