Abstract TBD

Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some `topological' features: they support fractional bulk excitations (dubbed fractons), and a ground state degeneracy that is robust to local perturbations. However, because previous fracton models have only been defined and analyzed on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this talk, we demonstrate that the X-cube model, a prototypical type-I fracton model, can be defined on general three-dimensional manifolds. Our construction revolves around the notion of a singular compact total foliation of the spatial manifold, which constructs a lattice from intersecting stacks of parallel surfaces called leaves. We find that the ground state degeneracy depends on the topology of the leaves and the pattern of leaf intersections. We further show that such a dependence can be understood from a renormalization group transformation for the X-cube model, wherein the system size can be changed by adding or removing 2D layers of topological states. Our results lead to an improved definition of fracton phase and bring to the fore the topological nature of fracton orders.

ABSTRACT TBD

I describe a generalization of the Jordan-Wigner transformation for lattice systems of dimension two and higher. It maps any fermionic system on a D-dimensional lattice to a (D-1)-form Z_2 gauge theory. The map preserves locality of the Hamiltonian.

I look back at some classic work on electric-magnetic duality that relates the partition function of 4d gauge theories to 2d conformal field theories. I will explain how the correspondence becomes richer when one studies 4d gauge theories with gapped fermionic charges and monopoles, and how electric-magnetic duality in this context is related to modularity for 2d spin conformal field theories.

Matrix models offer solvable, many-body quantum Hamiltonians for quantum Hall systems, both Abelian and non-Abelian. I'll describe the basics of these models and explain some of the results one can extract from them.

Progress in the detection of anyonic quasiparticles in the paired Hall state at 5/2 filling will be reported.

Nonabelian bosonization may provide insight into the apparent superuniversality observed at various quantum Hall plateau transitions.

ABSTRACT TBD

We present numerical evidence for an unconventional continuous transition in anisotropic S=1/2 magnets. The field theory for such a transition could be the O(4) sigma model with a topological term and theta=pi

Fermions on a double layer honeycomb lattice display a rich phase diagram that includes semi-metallic and Mott insulating phases, as well as bosonic symmetry protected topological (BSPT) phases. Here we present a unified framework that captures these phases and the accompanying quantum phase transitions. First, we show that the bosonic topological transition (BTT) between an SO(4) BSPT phase and the trivial phase can be described by a compact QED_3 theory with N_f=4 massless Dirac fermions. Within a systematic large-N_f RG analysis, we identify the BTT fixed point with an emergent O(4) symmetry and all expected relevant operators. Then by merging two BTT theories together, we arrived at a field theory description for the symmetric mass generation (SMG), which is an SU(2) QCD-Higgs theory with N_f=4 flavors of SU(2) fundamental fermions and one SU(2) fundamental Higgs boson. To be compatible with the SO(4) symmetry of the adjacent BSPT phase, the SMG analyzed in this work has a microscopic symmetry SO(4)xSO(3). Finally, we use their relationship to justify our previously postulated SMG mechanism by showing that the proposed mass generation naturally gives rise to the BTT theory.

The "standard" deconfined quantum critical point (dQCP) is a direct second order transition on the square lattice between the antiferromagnetic Neel order with a ground state manifold (GSM) S^2 and a four fold degenerate valence bond solid (VSB) state whose GSM can be approximated by S^1 near the critical point. Connection of this dQCP to the 3d bulk symmetry protected topological (SPT) states has also been discussed recently, and a miniweb of dual descriptions of the easy-plane dQCP has been developed. In this talk we will propose a dQCP on the triangular lattice, which is a direct unfine-tuned second order transition between the standard 120 degree noncollinear antiferromagnetic order and a VBS order. Near the dQCP, the system has a "self-duality" structure, namely both the noncollinear order and the VBS order have GSM SO(3) (or approximate SO(3)). Our proposal is based on a controlled RG calculation. We will derive a topological term that captures the "intertwinement" between the noncollinear magnetic order and the VBS order. We will also show that this dQCP is analogous to the boundary of a 3d bosonic SPT state.

I will describe the recently developed bimetric theory of fractional quantum Hall states. This effective theory includes the Chern-Simons theory that describes the topological properties of the fractional quantum Hall states and an action a la bimetric gravity that describes the massive Girvin-MacDonald-Platzman mode. The theory reproduces the universal features of chiral lowest Landau level (LLL) FQH states which lie beyond the TQFT data, such as the projected static structure factor and the GMP algebra. The LLL projection and particle-hole "symmetry" are particularly transparent. Familiar quantum Hall observables acquire a curious geometric interpretation in the bimetric language.

It has been conjectured Chern Simons gauged fermions are level rank dual to Chern Simons gauged bosons. In particular this conjecture predicts that the thermal partition functions of the theories on either side of this duality are identical. This prediction has been explicitly verified in the large N limit when the boson mass is positive (`paramegnetic'), In this talk we compute the large N boson partiion function at negative (`ferromagnetic') boson mass. Once again our final result agrees perfectly with the fermionic partition function, in agreement with the predictions of duality. The bosonic partition function in this phase is obtained by performing loop integrals of the physical exictations in this phase - which turn out to be spin 1 W bosons.

I will discuss conjectures for various new non-supersymmetric dualities in three dimensions, some involving quiver gauge theories, and others involving theories with both fundamental fermions and bosons.

Numerical results suggest that the quantum Hall effect at {\nu} = 5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau level mixing are strong. We show that the PH-Pfaffian topological order is consistent with all experiments. The absence of the particle-hole symmetry at {\nu} = 5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.

ABSTRACT TBD

I will discuss a holographic model in which a U(1) symmetry and translational invariance are broken spontaneously at the same time. The symmetry breaking leads to a scalar condensate and a charge density which are spatially modulated and exhibit stripe order, realizing the key features of pair density wave order. The model also admits a phase with co-existing superconducting and charge density wave orders, in which the scalar condensate has a uniform component. In our construction the various orders are intertwined with each other and have a common origin. Moreover, the striped features are a relevant deformation of the UV field theory. After discussing details of the model I will conclude with comments on ongoing work.

I will review the current status of the relevance of modular invariance and duality for quantum critical systems in 2+1 dimensions.

Quantum spin liquids (QSL) are phases that host fractionalized excitations. It is difficult for local probes to characterize QSL, whereas entanglement can serve as a powerful diagnostic tool due to its non-locality. Here, we study the kagome Heisenberg model using large-scale density-matrix renormalization group simulations. The entanglement entropy reveals the presence of interacting gapless Dirac fermions, supporting the claim that the kagome QSL is described by QED3. We benchmark our methods on a quantum critical point between a Dirac semimetal and a charge ordered phase, described by the Gross-Neveu-Yukawa CFT. Field theory calculations will be discussed.

Quantum fluctuations of the N ́eel state of the square lattice antiferromagnet are usually described by a CP1 theory of bosonic spinons coupled to a U(1) gauge field. Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge 2 Higgs fields, deconfined phases with Z2 topological order possibly intertwined with discrete broken global symmetries. We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in π-flux in each square lattice plaquette. Fluctuations about this π-flux state are described by 2+1 dimensional quantum chromodynamics (QCD3) with a SU(2) gauge group and Nf = 2 flavors of massless Dirac fermions. It has recently been argued by Wang et al. (arXiv:1703.02426) that this QCD3 theory describes the Neel- VBS quantum phase transition. We introduce adjoint Higgs fields in QCD3, and obtain fermionic dual descriptions of the phases with Z2 topological order obtained earlier using the bosonic CP1 theory. We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state. The global phase diagram of these phases contains multi-critical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.

