6. Auxiliary field continuous time quantum Monte Carlo

Transcription

1 6. Auxiliary field continuous time quantum Monte Carlo The purpose of the auxiliary field continuous time quantum Monte Carlo method 1 is to calculate the full Greens function of the Anderson impurity model H µ n +Un n + k t k a k a k + k V k a k c k +H.c. 6.1 Diagrammatic Monte Carlo In order to evaluate the partition function, we write the Hamiltonian as H H + V so that we have { β } e βh e βh exp T τ dτ ˆVτ 6.2 with ˆVτ in the interaction picture ˆVτ e +τh Ve τh. and the usual imaginary time ordering operatore T τ. In Diagrammatic Monte-Carlo, the Feynman diagrams of the perturbation expansion are sampled by Monte Carlo. Then, the partition function is Tr Z Tr e βh Z with Z Tre βh. Thus Z 1 { β Tr e βh exp T τ Z Z { } e βh β exp T τ dτ ˆVτ Tre βh dτ ˆVτ } { β exp T τ 6.3 dτ ˆVτ } 1 This chapter is based on a course given by Aaram J. Kim. The master thesis Topological Phases of Interacting Fermions in Optical Lattices with Artificial Gauge Fields by Michael Buchhold is used as a reference. 77

2 6.4 This expectation value will be calculated with quantum Monte Carlo. First, we write more explicitly { β exp T τ dτ ˆVτ } 1 β β dτ k... dτ 1 Vτk... Vτ 1 k! k β β β dτ k... dτ 2 dτ 1 Vτk... Vτ 1 τ k 1 τ 1 k because if τ e +τh e τh, then 6.5 τ τ e +τh H He τh e +τh Ve τh τ Vτ τ This is formally solved by integration: β Iteration via τ β τ 6.6 dτvττ 6.7 dτ Vτ τ 6.8 leads to the expression 6.5. Then Z { β exp T τ dτ Z ˆVτ } 1 β β dτ k... dτ 1 ˆVτ k... ˆVτ 1 k! k Wx dxwx 6.9 x Here, we interpret Wx as a weight. In order to be able to interpret Wx as a probability, we assume it is positive. In the continuous-time auxiliary field quantum Monte-Carlo algorithm, an auxilary field decomposition of the interaction part of the Anderson impurity model is used. In CT-AUX, the choice of interaction is V U n n n + n 2 K β

3 where the second term is a Hartree like term, and the third term is a control parameter for the expansion order. This means that the noninteracting Hamiltonian is modified as H H V 6.11 The choice of V yields the following values for empty, singly occupied and doubly ocupied impurity: K β K β U 2 K β U 2 K β 6.12 Now a Hubbard-tratonovich transformation is applied to V: V U n n n + n K 2 β K e γsn n s± The identity can be checked by applying the last expression on the basis states in For example, for, n n, and the expression yields K β. For, n 1, n V K β cosh γ cosh γ 1 + Uβ 2K 6.14 s represents an Ising type boson that was introduced, an auxiliary spin. Eventually there will be k auxiliary spins over which we sum: Z + K k [ e γs kn τ k... e γs 1n τ 1 ] 6.15 Z {s i } Here, is the physical spin, with numerical value +1 for and 1 for. Then, for Monte Carlo sampling we identify Z dτ k... dτ 1 Wx 6.16 Z k {s i } 79

4 s 2 1 s 4 +1 s 1 +1 s 3 +1 s 5 1 β τ 1 τ 2 τ 3 τ 4 τ 5 Figure 6.1: Example of a CT-AUX configuration at perturbation order 5. The configuration space consists of perturbation order k, time indices τ and auxiliary spin indices s i ; there are k time points. The random walker needs to roam the entire configuration space. CT-AUX is a generalization of the discrete time quantum Monte Carlo algorithm Hirsch-Fye. Explicit calculation of the matrix We now prepare to write the action in terms of Grassmann variables by rewriting the interaction V K e γsn n K e γsn 6.17 s±1 s±1 further. Taylor expansion of the exponential yields because n 2 n e γsn 1 + γsn + 1 2! γs2 n n + n + γsn + 1 2! γs2 n n + e γs n 1 1 e γs n 1 1 e γs 1 c c 1 [ ] 1 e γs c c + e γs c c e γs e γs 1c c e γs 1 1 e γs c c 6.18 where Fermion antiperiodicity c c 1 c c was used. Thus, we have found that V K e γsn n K [ ] e γs e γs 1c c 6.19 s±1 s±1, Now we can go from Hamiltonian to action formalism, writing for the average of an operator A with respect to the noninteracting Hamiltonian 8

