**Leon Balents**, Kavli Institute for Theoretical Physics

*Quantum Spin Liquids, Density Matrix Renormalization Group, and Entanglement *

I will review recent work in our group using Density Matrix Renormalization Group (DMRG) to search for and study quantum spin liquid and topologically ordered states in two dimensional model Hamiltonians. This proves an efficient way to study these phases in semi-realistic situations. I will try to draw lessons from several studies and theoretical considerations.

**Maissam Barkeshli**, Stanford University

*Defects in Topologically Ordered Quantum Matter*

I will discuss recent advances in our understanding of extrinsic defects in topologically ordered states. These include line defects, where I will discuss recent developments in the classification of gapped boundaries between Abelian topological states, and various kinds of point defects, which host a rich set of topological physics. The extrinsic point defects provide a new way of realizing topologically protected ground state degeneracies, they carry projective non-abelian statistics even in an Abelian topological state and provide a new path towards universal topological quantum computation, they host a general class of topologically protected "parafermion" zero modes, and they provide an avenue towards distinguishing various symmetry-enriched topological phases. I will discuss several novel physical realizations of such point defects, and also a recent experimental proposal to realize such defects in conventional bilayer fractional quantum Hall systems.

**Ganapathy Baskaran**, Institute of Mathematical Sciences

*Emergent Fermionic Strings in Bosonic He4 Crystal*

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.

**Erez Berg**, Harvard University

*Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states*

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.

**Hector Bombin**, Perimeter Institute

*Topological Order with a Twist: Ising Anyons from an Abelian Model*

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.

**Fiona Burnell**, University of Oxford

*3D topological lattice models with topologically ordered surface states*

I will discuss a family of solvable 3D lattice models that have a ``trivial" bulk, in which all excitations are confined, but exhibit topologically ordered surface states. I will discuss perturbations to these models that can drive a phase transition in which some of these excitations become deconfined, driving the system into a phase with bulk topological order.

**Xie Chen**, Massachusetts Institute of Technology

*Impossible symmetry enriched topological phases in 2D and their realization on 3D surface*

In quantum systems with symmetry, the same topological phase can be enriched by symmetry in different ways, resulting in different symmetry transformations of the superselection sectors in the phase. However, not all symmetry transformations are allowed on the superselection sectors in topological phases in purely 2D systems. In this talk, I will discuss some examples of such symmetry enrichment of topological phases, which seem to be consistent with the fusion and braiding rules of the superselection sectors in the theory but are nonetheless impossible to realize in 2D. Interestingly, we show further that they can be realized on the surface of a 3D gapped system with a topologically trivial bulk.

**Lukasz Cincio**, Perimeter Institute

*Characterizing topological order form a microscopic lattice Hamiltonian*

In this talk I will show how to obtain a detailed characterization of the emergent topological order starting from microscopic Hamiltonian on a two dimensional lattice. A key step is to obtain a tensor network representation for a complete set of ground states of the Hamiltonian, ﬁrst on an inﬁnite cylinder and then on a ﬁnite torus. As an application of the method I will study lattice Hamiltonians that give rise to selected anyon models, namely chiral semion, Ising as well as Z_3 and Z_5 models.

**Philippe Corboz**, ETH Zuriich

*Spin-orbital quantum liquid on the honeycomb lattice*

The symmetric Kugel-Khomskii can be seen as a minimal model describing the interactions between spin and orbital degrees of freedom in certain transition-metal oxides with orbital degeneracy, and it is equivalent to the SU(4) Heisenberg model of four-color fermionic atoms. We present simulation results for this model on various two-dimensional lattices obtained with infinite projected-entangled pair states (iPEPS), an efficient variational tensor-network ansatz for two dimensional wave functions in the thermodynamic limit. We find a rich variety of exotic phases: while on the square and checkerboard lattices the ground state exhibits dimer-N\'eel order and plaquette order, respectively, quantum fluctuations on the honeycomb lattice destroy any order, giving rise to a spin-orbital liquid. Our results are supported from flavor-wave theory and exact diagonalization. Furthermore, the properties of the spin-orbital liquid state on the honeycomb lattice are accurately accounted for by a projected variational wave-function based on the pi-flux state of fermions on the honeycomb lattice at 1/4-filling. In that state, correlations are algebraic because of the presence of a Dirac point at the Fermi level, suggesting that the ground state is an algebraic spin-orbital liquid. This model provides a possible starting point to understand the recently discovered spin-orbital liquid behavior of Ba_3CuSb_2O_9. The present results also suggest to choose optical lattices with honeycomb geometry in the search for quantum liquids in ultra-cold four-color fermionic atoms.

