Combining symmetry collective states with coupled-cluster theory: Lessons from the Agassi model Hamiltonian

The failures of single-reference coupled-cluster theory for strongly correlated many-body systems is flagged at the mean-field level by the spontaneous breaking of one or more physical symmetries of the Hamiltonian. Restoring the symmetry of the mean-field determinant by projection reveals that coupled-cluster theory fails because it factorizes high-order excitation amplitudes incorrectly. However, symmetry-projected mean-field wave functions do not account sufficiently for dynamic (or weak) correlation. Here we pursue a merger of symmetry projection and coupled-cluster theory, following previous work along these lines that utilized the simple Lipkin model system as a test bed [J. Chem. Phys. 146, 054110 (2017)]. We generalize the concept of a symmetry-projected mean-field wave function to the concept of a symmetry projected state, in which the factorization of high-order excitation amplitudes in terms of low-order ones is guided by symmetry projection and is not exponential, and combine them with coupled-cluster theory in order to model the ground state of the Agassi Hamiltonian. This model has two separate channels of correlation and two separate physical symmetries which are broken under strong correlation. We show how the combination of symmetry collective states and coupled-cluster theory is effective in obtaining correlation energies and order parameters of the Agassi model throughout its phase diagram.

Figure 1

A representative state of the Agassi model with $j=4$. Starting from the noninteracting ground state, in which all lower levels and no upper levels are occupied [see Eq. (24)], this state is reached by the action of two $J_{+}$ excitation operators at $m=-4$ and $m=1$, one $A_{-1}$ annihilation operator at $m=2,-2$, and one $A_{+1}^{†}$ creation operator at $m=3,-3$.

Figure 2

The mean-field phase diagram of the Agassi model. The two parameters $q_{\mathrm{L}}$ and $q_{\mathrm{P}}$ define the general mean-field wave function of the Agassi model [Eqs. (6) and (23)]. Along the ray where $\chi ={\mathrm{\Sigma }}_0>1$, both $q_{\mathrm{L}}$ and $q_{\mathrm{P}}$ are nonzero.

Figure 3

Ground-state energy errors of the Agassi model with $j=20$ calculated by RCCD [Eqs. (1) and (27)], RHF [Eq. (24)], UHF [Eqs. (6) and (23)], and PHF [Eqs. (12) and (28)]. The inset shows the lines through the phase diagram along which the energies were calculated, and $x$ is the larger of $\chi$ or ${\mathrm{\Sigma }}_0$. No real solutions to the RCCD equations were found for $x>1.1$.

Figure 4

The fractional correlation energy ($E_c$) error from FCI for an Agassi model with 40 particles, calculated variationally with the RCCD ($|T_2\rangle$), PHF ($|Q_1\rangle$), and $|Q_1{Q}_2\rangle$ wave functions. (a) RCCD ($|T_2\rangle$) wave function. (b) PHF ($|Q_1\rangle$) wave function. The color axis is slightly shifted and an ellipse inscribed to guide the eye towards the region of the phase diagram where $|Q_1\rangle$ results differ from $|Q_1{Q}_2\rangle$ results. (c) $|Q_1{Q}_2\rangle$ wave function. The color axis is shifted and an ellipse inscribed as in (b).

Figure 5

The fractional correlation energy ($E_c$) error from FCI for an Agassi model with 40 particles, calculated variationally with the PRCC ($|T_2{Q}_1\rangle$) and PQCC ($|T_2{Q}_1{Q}_2\rangle$) wave functions. (a) PRCC ($|T_2{Q}_1\rangle$) wave function. (b) PQCC ($|T_2{Q}_1{Q}_2\rangle$) wave function.

Figure 6

Three order parameters for the 40-particle Agassi model calculated from the FCI wave function. (a) Order parameter $n$ [Eq. (37)]. (b) Order parameter $J$ [Eq. (40)]. (c) Order parameter $\mathrm{\Delta }$ [Eq. (41)].

Figure 7

Fractional errors from the exact result of order parameters for the 40-particle Agassi model obtained from various wave functions, plotted along two lines through the phase diagram, ${\mathrm{\Sigma }}_0=x$ and $\chi =a-x$ with various $a$. The lines are displayed in insets: (a) $a=1.5$ and (b) $a=3.$

Figure 8

The absolute value of the energy error from FCI for an Agassi model with 40 particles, calculated with the PRCC ($|T_2{Q}_1\rangle$) and PQCC ($|T_2{Q}_1{Q}_2\rangle$) methods using a similarity-transformation formalism. (a) PRCC ($|T_2{Q}_1\rangle$) method. (b) PQCC ($|T_2{Q}_1{Q}_2\rangle$) method.