Abstract. We address the question of the classification of gapped ground states in one dimension that cannot be distinguished by a local order parameter. We introduce a family of quantum spin systems on the one-dimensional chain that have a unique gapped ground state in the thermodynamic limit that is a simple product state but which on the left and right half-infinite chains, have additional zero energy edge states. The models, which we call Product Vacua with Boundary States (PVBS), form phases that depend only on two integers corresponding to the number of edge states at each boundary. They can serve as representatives of equivalence classes of such gapped ground states phases and we show how the AKLT model and its SO(2J + 1)-invariant generalizations fit into this classification.
Abstract. We address the question of the classification of gapped ground states in one dimension that cannot be distinguished by a local order parameter. We introduce a family of quantum spin systems on the one-dimensional chain that have a unique gapped ground state in the thermodynamic limit that is a simple product state but which on the left and right half-infinite chains, have additional zero energy edge states. The models, which we call Product Vacua with Boundary States (PVBS), form phases that depend only on two integers corresponding to the number of edge states at each boundary. They can serve as representatives of equivalence classes of such gapped ground states phases and we show how the AKLT model and its SO(2J + 1)-invariant generalizations fit into this classification.
We introduce a family of quantum spin chains with nearest-neighbor interactions that can serve to clarify and refine the classification of gapped quantum phases of such systems. The gapped ground states of these models can be described as a product vacuum with a finite number of particles bound to the edges. The numbers of particles, nL and nR, that can bind to the left and right edges of the finite chains serve as indices of the particular phase a model belongs to. All these ground states, which we call product vacua with boundary states (PVBS), can be described as matrix product states (MPS). We present a curve of gapped Hamiltonians connecting the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to its representative PVBS model, which has indices nL=nR=1. We also present examples with nL=nR=J, for any integer $J\ensuremath{\ge}1$, that are related to a recently introduced class of SO(2J+1)-invariant quantum spin chains.
Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature [Formula: see text] particle species on [Formula: see text]. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary species number [Formula: see text] and in arbitrary dimension [Formula: see text] in the perturbative regime of small anisotropy parameters. Instead of using the more common martingale method, the proof verifies a finite-size criterion in the spirit of Knabe.
Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for strongly interacting systems. Here we make an attempt to answer these questions for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range-correlated states in the same phase, we classify possible quantum phases for one-dimensional (1D) matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if a certain symmetry is required, many phases exist with different symmetry-protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.
Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for strongly interacting systems. Here we make an attempt to answer these questions for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range-correlated states in the same phase, we classify possible quantum phases for one-dimensional (1D) matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if a certain symmetry is required, many phases exist with different symmetry-protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.
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.
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages