- CMSA EVENT: CMSA Interdisciplinary Science Seminar: Mechanics of biomolecular assemblies
The mechanical properties of biomolecular assemblies play pivotal roles in many biological and pathological processes. In this talk, I’ll focus on two different self-assembled structures in cells: 1) the plasma membrane, which defines the boundary of a cell; and 2) protein condensates, which arise from liquid-liquid phase separation (LLPS) inside cells.
In the first part, I’ll discuss recent findings on how cell membranes respond to local mechanical perturbations. In most non-motile cells, local perturbations to membrane tension remain highly localized, leading to subcellular Ca2+ influx and vesicle fusion events. Membrane-cortex attachments are responsible for impeding the propagation of membrane tension. Exception to this rule can be found in the axon of neurons, where a rapid propagation of membrane tension coordinates the growth and branching of the axon.
In the second part, I’ll discuss the development of quantitative techniques to measure the surface tension and viscosity of liquid protein condensates. Our results highlight a common misconception about LLPS in biology: ‘oil droplets in water’ is often used to give an intuition about protein condensates in cells. However, oil droplets and protein condensates represent two extremes in the realm of liquid properties. The unique properties of protein condensates have important implications in achieving molecular and functional understanding of LLPS.
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: From Fractional Quantum Hall to higher rank symmetry
Electron gas in 2+1D in a strong magnetic field forms fractional quantum Hall states. In this talk, I will show that electrons in the lowest Landau level limit of FQH enjoy the area-persevering diffeomorphism symmetry. This symmetry is the long-wavelength limit of W-infinity symmetry. As a consequence of the area-preserving diff symmetry, the electric dipole moment and the trace of quadrupole moment are conserved, which demonstrates the fractonic behaviour of FQH systems. Gauging the area-preserving diff gives us a non-abelian higher-rank gauge theory whose linearized version is the traceless symmetric tensor gauge theory proposed by Pretko. Using the traceless symmetric tensor gauge formalism, I will derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term. I will extend the discussion to the area-preserving diff in 3+1D, the physical system that realizes this symmetry is skyrmions in ferromagnets.
- CMSA EVENT: CMSA Interdisciplinary Science Seminar: Modeling invertible topological phases of matter using homotopy theory
Condensed-matter theorists have discovered examples of physical systems with unusual behavior, such as pointlike excitations that behave neither as bosons nor as fermions, leading to the idea of topological phases of matter. Classifying the possible topological phases has been the focus of a lot of research in the last decade in condensed-matter theory and nearby areas of mathematics. In this talk, I’ll focus primarily on the special case of invertible phases, also called symmetry-protected topological (SPT) phases, whose classification uses techniques from homotopy theory. I will discuss two different approaches to this, due to Kitaev and Freed-Hopkins, followed by details of the homotopy-theoretic classifications. The latter includes work of Freed-Hopkins and of myself.
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Solvable Lattice Hamiltonians with Fractional Hall Conductivity
We construct a class of bosonic lattice Hamiltonians that exhibit fractional Hall conductivity. These Hamiltonians, while not being exactly solvable, can be reliably solved in their low energy sectors through a combination of perturbative and exact techniques. Our construction demonstrates a systematic way to circumvent the Kapustin-Fidkowski no-go theorem, and is applicable to more general cases including fermionic ones.
References: Zhaoyu Han and Jing-Yuan Chen, [2107.0xxxx] Jing-Yuan Chen, [1902.06756].
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Hybrid Fracton Orders
I will introduce a family of gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders called “Hybrid Fracton Orders”. First, I will present the simplest example of such an order: the “Hybrid X-cube” model, where excitations can be labeled identically to those of the Z2 toric code tensored with the Z2 X-cube model, but exhibit fusion and braiding properties between the two sets of excitations. Next, I will provide a general construction of hybrid fracton orders which inputs a finite group G and an abelian normal subgroup N and produces an exactly solvable model. Such order can host non-abelian fracton excitations when G is non-abelian. Furthermore, the mobilities of a general excitation is dictated by the choice of N, from which by varying, one can view as “interpolating” between a pure 3D topological order and a pure fracton order.
