Mean-Field Theory: HF and BCS

Mean-Field Theory: HF and BCS Erik Koch Institute for Advanced Simulation, Jülich N (x) = p N! ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )........ ' (x N ) ' 2 (x N ) ' N (x N ) Slater determinant c 0i =0 c,c =0= c,c h0 0i = c,c = h i N i = c N c 2 c 0i product state in Fock space 2.5 HF nonint HF nonint 2.5 =0 =0. =0 =0. ε k - ε kf 0.5 ε k - ε kf 0.5 0 0-0.5-0.5-0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states - 0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states

Standard Model: Elementary Particles en.wikipedia.org/wiki/standard_model

indistinguishable particles notion of elementary particle change over time/length/energy-scale https://community.emc.com/people/ble/blog/20/

what does indistinguishable mean? observable for N indistinguishable particles M(x) =M 0 + X i M (x i )+ 2! X M 2 (x i,x j )+ 3! X M 3 (x i,x j,x k )+ = M 0 + X i i6=j M (x i )+ X i<j M 2 (x i,x j )+ i6=j6=k X M 3 (x i,x j,x k )+ i<j<k operators must be symmetric in particle coordinates, if not they could be used to distinguish particles...

indistinguishability and statistics N-particle systems described by wave-function with N particle degrees of freedom (tensor space): Ψ(x,..., xn) introduces labeling of particles indistinguishable particles: no observable exists to distinguish them in particular no observable can depend on labeling of particles probability density is an observable consider permutations P of particle labels P (x,x 2 )= (x 2,x ) with (x,x 2 ) 2 = (x 2,x ) 2 P (x,x 2 )=e i (x,x 2 ) when P 2 = Id e iɸ = ± (Ψ (anti)symmetric under permutation) antisymmetric: Ψ(x, x2 x) = 0 (Pauli principle)

Do we need the wave-function? we use the wave-function as a tool for calculating observables M = = N expectation value dx dx N (x,...,x N ) X M (x i ) i (x,...,x N ) dx M (x ) dx 2 dx N (x,...,x N ) (x,...,x N ) {z } = () (x ) M = = N dx dx N dx for non-local operators, e.g. M(x) = ½ lim x 0!x (x,...,x N ) X M (x i ) (x,...,x N ) i M (x ) dx 2 dx N (x 0,...,x N) (x,...,x N ) {z } = () (x 0 ;x )

reduced density matrices (p) (x 0,...,x 0 p; x,...,x p )= p-body density matrix of N-electron state for evaluation of expectation values of Mp N p dx p+ dx N (x 0,...,x0 p,x p+,...,x N ) (x,...,x p,x p+,...,x N ) Hermitean (x x) and antisymmetric under permutations of the xi (or xi ) N normalization sum-rule dx dx (p) p (x,...,x p ; x,...,x p )= p allows evaluation of expectation values of observables Mq with q p: recursion relation (p) (x,...,x 0 p; 0 x,...,x p )= p + dx (p+) p+ (x 0 N p,...,xp,x 0 p+ ; x,...,x p,x p+ )

Coulson s challenge * X + external potential hv i = V (r i ) = dx V (r) () (x; x) kinetic energy ht i = * i 2 X i r i + = 2 dx r () (x 0 ; x) x 0 =x Coulomb repulsion hui = * X i<j r i r j + = dx dx 0 (2) (x,x 0 ; x,x 0 ) r r 0 minimize Etot = T + V + U as a function of the 2-body density matrix Γ (2) (x,x2 ; x,x2) instead of the N-electron wave-function Ψ(x,..., xn) representability problem: what function Γ(x,x2 ; x,x2) is a fermionic 2-body density-matrix?

antisymmetric wave-functions (anti)symmetrization of N-body wave-function: N! operations S ± (x,...,x N ):= p X (±) P x p(),...,x p(n) N! computationally hard! antisymmetrization of products of single-particle states P S ' (x ) ' N (x N )= p N! ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )....... ' (x N ) ' 2 (x N ) ' N (x N ) much more efficient: scales only polynomially in N Slater determinant: N (x,...,x N )

