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

Abstract

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 both in the absence and 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, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. 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 labelled 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 twodimensional 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.

Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

“…Similar terminology is used for tensor product states discussed later. The local action of time-reversal symmetry on matrix product states has been discussed extensively in the study of 1D symmetry-protected topological phases [17][18][19][20]. Here, we review the procedure and discuss the notion of time-reversal flux and time-reversal twists based on such a formalism.…”

“…The result remains the same if we insert flux by changing the matrices on site N as A i → A i M −1 . This corresponds exactly to the procedure of extracting SPT order from the MPS representation of a gapped symmetric state [17][18][19][20]. Here, we are merely reinterpreting the procedure as finding the projective composition rule of time-reversal twists induced by inserted timereversal fluxes.…”

“…That is, two time-reversal twists compose into identity up to a universal phase factor characterizing the underlying SPT order of the state. In 1D, the distinction between different time-reversal SPT phases has been well understood in the matrix product formalism [ [17][18][19][20]. Our discussion is just a reinterpretation of that procedure in terms of local time-reversal symmetry action and time-reversal twists.…”

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

“…This realization led to a complete classification of all (1+1)D gapped bosonic quantum phases [42][43][44].…”

“…Some examples of two-dimensional (2D) SPT phases protected by translation and some other symmetries were discussed in Refs. [41][42][43].…”