Dynamic Correlations in Disordered Systems: Implications for High-Temperature Superconductivity

Abstract

Liquids and gases are distinct in their extent of dynamic atomic correlations; in gases, atoms are almost uncorrelated, whereas they are strongly correlated in liquids. This distinction applies also to electronic systems. Fermi liquids are actually gas-like, whereas strongly correlated electrons are liquid-like. Doped Mott insulators share characteristics with supercooled liquids. Such distinctions have important implications for superconductivity. We discuss the nature of dynamic atomic correlations in liquids and a possible effect of strong electron correlations and Bose–Einstein condensation on the high-temperature superconductivity of the cuprates.

1. Introduction

At ambient pressure, liquid evaporates to gas through a first-order phase transition. So, the common perception is that liquids and gases represent different phases of matter. However, the distinction between liquids and gases is not as obvious as it may appear. For instance, liquid changes into gas continuously under high pressure beyond the tri-critical point [1]. From a more fundamental point of view, the crucial difference between gases and liquids is in the way that the interatomic potential, V(r), affects the dynamics [2,3,4]. In an ideal gas, atoms interact with each other only through occasional collisions, so that the average potential energy, <U>, is practically zero. This happens at a low density, or equivalently, at a high temperature where the effective atomic radius, R_eff, defined by $V\left(2R_{eff}\right)=k_{B}T$, is much smaller than the radius determined by the minimum in V(r). On the other hand, atoms in a liquid are always influencing each other through the interatomic potential, so <U> is non-zero and plays a major role in the dynamics.

The distinction between liquids and gases can be found most clearly in the specific heat, C_V. In an ideal gas, C_V = (3/2)k_B, whereas in liquid, C_V = 3 k_B [5,6]. This is because in gas, <U> ≅ 0, so that the specific heat originates only from the kinetic energy, <T> = (3/2)k_BT. In contrast, atoms in a liquid are always influencing each other through the interatomic potential. Thus, through the equipartition theorem, <U> = <T>, resulting in C_V = 3k_B, known as Dulong–Petit law for solids [7]. In other words, liquid is condensed matter, not a high-density gas. Indeed, the physical density of a liquid is almost as high as that of a solid [8]. In gas, atom–atom interactions are collisional. In liquid, atoms are held together by cohesive forces, just as in a solid, and atomic motions are vibrational [9].

However, it is quite challenging to formulate theories of such strongly and randomly interacting atoms. In crystals, the lattice periodicity greatly simplifies our understanding, and the theories of condensed matter physics are well established [7]. In contrast, to study liquids, we have to face the dynamic many-body problem directly. For this reason, the progress in the science of liquids has been slow, despite a large amount of effort over many years [2,3,4,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. In the absence of an obvious starting point, many early ideas and theories of liquids were based upon the models of high-density gases, in spite of the important differences between liquids and gases. For instance, the hard-sphere (HS) model [12,13,14] is a gas model with <U> = 0. Widely used concepts, such as free-volume [15,16,17] and jamming [37,38], stem from the HS model, and thus, they assume that a liquid is basically a high-density gas. The Weeks–Chandler–Andersen principle [43] states that at high temperatures, only the repulsive part of the potential is relevant, thus the HS model is always justified. Here, the role of the potential is essentially neglected. However, as we discuss below, the potential energy plays a critical role in controlling the properties of a liquid, supercooled liquid in particular.

Because there are fundamental and qualitative differences between the gas model and the liquid/solid model, we believe that it is better to start with an elastic model in which the harmonic interaction is assumed from the beginning, instead of starting with the gas model and increasing the density to describe a liquid. Such an approach, based on the concept of atomic-level stresses [44] and the density wave instability [45], had major success in describing the evolution of the structure of a liquid with temperature [46] and in elucidating the glass transition [47]. In this paper, we briefly describe this theory and discuss the atomic dynamics in superfluid ⁴He observed by inelastic neutron scattering [48] and the possible implication of this observation on the mechanism of the high-temperature superconductivity of the cuprates [49].