Slater determinants N (x) = p N! simple examples ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )...... ' (x N ) ' 2 (x N ) ' N (x N ) N=: (x )=' (x ) N=2: 2 (x) = p ' (x )' 2 (x 2 ) ' 2 (x )' (x 2 ) 2 expectation values need only one antisymmetrized wave-function: dx (S ± a (x)) M(x) (S ± b (x)) = corollary: overlap of Slater determinants: dx dx N N (x,...,x N ) N (x,...,x N )=det p dx N! a (x) M(x) (S ± b (x)) remember: M(x,..., xn) symmetric in arguments h' n ' m i

reduced density-matrices: p= N (x,...,x N )= p N Laplace expansion N X n= ( ) +n ' n (x ) i6=n (x 2,...,x N ) () (x 0 ; x) = N X n,m ( ) n+m ' n (x 0 ) ' m (x) det(h' j6=n ' k6=m i) det(h' j ' k i) () (x 0 ; x) = X n for orthonormal orbitals ' n (x 0 ) ' n (x) and n(x) = X n ' n (x) 2

reduced density-matrices expansion of determinant in product of determinants N (x) = q X ( ) +P i n i n (x n p,...,x p ) (x i62{n N,...,np } p+,...,x N ) p n <n 2 < <n p p-electron Slater det (N p)-electron Slater det express p-body density matrix in terms of p-electron Slater determinants: () (x 0 ; x) = X n (2) (x 0 x 0 2; x,x 2 )= X n<m ' n (x 0 ) ' n (x) and n(x) = X n n, m (x 0,x0 2 ) n, m (x,x 2 ) ' n (x) 2 and n(x,x 2 )= X n,m n, m (x,x 2 ) 2 in particular n(x,x 2 )= X n,m = X n,m 2 p ' n (x )' m (x 2 ) ' m (x 2 )' n (x ) 2 ' n (x ) 2 ' m (x 2 ) 2 ' n (x )' m (x ) ' m (x 2 )' n (x 2 )

Slater determinants Hartree-Fock method: know how to represent 2-body density matrix derived from Slater determinant (2) (xx 0 2; 0 x,x 2 )= X n, m (x 0,x0 2 ) n, m (x,x 2 ) n<m minimize (á la Coulson) could generalize reduced density matrices by introducing density matrices for expectation values between different Slater determinants see e.g. Per-Olov Löwdin, Phys. Rev. 97, 474 (955) still, always have to deal with determinants and signs. there must be a better way...

second quantization: motivation keeping track of all these signs... Slater determinant (x,x 2 )= p 2 (' (x )' (x 2 ) ' (x )' (x 2 )) corresponding Dirac state, i = p 2 ( i i i i) use operators, i = c c 0i position of operators encodes signs c c 0i =, i =, i = c c 0i product of operators changes sign when commuted: anti-commutation anti-commutator {A, B} := AB+ BA

second quantization: motivation specify N-electron states using operators N=0: 0i (vacuum state) normalization: h0 0i = N=: i = c 0i (creation operator adds one electron) normalization: overlap: h i = h0 c c 0i h i = h0 c c 0i N=2:, i = c c 0i adjoint of creation operator removes one electron: annihilation operator c 0i =0andc c = ±c c + h i antisymmetry: c c = c c

second quantization: formalism vacuum state 0 and set of operators cα related to single-electron states φα(x) defined by: c 0i =0 c,c =0= c,c h0 0i = c,c = h i creators/annihilators operate in Fock space transform like orbitals!

second quantization: field operators creation/annihilation operators in real-space basis ˆ (x) with x =(r, ) then c = creates electron of spin σ at position r dx ' (x) ˆ (x) {' n (x)} complete orthonormal set put electron at x with amplitude φa(x) X j ' j (x) ' j (x 0 )= (x x 0 ) ˆ(x) = X n ' n (x) c n they fulfill the standard anti-commutation relations ˆ(x), ˆ(x 0 ) =0= ˆ (x), ˆ (x 0 ) ˆ(x), ˆ (x 0 ) = (x x 0 )

