The success of polynomial-time tensor network methods for computing ground states of certain quantum local Hamiltonians has recently been given a sound theoretical basis by Arad et al. [Math. Phys. 356, 65 (2017)]. The convergence proof, however, relies on ``rigorous renormalization group'' (RRG) techniques which differ fundamentally from existing algorithms. We introduce a practical adaptation of the RRG procedure which, while no longer theoretically guaranteed to converge, finds matrix product state ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians in realistic situations. In contrast to other schemes, RRG does not utilize variational methods on tensor networks. Rather, it operates on subsets of the system Hilbert space by constructing approximations to the global ground space in a treelike manner. We evaluate the algorithm numerically, finding similar performance to density matrix renormalization group (DMRG) in the case of a gapped nondegenerate Hamiltonian. Even in challenging situations of criticality, large ground-state degeneracy, or long-range entanglement, RRG remains able to identify candidate states having large overlap with ground and low-energy eigenstates, outperforming DMRG in some cases.
We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges
Q and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter
g. When the parameter
g vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-
g limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when
g>0. We also rigorously prove that for sufficiently large
g the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.
We have applied a numerical renormalization-group technique to study the Hubbard-Peierls model for (CH
)x. The method provides accurate many-body wave functions and energies for the ground and low-lying excited states of finite polyenes. We report tests of the method for finite chains (N=16) which show that the technique accurately reproduces known results in the noninteracting particle limit (U=0) and in the highly correlated limit (U/4t\ensuremath{\gg}1) and agrees well with Monte Carlo data where available for intermediate U. The model is applied to study correlation effects on the ground-state dimerization amplitude and to the ordering of the low-lying excited states of the rigid-lattice spectrum. Extension of the method to even longer systems appears to be quite practical.