2. Local Structure of Liquid and Glass

In 1952, Frank [50] suggested that icosahedral clusters of atoms (Figure 1) may exist in a liquid to explain the high degree of supercooling observed by Turnbull [51]. He argued that for the Lennard–Jones potential, an icosahedral cluster has an 8.4% lower energy than the close-packed f.c.c. cluster. Icosahedral clusters cannot be the basis of a simple crystalline phase because of the predominant five-fold symmetry, whereas it can exist as a short-range order (SRO) in an aperiodic structure of a liquid. Frank argued that for this reason, crystallization from a liquid requires significant structural reorganization, and this presents the energy barrier for the transformation from a liquid to a crystal, resulting in supercooling [50]. Soon, Bernal discovered a high density of icosahedral clusters in his random-packed HS model [12,13]. Since then, the idea of regarding the icosahedral cluster or similar high-symmetry clusters as the building block to form a glass structure became the standard approach in the effort to elucidate the structure of metallic glass [52,53,54,55,56,57,58].

The structure of liquid and glass is usually expressed in terms of the atomic pair-distribution function (PDF), $g\left(r\right)$, given by

$$g\left(r\right)=\frac{1}{N}\sum _{i,j}\delta \left(r-\left|{\mathit{r}}_{\mathit{i}}-{\mathit{r}}_{\mathit{j}}\right|\right)$$

(1)

where ${\mathit{r}}_{\mathit{i}}$ is the position of the i-th atom, and N is the total number of atoms, or $G\left(r\right)=4\pi r{\rho }_0\left[g\left(r\right)-1\right]$, where ${\rho }_0$ is the atomic number density [2,3,4]. The PDF describes only the two-body correlation, whereas the full description of the real structure requires high-order correlations as well [2,3,4]. However, because the PDF can be directly determined by a diffraction experiment with X-ray or neutrons [59], it is an excellent starting point. Also, even though it is a two-body correlation function, its features reflect higher-order correlations. For instance, its n-th order derivative with r contains information on the correlations among 2 + n particles. In particular, the oscillations in the PDF beyond the first peak describe the medium-range order (MRO) in the coarse-grained density fluctuations, rather than the detailed position of each atom [60], as discussed below.

In 1914, Ornstein and Zernike (OZ) [10] proposed a scheme to connect the SRO and the MRO through a mean-field convolution equation. They divide the PDF into two parts,

$$g\left(\mathit{r}\right)=1+c\left(\mathit{r}\right)+h\left(\mathit{r}\right),$$

(2)

where c(r) describes the short-range correlation, and h(r) is the medium-range correlation. Beyond the first peak, c(r) = 0. The c(r) is directly influenced by the interatomic potential. Then, they propose a mean-field convolution equation,

$$h\left(\mathit{r}\right)=c\left(\mathit{r}\right)+{\rho }_0\int c\left(\left|\mathit{r}-\mathit{r}\prime \right|\right)h\left(\mathit{r}\prime \right)d\mathit{r}\text{'}.$$

(3)

Its Fourier-transform to the structure function

$$S\left(\mathit{Q}\right)=1+c\left(\mathit{Q}\right)+h\left(\mathit{Q}\right),$$

(4)

$$h\left(\mathit{Q}\right)=c\left(\mathit{Q}\right)+{\rho }_{0}c\left(\mathit{Q}\right)h\left(\mathit{Q}\right),h\left(\mathit{Q}\right)=\frac{c\left(\mathit{Q}\right)}{1-{\rho }_{0}c\left(\mathit{Q}\right)}$$

(5)

shows that the MRO is a direct consequence of the SRO. For instance, the first peak of c(Q) directly results in the first peak of h(Q) in the amplified manner. Even though it is a 100 year-old theory, the OZ theory is still widely used. However, as shown below, the SRO and the MRO are fundamentally different, with different origins and behaviors. The OZ theory may work for strongly disordered complex soft-matter [61], but it does not work well for supercooled metallic liquids in which atoms are dynamically correlated.

The S(Q) and g(r) are connected by [59]

$$g\left(r\right)=\frac{1}{2{\pi }^{2}r{\rho }_0}\int \left[S\left(Q\right)-1\right]sin\left(Qr\right)QdQ.$$

(6)

Therefore, a sharp peak in S(Q) results in an oscillating function in g(r), and vice versa. The widely used practice of the one-to-one correspondence between the first peak in S(Q) at Q₁ and the first peak in g(r) at r₁ by $r_1=2\pi /Q_1$ [62] does not work. Actually, as shown by Cargill long ago [63], the first peak of S(Q) describes the MRO as shown in Figure 2 [64]. Similarly, the first peak of g(r) that describes the SRO corresponds to the high-Q part of S(Q).

