Tensor Networks for Quantum Field Theories II

Conference Date:
Tuesday, April 18, 2017 (All day) to Friday, April 21, 2017 (All day)
Scientific Areas:
Condensed Matter
Quantum Foundations
Quantum Gravity
Quantum Information

Tensor networks have proven to be an extremely useful tool in examining quantum many-body systems.  More recently, they have also emerged in the study of the holographic principle in quantum gravity.  While these discussions and successes have referred to applying tensor networks to describe discrete lattice systems, there has been growing interest and also progress in extending these techniques to continuous quantum field theories (QFTs).

The purpose of this meeting is to discuss current research in this direction.  Continuous matrix product states and continuous multi-scale entanglement renormalization ansatz (cMERA) can tackle QFTs directly, without the need to put them on the lattice.  They offer a non-perturbative, wavefunctional-based, variational approach to QFT's, with a variety of potential applications, including the efficient simulation of relativistic and non-relativisitc continuous systems, and the study of their renormalization group flow.  On the other hand, hyperbolic tensor networks such as MERA, the exact holographic mapping, or holographic quantum error correction codes, are currently investigated for its conjectured relation to the AdS/CFT correspondence of quantum gravity.  The continuous versions of these constructions, such as the cMERA, are natural candidates to realizing the AdS/CFT correspondence more accurately.

Topics:  Non-relativisitc QFT's    Renormalization group    Conformal field theory    Holography

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• Bartek Czech, Institue for Advanced Study
• Glen Evenbly, University of Sherbrooke
• Martin Ganahl, Perimeter Institute
• Jutho Haegeman, University of Ghent
• Janet Hung, Fudan University
• Robert Leigh, University of Illinois at Urbana-Champaign
• Ashley Milsted, Perimeter Institute
• Robert Myers, Perimeter Institute
• *Tobias Osborne, University of Hannover
• Xiaoliang Qi, Stanford University
• Volker Scholz, Ghent University
• Miles Stoudenmire, University of Calfornia, Irvine
• Jamie Sully, McGill University
• Brian Swingle, MIT, Harvard University & Brandeis University
• Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
• Frank Verstraete, University of Ghent
• Guifre Vidal, Perimeter Institute
• Steven White, University of California, Irvine

*via teleconference

• Javier Arguello, Perimeter Institute
• Ganapathy Baskaran, Institute of Mathematical Sciences Chennai
• Lakshya Bhardwaj, Perimeter Institute
• Arpan Bhattacharyya, Fudan University
• Dean Carmi, Perimeter Institute
• Shira Chapman, Perimeter Institute
• Jordan Cotler, Stanford University
• Bartek Czech, Institute for Advanced Study
• Clement Delcamp, Perimeter Institute
• Bianca Dittrich, Perimeter Institute
• Glen Evenbly, University of Sherbrooke
• Matthew Fishman, California Institute of Technology
• Adrian Franco Rubio, Perimeter Institute
• Adil Gangat, National Taiwan University
• Martin Ganahl, Perimeter Institute
• Jutho Haegeman, University of Ghent
• Muxin Han, Florida Atlantic University
• Markus Hauru, Perimeter Institute
• Joshuah Heath, Boston College
• Michal Heller, Albert Einstein Institute
• Qi Hu, Perimeter Institute
• Janet Hung, Fudan University
• Nick Hunter-Jones, California Institute of Technology
• Robert Jefferson, Perimeter Institute
• Robert Leigh, University of Illinois at Urbana-Champaign
• Shengqiao Luo, Perimeter Institute
• Hugo Marrochio, Perimeter Institute
• Alex May, University of British Columbia
• Roger Melko, Perimeter Institute & University of Waterloo
• Ashley Milsted, Perimeter Institute
• Sebastian Mizera, Perimeter Institute
• Robert Myers, Perimeter Institute
• Xiaoliang Qi, Stanford University
• Jason Pye, University of Waterloo
• Hammam Qassim, Institute for Quantum Computing
• Julian Rincon, Perimeter Institute
• Burak Sahinoglu, California Institute of Technology
• Volker Scholz, Ghent University
• Didina Serban, Perimeter Institute
• Andrei Shieber, Perimeter Institute
• Vasudev Shyam, Perimeter Institute
• Joan Simon, University of Edinburgh
• Kevin Slagle, University of Toronto
• Barbara Soda, Perimeter Institute
• Miles Stoudenmire, University of Calfornia, Irvine
• Jamie Sully, McGill University
• Brian Swingle, MIT, Harvard University & Brandeis University
• Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
• Nick Van den Broeck, Perimeter Institute
• Guillaume Verdon-Akzam, Institute for Quantum Computing
• Frank Verstraete, University of Ghent
• Guifre Vidal, Perimeter Institute
• Steven White, University of California, Irvine
• Gabriel Wong, University of Virginia
• Shuo Yang, Perimeter Institute
• Beni Yoshida, Perimeter Institute
• Jose Zapata, Centro de Ciencias Matematicas
• Yijian Zou, Perimeter Institute

Tuesday, April 18, 2017

 Time Event Location 9:00 – 9:30am Registration Reception 9:30 – 9:35am Guifre Vidal, Perimeter InstituteWelcome and Opening Remarks Bob Room 9:35 – 10:35am Steven White, University of CaliforniaDiscretizing the many-electron Schrodinger Equation Bob Room 10:35 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Ashley Milsted, Perimeter InstituteEmergence of conformal symmetry in critical spin chains Bob Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 2:40pm Miles Stoudenmire, University of CaliforniaApplying DMRG to Non-relativistic Continuous Systems in 1D and3D Bob Room 2:40 – 3:20pm Martin Ganahl, Perimeter InstituteSolving Non-relativistic Quantum Field Theories with continuous Matrix Product States Bob Room 3:20 – 3:50pm Coffee Break Bistro – 1st Floor 3:50 – 4:30 pm Jutho Haegeman, University of GhentBridging Perturbative Expansions with Tensor Networks Bob Room

Wednesday, April 19, 2017

 Time Event Location 9:30 – 10:30am Guifre Vidal, Perimeter InstituteThe continuous multi-scale entanglement renormalization ansatz (cMERA) Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Robert Leigh, University of Illinois at Urbana-ChampaignUnitary Networks from the Exact Renormalization of Wavefunctionals Bob Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 2:40pm Brian Swingle,Massachusetts Institute of TechnologyHarvard UniversityBrandeis UniversityTensor networks and Legendre transforms Bob Room 2:40 – 3:20pm Volkher Scholz, University of GhentAnalytic approaches to tensor networks for field theories 3:20 – 3:50pm Coffee Break Bistro – 1st Floor 3:50 - 4:50pm Frank Verstraete, University of GhentTensor network renormalization and real space Hamiltonian flows Bob Room 5:00 – 6:00pm Poster Session Atrium 6:00pm Banquet Bistro – 2nd Floor

Thursday, April 20, 2017

 Time Event Location 9:30 – 10:30am Jamie Sully, McGill UniversityTensor Networks and Holography Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Tadashi Takayanagi, Yukawa Institute for Theoretical PhysicsTwo Continous Approaches to AdS/Tensor Network duality Bob Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Robert Myers, Perimeter InstituteComplexity, Holography & Quantum Field Theory Bob Room 3:00 – 3:30pm Coffee Break Bistro – 1st Floor 3:30 – 4:10pm Bartek Czech, Institute for Advanced StudyHow Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity Bob Room

Friday, April 21, 2017

 Time Event Location 9:00 – 10:00am Xiaoliang Qi, Stanford UniversityRandom tensor networks and holographic coherent states Bob Room 10:00 – 10:30am Coffee Break Bistro – 1st Floor 10:30 – 11:10am Tobias Osborne, University of Hannover [via teleconference]Dynamics for holographic codes Bob Room 11:10 – 11:50am Janet Hung, Fudan UniversityTensor network and (p-adic) AdS/CFT Bob Room 11:50 – 12:30pm Glen Evenbly, University of SherbrookeHyper-invariant tensor networks and holography Bob Room 12:30pm Lunch Bistro – 2nd Floor