second quantization: Slater determinants 2... N (x,x 2,...,x N )= p N! 0 ˆ(x ) ˆ(x 2 )... ˆ(x N ) c N...c 2 c 0 proof by induction N=: 0 ˆ(x ) c 0 = 0 ' (x ) c ˆ(x ) 0 = ' (x ) using { ˆ(x),c } = R dx 0 ' (x 0 ) ˆ(x), ˆ (x 0 ) = ' (x) N=2: 0 ˆ(x ) ˆ(x 2 ) c 2 c 0 = 0 ˆ(x ) ' 2 (x 2 ) c 2 ˆ(x 2 ) c 0 = 0 ˆ(x )c 0 ' 2 (x 2 ) 0 ˆ(x )c 2 ˆ(x 2 )c 0 = ' (x )' 2 (x 2 ) ' 2 (x )' (x 2 )

second quantization: Slater determinants general N: commute Ψ(xN) to the right D E 0 ˆ(x )... ˆ(x N ) ˆ(x N ) c N c N...c 0 = +. D E 0 ˆ(x )... ˆ(x N ) c N...c 0 D 0 ˆ(x )... ˆ(x N ) Q E n6=n c n 0 E ( ) D0 N ˆ(x )... ˆ(x N ) c N... c 2 0 ' N (x N ) ' N (x N ) ' (x N ) Laplace expansion in terms of N- dim determinants wrt last line of = ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )........ ' (x N ) ' 2 (x N ) ' N (x N )

second quantization: Dirac notation product state c N c 2 c 0i corresponds to Slater determinant 2... N (x,x 2,...,x N ) as Dirac state i corresponds to wave-function ' (x)

second quantization: expectation values expectation value of N-body operator wrt N-electron Slater determinants dx dx N (x N,,x N )M(x,,x N ) N (x,,x N ) D E = 0 c c N ˆMc N c 0 E dx dx N p D0 c N! c ˆ (x N N ) ˆ (x ) 0 = 0 c c N N! M(x,,x N ) p 0 ˆ(x ) ˆ(x N ) c N! N c 0 dx dx N ˆ (x N ) ˆ (x )M(x,,x N ) ˆ(x ) ˆ(x N ) c N c 0 0ih0 = on 0-electron space collecting field-operators to obtain M in second quantization: ˆM = dx x N ˆ (x N ) ˆ (x ) M(x,,x N ) ˆ(x ) ˆ(x N ) N! apparently dependent on number N of electrons!

second quantization: zero-body operator zero-body operator M0(x,...xN)= independent of particle coordinates second quantized form for operating on N-electron states: ˆM 0 = dx dx 2 x N ˆ (x N ) ˆ (x 2 ) ˆ (x ) ˆ(x ) ˆ(x 2 ) ˆ(x N ) N! = dx 2 x N ˆ (x N ) ˆ (x 2 ) ˆN ˆ(x 2 ) ˆ(x N ) N! = dx 2 x N ˆ (x N ) ˆ (x 2 ) ˆ(x 2 ) ˆ(x N ) N! only(!) when operating on N-electron state. = N! 2 N = using dx ˆ (x) ˆ(x) = ˆN result independent of N

second quantization: one-body operators one-body operator M(x,...,x N )= P j M (x j ) ˆM = N! dx dx N ˆ (x N ) ˆ (x ) X j M (x j ) ˆ(x ) ˆ(x N ) = X N! j = X N j = dx j ˆ (x j ) M (x j )(N )! ˆ(x j ) dx j ˆ (x j ) M (x j ) ˆ(x j ) dx ˆ (x) M (x) ˆ(x) result independent of N expand in complete orthonormal set of orbitals ˆM = X n,m dx ' n (x) M(x) ' m (x) c n c m = X n,mh n M m i c n c m transforms as -body operator

