【General Seniority】学习笔记(1): BCS

【General Seniority】学习笔记(1): BCS

General Seniority 学习笔记(1): BCS

我学习了参考文献 [1,2]，把里面的核心公式推导核对整理了，因为觉得有点意思。做完笔记我顺手写了个代码，还没来得及核对。

下一步可以考虑投影 broken-pair 的优化和投影，即 seniority 取次极小的内秉态做投影。

参考文献：

[1] 贾力源，"Application of the variational principle to a coherent-pair condensate: The BCS case", PRC99, 014302 (2019).

[2] 俞一信 et al., "Nucleon pair truncation of the shell model for medium-heavy nuclei", to be submitted.

1. 时间反演轨道上的配对凝聚

所有轨道和自己的时间反演轨道，进行两两配对 ((alpha, bar{alpha}))，然后定义时间反演轨道上的配对：

[hat{P}^dagger_alpha = hat{c}^dagger_alpha hat{c}^dagger_bar{alpha}. ]

1.1 配对与配对凝聚

配对定义为

[hat{P}^dagger = sum_{alpha in Theta} nu_alpha hat{P}^dagger_alpha, ]

其中 (Theta) 表示非集体对序号集合，序号个数是单粒子轨道数的一半。单粒子轨道可以是形变基，也可以是球形基，只要两两配对即可。General Seniority 的全部变化，就是这样的配对的凝聚、对破缺。

一般假定基态对应着配对凝聚，

[|phi_N rangle = frac{1}{ sqrt{chi_N} } (hat{P}^dagger)^N | 0 rangle, ]

其中 (chi_N) 是归一化因子，

[chi_N = langle 0 | hat{P}^N (hat{P}^dagger)^N | 0 rangle. ]

Block 掉非集体对 (alpha) 以后，态矢变为

[|phi^{[alpha]}_N rangle = frac{1}{sqrt{chi^{[alpha]}_N}}( hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^N | 0 rangle, ]

其中，归一化因子

[chi^{[alpha]}_N = langle 0 |hat{P}^N hat{P}_alpha hat{P}^dagger_alpha(hat{P}^dagger)^N | 0 rangle, ]

中间加上 (hat{P}_alpha hat{P}^dagger_alpha) 起的作用就是 block 掉非集体对 (alpha)。

1.2 归一化因子的计算

如果已经 block 掉 (gamma_1, gamma_2, cdots, gamma_r)，则有

[chi^{[gamma_1 cdots gamma_r]}_N = N sum_alpha nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1},　\ chi^{[gamma_1 cdots gamma_r]}_N - chi^{[alphagamma_1 cdots gamma_r]}_{N} = N^2 nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}. ]

另外，因为 (chi^{[alphagamma_1 cdots gamma_r]}_{0} = 0)，所以可以构造迭代：

[left{chi^{[alphagamma_1 cdots gamma_r]}_{0} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{1} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{1} right} rightarrow cdots rightarrowleft{chi^{[alphagamma_1 cdots gamma_r]}_{N-1} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{N} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right}. ]

也就是说，给定 (gamma_1, cdots, gamma_r)，任意粒子 (N) 的归一化因子都可以如此迭代计算。因为迭代式非常简单，所以预计是秒出。可以定义矢量

[vec{chi}^{[gamma_1, cdots, gamma_r]}_N equiv left( chi^{[gamma_1 cdots gamma_r]}_{N},left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right} right) ]

那么上面的迭代即

[vec{chi}^{[gamma_1, cdots, gamma_r]}_0 = (1,1,cdots, 1), \ vec{chi}^{[gamma_1, cdots, gamma_r]}_{N-1} rightarrow vec{chi}^{[gamma_1, cdots, gamma_r]}_N. ]

test case: (nu_i = 1, i in Theta)

此时 (chi_N) 很容易解析地得到：

[chi_N = C^N_Omega (N!)^2 = frac{Omega!N!}{(Omega-N)!}. ]

2. 矩阵元的计算

通过 many-pair density matrix，可以计算由对算符构造的任意算符矩阵元。

体系哈密顿量为

[hat{H} = sum_{alpha beta} epsilon_{alpha beta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_gamma hat{c}_delta, ]

