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We elucidate the vector space (twisted relative cohomology) that is Poincar\&#39;e dual to the vector space of Feynman integrals (twisted cohomology) in general dimension. These spaces are paired via an inner product called the intersection number -- an invariant which can be computed algebraically. In this language, reduction of an integrand modulo integration-by-part identities is simply its projection onto a chosen basis. The dual-forms turn out to be far simpler than their Feynman counterparts; they are uplifts of lower-dimensional forms supported on the maximal cuts of various sub-topologies (boundaries). The intersection numbers are then computed by taking residues around cuts where various subsets of propagators become on-shell, giving a systematic approach to generalized unitarity. Moreover, the dual integrals satisfy the transposed differential equation to their Feynman counterparts. Since the dual-forms are localized to cut the transposed differential equation is a much simpler to construct.

CAP Virtual Congress 2021

Generalized Unitarity and the Poincaré Duals of Feynman Integrals

algebraic geometry

intersection theory

integration-by-parts identities

generalized unitarity

amplitudes

feynman integrals

The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. The program featured plenary speakers, sessions, workshops and many special activities.

For more information please visit the 2020 AAPT Virtual Summer Meeting page at: https://www.aapt.org/Conferences/SM2020/index.cfm.


AAPT SM 20

The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. 

We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general dimension. These spaces are paired via an inner product called the intersection number -- an invariant which can be computed algebraically. In this language, reduction of an integrand modulo integration-by-part identities is simply its projection onto a chosen basis. The dual-forms turn out to be far simpler than their Feynman counterparts; they are uplifts of lower-dimensional forms supported on the maximal cuts of various sub-topologies (boundaries). The intersection numbers are then computed by taking residues around cuts where various subsets of propagators become on-shell, giving a systematic approach to generalized unitarity. Moreover, the dual integrals satisfy the transposed differential equation to their Feynman counterparts. Since the dual-forms are localized to cut the transposed differential equation is a much simpler to construct.

technical paper

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Laser-induced rotational wavepacket of hydrogen molecules has been experimentally observed in real time by using two sequential 25-fs laser pulses (pump-probe scheme) and a COLTRIMs spectrometer. By measuring the time-dependent yield of the above-threshold dissociation and the enhanced ionization of the molecule, we observed a few-femtosecond time delay in between the two dissociation pathways for both H2 and D2. The delay was understood and interpreted by a classical model considering enhanced ionization requires extra interaction with the laser field. We demonstrate the molecule rotational clock in hydrogen molecule is a straightforward method for timing ultrafast molecular dissociation dynamics.

Clocking enhanced ionization of hydrogen molecule using molecular rotational wavepackets

Currently, plasmonic nanofibers doped with semiconductor quantum dots, organic dye quantum emitters (QEs), and metallic nanoparticles (MNPs) have attracted much attention due to their wide range of applications including waveguides, light-sources, and optical sensors. These nanofibers doped with QEs and MNPs have been fabricated using a variety of metals and emitters. For example, Hu et al. [1] have studied the fabrication of a plasmonic random fiber from gold MNPs and pyrromethene dye molecules (QEs) embedded in the liquid core optical fiber. They found that a narrower and sharper photoluminescence (PL) spectrum can be more easily obtained when there is greater overlap between the plasmonic resonance of the gold-MNP and the dye molecules. Here we have developed a theory of photoluminescence for plasmonic nanofibers [2]. When probe light propagates inside the nanofiber, it induces surface plasmon polariton (SPPs) and electric dipoles in metallic nanoparticles. These dipoles interact with each other via the dipole-dipole interaction (DDI) [3]. The energy of photonic bound states in the presence of the SPP and DDI fields is then calculated. We have demonstrated that the number of bound states can be controlled by changing the strength of the SPP and DDI couplings. The expression of photoluminescence has been calculated using the density matrix method in the presence of the DDI coupling. We found that the intensity of the PL spectrum depends on the quality called quantum efficiency, which depends on the radiative and non-radiative decay rates. We have found that the quantum efficiency is enhanced when the exciton energy is in resonance with the bound photon energy. Further, we predicted that the PL intensity is also enhanced due to the DDI coupling. The enhancement of the PL spectrum can be used to fabricate plasmonic nanosensors. 

