Solving Condensed-Matter Ground-State Problems by Semidefinite Relaxations

We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green’s functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for the study of strongly correlated fermionic and frustrated spin systems in $D>1$ spatial dimensions. It is demonstrated for the $XXZ$ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo calculations, and Anderson bounds, showing the competitiveness of the SDP method.

Figure 1

Lower bounds $E$ to the ground-state energy and approximations to correlators for the $XXZ$ chain. The Anderson bound (2) was calculated with clusters of 25 sites. The subsystems ${\Omega }_k$ for the SDP method (12) are chosen to be clusters of contiguous sites with sizes $L_1$, $L_2$, $L_3$, and $L_4$ for constraint operators of polynomial degree 1, 2, 3, and 4, respectively, as specified in the legend ($K=4$). The Bethe ansatz yields the exact ground-state energy $E^0$ and short-range correlators [38, 39].

Figure 2

Lower energy bounds and correlators for the 2D $XXZ$ model, complemented by QMC data calculated for a square lattice of $16\times 16$ sites with periodic boundary conditions and inverse temperature $\beta =96$. The QMC energies for $J_z=0,1$ coincide with earlier results (◆) from Refs. 43, 44. As SDP constraint subsystems ${\Omega }_k$ [Eq. (12)], we chose $L_k\times L_k$ squares with $K=4$ and $L_1$, $L_2$, $L_3$, and $L_4$, as specified in the legend.

Figure 3

Lower energy bounds for the 2D $t-V$ Hubbard model of spinless fermions on a square lattice, compared to exact energies for $4\times 4$ and $6\times 6$ lattices with periodic boundary conditions (PBC) as well as the thermodynamic limit at $V=0$.