Title:
Spectral and topological properties of discrete operators with prescribed spatial asymptotics
Abstract:
Discrete Hamiltonians and discrete time-step quantum walk unitaries can often be modeled by combinations of shifts and multiplication by bounded functions. By fixing the spatial asymptotics of the functions under consideration, I will introduce an operator algebraic method to understand the essential spectrum. This method gives us conditions to check if an operator is Fredholm and, in some cases, techniques to compute the Fredholm index. I will try and explain these results with minimal assumed knowledge of operator algebras. This is joint work with S. Richard.
Title:
Contour methods for high-dimensional Long-Range Ising Model
Abstract:
On the $d$-dimensional lattice $\mathbb{Z}^d$ with $d\ge 2$, the phase transition of the nearest-neighbor ferromagnetic Ising model can be proved by using Peierls argument, that requires a notion of contours, geometric curves on the dual of the lattice to study the spontaneous symmetry breaking.
It is known that the one-dimensional nearest-neighbor ferromagnetic Ising model does not undergo a phase transition at any temperature. On the other hand, if we add a polynomially decaying long-range interaction given by $J_{xy}=|x-y|^{-\alpha}$ for $x,y\in \mathbb{Z}$, the works by Dyson and Fr\"{o}hlich-Spencer show the phase transition at low temperatures for $1<\alpha\le 2$. Moreover, Fr\"{o}hlich and Spencer defined a notion of contours for $\alpha=2$. There are many other works that study and extend the notion of contours for other $\alpha$.
In this talk, we define a notion of contours for $d$-dimensional long-range Ising model with $d\ge 2$, where the interactions are given by $J_{xy}=|x-y|^{-\alpha}$ where $\alpha>d$ and $x,y\in \mathbb{Z}^d$. As an application, we add non-homogeneous external fields $h_x=|x|^{-\gamma}$ in the Hamiltonian and give conditions for $\gamma$ and $\alpha$ so that the model undergoes a phase transition at low temperatures.
Joint work with Lucas Affonso, Rodrigo Bissacot, and Satoshi Handa.
Title:
Eigenvalues, overlaps and tensor-valued processes associated with the non-Hermitian matrix-valued Brownian motion
Abstract:
In this talk, we consider stochastic processes associated with eigenvalues and eigenvectors of the non-Hermitian matrix-valued Brownian motion (nHBM). The nHBM is a dynamical $N \times N$ random matrix model whose entries are given by i.i.d. complex Brownian motions. For nHBM, the bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for the scale-transformation invariance of the system. The eigenvalue processes associated with the nHBM are related to the Ginibre ensemble. It is difficult to give the SDE of the eigenvalue processes of the nHBM using only themselves. Each eigenvalue process is coupled with the eigenvector-overlap process, which is a Hermitian matrix-valued process with entries given by products of overlaps of the right and left eigenvectors.
To analyze the dynamics of the eigenvalues, we think it is important to treat time-dependent point processes of eigenvalues and their variations weighted by the diagonal elements of the eigenvector-overlap processes. In this talk, we introduce tensor-valued processes associated with left and right eigenvectors and related time-dependent point processes. In addition, we discuss a relation to the time-dependent point process related to eigenvector-overlap processes. If time permits, I will also talk about a one-parameter family of non-Hermitian matrix-valued processes, which can be regarded as a dynamical extension of Girko's ensemble interpolating the GUE and the Ginibre ensemble of random matrices. The present talk is based on the joint work with Makoto Katori (Chuo University), Jacek Malecki (Wroclaw University of Science and Technology) and Satoshi Yabuoku (Fukuoka University).
Title:
Stochastic nonlinear Schrodinger equations
Abstract:
This talk will focus on the solvability of nonlinear Schrodinger equations perturbed by an additive noise. The theory of regularity structure would not apply directly to the nonlinear Schrdinger equation, thus this question is an open problem and very challenging in the field of SPDEs. On the other hand, historically there are mainly two lines of research:
(1) Nonlinnear Schrodinger equations with random initial condition
(2) Stochastically forced nonlinear Schrodinger equations
Despite the different subjects, the interplay of randomness and nonlinear effects may result in unexpected connections and analogies between the various approaches and results presented. We succeeded to give a meaning to the 1-d quadratic Schrodinger equation (the model of (2)) employing the methods used in (1); key words of our methods are, explicit renormalization and random operators. Joint work with Aurelien Deya (Nancy) and Laurent Thomann (Nancy).
