Floating Wigner Crystal and Periodic Jellium Configurations

Extending on ideas of Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127, (2019)) we present a modified"floating crystal"trial state for Jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the Jellium energy coincide in dimensions $d\geq 2$, thus extending the result of Cotar and Petrache (arXiv: 1707.07664) and Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127, (2019)) that the three definitions coincide in dimension $d \geq 3$. We show that the Jellium energy is also equivalent to a"renormalized energy"studied in a series of papers by Serfaty and others and thus, by work of B\'etermin and Sandier (Constr Approx, 47:39-74, (2018)), we relate the Jellium energy to the order $n$ term in the logarithmic energy of $n$ points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the Jellium energy of periodic configurations.


Introduction
The Jellium model is an important and very simple model, which models electrons in a uniformly charged background. In this paper we discuss three different definitions of the Jellium energy density Jel . These are known to coincide in dimensions ≥ 3 [10,17] and differ in dimension = 1 [1,8,9,14,17]. We show that they coincide in dimension ≥ 2 using a similar method as in [17]. This verifies a conjecture by Cotar and Petrache [10,Remark 1.7], that the definitions of the Jellium energy density coincide in dimensions = 2. The main difference in our argument as compared to that of [17] is the choice of a slightly different "floating crystal" trial state, for which the density is a characteristic function. In dimension = 3 the thermodynamic limit of the Uniform Electron Gas exists under weaker conditions by the Graf-Schenker inequality [12] as discussed in [17,18]. We do not have this stronger form of the thermodynamic limit in dimensions ≠ 3. Hence we need a trial state, for which the density is a characteristic function.
Secondly we consider the question of periodic Jellium configurations, where the electrons are confined to sites of a lattice. Here we find that the energy is given by the Epstein (or lattice) -function associated to the lattice, on which the electrons sit.
Thirdly we relate the evaluation of the Jellium energy to that of a "renormalized energy" studied in [3,4,10,15,[24][25][26][27]. Here we show that min A 1 = 2 Jel (notation explained in section 4). This gives another equivalent definition of the Jellium energy. Bétermin and Sandier [3] have shown that the logarithmic energy on points on S 2 has a term of order given by log = 1 min A 1 + log 4 2 , (see Remark 4.6). Hence we get yet another equivalent definition of the Jellium energy. The relation of Jel and min A 1 carry over the known bounds for the Jellium energy. The lower bound of Jel ≥ −0.66118 by Lieb and Narnhofer [21] and Sari and Merlini [28] has been known for many years. It improves upon known bounds for the constants log and min A 1 . In particular it gives the bound −0.0569 ≤ log ≤ −0.0556 improving on the best known lower bound of log ≥ −0.0954 due to Steinerberger [29]. Since the proof of the lower bound in [28] is not very detailed, we give the proof in the appendix.

