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Axiverse Cosmology and the Energy Scale of Inflation
David Marsh King's College London
Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
Erez Berg Weizmann Institute of Science
A 3d Boson Topological Insulator and the “Statistical Witten Effect”
Matthew Fisher University of California, Santa Barbara
Asymmetry protected emergent E8 symmetry
Brian Swingle Brandeis University
Protected edge modes without symmetry
Michael Levin University of Chicago
Emergent Fermionic Strings in Bosonic He4 Crystal
Baskaran Ganapathy Institute of Mathematical Sciences
Quantum spin liquid phases in the absence of spin-rotation symmetry
Yong-Baek Kim University of Toronto
TQFTs and Topological Phases of Matter
Zhenghan Wang Microsoft Corporation
3d boson topological insulators and quantum spin liquids
Senthil Todadri Massachusetts Institute of Technology (MIT) - Department of Physics
Majorana Ghosts: From topological superconductor to the origin of neutrino mass, three generations and their mass mixing
Zheng-Cheng Gu Chinese University of Hong Kong
The existence of three generations of neutrinos and their mass mixing is a deep mystery of our universe. On the other hand, 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=-1 time 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 Majorana zero modes. Interestingly, we find that a topological Majorana particle will realize a P^4=-1 parity symmetry as well. It even realizes 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 is 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 types of topological Majorana zero modes protected by CPT symmetry. Such an assumption leads to an S3 symmetry in the generation space and uniquely determines 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}.Axiverse Cosmology and the Energy Scale of Inflation
David Marsh King's College London
Ultra-light axions (m_a<10^{-18} eV), motivated by string theory, can be a powerful probe of the energy scale of inflation if they exist as a sub-dominant component of the Dark Matter. In contrast to heavier axions the isocurvature modes in the ultra-light axions can coexist with observable gravitational waves. Here it is shown that existing (2005) large scale structure constraints severely limit the parameter space for axion mass, density fraction and isocurvature amplitude. It is also shown that radically different CMB observables for the ultra-light axion isocurvature mode additionally reduce this space. The results of a new, accurate and efficient method to calculate this isocurvature power spectrum are presented, and can be used to constrain ultra-light axions and inflation.I will also present preliminary results of constraints to this model using up-to-date cosmological observations, which verify the above picture. The parameter space is interesting to explore due to a strongly mass dependent covariance matrix, motivating comparisons between Metropolis-Hastings and nested sampling. Finally I discuss fine-tuning and naturalness in these models.Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
Erez Berg Weizmann Institute of Science
We study the non-abelian statistics characterizing systems where counter-propagating gapless modes on the edges of fractional quantum Hall states are gapped by proximity-coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of $\nu= 1/m$, while electrons of the opposite spin occupy a similar state with $\nu = -1/m$. However, we also propose other examples of such systems, which are easier to realize experimentally. We find that each interface between a region on the edge coupled to a superconductor and a region coupled to a ferromagnet corresponds to a non-abelian anyon of quantum dimension $\sqrt{2m}$. We calculate the unitary transformations that are associated with braiding of these anyons, and show that they are able to realize a richer set of non-abelian representations of the braid group than the set realized by non-abelian anyons based on Majorana fermions. We carry out this calculation both explicitly and by applying general considerations. Finally, we show that topological manipulations with these anyons cannot realize universal quantum computation.A 3d Boson Topological Insulator and the “Statistical Witten Effect”
Matthew Fisher University of California, Santa Barbara
Electron topological insulators are members of a broad class of “symmetry protected topological” (SPT) phases of fermions and bosons which possess distinctive surface behavior protected by bulk symmetries. For 1d and 2d SPT’s the surfaces are either gapless or symmetry broken, while in 3d, gapped symmetry-respecting surfaces with (intrinsic) 2d topological order are also possible. The electromagnetic response of (some) SPT’s can provide an important characterization, as illustrated by the Witten effect in 3d electron topological insulators. Using a 3d parton-gauge theory construction, we have recently developed a dyon condensation approach to access exotic new phases including some 3d bosonic SPT’s. A bosonic SPT with both time-reversal and charge conservation symmetries, is thereby obtained, a phase which supports a gapped, symmetry-unbroken 2d surface with topological order - a toric code with charge one-half anyons. The 3d electromagnetic response of this bosonic SPT phase is quite remarkable - an external magnetic monopole can remain charge neutral, but is statistically transmuted becoming a fermion - a “statistical Witten effect” that characterizes the phase.Asymmetry protected emergent E8 symmetry
Brian Swingle Brandeis University
