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ment and couples only to the site density N$_{i}$ , reflecting the physics of dynamical screening. It is generated by a screening spectral function Im W ( ω ): 

with W$_{ω}$$_{0}$ ( τ ) = cosh [( τ − β 2 ) ω$_{0}$ ] / sinh [ ω 0 β 2 ] for 0 ≤ τ ≤ β and W$_{ω}$$_{0}$ ( τ ) = W$_{ω}$$_{0}$ ( τ + nβ ) ( n integer). Other interac- tion parameters acquire only a very weak frequency de- pendence [5] because they involve operators correspond- ing to higher multipoles which are weakly screened. We recast the time dependent interaction in Hamilto- nian form by using Hubbard-Stratonovich techniques to introduce boson operators whose spectrum reproduces Im W . We define λ 2 ω 0 = − π Im W ( ω$_{0}$ ) and at each site i and each frequency ω$_{0}$ we employ the identity (“*” denotes integration over time) e 1 $_{2}$λ$_{ω}$$_{0}$ $^{N}$∗$^{W$_{ω}$$_{0}$}$∗Nλ$_{ω}$$_{0}$ = ∫ D φ$_{ω}$$_{0}$$_{e}$− $^{φ$_{ω}$$_{0}$}$∗W − 1 ω$_{0}$ ∗ $^{φ$_{ω}$$_{0}$}$− √$_{2}$$_{φ}$$_{ω}$$_{0}$ ∗ $^{Nλ$_{ω}$$_{0}$}$. Noting that W − 1 ω 0 = ( − ∂ 2 τ + ω 2 $_{0}$) / (2 ω$_{0}$ ) (the periodicity under τ → τ + nβ implies a derivative discontinuity at τ = nβ ) and identi- fying ∂$_{τ}$ φ$_{ω}$$_{0}$ as ω$_{0}$ times the momentum Π$_{ω}$$_{0}$ conjugate to φ$_{ω}$$_{0}$ we see that the time dependent interaction is equiv- alent to a Hamiltonian model in which the site density is coupled via λ$_{ω}$$_{0}$ to a set of oscillators, 

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To solve the Hamiltonian H = H$_{band}$ + H$_{int}$ + H$_{screen}$ we employ dynamical mean field theory [6, 11] which computes the solution to the correlated electron prob- lem from the solution of an auxiliary quantum impurity model with interactions given by the local interactions of the original model and an impurity-bath hybridiza- tion term H$_{hyb}$ = c † V d + H.c. with V determined by a self-consistency condition. Following Ref. [9] we solve the impurity model by expanding the partition function in powers V , collecting diagrams into determinants of matrices of hybridization functions and using a Monte Carlo procedure to sample the resulting sum of determi- nants. As in Ref. [12] we treat the electron-boson cou- pling by a canonical transformation which shifts ˆ φ$_{ω}$$_{0}$ by √$_{2}$$_{λ}$$_{ω}$$_{0}$ ˆ N/ω$_{0}$ , changes the instantaneous interaction U to the screened value 

and shifts the chemical potential µ to µ$_{scr}$ = µ + ∫ ∞ 0 dω 0 π Im W ( ω 0 ) ω 0 . The transformation also multiplies d , d † by e is ∫ ∞ 0 dω 0 $_{Π$_{ω}$$_{0}$}$√$_{2}$$_{λ}$$_{ω}$$_{0}$$_{/ω}$$_{0}$ ( s = 1 for d † and s = − 1 for d ). The result is that a term in the hybridiza- tion expansion with 2 n hybridization events at times 0 ≤ τ$_{1}$ < τ$_{2}$ < . . . < τ$_{2}$$_{n}$ < β acquires an extra weight factor w$_{screen}$ ( { τ$_{i}$ } ) given by an exponential of correlators 

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of noninteracting boson operators, which evaluates to: 

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It follows from Eqs. (4) and (5) and the illustration in Fig. 1 that the screening contribution amounts to a non- local interaction between all pairs of hybridization events. Since the computational bottleneck is the handling of the hybridization matrix determinants, the additional weight factor w$_{screen}$ does not significantly slow down the simu- lation, which remains very efficient [13]. 

We now illustrate the method with single-site, single- orbital DMFT calculations for a semi-circular density of states of bandwidth D = 4 and inverse temperature β = 50. All calculations are performed in the para- magnetic phase and for half filling. We consider two screening functions: (i) K ( τ ) = − ( λ/ω$_{0}$ ) $^{2}$(cosh(( β/ 2 − τ ) ω$_{0}$ ) − cosh( βω$_{0}$/ 2)) / sinh( βω$_{0}$/ 2) corresponding to a delta-function Im W ( ω ) = − λ $^{2}$π ( δ ( ω − ω$_{0}$ ) − δ ( ω + ω$_{0}$ )) and (ii) K ( τ ) = α ln[1+ βω$_{c}$ sin( πτ/β ) /π )] corresponding to “Ohmic” screening Im W ( ω ) ∼ − απω at small ω , with an ultraviolet cutoff at ω$_{c}$ . Model (i) implies that the screened interaction is U$_{scr}$ = U − 2 λ $^{2}$/ω$_{0}$ and Re W ( ω ) = 2 λ $^{2}$ω$_{0}$ / ( ω 2 − ω 2 $_{0}$); for model (ii) U$_{scr}$ = U − 2 αω$_{c}$ and Re W ( ω ) = αω ln | ( ω$_{c}$ + ω ) / ( ω$_{c}$ − ω ) | − 2 αω$_{c}$ . In princi- ple the coupling strengths can be made large enough to drive U$_{scr}$ negative (overscreening), but in the absence of phonons overscreening is not believed to occur in real ma- terials, limiting the range of physically relevant couplings to λ < √ ω$_{0}$U/ 2 or α < U/ (2 ω$_{c}$ ). Model (i) is a rough represention of results obtained from constrained RPA calculations for Gd and SrVO$_{3}$ and model (ii) captures a characteristic feature of paramagentic Ni [3–5]. 

The calculated metal-insulator phase diagrams for our two models are shown in the left-hand panels of Fig 2. At our simulation temperature the critical U for the metal insulator transition of the unscreened model is U$_{c}$$_{2}$ ( β = 50) ≈ 5 . 1 and the transition is first-order [6]. As expected on physical grounds, increasing the strength of the screening shifts the metal-insulator transition to larger values of the bare interaction. The transition re- mains first order but the coexistence region becomes nar- 

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