Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

Csirik, Mihály A.; Laestadius, Andre; Tellgren, Erik I.

doi:10.1063/5.0198786

Although the concept of the uniform electron gas is essential to quantum physics, it has only been defined recently in a rigorous manner by Lewin, Lieb and Seiringer. We extend their approach to include the magnetic case, by which we mean that the vorticity of the gas is also held constant. Our definition involves the grand-canonical version of the universal functional introduced by Vignale and Rasolt in the context of current-density-functional theory. Besides establishing the existence of the thermodynamic limit, we derive an estimate on the kinetic energy functional that also gives a convenient answer to the (mixed) current-density representability problem.

I. INTRODUCTION

Following Ref. 1, we view the uniform electron gas (UEG) as a continuous system of (interacting or not) electrons with prescribed constant density everywhere. In the non-interacting case, one can readily write down the ground state of the UEG via a fully occupied Fermi sphere. One can consider the classical UEG by neglecting the kinetic energy of the electrons.¹

In contrast, the Jellium is a continuous system of interacting electrons in a constant neutralizing potential.² In the classical case, it was recently shown that the Jellium and the UEG energies are the same.³ In the quantum case, the problem is still open. We will solely focus on the UEG in this article.

Reference 1 pioneered the use of universal density functionals of density-functional theory (DFT) as a theoretical tool in quantum statistical mechanics, namely for the definition of the UEG. In a later article,⁴ the same authors established an even more elegant (and equivalent) definition of the UEG, and we will restrict ourselves to this latter approach.

When magnetic fields are present, the situation is much more delicate.⁵ Even the noninteracting UEG is more difficult to analyze in the magnetic case,^6–8 although the ground state can be given exactly. The celebrated Laughlin wave function has also received attention from the mathematics community,^9,10 which is a rather precise trial state (with almost constant density in the thermodynamic limit) for the Jellium in a constant magnetic field.

II. SETTING

A. States

In the first half of this section, we will fix the particle number $N \in N$ ⁠. For simplicity of notation, we disregard spin in this work. The N-particle fermionic Hilbert space is the antisymmetric power $H^{N} = ⋀^{N} H$ (endowed with the usual inner product), where $H = L^{2} (R^{d})$ is the one-particle Hilbert space. Note that $H^{1} = H$ ⁠.

Generally speaking, a density matrix (or state) on a Hilbert space is self-adjoint, non-negative trace class operator that has trace one. We first consider the Hilbert space to be the fermionic N-particle space

H^{N}

⁠. To make notations more compact, we henceforth restrict our attention to states with finite kinetic energy, which we call H¹-regular states. The set of H¹-regular (fermionic) N-particle states is defined as

D_{N} = \{\begin{aligned} Γ & \in S_{1} (H^{N}) : Γ = \sum_{j ⩾ 1} μ_{j} | Ψ_{j} 〉 〈 Ψ_{j} |, for some μ_{j} ⩾ 0 with \sum_{j ⩾ 1} μ_{j} = 1, and \\ {Ψ_{j}} \subset H^{N} \cap H^{1} (R^{d N}) is L^{2} - orthonormal with \sum_{j ⩾ 1} μ_{j} ‖ \nabla Ψ_{j} ‖^{2} < \infty \end{aligned}\},

where

S_{1}

the trace class. Then clearly 0⩽Γ = Γ^† and

{T r}_{H^{N}} Γ = 1

for all

Γ \in D_{N}

⁠. It is not hard to show that the operator Γ can be given by the Hermitian integral kernel

Γ : R^{d N} \times R^{d N} \to C

(denoted with the same symbol),

Γ (X; Y) = \sum_{j ⩾ 1} μ_{j} Ψ_{j} (X) \bar{Ψ_{j} (Y)},

so that its action on an element

Φ \in H^{N}

is given by ΓΦ(X) = ∫Γ(X; Y)Φ(Y) dY.

Let

A : H^{N} \to R

be a nonnegative quadratic form with domain in

H^{N} \cap H^{1} (R^{d N})

⁠, and let

A : H^{N} \to H^{N}

be its self-adjoint realization. Then for any

Γ \in D_{N}

⁠, we define

{T r}_{H^{N}} A Γ ≔ \sum_{j ⩾ 1} μ_{j} A (Ψ_{j}),

which is a nonnegative number or +∞. Choosing A = −Δ, i.e.,

A (Ψ) = ‖ \nabla Ψ ‖^{2}

⁠, we find that

{T r}_{H^{N}} (- Δ Γ) = \sum_{j ⩾ 1} μ_{j} ‖ \nabla Ψ_{j} ‖^{2} < \infty

for any

Γ \in D_{N}

⁠.

Next, we consider states on Fock space. For convenience, we introduce the notation

H^{0} = C

⁠. The fermionic Fock space

F

is then the Hilbert space given by direct sum of the Hilbert spaces

H^{n}

⁠,

F = H^{0} \oplus H^{1} \oplus H^{2} \oplus \dots

endowed with the usual inner product. A Fock space state Γ is a positive self-adjoint, trace-class operator on

F

with unit trace:

Γ \in S_{1} (F)

⁠, 0⩽Γ = Γ^† and

{T r}_{F} Γ = 1

⁠. We will not need this generality, however, and it will suffice to consider states Γ which commute with the number operator. It is easy to show by induction that such states are block-diagonal:

Γ = Γ_{0} \oplus Γ_{1} \oplus \dots,

where

Γ_{n} \in S_{1} (H^{n})

is such that

0 ⩽ Γ_{n} = Γ_{n}^{†}

⁠. Note in particular that

Γ_{0} \in R_{+}

⁠, and that the normalization constraint now reads

\sum_{n ⩾ 0} {T r}_{F} Γ_{n} = 1

⁠. Again, we will further restrict attention to states with finite kinetic energy, which we called H¹-regular states. All this leads to the definition

D = \{\begin{aligned} Γ = Γ_{0} \oplus Γ_{1} \oplus \dots \in S_{1} (F) : & Γ_{n} \in S_{1} (H^{n}) and 0 ⩽ Γ_{n} = Γ_{n}^{†} for all n ⩾ 0, \\ \sum_{n ⩾ 0} T r Γ_{n} = 1 and \sum_{n ⩾ 0} {T r}_{H^{n}} (- Δ Γ_{n}) < \infty \end{aligned}\}

of the set of H¹-regular block-diagonal Fock space states. Each Γ_n may be given by an integral kernel Γ_n(X_n; Y_n), like before (of course these will not be normalized to have unit trace, in general). Note also that by linearity, the one-, and the two-particle reduced density matrices of

Γ = Γ_{0} \oplus Γ_{1} \oplus \dots \in D

are given by

γ_{Γ} = \sum_{n ⩾ 0} γ_{Γ_{n}}, and γ_{Γ}^{(2)} = \sum_{n ⩾ 0} γ_{Γ_{n}}^{(2)} .

We note in passing that all derived quantities, such as ρ_Γ, may be defined by linearity.

B. One-particle quantities

In this section, we introduce various quantities which may be defined using the one-particle density matrix. The one-particle reduced density matrix

γ \in S_{1} (H)

of

Γ \in D_{N}

is defined as (N times) the partial trace of Γ via duality:

{T r}_{H} (γ B) = N {T r}_{H^{N}} (Γ (B \otimes 1_{H^{N - 1}})) for all B \in B (H) .

In symbols, this operation is sometimes written as

γ = N {T r}_{H^{N - 1}} Γ

⁠. Note the choice of normalization, so that we have

{T r}_{H} γ = N

⁠. It is easy to see that γ can be given by the kernel

γ (x, y) = N \int_{R^{d (N - 1)}} Γ (x, x_{2}, \dots, x_{N}; y, x_{2}, \dots, x_{N}) d x_{2} \dots d x_{N} .

It is well-known that γ is a trace class operator on the one-particle Hilbert space

H

such that

0 ⩽ γ = γ^{†} ⩽ 1

⁠. By the spectral theorem γ has the decomposition γ = ∑_j⩾1λ_j|φ_j⟩⟨φ_j|, which in terms of its kernel can be written as

γ (x, y) = \sum_{j ⩾ 1} λ_{j} φ_{j} (x) \bar{φ_{j} (y)}

for some 0⩽λ_j⩽1 with ∑_j⩾1λ_j = N and

{φ_{j}}_{j ⩾ 1} \subset H^{1} (R^{d}) L^{2}

-orthonormal. It is clear that γ has finite kinetic energy, i.e., it is H¹-regular: ∑_j⩾1λ_j‖∇φ_j‖² < ∞.

The density

ρ_{γ} \in L^{1} (R^{d})

of the one-particle density matrix γ may also be defined via duality, i.e., by requiring that

\int_{R^{d}} ρ_{γ} φ = {T r}_{H} (γ φ)

for all

φ \in L^{\infty} (R^{d})

⁠. This gives

ρ_{γ} (x) = \sum_{j ⩾ 1} λ_{j} | φ_{j} (x) |^{2} .

The kinetic energy

{T r}_{H} (- Δ γ) ≔ \sum_{j ⩾ 1} λ_{j} ‖ \nabla φ_{j} ‖^{2} < \infty

may be expressed using the kernel γ(x, y) as

T r (- Δ γ) = \int_{R^{d}} \nabla_{x} \cdot \nabla_{y} γ (x, y) |_{y = x} d x .

Here, the integrand τ_γ(x)≔∇_x ⋅∇_yγ(x, y)|_y=x is called the kinetic energy density, and more generally

τ_{γ} (x) ≔ \nabla_{x} \otimes \nabla_{y} γ (x, y) |_{y = x} = \sum_{j ⩾ 1} λ_{j} \nabla φ_{j} (x) \otimes \bar{\nabla φ_{j} (x)}

the kinetic energy tensor. Clearly, 0⩽τ_γ(x) = τ_γ(x)^† and

{T r}_{R^{3}} τ_{γ} (x) = τ_{γ} (x)

⁠.

A further useful quantity is the complex current density

ζ_{γ} : R^{d} \to C^{d}

⁠, which is given by

ζ_{γ} (x) ≔ \nabla_{x} γ (x, y) |_{y = x} = \sum_{j ⩾ 1} λ_{j} \nabla φ_{j} (x) \bar{φ_{j} (x)} .

(1)

A straightforward calculation shows that

R e ζ_{γ} (x) = \frac{1}{2} \nabla ρ_{γ} (x), and j_{γ}^{p} (x) ≔ I m ζ_{γ} (x) = I m \nabla_{x} γ (x, y) |_{y = x},

(2)

where

j_{γ}^{p} : R^{d} \to R^{d}

is called the paramagnetic current density.

The following theorem is based on,^11–13 the proof may be found toward the end of the paper.

Proposition II.1.

Suppose that γ is fermionic one-particle density matrix composed of sufficiently smooth orbitals. Then the following hold true.

$| \nabla \sqrt{ρ_{γ}} (x) |^{2} + \frac{| j_{γ}^{p} (x) |^{2}}{ρ_{γ} (x)} + \frac{1}{\sqrt{d}} ρ_{γ} (x) |D_{a} \frac{j_{γ}^{p} (x)}{ρ_{γ} (x)}| ⩽ τ_{γ} (x)$ for all $x \in R^{d}$ ⁠.
$| j_{γ}^{p} (x) | ⩽ | ζ_{γ} (x) | ⩽ τ_{γ} {(x)}^{1 / 2} ρ_{γ} {(x)}^{1 / 2}$ for all $x \in R^{d}$ , hence $j_{γ}^{p} \in L^{1} {(R^{d})}^{d}$ ⁠.

Here, the symmetric-, and antisymmetric derivatives D_s and D_a will be used at some occasions, given by

D_{s} u = \frac{D u + {(D u)}^{⊤}}{2} and D_{a} u = \frac{D u - {(D u)}^{⊤}}{2},

for any differentiable vector field

u : R^{d} \to R^{d}

and

D u : R^{d} \to R^{d \times d}

is its derivative. Further, we denote by A: B = Tr(A^⊤B) the Frobenius inner product, and by

| A | = \sqrt{A : A}

the Frobenius norm.

Let us consider the case d = 3. Then

|D_{a} u| = \frac{\sqrt{2}}{2} | r o t u |

and part (i) improves the usual Hoffmann–Ostenhof bound¹⁴ as

T r (- Δ γ) ⩾ \int_{R^{3}} | \nabla \sqrt{ρ_{γ}} |^{2} + \int_{R^{3}} \frac{| j_{γ}^{p} |^{2}}{ρ_{γ}} + \frac{1}{\sqrt{6}} \int_{R^{3}} ρ_{γ} |ν_{γ}|,

(3)

for any H¹-regular density matrix γ, where

ν_{γ} = r o t \frac{j_{γ}^{p}}{ρ_{γ}}

is the (gauge-invariant, see below) vorticity. We call the terms on the r.h.s. the von Weizsäcker-, the gauge-, and the vorticity term, respectively. Note that implies that for H¹-regular density one-particle matrices γ, we have

\int_{R^{3}} | \nabla \sqrt{ρ_{γ}} |^{2} < \infty

⁠,

\int_{R^{3}} \frac{| j_{γ}^{p} |^{2}}{ρ_{γ}} < \infty

and

\int_{R^{3}} ρ_{γ} |ν_{γ}| < \infty

⁠, quantities which play important roles later. Finally, we note that for d = 3 the constant in front of vorticity term in can be improved to 1, see Ref. 12.

Remark 1.

Part (ii) of Proposition II.1 implies that the support of j^p is contained in the support of ρ, in particular, the quotients $\frac{j^{p}}{\sqrt{ρ}}$ and $\frac{| j^{p} |^{2}}{ρ}$ are meaningful on suppρ. Moreover, if τ ≡ 0 in some ball, then by (i), ρ is constant and j^p ≡ 0 on that ball.

C. Gauge transformation

We now discuss a very important tool in the study of magnetic systems. Given an N-particle state

Γ \in D_{N}

⁠, consider the gauge-transformed state

\tilde{Γ} (X, Y) = e^{i \sum_{1 ⩽ j ⩽ N} (g (y_{j}) - g (x_{j}))} Γ (X, Y)

where g is called the gauge function. The following is immediate.