second quantization: two-body operators two-body operator M(x,...,x N )= P i<j M 2(x i,x j ) ˆM 2 = dx dx N ˆ (x N ) ˆ (x ) X M 2 (x i,x j ) ˆ(x ) ˆ(x N ) N! i<j = N! = = X i<j N(N ) dx i dx j ˆ (x j ) ˆ (x i ) M 2 (x i,x j )(N 2)! ˆ(x i ) ˆ(x j ) X i<j 2 dx i dx j ˆ (x j ) ˆ (x i ) M 2 (x i,x j ) ˆ(x i ) ˆ(x j ) dx dx 0 ˆ (x 0 ) ˆ (x) M 2 (x,x 0 ) ˆ(x) ˆ(x 0 ) result independent of N expand in complete orthonormal set of orbitals ˆM 2 = X dxdx 0 ' n 0 (x 2 0 )' n (x) M 2 (x,x 0 ) ' m (x)' m 0 (x 0 ) c c n 0 n c m c m 0 n,n 0,m,m 0 = X h n n 0 M 2 m m 0i c 2 c n 0 n c m c m 0 n,n 0,m,m 0

expectation values h M (2) i = h M () i = X n,m X n 0 >n, m 0 >m M () nm h c nc m i {z } =: () nm =Tr M (2) nn 0,mm 0 h c n 0 c nc m c m 0 i {z } =: (2) nn 0,mm 0 () M () =Tr (2) M (2) i = c N c 0i Slater determinant P = X n n ih n (p) n n p,m m p -DM: projector on occupied subspace () nm = h' m P ' n i = h c n p c n c m c mp i =det 0 B @ () () n m p n m..... () n p m () n p m p C A

many-body problem H i = E i introduce (orthonormal) orbital basis induces an orthonormal basis in N-electron space { ' n i n =,...,K} n n N i n < <n N i = X CI expansion n < <n N a n,...,n N n,...,n N i = X n i a ni n i i matrix eigenvalue problem 0 0 0 h n H n ih n H n 2 i a n B @ h n2 H n ih n2 H n 2 i CB A@ a n2 C B A = E @............ a n a n2... C A

dimension of Hilbert space number of ways to pick N different indices out of K K (K ) (K 2) (K (N )) pick one ordering of the set of indices: /N! dim H (N) K = K! N!(K N)! = K N > bc bc.06 Copyright 99-994, 997, 998, 2000 Free Software Foundation, Inc. This is free software with ABSOLUTELY NO WARRANTY. For details type `warranty'. define f(n) { if (n==0) return else return n*f(n-) } define b(k,n) { return f(k)/f(n)/f(k-n) } b(00,25) 24259269720337205504 b(00,25)*8/024/024/024 # memory in GB 806909365358480

non-interacting electrons Ĥ = X n,m H nm c nc m apply to single Slater determinant: linear combination of single-excitations choose orbitals that diagonalize single-electron matrix H Ĥ = X n,m " n n,m c nc m = X n " n c nc n N-electron eigenstates ni = c n N c n 0i X " n c nc n c n N c n 0i = NX " ni! c n N c n 0i n i=

Hartree-Fock variational principle on manifold of Slater determinants E HF = min h Ĥ i h i unitary transformations among Slater determinants Û( )=e i ˆM with ˆM = X, M c c hermitian energy expectation value E( )=h e i ˆM Ĥe i ˆM i = h Ĥ i + i h [Ĥ, ˆM] i + (i )2 2 h [Ĥ, ˆM], ˆM i + variational equation h HF [Ĥ, ˆM] HF i =0

unitary transformations on Slater manifold Û( )=e i ˆM with ˆM = X, M c c hermitian e i ˆM c N c 0i = e i ˆM c N e i ˆM {z } P (e i M ) N c e i ˆM e i ˆM e i ˆM c e i ˆM e i ˆM 0i {z } = 0i d d =0 d 2 d 2 =0 d n d n =0 e i ˆM c e i ˆM = e i ˆM i [ ˆM, c ] e i ˆM e i ˆM c e i ˆM = d d =0. e i ˆM i X 0 c 0 M 0! =0 e i ˆM = i 2 X e i ˆM c e i ˆM = i n X c = i X X c M 0 {z } M 0M 0 (M 2 ) c (M n ) [c c,c ]=c {c,c } {c,c }c = c,