[1] Hu, Z et al., Gold nanoparticle-based plasmonic random fiber laser. J. Opt. 2015, 17, 35001.
[2] Singh, M. R.; Brassem, G.; Yastrebov, S. G. Optical quantum yield in plasmonic
Nanowaveguide. Annalen der Physik in press, 2021.
[3] Singh, M. R. The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials. Journal of Modern Optics 2007, 54, 1739.

Study of photoluminescence in plasmonic nanoparticles

Graphene is an interesting system for condensed matter physicists because of many potential technological applications. It is interesting to theorists as an abelian strongly coupled system that has some of the interesting properties of QCD.

We are particularly interested in studying the effects of anisotropic strain of the graphene
lattice on the critical coupling, which determines the transition point for graphene
from a conductor to an insulator. We use a low energy effective theory and a Schwinger-Dyson approach. The full set of coupled equations is numerically very difficult to solve, and so previous calculations have made many simplifying assumptions in order to make the procedure tractable. These simplifications have led to disagreement about the effect anisotropy has on the critical coupling constant. We have studied the effect of some common approximations including polarization tensor approximations and vertex ansaetze. In this talk we discuss the motivation for several of these approximations, and show that in some cases their effect on the critical coupling is significant. We discuss more accurate ways to solve the non-perturbative low energy effective theory, and its ability to predict the critical coupling of the phase transition.

Semi-metal insulator phase transitions in graphene - the role of anisotropy

Organic molecules capable of forming strongly bound self-assembled monolayers (SAMs) offer a promising route to surface modification and functionalization. Technological advances including the protection and stabilization of nanoparticles and the development of lab-on-chip sensors have been largely realized by the adsorption of alkanethiols. *N*-heterocyclic carbenes (NHCs) were recently introduced as alternative anchors in the preparation of SAMs on metal surfaces and may be superior to thiol-based systems. NHCs have important advantages compared to thiol-based systems, most notably their chemical tunability and ability to form strongly bound monolayers with improved stability. Given their novelty in the field of materials science, many fundamental properties of NHC-based SAMs are not well-understood, including factors that control the adsorption and self-assembly processes.

By means of scanning tunneling microscopy and further supplemented by *ab initio* (DFT) and Monte Carlo simulations, we investigate the mechanisms by which NHCs bind and self-assemble on Au(111). Our investigation demonstrates a critical dependence of the binding mode on NHC structure, surface coverage, and substrate temperature. The dependence on these factors arises due to the minimization of molecule-surface steric interactions, the production of Au adatoms by the effect of NHC adsorption on the underlying gold substrate, and the reactivity and stability of the ligand, respectively. The main effect on the adsorption process is a competition between upright attachment geometries and the formation of flat-lying bis-NHC-metal complexes. These species form a rich variety of self-assembled structures, including in particular a zig-zag lattice formed by adatom-bound NHCs and a non-porous chiral Kagome lattice containing trapped NHC monomers. These lattices are analyzed within our three-pronged methodology, providing valuable insights into molecule--molecule and molecule-substate interactions, as well as the role of surface modifications and on-surface chemical reactions in the formation of NHC SAMs. Efforts to understand these processes were crucial in advancing the chemistry and applications of thiol-based SAMs, and will similarly offer critical strategies for the design of upright NHC SAMs and their applications.

N-heterocyclic carbene adsorption and self-assembly on Au(111): Fine-tuning the binding mode

One of the more exciting things to emerge from black hole thermodynamics in the past 10 years is the understanding that black holes can undergo a broad range of chemical-like phase transitions, including liquid-gas phase transitions, triple points, superfluid transitions, polymer-type transitions, and exhibit critical behaviour. It is even possible to consider black holes as the working material for heat engines. The efficiencies for a variety of black holes can be calculated and compared against each other.

In this talk I will discuss the connection between critical behaviour and the efficiency of black hole heat engines. I first consider the heat capacity of static black holes at constant volume such that Cv=0 Using the near critical expansion of the equation of state, the coefficients appearing in this expansion can be found from an engine cycle placed along a critical point on a PV plot. 

I will discuss the importance and applications of the simplifications made, along with how this result allows one to go from the near critical expansion of the equation of state directly to a conclusion about the behaviour of a heat engine near the critical point.