Proposition II.2.

Let γ be a fermionic one-particle density matrix. Then the gauge transformation of γ is given by

\tilde{γ} (x, y) = e^{i (g (y) - g (x))} γ (x, y) .

In particular,

ρ_{\tilde{γ}} = ρ_{γ}

. Furthermore,

j_{\tilde{γ}}^{p} = j_{γ}^{p} - ρ_{γ} \nabla g

and

T r (- Δ \tilde{γ}) = T r (- Δ γ) - 2 \int_{R^{d}} j_{γ}^{p} \cdot \nabla g + \int_{R^{d}} ρ_{γ} | \nabla g |^{2} .

(4)

We did not specify what function space the gauge function g is coming from. In order for the r.h.s. of (4) to make sense, it is enough to assume that $\nabla g \in L_{ρ_{γ}}^{2} {(R^{d})}^{d}$ ⁠, using the Cauchy–Schwarz inequality and (3). Here and onwards, $L_{ρ}^{p}$ stands for the ρ-weighted L^p space.

Note that the gauge term

\int_{R^{d}} \frac{| j^{p} |^{2}}{ρ}

transforms exactly like the kinetic energy . More precisely,

\int_{R^{d}} \frac{| j_{\tilde{γ}}^{p} |^{2}}{ρ_{\tilde{γ}}} = \int_{R^{d}} \frac{| j_{γ}^{p} |^{2}}{ρ_{γ}} - 2 \int_{R^{d}} j_{γ}^{p} \cdot \nabla g + \int_{R^{d}} ρ_{γ} | \nabla g |^{2},

therefore the inequality is “consistent” with a gauge transformation in the sense that all the terms involving g will cancel. We call such inequalities gauge-invariant.

D. Representability

Fix

N \in N

⁠. Recall that the set of pure (or mixed) state N-representable densities may be described as

I_{N} = \{ρ \in L^{1} (R^{d}; R_{+}) : \nabla \sqrt{ρ} \in L^{2} {(R^{d})}^{d}, \int_{R^{d}} ρ = N\} .

according to Refs. 15 and 16. For later purposes, we also introduce the set of Fock space-representable densities as

I = \{ρ \in L^{1} (R^{d}; R_{+}) : \nabla \sqrt{ρ} \in L^{2} {(R^{d})}^{d}\} .

For density-paramagnetic current density pairs, the situation is more complicated. Consider the set

J_{N}^{p u r e} = \{(ρ, j^{p}) : ρ = ρ_{Ψ}, j^{p} = j_{Ψ}^{p}, for some Ψ \in H^{N} \cap H^{1} (R^{d N}), ‖ Ψ ‖ = 1\}

of all pure state N-representable density-current pairs, and

J_{N}^{m i x e d} = {(ρ, j^{p}) : ρ = ρ_{Γ}, j^{p} = j_{Γ}^{p}, for some Γ \in D_{N}},

the set of mixed state N-representable density-current pairs. In this more general setting it is yet unknown whether

J_{N}^{p u r e}

is equal to

J_{N}^{m i x e d}

or not. Unfortunately, no explicit characterization of either

J_{N}^{p u r e}

or

J_{N}^{m i x e d}

exists up to date. We can specify a larger set by setting

J_{N}^{'} = \{(ρ, j^{p}) : ρ \in I_{N}, j^{p} \in L^{1} {(R^{d})}^{d}, \int_{R^{d}} \frac{| j^{p} |^{2}}{ρ} < \infty\},

(5)

so that by ,

J_{N}^{p u r e}, J_{N}^{m i x e d} \subset J_{N}^{'}

⁠. It can be shown that

J_{N}^{'}

is also convex.¹⁷ We remark that the finiteness of the vorticity term in may also be added to the definition of

J_{N}^{'}

⁠.

Whether any $(ρ, j^{p}) \in J_{N}^{'}$ can be represented with a Slater determinant $Φ \in H^{N} \cap H^{1} (R^{3 N})$ such that ρ_Φ = ρ and $j_{Φ}^{p} = j^{p}$ (the determinantal density-current N-representability problem) is still not settled completely. The following result gives a partial answer, however.

Theorem II.3.

(Lieb–Schrader¹⁸). Let d = 3 and suppose that $(ρ, j^{p}) \in J_{N}^{'}$ and set $ν = r o t \frac{j^{p}}{ρ}$ . Then the following holds true.

(Zero vorticity) If ν = 0, then there exists a Slater determinant $Φ \in H^{N} \cap H^{1} (R^{3 N})$ , such that ρ_Φ = ρ, $j_{Φ}^{p} = j^{p}$ and
$‖ \nabla Φ ‖^{2} ⩽ C_{N} \int_{R^{3}} | \nabla \sqrt{ρ} |^{2} + \int_{R^{3}} \frac{| j^{p} |^{2}}{ρ}$
for some constant C_N > 0.
If N⩾4, then there exists a Slater determinant $Φ \in H^{N}$ , such that ρ_Φ = ρ, $j_{Φ}^{p} = j^{p}$ . Moreover, if there is a δ > 0 such that the growth conditions
$\sup_{x \in R^{3}} f (x) | ν (x) | < \infty, and \sup_{x \in R^{3}} f (x) | D ν (x) | < \infty$
hold true, then ‖∇Φ‖ < ∞. Here, we have set $f (x) = {(1 + x_{1}^{2})}^{\frac{1 + δ}{2}} {(1 + x_{2}^{2})}^{\frac{1 + δ}{2}} {(1 + x_{3}^{2})}^{\frac{1 + δ}{2}}$ ⁠.
Moreover, Ref. 18 also exhibits an example (ρ, j^p) in the case N = 2 (and of course ν ≠ 0), when there is no C¹-solution to the representability problem. Unfortunately, the restriction N⩾4 and the fact that no control over the kinetic energy is retained in (ii) renders the preceding result inapplicable to us.

The following theorem establishes mixed representability under different assumptions, now for general $N \in N$ ⁠.

Theorem II.4.

(Tellgren–Kvaal–Helgaker¹⁹ ). Let d = 3 and suppose that

(ρ, j^{p}) \in J_{N}^{'}

and that

\int_{R^{3}} (1 + | x |^{2}) ρ {|D \frac{j^{p}}{ρ}|}^{2} d x < \infty .

(6)

Then there exists a constant C > 0 and a fermionic one-particle density matrix γ such that ρ_γ = ρ,

j_{γ}^{p} = j^{p}

and

T r (- Δ γ) ⩽ \int_{R^{3}} | \nabla \sqrt{ρ} |^{2} + C N + \int_{R^{3}} \frac{| j^{p} |^{2}}{ρ} + C \int_{R^{3}} (1 + | x |^{2}) ρ {|D \frac{j^{p}}{ρ}|}^{2} d x .

(7)

Representability holds also true without the integer particle number constraint on the density, i.e., when

\int_{R^{d}} ρ \notin N

⁠, but the representing state will live in the Fock space

F

⁠. Analogously to the N-particle case, we define the set of Fock space-representable density-current pairs as

J ≔ {(ρ, j^{p}) : ρ = ρ_{Γ}, j^{p} = j_{Γ}^{p} for some Γ \in D} .

Then the conditions of Theorem II.4 provide a smaller set, but we will use an improved version below.

E. Definition of the grand-canonical current-density functionals

In this section, we define the current-density functionals relevant to our study. These are well-known in CDFT, but we will also employ them in the definition of the uniform electron gas. In this section we restrict ourselves to the physically most relevant 3D case.

We begin by introducing the grand-canonical Vignale–Rasolt functional²⁰ via

\begin{matrix} F (ρ, j^{p}) ≔ \inf_{\begin{array}{c} Γ \in D \\ ρ_{Γ} = ρ \\ j_{Γ}^{p} = j^{p} \end{array}} \sum_{n ⩾ 1} {T r}_{H^{n}} (- \sum_{1 ⩽ j ⩽ n} Δ_{x_{j}} + \sum_{1 ⩽ j < k ⩽ n} \frac{1}{| x_{j} - x_{k} |}) Γ_{n} . \end{matrix}

(8)

Omitting the Coulomb interaction in the above definition, we obtain the mixed-state kinetic energy functional T(ρ, j^p). By definition, T and F is (jointly) convex on

J

⁠. In our analysis we will consider the indirect energy

\begin{matrix} E (ρ, j^{p}) ≔ F (ρ, j^{p}) - D (ρ) \end{matrix}

(9)

where for any finite measures μ₁ and μ₂, we define the Coulomb inner product via

D (μ_{1}, μ_{2}) ≔ \frac{1}{2} \iint_{R^{3} \times R^{3}} \frac{μ_{1} (d x) μ_{2} (d y)}{| x - y |},

and additionally, as an abuse of notation, we set D(μ)≔D(μ, μ) for the direct energy of μ. If the measures μ_j come from densities ρ_j, i.e., dμ_j = ρ_j dx (j = 1, 2), then we will use the notations D(ρ₁, ρ₂) and D(ρ).

For convenience, we introduce the gauge-corrected indirect energy

\begin{matrix} \bar{E} (ρ, j^{p}) ≔ E (ρ, j^{p}) - \int_{R^{3}} \frac{| j^{p} |^{2}}{ρ} \end{matrix}

and similarly

\bar{T} (ρ, j^{p})

⁠. We will also consider the exchange-correlation functional

\begin{matrix} E^{x c} (ρ, j^{p}) ≔ E (ρ, j^{p}) - T (ρ, j^{p}) \end{matrix}

which is more common in the physics and chemistry literature.

The reason for the introduction of the “gauge-corrected” quantities is the following. Note that under a gauge transformation

\tilde{j^{p}} = j^{p} - ρ \nabla g

⁠, the functional F transforms as

F (ρ, \tilde{j^{p}}) = F (ρ, j^{p}) - 2 \int_{R^{3}} j^{p} \cdot \nabla g + \int_{R^{3}} ρ | \nabla g |^{2},

(10)

i.e., like the kinetic energy (see Proposition II.2). Analogous relations holds for T(ρ, j^p) and E(ρ, j^p). It follows that the gauge-corrected indirect-, and kinetic energy

\bar{E} (ρ, j^{p})

and

\bar{T} (ρ, j^{p})

⁠, and exchange-correlation functional E^xc(ρ, j^p) are all gauge-invariant.

By replacing the N-particle Hilbert space with the corresponding Fock space, the next result follows along similar lines as the one in Ref. 21.

Theorem II.5.

Let $(ρ, j^{p}) \in J$ . Then the infima in the definitions (8) of the Vignale–Rasolt functional F and the kinetic energy functional T are both attained.

The grand-canonical Levy–Lieb functional F(ρ) (no paramagnetic current density argument) and the kinetic energy functional T(ρ) is defined similarly to F(ρ, j^p) and T(ρ, j^p), only without the constraint on $j_{Γ}^{p}$ in the infimum.

Proposition II.6.

(Correspondence with the nonmagnetic case). For any

ρ \in I

, the relations T(ρ, 0) = T(ρ) and F(ρ, 0) = F(ρ) hold true. In particular,

E (ρ, 0) = E (ρ) and E^{x c} (ρ, 0) = E^{x c} (ρ),

where E(ρ) = F(ρ) − D(ρ) and E^xc(ρ) = E(ρ) − T(ρ).

Proof.

By convexity and time-reversal symmetry, we find $F (ρ, 0) ⩽ \frac{1}{2} F (ρ, j^{p}) + \frac{1}{2} F (ρ, - j^{p}) = F (ρ, j^{p})$ ⁠. Hence, $F (ρ) ⩽ F (ρ, 0) ⩽ \inf_{j^{p}} F (ρ, j^{p}) = F (ρ)$ as desired. The proof of the relation between the kinetic energy functionals is similar. □

III. MAIN RESULTS

A. The magnetic uniform electron gas

This section is devoted to the precise definition of the various energetic quantities of the 3D uniform electron gas (UEG) with constant density and constant vorticity.

Recall that the vorticity of the ground state of the noninteracting UEG in a constant magnetic field B₀ is ν₀ = −B₀. Physically, we can generate a constant vorticity ν₀ in the noninteracting UEG by subjecting it to an external magnetic field −B₀. Moreover, linear response theory suggests the same holds true to the first order in the interacting case⁵. Instead of referring to the external magnetic field, we will simply demand the UEG to have constant density and constant vorticity.

A function $0 ⩽ η \in C_{c}^{\infty} (R^{3})$ such that suppη ⊂ B₁, $\int_{R^{3}} x η (x) d x = 0$ and $\int_{R^{3}} η = 1$ will be called a regularization function. Note that this is a slightly more general definition than the one in Refs. 1 and 4, where it is required that a regularization function be radial. We also introduce the notation η_δ(x) = δ⁻³η(δ⁻¹x).

Define the indirect gauge-corrected energy per volume of the (interacting) UEG at constant density

ρ_{0} \in R_{+}

and constant vorticity

ν_{0} \in R^{3}

inside a bounded domain

Ω \subset R^{3}

smeared with η as

\begin{matrix} e_{Ω, η} (ρ_{0}, ν_{0}) ≔ \frac{1}{| Ω |} \bar{E} ((1_{Ω} * η) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) \end{matrix}

(11)

Note that

r o t (\frac{1}{2} ν_{0} \times x) = ν_{0}

and that e_Ω,η(0, ν₀) = e_Ω,η(0, 0) = 0. The smearing of the indicator

1_{Ω}

is necessary, since

\int_{R^{3}} | \nabla \sqrt{1_{Ω}} |^{2} = \infty

and this would render the energy E infinite [see ]. Recall that

\bar{E}

contains a term of the form

- \int_{R^{3}} \frac{| j^{p} |^{2}}{ρ}

⁠, which we call the gauge correction term, and it needs to be subtracted otherwise the energy per volume would diverge in the thermodynamic limit. In fact, the gauge correction term grows faster than the volume,

\frac{1}{| ℓ Ω |} \int_{R^{3}} (1_{ℓ Ω} * η) (x) | ν_{0} \times x |^{2} d x = C_{Ω, η, ν_{0}} ℓ^{2},

where

C_{Ω, η, ν_{0}} > 0

is a constant depending only on Ω, η and ν₀. Hence, by , the energy per volume diverges as ℓ → ∞. Notice that replacing

\frac{1}{2} ρ_{0} ν_{0} \times \cdot \to \frac{1}{2} ρ_{0} ν_{0} \times \cdot + ρ_{0} \nabla g

in the definition above would yield the same value for

\bar{E}

due to the gauge transformation rule . In particular, one can choose the Landau gauge instead of the symmetric gauge

\frac{1}{2} ν_{0} \times x

above without changing anything.