3. Density Wave Theory of Medium-Range Order

The PDF of metallic liquids and glasses shows prominent exponentially decaying oscillations around the average density with a well-defined periodicity (Figure 3 [65]). The structural coherence length, ${\xi }_s$, defined by the exponential decay of G(r),

$$G\left(r\right)=G_0\left(r\right)exp\left(-r/{\xi }_s\right),$$

(7)

obeys the Curie–Weiss law in its temperature dependence [46,65],

$${\xi }_s\left(T\right)=\frac{C}{T-T_{IG}}.$$

(8)

Equation (8) diverges at T_IG; although, it never happens because T_IG < 0. The state reached by the extrapolation to T_IG is characterized by the long-range density wave (DW) correlation [60,65]. This divergence suggests that there is a driving force toward this state, the DW state. The right theoretical framework to study this state is the density wave theory, in which the atomic density, $\rho \left(\mathit{r}\right)$, is given by

$$\rho \left(\mathit{r}\right)=\int \rho \left(\mathit{q}\right)e^{i\mathit{q}·\mathit{r}}d\mathit{q}.$$

(9)

The Fourier-transform of the interatomic potential, $\varphi \left(\mathit{q}\right)$, is dominated by the diverging part of the interatomic potential $\varphi \left(\mathit{r}\right)$ close to r = 0 and is positive up to large q. However, the diverging part of $\varphi \left(\mathit{r}\right)$ is never relevant because atoms do not come so close to each other. Therefore, we can remove the diverging part to define the pseudopotential as

$$\varphi \left(r\right)={\varphi }_{pp}\left(r\right)+{\varphi }_R\left(r\right)$$

(10)

where ${\varphi }_{pp}\left(\mathit{r}\right)$ is the pseudopotential defined by ${\varphi }_{pp}\left(r\right)=\varphi \left(r_c\right)$ for r < r_c and ${\varphi }_{pp}\left(r\right)=\varphi \left(r\right)$ for r ≥ r_c [45]. The cutoff distance, r_c, is chosen such that the cutoff temperature, $k_B{T}_u=\varphi \left(r_c\right),$ is well above the actual temperature, and no pairs of atoms are found at distances of r < r_c. The potential energy has two parts,

$$U=U_{pp}+U_R,$$

(11)

where

$$U_{pp}=\int \rho \left(\mathit{r}\right){\rho }^{*}\left(\mathit{r}\prime \right){\varphi }_{pp}\left(\left|\mathit{r}-\mathit{r}\prime \right|\right)d\mathit{r}d\mathit{r}\prime ,$$

(12)

$$U_R=\int \rho \left(\mathit{r}\right){\rho }^{*}\left(\mathit{r}\prime \right){\varphi }_R\left(\left|\mathit{r}-\mathit{r}\prime \right|\right)d\mathit{r}d\mathit{r}\prime .$$

(13)

Because no pair of atoms are found within r_c, U_R = 0. Thus, we only need to evaluate U_pp to determine the structure, with the density wave theory in q space.

The Fourier-transform of the pseudopotential, ${\varphi }_{pp}\left(\mathit{q}\right)$, was found to have a strong minimum close to the wavenumber expected for the DW state, ${\mathit{q}}_{\mathit{D}\mathit{W}}$. Thus, we theorize that ${\varphi }_{pp}\left(\mathit{q}\right)$ is driving the system to the DW state, resulting in the increase in ${\xi }_s$ as the temperature is reduced [46,65,66]. However, the SRO of the DW state was found to be quite poor, with almost no icosahedral local structure [65]. Therefore, the DW state, obtained by minimizing the energy in q space, and the SRO, obtained by minimizing the energy in r space, are incompatible with each other. For this reason, the DW state cannot be reached even at T = 0. The value of T_IG in Equation (8) indicates the extent of this conflict. The ratio, $T_{IG}/T_g$, is related to liquid fragility [67]. Liquid fragility [28] is defined by

$$m={\left.\frac{dlog\eta \left(T\right)}{d\left(T_g/T\right)}\right|}_{T_g}$$

(14)

where η(T) is viscosity, and m describes how rapidly the viscosity changes near T_g. Liquids with larger values of m are called “fragile”, whereas those with low m are called “strong”. It reflects the degree of covalency of atomic bonds. Thus, T_IG increases with increased covalency.

In addition, local thermal fluctuations disrupt the long-range nature of the DW state. Such local thermal fluctuations can be described in terms of the atomic-level pressure or volume fluctuations. It can be shown [46] that

$$\frac{a}{{\xi }_s}=\frac{10{\pi }^3}{9}〈{\left({\epsilon }_V^R\right)}^2〉,$$

(15)

where a is the nearest neighbor distance. The local volume fluctuation, $〈{\left({\epsilon }_V^R\right)}^2〉$, has two components,

$$〈{\left({\epsilon }_V^R\right)}^2〉=〈{\left({\epsilon }_V^{R,mf}\right)}^2〉+〈{\left({\epsilon }_V^{R,th}\right)}^2〉,$$

(16)

where $〈{\left({\epsilon }_V^{R,mf}\right)}^2〉$ is the atomic-level misfit strain, and $〈{\left({\epsilon }_V^{R,th}\right)}^2〉$ is the thermal volume fluctuation. Because of the equipartition theorem for the atomic-level pressure fluctuations [68], the thermal volume fluctuation is given by

$$〈{\left({\epsilon }_V^{R,th}\right)}^2〉=\frac{{\left(K_{\alpha }-1\right)}^2}{K_{\alpha }}\frac{k_{B}T}{2BV}$$

(17)

where B is bulk modulus, V is atomic volume, and K_α is the Eshelby parameter [69],

$$K_{\alpha }=\frac{3\left(1-\nu \right)}{2\left(1-2\nu \right)}$$

(18)

and $\nu$ is Poission’s ratio. Therefore,

$$\frac{{\xi }_s}{a}=C\frac{T_g}{T-T_{IG}}$$

(19)

where

$$C=\frac{9}{10{\pi }^3}\frac{K_{\alpha }}{{\left(K_{\alpha }-1\right)}^2}\frac{2BV}{k_B{T}_g}$$

(20)

$$T_{IG}=\frac{K_{\alpha }}{{\left(K_{\alpha }-1\right)}^2}\frac{2BV}{k_B}〈{\left({\epsilon }_V^{R,mf}\right)}^2〉$$

(21)

The MRO, however, freezes at the glass transition. The glass transition temperature is given by

$$T_{IG}=\frac{2BV}{k_B{K}_{\alpha }}〈{\left({\epsilon }_V^{T,crit}\right)}^2〉$$

(22)

where ${\epsilon }_V^{T,crit}=0.095$ is the universal critical strain determined by the percolation of liquid-like atomic sites [47]. This theory describes the results of an experiment and simulation with high quantitative accuracy for metallic liquids and provides a reasonable explanation of other properties [46,66].

4. Dynamic Correlation in Superfluid ⁴He

We now turn our focus to the atomic dynamics in ⁴He, which becomes superfluid due to the Bose–Einstein (BE) condensation. The BE condensation is well-defined for an ideal gas, whereas He atoms interact through the Lennard–Jones potential. As a result, only about 7% of He atoms condense to the BE state [70]. Thus, the PDF of ⁴He determined by neutron scattering which shows the average interatomic distance of 3.6 Å [71] describes essentially the structure of uncondensed atoms. To determine the structure of the BE condensed atoms, we measured the dynamic structure factor, S(q, E), of ⁴He by inelastic neutron scattering and transformed it to the energy-resolved PDF, g(r, E) [48], defined by

$$g\left(r,E\right)=\frac{1}{N}{\displaystyle {\sum }_{i,j}}\int \delta \left(r-\left|{\mathit{r}}_{\mathit{i}}\left(0\right)-{\mathit{r}}_{\mathit{j}}\left(t\right)\right|\right)exp\left(i\omega t\right)dt,$$

(23)

where ${\mathit{r}}_{\mathit{i}}\left(t\right)$ is the position of the i-th atom at time t and $E=\hslash \omega$.