**Glen Evenbly**, California Institute of Technology

*Directed influence in the RG Flow*

Given two lattice Hamiltonians H_1 and H_2 that are identical everywhere except on a local region R of the lattice, we propose a relationship between their ground states psi_1 and psi_2. Specifically, assuming the states can be represented as multi-scale entanglement renormalization ansatz (MERA), we propose a principle of directed influence which asserts that the tensors in the MERA’s that represent the ground states can be chosen to be identical everywhere except within a specific, localized region of the tensor network. The validity of this principle is justified by demonstrating it to follow from Wilson's renormalization ideas towards systems with manifestly separated energy scales. This result is shown, through numerical examples, to have practical applications towards the efficient simulation of systems with impurities, boundaries and interfaces, and also argued to provide useful insights towards holographic representations of quantum states.

**Matthew Fisher**, Kavli Institute for Theoretical Physics

*A 3d Boson Topological insulator and the “Statistical Witten effect*

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.

**Tarun Grover**, Kavli Institute for Theoretical Physics

*Highly Entangled Quantum Matter*

Strong correlations can lead to new phases of quantum matter with striking features, such as emergent fermions and photons in a system composed only of bosons, or even excitations that are neither bosons nor fermions ("anyons"). In this talk, I will illustrate the unique view provided by many-body quantum entanglement on such intriguing phases of matter. In particular, I will show that the quantum entanglement can be used to extract the universal properties associated with anyons, including their braiding statistics. I will also explain how one may exploit recently discovered constraints on the renormalization group flows, such as entanglement monotonicity, to determine the stability of phases which are described by strongly interacting gauge theories.

**Zhengcheng Gu**, California Institute of Technology

*Majorana Ghosts: From topological superconductor to the origin of neutrino mass, three generations and their mass mixing*

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}.

**Duncan Haldane**, Princeton University

*Geometry and the entanglement spectrum in the fractional quantum Hall effect.*

Fractional quantum hall states with nu = p/q have a characteristic geometry defined by the electric quadrupole moment of the neutral composite boson that is formed by "flux attachment" of q "flux quanta" (guiding-center orbitals) to p charged particles. This characterizes the "Hall viscosity". For FQHE states described by a conformal field theory with a Euclidean metric g_ab, the quadrupole moment is proportional to the "guiding-center spin" of the composite boson and the inverse metric. The geometry gives rise to dipole moments at external edges or internal "orbital entanglement cuts", and can be seen in the entanglement spectrum.

**Yong Baek Kim**, University of Toronto

*Quantum spin liquid phases in the absence of spin-rotation symmetry*

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.

**Sung-Sik Lee**, McMaster University

*Quantum renormalization group and AdS/CFT*

In this talk, I will discuss about the notion of quantum renormalization group, and explain how (D+1)-dimensional gravitational theories naturally emerge as dual descriptions for D-dimensional quantum field theories. It will be argued that the dynamical gravitational field in the bulk encodes the entanglement between low energy modes and high energy modes of the corresponding quantum field theory.

**Michael Levin**, University of Maryland

*Protected edge modes without symmetry*

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.

**Hong Liu**, Massachusetts Institute of Technology

*Propagation of entanglement in strongly coupled systems from gravity*

**John McGreevy**, Massachusetts Institute of Technology

*A gauge theory generalization of the fermion-doubling theorem*

This talk is about obstructions to symmetry-preserving regulators of quantum field theories in 3+1 dimensions. New examples of such obstructions can be found using the fact that 4+1-dimensional SPT states are characterized by their edge states.
(Based on work in progress with S.M. Kravec.)

**Roger Melko**, University of Waterloo

*Entanglement at strongly-interacting quantum critical points in 2+1D*

In two or more spatial dimensions, leading-order contributions to the scaling of entanglement entropy typically follow the "area" or boundary law. Although this leading-order scaling is non-universal, at a quantum critical point (QCP), the sub-leading behavior does contain universal physics. Different universal functions can be access through entangling regions of different geometries. For example, for polygonal shaped regions, quantum field theories have demonstrated that the subleading scaling is logarithmic, with a universal coefficient dependent on the number of vertices in the polygon. Although such universal quantities are routinely studied in non-interacting field theories, it often requires numerical simulation to access them in interacting theories. In this talk, I discuss quantum Monte Carlo (QMC) and numerical Linked-Cluster Expansion (NLCE) calculations of the Renyi entropies at the transverse-field Ising model QCP on the 2D square lattice. We calculate the universal coefficient of the vertex-induced logarithmic scaling term, and compare to non-interacting field theory calculations by Casini and Huerta. Also, we examine the shape dependence of the Renyi entropy for finite-size toroidal lattices with smooth boundaries. Such geometries provide a sensitive probe of the gapless wave function in the thermodynamic limit, and give new universal quantities that could be examined by future field-theoretical studies in 2+1D.

