## Student Poster Sessions

Session #1 |
Session #2 |
Session #3 |

### Session #1

**David Aasen **(Caltech)

**Title**: Topological Defects on the Lattice

**Abstract**: We construct topological defects in two-dimensional classical lattice models and one-dimensional quantum chains. The defects satisfy commutation relations guaranteeing the partition function depends only on topological properties of the defects. One useful consequence is a generalization of Kramers-Wannier duality to a wide class of height models, applicable on any surfaces.

Another is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization, and hence the spin of the associated conformal field. We extend this connection between microscopic and macroscopic properties to show that the resulting splitting and joining properties of our defect lines are exactly those of chiral operators in conformal field theory and topological quantum field theory. This enables us to drive modular transformations directly from lattice considerations. We develop this construction in detail in a variety of examples, including the height models of Andrews, Baxter and Forrester and their quantum limits.

**Victor Albert **(Yale University)

**Title**: Geometrical, response, and gap properties of Lindbladians

**Abstract**: Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful tool in studies of quantum matter and quantum information processing. This tool can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. More generally, this tool can be used to drive a quantum system to a unitarily-evolving subspace, which can be used to store, protect, and process quantum information. We show that the quantum information in all types of subspaces can be successfully manipulated by both small Hamiltonian perturbations and adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected steady states for both the unique state and subspace cases. We derive a Kubo formula governing linear response of the subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zero-frequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either corrections to adiabatic evolution or Hamiltonian perturbations, is different from the conventional Lindbladian dissipative gap. In certain cases, this scale is equivalent to the excitation gap of a related Hamiltonian.

**Jan Behrends **(Max-Planck-Insitut für Physik komplexer Systeme)

**Title**: Visualizing the chiral anomaly in Dirac and Weyl semimetals with photoemission spectroscopy

**Abstract**: Quantum anomalies are the breaking of a classical symmetry by quantum fluctuations. They dictate how physical systems of diverse nature, ranging from fundamental particles to crystalline materials, respond topologically to external perturbations, insensitive to local details. In the solid state, it fundamentally affects the properties of topological Weyl and Dirac semimetals. In this work we propose that the most identifying consequence of the chiral anomaly, the charge density imbalance between fermions of different chirality induced by non-orthogonal electric and magnetic fields, can be directly observed in these materials with the existing technology of photoemission spectroscopy. With angle resolution, the chiral anomaly is identified by a characteristic note-shaped pattern of the emission spectra, originating from the imbalanced occupation of the bulk states and a previously unreported momentum dependent energy shift of the surface state Fermi arcs. We further demonstrate that the chiral anomaly likewise leaves an imprint in angle averaged emission spectra, facilitating its experimental detection. Thereby, our work provides essential theoretical input to foster the direct visualization of the chiral anomaly in condensed matter.

**Zhen Bi **(University of California, Santa Barbara)

**Title**: Bilayer Graphene as a platform for Bosonic Symmetry Protected Topological States

**Abstract**: Bosonic symmetry protected topological (BSPT) states, i.e. bosonic analogue of topological insulators, have attracted enormous theoretical interests and efforts in the last few years. Although the BSPT states have been successfully classified with various approaches, there has been no successful experimental realization of BSPT states yet in two and higher dimensions. In this paper, we propose that the two dimensional BSPT state with U(1) × U(1) symmetry can be realized in a bilayer graphene under an out-of-plane magnetic field, where the two U(1) symmetries stand for the total spin Sz and total charge conservation respectively. The Coulomb interaction plays a central role in this proposal: 1. it gaps out all the fermions at the boundary of the system, hence the remaining symmetry protected gapless boundary states only have bosonic charge and spin degrees of freedom; 2. based on the conclusion above, we propose that the bulk quantum phase transition between the BSPT and trivial phase, which can be driven by a competition between the out-of-plane magnetic field and electric field, under strong interaction can become a “bosonic phase transition”, i.e. only bosonic modes close their gap at the transition.

**Samuel Boutin **(Université de Sherbrooke)

**Title**: Tight-binding theory of NMR shifts in topological insulators Bi2Se3 and Bi2Te3

**Abstract**: Motivated by recent nuclear magnetic resonance (NMR) experiments, we present a microscopic sp3 tight-binding model calculation of the Knight shift and the orbital shift in bulk Bi2Se3 and Bi2Te3. As byproducts, we obtain approximate values for the contact hyperfine couplings and the electronic g factors. Overall, our study unveils a number of points that may guide the interpretation of NMR measurements in topological materials. First, the contact Knight shift in layered crystals has a large uniaxial anisotropy. Second, dipolar interactions make a significant contribution to the isotropic NMR shift. Third, the contribution of the Van Vleck spin susceptibility to the Knight shift is significant. Fourth, the carrier-density-dependent part of the orbital shift is comparable to that of the contact and dipolar shifts. The first three of the preceding statements are hallmarks of strong spin-orbit interactions. In addition, we find that the contact hyperfine interaction makes a significant contribution to the isotropic shift reported in 209Bi NMR experiments, even though the electronic states at the Fermi level have a rather weak s-orbital character. In contrast, the contribution from the contact hyperfine interaction to the NMR shift of 77Se and 125Te is weak compared to the dipolar and orbital shifts therein.

**Daniel Bulmash **(Stanford University)

**Title**: Unconventional Magnetic Field Effects in Weyl and Dirac Semimetal Ultra-Thin Films

**Abstract**: We show that a thin film of Weyl or Dirac semimetal with a strong in-plane magnetic field becomes a novel two-dimensional Fermi liquid with interesting properties. The Fermi surface in this system is strongly anisotropic, which originates from a combination of chiral bulk channels and the Fermi arcs. The area enclosed by the Fermi surface depends strongly on the in-plane magnetic field component parallel to the Weyl/Dirac node splitting, which leads to unusual behavior in quantum oscillations when the magnetic field is tilted out of the plane. We also investigate the effects of interactions, which become anisotropic and magnetic field-tuned when projected to this Fermi surface.

