Configurations, Weights and Observables

To find the configurations and weights, we follow an opposite way as for the CT-AUX solver, namely, we start with a functional integral formulation of the AIM and switch back to operator notation after an expansion of the effective action exponential in the hybridization function. The effective action of the AIM reads (3.6)

Seff = Z

dτ

"

X

σ

c^†_σ(τ)(∂τ−µσ)c_σ(τ) +U n_↑(τ)n_↓(τ)

#

− Z

dτ dτ⁰X

σ

c^†_σ(τ)Γσ(τ−τ⁰)c_σ(τ⁰)

= S0− Z

dτ dτ⁰X

σ

c^†_σ(τ)Γσ(τ−τ⁰)c_σ(τ⁰) (3.76)

where the first part (i.e.S0) is local in time, i.e. has a direct operator correspondence. The second part, the hybridization, is non-local in time since it results from the integration over the infinite bath. The idea is now, to expand the partition function in terms of the hybridization functionΓ. In the expansion,

we denotenσas the perturbation order of spinσandn=n_↑+n_↓as the total perturbation order. The partition function in the functional integral representation is given by the integral

Z =

Z

D[c^†, c ]e^−Seff

= Z

D[c^†, c ]e^−S0



 Y

σ=↑↓

∞

X

nσ=0

1 n_σ!

n_σ

Y

l_σ=1

Z dτ^c

† σ

l dτ_l^cσ c^†_σ(τ^c

† σ

l )Γσ(τ^c

† σ

l −τ_l^cσ)c_σ(τ_l^cσ)





=

∞

X

n_σ=0

1 n_↑!n_↓!





n_σ

Y

σ=↑↓,l_σ=1

Z dτ^c

† σ

l dτ_l^cσΓσ(τ^c

† σ

l −τ_l^cσ)





× Z

D[c^†, c ]e^−S0

n_σ

Y

σ=↑↓,l_σ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ). (3.77)

The functional integral on the right side of this equation has a direct operator correspondence and we can rewrite the integral as an time-ordered operator averageh...i0with respect to the Hamiltonian

H₀=−X

σ

µ_σn_σ+U n_↑n_↓, (3.78)

resulting in the expression

Z =

∞

X

nσ=0

1 n_↑!n_↓!





n_σ

Y

σ=↑↓,lσ=1

Z dτ^c

† σ

l dτ_l^cσ Γσ(τ^c

† σ

l −τ_l^cσ)





* T

n_σ

Y

σ=↑↓,lσ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ) +

0

=

∞

X

n_σ=0

1 (n_↑!n_↓!)²





n_σ

Y

σ=↑↓,lσ=1

Z dτ^c

† σ

l dτ_l^cσ det

M_σ⁻¹({τ_l^c†σ, τ_l^cσ})





×

* T

n_σ

Y

σ=↑↓,lσ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ) +

0

. (3.79)

In the last step, the matrixM_σ⁻¹({τ_l^c†σ, τ_l^cσ})has been introduced, which contains the elements

M_σ⁻¹({τ_l^c†σ, τ_l^cσ})

ij

= Γσ(τ^c

† σ

i −τ_j^cσ), (3.80)

which leads ton_σ!products instead of a single one and enforces the additional factor1/(n_σ!)in equation (3.79). Note that the sign of the determinant exactly cancels with the sign that results from the time-ordering operator and so there is no sign problem. The integrals and the two sums over the perturbation order for spin-up and spin-down operators in (3.79) are supposed to be sampled by the Monte-Carlo pro-cess, which identifies the configurationsC = (n_↑, n_↓,{(τ_l^c†σ, τ_l^cσ

σ)|lσ = 1, ..., nσ, σ =↑↓}, consisting of the perturbation ordersn_↑, n_↓for spin up, spin down and the2n_σtimes(τ_l^c†σ, τ_l^cσ

σ)for the respective cre-ation and annihilcre-ation operators. In the Monte-Carlo procedure, the crecre-ation operators will be integrated in a time-ordered form, as well as the annihilation operators but the ordering of creation operators with respect to annihilation operators will not be performed. This cancels exactly the factors1/(n_σ)² from equation (3.79) above and leads to the formula for the weights

dCwC =Y

σ

(dτ)^2nσdet

M_σ⁻¹({τ_l^c†σ, τ_l^cσ})

* _nσ Y

σ=↑↓,lσ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ) +

0

. (3.81)

The next step is then to find an analytic expression for the trace

* _nσ Y

σ=↑↓,lσ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ) +

0

= X

n_↑,n_↓=0,1

hn↑, n↓|e^−βH0





n_σ

Y

σ=↑↓,lσ=1

c^†_σ(τ^c

† σ

l )c_σ(τ_l^cσ)



|n↑, n↓i, (3.82)

3.3. Continuous-Time Hybridization Expansion Quantum Monte-Carlo Algorithm 57

which is rather easy, since for any configurationC, only of the four summands in the trace will be non-zero. As mentioned above, the creation operators stay time-ordered with respect to annihilation operators and therefore in the product in (3.82) either all operators for a given spin component are ordered as c_σ(τn^cσσ)c^†_σ(τ^c

†

nσσ), ..., c_σ(τ₁^cσ)c^†_σ(τ^c

† σ

1 )or asc^†_σ(τ^c

†

nσσ)c_σ(τn^cσσ), ..., c^†_σ(τ^c

† σ

1 )c_σ(τ₁^cσ), where the latter introduces a factor of(−1)because of the changed ordering. A trace of the first product in the space of spinσ, i.e.

nσ ={0,1}, will have only one non-zero contribution, namely from the state|nσ = 1i, whereas in the second product the non-vanishing contribution results from the state|nσ = 0i, which clearly illustrates for the statement above that only a single state|n↑, n↓icontributes to the trace (3.82). The configurations Care best understood in terms of a segment picture of the impurity, which we now introduce.

The non-zero trace-factor for the productc_σ(τn^cσ_σ)c^†_σ(τ^c

†

nσ_σ), ..., c_σ(τ₂^cσ)c^†_σ(τ^c

† σ

2 )c_σ(τ₁^cσ)c^†_σ(τ^c

† σ

1 )belongs to the state|nσ = 0i, i.e. at timeτ = 0the impurity is unoccupied. Then at timeτ = τ^c

† σ

1 a spin-σ particle is inserted and stays in the impurity until it is removed atτ = τ₁^cσ and the impurity is empty again until at τ^c

† σ

2 a particle is inserted again and so on. These time intervals when the impurity is occupied by a spin up or spin down particle are called segments and uniquely define the configurations C. When the trace factor belongs to the occupied state|n_σ= 1i, we can interpret the respective product c^†_σ(τ^c

†

nσ_σ)c_σ(τn^cσ_σ), ..., c^†_σ(τ^c

† σ

1 )c_σ(τ₁^cσ)as starting with an occupied impurity, i.e. with an occupied interval [0, τ₁^cσ], and also end with an occupied impurity, i.e. with the interval[τ^c

†

nσ_σ, β]. Combining these two intervals, the result is a single interval[τ^c

†

nσσ, τ₁^cσ], which belongs to a segment that winds around and crosses the interval boundaries fromβ to zero. For such segments, an additional factor(−1)has to be introduced, as explained above, because of exchanged ordering.

In the segment picture, the trace-factor can be determined in terms of segment properties and is given by

Figure 3.7: Illustration of a configurationC in the segment picture. There are two pairs of annihilation and creation operators for spin up and three pairs of annihilation and creation operators for spin down occupying the impurity for different times. The filled squares stand for annihilation operators, whereas the empty squares denote creation operators. Colored lines depict a filled impurity. The prefactors for the trace of this configuration would be(−1)since for the down spins one segment crosses theβ,0line. The occupation timeτ_σ^occfor spinσwould be the total length of all lines with the corresponding color and the double occupation timeτ^doubleis the total width of the violet blocks, which indicate the time spans when the impurity is doubly occupied.

* _nσ Y

σ=↑↓,lσ=1

c^†_σ(τ_l^c†σ)c_σ(τ_l^cσ) +

0

= (−1)^Pσsσe^{µPστσocc−U τdouble}, (3.83) wheres_σis the number of segments for spinσ, which cross the0, βboundary (s_σ = 0,1),τ_σ^occis the total length of all segments for spinσandτ^doubleis the total overlap of the occupied segments of the different spins or in other words the total time in which the impurity is doubly occupied. One should note that for the case whenn_σ = 0, i.e. at zero perturbation order we must also distinguish between a completely filled or a completely empty impurity in order to always have a single contribution from the trace above as for all higher perturbation orders. This can be done by decomposing the corresponding Monte-Carlo configurationCinto two distinct configuration, i.e. a “filled” impurity and an “empty” impurity. A typical configuration is illustrated in Fig. 3.7, explaining the trace-factor’s ingredients.