5 H A Tr[ ] Ae βh Tr [ ] e βh D[c + c]ae D[c+ c]e 6.2 While in the first expression, c, c are operators, in the second, c and c + are Grassmann variables. We now come back to the effective action of the previous chapter eff dτdτ c + τg 1 τ τ cτ + dτvτ 6.21 where the Weiss function G encodes the result of integrating out the bath degrees of freedom. We will use a number of identities for Grassmann variables: D[c + c]e ij c+ i ijc j det[] D[c + c]c x c + y e ij c+ i ijc j D[c+ c]e ij c+ i ijc j 1 xy e ac+c 1 ac + c 6.22 We now introduce the matrix elements ij of the action with i, j indices for time, spin, lattice sites, etc., i, j,..., M where M ; as we have passed from continous imaginary time τ integrals to sums, the τ i have to be infinitely dense. There are infinitely more times τ i than the k times we have in a given configuration of perturbation order k. Thus D[c + c]e D[c + c]e ij c+ i ijc j with ij G 1 ij τ i τ j 6.23 For the interaction, we introduce the matrices A and B with { δ ij e γs i i for τ i {τ 1... τ k } A ij δ ij otherwise {δ ij 1 e γs i i for τ i {τ 1... τ k } B ij otherwise 6.24 The time mesh for the Feynman path integral now has the imaginary times τ 1 to τ k as well as an infinite number of additional time indices τ i. A has, 81

6 for perturbation order k, k exponentials on the diagonal for times τ 1, τ 2,..., τ k, and otherwise 1: 1 e γs 1 A e γs 2... and consequently, B is 1 e γs 1 B 1 e γs Then we can write the average of Eq. 6.9 as V... V deta D[c + c]e c+ i B ij c j e c+ i ij c j det 6.27 The product over spin reflects the fact that the noninteracting part of the Hamiltonian H contains only quadratic operators with index so that H commutes with H so that integration weights Wx are products of weights for the individual spin components,. If we now rewrite the exponentials as e c+ i B ij c j we find for the integral we need to compute I deta det B det 6.28 Due to the M index, matrices A, are infinite dimensional. The idea is now to use the zeros in the B matrix to make things finite-dimensional by reordering. det B detdet1 B

7 because B 1 B 1, and the determinant of a product splits into a product of determinants. We now exploit that at perturbation order k, only k entries in the B matrix are nonzero: b 1 b 1 B... B b 2... b k b k This rearrangement doesn t produce a sign as rows and columns are exchanged. In the same way, 1 is rearranged into 1 only that this doesn t produce empty submatrices. Then and B 1 x B 1 x x 1 B 1 T 1 B 1 T 1 but in the determinant, only 1 B 1 remains: det1 B 1 det1 B In Eq. 6.28, det cancels with the denominator, so that we have I deta det1 B 1 detã det1 B because the rearranged à has only entries of 1 beyond the reduced space of dimension k. Now we arrive at the final expression I det à à B 1 e Γ e Γ 1G detn where the k k matrix G, with k numbering the time points, was introduced, and the entire matrix that needs to be manipulated is now called N 1 matrix N matrix. The notation for à is à e Γ diag e γs1,..., e γs k 83

8 Later, the Weiss function G will be represented by a very large mesh with many more than k points; it will be interpolated linearly or by cubic methods from k points. Insertion and removal Now, we will assume that we know the N 1 matrix at a given perturbation order k. To implement a random walk through configuration space, we need to either insert a time point or remove a time point, based on the Metropolis algorithm. a k4 β τ 1 τ 2 τ 3 τ 4 b k5 insertion c k3 removal β τ 1 τ 2 τ 3 τ 4 τ β τ 1 τ 2 τ 3 Figure 6.2: chematic of insertion and removal updates to a configuration. Insertion.- The decision to insert an auxiliary spin into the configuration is taken based on a uniform random number r in the range [, 1]. If r > min 1, f W k+1 W k 6.36 where f is factor explained later, the update is accepted, otherwise rejected. Thus, to take the decision, the ratio of weights needs to be calculated, specifically: det N k+1 1 det N k 1 det N k 1 det N k N k Q det RN k Q 6.37 R Nk 1 Nk 1 R Q 84

9 This uses the fact that N k and N k 1 are inverses of each other, and matrices are multiplied using the properties of determinants. During the simulation N k rather than N k 1 is stored. For the enlarged matrix N k 1 we have N k+1 1 N k 1 Q 6.38 R with k 1 vector Q, 1 k vector R and defined by 1 l n: Q l e γs l 1G τ l τ R l e γs 1G τ τ l e γs e γs 1G In case the insertion move is accepted, we now need the missing elements of the enlarged matrix N k+1, expressed in terms of the additional elements Q, R, of N k+1 1 which we know: N k+1[ N k+1] 1 P Q P Q 1I R 6.4 R where we write P for N k 1. This yields the equations PP + QR 1I PQ + Q RP + R T RQ which we have to solve for PP, Q, R, and so on. The third equation yields RP R R RP and inserting into the fourth equation RP 1 Q + 1 RP 1 Q 1 1 RP 1 Q 6.43 with P 1 N k. is a number which we don t need to replace so that we have from the third equation R RN k