By replacing

\bar{E}

with

\bar{T}

on the r.h.s. of , we can also define the kinetic energy per volume τ_Ω,η(ρ₀, ν₀). Finally, we define the exchange-correlation energy per volume

e_{Ω, η}^{x c} (ρ_{0}, ν_{0}) ≔ e_{Ω, η} (ρ_{0}, ν_{0}) - τ_{Ω, η} (ρ_{0}, ν_{0}) .

Note that, in this case the gauge correction terms cancel.

Following Fischer, we say that a sequence of domains

{Ω_{N}}_{N \in N} \subset R^{3}

has a uniformly regular boundary if

| \partial Ω_{N} + B_{r} | ⩽ C r | Ω_{N} |^{2 / 3} for all r ⩽ \frac{| Ω_{N} |^{2 / 3}}{C},

for some C > 0. For instance, one may take Ω_N = NΩ for some convex, bounded and open set.²²

The following theorem is the main result of the paper.

Theorem III.1.

(Uniform electron gas). Let

ρ_{0} \in R_{+}

,

ν_{0} \in R^{3}

and let

{Ω_{N}}_{N \in N} \subset R^{3}

be a sequence of bounded domains such that |Ω_N| → ∞ and it has a uniformly regular boundary. Let δ_N > 0 be any sequence such that δ_N/|Ω_N|^1/3 → 0 and δ_N|Ω_N|^1/3 → ∞. Then the following thermodynamic limit exists

e_{U E G} (ρ_{0}, ν_{0}) ≔ \lim_{N \to \infty} e_{Ω_{N}, η_{δ_{N}}} (ρ_{0}, ν_{0})

and is independent of the choice of the sequences {Ω_N}, {δ_N} and the regularization function η. Analogous results hold true for τ_UEG(ρ₀, ν₀) and

e_{U E G}^{x c} (ρ_{0}, ν_{0})

⁠.

We see using Corollary III.4 (see below) for the upper bound and the Lieb–Thirring and Lieb–Oxford inequalities for the lower bound (see Corollary IV.2 and Theorem IV.3 below),

| e_{U E G} (ρ_{0}, ν_{0}) | ⩽ c_{T F} ρ_{0}^{5 / 3} + c_{L O} ρ_{0}^{4 / 3} + c^{'} ρ_{0} ν_{0} .

Note that in e_UEG(ρ₀, ν₀) only the magnitude of ν₀ matters due to rotational invariance in the thermodynamic limit [see Lemma IV.5(ii)]. It is clear from Proposition II.6 that we recover the nonmagnetic case by putting ν₀ = 0, i.e., e_UEG(ρ₀, 0) = e_UEG(ρ₀). Moreover, and the Lieb–Oxford inequality implies that

e_{U E G} (ρ_{0}, ν_{0}) ⩾ c ρ_{0} ν_{0} - c_{L O} ρ_{0}^{4 / 3}

⁠, so e_UEG(ρ₀, ν₀) → +∞ as ν₀ → ∞ for every fixed ρ₀ > 0.

Remark 2.

Unfortunately, we cannot offer a definition of the classical UEG in the magnetic case, akin to Ref. 1 in the nonmagnetic case. The reason is that j^p uses the phase of the wavefunction in an essential way, and we have no obvious classical probabilistic interpretation of that.

B. Upper bound on the kinetic energy functional and representability

In this section, we describe a technical tool that is needed in the Proof of Theorem III.1, but it is also of independent interest mainly because it improves previous results on the (mixed) representability problem for density-current density pairs (see Sec. II D).

If ρ_γ and

j_{γ}^{p}

comes from an H¹-regular one-particle density matrix γ, we know from that we necessarily have that

\int_{R^{d}} ρ_{γ} |D_{a} \frac{j_{γ}^{p}}{ρ_{γ}}| < \infty .

Note that if d = 3, the the condition is simply

\int_{R^{3}} ρ_{γ} | ν_{γ} | < \infty

⁠, where

ν_{γ} = r o t \frac{j_{γ}^{p}}{ρ_{γ}}

is the vorticity.

Using the method of Proof of Theorem 3 in Ref. 4, we obtain the following.

Theorem III.2.

Fix $ρ \in L^{1} (R^{d}; R_{+})$ such that $\nabla \sqrt{ρ} \in L^{2} {(R^{d})}^{d}$ . Let $v \in L_{ρ}^{2} {(R^{d})}^{d}$ and suppose that v admits a decomposition v = ∇g + w, where $\nabla g \in L_{ρ}^{2} (R^{d})$ , $w \in L_{ρ}^{2} {(R^{d})}^{d}$ , $D w \in L_{ρ}^{1} {(R^{d})}^{d \times d}$ ⁠.

Then there exists a one-particle density matrix γ with finite kinetic energy, such that ρ_γ = ρ,

j_{γ}^{p} = j^{p} ≔ ρ v

and

T r (- Δ γ) ⩽ (1 + κ_{1} ϵ) c_{T F} \int_{R^{d}} ρ^{1 + 2 / d} + \frac{κ_{2} {(1 + \sqrt{ϵ})}^{2}}{ϵ} \int_{R^{d}} | \nabla \sqrt{ρ} |^{2} + \int_{R^{d}} \frac{| j^{p} |^{2}}{ρ} + C_{d} \int_{R^{d}} ρ | D w |

(12)

for any ϵ > 0, and for some universal constants κ₁, κ₂ > 0. Here, the (dimension-dependent) Thomas–Fermi constant c_TF is given by

c_{T F} ≔ \frac{d}{d + 2} \frac{4 π^{2}}{| B_{1} |^{2 / d}}

and

C_{d} = 1 + \frac{d^{3}}{4}

⁠.

The inequality (12) is not gauge-invariant, to see this, take v(x) = (0, x₁, 0) (Landau gauge) and $w (x) = \frac{1}{2} e_{3} \times x$ (symmetric gauge), with $g (x) = \frac{1}{2} x_{1} x_{2}$ and some appropriate ρ. Then D_av = D_aw, D_sv ≠ 0 but D_sw = 0. Clearly, w = v and g = 0 is also a possible choice. The lack of gauge invariance in (12) solely comes from the presence of the symmetric derivative D_sw, which we discuss below. The utility of the formulation of Theorem III.2 is that with a proper choice of gauge, we can hope to make D_sw vanish.

The condition (6) of Theorem II.4 is easily seen to imply $D w \in L_{ρ}^{1} {(R^{d})}^{d \times d}$ (up to a gauge) via the Cauchy–Schwarz inequality. We also note that for v = 0, we exactly recover the statement (and proof) of Theorem 3 in Ref. 4, since that construction actually yields a state γ with $j_{γ}^{p} = 0$ ⁠. Based on the kinetic energy of the ground state of the noninteracting electron gas, we conjecture that the optimal constant in 3D is C₃ = 1.

The last term on the r.h.s. of has a similar role as the von Weizsäcker term. It can be rewritten as

\int_{R^{d}} ρ | D w | = \int_{R^{d}} ρ \sqrt{| D_{s} w |^{2} + | D_{a} w |^{2}} = \int_{R^{d}} ρ \sqrt{| D_{s} w |^{2} + | D_{a} v |^{2}},

where we used the fact D_a∇p = 0 for any C² scalar field

p : R^{d} \to R

⁠, due to Young’s theorem. Hence

\int_{R^{d}} ρ | D w | ≃ \int_{R^{d}} ρ | D_{s} w | + \int_{R^{d}} ρ | D_{a} v |,

in the sense of norm equivalence. In 3D, the second term on the r.h.s. is proportional to

\int_{R^{3}} ρ | ν |

⁠, which is gauge-invariant.

The first term on the r.h.s. of the preceding equation (with w = w_γ)—which we call the strain term—is not necessarily finite for H¹-regular γ. However, if we assume that γ is H²-regular (and the gauge function has $D^{2} g \in L_{ρ}^{1} {(R^{d})}^{d \times d}$ ⁠), then it is not hard to show that the strain term is finite. Recall that the eigenstates of the magnetic Schrödinger Hamiltonian are in H². Although Theorem III.2 does not provide the complete solution to the mixed density-current N-representability problem (it also does not provide an H²-regular γ without further assumptions, sadly), it seems to provide weak enough assumptions that any practically relevant (ρ, j^p) pair verifies it. That being said, it is our belief that the presence of the strain term is only an artifact of the method of the proof of both Theorem II.4 and Theorem III.2. It remains a challenge to devise a gauge-invariant analogue of (12).

Motivated by Theorem III.2, we define the smaller set

\tilde{J} \subset J

⁠, via

\tilde{J} ≔ \{\begin{aligned} (ρ, j^{p}) : & ρ \in I, \int_{R^{d}} \frac{| j^{p} |^{2}}{ρ} < \infty, \int_{R^{d}} ρ |D w| < \infty, where \frac{j^{p}}{ρ} = \nabla g + w, \\ for some \nabla g \in L_{ρ}^{2} (R^{d}) and w \in L_{ρ}^{2} {(R^{d})}^{d} \end{aligned}\} .

With this notation at hand, we can provide an upper bound on the kinetic energy functional T(ρ, j^p).

Corollary III.3.

For any $(ρ, j^{p}) \in \tilde{J}$ , we have T(ρ, j^p)⩽r.h.s. of (12).

Moreover, using the fact²³ that for every one-particle density matrix γ there exists a quasi-free state Γ, such that γ_Γ = γ, we obtain the following.

Corollary III.4.

For any $(ρ, j^{p}) \in \tilde{J}$ , we have E(ρ, j^p)⩽r.h.s. of (12).

In fact, for a quasi-free state Γ, the Coulomb energy of Γ minus D(ρ_Γ) is always nonpositive.

IV. PROOFS

The rest of the paper is the devoted to proofs. Since Ref. 4 explains technical details in a rather complete manner, we will avoid repetition whenever possible.