HF variational condition h HF [Ĥ, ˆM] HF i =0 h HF [Ĥ, c nc m + c mc n ] HF i =0 8 n m orthonormal basis HF i = c N c 0i c nc m HF i = if n, m 2 {,...,N} 0 if m /2 {,...,N} n,m HF i simplifies variational condition to (Brillouin theorem) h HF c mc n Ĥ HF i =0 8 m 2 {,...,N}, n/2 {,...,N} applying Hamiltonian does not generate singly excited determinants

analogy to non-interacting problem Ĥ = X n,m c n T nm c m + X n>n 0,m>m 0 c nc n 0 U nn0,mm 0 U nn0,m 0 m c m 0c m T nm + X m 0 applen U nm0,mm 0 Brillouin condition U nm0,m 0 m c nc m HF i =0 8n >N m {z } =:F nm same condition as for non-interacting Hamiltonian Fnm (Fock-matrix) " HF m = T mm + X m 0 applen depends on Φ HF self-consistent problem! U mm0,mm 0 U mm0,m 0 m {z } =: mm 0 = T mm + X m 0 applen mm 0!

quasi-particle picture h HF Ĥ HF i = X mapplen T mm + X m 0 <m mm 0! total energy = X T mm + 2 mapplen X m 0 applen mm 0! = X mapplen " HF m 2 X m 0 applen mm 0! remove electron from eigenstate of Fmn (Koopmans theorem) h HF c a Ĥc a HF i h HF Ĥ HF i = T aa + 2 X m 0 applen am 0! 2 X m6=aapplen ma = " HF a electron-hole excitation (b > N a) " HF a!b = h HF a!b Ĥ HF a!bi h HF Ĥ HF i = " HF b " HF a ab ab = 2 ( ab + ba )= 2 electron-hole attraction ' a ' b ' b ' a r r 0 ' a ' b ' b ' a > 0

homogeneous electron gas Ĥ = X dk k 2 2 c k, c k, + X 2(2 ) 3, 0 dk dk 0 0 dq 4 q 2 c k q, c k 0 +q, 0c k0, 0c k, Slater determinant HF i = Y of plane waves c k 0i k <k F no single-excitations (Brillouin condition) n (r) =h HF ˆ (r) ˆ (r) { ˆ (r), c k, } = density of electrons of spin σ dr 0 HF i = k <k F dk e ik r (2 ) 3/2 e ik r (2 ) 3/2 { ˆ (r), ˆ (r 0 )} = 2 e ik r (2 ) 3/2 = k 3 F 6 2

dispersion & DOS " HF k, = k 2 2 4 2 quasiparticle energies dk 0 k 0 <k F k k 0 2 = k 2 2 k F + k 2 F k 2 2k F k ln k F + k k F k D HF (") =4 k 2 d" HF k, dk quasiparticle density-of-states! =4 k 2 k k F k k 2 F + k 2 2k F k ln k F + k k F k 2.5 HF nonint HF nonint ε k - ε kf 0.5 0-0.5-0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states

exchange hole diagonal of two-body density matrix h kf ˆ 0(r 0 ) ˆ (r) ˆ (r) ˆ 0(r 0 ) k F i =det () (r, r) () 0 (r 0, r) () () 0 (r, r 0 ) 0 0(r 0, r 0 )! (r, r 0 )= one-body density matrix for like spins k <kf dk e ik (r r 0 ) (2 ) 3 =3n sin x x cos x x 3 exchange hole g x (r,0) = h k F ˆ 0(r 0 ) ˆ (r) ˆ (r) ˆ 0(r 0 ) n (r)n (r 0 ) sin kf r k F r cos k F r 2 = 9 (k F r) 3

exchange hole g(0, ; r, ) = 9 sin(k F r) k F r cos(k F r) 2 (k F r) 6 0-0. -0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9 - Wigner-Seitz radius g x (r)- r 2 (g x (r)-)/r 0 0.5.5 2 2.5 3 r/r σ

exchange energy correction to Hartree energy due to antisymmetry E x = 2 dr n dr 0 n g x(r,r 0 ) r r 0 = 2 dr n {z } =N d r n g x( r,0) r exchange energy per electron of spin σ " x = 4 n 2 0 dr r 2 g(r,0) r = 9 4 n 2k 2 F (sin x x cos x)2 dx 0 x {z 5 } =/4 = 3k F 4

HF state as vacuum HF i = Y k <k F c k 0i c k c k k F i =0 for k <k F k F i = 0 otherwise. HF ground state acts as vacuum state for transformed operators c k = (k F k ) c k + ( k k F ) c k = ( c k c k for k <k F for k >k F c k HF i =0 c k, c k 0 0 =0= c k, c k 0 0 h HF HF i = c k, c k 0 0 = (k k 0 ), 0 note: vacuum state no longer invariant under basis transformations!

