PHYSICAL REVIEW B 89, 235128 (2014)
Self-energy and vertex functions from hybridization-expansion continuous-time quantum Monte Carlo for impurity models with retarded interaction
Hartmut Hafermann Institut de Physique Theorique´ (IPhT), CEA, CNRS, 91191 Gif-sur-Yvette, France (Received 25 November 2013; revised manuscript received 3 June 2014; published 24 June 2014)
Optimized measurements for the susceptibility, self-energy, as well as three-leg and four-leg vertex functions are introduced for the continuous-time hybridization-expansion quantum Monte Carlo solver for the impurity model in the presence of a retarded interaction. The self-energy and vertex functions are computed from impurity averages which involve time integrals over the retarded interaction. They can be evaluated efficiently within the segment representation. These quantities are computed within dynamical mean-field theory in the presence of plasmonic screening. In the antiadiabatic regime, the self-energy is strongly renormalized but retains features of the low energy scale set by the screened interaction. An explicit expression for its high-frequency behavior is provided. Across the screening driven and interaction driven metal-insulator transitions, the vertex functions are found to exhibit similar structural changes, which are hence identified as generic features of the Mott transition.
DOI: 10.1103/PhysRevB.89.235128 PACS number(s): 71.10.−w, 71.27.+a, 71.30.+h
I. INTRODUCTION determined self-consistently by relating the impurity and lattice susceptibilities. The continuous-time hybridization-expansion quantum The solver is further important for the implementation of Monte Carlo algorithm (CT-HYB) [1] is an important the recently proposed dual boson approach [11], which may numerical tool in the context of dynamical mean-field be viewed as a diagrammatic extension of EDMFT. Nonlocal theory (DMFT) [2]. Restricting the general two-fermion corrections to the EMDFT self-energy and polarization are interaction to density-density terms greatly simplifies the structure of the fermionic trace which enters the Monte included through a perturbation series whose elements contain Carlo weight. An individual Monte Carlo configuration the three-leg and four-leg vertices of the impurity problem. can diagrammatically be depicted in terms of segments Apart from this, impurity vertex functions are important for the indicating intervals of occupancy of the impurity. This evaluation of momentum resolved response functions within “segment picture” variant (CT-SEG) of the algorithm pro- DMFT [2] and its diagrammatic extensions [11–14]. vides extremely efficient sampling and measurements for In the present paper, improved measurements for the single-site multiorbital impurity models with density-density susceptibility, self-energy, and vertex functions of the impurity interaction. model are provided, which are relevant to the above applica- A further advantage of this method, which has been tions. The self-energy and vertex functions are computed from exploited more recently, is the fact that it can be applied to higher-order correlation functions using relations obtained problems which involve a coupling of the impurity charge from the equation of motion. The idea was first applied in to bosons. Using the Lang-Firsov transformation to eliminate the numerical renormalization group (NRG) context [15,16], the electron-boson coupling leads to a modified Monte Carlo and has also proven very useful for the CT-HYB algorithm weight which contains an interaction between all pairs of [17]. In the presence of an electron-boson coupling, these hybridization events and can be computed at essentially no improved estimators have to be modified. The coupling to additional computational cost [3]. The CT-HYB algorithm can bosons gives rise to additional correlation functions involving further be generalized to treat the coupling of the impurity spin the Bose operators [18]. Here it is shown that they can to a vector bosonic field [4]. solely be expressed in terms of impurity averages, which Integrating out the bosonic degrees of freedom leads involve time integrals over the retarded interaction. In the to a retarded interaction among the impurity electrons. By segment representation, these averages can be evaluated introducing an auxiliary bath of bosonic modes, an arbitrary without approximation and at small additional computational frequency dependence can be treated without approximation cost. [5]. This has been used in realistic simulations of correlated The paper is organized as follows: The Hamiltonian and materials (“LDA+DMFT”) to account for the screening effect action formulation of the impurity problem are introduced in through a generally complicated frequency dependence of the Sec. II. After briefly reviewing the CT-SEG algorithm in the retarded interaction [6]. presence of a retarded interaction in Sec. III, the test case for Another domain in which the method has been applied subsequent calculations is defined (Sec. IV). How to improve [7] is the extended dynamical mean-field theory (EDMFT). the individual measurements is the focus of Sec. V. Results EDMFT was developed in the context of spin glasses [8] for the self-energy and vertex functions in the presence of and strongly correlated electron systems to treat the effect plasmonic screening computed within DMFT on the Bethe of nonlocal Coulomb interaction [9,10]. The screening lattice are presented and discussed in Sec. VI. Following the effect due to the nonlocal interaction leads to a retarded conclusions and outlook in Sec. VII, a detailed derivation of local interaction described by a bosonic bath. The latter is the relations used in the paper is provided in the Appendices.
1098-0121/2014/89(23)/235128(18) 235128-1 ©2014 American Physical Society HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014)
II. IMPURITY MODEL where the atomic, hybridization, and retarded interaction parts read We consider a multiorbital Anderson impurity model with Hubbard-type density-density interaction. Since the technical β β =− ∗ Gat −1 − aspects discussed in this paper are relevant also for realistic Sat dτ dτ ci (τ) ij (τ τ )cj (τ ) 0 0 ij calculations involving multiple orbitals, the notation is kept general: The expressions are given for a multiorbital model 1 β + U dτ n (τ)n (τ), (8) with potentially off-diagonal hybridization. 2 ij i j ij 0 β β A. Hamiltonian formulation = ∗ − Shyb dτ dτ ci (τ)ij (τ τ )cj (τ ), (9) The CT-SEG algorithm can be formulated using the 0 0 ij action representation of the impurity model. The derivation β β of the improved estimators from the equation of motion 1 Sret = dτ dτ ni (τ)Uret(τ − τ )nj (τ ), (10) however is based on the Hamiltonian formulation, which we 2 0 0 ij state first. The Hamiltonian representation has the following form: and the atomic propagator in (8) is given by Gat = + − −1 ij (iνn) (iνn μ i ) δij . (11) H = Hat + Hbath,F + Hhyb,F + Hbath,B + Hcoupling,B, (1) The hybridization function and retarded interaction read in where terms of their spectral representations ∞ = − + 1 dω Im (ω ) Hat (εi μ)ni Uij ni nj , (2) (iν) =− , (12) 2 ij − i ij −∞ π iν ω ∞ † dω 2ω Hbath,F = ε˜pif fpi, (3) U (iω) =− Im U (ω ) , (13) pi ret ret 2 − 2 pi 0 π (iω) ω where = † + † ∗ Hhyb,F (ci Vpij fpj fpiVpij cj ), (4) 1 pij − Im (ω) = V δ(ω − ε˜ )V ∗ , (14) π ij pil pl plj H = ω b† b , (5) p l bath,B q q q q 1 − Im U (ω) = λ δ(ω − ω )λ . (15) π ret q q q = † + q Hcoupling,B (bq bq )λq ni . (6) q i The hybridization amplitudes Vpij and bath levelsε ˜pl,aswell as the couplings λq and boson frequencies ωq , may be chosen Here latin indices label flavor (spin-orbital) indices of the to produce a given frequency dependence of the hybridization impurity. Hat is the atomic part of the Hamiltonian, corre- function and retarded interaction. sponding to a free atom with static density-density interaction In terms of the Boson frequencies and coupling constants, Uij , energy levels εi , and chemical potential μ. Hbath describes the retarded interaction in (13) is explicitly given by a bath of noninteracting fermions with dispersionε ˜pi, where 2 p labels the fermionic bath states. The hybridization of the 2λ ωq U (iω) =− q (16) bath electrons with the impurity is mediated by Hhyb, where ret ω2 − (iω)2 q q Vpij are the hybridization matrices which allow an impurity electron of a given flavor to hybridize with bath electrons and the full frequency-dependent interaction of the model of any other. The term Hbath,B describes a bath of free reads bosons, with bath states labeled by q. For simplicity, the = + impurity electrons are assumed to couple to the bosons only Uij (iω) Uij Uret(iω). (17) through the total charge density of the impurity n = n ,as i i In the infinite frequency limit, or for large frequencies described by the last term. This restriction can be relaxed in compared to a characteristic frequency ω in case of a single the present algorithm, as described below. λ is the coupling q q dominant bosonic mode, the interaction U(iω) approaches strength. the bare interaction: U(iω →∞) = U + Uret(iω →∞) = U. This expresses the fact that screening becomes less effective B. Impurity action at high energies. On the other hand, in the static limit and for small frequencies compared to the characteristic