**Anffany Chen **(University of British Columbia)

**Title**: Superconducting proximity effect and Majorana flat bands in the surface of a Weyl semimetal

**Abstract**: We study the proximity effect between an s-wave superconductor (SC) and the surface states of a Weyl semimetal. An interesting two-dimensional SC forms in such an interface with properties resembling in certain aspects the Fu-Kane superconductor. In a Weyl semimetal with unbroken time reversal symmetry the interface SC supports completely flat Majorana bands in a linear Josephson junction with a {\pi} phase difference. In a Weyl semimetal with broken time reversal symmetry the minimal interface SC has p-wave symmetry and exhibits a single gapless two-dimensional Majorana mode with a relativistic dispersion.

**Aaron Chew **(Caltech)

**Title**: Classifying Topological Phases with Tensor Networks and Braided Fusion Categories

**Abstract**: Tensor Networks have seen incredible use in physics, ranging from numerically simulating the ground states of systems to studying the Ads/CFT correspondence. Of more relevance to condensed matter is the recent work done by Xiao-Gang Wen, Xie Chen, Zheng-Cheng Gu, Liang Kong, and many others, concerning the classification of quantum systems with tensor networks and braided fusion categories. Their work beautifully unifies many concepts in mathematics with physics, most notably those of group cohomology and category theory, and also provides new insights of how to perform renormalization transformations on lattices.

**Soonwon Choi **(Harvard University)

**Title**: Dissipative preparation of AKLT ground states in atomic arrays

**Abstract**: The ground states of the Affleck-Lieb-Kennedy-Tasaki (AKLT) Hamiltonian are exactly solvable and have played an important role in understanding various concepts such as valence bond solid order, symmetry protected topological order, and matrix product states. Furthermore, these states can be used as a resource for quantum state transfer as well as measurement based quantum computation. Hence, experimental realizations of such states are of great interest for both fundamental physics and practical applications. Here, we describe how the AKLT ground states can be prepared in an atomic array using engineered dissipation. We provide an explicit, symmetry-based scheme to couple atoms to an environment in such a way that the (non-unitary) time evolution leads to atoms in an AKLT ground state. We prove that our protocol works for arbitrary initial states and does not yield unwanted steady states.

**Yang-Zhi Chou **(University of Colorado, Boulder)

**Title**: Helical Quantum Edge Gears in 2D Topological Insulators

**Abstract**: A remarkable and as-yet-unexploited aspect of topological insulator (TI) physics is the topology of the edge states, i.e. the fact that the edge liquid of a 2D TI forms a closed, unbreakable loop in the absence of electrical contacts or magnetic fields. We propose a novel experimental setup in which edge loops rotate as interlocking “gears” through Coulomb drag, in TIs with Rashba spin-orbit coupling. We show that two-terminal transport can measure the Luttinger liquid parameter K, a quantity that is otherwise notoriously difficult to measure. In the low-temperature (T → 0) perfect drag regime, the conductance is (e /h)(2 K + 1)/(K + 1). At higher T we predict a conductivity ~T Our results should trigger new experiments and may open a new venue for edge gear-based electronic devices.

**Luca Delacretaz **(Stanford)

**Title**: Memory matrix theory of quantum fluctuating superconductivity (based on arxiv.org/abs/1602.08171)

**Abstract**: A hydrodynamic theory of transport in quantum mechanically phase-disordered superconductors is possible when supercurrent relaxation can be treated as a slow process. We obtain general results for the frequency-dependent conductivity of such a regime. With time-reversal invariance, the conductivity is characterized by a Drude-like peak, with width given by the supercurrent relaxation rate. Using the memory matrix formalism, we obtain a formula for this width (and hence also the dc resistivity) when the supercurrent is relaxed by short range Coulomb interactions. This leads to a new – effective field theoretic and fully quantum – derivation of a classic result on flux flow resistance. With strong breaking of time-reversal invariance, the optical conductivity exhibits what we call a ‘hydrodynamic supercyclotron’ resonance. We obtain the frequency and decay rate of this resonance for the case of supercurrent relaxation due to an emergent Chern-Simons gauge field. The supercurrent decay rate in this ‘topologically ordered superfluid vortex liquid’ is determined by the conductivities of the normal component of the liquid. Our work gives a controlled framework for low temperature metallic phases arising from phase-disordered superconductivity.

**Vatsal Dwivedi **(University of Illinois, Urbana-Champaign)

**Title**: Entanglement in phase space

**Abstract**: The calculation of entanglement spectra of bipartitions necessitates a choice of a “cut” in a basis for the Hilbert space. For a one dimensional system, two common choices are the position and momentum space cuts, i.e, along one of the axes in a (classical) phase space picture. Thus, we ask a “natural” question: can one define entanglement cuts along arbitrary lines (or Lagrangian subspaces in general) in the phase space?

We answer in affirmative, and study the entanglement spectrum for bipartitions of fermionic noninteracting many-body states for a family of cuts in phase space that interpolates continuously between the position and momentum space cuts. Wigner functions, fractional Fourier transforms and harmonic oscillator eigenstates count among the dramatis personae. For a single particle state, we show that there is always a cut along which the entanglement spectrum is degenerate, i.e, the entanglement is maximal. However, additional constraints are needed to get degeneracies in the entanglement spectrum for a multiparticle state.

**Dominic V. Else **(University of California, Santa Barbara)

**Title**: Floquet Phases of Matter

**Abstract**: Periodically driven (Floquet) systems provide a new setting for studying many-body physics. Familiar topological and symmetry-breaking phases can be realized in the Floquet context, but there are also entirely new phases of matter that do not exist in stationary systems. In this poster, I discuss some of the new phases that can result, including symmetry-protected topological (SPT) and symmetry-enriched (SET) phases protected by time-translation symmetry, and symmetry-breaking phases where the time-translation symmetry is spontaneously broken, known as “Floquet time crystals”.

**Han Fu **(University of Minnesota)

**Title**: Collapse of electrons to a donor cluster in SrTiO3

**Abstract**: It is known that when a nucleus has charge Ze where Z > 137, electrons collapse onto the nucleus resulting in a net charge Zn < Z due to the relativistic dispersion law. A similar effect is found for a donor cluster in SrTiO3 (STO), but with a different origin. At low temperatures, STO has an enormously large dielectric constant and the nonlinear dielectric response becomes dominant, which leads to the collapse of electrons into a charged spherical donor cluster with radius R when its total charge number Z exceeds a critical value Zc ≃ R/a where a is the lattice constant. The net charge eZn grows with Z until Z exceeds Z*≃ (R/a) ^(9/7) . After this point, the charge of the compact core Zn remains ≃ Z* , while the rest Z* electrons form a sparse Thomas-Fermi electron atmosphere around it. It is shown that the thermal ionization of such two-scale atoms easily strips the outer atmosphere while the inner core remains preserved. The results are extended to the case of long cylindrical clusters. It is also discussed how the predictions can be tested by measuring conductivity of chain of discs of charge on the STO surface.

**Omri Golan **(Weizmann Institute of Science)

**Title**: Effective action for the px +i*py superconductor from an analogy to gravity

**Abstract**: The low energy limit of the px +i*py superconductor lattice model is shown to be a relativistic Majorana fermion coupled to gravity with both curvature and torsion (Riemann-Cartan space-time), where the order parameter and electromagnetic potential take the role of geometric fields known as the vielbein and the spin connection. Known results on Dirac fermions coupled to curvature and torsion are then applied to obtain the low energy effective action for the electromagnetic potential and for both the Higgs and Goldstone modes of the order parameter. This effective action contains a gravitational Chern-Simons term which describes in this context a U(1) anomaly relating the bulk Chern number and the boundary central charge, and is intimately related to the electronic energy and momentum densities, and to the dynamics of the Higgs mode.

**Yingfei Gu **(Stanford University)

**Title**: Fractional Statistics and the Butterfly Effect (authors: Yingfei Gu, Xiao-Liang Qi)

**Abstract**: In this poster, we show a connection between quantum chaos, known as the “butterfly effect”, in (1+1)-dimensional rational conformal field theories and fractional statistics in (2+1)-dimensional topologically ordered states. This connection comes from the characteristics of the butterfly effect by the out-of-time-order-correlator proposed recently. We show that the late-time behavior of such correlators is determined by universal properties of the rational conformal field theory such as the modular S-matrix. Using the bulk-boundary correspondence between rational conformal field theories and (2+1)-dimensional topologically ordered states, we show that the late time behavior of out-of-time-order-correlators is intrinsically connected with fractional statistics in the topological order. We also propose a quantitative measure of chaos in a rational conformal field theory, which turns out to be determined by the topological entanglement entropy of the corresponding topological order.

**Ciarán Hickey **(University of Toronto)

**Title**: Haldane-Hubbard Mott Insulator: Chiral Magnetic and Topological Order

**Abstract**: Experimentalists have recently been able to realise the topological Haldane model using ultracold fermions in a honeycomb optical lattice. Motivated by ongoing experimental efforts to tune interactions in this system we study the Haldane model with a local Hubbard interaction in the strongly correlated Mott limit. Using classical Monte Carlo, exact diagonalisation and density matrix renormalisation group techniques we show that the Haldane-Hubbard model exhibits various chiral magnetic orders including a wide regime of triple-Q tetrahedral order. Incorporating third-neighbour hopping frustrates and ultimately quantum-melts this tetrahedral magnetic order. From analyzing low energy spectra, many-body Chern numbers, entanglement spectra, and modular matrices, we identify the molten state as a chiral spin liquid with gapped semion excitations. Our numerical results suggest that this frustration induced melting may lead to an exotic continuous phase transition. We formulate a Chern-Simons-Higgs theory to describe this spin crystallization transition from a chiral spin liquid, an SU(2) invariant state with topological order, to a tetrahedral spin crystal, a topologically trivial state with broken SU(2) symmetry.

### Session #2

**Yi-Ting Hsu **(Cornell University)

**Title**: Topological superconductivity in monolayer transition metal dichalcogenides

**Abstract**: Theoretically it has been long known that breaking spin-degeneracy to realize so-called spinless fermions is a promising path to topological superconductivity. However, topological superconductors are rare to date. We propose a new strategy for materializing spinless fermions by splitting the spin-degeneracy in momentum space. Specifically, we identify monolayer hole-doped (p-type) transition metal dichalcogenide(TMD)s as candidates that can materialize topological superconductors out of such momentum space split spinless fermions. In fact, superconductivity in electron-doped (n-type) TMDs is by now well established. However, light hole-doping puts these systems in an unusual state with spin-valley locking that is absent in the electron-doped side. Using a renormalization group analysis, we predict two possible topological pairing states to emerge from electron-electron repulsion with degenerate pairing interaction: an inter-pocket pairing state which is an equal-mixture of singlet and triplet, and an intra-pocket pairing with finite pair-momentum. We find the inter-pocket pairing state to dominate over the intra-pocket pairing state despite the degenerate pairing interaction due to the doubled pairing phase space. We urge superconducting test on lightly hole-doped monolayer TMDs, which could be the first realization of an atomically thin two-dimensional topological superconductor.

**Sheng-Jie Huang **(University of Colorado, Boulder)

**Title**: Topological phases protected by point group symmetry

**Abstract**: We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with ZP2 (reflection) symmetry, electronic topological crystalline insulators (TCIs) with U(1)×ZP2 symmetry, and bosonic pgSPT phases with C2v symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a Z8 × Z2 classification, where the Z8 corresponds to known states obtained from non-interacting electrons, and the Z2 corresponds to a “strongly correlated” TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.

