Tensor networks states allow one to find the low-energy states of local lattice Hamiltonians through variational optimization.Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally.However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically.In this paper we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic and quartic bosonic QFT Hamiltonians.We show that GCTNSs provide arbitrarily accurate approximations to the ground states of quadratic Hamiltonians and decent estimates for quartic ones at weak coupling.Since they capture the short distance behavior of the theories we consider exactly, GCTNSs even allow one to renormalize away simple divergences variationally.In the end our study makes it plausible that CTNSs are indeed a good manifold to approximate the low-energy states of QFTs.
Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state
ketPsicM with finite UV cut-off
Lambda can capture the spacetime symmetries of the ground state
ketPsi. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation
V that maps
ketPsi into
ketPsicM and show two main results. (i) Any spacetime symmetry of the ground state
ketPsi is also mapped by
V into a spacetime symmetry of the cMERA
ketPsicM. However, while in the CFT the stress-energy tensor
Tmunu(x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor
TmunucM(x)=VTmunu(x)Vdagger is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators
OalphacM(x) characterized by the exact same scaling dimensions
Deltaalpha, conformal spins
salpha, operator product expansion coefficients
Calphabetagamma, and central charge
c as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale
1/Lambda is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than
1/Lambda. An infinitesimal coarse-graining step is then generated by the combined action of a \\textit{rescaling} operator
mathfrakL and a \\textit{disentangling} operator
mathfrakK that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U(
N) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.
Continuous tensor networks give a variational ansatz for the ground state of the quantum field theories (QFTs). Notable examples are the continuous matrix product state (cMPS) and the continuous multiscale entanglement renormalization ansatz (cMERA). While cMPS is just adapted to the nonrelativistic QFTs, only the Gaussian cMERA is well understood, which we cannot use to approximate the ground state of the interacting relativistic QFTs. But, instead, cMERA also corresponds to a real-space renormalization group flow in the context of the wave functions. In this paper, we investigate the backward Gaussian cMERA renormalization group flow of the class of cMPS by putting the standard cMPS at the IR scale. At the UV scale, for the bosonic systems in the thermodynamic limit, we achieve the variational class of states that has been proposed recently, as the relativistic cMPS (RCMPS) is adapted to the relativistic QFTs without requiring one to introduce of any additional IR or UV cutoff. We also extend the RCMPS to fermionic systems and theories on a finite circle.
We describe a general procedure to give effective continuous descriptions of quantum lattice systems in terms of quantum fields. There are two key novelties of our method: firstly, it is framed in the hamiltonian setting and applies equally to distinguishable quantum spins, bosons, and fermions and, secondly, it works for arbitrary variational tensor network states and can easily produce computable non-gaussian quantum field states. Our construction extends the mean-field fluctuation formalism of Hepp and Lieb (developed later by Verbeure and coworkers) to identify emergent continuous large-scale degrees of freedom - the continuous degrees of freedom are not identified beforehand. We apply the construction to to tensor network states, including, matrix product states and projected entangled-pair states, where we recover their recently introduced continuous counterparts, and also for tree tensor networks and the multi-scale entanglement renormalisation ansatz. Finally, extending the continuum limit to include dynamics we obtain a strict light cone for the propagation of information.
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≥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).