Tensor Networks for Quantum Field Theories Conference - 2011

This introductory talk aims to answer a few basic questions (What is a tensor network? Under which circumstance is a tensor network useful?) and describe the tensor network states that will be discussed during the workshop (matrix product state [MPS], projected entangled pair states [PEPS], and the multi-scale entanglement renormalization ansatz [MERA]). I will then briefly describe the recent developments that motivated this workshop on

The study of fermionic and frustrated systems in two dimensions is one of the biggest challenges in condensed matter physics. Among the most promising tools to simulate these systems are 2D tensor networks, including projected entangled-pair states (PEPS) and the 2D multi-scale entanglement renormalization ansatz (MERA), which have been generalized to fermionic systems recently. In the first part of this talk I will present a simple formalism how to include fermionic statistics into 2D tensor networks. The second part covers recent simulation results showing that infinite PEPS (iPEPS) can compete with the best known variational methods. In particular, for the t-J model and the SU(4) Heisenberg model iPEPS yields better variational energies than obtained in previous variational- and fixed-node Monte Carlo studies. Future perspectives and open problems are discussed.

In this talk, I will present a first principle construction of a holographic dual for gauged matrix models that include gauge theories. The dual theory is shown to be a closed string field theory coupled with an emergent two-form gauge field defined in one higher dimensional space. The bulk space with an extra dimension emerges as a well defined classical background only when the two-form gauge field is in the deconfinement phase. Based on this, it is shown that critical phases that admit holographic descriptions form a novel universality class with a non-trivial quantum order.

In 1987, Feynman devoted one of his last lectures to highlighting three serious objections against the usefulness of the variational principle in the theory of relativistic quantum fields. In that same year, in a different branch of physics, Affleck, Kennedy, Lieb and Tasaki devised a quantum state that resulted in the development of a handful of different variational ansätze for lattice models over the last two decennia. These quantum states are known as tensor network states and invalidate at least two of Feynman's arguments. They could thus be used in a variational study of relativistic quantum field theories on a lattice. However, two classes of tensor network states, namely the matrix product state and the multi-scale entanglement renormalization ansatz, have recently been ported to the continuous setting, so that we now have direct access to variational wave functions for quantum field theories and are no longer restricted to a lattice regularization.

One might be confused by the proliferation of tensor network states, such as MPS, PEPS, tree tensor networks [TTN], MERA, etc. What is the main difference between them? In this talk I will argue that the geometry of a tensor network determines several properties of the state that is being represented, such as the asymptotic scaling of correlations and of entanglement entropy. I will also describe the relation between the MERA and the Renormalization Group, and will review Brian Swingle

The MERA offers a powerful variational approach to quantum field theory. While the continuous MERA may allow us to directly address field theories in the continuum, the MERA on the lattice has already demonstrated its ability to characterize conformal field theories. In this talk I will explain how to extract the conformal data (central charge, primary fields, and their scaling dimensions and OPE) of a CFT from a quantum spin chain at a quantum critical point. I will consider both homogeneous systems (translation invariant) and systems with an impurity (where translation invariance is explicitly broken). Key to the success of the MERA is the exploitation of both scale and translation invariance. I will show how translation invariance can still be exploited even in the presence of an impurity, even if the system is no longer translation invariant. This follows from an intriguing "causality principle" in the RG flow. I will also discuss the relation of these results with Wilson's famous resolution of the Kondo impurity problem.

Topological Quantum field theories(TQFTs) are a special class of QFTs. Their actions do not depend on the metric of the background space-time manifold. Thus, it is very natural to define TQFTs on an arbitrary triangulation of the space-time manifold and they are independent on the triangulation. More importantly, TQFTs defined on triangulations are always a finite theory associated with a well defined cut-off. A well known example is the Turaev-Viro states sum invariants. Essentially, the Turaev-Viro constructions are (local) tensor network representations of a special class of 1+2D TQFTs. In this talk, I will show a new class of TQFTs that can be derived based on the (local) tensor network representations in arbitrary dimensions. They can be regarded as the discrete analogy of topological Berry phase terms of (discrete) non-linear sigma models. The edge theory of such a new class of TQFTs can be regarded as the discrete analogy of WZW terms. This new class of TQFTs naturally classify (bosonic) symmetry protected topological orders in arbitrary dimensions. Finally, I will also discuss new classes of fermionic TQFTs based on the Grassmann tensor network representations and possible new route towards Quantum Gravity(QG).

In this talk I will describe how to generalize the multiscale entanglement renormalization ansatz to quantum fields. The resulting variational class of wavefunctions, cMERA, arising from this RG flow are translation invariant and exhibit an entropy-area law. I'll illustrate the construction for some example fields, and describe how to cover the case of interacting theories.

In this second presentation, we will revisit Feynman's first argument and discuss how it still strongly influences variational studies of relativistic field theories with MPS or cMPS. However, as we explain, this argument can be completely overcome by introducing different variational parameters for the different length scales in the system, a strategy that naturally results in the MERA for lattice systems, or its continuous version for field theories. We then illustrate how a cMERA representation for the ground state of free relativistic quantum field theories can be constructed and discuss the main properties of this representation.