**Yi-Ping Huang **(University of Colorado, Boulder)

**Title**: Theory of quantum Kagome ice

**Abstract**: We derived an effective odd Z2 gauge theory to explain the numerically observed zero temperature quantum disorder phase on XYZ Kagome lattice with Zeeman field. We analyzed the symmetry fractionalization pattern of such Z2 topological order and derived trivial and nontrivial fractionalization classes for its vison and spinon sector respectively. The nontrivial fractionalization for spinon sector cannot emerge from spin model with continuous spin rotation symmetry.

**Jason Iaconis **(University of California, Santa Barbara)

**Title**: Studying Highly Entangled Ground States with DMET

**Abstract**: Density Matrix Embedding Theory (DMET) is a newly developed numerical algorithm which can be thought of as a numerically efficient alternative to DMFT which works directly with the ground state wave function. I will show how this method can be adapted to study highly entangled phases of matter and discuss possible applications for studying spin liquid physics in the triangular lattice Hubbard model.

**Shenghan Jiang **(Boston College)

**Title**: Symmetric tensor networks – algorithms to sharply identify classes of quantum phases distinguishable by short-range physics

**Abstract**: Phases of matter are sharply defined in the thermodynamic limit. One major challenge of accurately simulating quantum phase diagrams of interacting quantum systems is due to the fact that numerical simulations usually deal with the energy density, a local property of quantum wavefunctions, while identifying different quantum phases generally rely on long-range physics. We develop a general method to construct fully symmetric quantum wavefunctions under certain assumptions using projected entangled pair states (PEPS). We find that quantum phases can be organized into crude classes distinguished by short-range physics, which is related to the fractionalization of both on-site symmetries and space-group symmetries. Based on the analytical work, we develop an efficient simulation algorithm, which is able to sharply determine crude classes in interacting quantum systems. Examples of these crude classes are demonstrated in half-integer quantum spin systems on the kagome lattice as well as on the honeycomb lattice. Limitations and generalizations of our results are discussed.

**Byungmin Kang **(University of California, Berkeley)

**Title**: Universal crossover from ground state to excited-state quantum criticality

**Abstract**: We study the non-equilibrium properties of a non-ergodic random quantum chain in which highly excited eigenstates exhibit critical properties usually associated with quantum critical ground states. The ground state and excited states of this system belong to different universality classes, characterized by infinite-randomness quantum critical behavior. Using strong disorder renormalization group techniques, we show that the crossover between the zero and finite energy density regimes is universal. We analytically derive a flow equation describing the unitary dynamics of this isolated system at finite energy density from which we obtain universal scaling functions along the crossover. We also conjecture a general monotonicity property, ala the c-theorem for conformal field theories, of renormalization group flows between infinite randomness fixed points.

**Jack Kemp **(University of Oxford)

**Title**: Existence of strong edge zero modes in interacting, non-integrable systems

**Abstract**: Strong edge zero modes are operators localised at the edge of a one-dimensional system which commute with the Hamiltonian, leading to a degeneracy in the entire energy spectrum. This is in contrast to weak edge zero modes which only induce a degeneracy in the zero energy density states. The well-known Majorana edge zero modes in the Kitaev chain are examples of a strong edge zero mode in a free system, and more recently strong edge zero modes have been shown to exist in the interacting but integrable XYZ chain (Fendley, 2015, arXiv:1512.03441).

In this poster I will describe how strong zero modes may be calculated perturbatively for non-integrable systems as well, and discuss possible problems with normalisability, focussing in particular on transverse-field Ising with four Majorana-fermion interaction terms. I will give numerical results showing how the existence of this (pseudo-)strong zero mode may yield long-lived boundary magnetisation, as well as significant pairing of states throughout the spectrum, indicating a possible violation of the eigenstate thermalisation hypothesis.

**Christina Knapp **(University of California, Santa Barbara)

**Title**: How quickly can anyons be braided?

**Abstract**: Topological phases of matter are a potential platform for the storage and processing of quantum information with intrinsic error rates that decrease exponentially with inverse temperature and with the length scales of the system, such as the distance between quasiparticles. However, it is less well-understood how error rates depend on the speed with which non-Abelian quasiparticles are braided. In general, diabatic corrections to the holonomy vanish at least inversely with the length of time for the braid, with faster decay occurring as the time-dependence is made smoother. We show that such corrections will not affect quantum information encoded in topological degrees of freedom, unless they involve the creation of topologically nontrivial quasiparticles. Moreover, we show how measurements that detect unintentionally created quasiparticles can be used to control this source of error.

**Thomas Kvorning **(Stockholm University)

**Title**: Geometric effects of chiral superconductors

**Abstract**: For conventional (s-wave) superconductors the order parameter has no directionality and therefore no magnetic response to geometry.

For order parameters with directionality there will, however, in general be a response to curvature and the Meissner effect will no longer put the magnetic field to zero, but rather a value related to the geometric curvature.

We will discuss this phenomenon for a 2d p-wave superconductor that only depends on the intrinsic geometry. In that case, as far as the phase of the order parameter is concerned, there is no difference between magnetic flux and scalar curvature. We will see that this changes the Meissner effect and makes the magnetic field proportional to the magnetic field, rather than zero. We will also see that this effect resolves the following question: Given that, in the absence of any flux, the ground state on the annulus does not support Majorana modes, while the one on the cylinder does, how come that the conical geometry can interpolate smoothly between the two?

**Ella Lachman **(Weizmann Institute of Science)

**Title**: Visualization of superparamagnetic dynamics in magnetic topological insulators

**Abstract**: Magnetically doped topological insulators have recently been shown to host a quantum anomalous Hall (QAH) state at low temperatures. Using scanning nanoSQUID magnetic imaging on a Cr-doped (Bi,Sb)_{2}Te_{3} thin film^{[1]}, we reveal that the magnetic structure of magnetically doped topological insulators is not ferromagnetic as assumed so far. In fact it is superparamagnetic, formed by weakly interacting magnetic domains. These domains have a characteristic size of a few tens of nanometers, and undergo random reversals which drive the electronic state from one Hall plateau to the other.

The superparamagnetic state is metastable, with small energy barriers to relaxation. We observe magnetic relaxation even at 300 mK, evident also in transport measurements. Unexpectedly, magnetic relaxation can also be induced by varying the gate voltage, and we propose a mechanism for the influence of the electronic phase on the magnetic relaxation. We speculate that the dynamic nature of magnetic disorder in QAH systems may contribute to the observed fragility of the QAH state at elevated temperatures.

[1] Lachman *et.al* Science Advances Vol. 1, no. 10, e1500740 (2015)

**Ethan Lake** (University of Utah)

**Title**: Dimensional reduction and anomaly cancellation in reflection-symmetric topological phases

**Abstract**: One of the central ideas regarding anomalous fractionalization patterns in d spacetime dimensions is that they imply the existence of nontrivial (d+1)-dimensional physics. Typically, an anomaly in d dimensions is cancelled by the presence of a bulk (d+1)-dimensional SPT state protected by the symmetry of the surface state.

We demonstrate that for certain spacetime symmetries, this may not always be a full picture of how the anomaly cancellation occurs. We show that some anomalies may actually cancelled by a d-dimensional SPT state, provided that it is embedded in a (d+1)-dimensional trivial gapped bulk. We illustrate this idea for the example of Z_N topological order with reflection symmetry, and along the way establish a concrete classification of anomalous reflection symmetry fractionalization patterns.

**Yonah Lemonik **(New York University)

**Title**: Non-Equilibrium States and Entanglement Spectrum

**Abstract**: There is growing interest in non-equilibrium states with topological or otherwise universal properties. However the definition and detection of such properties in a non-equilibrium setting is non-trivial. We employ the perspective of the entanglement spectrum, a generalization of entanglement entropy. We show that it is well suited to the analysis of non-equilibrium states by applying it to (i) define topological invariants in Floquet Chern Insulators and (ii) extract universal exponents in critical prethermalization states.

**Songci Li **(University of Washington, Seattle)

**Title**: Spiraling Fermi Arcs in Weyl Semimetals

**Abstract**: In Weyl materials the valence and conduction electron bands touch at an even number of isolated points in the Brillouin zone. In the vicinity of these points the electron dispersion is linear and may be described by the massless Dirac equation. This results in nontrivial topology of the Berry connection curvature. One of its consequences is the existence of peculiar surface electron states whose Fermi surfaces form arcs connecting projections of the Weyl points onto the surface plane. Band bending near the boundary of the crystal also produces surface states. We show that in Weyl materials band bending near the crystal surface gives rise to a spiral structure of energy surfaces of arc states. The corresponding Fermi surface has the shape of a spiral that winds about the projection of the Weyl point onto the surface plane. The direction of the winding is determined by the helicity of the Weyl point and the sign of the band-bending potential. For close valleys the arc state morphology may be understood in terms of the avoided crossing of oppositely winding spirals.

**Ian Mondragon **(University of Illinois, Urbana-Champaign)

**Title**: Disordered topological crystalline phases

**Authors**: Ian Mondragon-Shem and Taylor L. Hughes

Abstract: We study disordered topological crystalline phases for which the disorder is exactly symmetric under a given spatial symmetry operation. This requires that we go beyond the conventional momentum-space description of topological crystalline phases. Instead, we provide an understanding of the topology of the ground state in real-space. To do this, we discuss a mechanism wherein the topological invariant of the ground state is carried by bulk states which are localized within spatial regions that remain invariant under the relevant spatial symmetry. We illustrate our results with specific realizations of mirror-symmetric DIII topological superconductors in two and three spatial dimensions. Finally, we comment on how these results can provide a bulk characterization of disordered topological crystalline phases protected by average symmetries.

**Sergej Moroz **(University of Colorado, Boulder)

**Title**: From chiral p+ip superfluid to super Efimov effect

**Abstract**: Two-dimensional fermionic chiral superfluidity and superconductivity is an active area of experimental and theoretical research in condensed matter physics. It is of interest in diverse fields such as the physics of 3He, quantum Hall physics, unconventional superconductivity and topological quantum computing. In this poster I will summarize our hydrodynamic effective theory of a chiral p+ip superfluid at zero temperature. It naturally incorporates the parity and time reversal violating effects such as the Hall viscosity and the edge current. I will also show how a chiral p+ip superfluid can be put on a sphere. In addition, I will also describe the few-body aspects of this problem and introduce the super Efimov effect- a new type of three-body quantum universality manifesting itself in a tower of three-body bound states with a double-exponential scaling. These universal few-body states may be observed in ultracold atom experiments and should be taken into account in future many-body studies of p+ip paired states.

**Yang Peng **(Freie Universität Berlin)

**Title**: Signatures of topological Josephson junctions

**Abstract**: Quasiparticle poisoning and diabatic transitions may significantly narrow the window for the experimental observation of the 4π-periodic dc Josephson effect predicted for topological Josephson junctions. Here, we show that switching current measurements provide accessible and robust signatures for topological superconductivity which persist in the presence of quasiparticle poisoning processes. Such measurements provide access to the phase-dependent subgap spectrum and Josephson currents of the topological junction when incorporating it into an asymmetric SQUID together with a conventional Josephson junction with large critical current. We also argue that pump-probe experiments with multiple current pulses provide access to the quasiparticle poisoning rates of the topological junction. The proposed signatures are particularly robust, even in the presence of Zeeman fields and spin-orbit coupling, when focusing on short Josephson junctions. Finally, we also consider microwave excitations of short topological Josephson junctions which may complement switching current measurements.

**Abhishodh Prakash **(Stony Brook University)

**Title**: Detection of gapped 1D phases with on-site and spatial symmetries

**Abstract**: We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global onsite A4, translation and lattice inversion symmetries. We detect different gapped phases characterized by SPT order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the onsite and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.

**Abhiav Prem **(University of Colorado, Boulder)

**Title**: Topological order, symmetry, and Hall response of two-dimensional spin-singlet superconductors

**Abstract**: Fully gapped two-dimensional superconductors coupled to dynamical electromagnetism are known to exhibit topological order. In this poster, we develop a unified low-energy description for spin-singlet paired states by deriving topological Chern-Simons field theories for s-wave, d+id, and chiral higher even-wave superconductors. These theories capture the quantum statistics and fusion rules of low-energy excitations and incorporate global continuous symmetries - specifically, spin rotation and conservation of magnetic flux - present in all singlet superconductors. We compute the Hall response for these symmetries and investigate the physics at the edge.

**Raquel Queiroz **(Max Planck Institute for Solid State Physics)

**Title**: Dimensional hierarchy of fermionic interacting topological phases

**Abstract**: We present a dimensional reduction argument to derive the classification reduction of fermionic symmetry protected topological phases in the presence of interactions. The dimensional reduction proceeds by relating the topological character of a d-dimensional system to the number of zero-energy bound states localized at zero-dimensional topological defects present at its surface. This correspondence leads to a general condition for symmetry preserving interactions that render the system topologically trivial, and allows us to explicitly write a quartic interaction to this end. Our reduction shows that all phases with topological invariant smaller than n are topologically distinct, thereby reducing the non-interacting Z classification to Zn.

### Session #3

**Willem Rischau **(Ecole superieure de physique et chimie industrielles (ESPCI))

**Title**: Quantum interference detected in a macroscopic Van der Waals conductor

**Abstract**: Quantum corrections to charge transport can give rise to an oscillatory magnetoconductance, typically observed in mesoscopic samples with a length shorter than or comparable with the phase coherence length. Here, we report the observation of magnetoconductance oscillations periodic in magnetic ﬁeld with an amplitude of the order of e²/h in macroscopic samples of Highly Oriented Pyrolytic Graphite (HOPG). The observed effect emerges when all carriers are conﬁned to their lowest Landau levels. We track the origin of the quantum interference phenomenon to the presence of moiré superlattices with a discrete (devil’s-staircase) distribution in periodicity. According to our results, when the magnetic length, the Fermi wave length and the length scale of ﬂuctuations in local chemical potential are comparable in a layered conductor, quantum corrections can be detected over centimetric length scales.

**Joaquin Rodriguez Nieva **(MIT)

**Title**: Berry’s Phase and Giant Non-Reciprocity in Dirac Quantum Dots

**Abstract**: Recently, nanoscale pn-junction rings have been introduced as a vehicle for confinement of electronic states in Dirac materials [1]. Confined states in these Dirac quantum dots arise due to constructive interference of electronic waves incident at the pn junction at oblique angles and inward-reflected from the ring. What special features sets Dirac quantum dots apart from conventional quantum dots? We show [2] that Berry curvature induces a strongly non-reciprocal spectrum of quantum dot resonances under weak magnetic fields. Such effect is maximal for massless Dirac electrons, e.g. graphene, and is manifested in anomalously large splittings of the resonances which are degenerate at B=0 due to time reversal symmetry. This non-reciprocity effect overwhelms the conventional orbital and spin-induced non-reciprocity.

The predicted giant non-reciprocity is readily accessible by Faraday and Kerr optical rotation measurements as well as by scanning tunneling spectroscopy.

[1] Zhao, et al., Science 348,672(2015).

[2] JRN and L. S. Levitov, arXiv:1508.06609.

**Lucile Savary** (MIT)

**Title**: Quantum Loop States in Spin-Orbital Models on the Honeycomb and Hyperhoneycomb Lattices

**Abstract**: The search for truly quantum phases of matter is one of the center pieces of modern research in condensed matter physics. Quantum spin liquids are exemplars of such phases. In the quest for the latter, the challenges are many: neither is it clear how to look for nor how to describe them, and definitive experimental examples of quantum spin liquids are still missing. In this poster I show how to devise a realistic model on the honeycomb lattice whose ground state realizes Haldane chains whose physical supports fluctuate, hence naturally providing the hallmark “fractional excitations” of quantum spin liquids. When taken to the three-dimensional hyperhoneycomb lattice, the ground state becomes a full-fledged symmetry-enriched U(1) quantum spin-orbital liquid, “disordered” both in the spin and orbital channels. The phase diagram also contains an interacting bosonic topological insulator phase. Crucially, this model is expected to describe actual materials, and I provide a detailed set of material-specific constraints which may be readily used for an experimental realization.

**Bjoern Sbierski **(FU Berlin)

**Title**: Weyl node with random vector potential

**Abstract**: We study Weyl semimetals in the presence of generic disorder, consisting of a random vector potential as well as a random scalar potential. We derive renormalization group flow equations to second order in the disorder strength. These flow equations predict a disorder-induced phase transition between a pseudo-ballistic weak-disorder phase and a diffusive strong-disorder phase for sufficiently strong random scalar potential or for a pure three-component random vector potential. We verify these predictions using a numerical study of the density of states near the Weyl point and of quantum transport properties at the Weyl point. In contrast, for a pure single-component random vector potential we find no transition to a diffusive strong-disorder phase.

**Xueyang Song **(Peking University)

**Title**: Low-energy spin dynamics of honeycomb model beyond Kitaev limit

**Abstract**: This work illustrates that the generic low-nergy weight of the dynamical spin structure factors in gapless spin liquid phase of the Kitaev honeycomb model is non-vanishing. We search for how spins would expand in terms of fractionalized low-energy particles under perturbations. We provide a generic relation between spin operators and the dominant low-energy quasiparticle operators commensurate with the symmetries as well as the emergent Z2 gauge field structure. It’s found that the dynamical spin structure factor henceforth obeys power-law asymptotic behavior instead of possessing a spin gap in the low frequency regime, in the vicinity of two special points of the momentum space.

**Sergey Syzranov **(University of Colorado, Boulder)

**Title**: Non-Anderson disorder-driven transitions in Weyl semimetals and other systems

**Abstract**: It is usually believed that increasing disorder strength in a d>2-dimensional system leads to the Anderson localisation transition with universal properties depending only on the space dimensionality. I will demonstrate that systems with a power-law quasiparticle dispersion $\xi_{\bf k}\propto k^\alpha$ in dimensions $d>2\alpha$ exhibit another type of a disorder-driven quantum phase transition at the bottom of the band, that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states and non-Anderson critical behaviour of other observables, e.g., conductivity. The transition can be observed in Weyl semimetals, chiral superconductors, arrays of ultracold ions with long-range interactions, and other systems.

**Rina Takashima **(University of Kyoto)

**Title**: Quantum Nature of Magnetic Skyrmions in Chiral Magnet

**Abstract**: A magnetic skyrmion is a vortex-like spin texture, which is characterized by a topological number. We theoretically study the quantum effects on magnetic skyrmions in two-dimensional chiral magnets. We propose that the quantum fluctuation induces a new quantum phase ”skyrmion quantum liquid phase”, where skyrmions are not spatially localized. We also show that the critical behavior between the quantum liquid phase and the ferromagnetic phase depends on the spin size S and the topological number of a single skyrmion.

**Nicolas Tarantino **(Stony Brook University)

**Title**: Discrete spin structures and fSPT commuting projector models

**Abstract**: We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (fSPTs) with onsite unitary symmetry group Z_2. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface M on which our model is defined, namely a ‘Kasteleyn’ orientation of a certain graph associated with the lattice. In particular, our construction yields commuting projector models for all 8 members of the Z8 classification of 2d fermionic SPTs with G = Z_2.

**Alex Thomson **(Harvard University)

**Title**: Spectrum of conformal gauge theories on a torus

**Abstract**: Many model quantum spin systems have been proposed to realize critical points or phases described by 2+1 dimensional conformal gauge theories. On a torus of size L and modular parameter τ, the energy levels of such gauge theories equal (1/L) times universal functions of τ. We compute the universal spectrum of QED3, a U(1) gauge theory with Nf two-component massless Dirac fermions, in the large Nf limit. We also allow for a Chern-Simons term at level k, and show how the topological k-fold ground state degeneracy in the absence of fermions transforms into the universal spectrum in the presence of fermions; these computations are performed at fixed Nf /k in the large Nf limit.

**Yuxuan Wang **(University of Illinois, Urbana-Champaign)

**Title**: Topological superconductivity from parity fluctuations

**Abstract**: We analyze the superconducting instabilities in the vicinity of the quantum-critical point of an inversion symmetry breaking order. We first show that, as previous obtained by Kozii and Fu, the fluctuations of the inversion symmetry breaking order lead to two degenerate superconducting (SC) instabilities, one in the s-wave channel, and the other in a time-reversal invariant odd-parity pairing channel (the simplest case being the same as the of 3He-B phase). Remarkably, we find that unlike many well-known examples, the selection of the pairing symmetry of the condensate is independent of the momentum-space structure of the collective mode that mediates the pairing interaction. We found that this degeneracy is a result of the existence of a conserved fermionic helicity, χ, and the two degenerate channels correspond to even and odd combinations of SC order parameters with χ = ±1. As a result, the system has an enlarged symmetry U(1) × U(1), with each U(1) corresponding to one value of the helicity χ. We discuss how the enlarged symmetry can be lifted by small perturbations, such as the Coulomb interaction or Fermi surface splitting in the presence of broken inversion symmetry, and we show that the resulting superconducting state can be topological or trivial depending on parameters. We present a global phase diagram of the superconducting states and discuss possible experimental implications.

**Xueda Wen **(University of Illinois, Urbana-Champaign)

**Title**: Topological entanglement negativity in Chern-Simons theories

**Authors**: Xueda Wen, Po-Yao Chang and Shinsei Ryu

Abstract: We compute directly the topological entanglement negativity between two spatial regions in (2+1) dimensional Chern-Simons gauge theories by using replica trick and surgery method. For a bipartite/tripartite manifold, we study how the topological entanglement negativity depends on the distribution of quasiparticles as well as the choice of ground state. In particular, for two adjacent non-contractible regions on a tripartite torus, the entanglement negativity provides a simple way to distinguish Abelian and non-Abelian theories. Our method applies to a Chern-Simons gauge theory defined on an arbitrary oriented (2+1) dimensional spacetime manifold. Our results agree with the edge theory approach in a recent work (X. Wen, S. Matsuura and S. Ryu, arXiv:1603.08534) for various cases.

**Julie Wildeboer **(Florida State University)

**Title**: Entanglement Entropy and Topological Order in Resonating Valence-Bond Quantum Spin Liquids

**Abstract**: On the triangular and kagome lattices, short-ranged resonating valence bond (RVB) wave functions can be sampled without the sign problem using a recently-developed Pfaffian Monte Carlo scheme [1]. In this poster [2], we present a study of the Renyi entanglement entropy in these wave functions using a replica-trick method [3]. Using various spatial bipartitions, including the Levin-Wen construction, our finite-size scaled Renyi entropy gives a topological contribution consistent with $\gamma = ln(2)$, as expected for a gapped Z_2 quantum spin liquid. We prove that the mutual statistics are consistent with the toric code anyon model and rule out any other quasiparticle statistics such as the double semion model.

[1] J. Wildeboer and A. Seidel, PRL {\bf 109}, 147208 (2012).

[2] J. Wildeboer, A. Seidel, and R. G. Melko, submitted to PRL.

[3] M. B. Hastings, I. Gonzalez, A. B. Kallin, and R. G. Melko, PRL {\bf 104}, 157201 (2010).

**Justin Wilson **(Caltech)

**Title**: Persistent Hall response in a quantum quench

**Abstract**: Out-of-equilibrium systems can host phenomena that transcend the usual restrictions of equilibrium systems. Here we unveil how out-of-equilibrium states, prepared via a quantum quench, can exhibit a non-zero Hall-type response that persists at long times, and even when the instantaneous Hamiltonian is time reversal symmetric; both these features starkly contrast with equilibrium Hall currents. Interestingly, the persistent Hall effect arises from processes beyond those captured by linear response, and is a signature of the novel dynamics in out-of-equilibrium systems. We propose quenches in two-band Dirac systems as natural venues to realize persistent Hall currents, which exist when either mirror or time-reversal symmetry are broken (before or after the quench). Its long time persistence, as well as sensitivity to symmetry breaking, allow it to be used as a sensitive diagnostic of the complex out-equilibrium dynamics readily controlled and probed in cold-atomic optical lattice experiments.

