We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model HA in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of HA on a lattice of lattice spacing a to the tensor product of ground spaces of two independent Hamiltonians HA and HB on lattices of lattice spacing 2a. We further find a disentangling unitary for the ground space of HB with the lattice spacing a to show that it decomposes into two copies of itself on the lattice of the lattice spacing 2a. The disentangling transformations yield a tensor network description, a branching MERA, for the ground state of the cubic code model.

### Monday, April 21st, 2014

### Tuesday, April 22nd, 2014

We shall present the construction of Tensor Networks using the tools of Conformal Field Theory (CFT).

The basic idea of this approach is to replace the finite dimensional tensors, used to describe the low energy states of local lattice Hamiltonians, by the primary fields of CFT. In doing so the ancilla space, that supports the entanglement, becomes infinite dimensional and is given by the representation spaces of the underlying CFT.

The construction shares strong similarities with the use of CFT in the study of the Fractional Quantum Hall wave functions. This fact allows the construction of lattice versions of the Laughlin states, the Moore-Read states, etc.

We shall give an overall account of this approach and discuss future developments.

Holographic duality is a duality between quantum many-body systems (boundary) and gravity systems with one additional spatial dimension (bulk). In this talk, I will describe a new approach to holographic duality for lattice systems, called the exact holographic mapping. The key idea of this approach can be summarized by two points: 1) The bulk theory is nothing but the boundary theory viewed in a different basis. 2) Space-time geometry is determined by the structure of correlations and quantum entanglement in a quantum state. For free fermion boundary theories, I will show how different bulk geometries including AdS space, black holes and worm-holes emerge. I will also discuss the generalization of this approach in more generic interacting systems.

Reference: Xiao-Liang Qi, Exact holographic mapping and emergent space-time geometry, arxiv:1309.6282 (2013)

It is well know that unitary symmetries can be 'gauged', i.e. defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an anti-unitary symmetry. Here we discuss a route to gauging time reversal symmetry that applies to gapped quantum ground states. We show how time reversal can be applied locally and also describe time reversal symmetry twists which act as gauge fluxes through nontrivial loops in the system. The procedure is based on the tensor network representation of quantum states which provides a notion of locality for the wave function coefficient. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain symmetry protected topological phases in $D=1,2$ are readily extracted using these ideas and also discuss how they help capture a subtle distinction between time reversal symmetric $Z_2$ gauge theories.

### Wednesday, April 23rd, 2014

I argue that, in the presence of a complete set of local integrals of motion, the whole spectrum of eigenvectors of a local Hamiltonian can be expressed efficiently in terms of a spectral tensor network. In D=1 dimensions, one can then use efficient tensor network manipulation techniques to compute a number of properties from the spectral MPS. These include the expectation value of any local observable for any energy eigenstate and (through perfect sampling) the estimation of time evolution, for arbitrarily long times, of any state that can be initially represented efficiently as a matrix product state.

We will analyze in this talk how to characterize in the local tensor of a PEPS the fact that it is a fixed point of a renormalization group (RG) flow. We will discuss why this could be an important step in the classication of quantum phases in 2D.

It is well known that many RG-fixed point states with topological order such as the quantum double models of Kitaev and the string net models of Levin and Wen can exactly be represented as PEPS. In this talk we will discuss how the topologically ordered properties of the state manifest themselves in the PEPS representation, even for states that are not exactly at the RG-fixed point. For the case of the quantum doubles, it is well known that this gives rise to the notion of G-injectivity, as was first introduced by Schuch et al. In this case, the PEPS tensors are invariant under the action of the discrete group G at the virtual level. This construction was recently generalised to models where the symmetry group is twisted by a 3-cocylce by Buerschaper. We further extend this formalism by showing how to suitably redefining the notion of injectivity when the invariance of the PEPS tensors is encoded in general MPOs, and we illustrate that the general Levin-Wen string nets are exactly of this form.

I will discuss recent exact solutions for non-equilibrium steady state density operators of several boundary driven quantum chains, namely XXZ spin 1/2, Fermi-Hubbard, and the Lai-Sutherland spin-1 chains, with the aim of establishing a unifying framework. The infinite bond-dimension matrix product operator for the steady states in all cases can be neatly encoded in terms of operator sums of walks over particular infinite graphs. In some cases, (local) Lax structure can be identified, corresponding to the integrability of the problem.

I will describe joint work with Zeph Landau and Umesh Vazirani (arXiv:1307.5143) in which we give a provably polynomial-time classical randomized algorithm for finding ground states of gapped local Hamiltonians in 1D. The algorithm is based on several ingredients, including the notions of boundary contraction and decoupling for matrix product states and a construction of approximate ground state projection (AGSP) based on random sampling. I will give a self-contained presentation of the algorithm that emphasizes these tools and their potential usefulness in the study of matrix product state representations.

### Thursday, April 24th, 2014

The question whether Anderson insulators can persist to finite-strength interactions–a scenario dubbed many-body localization–has recently received a great deal of interest. In this talk, I will discuss our recent work on defining such a many-body localized phase and exploring it through its entanglement properties. We formulate a precise sense in which a many-body localized system can be connected adiabatically to an Anderson insulator. This connection turns out to rely on many of the same concepts as tensor network states. The most striking consequence of our definition is an area law for the entanglement entropy of highly excited states in such a system. We present the results of numerical calculations for a one-dimensional system of spinless fermions, which are consistent with an area law and, by implication, many-body localization for weak enough interactions and strong disorder. Furthermore, we discuss the implications that many-body localization may have for topological phases and self-correcting quantum memories. We find that there are scenarios in which many-body localization can help to stabilize topological order at non-zero energy density, and we propose potentially useful criteria to confirm these scenarios.

Gapped Z_2 spin liquids have been proposed as candidates for the ground-state of the S=1/2 quantum antiferromagnet on frustrated lattices (like the Kagome lattice). We use Projected Entangled Pair States (PEPS) to construct (on the cylinder) Resonating Valence Bond (RVB) states. By considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB Minimally Entangled States. The spinon and vison coherence lengths are then extracted from a finite size scaling w.r.t the cylinder perimeter of the energy splittings of the four sectors.

A large enough magnetic field can generically induce "doping" of polarized S=1/2 spinons. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed spinons (RVB) superfluids with transverse staggered (Néel) magnetic order. On the Kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e. spinon bound pairs) yields uncondensed Bose liquids preserving U(1) spin-rotation symmetry.

Lastly, considering fermoionic PEPS, we construct doped fermionic RVB superconductors and investigate their properties and relevance to frustrated t-J models.

The entanglement spectra and hamiltonians of the corresponding PEPS on a partitioned (infinite) cylinder are also discussed.

The algebraic Bethe Ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard in general to extract properties from the states, like, for example, correlation functions. In our work, we apply the tools of Tensor Network States to describe the eigenstates approximately as Matrix Product States. From the Matrix Product State expression, we then obtain observables like correlation functions directly.

A topological phase is a phase of matter which cannot be characterized by a local order parameter. We introduce non-local order parameters that can detect symmetry protected topological (SPT) phases in 1D systems and then show how to generalize the idea to detect certain symmetry enriched topological (SET) phases in 2D. The proposed approaches are particularly useful if the quantum states are represented in terms of tensor product states. As concrete examples, we examine two model states which exhibit SET order: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond state on a kagome lattice.