4.3 A Renormalization Group Approach to DFT
4.3.1 Derivation of the DFT-RG Flow Equation
96 Renormalization Group and Density Functional Theory
with the energy density of the homogeneousFermigasϵHFG. In comparison to other approximations forExc[n], the LDA has the advantage that it is free of any parameters. Note also that the LDA represents the leading-order term of agradient expansionof the energy density functional. From Eq. (4.26), we can now compute the exchange-correlationcontribution for the KS potentialvKSin LDA. For this, we have to compute the functional derivative of Eq. (4.26) which yields
vxcLDA[n](x) =ϵHFG(n(x)) +n(x)∂ϵHFG(n(x))
∂n(x) . (4.27)
The latter then enters the self-consistent KS equations (4.14). In this section, we only presented a tiny part of all truncation schemes which are employed in DFT these days. For more details on thegradient expansionand a general overview of other approximation schemes in DFT, we refer to Ref. [320].
4.3 A Renormalization Group Approach to DFT 97
where the flow parameter is defined on the finite intervalλ ∈ [0,1]. Note that the external potential does also depend on the regulator choiceRλ. Moreover, the boundary conditions forVλcorrespond to the particular system under consideration and shall be discussed below for the different examples separately.
The regulator functionRλshall gradually turn on the interaction and modify the shape of the external potential in a predefined fashion. For instance, in case ofλ= 0, the system does only consist of non-interacting fermions confined in, e.g., a box with (anti)periodic boundary conditions or in a harmonic trap, respectively. By increasing the auxiliary coupling constantλfrom zero to one, the fermions begin to interact via the two-body potentialU2b. Atλ = 1the interaction is fully turned on and the shape of the confining potentialV is modified according to the choice of boundary conditions. For example, the confining potentialV can be switched off to study selfbound systems. Note that we use throughout this thesisRλ≡λ. However, in future studies one may also consider different types of regulator functions. As we know from relativistic studies, the choice of the regulator function encodes the specific form of the momentum (shell) integrations in a given theory. Depending on the system under consideration, it could be helpful to adapt the regulator function according to the specific problem of interest to increase, e.g., the speed of convergence, see Ref. [110] for a discussion.
In the previous section, we briefly discussed the basic ideas ofcoupling-constantintegration. If we now compare how the parameterλenters the classical actionSλ, we observe similarities. For example, the coupling constant inSλshall also turn on the two-body interaction gradually which is also the case incoupling-constantintegration techniques. As we shall see below, however, our field-theoretical approach directly provides us with information on the ground-state energy, density and higher correlation functions. Furthermore,exchange-correlationeffects are systematically included. The latter are encoded in then-density correlation functions of our theory. Again, the RG flow of the external potential depends on the particular system under consideration. For instance, it is also possible that the external potential remains constant and independent on the flow parameterλ. For example, this is the case in our study of one-dimensional fermions in a periodic box in the next subsection. We also add that the actionSλ, we consider in the present study, contains only a two-body potentialU2b. However, it is possible to include higher N-body interactions as it would be necessary if we, e.g., study nuclei with chiral three-nucleon forces, see Ref. [70]
for a review.
Let us now turn to the derivation of the DFT-RG flow equations. In analogy to the derivation of theWetterich equation, we begin with the partition function
Zλ[J]∼
∫
Dψ∗Dψe−Sλ[ψ∗,ψ]+
∫
τ
∫
xJ(τ,x)(ψ∗(τ,x)ψ(τ,x))
≡eWλ[J], (4.30)
where we coupled a space- and time-dependent sourceJ(τ, x)to the density-type fieldρ ∼ψ∗ψ. Moreover, in-stead of introducing a finite chemical potential, we fix the particle numberNby employing appropriate boundary conditions for the initial conditions of the density correlation functions, see Ref. [321]. Note further that the particle numberNis conserved by the RG flow, see Ref. [110]. We also stress that the external background potentialV(x) in the effective actionSλcan be absorbed completely by the source termJ with the shiftJ → J +V. From a DFT point of view, this is nothing but theuniversalityof the energy density functional as stated byHohenbergand Kohn[65].