The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid featureless phases from arising in certain spin systems. In this talk, we address the question how the LSM theorems are reflected at quantum critical points in the spin systems. In the field theory that captures the vicinity of these critical points, both the spatial and spin rotation symmetries act as on-site symmetries on the low energy degrees of freedom. The consequence of the LSM theorems is such that the symmetry action in the field theory is anomalous and can be identified as that in the boundary state of a symmetry-protected topological state in one higher dimension. Such a consequence can, in turn, serve as an indicator for the existence of generalized LSM theorems. From this perspective, we propose a new set of generalized LSM theorems for the 1D chain, 2D square lattice, 2D honeycomb lattice and 3D cubic lattice with SU(N) and SO(N) spin rotation symmetries.

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We discuss several bosonic topological phases in (3+1) dimensions enriched by a global Z2 symmetry, and gauging the Z2 symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular S and T matrices and compare them with (2+1)d topological phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in (3+1) dimensions described by the BF topological quantum field theories, with abelian exchange statistics between point-like and loop-like quasiparticles. Once the Z2 charge conjugation symmetry is gauged, the Z2 flux becomes non-abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-abelian group in (3+1) dimensions.

Recent developments in the theory of composite fermions (CFs) have led to a prediction of a π Berry phase for the CFs circling around the Fermi surface at half-filling ν=1/2 (where ν is the Landau level filling factor). In this talk, we provide the first experimental evidence for the detection of the Berry phase of CFs in the fractional quantum Hall effect by studying their density-dependent magnetotransport in heterojunction insulated-gate field-effect transistors (HIGFETs), in which the electron density can be tuned over a large range. Our measurements of the Shubnikov-de Haas oscillations of CFs as a function carrier density at a fixed magnetic field provide a strong support for an existence of a π Berry phase at ν = 1/2. We also discover that the conductivity of composite fermions at ν = 1/2 displays an anomalous linear density dependence, whose origin remains mysterious yet tantalizing. The work presented here was performed in collaboration with Woowon Kang, Kirk Baldwin, Ken West, Loren Pfeiffer, and Daniel Tsui (Nature Physics 13, 168–1172 (2017)).

We develop a vortex metal theory for partial filled Landau Level at 1/(2n) filling, whose ground state contains a composite Fermi surface formed by the vortex of electrons. In the projected Landau Level limit, the composite Fermi surface contains (−π/n) Berry phase. Such fractional Berry phase is a consequence of LL projection which produces the GMP guiding center algebra and embellishes an anomalous velocity to the equation of motion for the vortex metal. Further, we investigate a particle-hole symmetric bilayer system with filling 1/(2n) and 1−1/(2n) at each layer, and demonstrate that the (−π/n) Berry phase on the composite Fermi surface leads to the suppression of 2k_f back-scattering between the PH partner bilayer, which could be a smoking gun to detect the fractional Berry phase. We also mention various instabilities and competing orders in such bilayer system including a Z_{4n} topological order phase driven by quantum criticality.

Recent work on a family of boson-fermion mappings has emphasized the interplay of symmetry and duality: Phases related by a particle-vortex duality of bosons (fermions) are related by time-reversal symmetry in their fermionic (bosonic) formulation. I will present exact mappings for a number of concrete models that make this property explicit on the operator level. I will illustrate the approach with one- and two-dimensional quantum Ising models, and then similarly explore the duality web of complex bosons and Dirac fermions in (2+1) dimensions.

Recently there has been much interest in an IR "Boson-Fermion duality" in 3 spacetime dimensions. This duality generates many more dualities, forming the so-called "duality web"; some of these dualities have been extremely helpful in understanding intricate problems such as half-filled Landau level and surface of strongly interacting topological insulator, etc. Despite the usefulness of these dualities, a solid foundation of them was in need. In this talk I will present how the elementary Boson-Fermion duality can be UV-completed as an exact mapping of lattice gauge theories, thereby providing a simple, non-perturbative proof of the IR Boson-Fermion duality and hence the entire duality web.

