Classifying quantum phases using matrix product states and projected entangled pair states

We give a classification of gapped quantum phases of one-dimensional systems in the framework of matrix product states (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states and in both the absence and the presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground-state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, that is, the equivalence classes of its projective representations, a result first derived by Chen, Gu, and Wen [Phys. Rev. B 83, 035107 (2011)]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labeled by both the permutation action of the former and the cohomology class of the latter. Using projected entangled pair states (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular, systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS, which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus we can focus on to the classification of isometric forms.

Figure 1

MPS are constructed by applying a linear map $\mathcal{P}$ to maximally entangled pairs $|{\omega }_D\rangle :={\sum }_{i=1}^D|i,i\rangle$ of bond dimension $D$.

Figure 2

Construction of the interpolating path for MPS and parent Hamiltonians. Instead of interpolating between the MPS $|{\psi }_0\rangle$ and $|{\psi }_1\rangle$ directly (dotted line), we first show how to interpolate each of the two states toward a standard from $|{\widehat{\psi }}_p\rangle$, the isometric form, and then construct an interpolating path between the isometric forms. Note that using the parent Hamiltonian formalism, any such path in the space of MPS yields a path in the space of Hamiltonians right away.

Figure 3

Isometric form of an MPS. (a) The MPS projector $\mathcal{P}$ can be decomposed into a positive map $Q$ and an isometric map $W$. (b) By removing $Q$, one obtains the isometric form $W$ of the MPS. (c) Interpolation to the isometric form is possible by letting $Q_{\gamma }=\gamma Q+(1-\gamma )\mathbb{1}$.

Figure 4

The isometric form of an injective MPS consists of maximally entangled states $|{\omega }_D\rangle$ with bond dimension $D$ between adjacent sites, where each site is a $D^2$-level system. The local Hamiltonian terms $h^D$ [Eq. (4)] acts only on the degrees of freedom corresponding to the central entangled pair.

Figure 5

Structure of the isometric form for noninjective MPS. The state consists of a GHZ state ${\sum }_{\alpha }|\alpha ,\cdots ,\alpha \rangle$ (gray) and maximally entangled pairs between adjacent sites $|{\omega }_{D_{\alpha }}\rangle$, where the bond dimension $D_{\alpha }$ can couple to the value $\alpha$ of the GHZ state.

Figure 6

PEPS are constructed analogously to MPS by applying linear maps $\mathcal{P}$ to a 2D grid of maximally entangled states $|{\omega }_D\rangle$.