MERA - Multiscale Entanglement Renormalisation Ansatz MPS - Matrix Product State
AdS - Anti-de Sitter
CFT - Conformal Field Theory
PEPS - Projective Entangled Pair States
cMERA - continuous MERA (continuous multiscale entanglement renormalisation ansatz)
Chapter 1 Introduction
Drawing ideas from one field of physics into another field has lead to significant advances in multiple fields, quickly producing new knowledge as one concept’s developed ideas are applied to anothe field. Examples of this extend from applying entropy to black holes resulting in black hole thermodynamics [1], to applying statistical mechanics (amongst other tools) to biology, giving rise to the field of biophysics[2]. There is a particularly strong correspondence of this type between research in high energy physics and condensed matter physics. For example, Greens functions, phase transitions and topological ideas appeared first in one field before migrating across to great success. In quantum gravity an approach that has risen to prominence is the holographic principle [3, 4], realised by the AdS/CFT correspondence, also referred to as gauge/gravity duality, that proposes a theory of quantum gravity is exactly analogous by a theory of quantum mechanics in one lower dimension. This allows us to study quantum gravity with the well understood tools of quantum mechanics. There have also been a few attempts to try to reverse the process, trying to understand strongly coupled quantum systems in terms of some sort of limit of quantum gravity (i.e. general relativity) [5, 6]. These approaches are often considered questionable because it is not clear that interesting, strongly coupled systems correspond to simplifying limits of the graviational theory [7, 8].
In the last several years a different connection has emerged. In 2009 Brian Swingle proposed that a condensed matter tool called the Multiscale Entanglement Renormal-isation Ansatz (MERA) - a quantum information/condensed matter tool designed to model discrete strongly coupled quantum systems in 1D - is a discrete representation of the AdS/CFT correspondence [9] (published 2012). This proposal produced a flurry of work on tensor networks in AdS/CFT, and their general connection to holographic ideas
[10, 11, 12, 13, 14], of which the MERA representation is an example. Most approaches have focused on special cases, such as a network of random or perfect tensors [10, 14] and so is lacking is a general approach which could be applied to arbitrary MERA and still demonstrate a correspondence.
The work I have done in my PhD over the past four years is aimed at answering this ques-tion. Is it possible to take a tensor network description of a quantum state and construct a holographic network from it without losing the properties of the original quantum state?
The approach I took was to focus on taking a tensor network approximation (to arbitrary precision) of any given quantum state and developing a numerical approximation to a bulk state. If such a statement can be confirmed in the affirmative then this opens holog-raphy research to using the simpler numerical tools of tensor networks. Further, such tools would also allow us to explore arbitrary quantum states with respect to holography by removing fears that the correspondence is invalid within this regime.
Tensor networks are a numerical tool that have developed a very strong association with quantum information in the last couple of decades, having found much use in numerical many-body and condensed matter calculations. This is because the most straightforward way to model quantum states is to either describe them in terms of functions and solve analytically or for a more numerical flavour spatially discretise the quantum state. Both these methods have problems, the first assumes an analytic solution exists and can be found by people, the second requires that the computer stores a number of values that grows exponentially with the number of sites of a many-body state. Hybrid approaches where an ansatz, an educated guesses, is made about the structure of the correct state has had much success at solving many-body questions [15, 16, 17, 18]. These use analytic intuition about the structure of the state to reduce the degrees of freedom to be tractable for numerical methods. Tensor networks are a general method of using this kind of idea.
When describing them, tensor networks can be viewed as graphs, a collection of vertices connected by edges. Each vertex corresponds to a tensor in the network, and edges to the indices of said tensors, edges connecting two vertices corresponding a contraction of a pair of indices between the tensors [19, 20]. Each open edge on the tensor network (an edge which is only connected to only one vertex) corresponds to degrees of freedom labelled by an uncontracted index. Assigning these open edges to physical sites, the tensor network then becomes an exact or approximate representation of a quantum many body state depending on the tensor network graph and the quantum state in question. When
look-ing for numerical approximations the most commonly used tools tend to grow linearly, rather then exponentially, with the number of sites. This drastically reduces the memory costs associated with storing the tensor network state, in turn this reduces the time costs associated with updating the approximation if variational techniques were used. These approximations are achieved by placing restrictions on the kinds of correlations that the ansatz can represent. This is built into the structure of the graph and enforces a sense of locality not naturally built into the naive approaches to representing quantum states.