Title:
Large deviations for the symmetric harmonic model
Abstract:
We discuss large deviations for the symmetric harmonic model, i.e. the interacting particle system that arises from the integrable XXX chain with hyperbolic spins. This model can be considered as the "integrable version of the KMP model". Under diffusive space-time rescaling, the large deviations of symmetric systems around the hydrodynamic limit are expected to be described by the macroscopic fluctuation theory (MFT), developed by Jona-Lasinio et al. in early 2000. In particular, the symmetric harmonic model has a constant diffusivity coefficient and a mobility coefficient that is a quadratic convex function of the particle density, thus the formulas predicated by MFT for the large deviation functions of several observables are rather explicit.
In this presentation, by exploiting the integrability of the harmonic model, we discuss both static large deviations (e.g. large deviation of the density in a non-equilibrium steady state) and dynamic large deviations (e.g. large deviation of the height function for a step initial condition). For the former, we follow [arXiv:2307.14975] and use an explicit microscopic characterization of the stationary measure to prove the MFT formula. For the latter, we rely on a novel approach in the limit of a large spin (work in progress with T. Sasamoto). The novel approach is applicable to interacting particle systems described by spin chains and allows the definition of a discretized version of MFT. For the exact solution of the time-dependent problem (work in progress with K. Mallick, H. Suda, T. Sasamoto) we refer to Hayate's talk.
Title:
Integrable SYK-like models
Abstract:
We introduce four disorder-free variants of the Sachdev-Ye-Kitaev (SYK) models, each consisting of Majorana or complex fermions with all-to-all interactions: (i) clean Majorana SYK model, (ii) clean Majorana SYK model with N=1 supersymmetry (SUSY), (iii) clean complex SYK model, and (iv) clean complex SYK model with N=2 SUSY. All of these models are integrable and can thus be diagonalized exactly. In Models (i) and (ii), the Hamiltonian commutes with a quadratic Hamiltonian. In Models (iii) and (iv), the Hamiltonian can be mapped to a specific case of the Richardson-Gaudin model. Exploiting the integrability of Model (i), we analyze its static and dynamical properties in detail. In particular, we find that the out-of-time-order correlators exhibit exponential growth at early times, resembling the behavior observed in the original disordered SYK model.
References:
[1] S. Ozaki and H. Katsura, arXiv:2402.13154 [cond-mat.str-el] (2024).
[2] E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018).
Title:
Critical point for oriented percolation
Abstract:
We consider nearest-neighbor oriented percolation defined on the lattice $\mathbb Z^d \times \mathbb N_{0}$. For each pair of points $(x,t), (y,s) \in \mathbb Z^d \times \mathbb N_{0}$, we can define a bond by $(x-y,t-s)$, which is independently open with probability $pD(x-y)\mbox{1}\hspace{-0.25em}\mbox{l}\{|t-s|=1\}$ with $D(x)=\frac{\mbox{1}\hspace{-0.25em}\mbox{l}\{|x|=1\}}{2d}$. It is well known that oriented percolation exhibits a phase transition as the parameter $p$ varies around a critical point $p_c$ which is model-dependent. As d to infinity, $p_c$ coverges to1. However, the best estimate for $p_c$ provided by Cox and Durret (Math. Proc. Camb. Phil. Soc. (1983)) give upper and lower bounds, but do not yield an explicit expression for $p_c$. In this talk, we investigate the exact expression for $p_c$ when $d>4$, in a way that $p_c=1+C_1d^{-2}+C_2d^{-3}+C_3d^{-4}+O(d^{-5})$, where $C_1$ to $C_3$ are constants. The proof relies on the lace expansion, which is one of the most powerful tool to analyze the mean-field behavior of statistical-mechanical models in high dimensions.
Title:
On the time constant of first-passage percolation in high dimensions
Abstract:
We consider first-passage percolation on the d-dimensional integer lattice. The model, introduced by Hammersley and Welsh in the 60s, describes how a fluid penetrates in a porous medium. In this model, one puts i.i.d. nonnegative weights on the nearest-neighbor edges of $\mathbb{Z}^d$, and studies the induced (weighted) graph distance $T$. By Kingman's theorem, under a mild assumption, for any $x \in \mathbb{Z}^d$, $\lim_{n\to\infty} T(0, nx)/n$ exists almost surely; this limit is nonrandom and is known as the time constant. In this talk, I will focus on the asymptotic behavior of the time constant as $d\to\infty$. I will first survey some previously known results, and then I will talk about an ongoing project with Chengkun Guo, Te-Lun Lu and Si Tang, in which we extend the previous results to more general directions and distributions.
