Calendar

In this talk, I will discuss another possibility that arises from the interplay of integrability and symmetry; in integrable one dimensional quantum magnets with complex symmetries, spin transport is neither ballistic nor diffusive, but rather superdiffusive. Using a novel method for the simulation of quantum dynamics (termed Density Matrix Truncation), I will present a detailed analysis of spin transport in a variety of integrable quantum magnets with various symmetries. Crucially, our analysis is not restricted to capturing the dynamical exponent of the transport dynamics and enables us to fully characterize its universality class: for all superdiffusive models, we find that transport falls under the celebrated Kardar-Parisi-Zhang (KPZ) universality class.

Finally, I will discuss how modern atomic, molecular and optical platforms provide an important bridge to connect the microscopic interactions to the resulting hydrodynamical transport dynamics. To this end, I will present recent experimental results, where this KPZ universal behavior was observed using atoms confined to an optical lattice.

[1] Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
B Ye†, FM*, J Kemp*, RB Hutson, NY Yao
(PRL in press) – arXiv:2205.02853

[2] Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion
D Wei, A Rubio-Abadal, B Ye, FM, J Kemp, K Srakaew, S Hollerith, J Rui, S Gopalakrishnan, NY Yao, I Bloch, J Zeiher
Science (2022) — arXiv:2107.00038

Electromagnetic fields in a magneto-electric medium behave in close analogy to photons coupled to the hypothetical elementary particle, the axion. This emergent axion electrodynamics is expected to provide novel ways to detect and control material properties with electromagnetic fields. Despite having been studied intensively for over a decade, its theoretical understanding remains mostly confined to the static limit. Formulating axion electrodynamics at general optical frequencies requires resolving the difficulty of calculating optical magneto-electric coupling in periodic systems and demands a proper generalization of the axion field. In this talk, I will introduce a theory of optical axion electrodynamics that allows for a simple quantitative analysis. Then, I will move on to discuss the issue of the Kerr effect in axion antiferromagnets, refuting the conventional wisdom that the Kerr effect is a measure of the net magnetic moment. Finally, I will apply our theory to a topological antiferromagnet MnBi2Te4.

Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between the two layers can invert the many-body gap of a charge-transfer Mott insulator, resulting in a continuous phase transition to a quantum anomalous Hall insulator with a chiral spin structure. Experimental results will be discussed and compared with theoretical predictions.

Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not previously been addressed. In this talk, I discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula, which can be proved in some cases. Time permitting, I will explain my proof of these conjectures in the case of unitary Friedberg–Jacquet periods.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 2nd.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

The wonderful variety of a realizable matroid and its Chow ring have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. Yet, its $K$-rings are underexplored until recently. I will be sharing with you some discoveries on the $K$-rings of the wonderful variety associated with a realizable matroid: an exceptional isomorphism between the $K$-ring and the Chow ring, with integral coefficients, and a Hirzebruch–Riemann–Roch-type formula. These generalize a recent construction of Berget–Eur–Spink–Tseng on the permutohedral variety. We also compute the Euler characteristic of every line bundle on wonderful varieties, and give a purely combinatorial formula. This in turn gives a new valuative invariant of an arbitrary matroid. As an application, we present the $K$-rings and compute the Euler characteristic of arbitrary line bundles of the Deligne–Mumford–Knudsen moduli spaces of rational stable curves with distinct marked points. Joint with Matt Larson, Sam Payne and Nick Proudfoot.

For more information on the speaker, please see: http://www.shiyue.li

Collective cell migration drives numerous physiological processes such as tissue morphogenesis, wound healing, tumor progression and cancer invasion. However, how the interplay of mechanical interactions and the modes of collective self-organization among cells informs such processes is yet to be established. In this talk, I will focus on the role of three-dimensional force transmission, from a theoretical and computational perspective, on two phenomena: (1) cell extrusion from a cellular monolayer and (2) density-independent solid-like to fluid-like transition of active cell layers. For the first topic, I will focus on how increasing cell-cell adhesion relative to cell-substrate adhesion enables cells to collectively exploit distinct mechanical pathways – leveraging defects in nematic and hexatic phases associated with cellular arrangement – to eliminate an unwanted cell. For the second topic, I will show how solid-like to fluid-like transition in active cell layers is linked to the percolation of isotropic stresses. This is achieved via two distinct and independent paths to model this transition by increasing (a) cell-cell adhesion and (b) active traction forces. Additionally, using finite-size scaling analyses, the phase transition associated with each path is mapped onto the 2D site percolation universality class. Our results highlight the importance of force transmission in informing the collective behavior of living cells and opens the door to new sets of questions for those interested in connecting the physics of cellular self-organization to the dynamics of biological systems.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/active-matter-seminar

The pentagram map was introduced by Schwartz as a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of real tori. We will explain how the real, complex, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties (except possibly in characteristic 2). The second hour will get into the details of the proof, especially the definition of the Lax representation and the spectral curve.

Bordered Floer homology is an extension of Heegaard Floer homology to manifolds with parametrized boundary, and in the case of manifolds with torus boundary knot Floer homology gives another such extension. In earlier joint work with J. Rasmussen and L. Watson, it was shown that in this setting the bordered Floer invariant, which is equivalent to the UV=0 truncation of the knot Floer complex, can be encoded geometrically as a collection of immersed curves in the punctured torus and a pairing theorem recovers HF-hat (the simplest version of Heegaard Floer homology) of a glued manifold via Floer homology of immersed curves. In this talk, we will survey some applications of this result and then discuss a generalization that encodes the full knot Floer complex of a knot as a collection of decorated immersed curves in the torus. When two manifolds with torus boundary are glued, a pairing theorem computes HF^- of the resulting manifold as the Floer homology of certain immersed curves associated with each side. We remark that the curves we describe are invariants of knots, but we expect they are in fact invariants of the knot complements; if this is true, they may be viewed as defining a minus type bordered Floer invariant for manifolds with torus boundary.