Tensor Network

Tensor network/spacetime correspondences explore the exciting idea that geometric information about a quantum state might be related to the actual geometry that the state describes in a quantum gravitational setting. I will give an overview of a new type of correspondence between global de Sitter spacetime and the MERA. This simple correspondence is already enough to see several features of de Sitter gravity emerge, such as cosmic no-hair and horizon complementarity. I will also comment on some more speculative topics like complexity = action and possible future directions.

I will describe our recent work from 1709.07460, where we introduce a new renormalization group algorithm for tensor networks. The algorithm is based on a novel understanding of local correlations in a tensor network, and a simple method to remove such correlations from any network. It performs comparably with the best competing algorithms on 2D/(1+1)D systems, but is significantly simpler to implement, and easier to generalize to different lattices and graphs, including to higher dimensions. I will begin the talk by discussing renormalization group methods for tensor networks in general, then describe our algorithm and its advantages, show some benchmark results, and finally comment on the status of implementing real-space renormalization for 3D tensor networks.

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.

arXiv:1703.07782

In recent years there has been quite some effort to apply Matrix Product States (MPS) and more general Tensor Networks (TN) to lattice gauge theories. Contrary to the standard Euclidean-time Monte Carlo approach, which faces a major obstacle in the sign problem, numerical methods based on TN are free from the sign problem and allow to some extent simulating time evolution. Moreover, TN are also a suitable tool to explore proposals for potential future quantum simulators for lattice gauge theories.

In this talk I am going to present some examples where these possibilities allow novel insight into lattice gauge theories. After briefly introducing MPS, I will mainly focus on two models: The first part of the talk is going to be about the Schwinger model. I will show how MPS can help to explore proposals for potential future quantum simulators for this model by studying their spectral properties and simulating adiabatic preparation protocols for the interacting vacuum.

Furthermore, I will show an explicit example where TN allow to overcome the Monte Carlo sign problem in a lattice calculation by studying the zero-temperature phase structure for the two-flavor case at non-zero chemical potential with MPS.

In the second part, I am focusing on a non-Abelian gauge model, namely a 1+1 dimensional SU(2) lattice gauge theory. Using MPS, the phenomenon of string breaking in this theory can be studied in real time, thus allowing to gain new insight into this process. Moreover, I will show how the gauge field can be integrated out for systems with open boundary conditions and how to obtain a formulation which allows to address the model more efficiently with MPS.

In this talk I will give a short introduction into Projected Entangled-Pair States (PEPS), and their infinite variant iPEPS, a class of tensor network Ansatz targeted at the simulation of 2D strongly correlated systems. I will present work on two recent

projects: the first will be an application of the iPEPS algorithm to a Kitaev-Heisenberg model, a model which through-out recent years has received a lot of attention due to its potential connection to the physics of a subclass of the so-called Iridate compounds. The second will be work related to the development of the iPEPS method to specifically target cylindrical geometries. Here I will present some preliminary results where we apply the methods to the Heisenberg and Fermi-Hubbard models and evaluate their performance in comparison to infinite Matrix Product States. As a final part of my talk I will, depending on time, elaborate somewhat on potential future topics including (but not restricted to):  the main challenges of iPEPS simulations from a numerical perspective and what pre-steps we have experimented with to tackle these, the possibility of applying recent proposals for finite-temperature calculations within the PEPS framework to frustrated spin systems and the use of Tensor Network Renormalization for the study of RG flows.

In order to create ansatz wave functions for models that realize topological or symmetry protected topological phases, it is crucial to understand the entanglement properties of the ground state and how they can be incorporated into the structure of the wave function.

In this first part of this talk, I will discuss entanglement properties of models of topological crystalline insulators and spin liquids and show how to incorporate topological order, symmetry fractionalization, and lattice symmetry protected topological order into tensor network wave functions.

In the second part of this talk, I will discuss intrinsically fermionic topological phases and an exactly solvable model we built to elucidate the structure of the ground state wave functions in these phases.

References:

https://arxiv.org/abs/1507.00348

https://arxiv.org/abs/1605.06125

In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study local quantum many-body systems at low energies. 

In this talk, I will describe how a tensor network representation of a quantum many-body ground state also encodes, in a natural way, another quantum many-body state whose properties must be related to the ground state in a systematic way. One can apply this tensor network state correspondence to the multi-scale entanglement renormalization ansatz (MERA) representation of the ground state of a one dimensional (1D) quantum lattice system to obtain a quantum many-body state of a 2D hyperbolic quantum lattice, whose boundary is the original 1D lattice. I propose that this bulk/boundary correspondence could potentially be a candidate implementation of the holographic correspondence of String theory on a lattice using the MERA.

For the MERA representation of a critical ground state I will show how the critical properties can be obtained from the corresponding bulk state, in particular, illustrating how point-like boundary operators are identified with extended operators in the bulk. I will also describe the entanglement and correlations in the bulk state, present numerical results that illustrate that the bulk entanglement may depend on the boundary critical charge, and describe how the bulk state can be described in terms of "holographic screens". If the boundary state has a global symmetry, the corresponding bulk state has a local gauge symmetry (described by the same group). In fact, the bulk state decomposes in terms of spin networks as they appear in lattice gauge theory, where they describe the gauge-invariant sector of the theory (here, the bulk). This decomposition also reveals entanglement between gauge degrees of freedom in the bulk, which are dual to a global symmetry at the  boundary, and remaining bulk degrees of freedom which may potentially include gravitational degrees of freedom in a holographic interpretation of the MERA.

I discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. I review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. I describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the fluctuations of the boundary low-energy subspace. In this way we obtain, for every quantum system, an Einstein-like equation of motion for what might be interpreted as a bulk gravity theory dual to the boundary system. If time permits I will comment on the link to Brownian quantum circuits and tensor networks. 

Entanglement is fundamental to quantum mechanics. It is central to the EPR paradox and Bell’s inequality. Tensor network states constructed with explicit entanglement structures have provided powerful new insights into many body quantum mechanics. In contrast, the saddle points of conventional Feynman path integrals are not entangled, since they comprise a sequence of classical field configurations. The path integral gives a clear picture of the emergence of classical physics through the constructive interference between such sequences, and a compelling scheme for adding quantum corrections using diagrammatic expansions.  We combine these two powerful and complementary perspectives by constructing Feynman path integrals over sequences of matrix product states, such that the dominant paths support a degree of entanglement. We develop a general formalism for such path integrals and give a couple of simple examples to illustrate their utility [arXiv:1607.01778]. 

In the rst part of this talk I will give a brief introduction to the variational class of continuous
matrix product states (cMPS). Then I will present a time evolution algorithm for cMPS with periodic
boundary conditions. In the second part, I will explain how to apply this method to simulate
atomtronic circuits. In particular, I will show results for persistent currents in an interacting bose
gas rotating in a ring shaped trap in the presence of an arti cial U(1) gauge potential.

In this talk, I present a new framework for topologically ordered gapped ground states in 2+1 spacetime dimensions (which generalizes to higher dimensions) using tensor networks. We will see that topological order can exist in tensor network states (TNS), if the local tensor satisfies certain axioms which we call MPO (matrix product operator)-injectivity and pulling through. We then continue with examples, and see how renormalization fixed point models in the literature (Levin-Wen models, etc.) can be covered in this framework. If time permits, we finish with a discussion of excitations and the generalization to higher dimensions.

I will describe how to define a proper RG flow in the space of
tensor networks, with applications to the evaluation of classical
partition functions, euclidean path integrals, and overlaps of tensor
network states.