frequency, Integrating out the fermionic and bosonic bath degrees of the interaction is given by a smaller screened value freedom (see Appendix A) results in an action of the following form: 2λ2 U scr ≡ U + U (iω = 0) = U − q 235128-2 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) n (τ ; τ) 1 around zero, i.e., K(−τ) = K(τ), while its derivative K (τ)is ↑ 0 μscr antisymmetric around zero. K(τ) correspondingly has a slope ↑ discontinuity at τ = 0. Taking this into account and imposing + β Uscr 0 the boundary conditions K(β) = K(0 ) = 0, the weight due K(τ − τ ) ↓ to the retarded interaction is [5] = 1 αi αj − n (τ ; τ) 2 ij 2k α ,α >0 sαi sαj K(ταi ταj ) ↓ wret(τ) = e i,j i j + i =j + + × 2K (0 ) ij lij K (0 ) i lii FIG. 1. (Color online) Illustration of a configuration in the CT- e , (19) SEG algorithm. There is a time line for each flavor (spin) and segments where the second sum in the first line is over the 2ki operators mark the time intervals where the impurity is occupied. Creation at positions τα in channel i. The sign sα =±1 is positive operators are marked by closed circles and annihilators by open ones. i if the operator at time τα is a creator and negative for The realization of the density is piecewise constant and changes an annihilator. The second line has the same form as the between 0 (unoccupied) and 1 (occupied) at the operator positions. weight w . With U scr = U + U (iω = 0) and U (iω = Blue shaded areas indicate the time intervals for which the impurity is at ij ij ret ret − = β =− + doubly occupied. The retarded interaction kernel K(τ τ ) connects 0) 0 dτUret(τ) 2K (0 ), one therefore finds that the a given operator with all other operators in the configuration, as chemical potential and static interaction have to be shifted indicated by the dashed lines. internally in the solver according to scr Uij − U μ → μsolver = μ + ij = μ + K(0+), (20) III. CT-SEG ALGORITHM 2 The basic idea underlying the CT-SEG algorithm [1] → solver = scr = − + is to expand the partition function in the impurity-path Uij Uij Uij Uij 2K (0 ). (21) = ∞ τ τ hybridization term Z Zat k=0 d w( ), where k denotes The electron-boson coupling (6) through the full density τ = the perturbation order. Here (τ2k,...,τ1) is a time-ordered is sufficiently general for a number of applications involving sequence of imaginary time points which specify the operator bosons: This includes phonons or the problem of screening positions where the hybridization events occur. In order to fully through a retarded interaction with a given frequency depen- specify a Monte Carlo configuration, which we symbolically † dence. A spin-boson coupling of the form (b + b )λ S denote by τ, we additionally need to keep track of which q q q q z requires us to differentiate between the couplings to individual flavor a given time corresponds to. A configuration may densities n . This case can be treated within the present then be depicted by a number of “segments” for each flavor, i algorithm and the generalization is straightforward.2 The representing the time intervals during which the impurity is treatment of the coupling of the impurity spin to a vector occupied and hence the name of the algorithm. Assuming w(τ) bosonic field requires a secondary expansion in the bosonic is positive {otherwise we rewrite w(τ) = sgn[w(τ)]|w(τ)|}, bath in addition to the expansion in the hybridization [4] and we can sample configurations using a Metropolis algorithm. is beyond the scope of the present work. The Monte Carlo average of an observable O is given = ∞ τ τ τ ∞ τ τ by O MC : k=0 d O( )w( )/ k=0 d w( ), where O(τ) denotes the realization of the observable O inagiven IV. TEST CASE τ = configuration. The Monte Carlo weight is the product w( ) In order to test the improved estimators and to study the τ τ τ τ whyb( )wat( )wret( ), where whyb( ) can be expressed as a effect of the retarded interaction on the self-energy and vertex determinant of hybridization functions. The remaining weights functions, we will use the following test case throughout the are computed as a trace over the atomic states. For a given paper: We consider the Hubbard model with static interaction configuration (see Fig. 1), the realization of the density is U and in the presence of plasmonic screening within DMFT piecewise constant and changes at the operator positions on the Bethe lattice. The full bandwidth is W/t = 4 and the (kinks). Exploiting this fact makes the algorithm particularly temperature is fixed at T/t = 0.02. Energies will be measured efficient. For example, the evaluation of the atomic part of in units of the hopping t. For the model with static interaction the weight simply amounts to counting the length of the only, the Mott transition occurs at U/t ≈ 5.1. − 1 Uij lij μ lii segments lii and their overlap lij : wat(τ) = e 2 ij e i . It is instructive to consider a retarded interaction originating For the retarded part one has to evaluate the exponential from a single plasmon (or phonon) mode with a character- 1 2 of the double integral in (10) for a given configuration. To istic screening frequency ω0: − Im Uret(ω) = λ δ(ω − ω0). π this end, it is convenient to define the retarded interaction With the definition K (τ) = Uret(τ) and boundary conditions kernel K(τ) such that its second derivative yields the retarded interaction.1K(τ) and its derivatives obey bosonic symmetry, − = − i.e., K( τ) K(β τ). K(τ) and U(τ) are also symmetric 2 In this case, the boson operators acquire a flavor index and a bosonic = † bath is introduced for each flavor, i.e., Hbath,B qi ωqibqi,bqi. The coupling constants become matrices λqij and the spectral 1 − U = − While simply being defined here as the twice-integrated retarded density is given by (1/π)Im (ω) q l λqilδ(ω ωql)λqlj. interaction, K(τ) emerges naturally when performing an expansion Final expressions (e.g., for the improved estimators) are generalized in the electron phonon coupling Hcoupling and evaluating the thermal by simply attaching orbital indices to the retarded interaction in average over products of phonon fields [4]. complete analogy to the static case. 235128-3 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) K(0) = K(β) = 0, the retarded interaction kernel and its in imaginary time, it is binned on a fine grid and measured at derivative for this case are given by the expressions [see k2 time differences, where k is the current perturbation order. Eqs. (A10) and (A11)] Hence the algorithm does not scale with the grid size which can (and should) be chosen large. λ2 cosh[ω (τ − β/2)] − cosh(ω β/2) K(τ) =− 0 0 , (22) The density-density correlation function, on the other hand, 2 4 ω0 sinh(ω0β/2) is most simply measured as a ratio of time-ordered traces. In the segment picture, the ratio of the traces with an operator n(τ) λ2 sinh[ω (τ − β/2)] K(τ) = 0 . (23) inserted at time τ to the trace without this operator is simply 1 ω0 sinh(ω0β/2) if a segment is present (i.e., τ follows a creation operator) and The case where the static bare (unscreened) interaction is 0 otherwise. The density for a particular realization of a Monte large and the screened interaction is significantly smaller is Carlo configuration defined by τ is hence a piecewise constant particularly interesting. In the following, the static interaction function which changes at the operator positions (kinks), as is kept fixed to U = 8, which is equal to twice the bandwidth. illustrated in Fig. 1. We denote it by n˜i (τ; τ). Without screening, the system would hence be insulating. The In a time measurement, the contribution to the susceptibility τ screening frequency ω√0 is varied while choosing the electron- is computed in two steps. First, n˜( ; τ) is computed on a grid. boson coupling λ = (U − Uscr)ω0/2 such that the screened This operation is linear in the number of imaginary time bins interaction is fixed to Uscr = 3 3It may, in principle, be measured as a ratio of traces, which in general is not ergodic: Nonzero contributions to Green’s function exist 4It can also be measured using the shift operator method [29]. which can only be obtained by inserting operators into a configuration 5Using antiperiodic boundary conditions, segments are allowed to which has a vanishing weight and hence is never sampled. overlap from τ<βto τ > 0. 235128-4 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) 0 in the evaluation of Dyson’s equation is twofold. Firstly, by forming the difference between two functions, their absolute error propagates. Since both Green’s functions have numerical ω0 =1 -0.05 errors from different sources,6 one cannot expect these errors ω =4 0 to cancel. Secondly, the Green’s function decays as 1/iν,so that the absolute error of the inverses increases rapidly. In ) -0.1 ω0 =8 m interaction expansion continuous-time quantum Monte Carlo ω (i (CT-INT) the problem does not exist. The reason is that in χ CT-INT the Green’s function is measured as a correction to -0.15 ω0 =14 the noninteracting Green’s function G0 [19] (omitting indices): TI NTI = + G G0 G0 M MC G0, (28) -0.2 NTI CPUx25 where M is the inverse of the matrix of noninteracting 0 1 2 3 4 5 6 Green’s functions. The measured correction decays at least as 2 ωm 1/(iν) . Comparing Eq. (28) to Dyson’s equation, one sees that = M MC G0 G so that the CT-INT provides direct access FIG. 2. (Color online) Impurity charge susceptibility χ(iω)for to the product G. The self-energy can be determined from a different screening frequencies ω0. Results are shown for the ratio of observables, i.e., translationally invariant (TI) measurement (heavy solid lines, closed MG symbols) and without using time translational invariance (NTI, thin = 0 MC . (29) dashed lines, open symbols), for a runtime of 300 s on 16 cores. G While the TI results are converged (for these results the error bar is Note that it is because of (28) that in CT-INT this is equivalent smaller than the symbol size), the NTI results exhibit large errors in to determining it directly from Dyson’s equation,7 in contrast particular at low frequencies. Results for the NTI measurement with to CT-HYB. = the runtime increased by a factor of N 25 still exhibit substantial In general, for any method which yields Green’s functions errors at low frequencies, illustrating the slow convergence.√ All results afflicted with numerical errors, the self-energy should always are consistent with a decrease in error bar by a factor of N, except be determined as a ratio. However, due to the expansion in ω = for the NTI measurement at 0 4, relatively close to the Mott the impurity-bath hybridization, it is less obvious how to transition, showing that the NTI measurement is particularly sensitive measure the product Gin CT-HYB.The solution is to express to autocorrelation. G in terms of a higher-order correlation function which follows from the equation of motion for Green’s function. This The frequency measurement is significantly faster and technique was first applied successfully in NRG [15,16]. In a should be preferred over the imaginary-time measurement. previous publication [17] it has been shown that computing Note that the imaginary time and frequency measurements the self-energy using this improved estimator also proves very are equivalent in the sense that they encode, as long as useful for the CT-HYB as it yields substantially more accurate the same configurations are sampled, the same information results than the naive approach using Dyson’s equation. It albeit in a different basis (the two operations, the Fourier should therefore be the method of choice for the determination transform and the Monte Carlo sampling commute). For finite of the self-energy in CT-HYB. resolution in imaginary time, the Nyquist theorem ensures In the present work, the improved estimators for the self- that a function binned on Nτ grid points will reconstruct the energy and vertex functions are generalized for the impurity function in the frequency domain up to the Nyquist frequency model with retarded interaction. It is important that the 1/(2τ) = Nτ /(2β). Therefore, to compute the function at Nω resulting expressions can be written solely in terms of impurity frequencies from the time measurement, one needs a grid size averages. They can hence be evaluated without approximation. of at least Nτ ≈ 4πNω, so the speedup of using the frequency Despite the retarded character of the interaction, the resulting measurement is considerable. correlation functions can be evaluated efficiently within the Finally, note that measuring −ni (τ)nj (0), i.e., not ex- segment representation. ploiting time translational invariance, effectively does not In the following, it will be convenient to switch between the speed up the calculation. This is illustrated in Fig. 2, where representation of time-ordered averages in terms of operators † the equivalent frequency measurement −ni (iω)nj (τ = 0) is (which are denoted c,c ) and the path integral representation, plotted together with the results obtained using (27), both where the Grassmann numbers are denoted as c,c∗.Forthe measured in the same simulation. Because the former does latter, time ordering is not explicitly indicated as it is implicit not use time-translational invariance, it converges much more in the construction of the path integral. slowly than the latter at low frequencies. B. Fermionic self-energy 6The noninteracting Green’s function may or may not (as in DMFT In the CT-HYB algorithm, the extraction of the self-energy calculations) be known up to machine precision. 7 from Dyson’s equation leads to large numerical errors at This is true as far as the Monte Carlo error in M MC is concerned: intermediate to high frequencies. A similar problem appears The error propagation is the same because the Taylor expansions of for the vertex function. The origin of the numerical problems both expressions in M MC are identical. 235128-5 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) nj(τ) nj (˜τ) ret st Fab (τ − τ )= Fab(τ − τ )= Uaj U(˜τ − τ) a a b b τ τ τ τ FIG. 4. Illustration of the retarded part of the improved estimator, FIG. 3. Illustration of the static part of the improved estimator Eq. (33). As in the static case, one samples Hartree-like contributions F = G,Eq.(32). The measurement of this correlation function to G. Because the interaction is retarded, one has to integrate over corresponds to accumulating Hartree-like contributions to the product all timesτ ˜, as indicated by the arrows. G. For a static interaction, the self-energy can be expressed in the expression matrix form as [15,17] β F ret(τ − τ ) =− dτ˜ n (˜τ)U (˜τ − τ)c (τ)c∗(τ ). = −1 ab i ret a b (iν) F (iν)G (iν). (30) 0 i The equation for F = G is obtained by considering the (34) equation of motion for Green’s function: This form could have been anticipated from the static result − =− − − † (32): As illustrated in Fig. 4, the contributions are also Hartree- ∂τ Gab(τ τ ) δ(τ τ )δab Tτ [∂τ ca(τ)]cb(τ ) , like. Due to the retarded nature of the interaction, Uret has to (31) be integrated over all time differencesτ ˜ − τ. The impurity self-energy in the presence of a retarded interaction is still which involves the commutator with the Hamiltonian given by (30), where now ∂τ ca(τ) = [H,ca](τ). The time-ordered thermal average of st ret operators Tτ ··· gives rise to the correlation function F . F (iν) = F (iν) + F (iν) (35) In the case of a static interaction, F essentially stems from is the sum of the static and retarded contributions. In order to the commutator of ca with the interaction term ij Uij ni nj . Switching to the path integral representation, the resulting see how the improved estimators are measured, first consider correlation function is given by [17] the measurement for Green’s function. When measured as a ratio of determinants of hybridization functions, Gab(τ − st − =− ∗ =− ∗ Fab(τ τ ) nj (τ)Ujaca(τ)cb(τ ) . (32) τ ): ca(τ)cb(τ ) is obtained as follows [1]: j k 1 − =− e − s The equation of motion obtained by taking the derivative Gab(τ) Mβαδ τ,τα τβ δa,λαδb,λ . β β = with respect to τ generates the corresponding equation α,β 1 MC = for F : G (which is the same as F for a diagonal (36) basis). Diagrammatically, the correlation function (32) has the interpretation illustrated in Fig. 3: When accumulated in Here Mαβ is an element of the inverse of the matrix of the Monte Carlo process, one essentially samples Hartree-like hybridization functions for a given configuration, k is the s e contributions to G. current perturbation order, τβ (τα ) are the times associated with In the dynamic case, the bath of auxiliary bosons couples to the creators (annihilators) and mark the segment start (end) † the charge density n through (bq + bq )λq ni . Hence the times, and λα denotes the spin-orbital index associated with the q i − = − − − commutator of H with ca generates an additional term which matrix index α. δ (τ,τ ): sgn(τ )δ[τ τ θ( τ )β]isan gives rise to the correlation function antisymmetrized δ function, which transforms a measurement e − s with a negative time difference τα τβ < 0toonewitha ret − =− † + † Fab (τ τ ) λq Tτ [bq (τ) bq (τ)]ca(τ)cb(τ ) . positive time according to the identity G(−τ) =−G(β − τ). q The density is formally measured from a ratio of traces, as (33) discussed before. The static part of the improved estimator can be measured as a combination of the two, i.e., This result has been obtained previously in the context of NRG for the Anderson-Holstein impurity model. Since NRG 1 k is a Hamiltonian based method, the matrix elements involving F st (τ) =− n τ e ab β j α boson operators have to be computed explicitly [18]. Such a α,β=1 j treatment involves a truncation of the infinite boson Hilbert space. In the present algorithm, the improved estimator can − e s × UjaMβαδ τ,τ − τ δa,λαδb,λ . (37) be evaluated without approximation: Since the bosonic bath is α β β noninteracting, it is possible to express the correlation function MC e solely in terms of impurity averages. In Appendix A2 it is The density nj (τα ) needs to be evaluated at all segment shown that integrating out the bosonic bath from (33) leads to end times of segments with flavors λα = j.Forλα = j the 235128-6 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) = − e − s e − s contribution is zero since Ujj 0. Hence there is no ambiguity iν and δ (τ,τα τβ )byexp[iν(τα τβ )] in Eqs. (36), (37), in how to evaluate the density exactly at the operator position. and (38). The measurement formula for the retarded part of the For low frequencies, the improved estimator can be tested improved estimator is by comparing the results to the ones obtained from Dyson’s equation. For high frequencies, the correctness is difficult to k ret 1 e − e s judge due to the noise in the latter result. The high-frequency F (τ) =− I τ Mβαδ τ,τ − τ δa,λαδb,λ , ab α α β β behavior of the self-energy can however be computed ex- β = α,β 1 MC plicitly. As shown in Appendix B, the asymptotic behavior (38) of the self-energy in the presence of the retarded interaction = + e Uij (τ): Uij δ(τ) Uret(τ) is governed by where the interaction-density integral I(τα ) can be evaluated explicitly for any configuration as follows: 1 1 (iν) = 0 + a + O , (40) a a 2 β iν (iν) e = − e I τα : dτn˜ j (˜τ)Uret τ˜ τα where j 0 β + 0 =− − − e = dτUaj (τ)nj , (41) 2K (0 ) sβj K τβj τα . (39) a j 0 j βj β β Here it has again been used that the density changes its 1 =−U (0+) + dτ dτU (τ)U (τ ) value only at the kink positions and is piecewise constant. a ret ai aj ij 0 0 The second sum in the second line is over all operators × [n (τ)n (τ )−n n ]. (42) with flavor j and the sign sβj is positive (negative) for a i j i j creator (annihilator) as before. Since the retarded interaction = = From the Hartree shift and U(iω 0) Uscr one finds the also couples two segments with the same flavor, the sum in = 1 scr condition for half-filling, i.e., μ1/2 2 j Uaj . The first term the second line contains one potentially ambiguous term: for 1 in stems from the bosonic bath and is typically the dominant τ = τ e the argument of K vanishes and K is discontinuous βj α contribution. Note that for a single-band model with static at zero. The ambiguity is resolved by noting that the integral 1 = − e interaction U,Eq.(40) reduces to the familiar result ↑ is over segments and therefore the time differenceτ ˜ τα − 2 − is always negative. Hence the term K (0) means K (0 ). U n↓ (1 n↓ ). Since it is more convenient for implementation purposes, First benchmark results for the imaginary part of the the above formula has been written such that the K(0) term self-energy obtained from the improved estimator measured is to be interpreted as K(0+). Consequently, the difference in the Legendre basis [20] are shown in Fig. 5. The Legendre K(0−) − K(0+) =−2K(0+) has been separated explicitly filter eliminates the residual Monte Carlo noise and the results are seen to accurately reproduce the asymptotic behavior of in this formula. The term stems from the fact that nj does not commute with the operator c(τ e) on the same orbital. It the self-energy at high frequencies. The tail computed from α 2 − corresponds to the static (iω = 0) component of the retarded the expression U nˆ↓ (1 nˆ↓ ) is plotted for comparison for U = 8. As expected from Eq. (40), one can see that the tail part of the interaction Uret(iω). According to Eq. (21), it may be taken into account by replacing the bare value U in F st, in the presence of the screened interaction is considerably enhanced compared to the one for the bare interaction and Eq. (32), by its screened value Uscr. e that the enhancement increases with increasing screening The integral I(τα )(39) can be computed efficiently for a given configuration from the derivative K of the retarded frequency ω0. The regime where the self-energy approaches its interaction kernel K. Hence the function K as well as K are asymptotic behavior moves to larger frequencies with increas- passed as input to the solver.8 The same integral appears in ing ω0. For the largest ω0, the self-energy is clearly metallic and the improved estimators for the vertex functions, as discussed exhibits a hump structure at small Matsubara frequencies. With below. It is therefore convenient to precompute it for all decreasing screening frequency, it is strongly renormalized e at small frequencies and finally exhibits insulating behavior times τα of a configuration for which a measurement is to be performed and reuse it in the different measurements. for the smallest screening frequency. We will discuss these Note that the Green’s function and the improved estimators features in more detail in Sec. VI A. can be measured directly in any basis, such as in imaginary The inset of Fig. 5 shows the real part of the frequency time, on Matsubara frequencies, or in terms of Legendre dependent interaction for this model on real frequencies. For polynomials, by appropriately transforming the measurement small energies, the interaction approaches its screened value = rules. For example, in order to measure the correlation function Uscr 3, while for high frequencies, the interaction converges = on Matsubara frequencies, the measurement rules can be to the unscreened value U 8. The two regimes are separated Fourier transformed. This simply amounts to replacing τ by by a pole at the respective screening frequency ω0. Figure 6 compares the result from the improved estimator to the one obtained from Dyson’s equation. The improved estimator and Green’s function have been measured directly 8While K can be computed from the knowledge of K, it should be on Matsubara frequencies and within the same simulation. avoided to compute it inside the solver for accuracy reasons, since K The large noise in the results obtained from Dyson’s equation is usually represented on a discrete grid. is apparent. One can also see that the error of those results 235128-7 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) 0 0 8 Re U(ω) -0.5 3 -0.5 0 5 10 15 20 ω ) -1 ) -1 n n ν ν -1.5 -1.5 Im Σ(i Im Σ(i ω0 =14 ω0 =8 ω0 =14 -2 ω0 =4 -2 ω0 =8 ω0 =2 ω0 =4 static tail ω0 =2 -2.5 -2.5 0 10 20 30 40 50 60 0 10 20 30 40 50 60 νn νn FIG. 5. (Color online) Self-energies measured using the im- FIG. 6. (Color online) Comparison of self-energies obtained proved estimator in the Legendre basis for different screening from the improved estimator (solid lines) and Dyson’s equation frequencies ω0 at fixed screened interaction Uscr = 3. The static (points) for the same model and parameters as in Fig. 5.The interaction was chosen to be U = 8 and the temperature T = 0.02. correlation functions have been measured in the same simulation Thin lines show the high-frequency tails obtained from Eq. (42). The and on Matsubara frequencies in both cases. Comparison with the 2 self-energy tail for a purely static interaction U nˆ↓(1 −nˆ↓) with high-frequency tails (dotted lines) reveals that the use of Dyson’s U = 8 is shown for comparison. The inset shows the real part of the equation introduces large errors in a region where the self-energy retarded interaction on real frequencies. Energies are given in units clearly has not reached its asymptotic behavior. of the hopping t. model. The corresponding action also appears as the EDMFT is significant in a regime where the self-energy has not yet effective action [7,21]. By integrating out the bosonic field, it reached its asymptotic behavior. Hence the error cannot be can be brought into the form (10), but with ni replaced by n¯i eliminated by replacing the self-energy by its tail at high (this merely amounts to a shift in the chemical potential). For frequencies. While these results illustrate that one can extract this action, the boson propagator is defined as more accurate results for the self-energy by sampling over the same Monte Carlo configurations using the improved D(τ − τ ):=−φ˜(τ)φ˜(τ ). (44) estimator, in practice one is interested in highest accuracy for In Appendix C it is shown that the associated self-energy given runtime. Therefore, one needs to take into account that = D−1 − D−1 (the impurity polarization) is given by the measurement of the improved estimator and in particular the evaluation of the interaction-density integral, Eq. (39), χ(iω)D(iω) (iω) = , (45) slow down the simulation (depending on the perturbation D(iω) order). However, in practice the qualitative picture remains very similar as in Fig. 6 and hence the use of the improved with the susceptibility defined in (24). Hence the bosonic self- estimator outweighs the slowdown. When measuring vertex energy is computed as a ratio of observables. functions, the additional overhead is completely negligible. While the improved estimator is generally more accurate at D. Improved estimator for the two-particle vertex intermediate to high energies, it should be noted that deviations The impurity model two-particle vertex function finds may occur at low frequencies in the insulating phase because its application in the computation of susceptibilities within the overlap between segments and hence the statistics for F are DMFT [2]. Diagrammatic extensions of DMFT [11–14]also suppressed. The statistical errors of F (and G) should therefore rely on the computation of a suitable (reducible or irreducible) be monitored. impurity vertex function. In this section we discuss the computation of the reducible two-particle impurity vertex C. Bosonic self-energy using improved estimators. Another quantity of interest is the bosonic self-energy of In order to unify the notation, the Green’s functions (and the impurity. Consider an action which is obtained from the vertices) are labeled by an index “(n),” where n denotes impurity action (7) by substituting the retarded part (10)byan the number of legs. For example, the two-particle (four-leg) action of the form Green’s function will hence be denoted G(4) and the associated (4) β β improved estimator F . 