Bartek Czech, Institute for Advanced Study

How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity

According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the Einstein-Hilbert action of a certain region in the dual space-time. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries. By applying Tensor Network Renormalization to the discretized Euclidean path integral that prepares the CFT state, I will justify the recent suggestion by Caputa et al. that the complexity of a path integral is quantified by the Liouville action. The Liouville field specifies the conformal frame in which the path integral is evaluated; in the most efficient / least complexity frame, the Liouville field is closely related to entanglement entropies of CFT2 intervals. Assuming the Ryu-Takayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual space-time. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesic-wise rewriting of the non-linear vacuum Einstein's equations in 3d with a negative cosmological constant. I emphasize that this is very much work in progress; I hope the audience will help me to sharpen the arguments.

Glen Evenbly, University of Sherbrooke

Hyper-invariant tensor networks and holography

I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multi-scale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for AdS/CFT. Novel holographic features of this tensor network class will be examined, such as an equivalence between the causal cone C[R] and the entanglement wedge E[R] of connected boundary regions R.

Martin Ganahl, Perimeter Institute

Solving Non-relativistic Quantum Field Theories with continuous Matrix Product States

Since its proposal in the breakthrough paper  [F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405(2010)], continuous Matrix Product States (cMPS) have emerged as a powerful tool for obtaining non-perturbative ground state and excited state properties of interacting quantum field theories (QFTs) in (1+1)d. At the heart of the cMPS lies an efficient parametrization of manybody wavefunctionals directly in the continuum, that enables one to obtain ground states of QFTs via imaginary time evolution. In the first part of my talk I will give a general introduction to the cMPS formalism. In the second part, I will then discuss a new method for cMPS optimization, based on energy gradient instead of the usual imaginary time evolution. This new method overcomes several problems associated with imaginary time evolution, and allows to perform calculations at much lower cost / higher accuracy than previously possible.

Jutho Haegeman, University of Ghent

Bridging Perturbative Expansions with Tensor Networks

We demonstrate that perturbative expansions for quantum many-body systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This approach leads to classes of tensor-network states parameterized by few parameters with a clear physical meaning, while still providing excellent variational energies. We also demonstrate how to construct perturbative expansions of the entanglement Hamiltonian, whose eigenvalues form the entanglement spectrum, and how the tensor-network approach gives rise to order parameters for topological phase transitions.

Janet Hung, Fudan University

We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the Bruhat-Tits tree as a tensor network, we recover (partially) features of the p-adic AdS/CFT dictionary discussed recently in the literature.

Robert Leigh, University of Illinois at Urbana-Champaign

Unitary Networks from the Exact Renormalization of Wavefunctionals

The exact renormalization group (ERG) for O(N) vector models at large N on flat Euclidean space admits an interpretation as the bulk dynamics of a holographically dual higher spin gauge theory on AdS_{d+1}. The generating functional of correlation functions of single trace operators is reproduced by the on-shell action of this bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. This structure arises because of an enormous non-local symmetry of free fixed point theories. In this talk, I will review the ERG construction and describe its extension to the RG flow of the wave functionals of arbitrary states of the O(N) vector model at the free fixed point. One finds that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Thus the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and cMERA. The ERG tensor network appears to share the general structure of cMERA but differs in important ways.

Ashley Milsted, Perimeter Institute

Emergence of conformal symmetry in critical spin chains

We demonstrate that 1+1D conformal symmetry emerges in critical spin chains by constructing a lattice ansatz Hn for (certain combinations of) the Virasoro generators Ln. The generators Hn offer a new way of extracting conformal data from the low energy eigenstates of the lattice Hamiltonian on a finite circle. In particular, for each energy eigenstate, we can now identify which Virasoro tower it belongs to, as well as determine whether it is a Virasoro primary or a descendant (and similarly for global conformal towers and global conformal primaries/descendants). The central charge is obtained from a simple ground-state expectation value. Non-universal, finite-size corrections are the main source of error. We propose and demonstrate the use of periodic Matrix Product States, together with an improved ground state solver, to reach larger system sizes. We uncover that, importantly, the MPS single-particle excitation ansatz accurately describes all low energy excited states.

