Mathematica, tensor networks, MERA and entanglement

Speaker(s): Wilke van der Schee

Date: 28/08/2015

*A lecture for the Mathematica Summer School 2015*

Tensor network renormalization

Speaker(s): Guifre Vidal

Date: 27/08/2015

*A lecture for the Mathematica Summer School 2015*

Mathematica, tensor networks, MERA and entanglement

Speaker(s): Wilke van der Schee

Date: 26/08/2015

*A lecture for the Mathematica Summer School 2015*

Tensor network renormalization

Speaker(s): Guifre Vidal

Date: 27/08/2015

*A lecture for the Mathematica Summer School 2015*

The multi-scale entanglement renormalization ansatz

Speaker(s): Guifre Vidal

Date: 26/08/2015

*A lecture for the Mathematica Summer School 2015*

Ground state entanglement and tensor networks

Speaker(s): Guifre Vidal

Date: 25/08/2015

*A lecture for the Mathematica Summer School 2015*

Holographic mapping, quantum error correction code and sub-AdS locality

Speaker(s): Xiao-Liang Qi

Date: 20/08/2015

Abstract: In recent years, tensor networks have been proposed as a useful framework for understanding holographic duality, especially the relation between quantum entanglement and space-time geometry. Most tensor networks studied so far are defined in the large scale compared with AdS radius. In this talk, I will describe a new tensor network approach which defines a holographic mapping that applies to a refined network with sub-AdS scale resolution, or even to a flat space. The idea of quantum error correction code plays an essential role in this approach. Using this new tensor network, we can study features of the bulk theory, such as how locality at sub-AdS scale emerges in a "low energy subspace" even though the whole theory is intrinsically nonlocal, as a quantum gravity theory should be.

On quotients of MERA (and black holes in AdS3)

Speaker(s): Bartek Czech

Date: 20/08/2015

Untitled

Speaker(s): James Sully

Date: 20/08/2015

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

Speaker(s): Fernando Pastawski

Date: 18/08/2015

Abstract: In this talk I will introduce a family of exactly solvable toy models of a holographic correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. The building block for these models are a special type of tensor with maximal entanglement along any bipartition, which gives rise to an exact isometry from bulk operators to boundary operators. The entire tensor network is a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the holographic correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. I will describe how bulk operators may be represented on the boundary regions mimicking the Rindler-wedge reconstruction.

Entanglement renormalization for quantum fields

Speaker(s): Jutho Haegeman

Date: 18/08/2015

Abstract: The Multiscale Entanglement Renormalization Ansatz has proven to capture the ground state properties of strongly correlated quantum lattice systems, both in gapped regimes and at critical points, and realizes a lattice version of the holographic principle. In this talk, I will review a construction of entanglement renormalization that applies in the continuum (i.e. to quantum fields) and discuss several aspects such as the renormalization group equation and scaling exponents, illustrated using free field theories as example

Tensor Network Renormalization and the MERA

Speaker(s): Glen Evenbly

Date: 18/08/2015

Abstract: I describe a class of non-perturbative renormalization group (RG) transformations which, when applied to the (discrete time) Euclidean path integrals of a quantum systems on the lattice, can give results consistent with conformal transformations of quantum field theories. In particular, this class of transformation, which we call Tensor Network Renormalization (TNR), is shown to generate a scale-invariant RG flow for quantum systems at a critical point. Applications of TNR towards study of quantum critical systems, and its relationship to the multi-scale-entanglement renormalization ansatz (MERA) for ground and thermal states of quantum systems, will be discussed.

Bulk Locality and Quantum Error Correction in AdS/CFT

Speaker(s): Daniel Harlow

Date: 02/03/2015 - 2:00 pm

Abstract: In this talk I will describe recent work with Almheiri and Dong, where we proposed a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. Time permitting, I will also discuss work in progress with Pastawski, Preskill, and Yoshida on a new class of stabilizer codes that explicitly realize many of the properties we argued the AdS/CFT error correcting code should have.