Title:
Periodic PushASEP model
Abstract:
It is a joint work with Axel Saenz (Oregon). We are interested in an interacting particle system called PushASEP model, which is a natural generalization of the TASEP model. Instead of studying the model on an infinite line, we look at a periodic ring, which brings us back to a finite-state Markov process.
More precisely, we are in the following setup. At time 0, N particles are distributed on a periodic ring of size L, and they move to the left and right according to specific rules. We want to understand the asymptotic behavior of such a system for large L and N with the ratio N/L fixed.
We will explain how to study the model using different approaches derived from contour integrals. We will also explain how to diagonalize the system. These methods allow us to establish results about fluctuations in the relaxation time-scale t ~ L^{3/2}, which can be described by distributions interpolating the Gaussian distribution and the Tracy-Widom distribution.
Title:
Lindbladian PT phase transition: Emergence of continuous time crystals and critical exceptional points
Abstract:
Abstract: Continuous time-translation symmetry is often spontaneously broken in open quantum systems, and the condition for their emergence has been actively investigated [1,2]. However, there are only a few cases in which its condition for appearance has been fully elucidated. In this talk, we show that a Lindladian parity-time (PT) symmetry can generically produce persistent periodic oscillations, including dissipative continuous time crystals, in one-collective spin models [3,4]. By making an analogy to non-reciprocal phase transitions, we demonstrate that a transition point from the oscillating phase is associated with spontaneous PT symmetry breaking and typically corresponds to a critical exceptional point. These results are established by proving that the Lindbladian PT symmetry at the microscopic level implies a non-linear PT symmetry and by performing a linear stability analysis near the transition point.
[1] C. Booker, B. Buca, and D. Jaksch, Non-stationarity and dissipative time crystals: spectral properties and finite-size effects, New J. Phys. 22 085007 (2020).
[2] G. Piccitto, M. Wauters, F. Nori, and N. Shammah, Symmetries and conserved quantities of boundary time crystals in generalized spin models, Phys. Rev. B 104, 014307(2021).
[3] YN, T. Sasamoto, Dissipative time crystals originating from parity-time symmetry, Phys. Rev. A 107, L010201(2023).
[4] YN, R. Hanai and T. Sasamoto, Continuous time crystals as a PT symmetric state and the emergence of critical exceptional points, arXiv:2406.09018.
Title:
Topological Levinson's theorem in presence of embedded thresholds and discontinuities of the scattering matrix
Abstract:
During this talk, we shall present the scattering theory for a family of Schroedinger operators. The spectrum of these operators exhibit changes of multiplicity, and some of them possess resonances at thresholds. We shall then construct C*-algebras which contain the corresponding wave operators. The quotient of these algebras by the ideal of compact operators will be studied, and an index theorem will be deduced from these investigations. This result corresponds to a topological version of Levinson's theorem in presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrix. Finally, the dependence of this result on some external parameters will be discussed. In particular, a surface of resonances will be exhibited, probably for the first time. No prior C*-algebraic knowledge is expected from the audience. This presentation is based on a joint work with V. Austen, D. Parra, and A. Rennie.
Title:
Dynamical relation between determinantal point processes and operator algebras
Abstract:
Point processes are mathematical descriptions of random particle systems, and they appear in fields such as statistical mechanics, random matrix theory, and representation theory. It is known that point processes on discrete spaces can be produced by operator algebras called CAR algebras, which are generated by operators satisfying canonical anti-commutation relations. More precisely, suitable linear functionals on CAR algebras give rise to random processes, with a particular subclass associated with determinantal point processes. In fact, the static relationship between (determinantal) point processes on discrete spaces and CAR algebras has almost been established.
In this talk, we will explore a dynamical extension of this relationship. Specifically, we will examine discrete orthogonal polynomial ensembles, which are fundamental examples of determinantal point processes. When the orthogonal polynomials are of hypergeometric type, there exists a second-order difference operator that can be diagonalized by these polynomials. We demonstrate that this operator generates a suitable dynamical system on the CAR algebra, leading to a Markov process with the orthogonal polynomial ensemble as an invariant measure. Additionally, we will discuss recent progress on this topic to the extent possible.
Title:
Solving the discrete MFT equation for Frassek-Giardinà-Kurchan process
Abstract:
The Frassek-Giardina-Kurchan process (FGK process) is an integrable version of the discrete Kipnis-Marchioro-Presutti model, containing a parameter called spin. In a large spin limit the process converges to a deterministic process on lattice, and the corresponding large deviations can be predicted by a discrete analogue of the Macroscopic fluctuating theory (discrete MFT) [Giardinà-Sasamoto, in progress, see Cristian’s talk], In general, with MFT, the large deviation problem becomes solving a coupled PDE with initial/final time conditions (MFT equation).