10 The second equation Q PQ is inserted into the first PP PQR 1 P P QR where Q is a column vector, R is a row vector. Now we use the herman- Morrison formula for an invertible square matrix A, with v T A 1 u 1 A + uv T 1 A 1 A 1 uv T A 1 1 v T A 1 u 6.47 which allows a cheap update of the already known inverse of A if A is modified by the rank 1 matrix uv T. Here, we have u Q, vt R and P P QR 1 [ P P 1 1 Q ] [ RP 1 ] 1 RP 1 Q [ N N k k Q ] [RN k] + RN k Q Nk + [ N k Q ] [RN k] 6.48 This is the most massive calculation in the update. Now we find for Q Q PQ Nk Q + [ N k Q ] [RN k] Q Nk Q 1 + RNk Q N k Q RN k Q RN k Q N k Q 6.49 Removal.- The decision on insertion versus removal is taken with 5% probability. For the deletion of an auxiliary spin s at time τ from the matrix N k, the situation is N k+1 1 N k 1 Q P N Q k+1 R R with RN k Q 1 Q N k Q R RN k P N k + [ N k Q ] [RN k]

11 The idea is now to use to solve the P equation for N k using the expressions for Q and R: P N k + Q R Nk + Q R N k P Q R 6.51 Monte Carlo procedure.- To summarize, we use the expansion of the partition function β τ k K Z dτ k dτ 1 Z k {si, τ i } 6.52 where k {s i } Z k {si, τ i } Z detn 1 {si, τ i } 6.53 For the specific formulation of the update, we need the condition of detailed balance Wxpx x Wx px x 6.54 Then in general, the Metropolis algorithm is to update, with a random number r, if r < min 1, Wx 6.55 Wx The detailed balance condition can be slightly modified by splitting the transition probability into proposal and acceptance probabilities: p p proposal p acceptance 6.56 For an insertion, the proposal probability is the probability p 1 of choosing a time τ from the interval [1, β], p 1 dτ β, times the probability p of choosing the auxiliary spin s from { 1, 1}, i.e. p proposal k k + 1 p 1 p 2 dτ 6.57 For deletion, the tuple τ i, s i must be chosen from the existing k + 1 auxiliary spins, so that the proposal probability is p proposal k + 1 k 1 n

12 Then, the Metropolis update decision is modified to r < min 1, Wx p proposal x x Wxp proposal x x 6.59 Detailed balance then yields for the acceptance probabilities p acceptance k k + 1 p acceptance k + 1 k because W k+1 dτ 1... dτ k dτ k+1 K W k dτ 1... dτ k K W k+1 K detn 1 k + 1dτ W k k + 1 k+1 detn 1 k + 1 k detn 1 k k + 1 detn 1 k 6.6 This means that p acceptance k k + 1 min p acceptance k + 1 k min { { 1, K k + 1 1, k + 1 K detn 1 k + 1 detn 1 k detn 1 k detn 1 k + 1 Measurement of the Greens function The Greens function, written for QMC without a minus sign, is G τ τ T τ c τc τ } } 6.61 D[c + c]c x c + ye ij c+ i B ijc j D[c+ c]e ij c+ i B ijc j [ B 1] xy 6.62 where x, y stand for arbitrary imaginary time indices. For the matrix inverse, we can write B 1 [ 1 B 1 ] B B 1 1[ 1 B 1 + B 1] B 1 1 B

13 Here we recapitulate the method of getting rid of the zero subspace: The matrices, B are infinite dimensional but B has many zeros: B B B B 1 1 T B 1 1 B 1 T 1 B B B B 1 1 B B 6.64 Now we go back to the definitions of the A, B and matrices: A e Γ B 1 e Γ 1 G 6.65 with the noninteracting Weiss function G ; thus is invertible. Now we insert Ã: 1 B B 1 Ã 1Ã 1Ã B B Ã B 1 1Ã B N {si, τ i } e Γ 1 M 6.66 Then, coming back to the matrix inverse 6.63 B 1 G + G N {si, τ i } e Γ 1 This means that the full Greens function can be evaluated as G 6.67 G τ τ G τ τ + G τ τ i M ijg τ j τ 6.68 where τ i, τ j are configuration time points. The k k matrix M is M N e Γ The Weiss function G does not depend on the configuration, but M does. Therefore, we only need to accumulate terms that depend on the configuration, in particular M ij G τ j τ. This is accumulated for many i, and only a vector needs to be stored. This also explains why the N matrix is stored; it is needed for the Greens function. The matrix update cost is of order Ok 2 because only matrix vector multiplications are needed. 89