Based on 2102.09555 and 2106.03842
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group G, we show how to construct a (3+1)D topologically invariant path integral for a fermionic G symmetry-protected topological state (G-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form Z_2 background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the Z_16 anomaly indicator for time-reversal symmetric topological superconductors with T^2=(−1)^F. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all Z_16 Pin+ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin+ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.
- CMSA EVENT: CMSA Interdisciplinary Science Seminar: Moduli spaces of stable pairs on algebraic surfaces
As a variant of Grothendieck’s Quot schemes, we introduce the moduli space of limit stable pairs. We show an example over a smooth projective algebraic surface where there is a virtual fundamental class. We are able to describe this class explicitly. We will also show an application towards moduli of sheaves.
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Emergent spacetime supersymmetry in topological phases of matter
No definitive evidence of spacetime supersymmetry (SUSY) that transmutes fermions into bosons and vice versa has been revealed in nature so far. One may wonder whether SUSY can be realized in quantum materials. In this talk, I shall discuss how spacetime SUSY may emerge, in the sense of renormalization group flow, in the bulk of Weyl semimetals or at the boundary of topological insulators. Moreover, we have performed large-scale sign-problem-free quantum Monte Carlo simulations of various microscopic lattice models to numerically verify the emergence of spacetime SUSY at quantum critical points on the boundary of topological phases. I shall mention some experimental signatures such as optical conductivity which can be measured to test such emergent SUSY in candidate systems like the surface of 3D topological insulators.
 Shao-Kai Jian, Yi-Fan Jiang, and Hong Yao, Phys. Rev. Lett. 114, 237001 (2015)
 Shao-Kai Jian, Chien-Hung Lin, Joseph Maciejko, and Hong Yao, Phys. Rev. Lett. 118, 166802 (2017)
 Zi-Xiang Li, Yi-Fan Jiang, and Hong Yao, Phys. Rev. Lett. 119, 107202 (2017)
 Zi-Xiang Li, Abolhassan Vaezi, Christian Mendl, and Hong Yao, Science Advances 4, eaau1463 (2018)
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Boundary criticality of the O(N) model in d = 3 critically revisited
It is known that the classical O(N) model in dimension d > 3 at its bulk critical point admits three boundary universality classes: the ordinary, the extra-ordinary and the special. The extraordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in d = 3, it is less clear what happens to the extra-ordinary and special fixed points when d = 3 and N is greater or equal to 2. I’ll show that formally treating N as a continuous parameter, there exists a critical value Nc > 2 separating two distinct regimes. For N < Nc the extra-ordinary fixed point survives in d = 3, albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of log r. For N > Nc there is no fixed point with order parameter correlations decaying slower than power law. I’ll discuss how these findings compare to recent Monte-Carlo studies of classical and quantum spin models with SO(3) symmetry. Based on arXiv:2009.05119.
- CMSA EVENT: CMSA Interdisciplinary Science Seminar: Joint moments of multi–species $q$–Boson
The Airy_2 process is a universal distribution which describes fluctuations in models in the Kardar–Parisi–Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for analogous fluctuations of multi–species models. Here, we will discuss one model in the KPZ universality class, the $q$–Boson. We will show that the joint multi–point fluctuations of the single–species $q$–Boson match the single–point fluctuations of the multi–species $q$–Boson. Therefore the single–point fluctuations of multi–species models in the KPZ class ought to be the Airy_2 process. The proof utilizes the underlying algebraic structure of the multi–species $q$–Boson, namely the quantum group symmetry and Coxeter group actions.
- CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: The nu=5/2 enigma: Recent insights from theory and experiment
Non-Abelian phases of matter have long inspired quantum physicists across various disciplines. The strongest experimental evidence of such a phase arises in quantum Hall systems at the filling factor 5/2 but conflicts with decades of numerical works. We will briefly introduce the 5/2 plateau and explain some of the key obstacles to identifying its topological order. We will then describe recent experimental and theoretical progress, including a proposal for resolving the 5/2 enigma based on electrical conductance measurements.