Tensor network states and geometry

TL;DR

The paper advocates a unifying geometric perspective on tensor network states, showing how MPS/PEPS realize the D-dimensional physical geometry while MERA encodes a D+1 dimensional holographic geometry. It demonstrates that correlation decay and entanglement entropy scaling are largely determined by network geometry rather than variational parameters, providing intuitive links to geodesics and minimal surfaces. The authors review gapped versus critical behavior, introduce finite-range MERA to model gapped systems, and discuss branching MERA as a means to violate the entropic boundary law in D>1. This geometric framework clarifies strengths and limitations of current networks and motivates new constructions to capture a wider class of entangled ground states.

Abstract

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law -- that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D>1 dimensions.

Figures (18)

• Figure 1: (Color online) A local Hamiltonian on a $D$-dimensional lattice defines a discrete $D$-dimensional geometry. To each ground state we can attach a $(D+1)$-dimensional geometry, where the additional dimension corresponds to length scale.
• Figure 2: (Color online) (i) Matrix product state (MPS) for the ground state of a local Hamiltonian $H$ in a one-dimensional lattice. The tensors are connected according to a one-dimensional array, in correspondence with the one-dimensional physical geometry dictated by the interactions in $H$. (ii) Projected entangled pair state (PEPS) for the ground state of a two-dimensional lattice. The tensors are connected into a network that reproduces the two-dimensional physical geometry.
• Figure 3: (Color online) Multi-scale entanglement renormalization ansatz (MERA) for the ground state of a local Hamiltonian $H$ in a one-dimensional lattice. The tensors form a two-dimensional holographic geometry. The horizontal direction reproduces the spatial dimension of the lattice model, whereas the vertical direction corresponds to the different length scales that are relevant to describing the structure of entanglement in the ground state of the system. More generally, the MERA for a system in $D$ dimensions spans a holographic geometry in $D+1$ dimensions.
• Figure 4: (Color online) As pointed out by Swingle Swingle09, the scale invariant MERA for the ground state of a quantum spin chain can be interpreted as a discrete realization of the AdS/CFT correspondence. The ground state of the one-dimensional lattice model corresponds to a discrete version of the vaccuum of a CFT$_{1+1}$, whereas the MERA spans a two dimensional geometry that corresponds to a discrete version of a time slice of AdS$_{2+1}$. The Figure shows a MERA similar to that of Fig. \ref{['fig:MERA']}, but from another perspective, with the scale parameter $z$ as a radial coordinate.
• Figure 5: (Color online) Homogeneous tensor network states for the ground state in an infinite lattice in $D=1$ spacial dimensions. (i) A homogeneous MPS is characterized by a single tensor that is repeated infinitely many times throughout the tensor network. (ii) A homogeneous scale invariant MERA is characterized by two tensors, a disentangler and an isometry, repeated throughout the tensor network, which consists of infinitely many layers.