Kazumasa A. Takeuchi (Tōkyō University, Japan)
Introduction to the KPZ equation and its experimental aspects
[slides]
Lectures synopsis
Elisabeth Agoritsas
Title: Interfaces in disordered systems and directed polymer
1. Introduction
- Generic theoretical frameworks: disordered elastic systems.
- Threefold motivation: experimental/fundamental/methodological.
- Basic phenomenology and regimes of interest.
2. Disordered elastic systems: Recipe
- Description: Hamiltonian, Langevin dynamics, dynamical action.
- Observables: geometrical fluctuations and center-of-mass dynamics.
3. Disordered elastic systems: Statics
- Static interface without disorder (thermal roughness).
- With disorder: roughness regimes and crossover scales. Larkin model.
- Example of model reduction: effective 1D interface starting from a 2D Ginzburg-Laudau description.
- Focus on the 1D interface: mapping to the 1+1 Directed Polymer. Connections to the 1D Kardar-Parisi-Zhang equation and universality class.
- Scaling analysis: power counting versus physical scalings
3. Disordered elastic systems: Dynamics
- Velocity-force characteristics.
- Fast-flow regime.
- Creep regime: how to recover 1/4 creep exponent from 2/3 KPZ exponent.
Ivan Corwin
Lecture 1. Extreme Diffusion or Was Einstein Wrong About Diffusion
In a system of many particles diffusing in a common environment, the first few particles are often quite important. How do they behave and what does that behavior tell us about the environment in which they have evolved? We will approach these problems by studying random walks in random environments and utilizing a connection with the KPZ equation and universality class. In particular, this first lecture will highlight some of the types of tools (stochastic analysis, replicas, Bethe ansatz) and results one can show mathematically about models in the KPZ universality class.
Lecture 2. Pot Of Gold At The End Of The KPZ Rainbow
When people say "KPZ universality class" they presumably mean that many systems converge to a single universal limit. What is that limit and what does it tell us? We will address this question from the perspective of stochastically growing interface models over one spatial dimension. In particular, we will focus on results about the scaling limit of the height function for the multispecies (i.e. multicolor) Asymmetric Simple Exclusion Process (ASEP). The final two lectures will build up the tools to understand and prove these results.
Lecture 3. Colorblind Analysis
We will focus on the single species (colorblind) ASEP and its discrete time relative, the stochastic six vertex (S6V) model. We will show how the Yang-Baxter equation relates the ASEP/S6V height functions to measures written in terms of q-Boson weights and discuss some ideas (determinantal methods, replica Bethe ansatz, and Gibbsian line ensembles) behind extract limits of these measures.
Lecture 4. Back To The Rainbow
Returning to the setting of colors we will show how a colored version of the Yang-Baxter equation allows us to analyze this more general model by leveraging the colored blind analysis in Lecture 3. We will also touch on how, by fusion, it is possible to extend our analysis to a more general class of stochastic vertex models that includes many examples including some touched on in Lecture 1.
Sebastian Diehl
Driven open quantum matter is defined by many-body systems in which coherent quantum dynamics, drive, and dissipation occur on an equal footing. They break the conditions of thermodynamic equilibrium on the microscopic scale, and the question arises to what extent this manifests in the macrophysics, i.e. in the phases and phase transitions of these systems. We will develop key theoretical concepts for their description, and we will discuss various applications — in particular, we will encounter manifestations of KPZ-related physics in many diverse circumstances.
1- From the Lindblad equation to the Lindblad-Keldysh functional integral
- Lindblad equation for driven open quantum matter
- construction of Lindblad-Keldysh functional integral, structural properties
2- KPZ equation in exciton-polariton condensates
- background: semiclassical limit, classifying equilibrium vs. non-equilibrium states
- from exciton-polaritons to KPZ: absence of algebraic order out of equilibrium
- compact KPZ and non-equilibrium phase transition
3- Macroscopic non-equilibrium phenomena from weak non-equilibrium drive
- non-equilibrium O(N) models: phase structure, limit cycles
- novel non-equilibrium criticality at onset of a limit cycle
- route towards KPZ via breaking of time translation symmetry
4- Principles of universality in driven open quantum matter
- the principles: equilibrium vs. non-equilibrium; pure vs. mixed states; weak vs. strong symmetries
- application: 1D KPZ via weak and strong symmetries
5- Quantum aspects: Topology in driven open quantum matter
- topological dark states in Lindblad evolution
- universality of topological response: pure states, mixed states
Tomaž Prosen
Title: Non-equilibrium lattice quantum systems, emergence of KPZ in spin chains
In the series of four lectures I will introduce and discuss several exactly solvable or integrable non-equilibrium or time-dependent paradigms of quantum and also classical lattice systems (spin chains) with a focus on lattice models in discrete time.
Lecture 1 - Integrable spin chains and quantum circuits
I will introduce Yang-Baxter quantum circuits with a specific focus on unitary six-vertex model (XXZ circuit and XXZ spin chain). With respect to the problem of transport, I will discuss two fruitful aspects: (quasi-local conservation laws and matrix product non-equilibrium steady states of boundary driven chains.
Lecture 2 - XXX circuit and Heisenberg spin chain
I will focus SU(2) symmetric circuit/spin chain and discuss emergent KPZ scaling of dynamical 2-point functions. This lecture will be phenomenology oriented, I will present and discuss state-of-the-art t-DMRG computations and quantum simulations.
Lecture 3 - The classical limit and super-universality of super-diffusion
I will discuss simple symplectic coupled map lattices, or classical circuits, which are integrable and which exhibit the same phenomenology as integrable quantum circuits. A family of these models can be defined w.r.t. almost arbitrary compact Lie Group symmetry and exhibit the same (KPZ) scaling of 2-point functions.
Lecture 4 - Full counting statistics
I will discuss the problem of full counting statistics in integrable spin chains/circuits with a conserved quantity and introduce the concepts of cumulant generating function, dynamical free energy and large deviation. I will present an exactly solvable model of full counting statistics in a classical interacting cellular automaton. Finally, I will return to quantum simulation of full counting statistics in XXX circuit.
Kazumasa A. Takeuchi
Title: Theoretical and experimental physics aspects of the KPZ class for growing interfaces
The KPZ equation and the associated universality class were originally proposed in the context of growing interfaces. The lecture will include a pedagogical introduction to basic properties of the KPZ equation, an illustrative outline of characteristic distribution and correlation properties revealed by exact solutions, and a survey of experimental studies on growing interfaces.
1. Introduction: why should we care this?
2. Scaling exponents and universality classes
3. Basic properties of the KPZ equation
4. Distribution and correlation properties: stationary & non-stationary cases
5. Experimental test of predictions from integrable models
6. Distribution properties for general cases and variational formula
Main reference: K. A. Takeuchi, "An appetizer to modern developments on the Kardar-Parisi-Zhang universality class", Physica A 504, 77-105 (2018). https://doi.org/10.1016/j.physa.2018.03.009
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