Three Definitions of the Jellium Energy
We now introduce the three models. We will give the argument only in dimension = 2, partly because this case is where the argument is most complicated and partly because the physically interesting cases are dimensions = 1, 2, 3, and the cases ≥ 3 are solved [10,17]. For dimensions ≥ 3 the argument is the same, only one should replace every occurrence of − log with | · | 2− .
The first model is what we will call Jellium. By scaling we may assume that the density of the background is = 1. Then the Jellium energy of particles in a domain Ω of size |Ω | = is The electrons are thought of as discrete classical particles in a uniform (positive) background, such that the entire system is neutral. The electrons and background all interact through Coulomb interaction, in 2 dimensions given by − log | |. The long range behaviour of the logarithm means that this setting is somewhat different from the 3-dimensional case. In [28] it is shown that the thermodynamical limit exists under fairly non-restrictive conditions on the sequence of domains Ω . For instance Ω = 1/2 Ω for a fixed convex set Ω of size |Ω| = 1.
The second model is that of periodic Jellium. Here the = ℓ 2 electrons live on a torus of side length ℓ in a uniform background of opposite charge. The Coulomb potential between the electrons is replaced by the periodic Coulomb potential, where the electrons interact with all the mirror images of the other electrons and the uniform background. The functional is defined as follows.
First, we define the periodic Coulomb potential ℓ as follows. ℓ ( ) = 1 ( /ℓ), where 1 is the one-periodic Coulomb potential, satisfying −Δ 1 = 2 ∈Z 2 − 1 and ∫ 1 1 d = 0, where 1 = (−1/2, 1/2) 2 . It corresponds to the potential generated by a point charge and all its images together with a uniform oppositely charged background. The background must be included for this not to diverge. Then Now, 1 ( ) + log | | has a limit as → 0, which we call mad . It is the Madelung constant, i.e. twice the energy per particle of the configuration with 1 particle in the unit cell -i.e. a square lattice configuration. The functional E per,ℓ may now be defined as The first term is what one gets if one just naively replaces the Coulomb interaction in the Jellium functional by the periodic version ℓ . Note that then the particle-background and background-background terms vanish due to the fact that ∫ ℓ ℓ d = 0. We then define The existence of this limit was established in [15,[23][24][25][26]. It will also follow from the proof of Theorem 2.1 that indeed this limit exists. The third model is what has been called the uniform electron gas (UEG) in [17,18]. For a complete description of this model see [18]. Here there is no background charge, and the electrons are no longer point particles. Instead the electrons are distributed according to a probability density P (meaning P is a probability measure on R 2 ) which we require to give a constant density P = ½ Ω , where P is the sum of all the marginals. The indirect energy of the distribution is then We are interested in keeping the density fixed, and so, for any density with ∫ Since the electrons are indistinguishable, we should in principle restrict to symmetric P's. This however gives the same minimum. Again, we are interested in the thermodynamic limit, and for a system of uniform density, i.e.
The existence of this was established in [18,Theorem 2.6]. Their proof is done in dimensions ≥ 3, but works without change also in dimensions = 1, 2. Now, our main theorem is Theorem 2.1. We have Jel = per = UEG . The analogous result in dimensions ≥ 3 was proven by Cotar and Petrache [10] using methods of optimal transport and later in dimension = 3 by Lewin, Lieb and Seiringer [17] using a "floating crystal" trial state, which our method builds on. Cotar and Petrache [10,Remark 1.7] note that the case of = 2 is an open problem. Our findings here thus solves this open problem.

LATTICE CONFIGURATIONS
One inequality is the following argument. Let P be any -particle probability measure with Optimising over P and taking the thermodynamical limit we thus get EUG ≥ Jel . In order to get the inequality UEG ≤ per ≤ Jel we will superficially introduce a crystal structure to the Jellium configuration. This is similar to (and inspired by) the floating crystal argument from [17]. We give the proof in sections 5 and 6.

Lattice Configurations
We now consider the Jellium energies of periodic configurations, when the electrons are positioned on a lattice. We will consider these configurations in any dimension and for general Riesz interactions. These we first define. For ∈ R the Riesz potential on R is given by Then −2 is the Coulomb potential in dimensions. With this we may define for < the Jellium energy in dimensions with potential , This function has a meromorphic continuation to all of C with a simple pole at = , see [6]. These more complicated -functions can oftentimes be expressed in terms of simpler functions, see [30]. We prove the following.
Many similar results exist in the literature. In [5,6] a similar result is shown for a slightly different energy functional in the case − 2 < < via analytic continuation of the Epstein -function. Extending upon these ideas a partial result for the Jellium energy is shown in [16,Appendix B]. Other formulations also exist, see for instance [3,7,13] for the case of logarithmically interacting points on the unit 2-sphere, and [3,27] for the case of logarithmically interacting points on the plane. As we have not found a complete proof of Theorem 3.1 in the literature, we give a straightforward proof in section 7. As an application of the theorem, we compute the energy density of Jellium in the triangular lattice (in 2 dimensions).

Example 3.2. The triangular lattice is given by
. Thus by [30] we have where is the Riemann zeta-function and 3 ( ) = ( , ) is the Dirichlet -series for the nontrivial character mod 3, i.e.
where ( , ) is the Hurwitz -function. The values of these functions and their derivatives can be found in [22]. We conclude that L Jel, In comparison, the best-known lower bound [21,28] is Jel, =0 ≥ − 3 8 + 1 4 log ≃ −0.66118. The triangular lattice, which is what we expect to be the ground state, is remarkably close to this lower bound.