BCS theory Ĥ BCS = X k BCS Hamiltonian " k c k c k X kk 0 G kk 0 c k" c k# c k 0 # c k 0 " Bogoliubov-Valatin operators mix creators & annihilators b k" = u k c k" v k c k# b k# = u k c k# + v k c k" canonical anticommutation relations {b k,b k 0 0} =0={b k,b k 0 0 } and {b k,b k 0 0} = (k k 0 ), 0 fulfilled for uk 2 + vk 2 = corresponding vacuum state?

BCS state obvious candidate (product state in Fock-space) BCSi / Y k b k 0i need only consider groups of operators with fixed ±k b k" b k# b k" b k# 0i = v k (u k + v k c k" c k# ) v k(u k + v k c k" c k# ) 0i normalizable? h0 (u k +v k c k# c k" )(u k +v k c k# c k" )(u k +v k c k" c k# )(u k +v k c k" c k# ) 0i = u4 k +2u 2 k v 2 k +v 4 k BCSi = Y k (normalized) vacuum v k b k 0i = Y k (u k + v k c k" c k# ) 0i contributions in all sectors with even number of electrons

electronic properties c k" = u k b k" + v k b k# c k# = u k b k# v k b k" momentum distribution hbcs c k" c k" BCSi = hbcs (u kb k" + v kb k# )(u k b k" + v k b k# ) BCSi = v 2 k BCS wave function has amplitude in all even-n Hilbert spaces pairing density hbcs c k" c k# BCSi = hbcs (u kb k" + v kb k# )(u k b k# v k b k," ) BCSi = u k v k

minimize energy expectation value hbcs Ĥ energy expectation value fix average particle number via chemical potential µ ˆN BCSi = X k (" k µ) v 2 k X k,k 0 G kk 0 u k v k u k 0v k 0 variational equations 4(" k µ) v k =2 X k 0 G kk 0 u k v k v k u k u k 0v k 0 N = X k 2v 2 k solve (numerically) for vk and µ

simplified model assume constant attraction only for electrons close to Fermi level := X X G kk 0 u k 0v k 0 = G u k v k k 0 k:close to FS momentum distribution! vk 2 = " k µ p 2 ("k µ) 2 + 2 = G 2 gap equation X k p ("k µ) 2 + 2 = N X k electron density! " k µ p ("k µ) 2 + 2 solve for Δ and µ

quasi electrons (unrelaxed) quasi-electron state k "i = u k c k" BCSi = b k" BCSi quasi-particle energy hk " Ĥ µ ˆN k "i hbcs Ĥ µ ˆN BCSi =sgn(" k µ) p (" k µ) 2 + 2 2.5 =0 =0. =0 =0. ε k - ε kf 0.5 0-0.5-0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states

summary p N! indistinguishable electrons ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )...... ' (x N ) ' 2 (x N ) ' N (x N ) HF i = Y k <k F c k 0i ε k - ε kf 2.5 0.5 0 HF nonint HF nonint (anti)symmetrization is hard Slater determinants to the rescue -0.5-0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states c 0i =0 c,c =0= c,c h0 0i = c,c = h i b k" = u k c k" v k c k# b k# = u k c k# + v k c k" 2.5 =0 =0. =0 =0. Ĥ = X n,m second quantization: keeping track of signs Dirac states extends to Fock space c n T nm c m + X nn 0,mm 0 c nc n 0 U nn0,mm 0 c m 0 c m BCSi / Y k ε k - ε kf b k 0i 0.5 0-0.5-0 0.5.5 k/k F 0 5 0 5 20 25 30 density of states