1 −1 SBoson =− dτ dτ φ˜(τ)D (τ − τ )φ˜(τ ) The four-leg vertex function is defined by 2 0 0 (4) β γabcd(iν,iν ,iω) + dτφ˜(τ)n¯ (τ), (43) i = −1 −1 + i 0 Gaa (iν)Gcc (iν iω) abcd with the bare bosonic propagator D and n¯i = ni − ni .This × (4),con −1 + −1 viewpoint is useful, e.g., when studying the Anderson-Holstein Gabcd (iν,iν ,iω)Gbb(iν iω)Gdd (iν ), (46) 235128-8 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) ν ν ν ν ν ν (4),ret Fabcd (τa,τb,τc,τd ) β γ(4) − + = − ∗ ∗ dτ˜ ni (˜τ)Uret(˜τ τa)ca(τa)cb(τb)cc(τc)cd (τd ) . 0 i ν + ω ν + ωνν + ω ν + ω ν (51) (4) con (4) disc Gνν ω Gνν ω The improved estimators for the vertex function are measured GνGν+ωχωλνω similarlytoEqs.(37) and (38). The main difference is that the ν ν ν matrix M in these formulas is replaced by an antisymmetrized ω n product of two M matrices. The measurement formula for the ω γ(3) − + static contribution to the improved estimator for the four-leg vertex function was given in Ref. [17]. The contribution from ν + ω ν + ω ν the retarded interaction is obtained by replacing the prefactor e e (3) con (3) disc nj (τ )Uja by the interaction density integral I(τ ). On Gνω Gνω j α α (4) Matsubara frequencies, Gabcd(iν,iν ,iω) is measured in two FIG. 7. Diagrammatic representation of the four-leg (three fre- steps. In a first step, the Fourier transform quency) correlation function G(4) [Eq. (48)] (upper panel) and three- (3) k leg (two frequency) correlation function G [Eq. (53)] (lower panel) iντe −iντ s Mab(iν,iν ) = Mβαe α e β δaλ δbλ (52) in terms of their connected and disconnected parts. Spin indices are α β = omitted. Straight lines with arrows denote fully dressed propagators. αβ 1 The wavy line represents the charge susceptibility. The juxtaposition is computed. The final measurement is constructed as the anti- illustrates the formal difference between the vertex functions γ (4) symmetrized product thereof and similarly for the correlators and λ. (50) and (51). With Nν being the number of fermionic/bosonic 2 2 + 3 where the connected part of the two-particle Green’s function frequencies, this measurement scales as k Nν Nν for the (4) 4 3 G (iν,iν ,iω)is two steps instead of k Nν . The improvement of the results by abcd using the improved estimator for the vertex is comparable to (4),con = (4) Gabcd (iν,iν ,iω) Gabcd(iν,iν ,iω) that reported in Ref. [17]. − β[Gab(iν)Gcd(iν )δω,0 E. Improved estimator for the electron-boson vertex − Gad(iν)Gcb(iν + iω)δνν ]. (47) The three-leg vertex is of interest because it contains The two-particle Green’s function is given by the impurity information on the effective electron boson interaction10 and average it is relevant for the recently proposed dual boson method G(4) (iν,iν,iω):=c (iν)c∗(iν + iω)c (iν + iω)c∗(iν). [11]. Here we discuss the computation of the three-leg vertex abcd a b c d function as defined in the dual boson approach. (48) The three-leg correlation function is defined as Its relation to the four-leg vertex function is depicted dia- G(3) (iν,iω):=−c (iν)c∗(iν + iω)n (iω). (53) grammatically in Fig. 7. Similarly as for the self-energy, it is abc a b c more reliable to compute the vertex from improved estimators In analogy to the four-leg vertex, its connected part is given by instead of using the above definition. Using the equation of (3),con = (3) − + motion, one can derive an expression for the connected part of Gabc (iν,iω): Gabc(iν,iω) [βGab(iν iω) nc δω,0 the two-particle Green’s function: − G (iν + iω)G (iν)]. (54) ac cb (4),con = (4) Gabcd (iν,iν ,iω) Gai(iν)Fibcd(iν,iν ,iω) In terms of the connected part of the three-leg correlation i function, the vertex λ in the charge channel as defined in − (4) Ref. [11] is computed as Fai(iν) Gibcd(iν,iν ,iω), (49) i 1 −1 −1 λab(iν,iω) = G (iν)G (iν + iω) = 9 (4) χ(iω) aa b b where F G as before. Fabcd(iν,iν ,iω) consists of a abc static and a retarded part: (3),con (4),st × G (iν,iω) − 1 . (55) F (τa,τb,τc,τd ) a b c abcd = ∗ ∗ nj (τa)Ujaca(τa)cb(τb)cc(τc)cd (τd ) , (50) The relation between these quantities is depicted diagrammati- j cally in the lower panel of Fig. 7. Although they are not defined 9For a diagonal basis, F = F . In general, F can be either computed from F and G, or measured directly. In the latter case, the density 10Integrating out the bosonic field from the correlation function and the interaction-density integral (39) have to be evaluated at the cc∗φ using (A18), leads to the correlation function cc∗n, from s creator times τβ . which the vertex is derived. 235128-9 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) 3.5 0 3 Nc =32 ω0 =14 m Nc =48 -0.2 ω0 =20 ,ω 2.5 Nc =64 n ω0 =25 ν 2 Nc = ∞ λ -0.4 ω0 =50 1.5 m =0 ω0 =100 Re -0.6 ω0 =200 1 ) static Uscr −8 −6 −4 −2 0 2 4 6 8 n ν νn -0.8 FIG. 8. (Color online) Three-leg vertex for fixed bosonic fre- Im Σ(i -1 quency ωm = 0 computed from the four-leg vertex using the identity -1.2 (56) for a given cutoff Nc of the frequency sum. The result is compared to the one obtained from the improved estimator (57) corresponding -1.4 to an infinite cutoff. The convergence with the cutoff is slow, leading -1.6 to a large systematic error for all frequencies. 0 10 20 30 40 50 60 νn completely analogously, the three-leg and four-leg vertices are FIG. 9. (Color online) Self-energy as a function of Matsubara closely related. For a single-band model, the identity between frequencies in the antiadiabatic regime. The parameters are otherwise the two vertices reads [11] the same as in Fig. 5. Note that the asymptotic regime, where the self-energy is described by Eqs. (40)–(42) lies outside the plot range = 1 1 (4) λσ (iν,iω) γσσσσ (iν,iν ,iω) for the largest screening frequencies. χ(iω) β ν σ e and the interaction-density integral I(τα ), respectively. The × + − Gσ (iν )Gσ (iν iω) 1 . (56) frequency measurement of the three-leg Green’s function can in turn be written as the product of the Fourier transform This relation is however problematic to use numerically, of the M matrix [Eq. (52)], and the Fourier transform because the frequency sum converges slowly, as illustrated of the density [Eqs. (25) and (26)]. The former is also in Fig. 8. The three-leg vertex is plotted for different cutoffs of required to measure the four-leg correlation functions and the the frequency sum in (56) and a direct determination from the latter for the measurement of the density-density correlation three-leg correlation function which corresponds to an infinite function. When these quantities are measured, the three-leg cutoff. One can see that above the cutoff frequency, the tail is function G(3) can therefore be obtained at negligible additional unreliable. Even for a relatively high cutoff, a large systematic computational cost. error remains. In addition, the results exhibit more noise than the determination of λ through its improved estimator, VI. RESULTS although an improved estimator was used to obtain the four-leg vertex. The three-leg vertex should therefore be computed In the following we present results obtained using the directly from the corresponding three-leg correlation function, improved measurements. The test case has been defined in as discussed in the following. Sec. IV. The derivation of the improved estimator for the three-leg vertex is sketched in Appendix A2. One obtains the following A. Self-energy relation for the connected part of the three-leg correlation In order to better understand the behavior of the self-energy function in complete analogy to Eq. (49): in the presence of a retarded interaction, it is useful to (3),con = (3) consider the opposite regimes of large and small screening Gabc (iν,iω) Gai(iν)Fibc(iν,iω) i frequency in more detail. The so-called antiadiabatic regime is characterized by ω0 >U− Uscr. This situation appears − (3) Fai(iν)Gibc(iν,iω), (57) to be common to plasmonic screening in real materials i (see, e.g., Ref. [22] and references therein). Results for the where F is defined as before. The static and retarded self-energy in this regime are shown in Fig. 9. For energies contributions to the improved estimator F (3) read small compared to the screening frequency ω0, the electrons essentially experience the screened interaction. In order to (3),st =− ∗ Fabc (τ1,τb,τc): Tτ nj (τa)Ujaca(τa)cb(τb)nc(τc) , see this, the result from a calculation with a purely static j interaction equal to Uscr is plotted in this figure for comparison. β This is precisely the situation one commonly considers in (3),ret =− − Fabc (τ1,τb,τc): dτ˜ Tτ ni (˜τ)Uret(˜τ τa) calculations for real materials which neglect the effect of a 0 i retarded interaction: The Hubbard interaction U is taken equal ∗ to the anticipated screened value. However, one can see that × c (τ )c (τ )n (τ ). (58) a a b b c c the static model delivers a rather poor description even of the As in the four-leg case, the above correlation functions low-energy behavior of the self-energy unless the screening are measured by multiplying the measurement of the corre- frequency is extremely high. The static approximation only e sponding Green’s function [(53) in this case] by j nj (τα )Uja becomes exact in the limit of infinite screening frequency. 