From the generating functional
Wλ[J] = lnZλ[J], (4.31)
we can computeconnecteddensity correlation functions by means of functional derivatives. Further, we may expand the functionalWλ[J]in the spirit of a vertex expansion in terms of correlation functions:
Wλ[J] =G(0)λ +
∫
τ
∫
x
G(1)λ (τ, x)J(τ, x) +1 2
∫
τ1
∫
x1
∫
τ2
∫
x2
G(2)λ (τ1, x1, τ2, x2)J(τ1, x1)J(τ2, x2) +. . . .(4.32)
98 Renormalization Group and Density Functional Theory
Here, we obtain the density correlation functionG(n)λ by performingnfunctional derivatives with respect to the source and set the latter to zero. For example, we compute the time-dependent ground-state density via
ρgs,λ(τ, x) :=G(1)λ (τ, x) = δWλ[J] δJ(τ, x) J=0
. (4.33)
The time-dependentn-density correlation function can then be computed via G(n)λ (τ1, x1, . . . , τn, xn) = δnWλ[J]
δJ(τ1, x1)· · ·δJ(τn, xn) J=0
. (4.34)
We stress that we require an infinite number of correlation functions even for the description of a non-interacting system. Note further that we define the time-independent ground-state density in the following way
ngs,λ(x) := lim
β→∞
1 β
∫ β/2
−β/2
dτ ρgs,λ(τ, x), (4.35)
whereβdefines the extent of the imaginary time interval withτ ∈[−β/2, β/2)and is, moreover, associated with the inverse temperatureβ = 1/T. Note that we consider throughout the present workβ → ∞. In the next step, we define the 2PPI effective action using the functional analogue of aLegendretransformation:
Γλ[ρ] = sup
J
{
−Wλ[J] +
∫
τ
∫
x
J(τ, x)ρ(τ, x) }
, (4.36)
where the supremum ensures the convexity of the density functionalΓλ[ρ], see Ref. [110]. Moreover, the 2PPI classical field can be computed from
ρ(τ, x) = δWλ[J]
δJ(τ, x). (4.37)
Note that this is again in correspondence to the 1PI classical field entering theWetterichequation. As indicated above, the 2PPI effective actionΓλ[ρ]is related to theHohenberg-Kohnenergy density functional. In contrast to conventional DFT, we use time-dependent source termsJ ≡ J(τ, x)implying time-dependent correlation func-tions, see Eq. (4.34). For field-theoretical approaches using time-independent sources, see, e.g., Refs. [66, 110, 321–
325]. The energy density functional can then be computed from Eλ[ρ] = lim
β→∞
1
βΓλ[ρ], (4.38)
where the ground-state energy can be extracted from the ground-state density:
Egs,λ= lim
β→∞
1
βΓλ[ρgs,λ] =− lim
β→∞
1
βWλ[0]. (4.39)
The expression above can be verified from the spectral decomposition of the partition functionZλ∼∑
ne−βEn,λ where we assumeEn,λ< En+1,λwithn≥0. It can further be shown that the functionalΓλ[ρ]does not depend on the sourceJ, as it should be:
δΓλ[ρ]
δJ ρ
= 0, (4.40)
4.3 A Renormalization Group Approach to DFT 99
for fixedρ. Moreover, it is straightforward to show that δΓλ[ρ]
δρ(x) =J(x), (4.41)
which is the 2PPI version of theHohenberg-Kohnvariational principle from which we deduce forJ →0the ground-state densityρgs,λ. The latter is required to compute theλ-dependent ground-state energyEgs,λin our formulation.
Further, one can show that two-density correlation functions associated with the functionalW[J]and the two-point function associated withΓ[ρ]are related in the following sense:
δ2W[J] δJ δJ =
(δ2Γ[ρ]
δρ δρ )−1
. (4.42)
Higher-order 2PPI correlation functions can then be computed from the connected ones accordingly, see Ref. [110].