Relation to the Renormalized Energy
We now relate the Jellium energy to the renormalized energy defined in [27]. This renormalized energy has appeared before in the study of Ginzburg-Landau theory and of Coulomb gases, see [4,10,15,[24][25][26][27]. We recall the definition here.
Then, for any function we define The renormalized energy is then defined as where denotes any cutoff-functions satisfying We recall a few properties of from [27].
• The renormalized energy ( ) does not depend on the choice of cut-off functions .
• min A 1 is the limit of a sequence of periodic configurations with period → ∞.
For periodic Λ we have the following result.
Then ( ) ≥ ( { } ) for any satisfying Equation (4.1). Note that { } is defined uniquely up to a constant, and thus { } is well-defined. Now, the relation of this renormalized energy to the Jellium energy is the following.
Corollary 4.4. The renormalized energy is given by Cotar and Petrache [10] have shown a similar result for more general Riesz interactions, but not including the = 2, = 0 case, which is the one considered here, see remark 4.7.
Proof. Since min A 1 is the limit of periodic configurations [27, Theorem 1] and any such periodic configuration clearly has energy at least min A 1 we have Hence by Theorem 2.1 Remark 4.5. With this, the known bounds on Jel carry over. The upper bound of where L is the triangular lattice was previously known [3,27]. The lower bound Jel ≥ − 3 8 + 1 4 log [21,28] gives the bound This is an improvement on previous known lower bounds. The previous known lower bound by Steinerberger [29], translated to this setting using the results of [3], is Remark 4.6. The Renormalized energy has also been used by Bétermin and Sandier [3] in the problem of optimal point-configurations on the sphere S 2 with a logarithmic energy functional, i.e. points 1 , . . . , The problem of minimising the logarithmic energy of points on the sphere has received much study, see [3,7,11,13,29], and is linked to Smale's 7th problem, see [2] for a review.
Remark 4.7. The Renormalized energy has also been defined for general Riesz potentials in [24], and Jellium and periodic Jellium in [17]. Let W be as defined in [24,Definition 1.3] (this differs from the considered above by a factor of 2 in the case = 2, = 0) and per ( , ) the periodic Jellium energy for Riesz potential with parameter and in dimension as defined in [17]. Exactly the same proof as above shows that min if max(0, − 2) < < , In [10,15,17,24] and Theorem 2.1 it is proved that Jel ( , ) = per ( , ) for the relevant , .
Thus we have min A 1 W = 2 , Jel ( , ). This result was previously shown in [10] only not including the case = 2, = 0.
We now turn to the proofs of Theorems 2.1 and 3.1.