The signature of the B–E condensate is the roton excitation, which has a sharp feature at 0.7 meV. So, if we focus on g(r, E) at 0.7 meV, we are primarily looking at condensed atoms. Particularly, by comparing g(r, E) above and below the BE condensation temperature, T_BE, we can extract the atomic correlations in the condensate. We discovered that the interatomic distance for the BE condensed atoms is 4 Å [48], much longer than that of the uncondensed atoms [71]. To illustrate this difference, we compare the total PDF (in red) against the integral of g(r, E) from E = 0.69 to 0.9 meV (in blue) in Figure 4. It is clear that upon BE condensation, the first peak shifts outward by 0.3 Å or more. We explained this difference in terms of atomic wavefunction overlap [49]. In the condensed state, the wavefunctions of different atoms cannot overlap, because it would lead to level-splitting. To achieve orthogonalization with spatially extended wavefunctions, the ground state has to be a macroscopic resonant state, just as in the resonant valence bond (RVB) state [72]. However, slow dynamics will not allow the system to reach such a massive resonant ground state, just as an antiferromagnet stays in the Néel state even though the true quantum ground state is the resonant singlet state [73]. The simplest alternative is to eliminate the wavefunction overlap by atoms becoming separated a little further. As a result, the interatomic distance is larger than in the uncondensed state where a small overlap is allowed, just as in the exchange hole compared to the correlation hole for electrons [74].

5. Dynamic Correlation among Electrons

Our method to observe dynamic correlations in real space and time using the energy-resolved PDF [48] and the Van Hove function [75,76],

$$G\left(r,t\right)=\frac{1}{N}{\displaystyle {\sum }_{i,j}}\delta \left(r-\left|{\mathit{r}}_{\mathit{i}}\left(0\right)-{\mathit{r}}_{\mathit{j}}\left(\mathit{t}\right)\right|\right),$$

(24)

can be applied to electrons to observe dynamic correlations among electrons. For atoms, the classical interpretation of the correlation functions, Equations (1), (23) and (24), are valid, but for electrons, we have to be more careful about the quantum effect. The inelastic X-ray scattering cross-section of electrons is given by [77],

$$\frac{d\sigma }{d\omega d\Omega }={\sigma }_{0}S\left(q,\omega \right)=\frac{{\sigma }_0}{2\pi }{\displaystyle {\sum }_{\mathit{k}.\mathit{k}\prime }}\int 〈c_{\mathit{k}}^{+}\left(t\right)c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)c_{k\prime }\left(0\right)〉e^{-i\omega t}dt,$$

(25)

where $c_{\mathit{k}}^{+}\left(t\right)$ creates a fermion with the momentum k at time t, ${\sigma }_0=e^2/m$, e is the electron charge, m is the electron mass, and <…> denotes quantum and thermal average. Usually, the decoupling by the random-phase approximation (RPA),

$$〈c_{\mathit{k}}^{+}\left(t\right)c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)c_{{\mathit{k}}^{\prime }}\left(0\right)〉\to 〈c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)〉〈c_{\mathit{k}}^{+}\left(t\right)c_{{\mathit{k}}^{\prime }}\left(0\right)〉,$$

(26)

is applied, and

$$〈c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)〉〈c_{\mathit{k}}^{+}\left(t\right)c_{{\mathit{k}}^{\prime }}\left(0\right)〉={\delta }_{\mathit{k},\mathit{k}\prime }\left(1-n_{\mathit{k}+\mathit{q}}\right)n_{\mathit{k}}{\delta \left(\omega -\left(E_{\mathit{k}+\mathit{q}}-E_{\mathit{k}}\right)/\hslash \right)e}^{i\omega t}$$

(27)

where $n_{\mathit{k}}=〈c_{\mathit{k}}^{+}{c}_{\mathit{k}}〉$. Thus, the dynamic structure factor, S(q, ω), describes the excitation of electrons from below the Fermi level to above it. It is also interpreted as the imaginary part of the dielectric response function [78].