235128-10 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) 0 0 ω0 =6 ω0 =8 ω =8 -0.2 ω =14 -0.2 0 0 ω0 =14 ω0 =25 ω =25 -0.4 ω =50 0 static0 U -0.4 ω0 =50 -0.6 scr static Uscr ) ) -0.8 n n ν -0.6 ν -1 -1.2 Im Σ(i -0.8 Im Σ(i -1.4 -1 -1.6 -1.2 -1.8 -2 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 νn νn FIG. 10. (Color online) Comparison of the self-energy for the FIG. 11. (Color online) Comparison of the self-energy for the model with dynamical interaction (solid lines) with the result for the model with dynamical interaction (solid lines, with symbols) with low-energy effective model with static interaction (dashed lines). The the DALA result (dashed lines). curve labeled “static Uscr” is the result for a static calculation without band renormalization, which corresponds to the exact result for ω0 →∞. The effective model description breaks down for ω0 8. is reduced by the same factor. The physical origin of this spectral weight transfer are processes involving the emission or absorption of one or multiple plasmons. Following the evolution of the self-energies with decreasing Spectral functions of the full model and the effective model ω , one can see that for all screening frequencies considered in 0 have been compared in Ref. [23]. Here we compare the two this figure, a feature reminiscent of the minimum in the self- models on the level of the self-energy. Figure 10 compares energy of the static calculation remains, which degenerates results of the original model with those of the effective model. to a hump structure for the smallest frequency considered. For a given ω , the self-energies approach each other for small This feature is likely to carry information on the scale of 0 frequencies. The approximation clearly breaks down for small the screened interaction U . One also observes a steep scr screening frequencies, as can be seen for the results with increase of the self-energy at small energies for smaller ω = 6. Comparing to the calculation with an unrenormalized screening frequencies as well as a minimum at the scale 0 band (Z = 1, labeled static U in Fig. 10), one can see of the screening frequency ω . These features translate to B scr 0 that the bandwidth renormalization is essential to approximate corresponding features previously reported for the spectral the low-frequency behavior. Note that the effective model functions for models including a retarded interaction [5,22,23]: description is restricted to low energy and in particular does a suppression of the spectral weight at low energy and the not reproduce the plasmon peaks at high energy. appearance of a plasmon satellite at (and at multiples of) the For high energies, the electrons experience a partially screening frequency. Note that some of the data in Figs. 9 and screened and hence much larger interaction. This leads to the 12 corresponds and may be directly compared to the spectral strong renormalization of the self-energy for intermediate to functions reported in the bottom panel of Fig. 3 of Ref. [5]. high frequencies. This behavior and the corresponding spectral One finally observes that all self-energies in this figure weight redistribution from low to higher energies can be extrapolate to very similar values at zero, showing that the described in the so-called dynamic atomic limit approximation scattering rate is essentially independent of the frequency (DALA) [22]. The approximation relies on the separation dependence of the interaction and rather determined by its of the low-energy scale set by U and the high-energy low-frequency screened value. The density of states at the scr physics governed by the retarded part of the interaction. It Fermi level hence remains essentially unchanged with respect is based on the following ansatz for Green’s function: G(τ) = to the static calculation, in agreement with previous results G (τ)B(τ). Here G is the Green’s function obtained from [22]. This is confirmed by inspection of the imaginary-time scr scr the calculation with a static interaction equal to U .The Green’s functions at τ = β/2, which almost coincide11 (not scr bosonic propagator B(τ) is evaluated in the dynamical atomic shown). limit, which yields B(τ) = exp[−K(τ)]. As shown in Ref. [23], for large screening frequencies, the DALA results are compared to the numerically exact results low-energy physics of the model is approximately governed on the level of self-energy in Fig. 11. The DALA reproduces by an effective model with a purely static interaction given by the high-energy features of the self-energy (in particular the U , but with an additional bandwidth reduction by a factor scr minimum at the screening frequency observed in Fig. 9) Z = exp(−λ2/ω2). The spectral weight at the Fermi level B 0 remarkably well. As expected, it works better the higher of the original model compared to that of the effective model the screening frequency, i.e., when the assumption of the separation of energy scales is well justified. At low frequencies the approximation deviates because of the importance of the 11In the low temperature limit, (β/2)G(β/2) may be used as an hybridization in determining the low-energy properties. As approximate measure of the density of states at the Fermi level. proposed in Ref. [22], the DALA can be combined with the 235128-11 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) 0 ω =14 7 0 8 ω0 =8 m 5 6 ,ω -1 n ν 4 λ 3 -2 2 Re 1 0 4 -3 ω0 =4 ) 20 2 ω0 =2 n ω0 =2.00 m ν 15 0 ,ω -4 ω =2.25 n 10 m =0 0 ν −2 m =2 λ 5 3 ω0 =2.50 −4 ×10 m =4 m =6 Im Σ(i -5 0 ω =2.75 Re −6 m =8 0 −5 -6 ω0 =3.00 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 ν ν ω0 =3.50 n n -7 ω0 =4.00 static tail FIG. 13. (Color online) Three-leg vertex λ for fixed bosonic -8 0 2 4 6 8 10 12 14 frequency as a function of the fermionic frequency and for different screening frequencies ω0. As the Mott metal-insulator transition is νn approached for decreasing ω0, the vertex develops structure at small FIG. 12. (Color online) Self-energy as a function of Matsubara fermionic frequencies. frequencies in the adiabatic regime. The static tail computed from the static unscreened interaction U is shown for comparison. screening frequency decreases. The difference is larger for finite transferred frequency than for ω = 0. For ω = 0, the Lang-Firsov approach, i.e., with the previously introduced vertex appears to coincide at the first Matsubara frequency for effective model, to cure this deficiency. all screening frequencies. Figure 12 shows the self-energy in the opposite adiabatic regime. Here the screening is inefficient already at rather C. Four-leg vertex small energies (of the order of the plasmon frequency), so Results for the two-particle Green’s function and the that the electrons experience the unscreened interaction. As a vertex function obtained from the improved estimator are result, the self-energy is strongly affected for small Matsubara plotted in Fig. 15 for different screening frequencies ω0 at frequencies and a signature on the energy scale of the screened half-filling. Here we focus on the spin-up-up components interaction is no longer visible. One can further see that the for visualization purposes. Similar conclusions apply for the high-frequency behavior is similar to that obtained in a static up-down components. Because of the particle-hole symmetry, calculation with the static interaction equal to the unscreened both quantities are purely real. The transferred frequency is value U. kept fixed at ω = 0 and results are plotted as a function of the Electrons at the Fermi level, on the other hand, experience two fermionic frequencies. For high screening frequencies the the screened interaction Uscr. As a consequence, the self- system is metallic and the two-particle Green’s function and energy displays an upturn for small frequencies and metallic vertex function exhibit the typical structures of the metallic behavior. If the screening frequency is sufficiently small, or phase. The structure of the two-particle Green’s function the temperature is sufficiently high, the upturn is no longer is mainly determined by its disconnected part defined in resolved on the discrete Matsubara frequencies and the system Fig. 7, since the vertex is comparatively small. The cross behaves as an insulator. Hence, as the screening frequency and diagonal structures of the vertex are also present in the decreases, a Mott metal-to-insulator transition takes place, spin (magnetic) and charge (density) components, where they which is first order [5]. have been observed previously [24,25]. For small coupling these structures of the vertex can be understood in terms of B. Three-leg vertex perturbation theory [24]. As the Mott transition is approached, Figure 13 shows results for the three-leg vertex determined from the improved estimator for various values of the screening ω0 =14 ω0 =25 m 3.0 frequency. The vertex is plotted as a function of the fermionic ω0 =50 ,ω n ω0 = 100 frequency for different bosonic frequencies. It exhibits a peak ν 2.0 ω = 200 λ 0 at small frequencies, the maximum of which is shifted to higher m =0 static Uscr Re 1.0 frequencies with increasing bosonic frequency. This feature 7.0 m =9 m 6.0 grows in magnitude as the screening frequency decreases and ,ω n ν 5.0 the Mott transition is approached. Sufficiently close to the λ 4.0 transition, the vertex changes sign. In the Mott insulating phase Re the vertex is considerably larger in magnitude and the structure −8 −6 −4 −2 0 2 4 6 8 νn is different, reflecting the corresponding changes in the four- leg vertex function (see below). FIG. 14. (Color online) Three-leg vertex for two different Figure 14 shows the three-leg vertex in the antiadiabatic bosonic frequencies as a function of the fermionic Matsubara regime. Similarly to the self-energy, the vertex is close to the frequency in the antiadiabatic regime. As for the self-energy, the result result for the calculation with a static U taken to be equal is similar to the one from a static calculation with the interaction equal to the screened value and is considerably enhanced as the to its screened value when the screening frequency is large. 