Special Session 99: Recent Advances in Mathematical Physics: A focus on (many-body) quantum systems and spectral theory.

Mathematics of Moire materials

Simon Becker ETH Zurich Switzerland Co-Author(s):

Abstract: I will review recent advancements in the field of twistronics. Twistronics investigates how varying the angle between layers of two-dimensional materials can significantly alter their electrical properties. Notably, twisted bilayer graphene exhibits a broad spectrum of different strongly correlated phases of matter, ranging from non-conductive to superconductive, depending on the specific twist angle between its layers. These remarkable effects are observed at particular angles known as magic angles.

Ground States of Spin Boson Models and Long Range Order in 1D Ising Models

Benjamin Hinrichs Paderborn University Germany Co-Author(s):

Abstract: The spin boson model describes a two-state quantum system (the spin) which is linearly coupled to a bosonic quantum field. Using a Feynman--Kac formula, the vacuum expectation of the semigroup generated by the Hamiltonian of the model can be described by a one-dimensional Ising model. In this talk, we discuss how the existence of ground states in the spin boson model can be investigated by studying the long-range behavior of the Ising correlation function. This especially includes the recent proof for a phase transition of the spin boson model with respect to the strength of the spin-field coupling.

Super-critical entanglement in strongly interacting simple models

Ramis Movassagh Google Quantum AI USA Co-Author(s):    Varun Menon, Andi Gu

Abstract: We discuss recent developments on classes of simple quantum models in one-dimension that have surprising properties. These models have a rich combinatorial ground state structure in terms of Motzkin walks and Brownian excursions, have a local interaction, and unique ground states. They are surprising in that they are not describable by conformal field theories in the limit, and have exponentially more amount of entanglement entropy than even quantum critical systems. We introduce these models and present recent work on their spontaneous U(1) symmetry breaking, correlation functions, and new analytical work on their t-deformations.

LTQO and spectral gap stability for the AKLT model on the hexagonal lattice

Bruno Nachtergaele University of California, Davis USA Co-Author(s):    Thomas Jackson (UC Davis) and Amanda Young (UIUC)

Abstract: Local Topological Quantum Order (LTQO) is a crucial ingredient in proofs of stability of the spectral gap above the ground state of frustration free quantum lattice systems. LTQO has been proved to hold quite generally in one space dimension. It is also known in higher dimensions for many commuting Hamiltonians. In this work we prove LTQO for the AKLT model on the hexagonal lattice, adding one item to a rather short list of results that apply to non-commuting Hamiltonians in two or more dimensions.

Atypical spectra and dynamics of non-locally finite crystals

Mostafa Sabri New York University Abu Dhabi United Arab Emirates Co-Author(s):    Joachim Kerner, Olaf Post, Mattias Taufer

Abstract: In this talk, we investigate the spectral theory of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These periodic graphs are shown to have rather intriguing behaviour. We construct a periodic graph whose Laplacian has purely singular continuous spectrum. We prove that motion remains ballistic along at least one layer under quite general assumptions. We construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This answers negatively an open question in this regard, in our setting. Concerning the point spectrum, we construct a graph with a partly flat band whose eigenvectors must have infinite support, in contrast to the locally finite case. We believe the present class of graphs can serve as a playground to better understand exotic spectra and dynamics in the future.

Entanglement entropy in the ground state of non-interacting massless Dirac fermions in dimension one

Wolfgang Spitzer FernUniversitaet in Hagen Germany Co-Author(s):    Fabrizio Ferro, Paul Pfeiffer

Abstract: We present a novel proof of a formula of Casini and Huerta for the entanglement entropy of the ground state of non-interacting massless Dirac fermions in dimension one localized to (a union of) intervals and generalize it to the case of R\`enyi entropies. At first, we prove that these entropies are well-defined for non-intersecting intervals. This is accomplished by an inequality of Alexander V.~Sobolev. Then we compute this entropy using a trace formula for Wiener--Hopf operators by Harold Widom. For intersecting intervals, we discuss an extended entropy formula of Casini and Huerta and support this with a proof for polynomial test functions.

Quantum-inspired framework for computational fluid dynamics

Egor Tiunov Technology Innovation Institute, Abu Dhabi, UAE United Arab Emirates Co-Author(s):    Raghavendra Dheeraj Peddinti, Stefano Pisoni, Alessandro Marini, Philippe Lott, Henrique Argentieri, Leandro Aolita

Abstract: Computational fluid dynamics is both a thriving research field and a key tool for advanced industry applications. However, the simulation of turbulent flows in complex geometries is a compute-power intensive task due to the vast vector dimensions required by discretized meshes. We present a complete and self-consistent full-stack method to solve incompressible fluids with memory and run time scaling logarithmically in the mesh size. Our framework is based on matrix-product states, a compressed representation of quantum states. It is complete in that it solves for flows around immersed objects of arbitrary geometries, with non-trivial boundary conditions, and self-consistent in that it can retrieve the solution directly from the compressed encoding, i.e. without passing through the expensive dense-vector representation. This framework lays the foundation for a generation of more efficient solvers of real-life fluid problems.

A weak limit theorem for quantum walks in 1-dimension

Kazuyuki Wada Hokkaido University of Education Japan Co-Author(s):    Masaya Maeda, Akito Suzuki

Abstract: We consider the weak limit theorem for discrete-time quantum walks corresponding to the central limit theorem for classical random walks. The first result of this theorem was derived by Konno. After that, Suzuki extended this result to short-range cases. The next step is whether we can extend this result to long-range cases. In this talk, we will report some results in this direction.

Exponential tail estimates for quantum lattice dynamics

Albert H. Werner QMATH - University of Copenhagen Denmark Co-Author(s):    Christopher Cedzich, Alain Joye, Reinhard F Werner

Abstract: We consider the quantum dynamics of a particle on a lattice for large times. Assuming translation invariance, and either discrete or continuous time parameter, the distribution of the ballistically scaled position Q(t)/t converges weakly to a distribution that is compactly supported in velocity space, essentially the distribution of group velocity in the initial state. We show that the total probability of velocities strictly outside the support of the asymptotic measure goes to zero exponentially with , and we provide a simple method to estimate the exponential rate uniformly in the initial state. Near the boundary of the allowed region the rate function goes to zero like the power 3/2 of the distance to the boundary.

Propagation bounds for long-range interacting bosons

Jingxuan Zhang Tsinghua University Peoples Rep of China Co-Author(s):    M. Lemm; C. Rubiliani; I. M. Sigal

Abstract: We study propagation properties of nonequilibrium lattice models with long-range bosonic interactions. We prove (A) microscopic particle transport bounds and (B) Lieb-Robinson bounds, which are all thermodynamically stable. For both (A) and (B), we require Bose-Hubbard type Hamiltonians with hopping matrix decaying as $|x-y|^{-\alpha}$, $\alpha>d+2$ and initial state with uniformly bounded density from above. For (B), we further assume initially no particle lies in the region separating the supports of the probing observables. The proofs are based on a combination of commutator method originated in scattering theory and novel monotonicity estimate for certain adiabatic observables that track the spacetime localization of evolving states.