Black Hole Heat Engines and Critical Behaviour

One of the more exciting things to emerge from black hole thermodynamics is that black holes can form the working material for heat engines. I explore the connection between the critical behaviour of black holes and their efficiency as heat engines over a range of dimensions and for a variety of theories of gravity. 

I first show that their efficiency as heat engines near the critical point can be written in general dimensions in terms of the variables characterizing the geometry of the cycle and the critical exponents. Engines near the critical point approach the Carnot efficiency, with the rate of approach determined by the universality class of the black hole. I will specifically consider a broad range of charged black holes, Lovelock black holes, and black holes with isolated critical points. I will then discuss work in progress exploring this formalism for black holes whose specific heat at constant volume is nonzero, applying it to examples such as rotating black holes and STU black holes.

Efficiency of Black Hole Heat Engines and Universality

The common picture of a binary black hole merger is the “pair of pants” diagram for the event horizon. However, in many circumstances, such as those encountered in numerical simulations, the event horizon may be ill-suited and it is more practical to work with quasi-local definitions of black hole boundaries, such as marginally outer trapped surfaces (MOTS). The analog of the pair of pants diagram for the apparent horizons remains to be fully understood. In this talk, I will discuss the complete picture for the merger of two axisymmetric black holes. I will begin by introducing new classes of MOTS present in Brill-Lindquist initial data. I will then discuss the role played by these and related surfaces in understanding the final fate of the apparent horizons of the initial two participants in the merger.

The fate of apparent horizons in a binary black hole merger

I describe the first investigation of the holographic complexity conjectures for rotating black holes. Exploiting a simplification that occurs for equal-spinning odd dimensional black holes, I demonstrate a relationship between the complexity of formation and the thermodynamic volume associated with the black hole. This result suggests that it is thermodynamic volume and not entropy that governs the complexity of formation in both the Complexity Equals Volume and Complexity Equals Action proposals. This proposal reduces to known results involving the entropy in settings where the thermodynamic volume and entropy are not independent, but has much broader scope. Assuming the validity of a conjectured inequality for thermodynamic volume, this result suggests the complexity of formation is bounded from below by the entropy for large black holes.

Holographic Complexity and Black Hole Thermodynamic Volume

We carry out the first investigation of the entanglement and mutual information harvesting protocols for detectors moving on freely falling trajectories that cross the horizon of a black hole. We consider two pointlike Unruh-DeWitt detectors in different combinations of free-falling and static trajectories in (1+1)-dimensional Schwarzschild black hole spacetime and compare the results. We find that (i) correlations harvested between free-falling and static detectors are always less than those of the familiar two-static-detector case (ii) there is a large kinematic component to the correlations; (ii) correlations can be harvested purely from the quantum vacuum, with evidence that this is possible even when the detectors are causally disconnected by the event horizon; (iii) the previously known `entanglement shadow' near the horizon is indeed absent for the two free-falling-detector case since the relative gravitational redshift of the detectors remains finite as the horizon is crossed, in accordance with the equivalence principle.

Harvesting Entanglement inside a Black Hole

Magnetic Resonance Imaging (MRI) detects signal from hydrogen nuclei in biological tissue. MRI requires a homogeneous static magnetic field to generate artifact-free images. The subject is spatially encoded with magnetic field gradients. The signal is acquired in the frequency domain and the image is reconstructed by inverse Fourier transform. Objects with high magnetic susceptibility, such as MRI-safe metallic implants, distort the surrounding magnetic field. This leads to severe artifacts that appear as signal voids in the conventional MRI images, due to rapid intravoxel dephasing, in addition to misregistration of frequencies to position.
Pure phase encoding techniques with short encoding times are largely immune to magnetic field inhomogeneity artifacts. This is because the constant signal evolution times can be sufficiently short that no appreciable dephasing has occurred. High quality artifact-free MRI images were acquired with pure phase encoding techniques, from which the magnetic field distribution around the metal was derived. This approach was compared with conventional MRI methods, which failed to map the magnetic field in high susceptibility regions. Although it is challenging to apply the proposed method in a routine clinical MRI scan, the measured magnetic field distribution could enable the development of novel nonlinear encoding techniques, where the metal induced magnetic field distortion is exploited to provide spatial information.

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Generalized Unitarity and the Poincaré Duals of Feynman Integrals

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Clocking enhanced ionization of hydrogen molecule using molecular rotational wavepackets

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