Robert Myers, Perimeter Institute

Complexity, Holography & Quantum Field Theory

I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.

Tobias Osborne, University of Hannover

Dynamics for holographic codes

In this talk I discuss the problem of introducing dynamics for holographic codes. To do this it is necessary to take a continuum limit of the holographic code. As I argue, a convenient kinematical continuum limit space is given by Jones’ semicontinuous limit. Dynamics are then furnished by a unitary representation of a discrete analogue of the conformal group known as Thompson’s group T. I will describe these representations in detail in the simplest case of a discrete AdS geometry modelled by trees. Consequences such as the ER=EPR argument are then realised in this setup. Extensions to more general tessellations with a MERA structure are possible, and will be (very) briefly sketched.

Xiaoliang Qi, Stanford University

Random tensor networks and holographic coherent states

Tensor network is a constructive description of many-body quantum entangled states starting from few-body building blocks. Random tensor networks provide useful models that naturally incorporate various important features of holographic duality, such as the Ryu-Takayanagi formula for entropy-area relation, and operator correspondence between bulk and boundary. In this talk I will overview the setup and key properties of random tensor networks, and then discuss how to describe quantum superposition of geometries in this formalism. By introducing quantum link variables, we show that random tensor networks on all geometries form an overcomplete basis of the boundary Hilbert space, such that each boundary state can be mapped to a superposition of (spatial) geometries. We discuss how small fluctuations around each geometry forms a “code subspace” in which bulk operators can be mapped to boundary isometrically. We further compute the overlap between distinct geometries, and show that the overlap is suppressed exponentially in an area law fashion, in consistency with the holographic principle. In summary, random tensor networks on all geometries form an overcomplete basis of “holographic coherent states” which may provide a new starting point for describing quantum gravity physics.

References
[1] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, Zhao Yang, JHEP 11 (2016) 009
[2] Xiao-Liang Qi, Zhao Yang, Yi-Zhuang You, arxiv:1703.06533

Volker Scholz, Ghent University

Analytic approaches to tensor networks for field theories

I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that both Matrix product states (MPS) as well as the multiscale renormalization Ansatz (MERA) do approximate conformal field theories. In particular, I will discuss the case of Wess-Zumino-Witten models.

based on joint work with Robert Koenig (MPS), Brian Swingle and Michael Walter (MERA)

Miles Stoudenmire, University of California, Irvine

Applying DMRG to Non-relativistic Continuous Systems in 1D and 3D

The density matrix renormalization group works very well for one-dimensional (1D) lattice systems, and can naively be adapted for non-relativistic continuum systems in 1D by discretizing real space using a grid. I will discuss challenges inherent in this approach and successful applications. Recently, the success of the grid approach for 1D motivated us to extend the approach to 3D by treating the transverse directions with a basis set. This hybrid grid/basis-set approach allows DMRG to scale much better for long molecules and we obtain state-of-the-art results with modest computing resources. A key component of the approach is a powerful algorithm for compressing long-range interactions into a matrix product operator which I will present in some detail.

James Sully, Stanford Linear Accelerator Center

Tensor Networks and Holography

Brian Swingle, MIT, Harvard University & Brandeis University

Tensor networks and Legendre transforms

Tensor networks have primarily, thought not exclusively, been used to the describe quantum states of lattice models where there is some inherent discreteness in the system. This raises issues when trying to describe quantum field theories using tensor networks, since the field theory is continuous (or at least the regulator should not play a central role). I'll present some work in progress studying tensor networks designed to directly compute correlation functions instead of the full state. Here the discreteness arises from our choice of where and how to probe the field theory. This approach is roughly analogous to studying a Legendre transform of the state. I'll discuss the properties of such networks and show how to construct them in some cases of interest, including non-interacting fermion field theories. Partly based on work with Volkher Scholz and Michael Walter.