**Linda Ye **(MIT)

**Title**: Anomalous Hall Effect in Bi-layer Kagome Ferromagnet Fe3Sn2

**Abstract**: The ferromagnetic kagome lattice is theoretically known to possess topological electronic [1] and magnonic [2] band structures. Previous polycrystal studies on Fe3Sn2, a ferromagnetic metal with underlying bi-layer kagome lattice, have shown unconventional magnetic and transport properties [3,4]. Here we present a first systematic magnetic and transport investigation on single crystalline Fe3Sn2. We identified a spin-reorientation transition with temperature from anisotropic magnetic susceptibility. Furthermore, the anomalous Hall conductivity in single crystalline Fe3Sn2 is found to be at the crossover between intrinsic regime and clean limit in the universal anomalous Hall effect classification [5]. The intrinsic anomalous Hall effect takes a value around 150 /Ohm cm, which is weakly temperature dependent.

[1] *Phys. Rev. Lett.* 106 236802 (2011)

[2] *Phys. Rev. B* 87 144101 (2013)

[3] *J. Phys*:* Cond. Mat*. 21 452202 (2011)

[4] *J. Phys: Cond. Mat*. 23 112205 (2011)

[5] *Phys. Rev. B* 77 165103 (2008)

**Yizhi You **(University of Illinois, Urbana-Champaign)

**Title**: Stripe melting, a transition between weak and strong symmetry protected topological phases

**Abstract**: In this poster, we construct a phase transition between weak and strong SPT phase in strongly interacting boson system. The starting point of our construction is the superconducting Dirac fermions with pair density wave(PDW) order in 2d. We first demonstrate that the nodal line of the PDW contains a 1dboson SPT phase. We further show that melting the PDW stripe and condensing the nodal line provoke the transition from weak to strong SPT phase in 2d. The phase transition theory contains an O(4) non-linear-σ-model(NLσM) with topological Θ-term emerging from the proliferation of domain walls bound to an SPT chain.

Similar scheme also applies to weak-strong SPT transition in other dimensions and predicts possible phase transition from 2d to 3d topological order.

**Yizhuang You** (University of California, Santa Barbara)

**Title**: Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States

**Authors**: Yi-Zhuang You, Andreas W. W. Ludwig, Cenke Xu

**Abstract**: We consider the Sachdev-Ye-Kitaev (SYK) model[1–3] as an effective theory arising on the 0D boundary of an interacting 1D many-body localized, fermionic symmetry protected topological (SPT) phase. We find that the boundary is thermalized and investigate how its boundary anomaly, dictated by the bulk SPT order, is encoded in the quantum chaotic eigenspectrum of the SYK model. We show that given the SPT symmetry class, the boundary many-body level statistics varies among those of the three different Wigner-Dyson random matrix ensembles with a periodicity in the index that matches the interaction-reduced classification of the bulk SPT states. We consider all three symmetry classes BDI, AIII, and CII, whose SPT phases are Z classified in the absence of interactions in 1D. For symmetry class BDI, we derive the periodicity of the Wigner-Dyson statistics by using Clifford algebras.

**Yue Zhang **(University of Utah)

**Title**: Spectral narrowing and spin echo for localized carriers with heavy-tailed Levy distribution of hopping times

**Abstract**: We study analytically the free induction decay and the spin echo decay originating from the localized carriers moving between the sites which host random magnetic fields. Due to disorder in the site positions and energies, the on-site residence times, \tau, are widely spread according to the Levy distribution. The power-law tail \propto \tau^{-1-\alpha} in the distribution of waiting times does not affect the conventional spectral narrowing for \alpha >2, but leads to a dramatic acceleration of the free induction decay in the domain 2>\alpha >1. The next abrupt acceleration of the decay takes place as the tail parameter, \alpha, becomes smaller than 1. In the latter domain the decay does not follow a simple-exponent law. To capture the behavior of the average spin in this domain, we solve the evolution equation for the average spin using the approach different from the conventional approach based on the Laplace transform. Unlike the free induction decay, the tail in the distribution of the residence times leads to the slow decay of the spin echo. The echo is dominated by realizations of the carrier motion for which the number of sites, visited by the carrier, is minimal.

**Liujun Zou **(MIT)

**Title**: Spurious Long-Range Entanglement and Replica Correlation Length

**Abstract**: Topological entanglement entropy has been regarded as a smoking-gun signature of topological order in two dimensions, capturing the total quantum dimension of the topological particle content. An extrapolation method on cylinders has been used frequently to measure the topological entanglement entropy. Here, we show that a class of short-range entangled 2D states, when put on an infinite cylinder of circumference $L$, exhibits the entanglement R’enyi entropy of any integer index $\alpha \ge 2$ that obeys $S_\alpha = a L - \gamma$ where $a, \gamma > 0 $. Under the extrapolation method, the subleading term $\gamma$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\gamma$ is always present if the 2D state reduces to a certain symmetry-protected topological 1D state, upon disentangling spins that are far from the entanglement cut. The internal symmetry that stabilizes $\gamma > 0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\gamma$ decays exponentially in $L$ with a characteristic length, termed as a replica correlation length, which can be {\em arbitrarily large} compared to the two-point correlation length of the 2D state. We propose a simple numerical procedure to measure the replica correlation length through replica correlation functions. We also calculate the replica correlation functions for representative wave functions of abelian discrete gauge theories and the double semion theory in 2D, to show that they decay abruptly to zero. This supports a conjecture that the replica correlation length being small implies that the subleading term from the extrapolation method determines the total quantum dimension.