Matrix Product States (MPS) and Tensor Networks provide a general framework for the construction of solvable models. The best-known example is the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, which is the ground state of a 2-body nearest-neighbor parent Hamiltonian. We show that such simple parent Hamiltonians for MPS models are, in fact, much more prevalent than hitherto known: The existence of a single example with a simple Hamiltonian for a given choice of dimensions already implies that any generic MPS with those dimensions possesses an equally simple Hamiltonian. We illustrate our finding by discussing a number of models with nearest-neighbor parent Hamiltonians, which generalize the AKLT model on various levels.
Matrix Product States (MPS) and Tensor Networks provide a general framework for the construction of solvable models. The best-known example is the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, which is the ground state of a 2-body nearest-neighbor parent Hamiltonian. We show that such simple parent Hamiltonians for MPS models are, in fact, much more prevalent than hitherto known: The existence of a single example with a simple Hamiltonian for a given choice of dimensions already implies that any generic MPS with those dimensions possesses an equally simple Hamiltonian. We illustrate our finding by discussing a number of models with nearest-neighbor parent Hamiltonians, which generalize the AKLT model on various levels.
The authors develop a general formalism to systematically construct parent Hamiltonians with quasiparticle excited states starting from a matrix product state (MPS) wave function. The resulting parent Hamiltonians are generically nonintegrable, and the quasiparticle excited states are thus examples of quantum many-body scars of the Hamiltonian. This method is used to recover well-known quantum scarred models, such as the AKLT spin chain, as well as to construct many new families of models, thus illustrating the close connection between parent Hamiltonians of MPS and quantum scarred models.
There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.
We propose and study a specific adiabatic path to prepare those tensor network states that are unique ground states of few-body parent Hamiltonians in finite lattices, which include normal tensor network states, as well as other relevant nonnormal states. This path guarantees a gap for finite systems and allows for efficient numerical simulation. In one dimension, we numerically investigate the preparation of a family of states with varying correlation lengths and the one-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) state and show that adiabatic preparation can be much faster than standard methods based on sequential preparation. We also apply the method to the two-dimensional AKLT state on the hexagonal lattice, for which no method based on sequential preparation is known, and show that it can be prepared very efficiently for relatively large lattices.
Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color SU(N) symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of SU(N). In particular, SU(2) spin-1 AKLT states with edge states of arbitrary spin s=1/2,1,3/2,⋯ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are unique ground states. This enables us to study the stability of the edge states by interpolating between exact AKLT Hamiltonians. As an example, in the case of spin-1 physical degrees of freedom, it is shown that a quantum phase transition of central charge c=1 separates the Symmetry Protected Topological (SPT) phase with spin-1/2 edge states from a topologically trivial phase with spin-1 edge states. We also address some specificities of the generalization to SU(N) with N>2, in particular regarding the construction of parent Hamiltonians. For the AKLT state of the SU(3) model with the 3-box symmetric representation, we prove that the edge states are in the 8-dimensional adjoint irrep, and for the SU(3) model with adjoint irrep at each site, we are able to construct two different reflection-symmetric AKLT Hamiltonians, each with a unique ground state which is either even or odd under reflection symmetry and with edge states in the adjoint irrep. Finally, examples of two-column and adjoint physical irreps for SU(N) with N even and with edge states living in the antisymmetric irrep with N/2 boxes are given, with a conjecture about the general formula for their correlation lengths.
Density matrix renormalization group (DMRG) or matrix product states (MPS) is the most effective and accurate method for studying one-dimensional quantum many-body systems. However, the application of DMRG to two-dimensional systems is not as successful because of the limited entanglement encoded in the wave-function ansatz. The fully augmented matrix product states (FAMPS), introduced recently in [Chin. Phys. Lett. 40, 057102 (2023)], extends MPS formalism to two dimensions and increases the entanglement in the wave-function ansatz, representing a significant advance in the simulation of two-dimensional quantum many-body physics. In the study of one-dimensional systems, the concept of a parent Hamiltonian for MPS has proven pivotal in the understanding of quantum entanglement. In this work, we extend this framework to two-dimensional systems. We illustrate the procedure to construct a two-dimensional Hamiltonian with given FAMPS as its exact ground state (the parent Hamiltonian for FAMPS). Additionally, through numerical simulations, we demonstrate the effectiveness of the algorithm outlined in [Chin. Phys. Lett. 40, 057102 (2023)] in precisely identifying the FAMPS for the constructed parent Hamiltonian. The introduction of FAMPS and its associated parent Hamiltonian provides a useful framework for the future investigations of two-dimensional quantum many-body systems.
Tensor network states form a powerful framework for both the analytical and numerical study of strongly correlated phases. Vital to their analytical utility is that they appear as the exact ground states of associated parent Hamiltonians, where canonical proof techniques guarantee a controlled ground space structure. Yet, while those Hamiltonians are local by construction, the known techniques often yield complex Hamiltonians which act on a rather large number of spins. In this paper, we present an algorithm to systematically simplify parent Hamiltonians, breaking them down into any given basis of elementary interaction terms. The underlying optimization problem is a semidefinite program, and thus the optimal solution can be found efficiently. Our method exploits a degree of freedom in the construction of parent Hamiltonians---the excitation spectrum of the local terms---over which it optimizes such as to obtain the best possible approximation. We benchmark our method on the AKLT model and the toric code model, where we show that the canonical parent Hamiltonians (acting on 3 or 4 and 12 sites, respectively) can be broken down to the known optimal two-body and four-body terms. We then apply our method to the paradigmatic resonating valence bond (RVB) model on the kagome lattice. Here, the simplest previously known parent Hamiltonian acts on all the 12 spins on one kagome star. With our optimization algorithm, we obtain a vastly simpler Hamiltonian: we find that the RVB model is the ground state of a parent Hamiltonian whose terms are all products of at most four Heisenberg interactions, and whose range can be further constrained, providing a major improvement over the previously known 12-body Hamiltonian.