Tadashi Takayanagi,  Yukawa Institute for Theoretical Physics

Two Continous Approaches to AdS/Tensor Network duality

In this talk, I would like to discuss how we can realize the correspondence between AdS/CFT and tensor network in quantum field theories (i.e. the continous limit). As the first approach I will discuss a possible connection between continuous MERA and AdS/CFT. Next I will introduce the second approach based on the optimization of Euclidean path-integral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.

Frank Verstraete, University of Ghent

Tensor network renormalization and real space Hamiltonian flows

We will review the topic of tensor network renormalization, relate it to real space Hamiltonian flows, and discuss the emergence of matrix product operator algebras as symmetries of the renormalization fixed points.

joint work with Matthias Bal, Michael Marien and Jutho Haegeman

Guifre Vidal, Perimeter Institute

The continuous multi-scale entanglement renormalization ansatz (cMERA)

The first half of the talk will introduce the cMERA, as proposed by Haegeman, Osborne, Verschelde and Verstratete in 2011 [1], as an extension to quantum field theories (QFTs) in the continuum of the MERA tensor network for lattice systems. The second half of the talk will review recent results [2] that show how a cMERA optimized to approximate the ground state of a conformal field theory (CFT) retains all of its spacetime symmetries, although these symmetries are realized quasi-locally. In particular, the conformal data of the original CFT can be extracted from the optimized cMERA.

[1] J. Haegeman, T. J. Osborne, H. Verschelde, F. Verstraete, Entanglement renormalization for quantum fields, Phys. Rev. Lett, 110, 100402 (2013), arXiv:1102.5524
[2] Q. Hu, G. Vidal, Spacetime symmetries and conformal data in the continuous multi-scale entanglement renormalization ansatz, arXiv:1703.04798

Steven White, University of California, Irvine

Discretizing the many-electron Schrodinger Equation

Large parts of condensed matter theoretical physics and quantum chemistry have as a central goal discretizing and solving the continuum many-electron Schrodinger Equation.  What do we want to get from these calculations?  What are key problems of interest? What sort of approaches are used?  I'll start with a broad overview of these questions using the renormalization group as a conceptual framework. I'll then progress towards our recent tensor network approaches for the many electron problem, discussing along the way issues of the area law, wavelet techniques and Wilson's related work, wavelets and MERA, and discretizations that combine grids and basis sets.

Arpan Bhattacharyya, Fudan University

AdS/CFT via Tensor Network : Bulk boundary Reconstruction

We will demonstrate , how to reconstruct bulk operator starting form the local boundary using our model of tensor network which is basically using being build form the perfect tensor plus some small perturbations away form it. We will show that it has the similar features as that of HKLL construction thereby making the connection with the holography (AdS/CFT) concrete. Also we will demonstrate the connection between the linear part of the operator reconstruction and the wavelet transformation. Further we will show that the non linear part of the reconstruction has the possibility of giving the "Geodesic Witten diagram ". At last , we will consider the example of p-adic tree where all these things can be written down explicitly.

Jordan Cotler, Stanford University

cMERA for Interacting Scalar Fields

We upgrade cMERA to a systematic variational ansatz and develop techniques for its application to interacting quantum field theories in arbitrary spacetime dimensions. By establishing a correspondence between the first two terms in the variational expansion and the Gaussian Effective Potential, we can exactly solve for a variational approximation to the cMERA entangler. As examples, we treat scalar ϕ^4 theory and the Gross-Neveu model and extract non-perturbative behavior. We also comment on the connection between generalized squeezed coherent states and more generic entanglers.

Matthew Fishman, California Institute of Technology

Improving the Corner Transfer Matrix Renormalization Group Method with Fixed Points

We present an explicitly translationally invariant version of the Corner Transfer Matrix Renormalization Group (CTMRG) method, which allows us to reformulate the method in terms of a set of fixed point equations. This leads to speedups in the convergence time of the algorithm, particularly for systems near criticality. To show the performance of the algorithm, we present various benchmarks for contracting 2D statistical mechanics models as well as 2D quantum models written as projected entangled pair states (PEPS).