Upper Bound for the Uniform Electron Gas Energy
We first show that The proof is very similar to the proof of the same result in dimension = 3 presented in [17]. The main difference is the different choice of trial state P. As discussed above, we cannot just use the trial state considered in [17], since in 2 dimensions we do not have the more relaxed formulation of the thermodynamic limit for the UEG, which is used in [17]. It is not enough that the density P is 1 in the bulk of Ω , 0 outside and bounded close to the boundary. Secondly, due to the long range behaviour of the logarithm some error bounds are slightly more complicated in 2 dimensions. The construction of the trial state P is similar to that of [17]. We consider a floating crystal immersed in a thin fluid layer. We want our density P to be a characteristic function. In particular, we do not allow for the fluid to ever be where the crystal might also be (under different translations), i.e. we make a hole in the fluid which is larger than the crystal. Additionally, the fluid layer is not chosen to have constant density 1, but instead have some density profile , which we describe below. Consider any arrangement of points 1 , . . . , in the cube ℓ of side length ℓ = 1/2 (centered at 0). Adding a background shifted by the center of mass = 1 We choose such that the fluid and the crystal never overlap. The crystal, when shifted around, gives some density in the region Ω + ℓ . We thus want that any shift + of doesn't overlap with this. Since is in general just some vector ∈ ℓ , this leads to our definition of = Ω + 3 ℓ + . We need ≥ ½ + for any shift , and so this leads to ≥ ½ , with ⊃ Ω + 5 ℓ . This is true by construction. We compute that Thus, P is a characteristic function as desired.
In principle, we could have taken any function with ½ ≤ ≤ ½ +½ * ½ ℓ since then the density P would satisfy P ≤ 1 and it is trivial to extend the thermodynamic limit [18, Theorem 2.6] also to this case. In the computations below we will use the description of P in terms of , as this will make the computations slightly nicer.
To compute the energy E Ind (P) we first introduce the notation for two (signed) measures , . This is the Coulomb interaction energy between charge distributions and . Mostly we will use this in the case where the measures are given by functions.
The analogous object in dimensions ≥ 3 has ( ) ≥ 0 for any function. This is however not true in dimension = 2. In general it is only true for functions with zero mean. Proof. By density we may assume that ∈ S, i.e. that is rapidly decreasing. Define # ( ) = (− ). Note that # =ˆ . First we show that as functions. By the assumption ∫ d = 0 we have that the right-hand-side actually stays bounded (and smooth) as → 0. We conclude thatˆ ∈ S and so ∈ S has a Fourier transform as a function. Now, with ·|· denoting application of a distribution we have Remark 5.2. In dimension = 1 an analogous statement also holds. Now, we may calculate the energy (with , for > denoting the points + ℓ for ≠ 0) Remark 5.3. In dimensions ≥ 3 the analogous term is ≤ 0 since ( ) ≥ 0 for any function .
For any denote by = supp ( − ½ + ). Then we have | | = 1/2 and diam = ( ) = 1/2 . Thus We are thus left with the error term Plugging in the value of P we may calculate this term as We claim that Equation (5.1) is ( 1/2 log ) and thus vanishes in the desired limit. This will follow from appropriate Taylor expansions of − log | · | and the following two propositions.
One should think that is either Lebesgue measure or a sum of appropriately distributed -measures. To show this, note that for any fixed ∈ we can bound the -integral by the integral over a ball of radius = 1/2 centered at (removing the ball of radius 1). Thus Proposition 5.5. Let , be as in Proposition 5.4 and be a probability measure supported in ℓ . Denote by the first moment of , i.e. = ∫ d ( ). Let be a function supported in , which is bounded uniformly in . Then Remark 5.6. In dimension ≥ 1 we similarly have ∫ ∫ Thus our argument also works in higher dimensions.
We postpone the proof of Proposition 5.5 to the appendix. Proposition 5.5 immediately gives that the second and third terms of Equation (5.1) are ( 1/2 log ). For the first term we use that by Taylor expansion for bounded and | | bounded from below. Thus for the term by Proposition 5.4. This gives the bound E Ind ( P ) ≤ E Ind (P) ≤ E Jel (Ω + , 1 , . . . , ) + ( ). Hence, by taking the thermodynamic limit we get the desired.
We now show that This argument is more or less the same as in [17]. There are slight differences in the case = 2 compared to the case ≥ 3, which is why we present the argument here. We really only need the bound ≤, and this is what we now show. Note that the inter-particle distance is bounded uniformly from below (since there are only finitely many particles in the "unit cell" ℓ ). Hence, by replacing the point charges by smeared out charges of some small radius smaller than all the inter-particle distances Newton's theorem says that all the particle-particle energies are preserved, but the particle-background interaction only increases (decreases in numerical size, but this energy is negative). Writing we thus have We now investigate the first term more closely. We may write Since is of compact support and satisfies ∫ d = 0 and ∫ ( ) d = 0 (this is where the zero dipole moment is used) we have thatˆ is smooth and satisfiesˆ ( ) = ( ) as → 0, thus |ˆ ( )| 2 2 vanishes at zero. Thus, we need to consider the behaviour of ℓ 2 in the limit → ∞ (i.e. → ∞). We have weakly, and so Plugging in the definition of and using that ∫ ℓ ℓ d = 0 we thus have Since ⇀ as → 0 the second term converges to 1 < ℓ ( − ). This is exactly the first term in the functional E per,ℓ as desired. We now deal with the other term in the limit → 0.
where →0 (1) vanishes as → 0 uniformly in , by the compact support of 1 . Hence Putting everything together we now conclude This proves the one inequality. To conclude the other inequality note that the error we made in replacing the point charges with smeared out ones can be bounded by ∫ − log | | ( ) d = (− 2 log ) per particle by Newton's theorem. This vanishes as → 0 and thus we conclude the equality This proves that UEG ≤ E per,ℓ ( 1 ,..., ) .

