SEW 39 min KAF37 AWA100/1 我們來形象的理解一下上面變換的意義。如果F(12;34)表示一對(duì)電子-空穴3-4散射成另外一對(duì)電子-空穴1-2,這是一個(gè)水平的散射過程,那么將C2和C4兩個(gè)湮滅算符后,就變成了從3-2散射成1-4,是一個(gè)垂直的散射過程。交換兩個(gè)算符的過程,可以形象的理解為將這兩個(gè)算符的頂角對(duì)應(yīng)的外腳相互交換,這兩個(gè)外線必然要cross,所以我們把這個(gè)對(duì)稱性稱為crossingsymmetry。對(duì)應(yīng)于上述交換C2和C4兩個(gè)湮滅算符,crossing-symmetry將particle-holehorizontalchannel中的頂角函數(shù)變到了particle-holeverticalchannel中。同樣的道理,如果我們交換一個(gè)產(chǎn)生一個(gè)湮滅算符,比如C2和C3,我們也可以把particle-holechannel與particle-particlechannel聯(lián)系起來。總而言之,費(fèi)米子的交換反對(duì)稱性,使得三個(gè)channel中的Bethe-Salpeter方程并不獨(dú)立,他們之間應(yīng)當(dāng)可以相互轉(zhuǎn)換。另外,因?yàn)橥耆斀呛瘮?shù)F是遵從crossingsymmetry的,當(dāng)我們考慮任何一個(gè)Bethe-Salpeter方程,左邊的F在crossing操作下變成了自己(符號(hào)相應(yīng)的改變),等式右邊自然也應(yīng)當(dāng)變成自身。我們以particle-holehorizontalchannel為例,等式右側(cè)的項(xiàng),在crossc_2和c_4后會(huì)從horizontal的連接方式變成了vertical的,而不是變成自身。因此,我們只能要求等式右邊的項(xiàng)Γ中應(yīng)當(dāng)有一項(xiàng),它能夠在crossc_2和c_4后變成等式的項(xiàng)。換句話說,Γ中應(yīng)該包含verticalchannel中等式右邊的項(xiàng),它在cross2和4后,會(huì)給出horizontalchannel等式右邊的項(xiàng)。同樣的道理,Γ中也應(yīng)當(dāng)含有particle-particlechannel中的連接方式,這樣才能保證無論怎樣交換兩個(gè)外腳,都可以保證F能夠變回自身,滿足crossingsymmetry。因此,我們熟悉的Bethe-Salpeter方程中的Γ應(yīng)該表示成如下的樣子:
這里將在每一個(gè)channel中不可約(不可約的意思是說,不能夠通過剪斷任意兩條內(nèi)部的格林函數(shù)線使其分成兩個(gè)獨(dú)立的部分)的Γ進(jìn)一步分解,其中Λ被稱為完全不可約頂角函數(shù),它在三個(gè)channel中都是一樣的,并且在任何一個(gè)channel中都是兩粒子不可約的。作為對(duì)比,這里我們強(qiáng)調(diào)一下,Γ_ph僅在particle-holehorizontalchannel里是不可約的,我們不能再進(jìn)一步通過剪斷兩條單粒子格林函數(shù)線把它分解成horizontal方向上的兩個(gè)獨(dú)立部分了。但是Γ_ph在其他channel就變成可約的了。比如在particle-holeverticalchannel里,我們要剪斷垂直的兩個(gè)格林函數(shù)線。按照上圖,這顯然是可以把Γ_ph等式右側(cè)的張圖分成兩部分。所以我們說,Γ_ph在verticalchannel里可約的。同樣的道理,Γ_ph在paritlce-particlechannel里也是可約的。上圖對(duì)于Γ的分解,構(gòu)成了一組新的方程,我們稱之為parquet方程。這樣的分解完全保證了兩粒子完全頂角函數(shù)F具有crossingsymmetry。這時(shí)候,我們就可以安全的對(duì)Λ做近似,而不會(huì)破壞crossingsymmetry了。這顯然比在Bethe-Salpeter方程中對(duì)三個(gè)Γ做近似要更有優(yōu)越性。對(duì)于如何求解完整的parquet方程,可以參考筆者的程序和方法[1,2]。我們可以把Λ用它的階近似U來替代,叫做parquetapproximation。2007年,K.Held組用動(dòng)力學(xué)平均場(chǎng)方法嚴(yán)格計(jì)算了局域完全不可約頂角函數(shù)Λ,并用動(dòng)力學(xué)平均場(chǎng)lattice格林函數(shù)來替換parquet方程中的格林函數(shù)線。這樣做,完全不可約頂角函數(shù)雖然是局域的,但是通過parquet方程及Bethe-Salpeter方程,我們可以計(jì)算得到非局域的完全頂角函數(shù)F,從而計(jì)算得到非局域的自能。因此,這也構(gòu)成了一種動(dòng)力學(xué)平均場(chǎng)的非局域擴(kuò)展方法,稱之為dynamicalvertexapproximation[3].這里我們不再進(jìn)一步闡述這個(gè)方法了,剩下的大部分都是數(shù)值計(jì)算的細(xì)節(jié)了。通過和DualFermion及non-localexpansion方法對(duì)比,我們可以看到,雖然這些方法的出發(fā)點(diǎn)不同,但是自能的非局域修正均來自于局域兩粒子頂角函數(shù)的非局域連接。任何單粒子層次上的非局域性,都無法進(jìn)入到自能中,都僅僅只是在由impurity格林函數(shù)變換成lattice格林函數(shù)的過程中起作用。自能,作為單粒子中的相互作用部分,其本質(zhì)是來自于兩體相互作用,因此任何非局域的修正來至于兩粒子頂角函數(shù)也是情理之中。基于這樣的思想,也可以通過局域兩粒子頂角函數(shù)的不同連接方式,構(gòu)造出全新的量子多體方法,對(duì)動(dòng)力學(xué)平均場(chǎng)的局域自能進(jìn)行非局域修正。這是一個(gè)嶄新的領(lǐng)域,雖然略有門檻,但是有很多可以嘗試和擴(kuò)展的東西。目前的非局域擴(kuò)展多集中在單軌道體系中,Hubbard相互作用,如何將非局域擴(kuò)展方法更為核心的部分抽出來,簡(jiǎn)化計(jì)算,使得它更好的適用于真實(shí)材料體系的研究,將會(huì)是下一個(gè)研究中心。
SEW 39 min KAF37 AWA100/1 歡迎聯(lián)系小歐,手機(jī)18030229050,我們是做AB,GE,西門子,施耐德,巴赫曼,ABB機(jī)器人,霍尼韋爾,力士樂,安川,庫卡,等品牌
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