Entanglement structure and UV regularization in cMERA

The continuous multi-scale entanglement renormalization ansatz or cMERA provides a variational ansatz for the ground state of a quantum field theory. Such states come equipped with an intrinsic length scale that acts as an ultraviolet cutoff. We provide evidence for the existence of this cutoff based on the entanglement structure of a particular family of cMERA states, namely Gaussian states optimized for free bosonic and fermionic CFTs. Our findings reflect that short distance entanglement is not fully present in the ansatz states, thus hinting at ultraviolet regularization.

Steady States of Infinite-Size Dissipative Quantum Chains via Imaginary Time Evolution

Directly in the thermodynamic limit, we show how to combine imaginary and real time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESS) of one-dimensional dissipative quantum lattices governed by the Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly corre- lated subspace of the NESS, after which real time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the dissipative transverse field quantum Ising chain. We show that a crossover of an order parameter shown to be smooth in previous finite-size studies remains smooth in the thermodynamic limit.

Markus Hauru, Perimeter Institute

Topological conformal defects with tensor network

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_{\epsilon}$ and the Kramers-Wannier duality defect $D_{\sigma}$.
These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function $Z_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions $\Delta_{\alpha}$ and conformal spins $s_{\alpha}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data $\{\Delta_{\alpha}, s_{\alpha}\}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix $M_{D}$ for the partition function $Z_D$. We present results from a recent paper (arXiv:1512.03846), where we investigate the use of tensor network techniques to both represent and coarse-grain the partition functions $Z_{D_\epsilon}$ and $Z_{D_\sigma}$ of the critical Ising model with either a symmetry defect $D_{\epsilon}$ or a duality defect $D_{\sigma}$. We also explain how to coarse-grain the corresponding transfer matrices $M_{D_\epsilon}$ and $M_{D_\sigma}$, from which we can extract accurate numerical estimates of $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\epsilon}}$ and $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\sigma}}$. Two key new ingredients of our approach are (i) coarse-graining of the defect $D$, which applies to any (i.e.\ not just topological) conformal defect and yields a set of associated scaling dimensions $\Delta_{\alpha}$, and (ii) construction and coarse-graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins $s_{\alpha}$.

Qi Hu, Perimeter Institute

Continuous Multi-scale Entanglement Renormalization Ansatz

The generalization of the multi-scale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA, is a variational ansatz for the ground state of quantum field theories. For a conformal field theory, it can capture the space-time symmetries of the ground state, and we can extract the conformal data from cMERA.

Matrix Product State Simulations of Quantum Fields in an Expanding Universe

The matrix product state (MPS) ansatz makes possible computationally-efficient representations of weakly entangled many-body quantum systems with gapped Hamiltonians near their ground states, notably including massive, relativistic quantum fields on the lattice. No Wick rotation is required to apply the time evolution operator, enabling study of time-dependent Hamiltonians. Using free massive scalar field theory on the 1+1 Robertson-Walker metric as a toy example, I present early efforts to exploit this fact to model quantum fields in curved spacetime. We use the ADM formalism to write the appropriate Hamiltonian witnessed by a particular class of normal observers. Possible applications include simulations of gravitational particle production in the presence of interactions, studies of the slicing-dependence of entanglement production, and inclusion of the expectation of the stress-energy tensor as a matter source in a numerical relativity simulation.

Alex May, University of British Columbia

Tensor networks for dynamic spacetimes

Existing tensor network models of holography are limited to representing the geometry of constant time slices of static spacetimes. We study the possibility of describing the geometry of a dynamic spacetime using tensor networks. We find it is necessary to give a new definition of length in the network, and propose a definition based on the mutual information. We show that by associating a set of networks with a single quantum state and making use of the mutual information based definition of length, a network analogue of the maximin formula can be used to calculate the entropy of boundary regions.

