Matrix product states represent ground states faithfully

We quantify how well matrix product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.

Figure 1

(Color online) Convex sets of the possible reduced density operators of translational invariant states in the $XX$-$ZZ$ plane: the big triangle represents all positive density operators, the inner parallellogram represents the separable states, the union of the separable cone and the convex hull of the full curved line is the complete convex set in the case of a 1D geometry, and the dashed lines represent extreme points in the 2D case of a square lattice. The singlet corresponds to the point with coordinates $(-1,-1)$.

Figure 2

(Color online) A one-dimensional spin chain with finite correlation length ${\xi }_{\mathrm{corr}}$; $l_{AB}$ denotes the distance between the block $A$ (left) and $B$ (right). Because $l_{AB}$ is much larger than the correlation length ${\xi }_{\mathrm{corr}}$, the state ${\rho }_{AB}$ is essentially a product state.

Figure 3

(Color online) Convex sets in the $XXZ$ plane: the inner diamond borders the set of separable states (see Fig. 1). Dash-dotted: extreme points of the convex set produced by MPS of $D=2$.