Abstracts
Federico Camia (NYU Abu Dhabi)
Title:
Conformal field theory from Brownian loops
Abstract:
The Brownian loop soup is a conformally invariant Poissonian ensemble of loops in two dimensions, with intensity measure proportional to the unique (up to a multiplicative constant) conformally invariant measure on simple planar loops. In this talk, I will discuss several “operators” that compute statistical properties of the loop soup and behave like conformal primary fields in conformal field theory. For some of these fields one can compute the four-point function explicitly and perform the Virasoro conformal block expansion, which reveals the existence of an infinite number of new primary fields. Among these, we are able to identify all the scalar (spin zero) fields in terms of the operators mentioned above. When the loop soup is restricted to the upper half-plane, we also identify the boundary stress-energy tensor of the theory. The picture that starts to emerge is that of a rich conformal field theory with novel features such as primary fields whose conformal dimensions depend periodically on a real parameter. The tools involved in the analysis include the O(n) model, the full scaling limit of critical percolation, and the Schramm-Loewner Evolution. The talk is based on recent joint work (https://arxiv.org/abs/2109.12116) with Alberto Gandolfi, Valentino Foit and Matthew Kleban.
Kohtaro Kato (Nagoya University)
Title:
Fusion rules of anyons from entanglement
Abstract:
Quantum phases (without symmetry) in two-dimensional (2D) gapped systems are expected to be classified into different topologically ordered phases. A vast amount of work has been dedicated to classify/characterize these new phases, and nowadays, it is widely believed that all bosonic topologically ordered phases can be classified by modular tensor category and chiral central charge. In this work, we show that one can systematically define topological charges and fusion rules from an area law of entanglement for 2D gapped spin systems. We then show our fusion rules satisfy the axioms from modular tensor category. We also discuss recent applications of this approach to S-matrix and central charges.
Hosho Katsura (University of Tokyo)
Title:
Experimental mathematical physics
Abstract:
In this talk, I will discuss how to discover nontrivial exact results for many-body systems using cutting-edge technologies on the internet like the On-Line Encyclopedia of Integer Sequences (OEIS) [1] and Inverse Symbolic Calculator (ISC) [2]. I will first review some classic examples where "guesswork" based on extensive numerical calculations was successful. Then, I will talk about miraculous experiences that I had while exploring quantum many-body systems [3, 4]. If time allows, I will also discuss some open problems that seem to be within reach of this approach.
[2] http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html
[3] E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018).
[4] N. Sannomiya, H. Katsura, and Y. Nakayama, Phys. Rev. D 95, 065001 (2017).
Taro Kimura (Université de Bourgogne)
Title:
Universal multi-critical fluctuation of random partition
Abstract:
In this talk, I will discuss universal multi-critical fluctuations in random partition, which is a natural discrete analog of random matrix. I will explore the asymptotic behavior of the wave function as a building block of the corresponding kernel, and then demonstrate multi-critical analogs of the Airy kernel and the Tracy-Widom distribution associated with the Painlevé II hierarchy. I will also discuss the limit shape of random partitions through the semi-classical analysis of the wave function. This talk is based on https://arxiv.org/abs/2012.06424 and https://arxiv.org/abs/2105.00509
Matteo Mucciconi (University of Warwick)
Title:
Integrable cellular automata, free fermions and KPZ solvable models.
Abstract:
I will discuss a new paradigm of solvability of models in the KPZ universality class that, through bijective combinatorial techniques, relates them to free fermions at positive temperature.