Hugo Marrochio, Perimeter Institute

Holographic complexity and related progress towards a cMERA realization

Julian Rincon, Perimeter Institute

Continuous matrix product representations for mixed states

The continuous matrix product states (cMPS) is a powerful variational ansatz for the ground state of interacting quantum field theories in 1+1 spacetime dimensions [F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405(2010)]. Here we propose a density matrix generalization of the cMPS, the continuous matrix product density operator (cMPDO), and investigate its suitability to represent thermal states and master equation dynamics. We show the existence of the cMPDO by taking the continuum limit of a lattice MPDO and characterize its mathematical properties. For thermal states of field theories, we find that the cMPDO offers an accurate description of their corresponding density matrix. We argue that these results can also be extended for the case of master equation dynamics.

Yijian Zou, Perimeter Institute

Extracting conformal data with periodic boundary matrix product states

We construct Virasoro generators on a finite critical lattice system with the periodic boundary condition, and use them to identify conformal towers. Ground state and excited states corresponding to scaling operators are found with periodic boundary matrix product states. Scaling dimensions and central charge are estimated with high accuracy from finite size scaling.

Hyper-invariant tensor networks and holography

Friday Apr 21, 2017
Speaker(s):

I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multi-scale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible.

Scientific Areas:

Friday Apr 21, 2017
Speaker(s):

We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the Bruhat-Tits tree as a tensor network, we recover (partially) features of the p-adic AdS/CFT dictionary discussed recently in the literature.

Scientific Areas:

Dynamics for holographic codes

Friday Apr 21, 2017
Speaker(s):

In this talk I discuss the problem of introducing dynamics for holographic codes. To do this it is necessary to take a continuum limit of the holographic code. As I argue, a convenient kinematical continuum limit space is given by Jones’ semicontinuous limit. Dynamics are then furnished by a unitary representation of a discrete analogue of the conformal group known as Thompson’s group T. I will describe these representations in detail in the simplest case of a discrete AdS geometry modelled by trees.

Scientific Areas:

Random tensor networks and holographic coherent states

Friday Apr 21, 2017
Speaker(s):

Tensor network is a constructive description of many-body quantum entangled states starting from few-body building blocks. Random tensor networks provide useful models that naturally incorporate various important features of holographic duality, such as the Ryu-Takayanagi formula for entropy-area relation, and operator correspondence between bulk and boundary. In this talk I will overview the setup and key properties of random tensor networks, and then discuss how to describe quantum superposition of geometries in this formalism.

Scientific Areas:

How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity

Thursday Apr 20, 2017
Speaker(s):

According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the Einstein-Hilbert action of a certain region in the dual space-time. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries.

Scientific Areas:

Complexity, Holography & Quantum Field Theory

Thursday Apr 20, 2017
Speaker(s):

I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.

Scientific Areas:

Two Continous Approaches to AdS/Tensor Network duality

Thursday Apr 20, 2017
Speaker(s):

In this talk, I would like to discuss how we can realize the correspondence between AdS/CFT and tensor network in quantum field theories (i.e. the continous limit). As the first approach I will discuss a possible connection between continuous MERA and AdS/CFT. Next I will introduce the second approach based on the optimization of Euclidean path-integral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.

Scientific Areas:

Tensor Networks and Holography

Thursday Apr 20, 2017
Speaker(s):
Scientific Areas:

Tensor network renormalization and real space Hamiltonian flows

Wednesday Apr 19, 2017
Speaker(s):

We will review the topic of tensor network renormalization, relate it to real space Hamiltonian flows, and discuss the emergence of matrix product operator algebras as symmetries of the renormalization fixed points.

joint work with Matthias Bal, Michael Marien and Jutho Haegeman

Scientific Areas:

Analytic approaches to tensor networks for field theories

Wednesday Apr 19, 2017
Speaker(s):

I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that both Matrix product states (MPS) as well as the multiscale renormalization Ansatz (MERA) do approximate conformal field theories.

Scientific Areas:

Pages

Scientific Organizers:

• Robert Myers, Perimeter Institute
• Tadashi Takayanagi, Yukawa Institute for Theoretical Physics
• Frank Verstraete, University of Ghent
• Guifre Vidal, Perimeter Institute
• Steven White, University of California, Irvine