A. Proof of Theorem III.2

Step 0. First, define the ground-state one-particle density matrix f_t(x − y) of the noninteracting uniform electron gas of density t⩾0, whose translation-invariant kernel is given by
$f_{t} (z) = \int_{| k |^{2} ⩽ \frac{d + 2}{d} c_{T F} t^{2 / d}} e^{i k \cdot z} \frac{d k}{{(2 π)}^{d}} .$
It is immediate that f_t is Hermitian and $0 ⩽ f_{t} ⩽ 1$ in the sense of operators. Also, its density is f_t(0) = t. Moreover, ∇_xf_t(x − y)|_y=x = ∇_yf_t(x − y)|_x=y = 0. Furthermore, the kinetic energy density is also constant
$\nabla_{x} \cdot \nabla_{y} f_{t} (x - y) |_{y = x} = c_{T F} t^{1 + 2 / d} .$
Step 1. As the first derivative of f_t vanishes on the diagonal, this state has zero paramagnetic current density. We would like to construct a state $f_{t}^{u} (x - y)$ which has a constant paramagnetic current density proportional to a fixed vector $u \in R^{d}$ ⁠. To achieve this, we simply shift the Fermi sphere $| k |^{2} ⩽ \frac{d + 2}{d} c_{T F} t^{2 / d}$ by u. More precisely, we consider the translation-invariant kernel $f_{t}^{u} (x - y)$ ⁠, where
$f_{t}^{u} (z) = \int_{| k - u |^{2} ⩽ \frac{d + 2}{d} c_{T F} t^{2 / d}} e^{i k \cdot z} \frac{d k}{{(2 π)}^{d}} = e^{i u \cdot z} f_{t} (z) .$
Clearly, $f_{t}^{u}$ is Hermitian and $0 ⩽ f_{t}^{u} ⩽ 1$ in the sense of operators. Its density is $f_{t}^{u} (0) = t$ ⁠, but now
$\begin{aligned} \nabla_{x} f_{t}^{u} (x - y) |_{y = x} & = i t u, \\ \nabla_{y} f_{t}^{u} (x - y) |_{y = x} & = - i t u . \end{aligned}$
(13)
Therefore, this state has paramagnetic current density
$j^{p} (x) = I m \nabla_{x} f_{t}^{u} (x - y) |_{y = x} = t u$
as desired. Further, the kinetic energy density is again constant, but has an additional term
$\nabla_{y} \cdot \nabla_{x} f_{t}^{u} (y - x) |_{y = x} = t | u |^{2} + c_{T F} t^{1 + 2 / d} .$
Step 2. We are now ready to construct a fermionic one-particle density matrix γ with prescribed density ρ(x) and paramagnetic current density of the form j^p = ρv. Let us first assume that v = w. Following,⁴ we define²⁴
$γ (x, y) = \int_{R^{d}} \int_{0}^{+ \infty} \sqrt{θ (u - v (x)) η (\frac{t}{ρ (x)})} f_{t}^{u} (x - y) \sqrt{θ (u - v (y)) η (\frac{t}{ρ (y)})} \frac{d t}{t} d u,$
where the weight function $η : R_{+} \to R_{+}$ is smooth, and satisfies
$\int_{0}^{+ \infty} η = 1, and \int_{0}^{+ \infty} \frac{η (t)}{t} d t ⩽ 1,$
and $θ : R^{d} \to R_{+}$ is also smooth and satisfies
$\int_{R^{d}} θ = 1, \int_{R^{d}} u θ (u) d u = 0, and \int_{R^{d}} \nabla θ (u) d u = 0 .$
(14)
These will be determined later in the proof. Then we have
$ρ_{γ} (x) = \int_{R^{d}} \int_{0}^{+ \infty} θ (u - v (x)) η (\frac{t}{ρ (x)}) d t d u = ρ (x),$
and
$0 ⩽ γ ⩽ \int_{R^{d}} \int_{0}^{+ \infty} θ (u - v (x)) η (\frac{t}{ρ (x)}) \frac{d t}{t} d u ⩽ 1 .$
Furthermore, we find
$j_{γ}^{p} (x) = I m \nabla_{x} γ (x, y) |_{y = x} = \int_{R^{d}} θ (u - v (x)) u d u \int_{0}^{+ \infty} η (\frac{t}{ρ (x)}) d t = ρ (x) v (x)$
as required. Next, we compute the kinetic energy density,
$\begin{array}{l} \nabla_{y} \cdot \nabla_{x} γ (x, y) |_{y = x} \\ = \int_{R^{d}} \int_{0}^{+ \infty} {|\nabla \sqrt{θ (u - v (x)) η (\frac{t}{ρ (x)})}|}^{2} d t d u + \int_{R^{d}} \int_{0}^{+ \infty} θ (u - v (x)) η (\frac{t}{ρ (x)}) (| u |^{2} + c_{T F} t^{2 / d}) d t d u \\ ≕ (I) + (I I) \end{array}$
since the sum of the first order derivatives of $f_{t}^{u}$ cancel on the diagonal, see (13). Here, using the first two relations of (14), we have
$(I I) = ρ (x) | v (x) |^{2} + ρ (x) \int_{R^{d}} θ (u) | u |^{2} d u + c_{T F} ρ {(x)}^{1 + 2 / d} \int_{0}^{+ \infty} η (s) s^{2 / d} d s .$
Moreover,
$(I) = ρ (x) \int_{R^{d}} \frac{| D v (x) \nabla θ (u) |^{2}}{4 θ (u)} d u + | \nabla \sqrt{ρ} (x) |^{2} \int_{0}^{+ \infty} \frac{t^{2} η^{'} {(t)}^{2}}{η (t)} d t + \frac{1}{2} D v (x) \frac{\nabla ρ (x)}{ρ (x)} \cdot \int_{R^{d}} \nabla θ (u) d u \underset{= 1}{\underset{︸}{\int_{0}^{+ \infty} t η^{'} (t) d t}} .$
The last term vanishes because $\int_{R^{d}} \nabla θ (u) d u = 0$ ⁠, by hypothesis. In summary,
$\begin{array}{l} \nabla_{y} \cdot \nabla_{x} γ (x, y) |_{y = x} & = c_{T F} ρ {(x)}^{1 + 2 / d} \int_{0}^{+ \infty} η (s) s^{2 / d} d s + | \nabla \sqrt{ρ} (x) |^{2} \int_{0}^{+ \infty} \frac{t^{2} η^{'} {(t)}^{2}}{η (t)} d t \\ + ρ (x) | v (x) |^{2} + \underset{(*)}{\underset{︸}{ρ (x) \int_{R^{d}} θ (u) | u |^{2} d u + ρ (x) \int_{R^{d}} \frac{| D v (x) \nabla θ (u) |^{2}}{4 θ (u)} d u .}} \end{array}$
To optimize η, we simply follow Ref. 4 because there are no mixed terms containing both η and θ. The last term can be bounded by
$ρ (x) | D v (x) |^{2} \int_{R^{d}} \frac{| \nabla θ (u) |^{2}}{4 θ (u)} d u .$
Clearly, it is enough to consider radial θ(u) = θ(r) for the optimization of the prefactors involving θ, where r = |u|. After switching to spherical coordinates, we obtain that with
$C (δ) ≔ d | B_{1} | \inf_{\begin{array}{c} θ ⩾ 0 \\ \int_{0}^{+ \infty} r^{d - 1} θ (r) d r = {(d | B_{1} |)}^{- 1} \\ \int_{0}^{+ \infty} r^{d + 1} θ (r) d r = {(d | B_{1} |)}^{- 1} δ \end{array}} \int_{0}^{+ \infty} \frac{r^{d - 1} θ^{'} {(r)}^{2}}{4 θ (r)} d r,$
the bound (∗)⩽δρ(x) + C(δ)ρ(x)|Dv(x)|² holds true. The function
$θ (r) = {(\frac{2 π δ}{d})}^{- d / 2} e^{- \frac{d}{2 δ} r^{2}}$
(15)
is admissible, and has
$C (δ) ⩽ d | B_{1} | \int_{0}^{+ \infty} \frac{r^{d - 1} θ^{'} {(r)}^{2}}{4 θ (r)} d r = \frac{d^{3}}{4 δ} .$
Hence,
$(*) ⩽ δ ρ (x) + \frac{d^{3}}{4 δ} ρ (x) | D v (x) |^{2} = C_{d} ρ (x) | D v (x) |,$
with the choice δ = |Dv(x)| and $C_{d} = 1 + \frac{d^{3}}{4}$ ⁠.
In summary, we end up with
$T r (- Δ γ) ⩽ (1 + κ_{1} ϵ) c_{T F} \int_{R^{d}} ρ^{1 + 2 / d} + \frac{κ_{2} {(1 + \sqrt{ϵ})}^{2}}{ϵ} \int_{R^{d}} | \nabla \sqrt{ρ} |^{2} + \int_{R^{d}} ρ | v |^{2} + C_{d} \int_{R^{d}} ρ (x) | D v (x) | d x,$
(16)
which completes the proof for v = w.
Step 3. For the general case v = −∇g + w, we apply the gauge transformation g on γ to get $\tilde{γ}$ ⁠. Proposition II.2 together with (16) (where v is replaced by w) gives
$\begin{array}{l} T r (- Δ \tilde{γ}) & = T r (- Δ γ) - 2 \int_{R^{d}} ρ w \cdot \nabla g + \int_{R^{d}} ρ | \nabla g |^{2} \\ ⩽ (1 + κ_{1} ϵ) c_{T F} \int_{R^{d}} ρ^{1 + 2 / d} + \frac{κ_{2} {(1 + \sqrt{ϵ})}^{2}}{ϵ} \int_{R^{d}} | \nabla \sqrt{ρ} |^{2} + \int_{R^{d}} ρ | v |^{2} + C_{d} \int_{R^{d}} ρ | D w |, \end{array}$
which finishes the Proof of Theorem III.2.

B. Further bounds

We start by recalling a simple version of the magnetic Lieb–Thirring inequality, which can be used to derive a gauge invariant bound. Let $A \in L_{l o c}^{2} {(R^{3})}^{3}$ and denote the corresponding magnetic Sobolev space as $H_{A}^{1} (R^{3})$ ⁠, see Ref. 25.

Theorem IV.1

(Magnetic Lieb–Thirring kinetic energy inequality).²⁶ Fix

A \in L_{l o c}^{2} (R^{3})

. For any fermionic one-particle reduced density matrix γ in

H_{A}^{1} (R^{3})

, the bound

T r {(- i \nabla + A)}^{2} γ ⩾ C_{L T} \int_{R^{3}} ρ_{γ}^{5 / 3}

holds true for some universal constant C_LT > 0.

Using this, the usual Lieb–Thirring kinetic energy inequality can be cast in a gauge-invariant form.

Corollary IV.2

(Gauge-invariant Lieb–Thirring inequality). For any fermionic one-particle reduced density matrix γ with finite kinetic energy, the bound

T r (- Δ γ) ⩾ c_{L T} \int_{R^{3}} ρ_{γ}^{5 / 3} + \int_{R^{3}} \frac{| j_{γ}^{p} |^{2}}{ρ_{γ}}

holds true for some universal constant c_LT > 0.

Proof.

For any ϵ > 0, we set

A_{ϵ} = - \frac{j_{γ}^{p}}{ρ_{γ} + ϵ \sqrt{ρ_{γ}}}

⁠. Then

A_{ϵ} \in L_{l o c}^{2} (R^{3})

since

\int_{R^{3}} \frac{| j_{γ}^{p} |^{2}}{ρ_{γ}} < \infty

⁠. Note that

\begin{array}{l} C_{L T} \int_{R^{3}} ρ_{γ}^{5 / 3} & ⩽ T r {(- i \nabla + A_{ϵ})}^{2} γ = T r (- Δ γ) + 2 \int_{R^{3}} A_{ϵ} \cdot j_{γ}^{p} + \int_{R^{3}} ρ_{γ} | A_{ϵ} |^{2} \\ = T r (- Δ γ) - \int_{R^{3}} | j_{γ}^{p} |^{2} \frac{ρ_{γ} + 2 ϵ \sqrt{ρ_{γ}}}{{(ρ_{γ} + ϵ \sqrt{ρ_{γ}})}^{2}} . \end{array}

An application of Fatou’s lemma finishes the proof. □

For any

Γ \in D

let us introduce the notations

T (Γ) = \sum_{n ⩾ 1} \sum_{1 ⩽ j ⩽ n} {T r}_{H^{n}} (- Δ_{x_{j}} Γ_{n}), C (Γ) = \sum_{n ⩾ 1} \sum_{1 ⩽ j < k ⩽ n} {T r}_{H^{n}} (\frac{1}{| x_{j} - x_{k} |} Γ_{n})

for the kinetic-, and the Coulomb energy of the state Γ. Of course,

T (Γ) = T r (- Δ γ_{Γ})

⁠, so Corollary IV.2 provides a gauge-invariant lower bound on

T (Γ)

⁠. The following famous estimate on the indirect Coulomb energy will be used repeatedly in the sequel (which is clearly gauge-invariant).

Theorem IV.3.

(Lieb–Oxford inequality). ^27–30 There exists a universal constant c_LO > 0 such that for any Fock space density matrix Γ (not necessarily fermionic or H¹-regular), the bound

C (Γ) - D (ρ_{Γ}) ⩾ - c_{L O} \int_{R^{3}} ρ_{Γ}^{4 / 3}

holds true.

C. Transformation properties

In this section, we discuss the behavior of the density functionals under various transformations of the density-current density pair.

First, we consider general affine transformations of states. Let

Γ = Γ_{0} \oplus Γ_{1} \oplus \dots \in D

and for any

M \in R^{3 \times 3}

nonsingular and

a \in R^{3}

⁠, set T(x) = Mx + a, and define the transformed state ϒ_TΓ by setting

(ϒ_{T} Γ_{n}) (X; Y) = {(\det M)}^{n} Γ_{n} (T (x_{1}), \dots, T (x_{n}); T (y_{1}), \dots, T (y_{n})) .

Then the reduced one-particle density matrix of ϒ_TΓ is given by

γ_{ϒ_{T} Γ} (x, y) = (\det M) γ_{Γ} (T (x), T (y))

⁠, where γ_Γ denotes the one-particle density matrix of Γ. We have

ρ_{ϒ_{T} Γ} (x) = (\det M) ρ_{Γ} (T (x))

and

j_{ϒ_{T} Γ}^{p} (x) = (\det M) M^{⊤} j_{Γ}^{p} (T (x))

⁠. Further,

γ_{ϒ_{T} Γ}^{(2)} (x, y; x, y) = {(\det M)}^{2} γ_{Γ}^{(2)} (T (x), T (y); T (x), T (y))

⁠. We immediately obtain that for any

Γ \in D

the following relations hold true:

T (ϒ_{T} Γ) = \int_{R^{3}} {T r}_{R^{3}} (M M^{⊤} (\nabla_{x} \otimes \nabla_{y}) γ_{Γ} (x, y) |_{y = x}) d x,

(17)

C (ϒ_{T} Γ) = \iint_{R^{3} \times R^{3}} \frac{γ_{Γ}^{(2)} (x, y; x, y)}{| M^{- 1} (x - y) |} d x d y .

(18)

Motivated by the above considerations, for any

(ρ, j^{p}) \in J

⁠, we define

\begin{matrix} ϒ_{T} (ρ, j^{p}) (x) ≔ (\det M) (ρ (T (x)), M^{⊤} j^{p} (T (x))) \end{matrix}

(19)

as a slight abuse of notation. Using and , and introducing the the gauge-corrected kinetic energy tensor

{\bar{τ}}_{γ} = τ_{γ} - \frac{j_{γ}^{p} \otimes j_{γ}^{p}}{ρ_{γ}},

the transformed gauge-corrected indirect energy may be expressed as

\bar{E} (ϒ_{M} (ρ, j^{p})) = \inf_{\begin{array}{c} Γ \in D \\ ρ_{Γ} = ρ \\ j_{Γ}^{p} = j^{p} \end{array}} [\int_{R^{3}} {T r}_{R^{3}} (M M^{⊤} {\bar{τ}}_{γ_{Γ}} (x)) d x + C (ϒ_{M} Γ)] - D (ϒ_{M} ρ),

(20)

which follows by a simple reparametrization of the infimum, using the fact that the map Γ↦ϒ_MΓ is invertible.

In ordinary DFT, the Levy–Lieb functional is invariant under isometries. In CDFT, we have the following transformation rule.

Lemma IV.4.

Let T(x) = Rx + a be an isometry for some R ∈ O(3) and

a \in R^{3}

. Then

ϒ_{T} (ρ, j^{p}) = (ρ ◦ T, R^{⊤} (j^{p} ◦ T))

and

T (ϒ_{T} (ρ, j^{p})) = T (ρ, j^{p}), and F (ϒ_{T} (ρ, j^{p})) = F (ρ, j^{p}) .

The proof follows from relations analogous to (20). As an application, we deduce how the indirect energy E and $e_{Ω, η_{δ}} (ρ_{0}, ν_{0})$ ⁠, $τ_{Ω, η_{δ}} (ρ_{0}, ν_{0})$ and $e_{Ω, η_{δ}}^{x c} (ρ_{0}, ν_{0})$ transform under isometries. The proof uses the gauge transformation rule (10) in an essential way.

