Abstract
The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.
Highlights
One of the hallmarks of critical behavior is the divergence of correlations, and the emergence of scale invariance; the low-energy behavior of the system seems to be of a similar nature on small and large scales
We presented exact hierarchical tensor networks for representing the ground state of the spin-1 Motzkin spinchain
The tile-based two-dimensional representation of each walk provides a bulk description of the spin chain: each valid bulk “picture” corresponds to a particular boundary state in the ground state superposition
Summary
One of the hallmarks of critical behavior is the divergence of correlations, and the emergence of scale invariance; the low-energy behavior of the system seems to be of a similar nature on small and large scales. A simple class of 1D tensor networks, useful for describing spin-chains with finite range correlations, are known as matrix product states (MPS) [9], and are the variational class of states used in White’s density matrix renormalization group (DMRG) numerical procedure [10], arguably the most successful tool for numerical investigation of quantum phases in 1D Another class of tensor network states, known as the multi-scale entanglement renormalization ansatz (MERA) [11], was proposed by Vidal [12] to be specially tailored for describing quantum critical points. The second network we propose has a MERA-like structure [30] and defines a natural renormalization process of Motzkin walk configurations
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