Another possible phase in liquid hydrogen

The Resonant Valence Bond state has been shown to describe accurately frustrated spin systems. It appears that in many systems superconductivity is realized when the system in the normal state is quite close to metal-insulator transition even in presence of strong disorder, see Ref. (100). The large mass difference between ions and electrons leads, in a good approximation, to two distinct dynamics. This allows us to consider always the electrons in the ground state of a disordered ionic system at finite temperature, and from this point of view it is "similar" to an average on disorder. Following this hypotheses we tried to study the possibility of stable superconducting phases driven by correlation close to the metal-insulator transition and solid-liquid transition. Moreover as in high-Tc superconductor, we expect that due to the small size of the Cooper pair a superconductive state can be more stable close to a disordered phase (101).
In order to detect superconductivity we have calculated the condensation energy on different configurations in the liquid phase, with VMC and DMC. The condensation energy, in a variational scheme, is defined as the energy difference obtained between the best Slater determinant, the normal state, and the AGP. In order to estimate the condensation energy we have reduced the rank of the pairing matrix to $N/2$ . In such limit the pairing determinant is equivalent to a Slater one, as shown in (2). We optimized the Slater wave-function on given configurations and we found a non-zero condensation energy (see figure 6.5), namely a gain in energy given by allowing pairing within a variational ansatz. At present we do not even know whether this gain in energy is macroscopic or is just a finite size effect even a the VMC level. The corresponding energy gain, obtained in this way, certainly overestimate a possible true condensation energy due to a real superconducting ground state for high pressure hydrogen. In fact the variational approach is certainly biased towards superconducting phases even though recently in the 2D Hubbard model there are other evidences of superconducting phase from non variational approaches (102).
As shown in the famous paper of Yang (103) a peculiar property of superconductors is the Off-Diagonal Long-Range Order (ODLRO) exhibited by the reduced density matrices in the coordinate space representation (103). For Fermions ODLRO corresponds to the appearance of an eigenvalue which scales with the number of particles in the two-body density matrix (103):

$$\rho_2(x_1,x_2;x'_1,x'_2) = \langle \Psi_N \vert a^+(x_1) a^+ (x_2) a(x_1') a(x_2') \vert \Psi_N \rangle$$ (6.2)

$$Tr \rho_2 = \frac{1}{2} \int \rho_2 (x_1,x_2;x_1,x_2) dx_1 dx_2 = \frac{N(N-1)}{2},$$ (6.3)

where $N$ is the number of particles. In a translational invariant system, ODLRO implies for the two-body density matrix the following asymptotic behavior:

$$\rho_2(x'_1,x'_2;x_1,x_2) = \alpha f^*(\vec x'_1-\vec x'_2)f( \vec x_1-\vec x_2)$$ (6.4)

$$\left\vert x_1-x_2 \right\vert,\left\vert x'_1-x'_2 \right\vert \leq \xi \left\vert x_1-x'_1 \right\vert \rightarrow \infty$$ (6.5)

where $\alpha/N$ is the pair condensate fraction and $\xi$ is the size of the pair defined by the pairing function $f$ . The function $f$ is zero for large separation $\left\vert x_1-x_2 \right\vert$ and is $\simeq 1/V^{1/2}$ for microscopic separation for $x_1$ and $x_2$ . In order to estimate the condensate fraction, following De Palo et al. (104), we resort the two-body density matrix to the projected density matrix:

$$h(x,\theta,\phi) = \frac{1}{N} \int{dx_1 dx_2 \rho_2(x'_1 + x, x'_2 + x; x_1, x_2)}$$ (6.6)

which tends to $\alpha$ in the large x limit. The presence of non positive eigenvalues in the $\lambda$ matrix, see figure 6.6, led us to investigate the presence of non s-wave superconductivity. Therefore we also evaluated $h(x,\theta,\phi)$ as a function of rotations angle $\theta,\phi$ of the electron pair in order to investigate the possibility of different symmetries.

$$\vec x_1' - \vec x_2' = U_{\theta,\phi} (\vec x_1 - \vec x_2) \frac{\vec x_1' +\vec x_2'}{2} = \frac{\vec x_1 + \vec x_2}{2} + \vec x$$ (6.7)

A simple estimator of $h(x,\theta,\phi)$ is given by

$$h(x,\theta,\phi)= \frac{1}{M_c} \sum_{i<j \{ r_{ij}<\xi \}} \frac{\Psi_t(r_1,r_2,...,r'_i,...,r'_j)}{\Psi(r_1,...,r_n)}$$ (6.8)

where $r_i',r_j'$ is an electron pair translated of $x$ and rotated of $\theta,\phi$ , and $M_c$ the number of pairs translated. A cutoff $\xi$ is introduced to speed up the calculation excluding contributions coming by pairs of far electrons that do not contribute to the ODLRO because the pairing is short range. We have verified that the cutoff used does not effect the final result. In practice, for each pair we generate a few translations $x$ uniformly distributed in the simulation box and, for better statistics, we also average over all pairs with the condition $r_{ij} \leq \xi$ .
We argue the possibility of a non s-wave symmetry in the ODLRO, see figure 6.7 and 6.8. Unfortunately the size of the studied system is too small to give a conclusive answer.

Figure 6.5: Variational and Diffusion Condensation Energy per atom

Figure 6.6: Eigenvalues of the $\lambda$ matrix for 16 hydrogen atoms at Rs=1.31 and 100K as function of the simulation time

Figure 6.7: Off-Diagonal Long Range Order for 16 hydrogen atoms at Rs=1.31 and 100K, in a box of size $L=5.3211$ , for $\theta =0$ as function of the distance $x$

Figure 6.8: Off-Diagonal Long Range Order for 54 hydrogen atoms at Rs=1.31 at 100K in a box of size $L=7.9817$ , for $\theta =0$ as function of the distance $x$