Lemma IV.5.

Fix $ρ_{0} \in R_{+}$ and $ν_{0} \in R^{3}$ . Let T(x) = Rx + a be an isometry for some R ∈ O(3) and $a \in R^{3}$ . Let $Ω \subset R^{3}$ be a bounded domain with barycenter 0, and let η be a regularization function.

The following formula holds true:
$E ((1_{T^{- 1} (Ω)} * (η_{δ} ◦ R)) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) = E ((1_{Ω} * η_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) + \frac{ρ_{0}}{4} | R ν_{0} \times a |^{2} \int_{R^{3}} (1_{Ω} * η_{δ}) .$
Further, the same relation holds with E replaced by T.
The relation $e_{T^{- 1} (Ω), η_{δ} ◦ R} (ρ_{0}, ν_{0}) = e_{Ω, η_{δ}} (ρ_{0}, R ν_{0})$ holds true. Moreover, the formula is also valid if e is replaced by τ and e^xc.

Proof.

First, we show (i). Clearly, T⁻¹(x) = R^⊤(x − a). Also,

D (((1_{Ω} * η_{δ}) ◦ T) ρ_{0}) = D ((1_{Ω} * η_{δ}) ρ_{0})

⁠. We may write the l.h.s. using Lemma IV.4 as

\begin{array}{l} E (((1_{Ω} * η_{δ}) ◦ T) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) & = E (x \mapsto (1_{Ω} * η_{δ}) (x) (ρ_{0}, \frac{1}{2} ρ_{0} R (ν_{0} \times T^{- 1} (x)))) \\ = E (x \mapsto (1_{Ω} * η_{δ}) (x) (ρ_{0}, \frac{1}{2} ρ_{0} R (ν_{0} \times R^{⊤} x) + \frac{1}{2} ρ_{0} R (ν_{0} \times R^{⊤} a)))) \\ = E (x \mapsto (1_{Ω} * η_{δ}) (x) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times x + \frac{1}{2} ρ_{0} R ν_{0} \times a)) \end{array}

where we used the identity R(u ×R^⊤v) = Ru ×v. Using the gauge transformation rule , we obtain

\begin{array}{l} E (((1_{Ω} * η_{δ}) ◦ T) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) = E ((1_{Ω} * η_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) \\ - \frac{ρ_{0}}{2} \int_{R^{3}} (1_{Ω} * η_{δ}) (x) (R ν_{0} \times x) \cdot (R ν_{0} \times a) d x + \frac{ρ_{0}}{4} | R ν_{0} \times a |^{2} \int_{R^{3}} (1_{Ω} * η_{δ}) (x) d x \\ = E ((1_{Ω} * η_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) + \frac{ρ_{0}}{4} | R ν_{0} \times a |^{2} \int_{R^{3}} (1_{Ω} * η_{δ}), \end{array}

as stated. Here, the cross term disappeared because

\int_{R^{3}} x (1_{Ω} * η_{δ}) (x) d x = 0

⁠, since the smeared set also has barycenter 0, due to

\int_{R^{3}} x η (x) d x = 0

⁠.

For part (ii), we may write based on part (i) that

\begin{array}{l} e_{T^{- 1} (Ω), η_{δ} ◦ R} (ρ_{0}, ν_{0}) & = \frac{1}{| Ω |} [E ((1_{Ω} * η_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) + \frac{ρ_{0}}{4} | R ν_{0} \times a |^{2} \int_{R^{3}} (1_{Ω} * η_{δ}) \\ - \frac{ρ_{0}}{4} \int_{R^{3}} (1_{Ω} * η_{δ}) (x) | ν_{0} \times (R^{- 1} (x - a)) |^{2} d x] = e_{Ω, η_{δ}} (ρ_{0}, R ν_{0}), \end{array}

where the mixed term in last term vanished as before. □

Choosing

M = λ^{1 / 3} 1

in and setting

δ_{λ} (ρ, j^{p}) (x) ≔ ϒ_{M} (ρ, j^{p}) (x) = (λ ρ (λ^{1 / 3} x), λ^{4 / 3} j^{p} (λ^{1 / 3} x)),

we deduce the following (the lower bound uses the Lieb–Oxford inequality as well).

Theorem IV.6.

For every

(ρ, j^{p}) \in J

and 0⩽λ⩽1, we have

\bar{E} (δ_{λ} (ρ, j^{p})) ⩽ λ^{1 / 3} \bar{E} (ρ, j^{p})

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and

\bar{E} (δ_{λ} (ρ, j^{p})) ⩾ λ^{2 / 3} \bar{E} (ρ, j^{p}) + (λ^{2 / 3} - λ^{1 / 3}) c_{L O} \int_{R^{3}} ρ^{4 / 3} .

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D. Decoupling lower bound

Let

C_{1} ≔ {(- \frac{1}{2}, \frac{1}{2})}^{3}

denote the origin-centered unit cube, and Δ_j = T_jΔ (j = 1, …, 24) be its tiling with 24 congruent tetrahedra, where T_jx = R_jx − z_j for some R_j ∈ SO(3) and z_j ∈ C₁. Here, Δ is a “reference” tetrahedron with barycenter 0. Then with C_ℓ = ℓC₁, we write

R^{3} = ⋃_{z \in Z^{3}} (C_{ℓ} + ℓ z) = ⋃_{z \in Z^{3}} ⋃_{j = 1}^{24} (ℓ T_{j} Δ + ℓ z),

where ℓ > 0 the scale of the tiling. The reason behind the use of this particular tetrahedral tiling is that the Graf–Schenker inequality is employed for the Coulomb interaction, see Refs. 1 and 4 and the original^31,32 for details.

The corresponding partition of unity is given by

\sum_{z \in Z^{3}} \sum_{j = 1}^{24} 1_{ℓ T_{j} Δ} (\cdot - ℓ z) \equiv 1 (a.e.) .

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The lower bound is based on well-known localization techniques, and for that, smoothing is needed. Let

0 ⩽ η \in C_{c}^{\infty} (R^{3})

be such that

\int_{R^{3}} x η (x) d x = 0

with suppη ⊂ B₁ and

\int_{R^{3}} η = 1

⁠. Define

η_{δ} (x) = {(\frac{10}{δ})}^{3} η (\frac{10 x}{δ}) .

Then

s u p p η_{δ} \subset B_{\frac{δ}{10}}

and

\int_{R^{3}} η_{δ} = 1

⁠. The functions

1_{ℓ Δ_{j}} * η_{δ}

form a partition of unity in the following sense:

\sum_{z \in Z^{3}} \sum_{j = 1}^{24} (1_{ℓ Δ_{j}} * η_{δ}) (\cdot - ℓ z) = \sum_{z \in Z^{3}} \sum_{j = 1}^{24} \int_{R^{3}} 1_{ℓ Δ_{j}} (\cdot - x - ℓ z) η_{δ} (x) d x = \int_{R^{3}} η_{δ} \equiv 1

by . The following facts will be used repeatedly. Fix

x \in R^{3}

⁠, if suppη_δ(x − ·) ∩ ∂(ℓΔ_j) = ∅, then clearly either supp η_δ(x − ·) ⊂ ℓΔ_j or else

s u p p η_{δ} (x - \cdot) \subset {(ℓ Δ_{j})}^{c}

⁠. In the former case,

(1_{ℓ Δ_{j}} * η_{δ}) (x) = 1

and in the latter,

(1_{ℓ Δ_{j}} * η_{δ}) (x) = 0

⁠. Since

s u p p η_{δ} (x - \cdot) \subset B_{\frac{δ}{10}} (x)

⁠, we obtain that

(1_{ℓ Δ_{j}} * η_{δ}) (x) \equiv \{\begin{cases} 1, & if B_{\frac{δ}{10}} (x) \subset ℓ Δ_{j} \\ 0, & if B_{\frac{δ}{10}} (x) \subset {(ℓ Δ_{j})}^{c} \end{cases}

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It follows that

1_{ℓ Δ_{j}} * η_{δ} \equiv 1

on

ℓ Δ_{j} \ (\partial (ℓ Δ_{j}) + B_{\frac{δ}{10}})

⁠. More conveniently,

1_{ℓ Δ_{j}} * η_{δ} \equiv 1

on (ℓ − δ)Δ_j, as this set is clearly contained in the former.

The following trivial observation shows that j^p behaves the same way as ρ with respect to localization. Let γ be a one-particle density matrix and fix 0⩽χ⩽1, where $χ \in C^{1} (R^{3})$ ⁠, and define γ|_χ = χγχ. Then $ρ_{γ |_{χ}} = χ^{2} ρ_{γ}$ and $j_{γ |_{χ}}^{p} = χ^{2} j_{γ}^{p}$ ⁠, where we used the fact that χ real-valued.

This implies that we may proceed exactly as in Refs. 1 and 4. We do not, however, discard the kinetic energy term $\frac{C δ}{ℓ} T (\dots)$ for the bound on E(ρ, j^p), as it will be required to cancel the gauge terms in Theorem IV.10. Consequently, we have

Theorem IV.7.

There is a constant C > 0 such that for any δ > 0 such that 0 < δℓ < 1/C and

(ρ, j^{p}) \in \tilde{J}

the bound

\begin{array}{l} E (ρ, j^{p}) \\ ⩾ \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} [(1 - \frac{C δ}{ℓ}) E ((1_{ℓ Δ_{j}} * η_{δ}) (R \cdot - ℓ z - τ) (ρ, j^{p})) + \frac{C δ}{ℓ} T ((1_{ℓ Δ_{j}} * η_{δ}) (R \cdot - ℓ z - τ) (ρ, j^{p}))] \\ - \frac{C}{ℓ} \int_{R^{3}} ((1 + δ^{- 1}) ρ + δ^{3} ρ^{2}) \end{array}

holds. Furthermore, for any δ > 0 and ℓ > 0 the bound

T (ρ, j^{p}) ⩾ – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} T ((1_{ℓ Δ_{j}} * η_{δ}) (R \cdot - ℓ z - τ) (ρ, j^{p})) - \frac{C}{δ ℓ} \int_{R^{3}} ρ

also holds true.

E. Decoupling upper bound

First, is regularized as follows. For any

δ ⩽ \frac{ℓ}{2}

⁠, consider the functions

{(1 - δ / ℓ)}^{- 3} 1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}

⁠. Here,

ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ = (ℓ - δ) R_{j} Δ + ℓ z_{j}

and

\int_{R^{3}} {(1 - δ / ℓ)}^{- 3} 1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ} = | ℓ Δ | .

To obtain a true partition of unity, these functions are averaged over the cubic tiling:

– \int_{C_{ℓ}} \sum_{z \in Z^{3}} \sum_{j = 1}^{24} {(1 - δ / ℓ)}^{- 3} (1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z) d τ \equiv 1 (a.e.) .

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The following result says that the (grand-canonical) Vignale–Rasolt functional decouples the same way as the Levy–Lieb functional.

Proposition IV.8.

For any

(ρ, j^{p}) \in \tilde{J}

, the bounds

\begin{array}{l} F (ρ, j^{p}) & ⩽ – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} F ({(1 - δ / ℓ)}^{- 3} (1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z) (ρ, j^{p})) \\ + – \int_{C_{ℓ}} d τ \sum_{\begin{array}{c} z \neq z^{'} \\ j \neq j^{'} \end{array}} {(1 - δ / ℓ)}^{- 6} D ((1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z) ρ, (1_{ℓ T_{j^{'}} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z^{'}) ρ) \end{array}

and

T (ρ, j^{p}) ⩽ – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} T ({(1 - δ / ℓ)}^{- 3} (1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z) (ρ, j^{p}))

hold true.

Proof.

The partition of unity allows us to write our density-current pair as

(ρ, j^{p}) = – \int_{C_{ℓ}} \sum_{z \in Z^{3}} \sum_{j = 1}^{24} (ρ_{τ, z, j}, j_{τ, z, j}^{p}) d τ,

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where we have set

(ρ_{τ, z, j}, j_{τ, z, j}^{p}) = {(1 - δ / ℓ)}^{- 3} (1_{ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{δ}) (\cdot - τ - ℓ z) (ρ, j^{p})

for any τ ∈ C_ℓ,

z \in Z^{3}

and j = 1, …, 24. We now claim that the density-current pairs

(ρ_{τ, z, j}, j_{τ, z, j}^{p})

are representable, i.e.,

(ρ_{τ, z, j}, j_{τ, z, j}^{p}) \in \tilde{J}

⁠. In fact, it is clear that

ρ_{τ, z, j} \in I

⁠, and with

v = \frac{j^{p}}{ρ}

and

v_{τ, z, j} ≔ \frac{j_{τ, z, j}^{p}}{ρ_{τ, z, j}} 1_{s u p p ρ_{τ, z, j}} = v 1_{s u p p ρ_{τ, z, j}}

there holds

\int_{R^{3}} ρ_{τ, z, j} | v_{τ, z, j} |^{2} ⩽ \int_{R^{3}} ρ | v |^{2} < \infty

⁠. Also,

\int_{R^{3}} ρ_{τ, z, j} | D v_{τ, z, j} | ⩽ \int_{R^{3}} ρ | D v | < \infty

⁠.

The rest of the proof is analogous to that of Proposition 1 in Ref. 4. We note that in the construction of the trial state Theorem II.5 is used. □

Next, we state a similar bound for the indirect energy defined in 9.

Theorem IV.9.

For any

(ρ, j^{p}) \in \tilde{J}

,

0 < δ < \frac{ℓ}{2}

and

0 < α ⩽ \frac{1}{2}

, the bound

\begin{array}{l} E (ρ, j^{p}) \\ ⩽ – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} E ({(1 - δ / ℓ)}^{- 3} (1_{t ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{t δ}) (R \cdot - t ℓ z - τ) (ρ, j^{p})) + C δ^{2} \log (α^{- 1}) \int_{R^{3}} ρ^{2} \end{array}

holds for a universal constant C. Here,

– \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} ≔ {(\int_{1 - α}^{1 + α} \frac{d t}{t^{4}})}^{- 1} \int_{1 - α}^{1 + α} \frac{d t}{t^{4}}

⁠.