In this talk, we will consider the large spin large deviations for the integrated current of the FGK process with step initial condition via the discrete MFT. We will show that the discrete MFT equation for this problem can be solved exactly by the inverse scattering method.
This is an ongoing work with Cristian Giardinà, Tomohiro Sasamoto and Kirone Mallick.
Title:
Class of operators that satisfy the eigenstate thermalization hypothesis in generic nonintegrable systems
Abstract:
Reconciling macroscopic laws with microscopic equations of motion has been a fundamental problem since the era of Boltzmann. The state-of-the-art experiments in ultracold atoms have revealed the relaxation of isolated quantum systems to thermal equilibrium [1]. The eigenstate thermalization hypothesis (ETH) [2,3] is believed to be the primary mechanism behind thermalization of isolated quantum systems. The ETH asserts that every energy eigenstate of a many-body system is thermal per se regarding “physical” operators. Yet, no evidence has been presented as to whether the ETH holds for all few-body operators.
In this talk, we give rigorous upper and lower bounds on m* such that all 𝑚-body operators with m<m* satisfy the ETH in fully chaotic systems [4]. For arbitrary dimensional 𝑁-particle systems subject to the Haar measure, we prove that there exist 𝑁-independent constants α_L and α_U such that α_L≤m*/N≤α_U. We numerically test the predictions from the Haar measure for locally interacting systems by exploiting the multi-core CPUs and GPUs. Our results reveal that relatively high-order moments of thermal and quantum fluctuations of any m-body operators up to α_L N/m thermalize in generic nonintegrable systems.
References:
[1] A. M. Kaufman et al., Science 353, 794 (2016).
[2] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). [3] M. Srednicki, Phys. Rev. E 50, 888 (1994).
[4] S. Sugimoto, R. Hamazaki, and M. Ueda, arXiv:2303.10069 (2023).
Title:
The Critical 2D Stochastic Heat Flow: disordered system meets singular SPDE
Abstract:
We will discuss recent progress in the study of the 2-dimensional stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation, which are critical singular stochastic partial differential equations (SPDEs) that lie beyond existing solution theories. Both the 2D SHE and KPZ undergo a phase transition, and the solution of the 2D SHE at the critical point leads to the so-called critical 2D stochastic heat flow. This provides a rare example of a model in the critical dimension and at the critical point with a non-Gaussian limit. Our approach is motivated by the scaling limits of disordered systems, in particular, the directed polymer model in random environment for which disorder is marginally relevant in 2D.
Title:
Non-Equilibrium Almost Stationary States and Response to Electric Fields in Extended Systems of Interacting Electrons
Abstract:
I briefly review the construction of non-equilibrium almost stationary states (NEASSs) in interacting many-body quantum systems and their connection to resonances in finite systems. I then present a new application to the response of insulators to constant external fields at zero temperature and in the thermodynamic limit. Using the NEASS approach, we prove that the direct conductivity vanishes at any order in the applied field and the Hall conductivity is given by a many-body version of the double-commutator formula without power-law corrections. The new result is based on joint work with Giovanna Marcelli, Tadahiro Miyao, Domenico Monaco and Marius Wesle.
Title:
Continuous symmetry breaking for rotators and dimers
Abstract:
We discuss two models of interest in statistical mechanics. The first of these models, the XY and the Villain models, are models which exhibit the celebrated Kosterlitz-Thouless phase transitions in two dimensions. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, spin correlations of these models are closely related to Gaussian free fields. I will discuss some recent progress on this conjecture in d>=3. The second is the dimer model and the monomer double dimer model. I will introduce a new (complex) spin representation for models belonging to this class, and derive a new proof of the Mermin-Wagner theorem which does not require the positivity of the Gibbs measure. Based on joint works with Paul Dario (CNRS) and with Lorenzo Taggi (Sapienza Univ. di Roma).
Title:
Uniqueness of steady-state solutions of the Gorini-Kossakowski-Sudarshan-Lindblad equation
Abstract:
The dynamics of Markovian open quantum systems are described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the presence of dissipation, quantum systems relax to a steady state. In the language of the GKSL equation, the steady state is an eigenmode with zero eigenvalue. There always exists at least one steady state in a finite-dimensional system, but its uniqueness depends on the system.
In this talk, we discuss a simple proof of a sufficient condition for the uniqueness of the steady state and demonstrate its applications using examples of open quantum many-body systems.