Self-Consistent Random Phase Approximation from Projective Truncation Approximation Formalism

TL;DR

This work develops a finite-temperature, self-consistent extension of RPA by deriving sc-RPA from PTA for the equation of motion of two-time Green's functions, thereby unifying dynamic and static correlations within a single framework. By employing natural orbitals and enforcing $N$-representability constraints, the method achieves self-consistent one- and two-particle densities and provides a calculational route that connects to Rowe's zero-temperature RPA. The authors demonstrate the approach on a one-dimensional spinless fermion model, showing good agreement with exact results and capturing Luttinger-liquid features, spectral continua, and bound states. The PTA-based formalism offers a flexible platform to extend RPA to higher-order correlations and to bridge between model Hamiltonians and first-principles calculations, with potential applications to finite-temperature and strongly correlated systems.

Abstract

We derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalize Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the N-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.

Figures (5)

• Figure 1: (color online) Average energy per site as a function of $V$, obtained from sc-RPA (circles) and ED (solid line). Inset: the relative error of sc-RPA result with respect to ED, as functions of $|V|$ in the positive $V$ (circles) and negative $V$ (squares) regimes. The green dashed line is the eye-guiding line for the $V^{2}$ behaviour. Parameters are $L=12$, $N=6$, $t=1.0$, $T=10^{-4}$, $\Delta = 10^{-5}$, and the convergence precision of sc-RPA $10^{-7}$.
• Figure 2: (color online) Fermion occupation $\langle n_k \rangle$ of the momenta $k= 2\pi m_k /L + \Delta$ for $m_k = 0$, $1$, and $2$ (from top to bottom). They are obtained from sc-RPA (symbols) and ED (solid lines), respectively. Parameters are $L=12$, $N=6$, $t=1.0$, $T=0.01$, $\Delta = 0.2$, and the convergence precision of sc-RPA $10^{-7}$.
• Figure 3: (color online) Absolute value of density-density correlation function $|C_{1j}| = |\langle n_1 n_j \rangle - \langle n_1 \rangle \langle n_j \rangle|$ as functions of $j-1$. (a) $V=1.0$, (b) $V=1.5$, and (c) $V=-1.0$. They are obtained from sc-RPA (black squares) and bosonization Lukyanov1 (red circles). The lines are for guiding eyes. The results of sc-RPA are obtained at parameters $L=192$, $N=96$, $t=1.0$, $T=10^{-3}$, $\Delta=10^{-5}$, while those of bosonization are at $L=2N=\infty$, $t=1.0$, $T=0$, and $\Delta=0$. The long-distance asymptotic power from Eqs.(\ref{['Eq7-1']}) and (\ref{['Eq7-2']}) are marked in the figure.
• Figure 4: (color online) The color map of the spectral function $\rho(q, \omega)$ for $V>0$. They are obtained at $L=192$, $N=96$, $t=1.0$, $T=0.01$, $\Delta=10^{-5}$, with broadening parameter $\eta=0.05$. The red dashed lines are the boundaries of the particle-hole excitation continuum obtained by Bethe ansatz Caux1Caux2.
• Figure 5: (color online) The color map of the spectral function $\rho(q, \omega)$ for $V<0$. Parameters are same as those of Fig.4. The red dashed lines are the boundaries of the particle-hole excitation continuum obtained by Bethe ansatz Caux1Caux2. The black, green, and pink dot-dashed lines are the bound state dispersions from Eq.(\ref{['Eq7-5']})-(\ref{['Eq7-6']}).