Proof.

We have

E (ρ, j^{p}) ⩽ – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} E (ρ_{τ, z, j}, j_{τ, z, j}^{p}) + – \int_{C_{ℓ}} D (ρ_{Γ_{τ}} - ρ) d τ .

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According to the Proof of Proposition 1 in Ref. 4 (and Lemma 1 ibid.), replacing ℓ → tℓ, δ → tδ and ρ → ρ◦R (and j^p → R^⊤(j^p◦R)), and averaging, the preceding quantity may be bounded as follows

– \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} D ((ρ_{Γ_{τ}} - ρ) ◦ R) d τ ⩽ C δ^{2} \log (α^{- 1}) \int_{R^{3}} ρ^{2},

for 0 < δ < ℓ/2. By applying the same replacements and averagings in , we may use the preceding estimate and we end up with

\begin{array}{l} E (ρ ◦ R, R^{⊤} (j^{p} ◦ R)) \\ ⩽ – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} E ({(1 - δ / ℓ)}^{- 3} (1_{t ℓ T_{j} (1 - \frac{δ}{ℓ}) Δ} * η_{t δ}) (\cdot - t ℓ z - τ) (ρ ◦ R, R^{⊤} (j^{p} ◦ R)) \\ + C δ^{2} \log (α^{- 1}) \int_{R^{3}} ρ^{2} \end{array}

Recalling Lemma IV.4 finishes the proof. □

F. Energy per volume in a tetrahedron

We now prove that the thermodynamic limit for the indirect-, and the kinetic energy density of the uniform electron gas defined in Sec. III A exists when the domain is taken to be a tetrahedron.

Theorem IV.10

(Convergence rate for tetrahedra). Fix $ρ_{0} \in R_{+}$ and $ν_{0} \in R^{3}$ . Let η and $\tilde{η}$ be regularization functions. Let ${\underset{̲}{e}}_{U E G} (ρ_{0}, ν_{0})$ and ${\bar{e}}_{U E G} (ρ_{0}, ν_{0})$ denote the liminf and the limsup of $e_{ℓ Δ, η_{δ}} (ρ_{0}, ν_{0})$ as ℓδ → ∞ and δ/ℓ → 0.

For ℓ/δ sufficiently large, we have
$\int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R ⩽ {\underset{̲}{e}}_{U E G} (ρ_{0}, ν_{0}) + \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0} + δ ρ_{0}^{2 / 3} + δ ν_{0})$
We have for all $0 < α ⩽ \frac{1}{2}$ ,
$– \int_{1 - α}^{1 + α} \int_{S O (3)} e_{t ℓ Δ, {\tilde{η}}_{t δ}} (ρ_{0}, R ν_{0}) d R \frac{d t}{t^{4}} ⩾ {\bar{e}}_{U E G} (ρ_{0}, ν_{0}) - \frac{C ρ_{0}}{ℓ} (δ ρ_{0}^{2 / 3} + ℓ^{- 1}) - C δ^{2} \log (α^{- 1}) ρ_{0}^{2}$
If $ℓ ρ_{0}^{1 / 3} ⩾ C$ , then
$e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, ν_{0}) ⩾ {\bar{e}}_{U E G} (ρ_{0}, ν_{0}) - \frac{C δ}{ℓ} (ρ_{0}^{5 / 3} + ρ_{0}^{4 / 3} + ρ_{0} ν_{0}) - C \frac{ρ_{0}^{23 / 15} + ρ_{0}^{18 / 15}}{ℓ^{2 / 5}},$
The thermodynamic limit
$\lim_{\begin{array}{c} δ / ℓ \to 0 \\ ℓ δ \to \infty \end{array}} e_{ℓ Δ, η_{δ}} (ρ_{0}, ν_{0}) = e_{U E G} (ρ_{0}, ν_{0})$
exists and is independent of the choice of the regularization function η.

Proof.

Proof of (i). The upper bound in (i) is proved by applying the lower bound of Theorem IV.7 with the choice

(ρ, j^{p}) ≔ (1_{ℓ^{'} Δ} * η_{δ^{'}}) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot),

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where ℓ′ ≫ ℓ. Recalling , we get

\begin{aligned} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) \\ ⩾ \frac{1}{| ℓ^{'} Δ |} \sum_{z \in Z^{3}} \sum_{j = 1}^{24} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ [(1 - C δ / ℓ) E ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) \\ + \frac{C δ}{ℓ} T ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot))] \\ - \frac{1}{| ℓ^{'} Δ |} \frac{ρ_{0}}{4} \int_{R^{3}} (1_{ℓ^{'} Δ} * η_{δ^{'}}) (x) | ν_{0} \times x |^{2} d x \\ - \frac{C}{ℓ | ℓ^{'} Δ |} \int_{R^{3}} ((1 + δ^{- 1}) (1_{ℓ^{'} Δ} * η_{δ^{'}}) ρ_{0} + δ^{3} {(1_{ℓ^{'} Δ} * η_{δ^{'}})}^{2} ρ_{0}^{2}) \\ ≕ (I) + (I I) + (I I I) \end{aligned}

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Here, (III) can be bounded simply by

- \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0})

⁠, using the trivial estimate:

\int_{R^{3}} {(1_{ℓ^{'} Δ} * η_{δ^{'}})}^{α} ⩽ \int_{R^{3}} (1_{ℓ^{'} Δ} * η_{δ^{'}}) = | ℓ^{'} Δ |,

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valid for any α⩾1 due to the fact that

1_{ℓ^{'} Δ} * η_{δ^{'}} ⩽ 1

⁠.

Considering (I), we can proceed as follows. Define the index set

J ≔ \{\begin{aligned} (z, j) \in Z^{3} \times {1, \dots, 24} : & s u p p ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ)) \cap s u p p (1_{ℓ^{'} Δ} * η_{δ^{'}}) \neq \emptyset, \\ for some R \in S O (3) and τ \in C_{ℓ} \end{aligned}\},

which collects all the small tetrahedra R^⊤(ℓΔ_j + ℓz + τ) that possibly intersect with the big smeared tetrahedron

s u p p (1_{ℓ^{'} Δ} * η_{δ^{'}})

⁠. Furthermore, define

J_{0} ≔ \{\begin{aligned} (z, j) \in Z^{3} \times {1, \dots, 24} : & s u p p ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ)) \subset (ℓ^{'} - δ^{'}) Δ, \\ for some R \in S O (3) and τ \in C_{ℓ} \end{aligned}\},

which contains small smeared tetrahedra which are well inside ℓ′Δ. See Fig. 1.

Then, we may decompose the double sum in (I) into a sum over J₀ and one over J\J₀. The latter part is treated around , and the former part reads

\begin{align} \frac{1}{| ℓ^{'} Δ |} \sum_{(z, j) \in J_{0}} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ [(1 - C δ / ℓ) E ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) \\ + \frac{C δ}{ℓ} T ((1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot))] . \end{align}

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Note first that

1_{ℓ Δ_{j}} * η_{δ} = (1_{ℓ Δ} * η_{δ}) ◦ T_{j, ℓ}^{- 1}

⁠, where T_j,ℓ(x) = R_jx + ℓz_j is the isometry that puts T_j,ℓ(ℓΔ) = ℓΔ_j. Setting

T (x) = T_{j, ℓ}^{- 1} ◦ (R x - ℓ z - τ) = R_{j}^{⊤} R x - R_{j}^{⊤} (ℓ z + τ + ℓ z_{j}) .

Using Lemma IV.5 (i), we have

E (((1_{ℓ Δ} * {\tilde{η}}_{δ}) ◦ T) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) = E ((1_{ℓ Δ} * {\tilde{η}}_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R_{j}^{⊤} R ν_{0} \times \cdot)) + | ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (ℓ z + τ + ℓ z_{j}) |^{2} .

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Further, the same relation holds with E replaced by T.

In other words, we expressed the energy in any (smeared) tetrahedron well inside ℓ′Δ in terms of the energy in the (smeared) reference tetrahedron ℓΔ, plus a gauge term, which is position-dependent and grows faster than the volume. More concretely, ³¹ can be written as

\begin{align} \frac{M_{ℓ}}{| ℓ^{'} Δ ‖ ℓ Δ |} \int_{S O (3)} d R [(1 - C δ / ℓ) E ((1_{ℓ Δ} * {\tilde{η}}_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) + \frac{C δ}{ℓ} T ((1_{ℓ Δ} * {\tilde{η}}_{δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot))] \\ + \frac{1}{| ℓ^{'} Δ |} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J_{0}} | ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (ℓ z + τ + ℓ z_{j}) |^{2}, \end{align}

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where

M_{ℓ} = | Δ | | J_{0} | .

The term involving

\frac{C δ}{ℓ} T (\dots)

can be bounded using (discarding all but the gauge term) as

\frac{M_{ℓ} C δ / ℓ}{| ℓ^{'} Δ ‖ ℓ Δ |} \frac{ρ_{0}}{4} \int_{S O (3)} \int_{R^{3}} (1_{ℓ Δ} * {\tilde{η}}_{δ}) (x) | R ν_{0} \times x |^{2} d x d R

(34)

The gauge correction term (II) in can be rewritten using the partition of unity

\int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} (1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R \cdot - ℓ z - τ) \equiv 1,

see , as

(I I) = - \frac{1}{| ℓ^{'} Δ |} \frac{ρ_{0}}{4} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J} \int_{R^{3}} (1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R x - ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (x) | ν_{0} \times x |^{2} d x .

(35)

Again, this sum may be split into two parts: the J₀ part can be written as

\begin{array}{l} - \frac{1}{| ℓ^{'} Δ |} \frac{ρ_{0}}{4} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J_{0}} \int_{R^{3}} (1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R x - ℓ z - τ) | ν_{0} \times x |^{2} d x \\ = - \frac{1}{| ℓ^{'} Δ |} \frac{ρ_{0}}{4} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J_{0}} \int_{R^{3}} (1_{ℓ Δ} * {\tilde{η}}_{δ}) (x) | ν_{0} \times R^{⊤} (R_{j} x + ℓ z + ℓ z_{j} + τ) |^{2} d x \\ = - \frac{M_{ℓ}}{| ℓ^{'} Δ ‖ ℓ Δ |} \frac{ρ_{0}}{4} \int_{S O (3)} \int_{R^{3}} (1_{ℓ Δ} * {\tilde{η}}_{δ}) (x) | R ν_{0} \times x |^{2} d x d R \\ - \frac{1}{| ℓ^{'} Δ |} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J_{0}} | ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (ℓ z + ℓ z_{j} + τ) |^{2}, \end{array}

where in the last step we used once more that the smeared tetrahedron has barycenter 0, due to

\int_{R^{3}} x η (x) d x = 0

⁠. Note that the last term cancels exactly with the second term of . In total, the J₀ part of (I) + (II) may be bounded by [using also ]

\frac{(1 - C δ / ℓ) M_{ℓ}}{| ℓ^{'} Δ |} \int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R .

Next, we need to bound the part J \ J₀ of (I) corresponding to tetrahedra at the boundary. We use the Lieb–Oxford inequality and (again dropping all but the gauge term) to get

\begin{aligned} \frac{1}{| ℓ^{'} Δ |} \int_{S O (3)} d R – \int_{C_{ℓ}} d τ \sum_{(z, j) \in J \ J_{0}} [- (1 - C δ / ℓ) c_{L O} ρ_{0}^{4 / 3} \int_{R^{3}} {(1_{ℓ Δ_{j}} * {\tilde{η}}_{δ})}^{4 / 3} (R \cdot - ℓ z - τ) {(1_{ℓ^{'} Δ} * η_{δ^{'}})}^{4 / 3} \\ + \frac{ρ_{0}}{4} \int_{R^{3}} (1_{ℓ Δ_{j}} * {\tilde{η}}_{δ}) (R x - ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (x) | ν_{0} \times x |^{2} d x] . \end{aligned}

(36)

Again, we see that the second term cancels exactly with the remaining, J \ J₀ part of the gauge correction term (II), see . The first term may be bounded by

- \frac{(1 - C δ / ℓ) M_{ℓ}^{'}}{| ℓ^{'} Δ |} c_{L O} ρ_{0}^{4 / 3}, M_{ℓ}^{'} = | Δ | | J \ J_{0} | .