Up to now, the configurationsCand their corresponding weightsw_C have been identified. The

config-Figure 3.8: Subset of diagrams contributing to the weightw_C of a configuration C = (n_↑ = 2, n_↓ = 3), ...). In front of every diagram, there is the corresponding prefactor(−1)^s+d which comes from the determinant (d) and from the trace factor (s) and can be determined by counting the number n_R of hybridization lines that are reversed in time, i.e.s+d=nR.

urationsC = (n↑, n↓,S₁^↑, ...,S_n^↑_↑,S₁^↓, ...,S_n^↓_↓)contain the perturbation ordernσ for spinσand thenσ

distinct segmentsS_i^σ = (τ^c

† σ

i , τ_i^cσ)for both spin components. The corresponding weight is then deter-mined by

dCw_C = (−1)^Pσsσe^{µPστσocc−U τdouble}Y

σ

(dτ)^2nσdet

M_σ⁻¹({τ_l^c†σ, τ_l^cσ})

. (3.84)

Note that the prefactor(−1)^sσin this expression does not cause an explicit sign problem, since it cancels with the sign of the determinantdet(Mσ)⁻¹, which is positive when no segment crossesβ,0and becomes negative otherwise. The last point results from the fact that

Γσ(τ)>0 for τ >0, and Γσ(−τ) =−Γσ(β−τ), (3.85) which is clear by definition ofΓin (3.15).

After the weights and configurations are found, the next step is to determine the observablesg_C, which we will not do directly but instead in a more subtle way. An arbitrary Green’s functiong(τ)can, by using the identityg(τ) =−g(β+τ), be re-expressed as

g(τ) = Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ)g(τ⁰−τ⁰⁰), (3.86) where we have introduced the delta-function

β∆(τ⁰−τ⁰⁰−τ) = δ(τ⁰−τ⁰⁰−τ), for τ⁰> τ⁰⁰

−δ(β+τ⁰−τ⁰⁰−τ), for τ⁰ < τ⁰⁰

. (3.87)

The Green’s function for spinσ¯is then determined by G_σ¯(τ) =

Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ)G_¯σ(τ⁰−τ⁰⁰)

= 1 Z

Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ)hTc_σ¯(τ⁰)c^†_σ¯(τ⁰⁰)i, (3.88) which, remembering the effective action in (3.76), can be formulated as

G_σ¯(τ) = 1 Z

Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ) ∂Z

∂Γσ¯(τ⁰⁰−τ⁰). (3.89)

3.3. Continuous-Time Hybridization Expansion Quantum Monte-Carlo Algorithm 59

With the definition ofg_C, it is immediately clear from this expression, that g¯σC(τ) =

Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ) 1 w_C

∂w_C

∂Γσ¯(τ⁰⁰−τ⁰)

= Z

dτ⁰dτ⁰⁰∆(τ⁰−τ⁰⁰−τ) det(Mσ¯({τ_l^c†¯σ, τ_l^cσ¯}))∂det(M_σ¯⁻¹({τ^c

†

¯ σ

l , τ_l^c¯σ})

∂Γ_σ¯(τ⁰⁰−τ⁰). (3.90) Combining the rules for functional differentiation, i.e.

∂Γσ(τi−τl)

∂Γ_σ¯(τ⁰⁰−τ⁰⁰)=δ(τi−τl−τ⁰⁰+τ⁰)δσ¯σ, (3.91) and matrix inversion

Mij =det( ˜M⁻¹)(ji)

det(M⁻¹) = 1 det(M⁻¹)

∂det(M⁻¹)

∂(M⁻¹)_ji , (3.92)

where we used the cofactors( ˜M⁻¹)(ji)for the matrix inversion, which has a possible representation as the derivative of the determinant with respect to the corresponding matrix element(M⁻¹)_ji, we finally obtain the formula for the observables:

g¯σC(τ) = ∆(τ_i^cσ¯ −τ^c

†

¯ σ

j −τ)(M¯σ({τ_l^c†¯σ, τ_l^c¯σ}))ji. (3.93) The delta-function in this formula must either be measured on a fine grid inτ-space, i.e. on a discretized time interval[0, β]or, as we will do, the observables have to be performed in another representation which is much more suitable for continuous, smooth functions (see chapter ...). The average occupation hnσifor theσ-component and the average double occupancyhn↑n_↓iare other observables, which can be directly measured in the Monte-Carlo sampling, which becomes clear when taking the derivatives

βhnσi= 1 Z

∂Z

∂µσ

and βhn_↑n_↓i=−1 Z

∂Z

∂U. (3.94)

As above, this leads to the observables nσC = 1

βw_C

∂wC

∂µσ

= τ_σ^occ

β and (n↑n↓)C =− 1 βw_C

∂wC

∂U =τ^double

β , (3.95)

which can be accumulated without any effort during the Monte-Carlo sampling and are determined more accurately than the Green’s functions since they are static variables.

With all the necessary ingredients, namely the configurations, the weights and the observables, we can skip and jump to the next section and set up the Markov-Chain and determine the acceptance probabilities for the sampling process.

Im Dokument Topological phases of interacting fermions in optical lattices with artificial gauge fields (Seite 59-63)