The E8 state of bosons is a 2+1d gapped phase of matter which has no topological entanglement entropy but has protected chiral edge states in the absence of any symmetry. This peculiar state is interesting in part because it sits at the boundary between short- and long-range entangled phases of matter. When the system is translation invariant and for special choices of parameters, the edge states form the chiral half of a 1+1d conformal field theory - an E8 level 1 Wess-Zumino-Witten model. However, in general the velocities of different edge channels are different and the system does not have conformal symmetry. We show that by considering the most general microscopic Hamiltonian, in particular by relaxing the constraint of translation invariance and adding disorder, conformal symmetry remerges in the low energy limit. The disordered fixed point has all velocities equal and is the E8 level 1 WZW model. Hence a highly entangled and highly symmetric system emerges, but only when the microscopic Hamiltonian is completely asymmetric.Protected edge modes without symmetry
Michael Levin University of Chicago
Some 2D quantum many-body systems with a bulk energy gap support gapless edge modes which are extremely robust. These modes cannot be gapped out or localized by general classes of interactions or disorder at the edge: they are "protected" by the structure of the bulk phase. Examples of this phenomena include quantum Hall states and 2D topological insulators, among others. Recently, much progress has been made in understanding protected edge modes in non-interacting fermion systems. However, less is known about the interacting case. A basic problem is to predict, for general interacting systems, when such edge modes are present or absent, and to identify the different physical mechanisms that underlie their stability. In this talk, I will discuss this problem in the simplest case: interacting fermion systems without any symmetry.Emergent Fermionic Strings in Bosonic He4 Crystal
Baskaran Ganapathy Institute of Mathematical Sciences
Large zero point motion of light atoms in solid Helium 4 leads to several anomalous properties, including a supersolid type behavior. We suggest an `anisotropic quantum melted' atom density wave model for solid He4 with hcp symmetry. Here, atoms preferentially quantum melt along the c-axis and maintain self organized crystallinity and confined dynamics along ab-plane. This leads to profound consequences: i) statistics transmutation of He4 atoms into fermions for c-axis dynamics, arising from restricted one dimensional motion and hard core repulsion, ii) resulting `fermionic strings' undergo Peierls instability (an atom density wave formation) in a staggered fashion and help regain the original hcp crystal symmetry, iii) `particle-hole' type excitations iv) emergence of `confined' `half atom' domain wall excitations, and so on. Known anomalies of solid He4 gets a natural qualitative explanation in the present scenario.Topological Order with a Twist: Ising Anyons from an Abelian Model
Hector Bombin PsiQuantum Corp.
Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons.Quantum spin liquid phases in the absence of spin-rotation symmetry
Yong-Baek Kim University of Toronto
We investigate possible quantum spin liquid phases in the presence of a variety of spin-rotational-symmetry breaking perturbations. Projective symmetry group analysis on slave-particle representations is used to understand possible spin liquid phases on the Kagome lattice. The results of this analysis are used to make connections to the exiting and future experiments on Herbertsmithites. Applications to other systems are also discussed.TQFTs and Topological Phases of Matter
Zhenghan Wang Microsoft Corporation
In two spatial dimensions, there is a good correspondence between TQFTs and topological phases of matter for spin systems. I will discuss this correspondence in one and three spatial dimensions for spin systems. If time permits, I will also discuss the situation for fermion systems.3d boson topological insulators and quantum spin liquids
Senthil Todadri Massachusetts Institute of Technology (MIT) - Department of Physics
I will discuss recent work on 3d Symmetry Protected Topological (SPT) phases of bosonic systems, and their implications for understanding the more exotic quantum spin liquid phases. First I will describe various characterizations of these 3d SPT phases, in particular their surface effective theories and (when applicable) bulk electromagnetic response. Next I will show how this understanding leads to several new insights into the theory of both 2d and 3d quantum spin liquids. Finally I will provide an explicit construction of several 3d SPT phases in a system of `coupled layers'. This includes a 3d SPT state that is beyond the existing cohomology classification of such states.From Pauli's Principle to Fermionic Entanglement
Matthias Christandl ETH Zurich
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected for decades, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Surprisingly, these constraints are linear: they cut out a geometric object known as a polytope. This is a beautiful mathematical result, but are there systems whose physics is governed by these constraints? In order to address this question, we studied a system of a few fermions connected by springs. As we varied the spring constant, the occupation numbers moved within the polytope. The path they traced hugs very close to the boundary of the polytope, suggesting that the generalized constraints affect the system. I will mention the implications of these findings for the structure of few-fermion ground states and then discuss the relation between the geometry of the polytope and different types of fermionic entanglement.