I will focus on two particular models: the periodic Schur measure and the q-Whittaker measure. The first is a model of free fermions at positive temperature. As such its correlations are determinantal and its asymptotic limits are relatively straightforward to study. On the other hand, the q-Whittaker measure is has found application in the rigorous description of KPZ models. It is not (manifestly) a determinantal point process and since mathematical properties of q-Whittaker polynomials are much more complicated than the Schur polynomials, the q-Whittaker measure is a more difficult object to handle. Using a bijective combinatorial approach we are able to relate the theories of Schur and q-Whittaker polynomials producing a remarkable correspondence between the two measures. Byproducts of our construction are determinantal and pfaffian formulas for stochastic systems, whose derivation bypasses complicated Bethe Ansatz calculations. Our arguments pivot around a combination of various theories, which had not yet been used in integrable probability, that include Kirillov-Reshetikhin crystals, Demazure modules, the Box-Ball system or the skew RSK correspondence.
This is a joint work with Takashi Imamura and Tomohiro Sasamoto.
Makoto Nakashima (Nagoya University)
Title:
Gaussian fluctuations of stochastic heat equation and KPZ equation in higher dimension in subcritical regime.
Abstract:
The solution $h$ of KPZ equation in $d=1$ is described by the solution $u$ of the multiplicative stochastic heat equation (SHE) via the Cole-Hopf transformation $h=\log u$. However, for higher dimensions, this argument does not work since SHE does not have a function-valued solution.
In some recent works, the Gaussian fluctuations of the solutions of stochastic heat equation and KPZ equation with space-regularized noise in higher dimensions have been studied. In this talk, we will give a quick overview of them and discuss the joint fluctuations of the solutions. The talk is based on joint work with Clément Cosco and Shuta Nakajima.
Mamoru Okamoto (Osaka University)
Title:
Stochastic quantization of the $\Phi^3_3$-model
Abstract:
We study the construction of the $\Phi^3_3$-measure. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-convergence of the truncated $\Phi^3_3$-measures in the strongly nonlinear regime. We also go over the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise. We first discuss briefly the paracontrolled approach for local well-posedness. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport. This is a joint work with Tadahiro Oh (University of Edinburgh) and Leonardo Tolomeo (University of Bonn).
Yoshinori Sakamoto (Nihon University)
Title:
Expansions for energy eigenstates in the Edwards-Anderson model perturbed by random transverse fields and random bond XY interactions
Abstract:
We consider the Edwards-Anderson (EA) model perturbed by random transverse fields and random bond XY interactions. We study energy eigenstates of the models in $d$-dimensional finite cubic lattice by using the Kirkwood-Thomas convergent expansion method developed by Datta-Kennedy. The perturbations in the models split the two fold degenerate energy eigenvalues due to the ${\mathbb Z}_2$ symmetry in the unperturbed EA model. We show that, for sufficiently weak perturbation, the energy gap between split energy eigenvalues is exponentially small in the system size. We provide some sufficient conditions on the perturbations for the absence of level crossing between arbitrary energy eigenstates in each sector of the ${\mathbb Z}_2$ symmetry. This talk is based on the joint work with C. Itoi, K. Horie and H. Shimajiri.
Makiko Sasada (University of Tokyo)
Title :
On Varadhan’s decomposition theorem in an abstract setting
Abstract :
We rigorously formulate and prove for a relatively general class of interactions the characterization of shift-invariant closed L2-forms for a large scale interacting system. Such characterization of closed forms has played an essential role in proving the hydrodynamic limit of nongradient systems. The universal expression in terms of conserved quantities was sought from observations for specific models, but a precise formulation or rigorous proof up until now had been elusive. To obtain this, we first prove the universal characterization of shift-invariant closed “local” forms in a completely geometric way, that is, in a way that has nothing to do with probability measures. Our result is applicable for many important models including generalized exclusion processes, multi-species exclusion processes, exclusion processes on crystal lattices and so on. This talk is based on a joint work (https://arxiv.org/abs/2009.04699) with Kenichi Bannai and Yukio Kametani.
