Dymarsky, Anatoly

Finite Size Scaling of Integrability to Chaos Transition

Student:

Ahmed Khalifa, graduate
Debarghya Chakraborty, graduate

Precision ETH Without Full Direct Diagonalization

The project is devoted to the study of thermalization of isolated quantum systems. Using one-dimensional spin-chains as a theoretical model describing a typical chaotic quantum system we will study the process of thermalization for systems in pure state undergoing unitary evolution. The goal of the project is to investigate several different aspects of the Eigenstate Thermalization Hypothesis, which provides a microscopic description of thermalization process. In particular we aim to investigate emergence and properties of the chaotic behavior when the investigated models approach integrable regime.

The default numerical approach to tackle quantum isolated system is a (full) direct diagonalization. Essentially this method calls for finding all eigenvalues/eigenvecotrs of certain (very large) matrix by numerically solving corresponding linear algebra problem. This method becomes extremely resource (memory) consuming for matrices of size 10 6 and larger. In this project we suggest an alternative approach: instead of performing direct diagonalization, we pursue to find individual eigenvalue/eigenvector pairs. Namely we aim to find a large number of such pairs which, at the same time, constitute only a very small fraction of all eigenpairs. Say, for a matrix 10 6x10 6 we aim to find 10 4 eigenpairs which is merely 1% of the total number. Because of stochastic nature, finding 10^4 eigenpairs is sufficient to investigate physical properties of interest. The advantage of this approach is that finding different eigenpairs are logically independent mathematical problems, hence this process is highly parallelizable and can be efficiently implemented on the DLX cluster.

Computational methods:

The employed algorithms rely on Intel Mathematical Kernel Library (all software is currently available and have been tested on the DLX cluster).

Software used:

Intel C compiler, Intel MKL.

Students:

Maksim Beketov, (Skoltech MA student)
Kaushik Borah, Grad Student

Collaborators:

Oleg Dubinkin

Publications

2016 - 2019

1. Mechanism of macroscopic equilibration of isolated quantum systems, Anatoly Dymarsky Phys. Rev. B 99, 224302 – Published 6 June 2019
2. New characteristic of quantum many-body chaotic systems, Anatoly Dymarsky and Hong Liu, Phys. Rev. E 99, 010102(R) – Published 4 January 2019
3. Subsystem eigenstate thermalization hypothesis Anatoly Dymarsky, Nima Lashkari, and Hong Liu, Phys. Rev. E 97, 012140 – Published 25 January 2018
4. Bound on Eigenstate Thermalization from Transport, