Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group T takes the role of the latter in the discrete setting, and representations of T are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields-in the continuum limit-a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones's no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.
Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group T takes the role of the latter in the discrete setting, and representations of T are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields-in the continuum limit-a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones's no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
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.
A bstract We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large- N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.
The study of critical quantum many-body systems through conformal field theory (CFT) is one of the pillars of modern quantum physics. Certain CFTs are also understood to be dual to higher-dimensional theories of gravity via the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. To reproduce various features of AdS/CFT, a large number of discrete models based on tensor networks have been proposed. Some recent models, most notably including toy models of holographic quantum error correction, are constructed on regular time-slice discretizations of AdS. In this work, we show that the symmetries of these models are well suited for approximating CFT states, as their geometry enforces a discrete subgroup of conformal symmetries. Based on these symmetries, we introduce the notion of a quasiperiodic conformal field theory (qCFT), a critical theory less restrictive than a full CFT and with characteristic multi-scale quasiperiodicity. We discuss holographic code states and their renormalization group flow as specific implementations of a qCFT with fractional central charges and argue that their behavior generalizes to a large class of existing and future models. Beyond approximating CFT properties, we show that these can be best understood as belonging to a paradigm of discrete holography.
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.