235128-12 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) (4) (4) FIG. 15. (Color online) Two-particle Green’s function Re G↑↑(νn,νn ,ω) (upper panel) and vertex function Re γ↑↑ (νnνn ω) (lower panel) as a function of Matsubara frequency indices n and n for fixed bosonic frequency ω = 0 and different screening frequencies ω0 = 8,4,3,2(from left to right). The parameters are otherwise the same as in Fig. 5. the contrast in the two-particle Green’s function diminishes depend on the particular form of the frequency-dependent part while the magnitude of the vertex function increases. Very of the interaction. close to the transition (ω0 = 3), the vertex functions develops In Fig. 16 the same quantities as in Fig. 15 are plotted, peak structures at low Matsubara frequencies, in particular on albeit obtained from calculations where the interaction was =− = = the secondary diagonal νn νn. In the insulator (ω0 2), taken to be static. That is, U(iω) U and U was varied across the vertex diverges at low frequency (for T → 0), while the the metal-insulator transition. The figure therefore shows the regions with highest intensity maintain a cross structure. The evolution of the two-particle Green’s function and vertex results are in qualitative agreement with the ones of Ref. [26], across the interaction driven Mott transition in contrast to the where an ohmic screening model was used instead of the screening frequency driven transition of Fig. 15. One can see plasmonic screening model employed here and hence do not that both quantities exhibit all the qualitative features observed (4) (4) FIG. 16. (Color online) Two-particle Green’s function Re G↑↑(νn,νn ,ω) (upper panel) and vertex function Re γ↑↑ (νn,νn ,ω) (lower panel) plotted as in Fig. 15, but computed for the model without screening for different values of the static interaction U = 3,4,5,6 (from left to right). 235128-13 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) 0 and four-leg vertex functions in the presence of a retarded ν = ν interaction have been discussed. The improved measurements -20 lead to substantially more accurate results for a given runtime. -40 This is useful for solving impurity models with retarded inter- =0) action as they arise in the treatment of dynamical screening or m -60 ω i phonons, or in the context of extended dynamical mean-field , n ν -80 i theory. It also opens the way to the numerical implementation , n ν of the recently proposed dual boson approach and thereby (i -100 ch to the calculation of properties of models with long-range γ -120 ω0 =8 ω =14 interaction. The computation of the vertex functions may Re 0 ω =25 -140 0 further be employed to obtain momentum resolved response ω0 =50 static Uscr functions in the context of dynamical mean-field theory. -160 Within dual fermion calculations, one can use this solver to -3 -2 -1 0 1 2 3 ν include the effects of dynamical screening while accounting n for dynamical spatial correlations at the same time. This will FIG. 17. (Color online) Comparison of the low-energy behavior be relevant for an accurate description of real materials. of the vertex function in the charge channel for the model with Results for the self-energy and vertex functions have been retarded interaction (solid lines) and the low-energy effective model obtained within dynamical mean-field theory including the (dashed lines) for iν = iν and iω = 0. effects of a retarded interaction. While in the antiadiabatic regime the self-energy is governed by a low-energy effective model at small energies and approximated by the so-called in the screening driven transition. In particular, the emergence dynamic atomic limit approximation at high frequencies, the of the peak structures on the secondary diagonal very close present solver provides an efficient and unbiased method for to the transition can be seen for U/t = 5 (the Mott transition a general retarded interaction. The three-leg vertex function occurs at U ≈ 5.1). The evolution of the two quantities is has been seen to be strongly enhanced when approaching the hence identified to be a generic feature of the Mott transition. Mott transition. Because of the fact that it develops significant The similarity between the two-particle quantities for the structure one may expect vertex corrections in extensions interaction and screening driven transitions also holds for of extended dynamical mean-field theory to be important, higher bosonic frequencies and the spin-up-down component in particular close to the Mott transition. The structures in as well. the three-leg and four-leg vertex functions in the vicinity of We have seen that an effective model with static interaction the Mott transition are found to be generic features of the and a renormalized bandwidth approximately captures the transition, which are not affected by the particular choice of low-energy behavior of the model with retarded interaction screening model or how the transition is approached. as shown for the self-energy in Fig. 10. This can be expected An implementation of the improved estimators for the self- to be the case also for the vertex function. Results for the energy and vertex function for multiorbital impurity models vertex should be compared in the limit where all frequencies with retarded interaction is provided as an open source code are taken to zero. An extrapolation from the discrete Matsubara [27], as part of the ALPS libraries [28]. representation is difficult, but one can observe the trend already without extrapolation. In Fig. 17 the vertex in the ↑↑↑↑ ↑↑↓↓ ACKNOWLEDGMENTS charge channel (γch := γ − γ ) is plotted for the two models for iω = 0 and the cut iν = iν, which consists of The author would like to thank Thomas Ayral, Silke contributions from the transverse particle-hole channel [24]. Biermann, Alexander Lichtenstein, Erik van Loon, Junya ↑↑↑↑ = Otsuki, Olivier Parcollet, and Philipp Werner for helpful dis- Note that along this cut γν,ν,ω=0 0, which, by antisymmetry, is ultimately a consequence of the Pauli principle. Hence cussions and comments. The simulations have been performed ↑↑↑↑ ↑↑↓↓ using an implementation based on the ALPS-libraries [28]. −γch = γsp := γ − γ along this cut. The vertices are evidently similar at low frequencies. One can further see that The author acknowledges support from the FP7/ERC, under they approach the limiting value for the model with a static Grant Agreement No. 278472-MottMetals. screened interaction and unrenormalized bandwidth as the APPENDIX A: IMPURITY ACTION screening frequency ω0 is increased. At high energies, the vertices are very different, as is the case for the self-energy. The impurity model action may be decomposed into the The high-frequency behavior of the vertex is enhanced for the following three parts: model with retarded interaction. This is expected, because the = + + high-frequency behavior is governed by the large unscreened S Sat SFermion SBoson, (A1) interaction. with β β S =− dτ dτ c∗(τ)G0 −1(τ − τ )c (τ ) VII. CONCLUSIONS AND OUTLOOK at i ij j 0 0 ij In this paper the technical modifications of the continuous- 1 β time hybridization-expansion algorithm required to accurately + U dτ n (τ)n (τ), 2 ij i j compute the impurity susceptibility, self-energy, and three- ij 0 235128-14 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) β β ∗ f −1 In imaginary time, the bosonic propagator reads SFermion =− dτ dτ f (τ)G (τ − τ )fpj (τ ) pi pij − 0 0 ωq τ ωq τ pij D = D˜ + D˜ − =− e − e q (τ) q (τ) q (β τ) ω β −ω β β e q − 1 1 − e q + dτ [c∗(τ)V f (τ) + f ∗ (τ)V ∗ c (τ)], − i pij pj pi pij j =−cosh[(τ β/2)ωq ] 0 pij . sinh(ωq β/2) β β (A10) =− ∗ D˜ −1 − SBoson dτ dτ bq (τ) q (τ τ )bq (τ ) 0 0 q Defining the retarded interaction as β ∗ + + Uret(τ − τ ):= λq Dq (τ − τ )λq (A11) dτ [bq (τ) bq (τ)]λq ni (τ). 