其中 (V_{alpha betagammadelta} = - langle alpha beta | hat{V} | gamma delta rangle)，有对称性 (epsilon_{alphabeta} = epsilon_{beta alpha}, V_{alphabetagammadelta} = - V_{betaalphagammadelta} = V_{gammadeltaalphabeta})。贾师兄在这里假定了 (epsilon_{alphabeta} = epsilon_{bar{beta}bar{alpha}}, V_{alphabetagammadelta} = V(bar{delta}bar{gamma}bar{beta}bar{alpha}))，据说这是假定 time-even 哈密顿量。他还假定 (epsilon_{alphabeta}) 和 (V_{alphabetagammadelta}) 是实数。

2.1 many-pair density matrix

若给定 (gamma_1, cdots, gamma_r)，要求 block 掉 ({ hat{P}^dagger_{gamma_i}}) ，然后在这个截断空间里计算 ((hat{P}^dagger)^{M-p} hat{P}^dagger_{alpha_1} cdots hat{P}^dagger_{alpha_p} |0rangle) 与 ((hat{P}^dagger)^{M-q} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} |0rangle) 的 overlap，

[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} equiv langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} (hat{P}^dagger)^{M-q} |0rangle, ]

这里我们约定 ({alpha_i}, {beta_j})没有共同元素，否则可以移到 ({gamma_i})里面去。这样的话，这么写也可以

[t^{[gamma_1 gamma_2 cdots gamma_r], M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = langle 0 | hat{P}^{M-p} hat{P}_{gamma_1} cdots hat{P}_{gamma_r} hat{P}^dagger_{beta_1} cdots hat{P}^dagger_{beta_q} hat{P}_{alpha_1}cdots hat{P}_{alpha_p} (hat{P}^dagger)^{M-q} | 0 rangle, ]

所以可以叫做 many-pair density matrix。容易看出，它可以表示成 (chi) 的表达式：

[t^{[gamma_1, cdots, gamma_r],M}_{alpha_1, cdots, alpha_p; beta_1, cdots, beta_q} = nu_{alpha_1} cdots nu_{alpha_p} nu_{beta_1} cdots nu_{beta_q} frac{(M-p)!(M-q)!}{(M-p-q)!(M-p-q)!} chi^{[alpha_1 cdots alpha_p beta_1 cdots beta_q gamma_1 cdots gamma_r]}_{M-p-q}. ]

2.2 单体算符矩阵元

这个比较简单，

[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}_beta (hat{P}^dagger)^N | 0 rangle = delta_{alphabeta} chi_N langle phi_N | hat{n}_alpha | phi_N rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}. ]

2.3 两体算符矩阵元

只有以下 3 种情况，两体矩阵元不为零：

[langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha (hat{P}^dagger)^N |0rangle = (Nnu_alpha)^2 chi^{[alpha]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta (hat{P}^dagger)^N | 0 rangle = N^2 nu_alpha nu_beta chi^{[alphabeta]}_{N-1}, \ langle 0 | hat{P}^N hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha (hat{P}^dagger)^N | 0 rangle = N^2(N-1)^2 nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2}. ]

2.4 同类核子哈密顿量期望值

哈密顿量定义为

[hat{H} = sum_{alphabeta} epsilon_{alphabeta} hat{c}^dagger_alpha hat{c}_beta + frac{1}{4} sum_{alphabetagammadelta} V_{alphabetagammadelta} hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_delta hat{c}_gamma. ]

[langle phi_N | hat{H} | phi_N rangle = sum_{alpha in Theta} 2 epsilon_{alpha alpha} langle phi_N | hat{c}^dagger_alpha hat{c}_alpha | phi_N rangle + sum_{alpha in Theta } V_{alpha bar{alpha} alpha bar{alpha} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{alpha} hat{c}_alpha | phi_N rangle \ + sum^{alpha neq beta}_{alpha, beta in Theta} V_{alpha bar{alpha} beta bar{beta} } langle phi_N | hat{c}^dagger_alpha hat{c}^dagger_bar{alpha} hat{c}_bar{beta} hat{c}_beta | phi_N rangle + sum^{alpha neq beta}_{alpha, beta in Theta} (2V_{alpha beta alpha beta } + 2V_{alpha bar{beta} alpha bar{beta} } ) langle phi_N |hat{c}^dagger_alpha hat{c}^dagger_beta hat{c}_beta hat{c}_alpha | phi_N rangle. ]

