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Index and binary values of all integers and the corresponding wave function |n 1 , n 2 , n 3 in the space of 3 pairs in 3 degenerate shells of ω 1,2,3 = {4, 2, 1}. The decimal points in the binary values are just for separating different orbitals. The decimal values shown displays the ascending order.
Source publication
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 diagona...
Contexts in source publication
Context 1
... the maximum. With five iterations, all six integers obtained are summarized in Table 1, in which the indexes are assigned in the order of generation. In PairDiagSph program, a 64-bit integer is used to represent a basis vector and all the generated integers are stored sequentially in an 1D array. ...Context 2
... all basis vectors in a p-pairs system with given degeneracy, We define N p d as the minimum number of iterations required to generate a binary-based vector with the d-th ( j ω j ≥ d ≥ p) digit occupied from the minimum vector. For the first 4 items in Table 1 For an arbitrary vector |i in a system, we can define for the i-th shell the p i = i j n j and d i = n i + i−1 j ω j , then the hash function i = f (|i) for the search can be written as In general, the hash search for a n-pairs system requires all possible coefficients N p d with p ≤ n and d ≤ j ω j , and there is no simple formula to calculate them directly. One feasible way to get these coefficients is to solve them backwards in the linear equations of all the hash functions in Eq. (10) with known indexes. ...Context 3
... first equation with index 1 contains no coefficient and each subsequent equation will introduce at most one unknown new coefficient, this means these linear equations can be easily solved in order from the second one till the last, and it is also undoubtedly correct when we use these coefficients to calculate the indexes back. Still taking the vectors in Table 1 as an example, each vector with its index corresponds to a linear hash equation, and all these equations listed in Table 2 can be solved easily in order. In PairDiagSph program, all the required coefficients are calculated during the generation of the basis vectors, and then these results are stored in a 2D array which will be used as a table in the later hash search. ...Context 4
... the different P and V in |i, the maximum number of such | j and also the non-zero H i, j is m(m−1). Still using the previous example in Table 1 with assigning single-particle energies ǫ 1,2,3 = {1, 2, 3} and the constant G i, j = −0.2 as the overall pairing interaction strength. The Hamiltonian can be expressed as a 6 × 6 real symmetric matrix. ...Context 5
... the maximum. With five iterations, all six integers obtained are summarized in Table 1, in which the indexes are assigned in the order of generation. In PairDiagSph program, a 64-bit integer is used to represent a basis vector and all the generated integers are stored sequentially in an 1D array. ...Context 6
... all basis vectors in a p-pairs system with given degeneracy, We define N p d as the minimum number of iterations required to generate a binary-based vector with the d-th ( j ω j ≥ d ≥ p) digit occupied from the minimum vector. For the first 4 items in Table 1, we can get in that 3-pairs system N 3 3 = 0, N 3 5 = 1, N 3 6 = 2, and N 3 7 = 3. Except for N p p = 0, the value of N p d with d > p is degeneracy dependent. ...Context 7
... first equation with index 1 contains no coefficient and each subsequent equation will introduce at most one unknown new coefficient, this means these linear equations can be easily solved in order from the second one till the last, and it is also undoubtedly correct when we use these coefficients to calculate the indexes back. Still taking the vectors in Table 1 as an example, each vector with its index corresponds to a linear hash equation, and all these equations listed in Table 2 can be solved easily in order. In PairDiagSph program, all the required coefficients are calculated during the generation of the basis vectors, and then these results are stored in a 2D array which will be used as a table in the later hash search. ...Context 8
... the different P and V in |i, the maximum number of such | j and also the non-zero H i, j is m(m−1). Still using the previous example in Table 1 with assigning single-particle energies 1,2,3 = {1, 2, 3} and the constant G i, j = −0.2 as the overall pairing interaction strength. The Hamiltonian can be expressed as a 6 × 6 real symmetric matrix. ...Context 9
... the maximum. With five iterations, all six integers obtained are summarized in Table 1, in which the indexes are assigned in the order of generation. In PairDiagSph program, a 64-bit integer is used to represent a basis vector and all the generated integers are stored sequentially in an 1D array. ...Context 10
... all basis vectors in a p-pairs system with given degeneracy, We define N p d as the minimum number of iterations required to generate a binary-based vector with the d-th ( j ω j ≥ d ≥ p) digit occupied from the minimum vector. For the first 4 items in Table 1 For an arbitrary vector |i in a system, we can define for the i-th shell the p i = i j n j and d i = n i + i−1 j ω j , then the hash function i = f (|i) for the search can be written as In general, the hash search for a n-pairs system requires all possible coefficients N p d with p ≤ n and d ≤ j ω j , and there is no simple formula to calculate them directly. One feasible way to get these coefficients is to solve them backwards in the linear equations of all the hash functions in Eq. (10) with known indexes. ...Context 11
... first equation with index 1 contains no coefficient and each subsequent equation will introduce at most one unknown new coefficient, this means these linear equations can be easily solved in order from the second one till the last, and it is also undoubtedly correct when we use these coefficients to calculate the indexes back. Still taking the vectors in Table 1 as an example, each vector with its index corresponds to a linear hash equation, and all these equations listed in Table 2 can be solved easily in order. In PairDiagSph program, all the required coefficients are calculated during the generation of the basis vectors, and then these results are stored in a 2D array which will be used as a table in the later hash search. ...Context 12
... the different P and V in |i, the maximum number of such | j and also the non-zero H i, j is m(m−1). Still using the previous example in Table 1 with assigning single-particle energies ǫ 1,2,3 = {1, 2, 3} and the constant G i, j = −0.2 as the overall pairing interaction strength. The Hamiltonian can be expressed as a 6 × 6 real symmetric matrix. ...