Abstract
We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions $d\geq 2$. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the $N$-point functions of local observables. While we discuss mostly the continuous tensor network states extending Projected Entangled Pair States (PEPS), we propose a generalization bearing similarities with the continuum Multi-scale Entanglement Renormalization Ansatz (cMERA).
Highlights
Tensor network states (TNSs) provide an efficient parametrization of physically relevant many-body wave functions on the lattice [1,2]
We propose a definition of continuous tensor network states (CTNSs) that naturally extends TNSs to the continuum
We put forward a new class of states for quantum fields that is obtained as a continuum limit of tensor network states and, carries the same fundamental properties
Summary
Tensor network states (TNSs) provide an efficient parametrization of physically relevant many-body wave functions on the lattice [1,2]. A natural way to carry out such a program is to take the continuum limit of a TNS, by letting the lattice spacing tend to zero while appropriately rescaling the tensors This task has been carried in one spatial dimension, d 1⁄4 1, where it yields continuous matrix product states (CMPSs) [33,34]. II takes the form of a functional integral over auxiliary scalar fields as advertised From this definition, which makes local Euclidean invariance manifest, we derive an operator representation similar to the one used for CMPSs. Importantly, we show in Sec. III how this ansatz can be obtained from a continuum limit of a discrete TNS.
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