将上一小节的结果代入上式，若记 (G_{alphabeta} = V_{alphabar{alpha}betabar{beta}}, Lambda_{alphabeta} = V_{alphabetaalphabeta} + V_{alpha bar{beta} alpha bar{beta} })，得到

[bar{E} equiv langle phi_N | hat{H} | phi_N rangle = frac{N^2}{chi_N} left( sum_{alpha in Theta} (2epsilon_{alphaalpha} + G_{alphaalpha}) nu^2_alpha chi^{[alpha]}_{N-1} + sum^{alpha < beta}_{alpha, beta in Theta} 2G_{alpha beta} nu_alpha nu_beta chi^{[alphabeta]}_{N-1} + (N-1)^2 sum^{alpha < beta}_{alpha, beta in Theta} 2 Lambda_{alphabeta} nu^2_alpha nu^2_beta chi^{[alphabeta]}_{N-2} right). ]

这个与贾力源的公式一致，只是我的后两个求和符号上有 (alpha < beta)，导致相应的有个 2 倍。最后一项有点 tricky，本就只需考虑 (sum^{alpha<beta}_{alpha,beta in Theta}) 的情况，(V_{alphabetabetaalpha}, V_{alpha bar{beta} alpha bar{beta} }, V_{bar{alpha} bar{beta} bar{alpha} bar{beta} }, V_{bar{alpha} beta bar{alpha} beta })，这四项+时间反演对称，得到 (2Lambda_{alphabeta})。

感觉可以设置 (nu_i = 1)，来做简单测试。

2.5 同类核子哈密顿量期望值的偏导数

先算 overlap：

[delta chi_N = 2N langle 0 | hat{P}^N hat{P}^dagger_alpha (hat{P}^dagger)^{N-1} |0rangle delta nu_alpha = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1} delta nu_alpha = frac{ 2chi_N}{ nu_alpha} langle phi_N | hat{n}_alpha |phi_Nrangledelta nu_alpha. ]

这可以推广： $ alpha notin { gamma_1, cdots , gamma_r }$ 时，

[delta chi^{[gamma_1 cdots gamma_r]}_N = 2 N^2 nu_{alpha}chi^{[alpha gamma_1 cdots gamma_r]}_{N-1} delta nu_alpha. ]

Anyway, (chi_N) 的偏导数有了

[frac{ partial }{partial nu_alpha} chi_N = 2 N^2 nu_{alpha}chi^{[alpha]}_{N-1}. ]

考虑哈密顿量矩阵元的偏导数，

[delta langle hat{P}^N | hat{H} |(hat{P}^dagger)^N rangle = 2N langle hat{P}^{N-1} hat{P}_alpha | hat{H} | (hat{P}^dagger)^N rangle delta nu_alpha， ]

下面计算 (langle hat{P}^{N-1} hat{P}_alpha | hat{H} |(hat{P}^dagger)^N rangle) ，可以如下从容计算。

• 先考虑 (hat{H}) 中不含 (alpha) 的项的贡献，得到

[N nu_alpha langle (hat{P} - nu_alpha hat{P}_alpha)^{N-1} | hat{H} | (hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^{N-1} rangle = N nu_alpha langle phi^{[alpha]}_{N-1} | hat{H} | phi^{[alpha]}_{N-1} rangle chi^{[alpha]}_{N-1}. ]

• 再分别考虑 (hat{H}) 中包含 (alpha) 的各项贡献。

  • 源于 (hat{c}^dagger_alpha hat{c}_alpha, hat{c}^dagger_bar{alpha} hat{c}_bar{alpha}, hat{P}^dagger_alpha hat{P}_alpha)形式的相互作用，与(2epsilon_{alpha alpha} + G_{alpha alpha}) 相关的贡献（对单粒子能）为

    [(2epsilon_{alpha alpha} + G_{alphaalpha}) N nu_alpha chi^{[alpha]}_{N-1}; ]

    General Seniority 学习笔记(1): BCS

    我学习了参考文献 [1,2]，把里面的核心公式推导核对整理了，因为觉得有点意思。做完笔记我顺手写了个代码，还没来得及核对。

    下一步可以考虑投影 broken-pair 的优化和投影，即 seniority 取次极小的内秉态做投影。

    参考文献：

    [1] 贾力源，"Application of the variational principle to a coherent-pair condensate: The BCS case", PRC99, 014302 (2019).

    [2] 俞一信 et al., "Nucleon pair truncation of the shell model for medium-heavy nuclei", to be submitted.

    1. 时间反演轨道上的配对凝聚

    所有轨道和自己的时间反演轨道，进行两两配对 ((alpha, bar{alpha}))，然后定义时间反演轨道上的配对：

    [hat{P}^dagger_alpha = hat{c}^dagger_alpha hat{c}^dagger_bar{alpha}. ]

    1.1 配对与配对凝聚

    配对定义为

    [hat{P}^dagger = sum_{alpha in Theta} nu_alpha hat{P}^dagger_alpha, ]

    其中 (Theta) 表示非集体对序号集合，序号个数是单粒子轨道数的一半。单粒子轨道可以是形变基，也可以是球形基，只要两两配对即可。General Seniority 的全部变化，就是这样的配对的凝聚、对破缺。

    一般假定基态对应着配对凝聚，

    [|phi_N rangle = frac{1}{ sqrt{chi_N} } (hat{P}^dagger)^N | 0 rangle, ]

    其中 (chi_N) 是归一化因子，

    [chi_N = langle 0 | hat{P}^N (hat{P}^dagger)^N | 0 rangle. ]

    Block 掉非集体对 (alpha) 以后，态矢变为

    [|phi^{[alpha]}_N rangle = frac{1}{sqrt{chi^{[alpha]}_N}}( hat{P}^dagger - nu_alpha hat{P}^dagger_alpha)^N | 0 rangle, ]

    其中，归一化因子

    [chi^{[alpha]}_N = langle 0 |hat{P}^N hat{P}_alpha hat{P}^dagger_alpha(hat{P}^dagger)^N | 0 rangle, ]

    中间加上 (hat{P}_alpha hat{P}^dagger_alpha) 起的作用就是 block 掉非集体对 (alpha)。

    1.2 归一化因子的计算

    如果已经 block 掉 (gamma_1, gamma_2, cdots, gamma_r)，则有

    [chi^{[gamma_1 cdots gamma_r]}_N = N sum_alpha nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1},　\ chi^{[gamma_1 cdots gamma_r]}_N - chi^{[alphagamma_1 cdots gamma_r]}_{N} = N^2 nu^2_alpha chi^{[alphagamma_1 cdots gamma_r]}_{N-1}. ]

    另外，因为 (chi^{[alphagamma_1 cdots gamma_r]}_{0} = 0)，所以可以构造迭代：

    [left{chi^{[alphagamma_1 cdots gamma_r]}_{0} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{1} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{1} right} rightarrow cdots rightarrowleft{chi^{[alphagamma_1 cdots gamma_r]}_{N-1} right} rightarrow chi^{[gamma_1 cdots gamma_r]}_{N} rightarrow left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right}. ]

    也就是说，给定 (gamma_1, cdots, gamma_r)，任意粒子 (N) 的归一化因子都可以如此迭代计算。因为迭代式非常简单，所以预计是秒出。可以定义矢量

    [vec{chi}^{[gamma_1, cdots, gamma_r]}_N equiv left( chi^{[gamma_1 cdots gamma_r]}_{N},left{chi^{[alphagamma_1 cdots gamma_r]}_{N} right} right) ]

    那么上面的迭代即

    [vec{chi}^{[gamma_1, cdots, gamma_r]}_0 = (1,1,cdots, 1), \ vec{chi}^{[gamma_1, cdots, gamma_r]}_{N-1} rightarrow vec{chi}^{[gamma_1, cdots, gamma_r]}_N. ]

    test case: (nu_i = 1, i in Theta)

    此时 (chi_N) 很容易解析地得到：

    [chi_N = C^N_Omega (N!)^2 = frac{Omega!N!}{(Omega-N)!}. ]

    2. 矩阵元的计算

    通过 many-pair density matrix，可以计算由对算符构造的任意算符矩阵元。

    体系哈密顿量为

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