- Programmes

**Brief description:**

When talking about different phases in physics, the first thing which comes to mind is the division in solid, liquid and gas, where temperature is the varying parameter which connects them through phase transition points. At zero (or close to zero) temperature, where quantum mechanics is the physical law which governs the system, there are also different phases interconnected via phase transitions. Now what usually varies is some parameter or parameters in the model under study. The exotic and unexpected properties of some of these quantum phases, like superconductivity, superfluidity, fractional statistics, topological dependency, etc. have attracted the attention of condensed matter physicists for many years.

Many numerical methods to simulate quantum systems have been proposed and developed during the last half-century. One of the most successful ones, though only for 1D systems, has been the Density Matrix Renormalization Group (DMRG) algorithm. Despite being used with extreme success for many years, the theoretical proof of the extreme accuracy of DMRG had to wait until the very recent work by Hastings. He proves that the variational family over which DMRG optimizes the energy, and which is known as Matrix Product States (MPS) approximates efficiently all ground states of local gapped 1D Hamiltonians. These results, together with many improvements on the DMRG behaviour came as a consequence of the understanding of DMRG in the language of quantum information, that is, analyzing the entanglement properties of the system.

The major consequence of the introduction of quantum information ideas in condensed matter physics has been the appearance of variational families in 2D (and higher spatial dimensions) which seem to approximate well the ground state of all local quantum Hamiltonians and hence encode the physics of all quantum spin systems. All these families (MPS, PEPS, MERA, etc) form the set of Tensor Network States (TNS). Though TNS were originally introduced to design better numerical algorithms to solve local quantum Hamiltonians, the explicit role played by entanglement in their definition has converted them into a powerful tool for analyzing the role of entanglement, and in general the physics, in 2D quantum many body systems.

**Aims of the program:**

1. Give useful characterizations of interesting quantum properties of TNS. Use this knowledge to give a full classification of all possible quantum phases. Collect and analyze all new interesting and exotic quantum properties of matter which emerge in the process.

2. Working directly with states, gives some freedom in choosing an associate local Hamiltonian for the given PEPS (Projected Entangled Pair States). One aim is to explore all the possibilities of such a choice, with particular emphasis in the spectral properties and, in particular in the presence or absence of an energy gap between the ground state and the first excited state in the thermodynamic limit.

3. Giving new theorems showing the appropriateness of the class of PEPS, which would allow to lift any result obtained in the context of PEPS to the general case.

4. One of the main difficulties one encounters a priori when trying to classify all quantum phases in 2D is the necessity of dealing with exotic quantum states, such as topologically ordered ones (i.e. Kitaev’s toric code) or spin liquids (i.e. the RVB state). In order to isolate the crucial properties which make them belong to one or a different phase, one needs to analyze them in detail. This analysis can have the extra benefit of shedding new light to the mechanisms giving rise to such phases.

5. Since PEPS are exactly the set of quantum states which appear as ground state of local Hamiltonians, asking about generic properties of PEPS is similar as asking about generic properties of quantum systems. In this way, one can understand certain laws and principles from a purely probabilistic point of view. That is, these principles hold because the states satisfying them are generic, that is, they are all states but an exponentially small family. For that we will have to push forward the use of random matrix theory in the context of PEPS.

**Website: ** www.mathqi.es

**Researchers:** **David Pérez García (Associate Professor, Fac. ****Math. UCM)**

Ignacio Villanueva (Associate Professor, Fac. Math. UCM)

Miguel Tierz (Post-doc Juan de la Cierva, Fac. Math. UCM)

Thomas Cooney(Post-doc, LSU, EEUU)

Toby S. Cubitt (Royal Society Fellow, Cambridge)

Carlos Hugo Jiménez (Post-doc, UCM)

David Elkouss (Post-doc, UCM)

Carlos Palazuelos (Ramón y Cajal researcher, UCM).

Carlos González Guillén (Assistant teacher, UPM)

Carlos Fernández González (Assistant teacher, UNED)

Angelo Lucia (PhD student., UCM)

Andrea Cadarso (PhD student, UCM-CSIC)

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