0 qi q In Fourier representation, the different propagators read yields the result (10). By introducing the operators and corresponding conjugate momenta f = − i −1 Gpij (iνn) iνn p δij , 1 † 1 † − = √ + = √ − G0 = + − 1 φq : (bq bq ),q : (bq bq ), (A12) ij (iνn) [iνn μ i ] δij , (A2) 2 i 2 D˜ iω = iω − ω −1, qij( m) [ m q ] which obey [φq ,q ] = iδqq , one obtains an alternative representation for the bosonic part of the Hamiltonian, Eqs. (5) where νn = (2n + 1)π/β is a fermionic Matsubara frequency and (6). Up to an irrelevant additive constant it can be rewritten and ωm = 2mπ/β is bosonic. The fermionic degrees of freedom can be integrated out in the following form: using the following identity for Grassmann variables: ω √ = q 2 + 2 + HBoson φq q 2φq λq ni . (A13) ∗ − ∗ + ∗ + ∗ ∗ 2 fi Hij fj ci bij fj fi bij cj q q i dfk dfke k Using the equation of motion for the Heisenberg operators ∗ − ∗ † c [bH 1b ] c ( ) = [det H]e i ij j , (A3) bq (τ), one finds that which gives rise to the hybridization function 2 =− 2 2 [∂τ φq (τ)] ωq q (τ), (A14) ∗ VpilV V G iν V ∗ = plj = iν . or 2 (iω) =−(iω)2φ2/ω2. Passing to the action formulation, pik pkl( ) plj − : ij ( ) (A4) q q q iν pl one therefore obtains pkl pl In complete analogy one may integrate out the free bosons −(iω )2 + ω2 = m q − using the identity SBoson φq (iωm) φq ( iωm) 2ωq ∗ m q − dbk dbk −b∗H b +J ∗b +J b∗ −1 J ∗H 1J √ e i ij j i i i i = [det H] e i ij j . + − 2πi 2φ(iωm)λq ni ( iωm) k m qi (A5) 1 − This leads to =− φ˜ (iω )D 1(iω )φ˜ (−iω ) 2 q m q m q m β β m q Sret = dτ dτ ni (τ)λq D˜ q (τ − τ)λq nj (τ ), (A6) + φ˜ (iω )λ n (−iω ), (A15) qij 0 0 q m q i m m qi where the noninteracting propagator of the field b is defined √ as where in the second line the rescaled fields φ˜ = 2φ have ∗ been introduced for convenience. The propagator in (A15)is D˜ (τ − τ ):=−b (τ)b (τ ) , (A7) q q q 0 thesameasin(A9), since φ˜ = b∗ + b. One may easily verify which is complex. It is more convenient to work with a that both formulations lead to the same results. In particular, propagator which is real, since in particular the retarded one may use the identity interaction is real. This is accomplished by considering the ∗ − + i dφi − 1 φ W 1φ ±φ n 1 n W n propagator of the real field b b ,forwhich e 2 i ij j i i = e 2 i ij j (A16) N D = D˜ + D˜ ∗ (2π) det W q (iω) q (iω) q (iω)(A8) holds. Note that when written in terms of this propagator, the to integrate out the φ˜ fields, which recovers (A9). retarded part of action carries a factor 1/2: β β 1. Generating function = 1 D − Sret dτ dτ ni (τ)λq q (τ τ)λq nj (τ ). For the following derivation, one needs the generating 2 0 0 qij function for correlation functions involving bosonic fields. (A9) Introducing sources and integrating out the fields using (A16) 235128-15 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) yields where the following correlation functions have been defined: − ˜ + ∗ ˜ † = D ˜ SBoson[φ] q Jq (τ) φq (τ) (3) GBoson[J ] [φ]e G (τ1,τb,τc):=−Tτ ca(τa)c (τb)nc(τc), abc b (3),f c aj † 1 [J (τ)−λ n (τ)]∗D (τ−τ )∗[J (τ )−λ n (τ )] G (τ ,τ ,τ ):=− V T f (τ )c (τ )n (τ ), = e 2 q ij q q i q q q j , abc 1 b c p τ pj a b b c c pj (A17) (3),st † F (τ1,τb,τc):=− Tτ nj (τa)Ujaca(τa)c (τb)nc(τc), where the ∗ denotes time integration. Integrating out a single abc b j field φ˜q (τ) from a given expression “···” is hence accom- (3),bc =− ˜ † plished by taking the corresponding functional derivative of Fabc (τ1,τb,τc): λq Tτ φq (τa)ca(τa)cb(τb)nc(τc) . the generating function: q (A24) δGBoson[J ] D[φ˜q ]φ˜q (τ) ···= ··· δJq (τ) J =0 The last correlation function is evaluated in analogy to the β = dτn˜ i (˜τ)Dq (˜τ − τ)λq ··· . foregoing (Appendix A2): i 0 (A18) β (3),ret =− − Fabc (τ1,τb,τc): dτ˜ ni (˜τ)Uret(˜τ τa) 0 i 2. Improved estimators × c (τ )c∗(τ )n (τ ). (A25) The derivation is similar to the one in Ref. [17]. The a a b b c c difference is that the commutator with the Hamiltonian (1) Taking the Fourier transform in Eq. (31) generates an additional term due to the coupling to f (iν,iω) = F[f (τa,τb,τc)] a bosonic bath: =− − aj − β β β [H,ca] εaca Vp fpj nj Ujaca 1 := dτa dτb dτcf (τa,τb,τc) pj j β 0 0 0 − + − † + × iντa i(ν ω)τb iωτc (bq bq )λq ca. (A19) e e e (A26) q (3),f c (3) and expressing Gabc in terms of Gabc through its equation This extra term gives rise to an additional correlation function of motion yields † F ret(τ − τ ) =− λ T φ˜ (τ)c (τ)c (τ ), (A20) ab q τ q a b [(iν − )δ − (iν)]G(3) (iν,iω) q a aj aj jbc j † where φ˜ has been substituted for b + b . The correlation q q q = F (3) (iν,iω) + βδ n δ − δ G (iν + iω). function can be further evaluated by switching to the path abc ab c ω ac cb (A27) integral representation and using Eq. (A18) to integrate out ˜ field φq . With the definition (A11) of the retarded interaction, Subtracting (iν)G(3) (iν,iω) on both sides, the left- j aj jbc one obtains the final expression (34). −1 (3) Similarly, the equation of motion generates an extra term hand side becomes j Gaj (iν)Gjbc(iν,iω). Hence multiply- for the improved estimator of the vertex function: ing both sides by G and defining the connected part (4),ret (3),con (3) F (τa,τb,τc,τd ) G (iν,iω) = G (iν,iω) − [βGab(iν) nc δω,0 abcd abc abc † † = ˜ − Gac(iν)Gcb(iν + iω)], (A28) λq Tτ φq (τa)ca(τa)cb(τb)cc(τc)cd (τd ) . (A21) q as well as using F := G, one finally obtains Integrating out the boson fields as before results in (51). (3),con = (3) For the three-leg vertex one uses the following identity: Gabc (iν,iω) Gai(iν)Fibc(iν,iω) † i ∂τa Tτ ca(τa)cb(τb)nc(τc) − F (iν)G(3) (iν,iω). (A29) = † + − ai ibc Tτ ∂τa ca(τa)cb(τb)nc(τc) δ(τa τb)δab nc i + − † δ(τa τc)δac Tτ cc(τc)cb(τb) . (A22) APPENDIX B: SELF-ENERGY TAILS The δ-function contributions stem from the discontinuities of this function. The rest of the derivation follows Ref. [17]. We are interested in the high-frequency behavior of the Inserting the commutator (A19) yields self-energy up to first order in 1/(iνn), −[∂ + ]G(3) (τ ,τ ,τ ) 1 τa a abc a b c (iν) = 0 + +··· . (B1) = (3),f c + (3) + (3),bc iν Gabc (τa,τb,τc) Fabc(τa,τb,τc) Fabc (τa,τb,τc) Expanding + δ(τa − τb)δab nc − δ(τa − τc)δacGcb(τa − τb), = − − − −1 (A23) G(iν) [(iν)1 εˆ (iν) (iν)] (B2) 235128-16 SELF-ENERGY AND VERTEX FUNCTIONS FROM . . . PHYSICAL REVIEW B 89, 235128 (2014) in 1/iν,using(B1) and the corresponding expansion for (iν) as well as the interacting Boson propagator up to first order, Dqq (τ − τ ):=−φ˜q (τ)φ˜q (τ ), (B12) ∗ V aiV ib 1 (iν) = p p = ab +··· (B3) by taking the second derivative of the generating function ab iν − iν pi p (A17) with respect to J : one obtains the following expression in matrix form: Dqq (τ − τ ) + 0 β β 1 εˆ G(iν) = + = D − − iν (iν)2 q (τ τ )δqq dτ1 dτ2 0 0 + 0 + 0 + 1 + 1 εˆ a (ˆε ) × D − D − + +··· , (B4) λq q (τ τ1) ni (τ1)nj (τ2) q (τ2 τ )λq . 3 (iν) ij whereε ˆab = (εa − μ)δab. The high-frequency expansion of (B13) the Green’s function is computed from the well-known ˜ expression The expectation values in (B6) and (B7) involving φq can now be expressed in terms of time ordered correlation functions ∞ † {[H,ca]{k},c } G (iν) = (−1)k b , (B5) ˜ ≡˜ + ab (iν)k+1 φq φq (0 ) , (B14) k=0 H,c { } k + where [ a] k denotes the -fold nested commutator with the φ˜ n =φ˜ (0 )n (0), (B15) Hamiltonian and {A,B} denotes the anticommutator of A and q j q j 1 2 { †} = B.Using(A19), defining Ca and Ca such that [H,ca],cb + † = 1 { } = 1 + 2 φ˜q φ˜q φ˜q (0 )φ˜q (0) . (B16) Ca δab and [H,[H,ca]],c Ca δab, one finds b Noting that =0 and substituting (B10)–(B13)into(B8), C1 =−ε − n U − λ φ˜ (B6) q a a j ja q q (B9), and using (A11), yields the final expressions j q β and 0 = + a Uaj nj ni dτUret(τ), (B17) C2 = 2 + 2 U n + U U n n j i 0 a a a aj j ai aj i j j ij 1 =− + + − a Uret(0 ) UaiUaj ( ni nj ni nj ) + 2 Uaj λq φ˜q nj +2 aλq φ˜q ij qj q β + 2 Uaj dτUret(τ)[ni (τ)nj (0)−ni nj ] − i ω λ ˜ + λ λ φ˜ φ˜ . (B7) q q q q q q q ij 0 q qq β β Comparing (B4) with (B5), one arrives at + dτ dτ Uret(τ)Uret(τ ) ij 0 0 0 = U n + λ φ˜ (B8) a aj j q q × − j q [ ni (τ)nj (τ ) ni nj ], (B18) and which can be brought into the compact form (40)–(42)interms = + 1 =− ˜ + − of the retarded interaction Uij (τ) Uij δ(τ) Uret(τ). a i ωq λq q UaiUaj ( ni nj ni nj ) q ij APPENDIX C: LOCAL BOSONIC PROPAGATOR + 2 U λ (φ˜ n −φ˜ n ) aj q q j q j In terms of the fields qj φ˜ := λq φ˜q (C1) + λq λq (φ˜q φ˜q −φ˜q φ˜q ). (B9) q qq In order to further evaluate this expression, one uses the one can rewrite the bosonic part of the action (A15)intheform generating function Eq. (A18) to integrate out the fields φ˜ 1 S =− φ˜(iω )D−1(iω )φ˜(−iω ) and to compute the correlation functions Boson 2 m m m m β ˜ = − φq (τ) ni dτ Dq (τ τ )λq , (B10) + φ˜(iωm)ni (−iωm). (C2) 0 i m i β The local bosonic propagator of the impurity model is then ˜ = − φq (τ)nj (0) dτ Dq (τ τ ) ni (τ )nj (0) λq , defined as follows: i 0 (B11) D(τ − τ ) =−φ˜(τ)φ˜(τ ). (C3) 235128-17 HARTMUT HAFERMANN PHYSICAL REVIEW B 89, 235128 (2014) = In terms of the propagator of the bosonic bath it can be where n i ni is the total charge density. When the retarded expressed as part of the action Eq. (A9) is written in terms of n − n instead of n itself, the average in the above equation becomes n¯(τ1)n¯(τ2). In this case, the above expression can be written − = − D(τ τ ) λq Dqq (τ τ )λq . (C4) in terms of the local charge susceptibility qq χ(τ1 − τ2) =− n¯i (τ1)n¯j (τ2) (C6) From the corresponding relation for the bare propagators, ij one sees that the retarded interaction (A11) plays the role of on Matsubara frequencies as the bare local bosonic propagator D =−φ˜(τ)φ˜(τ ) .Using 0 = D + D D (B13), the bosonic propagator can be written D(iω) (iω) (iω)χ(iω) (iω). (C7) From this relation the bosonic self-energy (impurity polariza- −1 −1 β β tion) = D − D is identified to be − = D − − D − D(τ τ ) (τ τ ) dτ1 dτ2 (τ τ1) χ(iω)D(iω) 0 0 (iω) = . (C8) D(iω) ×n(τ1)n(τ2)D(τ2 − τ ), (C5) [1] P. Werner, A. Comanac, L. de’Medici, M. Troyer, and A. J. [17] H. Hafermann, K. R. Patton, and P. Werner, Phys. Rev. B 85, Millis, Phys.Rev.Lett.97, 076405 (2006). 205106 (2012). [2] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, [18] A. C. Hewson and D. Meyer, J. Phys.: Condens. Matter 14, 427 Rev. Mod. Phys. 68, 13 (1996). (2002). [3] P. Werner and A. J. 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