Let us bound the quantities M_ℓ and

M_{ℓ}^{'}

⁠. If

\int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R ⩽ 0

⁠, then we can use the trivial bound M_ℓ/|ℓ′Δ|⩽1. The boundary tetrahedra indexed by J \ J₀ are at a distance

O (ℓ + δ + δ^{'})

from ∂(ℓ′Δ), and hence they fill a volume of

M_{ℓ}^{'} = O (ℓ^{' 2} (ℓ + δ + δ^{'}))

⁠. But then M_ℓ⩾|ℓ′Δ| − Cℓ′²(ℓ + δ + δ′), so we arrive at the bound

\begin{align} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) & ⩾ (1 - C σ \frac{δ}{ℓ} - C σ \frac{ℓ + δ + δ^{'}}{ℓ^{'}}) \int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R \\ - \frac{ℓ + δ + δ^{'}}{ℓ^{'}} ρ_{0}^{4 / 3} - \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0}), \end{align}

(37)

where we have set

σ = \{\begin{cases} 0 & if \int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R ⩽ 0 \\ 1 & otherwise \end{cases}

(38)

Next, we use Corollary III.4 to bound part of the l.h.s. of the preceding inequality. To this end, note that

v (x) = 1_{s u p p (1_{ℓ^{'} Δ} * η_{δ^{'}})} (x) \frac{1}{2} ν_{0} \times x

is clearly divergence-free and has

\int_{R^{3}} ρ | D v | = | ℓ^{'} Δ | \frac{\sqrt{2}}{2} ρ_{0} | ν_{0} |

⁠. We have

\begin{array}{l} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) = (1 - C σ δ / ℓ) e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) + \frac{C σ δ}{ℓ} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) \\ = (1 - C σ δ / ℓ) e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) + \frac{1}{| ℓ^{'} Δ |} \frac{C σ δ}{ℓ} (C ρ_{0}^{5 / 3} \int_{R^{3}} {(1_{ℓ^{'} Δ} * η_{δ^{'}})}^{5 / 3} + C ρ_{0} \int_{R^{3}} | \nabla \sqrt{1_{ℓ^{'} Δ} * η_{δ^{'}}} |^{2} + C | ℓ^{'} Δ | ρ_{0} ν_{0}) \\ ⩽ (1 - C σ δ / ℓ) e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) + \frac{C σ δ}{ℓ} (ρ_{0}^{5 / 3} + \frac{ρ_{0}}{ℓ^{'} δ^{'}} + ρ_{0} ν_{0}), \end{array}

where we also used the fact that

\frac{1}{ℓ^{' 3}} \int_{R^{3}} | \nabla \sqrt{1_{ℓ^{'} Δ} * η_{δ^{'}}} |^{2} = O (\frac{1}{ℓ^{'} δ^{'}})

⁠. We obtain

\begin{array}{l} (1 - C σ \frac{δ}{ℓ} - C σ \frac{ℓ + δ + δ^{'}}{ℓ^{'}}) \int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R \\ ⩽ (1 - C σ \frac{δ}{ℓ}) e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) + \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0} + δ ρ_{0}^{2 / 3} + \frac{δ}{ℓ^{'} δ^{'}} + δ ν_{0}) + \frac{ℓ + δ + δ^{'}}{ℓ^{'}} ρ_{0}^{4 / 3} . \end{array}

After taking the liminf as ℓ′ → ∞ and dividing by (1 − Cσδ/ℓ), we arrive at

\int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R ⩽ {\underset{̲}{e}}_{U E G} (ρ_{0}, ν_{0}) + \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0} + δ ρ_{0}^{2 / 3} + δ ν_{0})

as desired.

Proof of (ii). The proof of the lower bound is similar. We use the upper bound of Theorem IV.9 with the same choice . We have for any

0 < \tilde{δ} < \frac{ℓ}{2}

and

0 < α ⩽ \frac{1}{2}

⁠,

\begin{aligned} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) \\ ⩽ \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{z \in Z^{3}} \sum_{j = 1}^{24} E (β^{- 3} (1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R \cdot - t ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) \\ - \frac{1}{| ℓ^{'} Δ |} \frac{ρ_{0}}{4} \int_{R^{3}} (1_{ℓ^{'} Δ} * η_{δ^{'}}) (x) | ν_{0} \times x |^{2} d x + C δ^{2} \log (α^{- 1}) ρ_{0}^{2} \end{aligned}

(39)

where we used the notation

β = 1 - \tilde{δ} / ℓ

⁠. The smearing parameter

\tilde{δ}

is chosen so that

\tilde{δ} = β δ

⁠, i.e.,

\tilde{δ} = δ / (1 + δ / ℓ)

⁠.

Define J and J₀ analogously. First, we consider the summation over J₀ in the first term. Using

t ℓ T_{j} (1 - \tilde{δ} / ℓ) Δ = t ℓ β R_{j} Δ + t ℓ z_{j}

⁠, we may write

(1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R x - t ℓ z - τ) = (1_{t ℓ β Δ} * {\tilde{η}}_{t \tilde{δ}}) (T (x)) = (1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (β^{- 1} T (x)),

where T is defined as above except with tℓ in place of ℓ. Here, we used the scaling relation

(1_{ℓ / a Δ} * {\tilde{η}}_{δ / a}) (x / a) = (1_{ℓ Δ} * {\tilde{η}}_{δ}) (x)

(40)

valid for all a > 0. We have

\begin{aligned} \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J_{0}} E (x \mapsto β^{- 3} (1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (β^{- 1} T (x)) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times x)) \\ = & \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J_{0}} [E (x \mapsto β^{- 3} (1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (β^{- 1} x) (ρ_{0}, \frac{1}{2} ρ_{0} R_{j}^{- 1} R ν_{0} \times x)) \\ + | t ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (t ℓ z + τ + t ℓ z_{j}) |^{2}] \\ ⩽ & \int_{S O (3)} d R – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \frac{M_{t ℓ}}{| ℓ^{'} Δ ‖ t ℓ Δ |} β^{- 1} E (x \mapsto (1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (x) (ρ_{0}, \frac{1}{2} ρ_{0} β R ν_{0} \times x)) \\ + \frac{1}{| ℓ^{'} Δ |} \int_{S O (3)} d R – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J_{0}} | t ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (t ℓ z + τ + t ℓ z_{j}) |^{2} \end{aligned}

(41)

where in the first step we used Lemma IV.5, and in the second (with λ = β⁻³ and dropping the gauge terms).

Furthermore, note that the function

ν_{0} \mapsto E ((1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot))

is convex for all fixed ρ₀ and R, using the convexity of F. Hence, we find

β^{- 1} E ((1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (ρ_{0}, \frac{1}{2} ρ_{0} β R ν_{0} \times \cdot)) ⩽ E ((1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (ρ_{0}, \frac{1}{2} ρ_{0} R ν_{0} \times \cdot)) + \frac{1 - β}{β} E ((1_{t ℓ Δ} * {\tilde{η}}_{t δ}) ρ_{0}),

where used

0 < β < \frac{1}{2}

⁠. Here, we used that E(ρ, 0) = E(ρ), see Proposition II.6.

The gauge correction term in may be written similarly to . The J₀ part reads

\begin{array}{l} - \frac{ρ_{0}}{4} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \frac{M_{t ℓ}}{| ℓ^{'} Δ ‖ t ℓ Δ |} \int_{R^{3}} (1_{t ℓ Δ} * {\tilde{η}}_{t δ}) (x) | ν_{0} \times x |^{2} d x \\ - \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J_{0}} | t ℓ Δ | \frac{ρ_{0}}{4} | R ν_{0} \times (t ℓ z + t ℓ z_{j} + τ) |^{2} . \end{array}

We see that the second term here cancels with the last term of .

Next, we consider the J\J₀ part. We use Corollary III.4 to bound the first term of ,

\begin{array}{l} \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J \ J_{0}} E (β^{- 3} (1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R \cdot - t ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (ρ_{0}, \frac{1}{2} ρ_{0} ν_{0} \times \cdot)) \\ ⩽ \frac{1}{| ℓ^{'} Δ |} – \int_{1 - α}^{1 + α} \frac{d t}{t^{4}} \int_{S O (3)} d R – \int_{C_{t ℓ}} d τ \sum_{(z, j) \in J \ J_{0}} [C ρ_{0}^{5 / 3} β^{- 5} \int_{R^{3}} (1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R \cdot - t ℓ z - τ) \\ + C ρ_{0} β^{- 3} \int_{R^{3}} {|\nabla \sqrt{(1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R \cdot - t ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}})}|}^{2} \\ + β^{- 3} \frac{ρ_{0}}{4} \int_{R^{3}} (1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R x - t ℓ z - τ) (1_{ℓ^{'} Δ} * η_{δ^{'}}) (x) | ν_{0} \times x |^{2} d x \\ + C β^{- 3} ρ_{0} ν_{0} \int_{R^{3}} (1_{t ℓ T_{j} β Δ} * {\tilde{η}}_{t \tilde{δ}}) (R \cdot - t ℓ z - τ)] \end{array}

For the J\J₀ part of the gauge correction term, we use the partition of unity with holes to cancel the gauge term in the previous estimate. In total, we have obtained

\begin{aligned} e_{ℓ^{'} Δ, η_{δ^{'}}} (ρ_{0}, ν_{0}) & ⩽ (1 + C σ \frac{ℓ + δ + δ^{'}}{ℓ^{'}}) (– \int_{1 - α}^{1 + α} \int_{S O (3)} e_{t ℓ Δ, {\tilde{η}}_{t δ}} (ρ_{0}, R ν_{0}) d R \frac{d t}{t^{4}} + \frac{δ}{ℓ} – \int_{1 - α}^{1 + α} e_{t ℓ Δ, {\tilde{η}}_{t δ}} (ρ_{0}, 0) \frac{d t}{t^{4}}) \\ + C \frac{ℓ + δ + δ^{'}}{ℓ^{'}} ρ_{0} (ρ_{0}^{2 / 3} + {(ℓ δ)}^{- 1} + ν_{0}) + \frac{C ρ_{0}}{δ^{'} ℓ^{'}} + C δ^{2} \log (α^{- 1}) ρ_{0}^{2} \\ ⩽ (1 + C σ \frac{ℓ + δ + δ^{'}}{ℓ^{'}}) (– \int_{1 - α}^{1 + α} \int_{S O (3)} e_{t ℓ Δ, {\tilde{η}}_{t δ}} (ρ_{0}, R ν_{0}) d R \frac{d t}{t^{4}} + \frac{C ρ_{0}}{ℓ} (δ ρ_{0}^{2 / 3} + ℓ^{- 1})) \\ + C \frac{ℓ + δ + δ^{'}}{ℓ^{'}} ρ_{0} (ρ_{0}^{2 / 3} + {(ℓ δ)}^{- 1} + ν_{0}) + \frac{C ρ_{0}}{δ^{'} ℓ^{'}} + C δ^{2} \log (α^{- 1}) ρ_{0}^{2} . \end{aligned}

(42)

The proof of (ii) is finished after taking the limsup as ℓ′δ′ → ∞, δ′/ℓ′ → 0.

Proof of (iii). Finally, to show (iii), we substitute ℓ → tℓ and δ → tδ in , apply

– \int_{1 - α}^{1 + α} \frac{d t}{t^{4}}

to the inequality and use (ii), with the roles of η and

\tilde{η}

interchanged,

\begin{array}{l} e_{ℓ^{'} Δ, {\tilde{η}}_{δ^{'}}} (ρ_{0}, ν_{0}) & ⩾ (1 - C σ \frac{δ}{ℓ} - C σ \frac{ℓ + δ + δ^{'}}{ℓ^{'}}) – \int_{1 - α}^{1 + α} \int_{S O (3)} e_{t ℓ Δ, η_{t δ}} (ρ_{0}, R ν_{0}) d R \frac{d t}{t^{4}} \\ - C \frac{ℓ + δ + δ^{'}}{ℓ^{'}} ρ_{0}^{4 / 3} - \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0}) \\ ⩾ {\bar{e}}_{U E G} (ρ_{0}, ν_{0}) - C \frac{ℓ + δ + δ^{'}}{ℓ^{'}} (ρ_{0}^{5 / 3} + ρ_{0}^{4 / 3} + ρ_{0} ν_{0}) \\ - \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0} + δ ρ_{0}^{2 / 3} + δ ρ_{0} ν_{0}) - C δ^{2} ρ_{0}^{2} . \end{array}

Here, we have modified σ in suitable way [cf. ]. The choice

δ = ℓ^{- 1 / 3} ρ_{0}^{- 4 / 9}

leads to

\begin{array}{l} e_{ℓ^{'} Δ, {\tilde{η}}_{δ^{'}}} (ρ_{0}, ν_{0}) & ⩾ {\bar{e}}_{U E G} (ρ_{0}, ν_{0}) - C \frac{ℓ + ℓ^{- 1 / 3} ρ_{0}^{- 4 / 9} + δ^{'}}{ℓ^{'}} (ρ_{0}^{5 / 3} + ρ_{0}^{4 / 3} + ρ_{0} ν_{0}) \\ - C \frac{ρ_{0}^{13 / 9} + ρ_{0}^{10 / 9}}{ℓ^{2 / 3}} - C \frac{ρ_{0}^{11 / 9} + ν_{0} ρ_{0}^{14 / 9}}{ℓ^{4 / 3}} - \frac{C ρ_{0}^{2 / 3}}{ℓ^{2}} - \frac{C ρ_{0}}{ℓ} \end{array}

and setting

ℓ = {(ℓ^{'})}^{3 / 5} ρ_{0}^{- 2 / 15}

results in

e_{ℓ^{'} Δ, {\tilde{η}}_{δ^{'}}} (ρ_{0}, ν_{0}) ⩾ {\bar{e}}_{U E G} (ρ_{0}, ν_{0}) - \frac{C δ^{'}}{ℓ^{'}} (ρ_{0}^{5 / 3} + ρ_{0}^{4 / 3} + ρ_{0} ν_{0}) - C \frac{ρ_{0}^{23 / 15} + ρ_{0}^{18 / 15}}{{(ℓ^{'})}^{2 / 5}},

using

ℓ^{'} ρ_{0}^{1 / 3} ⩾ C

⁠.

Proof of (iv). To show that the limit exists, it is enough to prove that

{\bar{e}}_{U E G} (ρ_{0}, ν_{0}) ⩽ {\underset{̲}{e}}_{U E G} (ρ_{0}, ν_{0})

⁠. But this follows from (i) combined with (iii). It remains to prove that e_UEG(ρ₀, ν₀) is independent of the choice of the regularization function. Let

{\tilde{e}}_{U E G} (ρ_{0}, ν_{0})

denote e_UEG(ρ₀, ν₀), but with η_δ replaced with

{\tilde{η}}_{δ}

⁠. Using (iii) and then (i), we find that the following chain of inequalities hold true,

e_{U E G} (ρ_{0}, ν_{0}) ⩽ \int_{S O (3)} e_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, R ν_{0}) d R + (e r r o r) ⩽ e_{U E G} (ρ_{0}, ν_{0}) + (e r r o r),

from which we deduce

{\tilde{e}}_{U E G} (ρ_{0}, ν_{0}) = e_{U E G} (ρ_{0}, ν_{0})

⁠. □

By omitting the Coulomb interaction, we find the following.

FIG. 1.

$The tetrahedra corresponding to R=1 and τ = 0 included in the sets J0 and J\J0 are depicted in blue and red, respectively. The tiling is composed of translates of ℓΔj.$

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The tetrahedra corresponding to $R = 1$ and τ = 0 included in the sets J₀ and J\J₀ are depicted in blue and red, respectively. The tiling is composed of translates of ℓΔ_j.