Exact holographic mapping, tensor networks and space-time geometry

Speaker(s): Xiao-Liang Qi

Date: 27/02/2015 - 11:00 am

Abstract: Holographic duality is a duality between gravitational systems and non-gravitational systems. In this talk, I will propose a different approach for understanding holographic duality named as the exact holographic mapping. The key idea of this approach can be summarized by two points: 1) The bulk theory and boundary theory are related by a unitary mapping in the Hilbert space. 2) Space-time geometry is determined by the structure of correlations and quantum entanglement in a quantum state. When applied to lattice systems, the holographic mapping is defined by a unitary tensor network. For free fermion boundary theories, I will discuss how different bulk geometries are obtained as dual theories of different boundary states. A particularly interesting case is the AdS black hole geometry and the interpretation of the interior of a black hole. We will also discuss dual geometries of topological states of matter.

Tensor Networks from Integral Geometry

Speaker(s): Bartek Czech

Date: 24/02/2015 - 2:00 pm

Abstract: The analogy between Multi-scale Entanglement Renormalization Ansatz (MERA) and the spatial slice of three-dimensional anti-de Sitter space (AdS3) has motivated a great interest in tensor networks among holographers. I discuss a way to promote this analogy to a rigorous, quantitative, and constructive relation. A key quantitative ingredient is the way the strong subadditivity of entanglement entropy is encoded in MERA and in a holographic spacetime. The upshot is that the map between MERA and the spatial slice of AdS3 is mediated through an additional integral transform. Interpreted directly, MERA is a discretization not of the spatial slice of AdS3, but of the space of geodesics on the spatial slice of AdS3.

Geometry from Compression

Speaker(s): James Sully

Date: 03/02/2015 - 2:00 pm

Abstract: Recent research has suggested deep connections between geometry and entropy. This connection was first seen in black hole thermodynamics, but has been more fully realized in the Ryu-Takayanagi proposal for calculating entanglement entropies in AdS/CFT. We suggest that this connection is even broader: entropy, and in particular compression, are the fundamental building blocks of emergent geometry. We demonstrate how spatial geometry can be derived from the properties of a recursive compression algorithm for the boundary CFT. We propose a general algorithm for constructing MERA-like tensor networks and elucidate connections to the mathematical field of integral geometry.

Tensor Networks for nonabelian Gauge Theory

Speaker(s): Ashley Milsted

Date: 25/11/2014 - 3:30 pm

Abstract: We present an analytic, gauge invariant tensor network ansatz for the ground state of lattice Yang-Mills theory for nonabelian gauge groups. It naturally takes the form of a MERA, where the top level is the strong coupling limit of the lattice theory. Each layer performs a fine-graining operation defined in a fixed way followed by an optional step of adiabatic evolution, resulting in the ground state at an intermediate coupling. The ansatz is very much in the spirit of Kogut and Susskind's Hamiltonian approach to understanding confinement by starting from the strong coupling limit and perturbing, but exploiting a tensor network structure to go beyond perturbative approaches.

Self-assembling tensor networks and holography in disordered spin chains

Speaker(s): Andrew Goldsborough

Date: 16/09/2014 - 3:30 pm

Abstract: We show that the numerical strong disorder renormalization group algorithm (SDRG) of Hikihara et. al. [{Phys. Rev. B} {bf 60}, 12116 (1999)] for the one-dimensional disordered Heisenberg model naturally describes a tree tensor network (TTN) with an irregular structure defined by the strength of the couplings. Employing the holographic interpretation of the TTN in Hilbert space, we compute expectation values, correlation functions and the entanglement entropy using the geometrical properties of the TTN. We find that the disorder averaged spin-spin correlation scales with the average path length through the tensor network while the entanglement entropy scales with the minimal surface connecting two regions. Furthermore, the entanglement entropy increases with both disorder and system size, resulting in an area-law violation. Our results demonstrate the usefulness of a self-assembling TTN approach to disordered systems and quantitatively validate the connection between holography and quantum many-body systems.

Stripes and pairing in t-J and Hubbard models

Speaker(s): Steven White

Date: 11/09/2014 - 2:00 pm

Abstract: This has been a leading question in condensed matter physics since the discovery of the cuprate superconductors. In this talk I will review some of the DMRG and tensor network results for the ground states of these models. A key question I'll address is the issue of stripes: are the ground states striped? Do stripes compete with or induce d-wave superconductivity? Another question I'll address is: how well does 2D DMRG do in comparison with iPEPS and quantum Monte Carlo. I will also show recent results for a standard 3-band Hubbard model for the cuprates. The asymmetry between hole and electron doped systems which is seen in the cuprates arises naturally from this model.