However, the RPA decoupling in reciprocal space erases the phase relationship between the initial and final states in the transition, which contains the information on the real-space correlations. To avoid this loss of phase information and to observe the correlation in real space, we introduce a real-space description,

$$c_{\mathit{k}}^{+}\left(t\right)=\int c^{+}\left({\mathit{r}}^{\prime },t\right)e^{-i\mathit{k}·\mathit{r}\prime }d{\mathit{r}}^{\prime }, c_{\mathit{k}+\mathit{q}}\left(t\right)=\int c\left({\mathit{r}}^{″},t\right)e^{i\left(\mathit{k}+\mathit{q}\right)·\mathit{r}″}d{\mathit{r}}^{″}.$$

(28)

where $c^{+}\left(\mathit{r},t\right)$ is a fermion creator at r and t. Then,

$$\begin{gathered} c_{\mathit{k}}^{+}\left(t\right)c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)c_{{\mathit{k}}^{\prime }}\left(0\right)=\int c^{+}\left({\mathit{r}}^{\prime },t\right)c\left({\mathit{r}}^{″},t\right)c^{+}\left({\mathit{r}}^{‴},0\right)c\left({\mathit{r}}^{\prime \prime \prime \prime },0\right)\times \\ exp\left(i\left[-\mathit{k}{·\mathit{r}}^{\prime }+\left(\mathit{k}+\mathit{q}\right)·{\mathit{r}}^{″}-\left({\mathit{k}}^{\prime }+\mathit{q}\right)·{\mathit{r}}^{‴}+{\mathit{k}}^{\prime }·{\mathit{r}}^{\prime \prime \prime \prime }\right]\right)d{\mathit{r}}^{\prime }d{\mathit{r}}^{″}d{\mathit{r}}^{‴}d{\mathit{r}}^{\prime \prime \prime \prime }. \end{gathered}$$

(29)

The intermediate scattering function [79],

$$F\left(\mathit{q},t\right)=\int S\left(q,\omega \right)e^{i\omega t}d\omega ,$$

(30)

is given by

$$F\left(\mathit{q},t\right)={\displaystyle {\sum }_{\mathit{k}\prime \mathit{k}\prime }}〈c_{\mathit{k}}^{+}\left(t\right)c_{\mathit{k}+\mathit{q}}\left(t\right)c_{{\mathit{k}}^{\prime }+\mathit{q}}^{+}\left(0\right)c_{{\mathit{k}}^{\prime }}\left(0\right)〉.$$

(31)

In the real-space representation, Equation (29), the summation over k and k′ in Equation (31) yields r′ = r′′, r′′′ = r′′′′. Then,

$$\begin{gathered} F\left(\mathit{q},t\right)=\int 〈c^{+}\left({\mathit{r}}^{\prime },t\right)c\left({\mathit{r}}^{\prime },t\right)c^{+}\left({\mathit{r}}^{‴},0\right)c\left({\mathit{r}}^{‴},0\right)〉exp\left(i\mathit{q}·\left({\mathit{r}}^{\prime }-{\mathit{r}}^{‴}\right)\right)d\mathit{r}\prime d\mathit{r}‴ \\ =\int 〈\left({\mathit{r}}^{\prime },t\right)\left({\mathit{r}}^{‴},0\right)〉exp\left(i\mathit{q}·\left({\mathit{r}}^{\prime }-{\mathit{r}}^{‴}\right)\right)d\mathit{r}\prime d\mathit{r}‴, \end{gathered}$$

(32)

where

$$\rho \left(\mathit{r},t\right)=c^{+}\left(\mathit{r},t\right)c\left(\mathit{r},t\right)$$

(33)

is the local density operator. Thus, the Van Hove function is expressed as the density correlation function in real space and time,

$$G\left(\mathit{r},t\right)=\int F\left(\mathit{q},t\right)e^{i\mathit{q}·\mathit{r}}d\mathit{q}=\int 〈\rho \left({\mathit{r}+\mathit{r}}^{\text{'}},t\right)\rho \left({\mathit{r}}^{\prime },0\right)〉d{\mathit{r}}^{\prime }.$$

(34)

This proves that the classical interpretation of the double-Fourier-transformation of S(q, E) as the Van Hove correlation function is valid, even for electrons. The energy-resolved PDF is given by

$$g\left(r,\omega \right)=\frac{1}{2\pi }\int G\left(r,t\right)e^{-i\omega t}dt.$$

(35)

We measured the electronic S(q, E) for polycrystalline beryllium by inelastic X-ray scattering with the incident energy of 11.32 keV and energy resolution of 0.7 eV at the Beamline 27ID-B of the Advanced Photon Source of Argonne National Laboratory. The preliminary result [80] shows the exchange-correlation hole with the radius of about 2 Å. Surprisingly, at the plasmon energy of about 20 eV, the exchange-correlation hole is extended to about 5 Å. In a plasmon, electrons are dynamically correlated, resulting in larger separation among electrons than in electrons at large. This result demonstrates that the dynamic state of electrons affects electron correlations. The details of this observation will be reported elsewhere.