Early tensor network approximations often restricted themselves to the same geometry as the approximated state, so a D dimensional quantum state was approximated by a D dimensional tensor network [20]. However more complicated networks such as the MERA takes this further by making the tensor network exist in one dimension higher then the geometry of the many-body system they are describing [21, 22, 23]. In the case of the MERA this extra dimension is explained as the renormalisation length scale of the state that is being described. This is reminiscent of the ideas of holography which I discussed earlier, suggesting that certain tensor network ansatz may be implicitly using ideas of holography. If this is true then tensor networks may turn out to be the Rosetta stone to see how to translate holographic principles into arbitrary strongly correlated many-body systems.
One problem with this idea is that these higher dimensional spaces have no physical degrees of freedom in them. If we want to stay firmly within the framework of tensor networks then we must find a way to associate physical sites with this higher dimensional space, i.e. to introduce more open edges. So the first question that must be answered before any holography ideas can be approached is how to introduce these extra degrees of freedom into the bulk. Beyond that the question is how to introduce them without destroying the initial tensor network and corresponding quantum state we started from.
To approach this I looked at a novel ”lifting” of tensor networks, specifically MERA, to introduce new physical degrees of freedom in this higher dimensional space. With this proposal in mind I studied a number of features that would be expected from any holographic theory for two choices of lifting tensors. The first being the first solution uncovered for some physically motivated axioms early in this work, the second arising much later but matching the behaviour of the first solution. This was done for a variety of explicit quantum states computed numerically, demonstrating that this approach with tensor networks is naturally extendible to explicit examples of quantum states.
While doing this, one major thing that I focused on was guaranteeing that this approach extended to tensor networks describing global symmetries. Because of this approach, the numerical part of my project ended up producing a very general computational tool for manipulating tensor networks with local on-site symmetries. Due to its generality, this numerical tool has applications beyond holographic tensor networks to arbitrary tensor networks. And because of its user friendly nature, hiding most of the complicated cal-culations behind the scenes, it is easy to believe it could be practically applied in other approaches as well.
The work I have done on this topic is contained within this thesis and is split into 6 parts throughout 8 chapters. This first chapter (this chapter) introduces the problem and outlines the rest of the thesis. The next three introduce core background topics of the thesis, discussing important topics from condensed matter in chapter 2, and back-ground on the holographic principle and related topics in chapter 3. Chapter 4 takes these topics and discusses the explicit tensor network I will focus on, the MERA. This chapter also contains an introduction to tensor networks and previous work connecting them to the holographic principle.
The new work in this thesis first appears in chapter 5, which introduces a method to
”lift” the MERA to the bulk MERA. A method to take the MERA tensor network and promote it to a quantum state in a higher number of space-time dimensions. This is done using something we call the lifting tensors - defined as a tensor satisfying a pair of axioms - giving rise to a powerful structure in the bulk MERA and reproducing aspects of the holographic principle. One choice of lifting tensor, called the basis independent lifting tensor, reproduce additional properties of the holographic principle beyond those that arise just from the two axioms satisfied by the copy lifting tensor. The results of both of these lifting tensors are included in this chapter and later chapters due to the copy lifting tensor being well understood regardless of is weaknesses relative to the basis independent lifting tensor. In chapter 6 this procedure is extended to symmetric MERA, where the Hamiltonian for the state described by the MERA has a global on-site symmetry giving rise to a local symmetry in the bulk MERA. This chapter also has a quick introduction to ideas from group theory and representation theory and how they can be used to describe tensors and tensor networks with local on-site symmetries. This was then extended to the anyonic MERA which is discussed in chapter 7 including some calculations for an anyonic bulk MERA. Lastly chapter 8 concludes this thesis.
In addition to the main chapters there are four appendices which are related to the main aspect of the thesis but would be out of place in relevant parts of the thesis. Appendix A gives a list of other, relevant, tensor networks that do not appear in the main text.
Appendix B proves certain assumptions used in chapter 5 when deriving the basis inde-pendent lifting tensor, this is not included in the main text because it requires ideas from chapter 6 and 7 (and given a deeper analysis in appendix C) to prove.
The theory behind general symmetries using representation theory is extended upon in appendix C, giving a self contained discussion of these well understood mathematical structures parallel to my new work. In appendix D these ideas are used to create a general tensor network contraction and manipulation code library, used for all numerical calculations in the main chapters (5,6,7) of the thesis. This code is designed to allow numerical manipulations of arbitrary tensor networks with on-site symmetries, including anyonic symmetries. An explicit demonstration of how the MERA is implemented using this code is also given in appendix D to demonstrate the relative ease this library gives to coding symmetric tensor network algorithms, including anyonic algorithms.