Let us now discuss the construction of then-density correlation functions. For this, we note that the density correlation functions can be constructed from one-particle propagators. In our present work, we shall use this property to derive the two-density and three-density correlation function. In particular, we find for the one-particle propagator:
∆0(τ1, x1, τ2, x2) = − ⟨Tψ(τ1, x1)ψ∗(τ2, x2)⟩
= − ⟨ψ(τ1, x1)ψ∗(τ2, x2)⟩θσ(τ1−τ2) +⟨ψ∗(τ2, x2)ψ(τ1, x1)⟩θσ(τ2−τ1),(4.43) whereT denotes the time-ordering operator and
θσ(τ) =
{ 1 for τ >0 and τ→0+,
0 otherwise. (4.44)
The time-dependent ground-state density of the non-interacting theory can be extracted from the one-particle prop-agator which we have defined above:
ρgs,λ=0(τ1, x1) = lim
τ2→τ1+
∆0(τ1, x1, τ2, x1). (4.45) Note further that the most generaln-density correlation function can be derived from one-particle propagators via
G(n)λ=0(χ1, . . . , χn) = (−1)n+1 n
∑
(i1,...,in)∈Sn
∆0(χi1, χi2)∆0(χi2, χi3)· · ·∆0(χin−1, χin), (4.46)
withχ = (τ, x). Moreover,Sn describes all possible permutations of the indicesi1, . . . , in. For example, the (non-interacting) two- and three-density correlation functions read
G(2)λ=0(χ1, χ2) =−∆0(χ2, χ1)∆0(χ1, χ2), (4.47) and
G(3)λ=0(χ1, χ2, χ3) = ∆0(χ1, χ2)∆0(χ2, χ3)∆0(χ3, χ1) + ∆0(χ2, χ1)∆0(χ1, χ3)∆0(χ3, χ2). (4.48) Both identities play a crucial role in ourKohn-Sham-improved DFT-RG approach in which we automatically con-struct the two- and three-density correlation functions from one-particle propagators within our numerical set-up, see Sec. 4.3.3.
Let us now discuss the functional RG equation for the generating functionalWλ[J]. The latter can be derived
100 Renormalization Group and Density Functional Theory
straightforwardly from Eq.(4.31) by taking the derivative with respect to the flow parameterλ. From this, we find
∂λWλ[J] = −
∫
χ1
(∂λVλ(χ1))δWλ[J]
δJ(χ1) −1 2
∫
χ1
∫
χ2
δWλ[J]
δJ(χ1)U2b(χ1, χ2) (∂λRλ(χ1, χ2))δWλ[J] δJ(χ2)
−1 2
∫
χ1
∫
χ2
U2b(χ1, χ2) (∂λRλ(χ1, χ2))
( δ2Wλ[J]
δJ(χ2)δJ(χ1)−δWλ[J]
δJ(χ2)δ(χ2−χ1) )
, (4.49) where we use the shorthand notations∫
χ =∫
x
∫
τandδ(χ1−χ2) =δ(x1−x2)δ(τ1−τ2). We add that the DFT-RG flow equation (4.49) can be generalized to higher dimensions. Furthermore, fermions with internal degrees of freedom can be considered as well, see, e.g., Ref. [108] for a study of ultracold spin-1/2fermions in a box with periodic boundary conditions.
The flow equation above describes the change of the generating functional under a gradual variation of the flow parameterλ: One starts the RG flow atλ= 0where the theory is free and the interaction is switched off. At this point, the theory is well understood and then-density correlation functions we gave in Eq. (4.46) can be computed straightforwardly. However, we emphasize that an exact description of the non-interacting theory still requires an infinite number ofn-density correlation functions. The DFT-RG flow equation (4.49) then interpolates between the non-interacting system atλ = 0and the interacting many-body problem atλ = 1where the interaction is fully switched on. We stress that this is done in a non-perturbative fashion as we automatically include arbitrarily high orders of the interactionU2bby solving the evolution equation (4.49).