Proposition IV.11.

Fix $ρ_{0} \in R_{+}$ and $ν_{0} \in R_{+}$ . Let η and $\tilde{η}$ be regularization functions. Let ${\underset{̲}{τ}}_{U E G} (ρ_{0}, ν_{0})$ and ${\bar{τ}}_{U E G} (ρ_{0}, ν_{0})$ denote the liminf and the limsup of $τ_{ℓ Δ, η_{δ}} (ρ_{0}, ν_{0})$ as ℓδ → ∞ and δ/ℓ → 0.

For Cδ⩽ℓ, we have
$τ_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, ν_{0}) ⩽ {\underset{̲}{τ}}_{U E G} (ρ_{0}, ν_{0}) + \frac{C ρ_{0}}{ℓ} (δ^{- 1} + δ ρ_{0}^{2 / 3} + δ ν_{0})$
We have
$τ_{ℓ Δ, {\tilde{η}}_{δ}} (ρ_{0}, ν_{0}) ⩾ {\bar{τ}}_{U E G} (ρ_{0}, ν_{0}) - \frac{C ρ_{0}}{ℓ} (δ ρ_{0}^{2 / 3} + ℓ^{- 1})$
The thermodynamic limit
$\lim_{\begin{array}{c} δ / ℓ \to 0 \\ ℓ δ \to \infty \end{array}} τ_{ℓ Δ, η_{δ}} (ρ_{0}, ν_{0}) = τ_{U E G} (ρ_{0}, ν_{0})$
exists and is independent of the choice of the regularization function η.

G. Proof of Theorem III.1

The proof is very similar to the one in Ref. 4. Let {Ω_N} be a sequence of domains such that |Ω_N| → ∞, |∂Ω_N + B_r|⩽Cr|Ω_N|^2/3 for r⩽|Ω_N|^1/3/C and δ_N/|Ω_N|^1/3 → 0, δ_N|Ω_N|^1/3 → ∞.

We can define the index sets J and J₀ for the domain Ω_N in an obvious manner,

J ≔ \{\begin{aligned} (z, j) \in Z^{3} \times {1, \dots, 24} : & s u p p ((1_{ℓ Δ_{j}} * η_{δ}) (R \cdot - ℓ z - τ)) \cap s u p p (1_{Ω_{N}} * η_{δ_{N}}) \neq \emptyset, \\ for some R \in S O (3) and τ \in C_{ℓ} \end{aligned}\},

which collects all the small tetrahedra R^⊤(ℓΔ_j + ℓz + τ) that possibly intersect the smearing of Ω_N. Furthermore, define

J_{0} ≔ \{\begin{aligned} (z, j) \in Z^{3} \times {1, \dots, 24} : & s u p p ((1_{ℓ Δ_{j}} * η_{δ}) (R \cdot - ℓ z - τ)) \subset Ω_{N} \ (\partial Ω_{N} + B_{δ_{N}}), \\ for some R \in S O (3) and τ \in C_{ℓ} \end{aligned}\},

which contains small smeared tetrahedra which are well inside Ω_N.

The argument for the tetrahedron ℓ′Δ works exactly the same for the general domain Ω_N, we only need to bound the quantities M_ℓ and

M_{ℓ}^{'}

to obtain the analog of . If

e_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0}) ⩽ 0

⁠, then we can use the trivial bound M_ℓ/|Ω_N|⩽1. The boundary tetrahedra indexed by J \ J₀ are at a distance r = ℓ + δ + δ_N and hence, by the uniform regularity hypothesis, they fill a volume of

M_{ℓ}^{'} = C | \partial Ω_{N} + B_{r} | ⩽ C (ℓ + δ + δ_{N}) | Ω_{N} |^{2 / 3}

for sufficiently large N, since r⩽|Ω_N|^1/3/C by the convergence δ_N/|Ω_N|^1/3 → 0. But then M_ℓ⩾|Ω_N| − C(ℓ + δ + δ′)|Ω_N|^2/3, so we arrive at the bound

\begin{array}{l} e_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0}) & ⩾ (1 - C σ \frac{δ}{ℓ} - C σ \frac{ℓ + δ + δ_{N}}{| Ω_{N} |^{1 / 3}}) \int_{S O (3)} e_{ℓ Δ, δ} (ρ_{0}, R ν_{0}) d R \\ - \frac{ℓ + δ + δ_{N}}{| Ω_{N} |^{1 / 3}} ρ_{0}^{4 / 3} - \frac{C ρ_{0}}{ℓ} (1 + δ^{- 1} + δ^{3} ρ_{0}) . \end{array}

Taking δ fixed and ℓ = |Ω_N|^1/6, the r.h.s. goes to e_UEG(ρ₀, ν₀) as δ_N/|Ω_N|^1/3 → 0.

For the upper bound, we use the analog of , with α = 1/2,

\begin{array}{l} e_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0}) & ⩽ (1 + C σ \frac{ℓ + δ + δ_{N}}{| Ω_{N} |^{1 / 3}}) (– \int_{1 / 2}^{3 / 2} \int_{S O (3)} e_{t ℓ Δ, t δ} (ρ_{0}, R ν_{0}) d R \frac{d t}{t^{4}} + \frac{C ρ_{0}}{ℓ} (δ ρ_{0}^{2 / 3} + ℓ^{- 1})) \\ + C \frac{ℓ + δ + δ_{N}}{| Ω_{N} |^{1 / 3}} ρ_{0} (ρ_{0}^{2 / 3} + {(ℓ δ)}^{- 1} + ν_{0}) + \frac{C ρ_{0}}{δ_{N} | Ω_{N} |^{1 / 3}} + C δ^{2} ρ_{0}^{2} . \end{array}

Using now δ_N|Ω_N|^1/3 → ∞ as well as δ_N/|Ω_N|^1/3 → 0, and the choice ℓ = |Ω_N|^1/6, δ = |Ω_N|^−1/12, we see that the r.h.s. goes to e_UEG(ρ₀, ν₀), and the proof is finished.

The proof for the kinetic energy per volume $τ_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0})$ is very similar. The limit for $e_{Ω_{N}, δ_{N}}^{x c} (ρ_{0}, ν_{0})$ is obtained by subtracting $τ_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0})$ from $e_{Ω_{N}, δ_{N}} (ρ_{0}, ν_{0})$ and taking the limit term-by-term.

H. Further proofs

Proof of Proposition II.1.

Part (i). Following Ref. 12, it is convenient to consider the so-called intrinsic kinetic energy tensor,

ω_{γ} (x) ≔ (π_{x} \otimes \bar{π_{y}}) γ (x, y) |_{y = x} = τ_{γ} (x) - \frac{ζ_{γ} (x) \otimes \bar{ζ_{γ} (x)}}{ρ_{γ} (x)},

(43)

where

π_{x} ≔ - i \nabla_{x} - \frac{ζ_{γ} (x)}{ρ_{γ} (x)}

⁠. We immediately have that 0⩽ω_γ(x) = ω_γ(x)^†. Also, ω_γ is invariant under gauge transformations. For clarity, we omit the subscripts γ. Note first that

\begin{aligned} \frac{ω (x) - ω {(x)}^{⊤}}{2 i} = \frac{1}{2 i} (π_{x} \otimes {\bar{π}}_{y} - {\bar{π}}_{y} \otimes π_{x}) γ (x, y) |_{y = x} \\ = \frac{1}{2 i} \sum_{j ⩾ 1} λ_{j} (\nabla φ_{j} (x) \otimes \bar{\nabla φ_{j} (x)} - \bar{\nabla φ_{j} (x)} \otimes \nabla φ_{j} (x)) + \frac{1}{2 i} \frac{ζ (x) \otimes \bar{ζ (x)} - \bar{ζ (x)} \otimes ζ (x)}{ρ (x)} \\ = D_{a} j^{p} (x) - \frac{1}{2} \frac{\nabla ρ (x) \otimes j^{p} (x) - j^{p} (x) \otimes \nabla ρ (x)}{ρ (x)} \\ = ρ (x) D_{a} \frac{j^{p} (x)}{ρ (x)} . \end{aligned}

(44)

We find

ρ (x) |D_{a} \frac{j^{p} (x)}{ρ (x)}| ⩽ \sqrt{d} {T r}_{R^{d}} ω (x),

which upon expansion of the r.h.s. and using Theorem 6.17 in Ref. 25,

{T r}_{R^{d}} ω (x) = τ (x) - \frac{| ζ (x) |^{2}}{ρ (x)} = τ (x) - | \nabla \sqrt{ρ} (x) |^{2} - \frac{| j^{p} (x) |^{2}}{ρ (x)}

yields the desired bound.

Part (ii). Follows from the Cauchy–Schwarz inequality. □

Proof of (3). Let

γ (x, y) = \sum_{j ⩾ 1} λ_{j} φ_{j} (x) \bar{φ_{j} (y)}

⁠, for some 0⩽λ_j⩽1 with ∑_j⩾1λ_j = N and

{φ_{j}}_{j ⩾ 1} \subset H^{1} (R^{d}) L^{2}

-orthonormal. Let

{φ_{j}^{n}} \subset C_{c}^{\infty} (R^{3})

be such that

‖ φ_{j}^{n} ‖ = 1

and

φ_{j}^{n} \to φ_{j}

in H¹ as n → ∞. Denoting

ρ_{n} (x) = \sum_{j ⩾ 1} λ_{j} | φ_{j}^{n} (x) |^{2}

⁠, we have that

‖ \sqrt{ρ} - \sqrt{ρ_{n}} ‖ \to 0

and

‖ \nabla \sqrt{ρ} - \nabla \sqrt{ρ_{n}} ‖ \to 0

by an argument similar to Proof of Theorem 1.3 in Ref. 16. It is easy to see that we also have

‖ \frac{j_{n}^{p}}{\sqrt{ρ_{n}}} - \frac{j^{p}}{\sqrt{ρ}} ‖ \to 0

⁠. Using Proposition II.1 (i), we may write

\sum_{j ⩾ 1} λ_{j} ‖ \nabla φ_{j}^{n} ‖^{2} ⩾ \int_{R^{3}} | \nabla \sqrt{ρ_{n}} |^{2} + \int_{R^{3}} \frac{| j_{n}^{p} |^{2}}{ρ_{n}} + \frac{1}{\sqrt{6}} \int_{R^{3}} ρ_{n} | ν_{n} |,

where

j_{n}^{p}

and ν_n are defined in terms of the

φ_{j}^{n}

⁠. Using expansion ,

\begin{array}{l} \int_{R^{3}} |ρ_{n} ν_{n} - ρ ν| = \int_{R^{3}} |ρ_{n} r o t \frac{j_{n}^{p}}{ρ_{n}} - ρ r o t \frac{j^{p}}{ρ}| \\ ⩽ \sum_{j ⩾ 1} λ_{j} \int_{R^{3}} | \nabla φ_{j}^{n} \times \bar{\nabla φ_{j}^{n}} - \nabla φ_{j} \times \bar{\nabla φ_{j}} | + \int_{R^{3}} |\frac{\nabla ρ_{n}}{\sqrt{ρ_{n}}} \times \frac{j_{n}^{p}}{\sqrt{ρ_{n}}} - \frac{\nabla ρ}{\sqrt{ρ}} \times \frac{j^{p}}{\sqrt{ρ}}| \\ ⩽ \sum_{j ⩾ 1} λ_{j} (‖ \nabla φ_{j}^{n} ‖ + ‖ \nabla φ_{j} ‖) ‖ \nabla φ_{j}^{n} - \nabla φ_{j} ‖ + 2 [{(\int_{R^{3}} \frac{| j_{n}^{p} |^{2}}{ρ_{n}})}^{1 / 2} ‖ \nabla \sqrt{ρ_{n}} - \nabla \sqrt{ρ} ‖ + ‖ \nabla \sqrt{ρ} ‖ ‖\frac{j_{n}^{p}}{\sqrt{ρ_{n}}} - \frac{j^{p}}{\sqrt{ρ}}‖] \to 0, \end{array}

we obtain the desired bound. □

ACKNOWLEDGMENTS

A.L. and M.A.Cs. acknowledge funding through ERC StG REGAL under Agreement No. 101041487. M.A.Cs. is grateful to Mathieu Lewin and Giovanni Vignale for helpful discussions. A.L., M.A.Cs. and E.I.T. acknowledge funding from RCN through the Hylleraas Center under Agreement No. 262695. E.I.T. acknowledges funding from RCN under Agreement No. 287950. A.L. and M.A.Cs. also received partial funding from the Norwegian Research Council through Grant Nos. 287906 (CCerror) and 262695 (CoE Hylleraas Center for Quantum Molecular Sciences).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Mihály A. Csirik: Writing – original draft (equal). Andre Laestadius: Writing – original draft (equal). Erik I. Tellgren: Writing – original draft (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

I. INTRODUCTION

II. SETTING

A. States

B. One-particle quantities

C. Gauge transformation

D. Representability

E. Definition of the grand-canonical current-density functionals

III. MAIN RESULTS

A. The magnetic uniform electron gas

B. Upper bound on the kinetic energy functional and representability

IV. PROOFS

A. Proof of Theorem III.2

B. Further bounds

C. Transformation properties

D. Decoupling lower bound

E. Decoupling upper bound

F. Energy per volume in a tetrahedron

G. Proof of Theorem III.1

H. Further proofs

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

I. INTRODUCTION

II. SETTING

A. States

B. One-particle quantities

C. Gauge transformation

D. Representability

E. Definition of the grand-canonical current-density functionals

III. MAIN RESULTS

A. The magnetic uniform electron gas

B. Upper bound on the kinetic energy functional and representability

IV. PROOFS

A. Proof of Theorem III.2

B. Further bounds

C. Transformation properties

D. Decoupling lower bound

E. Decoupling upper bound

F. Energy per volume in a tetrahedron

G. Proof of Theorem III.1

H. Further proofs

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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