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)

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.

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.

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.

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.