Scientific Series

The entanglement spectrum, i.e. the logarithm of the eigenvalues of reduced density matrices of
quantum many body wave functions, has been the focus of a rapidly expanding research endeavor recently.
Initially introduced by Li & Haldane in the context of the fractional quantum Hall effect, its usefulness has been
shown to extend to many more fields, such as topological insulators, fractional Chern insulators, spin liquids,
continuous symmetry breaking states, etc.
After a general introduction to the field we review some of our own contributions to the field, in particular the
perturbative structure of the entanglement spectrum in gapped phases, the entanglement spectrum across the
Mott-insulator transition in the Bose-Hubbard model, and the relation of the entanglement spectrum of
(1+1) dimensional quantum critical systems to the operator content of their underlying CFT.

Recent research has suggested deep connections between geometry and entropy. This connection was first seen in black hole thermodynamics, but has been more fully realized in the Ryu-Takayanagi proposal for calculating entanglement entropies in AdS/CFT. We suggest that this connection is even broader: entropy, and in particular compression, are the fundamental building blocks of emergent geometry. We demonstrate how spatial geometry can be derived from the properties of a recursive compression algorithm for the boundary CFT. We propose a general algorithm for constructing MERA-like tensor networks and elucidate connections to the mathematical field of integral geometry.

After a brief motivation of this question, the presentation is divided in two parts. We first introduce the principle of quantum information causality, which states the maximum amount of quantum information that a transmitted quantum system can communicate as a function of its Hilbert space dimension, independently of any quantum physical resources previously shared by the communicating parties. The second part of the talk considers superdense coding within the framework of general probabilistic theories and addresses the question of why in quantum theory, no more than two bits can be communicated by transmission of a single, entangled, qubit. We introduce hyperdense coding in general probabilistic theories: superdense coding in which N

> 2 bits are communicated by transmission of a system  that locally

encodes at most one bit, and present protocols with N arbitrarily large. Our hyperdense coding protocols imply superadditive classical

capacities: two entangled systems can encode N > 2 bits, even though each system locally encodes at most one bit. Our protocols violate either a reversibility condition or tomographic locality.

This talk will cover a number of fun topics related to studying and understanding the Cosmic Microwave Background, including where it came from, how it is like an inside-out star, CMB numerology, how much information it contains and how it is evolving in time.

The tt* equations define a flat connection on the moduli spaces of 2d, N=2 quantum field theories. For conformal theories with c=3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. I will show that the non-holomorphic content of the tt* equations in the cases d=1,2,3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the anti-holomorphic derivatives of tt* and provides these with a finer and algebraic meaning. The generators of the differential rings of special functions are given by quasi-modular forms for d=1 and their generalizations in d=2,3. For d=3, this can be used to provide a purely Lie algebraic formulation of the higher genus topological string theory amplitudes and of the BCOV holomorphic anomaly equations which govern them.

We study a contracting universe composed of cold dark matter and radiation, and with a positive cosmological constant. Assuming that loop quantum cosmology captures the correct high-curvature dynamics of the space-time, we calculate the spectrum of scalar and tensor perturbations after the bounce, assuming initial quantum vacuum fluctuations. We find that the modes that exit the (sound) Hubble radius during matter-domination when the effective equation of state is slightly negative due to the cosmological constant will be nearly scale-invariant with a slight red tilt, in agreement with observations. The tensor perturbations are also nearly scale-invariant, and the predicted tensor-to-scalar ratio is small.  Finally, as this scenario predicts a positive running of the scalar index, it can be differentiated from inflationary models.

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations. The SPT phases in free fermion systems, like topological insulators, can be classified by K-theory. However, it is not yet known what SPT phases exist in general interacting systems. In this talk, I will first present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases. Just as group theory allows us to construct 230 crystal structures in three dimensions, we find that group cohomology theory allows us to construct many interacting bosonic SPT phases. In my talk, I shall show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians with explicit ground state wavefunctions. Next, I will discuss the generalization of the classifying scheme to interacting fermionic systems and a new mathematical framework – group supercohomology theory, which predicts a fermionic SPT phase that can neither be realized in free fermionic nor interacting bosonic systems.

Finally, I will briefly mention the deep relationship between SPT phases and chiral anomalies in high energy physics.

We consider the one dimensional, periodic spin chain with $N$ sites, similar to the one studied by Haldane \cite{hal}, however in the opposite limit of very large anisotropy and small nearest neighbour, anti-ferromagnetic exchange coupling between the spins, which are of large magnitude $s$.  For a  chain with an even number of sites we show that actually the ground state is non degenerate and given by a superposition of the two Néel states, due to quantum spin tunnelling.  With an odd number of sites, the Néel state must necessarily contain a soliton. The position of the soliton is arbitrary thus the  ground state is $N$-fold degenerate.  This set of  states reorganizes into a band. We show that this occurs at order $2s$ in perturbation theory.  The ground state is non-degenerate for integer spin, but degenerate for half-odd integer spin as is required by Kramer's theorem \cite{kram}. arXiv:1404.6706 , Phys.Lett. A378 (2014) 3066-3069; arXiv:1304.3734 Phys.Rev. B88 (2013) 22, 220403 

In this talk, I will prove the Landau-Ginzburg mirror symmetry conjecture for general quasi-homogenous singularities, i.e., the FJRW theory (LG A-model) of such polynomials is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial. This is joint work with Weiqiang He, Rachel Webb and Yefeng Shen. 

If dark matter is asymmetric, fermionic, and self-interacting, it may form black holes in pulsars at the galactic center. In this case, a measurable maximum attainable pulsar age would track the density of the dark matter halo, with the oldest pulsars being allowed in the least dense parts of the halo. This could explain a recent observation, that there are not as many pulsars in the galactic center as expected.