We further observe that by considering an expansion scheme like Eq. (4.32), we receive an infinite tower of coupled differential equations for the differentn-density correlation functions. For instance, we find that the m-density correlation functionG(m)λ depends, in general, on all density correlation functions with order1 ≤m ≤ (n+ 2)as it can be verified from the structure of the DFT-RG flow equation (4.49). In practice, we therefore need to truncate the tower of flow equations at a given order. For this, it can be shown that results are improved significantly by considering truncations which entailG(n+1)λ=0 andG(n+2)λ=0 within the flow equation forG(n)λ , even if we set∂λG(m)λ = 0form≥(n+ 1). The reason for this observation relies on the structure of the coupled set of DFT-RG flow equations which can also be related to many-body perturbation theory. Using truncations for the tower of RG flow equations as described above, we could notably reduce certain truncation artifacts, for example, spurious fermion self-interactions, see Ref. [110] for details.
We already mentioned that the functionalsWλ[J]andΓλ[ρ]are related and can be transformed into each other.
In fact, it is straightforward to compute a functional Renormalization Group equation for the density functional Γλ[ρ]. For the latter, we find
∂λΓλ[ρ] =
∫
χ1
(∂λVλ(χ1))ρ(χ1) +1 2
∫
χ1
∫
χ2
ρ(χ1)U2b(χ1, χ2) (∂λRλ(χ1, χ2))ρ(χ2) +1
2
∫
χ1
∫
χ2
U2b(χ1, χ2) (∂λRλ(χ1, χ2))
((δ2Γλ[ρ]
δρ δρ )−1
(χ2, χ1)−ρ(χ2)δ(χ2−χ1) )
. (4.50) Interestingly, the DFT-RG equation for the functionalWλ[J]as well as the flow equation for the density functional Γλ[ρ]look very similar. Nevertheless, we emphasize that the correlation functions which enter our flow equations are not identical, see Ref. [110]. The functionalWλ[J]relies on connected density correlation functions Wλ(n) entering our computations, whereas 2PPI correlation functions underlie the DFT-RG flow of the density functional Γλ[ρ]. We further stress that the computation of initial conditions for the 2PPI flow equation (4.50) requires an inversion of the two-density correlation functionG(2)λ . The latter can be computed from the one-particle propagator given in Eq. (4.43). However, while an inversion ofG(2)λ can be simple for theories obeying a translation symmetry in space and time, it can be involved as soon as translation invariance is broken because of, e.g., an external potential.
Therefore, throughout the present thesis, we use Eq. (4.49) and consider connectedn-density correlation functions as the building blocks in our coupled set of DFT-RG flow equations.
4.3 A Renormalization Group Approach to DFT 101
We observe that the structures of the flow equations (4.49) and (4.50) are related to the one we found for the Wetterichequation (1PI) (2.23) as they all exhibit a simple one-loop structure. As already indicated, however, this does by no means imply that only one-loop contributions in the sense of perturbation theory are present. Of course, there are also several differences on a technical level. For example, the regulator entering theWetterichequation is inserted in the kinetic part of the classical action which is not the case in our DFT-RG framework. Here, we find that the latter is inserted alongside the two-body interaction of the classical action.
Moreover, we note that theCallan-Symanzikequation [133, 134] and the DFT-RG flow equations above are also related. Originally, theCallan-Symanzikequation was derived to investigate the scaling behavior of the correlation functions under a variation of the renormalized mass parameter. In our case, we study the scaling relations of the density correlation functions under a variation of the scale parameterλ, see Ref. [110] for a detailed discussion.
Let us further analyze the DFT-RG equations (4.49) and (4.50). The part∼ ∫
χ1,χ2ρ(χ1)U(χ1, χ2)ρ(χ2) ap-pearing in both DFT-RG equations is often referred to as theHartreeterm. We already encountered such a con-tribution in our discussion of conventional DFT, see Eq. (4.16). There, we stated that the latter does not include exchange-correlationeffects as they are included inFock-type contributions. For this, we find in our present DFT-RG approach the term∼∫
χ1,χ2U(χ1, χ2)W(2)(χ2, χ1)encoding allexchangeandcorrelationcontributions of the many-body problem under consideration. Moreover, we also find a term∼∫
χ1,χ2ρ(χ2)δ(χ2−χ1)present in our evolution equation. The latter is important as it implements thePauliexclusion principle for identical fermions, see our discussion in Ref. [110].