Etsuo Segawa (Yokohama National University)
Title:
Factors of graph induced by quantum walks
Abstract:
We obtain stationary states of quantum walks on semi-infinite graphs [1]. The initial state of this quantum walk is parametarized by a z=+1 of -1 and uniformly bounded. In this talk, we attempt to extract some structures of the given graph from the stationary state. In particular, as a first trial, we focus on the scattering on the surface of the graph and the energy in the internal graph. If z=1, the stationary state is characterized by an electric network [2], and if z=-1, we newly find that the scattering depends on the bipartiteness of the graph, and we also find that the energy can be obtained by counting some spanning subgraphs. If I have time, some observations on the extended initial state represented by complex number z with |z|=1 will be shown.
[1] Yu. Higuchi, E. Segawa, Journal of Physics A: Mathematics and Theoretical 52(39) 395202 (2019).
[2] Yu. Higuchi , M. Sabri, E. Segawa, Journal of Statistical Physics 21, 603-617 (2020).
Daisuke Shiraishi (Kyoto University)
Title:
Uniform spanning tree in three dimensions
Abstract:
We show that the law of the three-dimensional uniform spanning tree (UST) converges under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also obtain various properties of the intrinsic metric and measure of the scaling limit, including showing that the Hausdorff dimension of such is given by 3/β, where β ≈ 1.624 . . . is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension and its spectral dimension. This talk is based on a joint work with Omer Angel, David Croydon and Sarai Hernandez-Torres.
Rongfeng Sun (National University of Singapore)
Title:
A New Correlation Inequality for Ising Models with External Fields
Abstract:
We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin-spin correlation is maximised when the external field vanishes. In particular, the random field Ising model on Z^d, d>=3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model on Z^d, d>=3, satisfies the conjectured strong spatial mixing property in the entire high temperature regime. Based on joint work with Jian Ding and Jian Song.
Hal Tasaki (Gakushuin university)
Title:
Elementary Index Theorem for One-Dimensional Interacting Topological Insulators
Abstract:
We present a mathematically rigorous index theorem for certain classes of one-dimensional interacting particle systems which include the Su-Schrieffer-Heeger (SSH) model as a special case. We prove, as has been expected, that the sign of the expectation value of the (suitably defined) twist operator gives a topological Z_2 index for a unique gapped ground state of an infinite system.
In this talk we do not assume any knowledge on topological insulators, and start by reviewing simple standard results on the SSH model.
Mayuko Yamashita (RIMS)
Title:
Differential pushforwards, Anderson duality and invertible QFT's
Abstract:
In this talk, I present a mathematical approach to classifications of invertible QFT's (Quantum Field Theories) in terms of generalized differential cohomology theories. Differential cohomology is a framework that refines generalized cohomology theories with differential geometric data on manifolds. First we explain the construction of a differential model of a generalized cohomology theory called "the Anderson dual to G-bordism theories", which is conjectured by Freed and Hopkins to classify invertible QFT's. This model is given by directly abstractizing properties of invertible QFT's. Next we explain that, using this model, many interesting examples of invertible QFT's can be understood as arising from "differential pushforwards (integrations)" in generalized differential cohomology theories. This is a joint work with Kazuya Yonekura (Tohoku).
Kazuki Yokomizo (RIKEN)
Title:
Non-Bloch band theory of non-Hermitian systems
Abstract:
Non-Hermitian systems are nonequilibrium systems which can be described by a non-Hermitian Hamiltonian. Interestingly, non-Hermitian systems with periodic structure have the non-Hermitian skin effect which induce the localization of bulk eigenstates [1]. Then, the non-Hermitian skin effect exhibits the difference between eigenspectra under a periodic boundary condition and those under an open boundary condition. While the eigenspectra under a periodic boundary condition can be obtained from the conventional Bloch band theory, it is unclear how to calculate the eigenspectral under an open boundary condition. In this talk, we present the non-Bloch band theory which can produce eigenspectra of non-Hermitian crystals with an open boundary condition [2]. We show that in the thermodynamic limit, the eigenspectra can be calculated by the generalized Brillouin zone. Remarkably, the eigenspectra are independent of any boundary conditions of an open chain.
[1] S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
[2] K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019).