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When a large number of quantum mechanical particles are put together and allowed to interact, various condensed matter phases emerge with macroscopic quantum properties. While conventional quantum phases like superfluids or quantum magnets can be understood as a simple collection of
single particle quantum states, recent discoveries of fractional quantum Hall or spin liquids states contain intrinsic entanglement among all the particles. To understand such unconventional phases requires unconventional methods. In this talk, I will discuss how the quantum information insights about many-body entanglement gives us a unique perspective and a powerful tool to study these
unconventional phases. In particular, starting from simple entanglement building blocks, we are able to construct new gapped quantum phases, classify all possible gapped phases in certain cases and obtain a better understanding of the structure of the phase diagram. With these progress, we expect the many-body entanglement point of view to play an important role in our effort to map the full quantum phase diagram, leading to breakthroughs in our understanding of gapless phases and phase transitions and in the development of numerical tools to simulate such systems.

The decomposition of the magnetic moments in spin ice into freely moving magnetic monopoles has added a new dimension to the concept of fractionalization, showing that geometrical frustration, even in the absence of quantum fluctuations, can lead to the apparent reduction of fundamental objects into quasi particles of reduced dimension [1]. The resulting quasi-particles map onto a Coulomb gas in the grand canonical ensemble [2]. By varying the chemical potential one can drive the ground state from a vacuum to a monopole crystal with the Zinc blend structure [3].

The condensation of monopoles into the crystallized state leads to a new level of fractionalization:
the magnetic moments appear to collectively break into two distinct parts; the crystal of magnetic charge and a magnetic fluid showing correlations characteristic of an emergent Coulomb phase [4].

The ordered magnetic charge is synonymous with magnetic order, while the Coulomb phase space is equivalent to that of hard core dimers close packed onto a diamond lattice [5]. The relevance of these results to experimental systems will be discussed.

[1] C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature 451, 42 (2008).
[2] L. D. C. Jaubert and P. C. W. Holdsworth, Nature Physics 5, 258 (2009).
[3] M. Brooks-Bartlett, A. Harman-Clarke, S. Banks, L. D. C. Jaubert and P. C. W. Holdsworth, In Preparation, (2013).
[4] C. L. Henley, Annual Review of Condensed Matter Physics 1, 179 (2010).
[5] D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 91, 167004 (2003).

Many of the topological insulators, in their naturally
available form are not insulating in the bulk. It has been  shown that some of these metallic compounds,
become superconductor at low enough temperature and the nature of their
superconducting phase is still widely debated. In this talk I show that even
the s-wave superconducting phase of doped topological insulators, at low
doping, is different from ordinary s-wave superconductors and goes through a
topological phase transition to an ordinary s-wave state by increasing the
doping. I show that the critical doping is determined using the
SU(2) Berry phase on the fermi surface of doped
topological insulator and can be modified by different tunable features of the
material. At the end I present the results of a recent experiment on the
Josephson junctions made of thin films of Bismuth selenide , which can be
explained using our theory of doping induced phase transition in topological
insulators.

The existence of three generations of neutrinos and their mass mixing is a deep mystery of our universe. Majorana's elegant work on the real solution of Dirac equation predicted the existence of Majorana particles in our nature, unfortunately, these Majorana particles have never been observed. In this talk, I will begin with a simple 1D condensed matter model which realizes a T^2=-1time reversal symmetry protected superconductors and then discuss the physical property of its boundary Majorana zero modes. It is shown that these Majorana zero modes realize a T^4=-1 time reversal doubelets and carry 1/4 spin. Such a simple observation motivates us to revisit the CPT symmetry of those ghost particles--neutrinos by assuming that they are topological Majorana particles made by four
Majorana zero modes. Interestingly, we find that Majorana zero modes will realize a P^4=-1 parity symmetry as well. It can even realize a nontrivial C^4=-1 charge conjugation symmetry, which is a big surprise from a usual perspective that the charge conjugation symmetry for a Majorana particle is trivial. Indeed, such a C^4=-1 charge conjugation symmetry can be promoted to a Z_2 gauge symmetry and its spontaneously breaking leads to the origin of neutrino mass. We further attribute
the origin of three generations of neutrinos to three distinguishable ways of defining two complex fermions from four Majorana zero modes.
The above assumptions lead to a D2 symmetry in the generation space and uniquely determine the mass mixing matrix with no adjustable parameters! In the absence of CP violation, we derive
\theta_12=32degree, \theta_23=45degree and \theta_13=0degree, which is intrinsically closed to
the current experimental results. We further predict an exact mass ratio of the three mass eigenstate with m_1/m_3=m_2/m_3=3/\sqrt{5}.