**Xiaoliang Qi**, Stanford University

*Momentum polarization: an entanglement measure of topological spin and chiral central charge*

Topologically ordered states are quantum states of matter with topological ground state degeneracy and quasi-particles carrying fractional quantum numbers and fractional statistics. The topological spin is an important property of a topological quasi-particle, which is the Berry phase obtained in the adiabatic self-rotation of the quasi-particle by . For chiral topological states with robust chiral edge states, another fundamental topological property is the edge state chiral central charge . In this paper we propose a new approach to compute the topological spin and chiral central charge in lattice models by defining a new quantity named as the momentum polarization. Momentum polarization is defined on the cylinder geometry as a universal subleading term in the average value of a "partial translation operator". We show that the momentum polarization is a quantum entanglement property which can be computed from the reduced density matrix, and our analytic derivation based on edge conformal field theory shows that the momentum polarization measures the combination of topological spin and central charge. Numerical results are obtained for two example systems, the non-Abelian phase of the honeycomb lattice Kitaev model, and the Laughlin state of a fractional Chern insulator described by a variational Monte Carlo wavefunction. The numerical results verifies the analytic formula with high accuracy, and further suggests that this result remains robust even when the edge states cannot be described by a conformal field theory. Our result provides a new efficient approach to characterize and identify topological states of matter from finite size numerics.

**Subir Sachdev**, Harvard University

*Entangled states of quantum matter*

Theorists have been studying and classifying entanglement in many-particle quantum states for many years. In the past few years, experiments on such states have finally appeared, generating much excitement. I will describe experimental observations on magnetic insulators, ultracold atoms, and high temperature superconductors, and their invigorating influence on our theoretical understanding.

**Norbert Schuch**, Aachen University

*Characterizing topological spin liquids using PEPS*

Projected Entangled Pair States (PEPS) provide a local description of correlated many-body states. I will discuss how PEPS can be used to characterize topological spin liquids, in particular Resonating Valence Bond states. On the one hand, I will show how the symmetries in the local PEPS description allow to identify that these states appear as topologically degenerate ground states of local Hamiltonians. On the other hand, I will discuss how from exact diagonalization of the transfer operator one can extract both the topological order and the spin liquid nature of the ground state.

**Brian Swingle**, Harvard University

*Asymmetry protected emergent E8 symmetry*

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.

**Tadashi Takayanagi**, Kyoto University

*Thermodynamical Property of Entanglement Entropy for Excited States*

We will point out that there is a universal thermodynamical property of entanglement entropy for excited states. We will derive this by using the AdS/CFT correspondence in any dimension. We will also directly confirm this property from direct field theoretic calculations in two dimensions. We will define a new quantity called entanglement density by taking derivatives of entanglement entropy with respect to the shape of subsystem. We will show that this quantity coincides with the energy density by taking the small subsystem limit and show that this is another equivalent statement of the thermodynamical property.

**Senthil Todadri**, Massachusetts Institute of Technology

*3d boson topological insulators and quantum spin liquids*

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.

**Frank Verstraete**, University of Vienna

*Emergence and Entanglement in Matrix Product States*

**Zhenghan Wang**, Microsoft

*TQFTs and Topological Phases of Matter*

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.

**Steve White**, University of California, Irvine

*Searching for Spin Liquids*

**William Witczak- Krempa**, Perimeter Institute

*Holographic insights into quantum critical transport: from branes to Bose-Hubbard*

We discuss the general features of charge transport of quantum critical points described by CFTs in 2+1D. Our main tool is the AdS/CFT correspondence, but we will make connections to standard field theory results and to recent quantum Monte Carlo data. We emphasize the importance of poles and zeros of the response functions. In the holographic setting, these are the discrete quasinormal modes of a black hole/brane; they map to the excitations of the CFT. We further describe the role of particle-vortex or S-duality on the conductivity, which is argued to obey two powerful sum rules.
References (with S. Sachdev): arXiv:1210.4166 (PRB 12); arXiv:1302.0847 (PRB 13)

**Cenke Xu**, University of California, Santa Barbara

*Field theory, Wave function, and Defects of Symmetry Protected Topological Phases*

**Peng Ye**, Perimeter Institute

*3D bosonic topological insulator and its exotic electromagnetic response*

Recently, many new types of bosonic symmetry-protected topological phases, including bosonic topological insulators, were predicted using group cohomology theory. The bosonic topological insulators have both U(1) symmetry (particle number conservation) and time-reversal symmetry, described by symmetry group $U(1)\rtimes Z_2^T$. In this paper, we propose a projective construction of three-dimensional correlated gapped bosonic state with $U(1)\rtimes Z_2^T$ symmetry. The gapped bosonic insulator is formed by eight kinds of charge-1 bosons. We show that, in our bosonic state, an {\it electromagnetic} monopole with a unit magnetic charge is fermionic while an {\it electromagnetic} dyon with a unit magnetic charge and a unit electric charge is bosonic. This indicates that the constructed bosonic state is a non-trivial bosonic topological insulator, since in a trivial bosonic Mott insulator, the monopole is bosonic while the dyon is fermionic. We also constructed a three-dimensional correlated gapless bosonic insulator with $U(1)\rtimes Z_2^T$ symmetry, that has two emergent gapless $U(1)$ gauge fields, and excitations with fractional gauge charges for both the emergent and electromagnetic gauge fields. Both bosonic insulators can have protected conducting surface states. The gapless boundary excitations of the gapless bosonic insulator can even be fermionic.