Exact renormalization in quantum spin chains

We introduce a real-space exact renormalization-group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with $S\le 4$ and nearest-neighbor interactions, as well as examples with $S=5$. We obtain two classes of solutions. One of them converges to the fixed points of renormalization group and the ground states are matrix-product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.

Figure 1

Real-space renormalization. (a) The real-space renormalization group yields matrix-product states. (b) In periodic boundary condition, the translationally invariant MPS are constructed from the fixed-point Kraus operators.

Figure 2

(Color online) Real-space renormalization with SU(2). (a) An isometry $A^{\left[S,M\right]}$ is decomposed as a matrix $T$ and a fusion of angular momentums. (b) The renormalization of two spins can be done in two successive steps. (c) The same input and output states with different intermediate fusion channels for two spins can be related by the $F$ symbol.

Figure 3

Injectivity. A MPS is injective with injectivity length $L_0$ if for the MPS constructed for $L_0$ sites, different boundary conditions (linearly independent), represented by the cones in the figure, give rise to different states (linearly independent), and this does not happen for $L_0-1$ spins.

Figure 4

Symmetry. The unitary $u$ applied on the physical level is reflected in the virtual level as pair of unitaries $U$, as shown in Ref. 21.

Figure 5

(Color online) Ground-state physical picture. (a) The fixed-point-type MPS solutions have a VBS picture. Each dot denotes a virtual spin representation. The wavy lines represent the valence-bond singlets between virtual spins and the circles indicate the projection of two virtual spins onto physical spin representations. In the exact renormalization calculations, the fixed-point representations are from the tensor product of two virtual spins (edge states). (b) The partially ferromagnetic states have a magnetization plateau. The arrows denote a fully polarized virtual spin $\left(S-1\right)$ in a spin $S$ partially ferromagnetic state.