Upper Bound for the Periodic Energy
We now show that This finishes the proof of Theorem 2.1. This argument is again more or less the same as in [17]. Again, there are some slight differences in the case = 2, which is why we give the argument here. First, we show that a version of Newton's theorem hold for the periodic potential ℓ . Namely that separated neutral radial charge densities have zero total interaction. More precisely, let be any compactly supported radial neutral charge distribution, i.e. supp ⊂ for some > 0, is radial and ∫ d = 0. Let be large enough so that ⊂ . Then we have that (Note the importance of being neutral, so * 1 = 0.) Since both and * are -periodic this shows that they differ by a periodic harmonic function, i.e. a constant. Moreover, by Newton's theorem we have that vanish on \ , thus we see that * is constant on \ , and so for another neutral radial charge distribution ′ supported in this region their interaction vanish, ∬ 1 We use the Swiss cheese theorem [20, sect. 14.5] to fill (most of) the cube with balls of integer volume ranging in sizes from some fixed ℓ 0 to a largest size of order ℓ. The ratio of the volume not covered by the balls is small in comparison to the volume of the cube, in the sense that if we take ℓ → ∞ after taking → ∞ this ratio vanishes.
We now construct a trial state using these balls. In each ball we place = | | particles in the optimal Jellium configuration for the ball . (Note that refers to the index of the ball, and not its radius.) The remaining = − | | particles are placed uniformly in the remainder = \ , meaning that we smear the particles out in this region. This yields where 1 , . . . , − denote the points in and we used that ∫ d = 0. Now, by rotating the charges inside each of the balls separately and taking the average over all such 1 Note that the analogous statement in [17] is wrong. There it is claimed that * = 0 on \ . This is not true in general. In general we only have * constant on \ . Since the result is only ever used for interactions between neutral charges (as we do here), their proof that UEG = per = Jel in dimension = 3 still works.
rotations we may use the modified Newton's theorem above to conclude that the balls don't interact with each other. Writing˜ for the rotational average of we thus get the upper bound As → ∞ we have that ( ) = − log | | + log + mad + (1). Plugging this into the bound above the 1 -term is (1) and the remaining log and mad -terms cancel. What we are left with is the bound E Jel , where the term →∞ (1) depends on ℓ, but not on . Dividing by = 2 and taking the consecutive limits → ∞, ℓ → ∞ and ℓ 0 → ∞ this gives Jel , by the existence of the thermodynamic limit for Jellium [28]. We conclude that We now give the proof of Theorem 3.1.
Proof of Theorem 3.1. First, we extend to complex-valued as follows. For ∈ C \ R we define ( ) = | | − . Now define the functions and˜ for any ∈ C by For > − 4 the sum ∈L\0 ( ) converges and so we may take the thermodynamic limit. Let now ∈ C, Re( ) > . We may then write We now want to write this in a form, that makes sense for all ≠ satisfying Re( ) > − 4. For any fixed > 0 such that Both of these terms are holomorphic for ≠ for any fixed > 0. For Re( ) < we may take → 0. Hence the analytic continuation of this term is − ∫ | | − d when Re( ) < . For the second term we write ∫ We now split this integral according to makes sense for < Re( ) < 2 and extends analytically to ≠ , 2 . The "poles" at = 2 in fact cancel out, so 2 is not a pole of L ( ).
For Re( ) < we may take , , → 0 in a suitable order. All these terms combined then give the limit Thus, for − 4 < Re( ) < we have that (for the analytic continuation) Thus for real − 4 < < with ≠ 0 we have This finishes the proof of Theorem 3.1.
A Proof of Proposition 5.5 Proof of Proposition 5.5. Let = | | 1/2 be the side length of , find an ℓ ′ ≥ 4ℓ of order 1 such that /ℓ ′ is an integer. Let denote the square of side length ℓ ′ centered at zero. Tile the plane with translates of such that, for the relevant translates, the centers lie on the boundary . That is, R 2 = ( + ) and if + intersect both and , then ∈ . Now, for any ∈ we have that ∈ + for some (unique) ∈ , see Figure  We now split the -integral in two according to whether is "close" to , namely if ∈ + 2 or if is "far" from , namely if ∉ + 2 . For the close 's we get the contribution