6. Implication on High-Temperature Superconductivity

The combination of the result on ⁴He and the observation of electron correlations by inelastic X-ray scattering have very significant implications on the high-temperature superconductivity of the cuprates. In the regular BCS superconductors, electrons are in the Fermi-liquid state, and they are gas-like as discussed above. Electrons avoid each other by forming exchange-correlation holes, and the density functional theory (DFT) [81] works. The Cooper pairs are large, up to 1000 Å, and they strongly overlap with each other. If they are stable up to high temperatures, they would B-E condense at temperatures of the order of 1000 K. Thus, the pairing temperature determines the superconducting critical temperature, T_c.

In contrast, in the cuprates, electrons are strongly correlated, just as in supercooled liquids in which the potential energy is as important as the kinetic energy. For this reason, it is interesting to note that it was recently found that in the overdoped cuprates, the periodicity of the charge density wave was independent of the Fermi momentum as the doping density was changed [82]. It is likely that the Coulomb repulsion energy is involved in the formation of the density waves, for instance, through the pseudopotential discussed above [45,66].

Furthermore, in the cuprates, the Cooper pairs are small, of the order of nm in size, and low in density. Thus, they are likely to be preformed above the superconducting transition temperature [83,84,85]. If that is the case, T_c is equal to T_BE, and it is controlled by the BE condensation rather than by pairing. We speculate that as in the case of ⁴He the separation between Cooper pairs is longer in the BE condensate than in the normal state to avoid wavefunction overlap. Then, the Coulomb repulsion energy is reduced in the BE condensate compared to the normal state, adding to the driving force for the BE condensation [49]. The difference is similar to the case of exchange and correlation effects for fermions, so the energy difference must be related to the exchange constant.

This effect is most likely negligible for ⁴He, because the interaction energy (van der Waals force) is small. However, for the cuprates, a small change in the repulsion energy can have a very significant effect. Because the hole density is low, the energy associated with the wavefunction overlap is a small percentage of U. Nevertheless, the on-site Coulomb repulsion, U, is nearly 10 eV for Cu [74], which is huge in contrast to k_BT_c~10 meV. Although the value of U itself may not be influenced by the BE condensation, slight changes in the extent of the p-d hybridization due to the BE condensation can affect the total repulsion energy. Thus, a tiny change in the repulsion energy can have a major effect on T_c. It is interesting to note that the pseudogap temperature, T*, extrapolates to zero just beyond the optimum doping [86]. If T* is the pair formation temperature, T_c should go down as well in the conventional pairing picture. It is possible that the repulsion energy reduction mechanism discussed here is maintaining the high value of T_c beyond the optimum doping. So far, the effort to increase T_c has focused on the pairing force. This new mechanism calls for a change in strategy, to increase T_c through the correlation effects.

To test this hypothesis, we plan to measure the g(r, E) for doped holes in the superconducting cuprates with inelastic X-ray scattering. The measurement is quite difficult because of the low intensity of the signal due to the high-energy resolution (better than 10 meV) required to observe the energy gap and due to the low doping density. However, this is within the realm of feasibility with advanced facilities.

7. Conclusions

Gases and liquids are fundamentally different in the role of the interatomic potentials. In gases, atomic interactions are collisional, and the average potential energy is negligibly small compared to the kinetic energy. In liquids, the potential energy is as important as the kinetic energy and results in complex atomic dynamics. An important consequence of the interatomic potential is the formation of the atomic medium-range order (MRO) which affects many properties. We propose a density wave theory to describe the behavior of the MRO. We suggest that this difference between gases and liquids can be instructive in discussing the difference in dynamics between nearly free electrons and strongly correlated electrons. We point out that electron correlation can be influenced by the dynamic state of electrons. In particular, the Bose–Einstein condensation of Cooper pairs could affect electron correlation energy, giving rise to an additional driving force to increase the superconducting transition temperature.