The conventional picture of possible ground states in a two-dimensional electron gas (2DEG) system, at zero temperature and in the presence of disorder, allows only superconducting or insulating phases (and in magnetic field also quantum Hall liquid phases). Tuning the disorder and/or magnetic field between superconducting and insulating ground states -- the so called superconductor-insulator transition (SIT) -- has received acute attention because they led to exploration of new ground states and appear to be broadly relevant to other quantum phase transitions (QPTs) and unsolved puzzles such as unconventional superconductivity in the high-Tc cuprates. In particular, detailed experimental studies of disordered superconducting thin-films near the magnetic-field tuned superconductor-insulator transition (SIT) have revealed several unexpected new ground states for films that otherwise superconduct at zero magnetic field. For weakly disordered films (with normal state resistivity small compared to the quantum of resistance, the superconducting state gives way to an anomalous metallic phase with a resistivity that extrapolates to a non-zero value as the temperature tends to zero. For highly disordered superconducting films, a direct SIT occurs at at some critical field, giving way to a boson-dominated insulator. By supplementing the longitudinal resistance with Hall resistance data, we show that the resistivity tensor at criticality approaches the universal value expected at a point of vortex-particle self-duality, while the insulating phase proximate to the SIT appears to be a Hall insulator where in the limit of T-->0, the longitudinal resistance tends to infinity while the Hall resistance is finite [PNAS 113, 280-285 (2016)]. These new results shade light on the nature of the SIT and bare important consequences to other QPTs in two-dimensional systems.

We present further evidence for the particle-hole symmetric response of Halperin, Lee, Read (HLR) theory. We explain why spatially inhomogeneous configurations of the half-filled Landau level as described by HLR theory exhibits particle-hole symmetry, and describe the relation between HLR and the Dirac formulations of composite fermions.

Using a simple four-fermion model constructed with staggered fermions on a space-time lattice we argue that fermions can become massive through the formation of a four-fermion condensate as opposed to the well known fermion bilinear condensate. In this mechanism of fermion mass generation fermion masses arise dynamically as opposed to the usual mechanism that requires spontaneous breaking of a symmetry. The massive phase is robust in all space-time dimensions. Using large scale lattice calculations we provide evidence that the unconventional massive phase is asymptotically free in two dimensions and connected to a fermionic quantum critical point in three dimensions, suggesting the presence of a continuum limit. In four dimensions however, we cannot rule out the old paradigm that the symmetric massive phase is a lattice artifact.

I will present some recent numerical results on three dimensional gauge theories coupled to massless fermions.

ABSTRACT TBD

Recent experiments in graphene hetro-structures have revealed the existence of Chern Insulators - integer and fractional Quantum Hall states made possible by the presence of a periodic substrate potential. Here we show that the new kinds of quantum critical points are enabled by the interplay of magnetic fields, interactions and the periodic potential. We discuss transitions between distinct quantized Hall states using both microscopic models and effective field theories, and highlight two important differences from the well studied disorder driven plateau transition. (i) First, the periodic potential, rather than disorder, enables a change in the Hall conductance at a fixed magnetic field and electron density and (ii) the presence of translation symmetry allows for direct transitions between quantum Hall states that otherwise require fine tuning. Transitions between particle-hole conjugate Jain states take the form of 'pure' QED3 with multiple flavors of Dirac fermions.

We consider a theory of a two-component Dirac fermion localized on a (2+1)-dimensional brane coupled to a (3+1)-dimensional bulk. Using the fermionic particle-vortex duality, we show that the theory has a strong-weak duality that maps the coupling e to (8π)/e. We explore the theory at e2 = 8π where it is self-dual. The electrical conductivity of the theory is a constant independent of frequency. When the system is at finite density and magnetic field at filling factor ν=1/2, the longitudinal and Hall conductivity satisfies a semicircle law, and the ratio of the longitudinal and Hall thermal electric coefficients is completely determined by the Hall angle. The thermal Hall conductivity is directly related to the thermal electric coefficients.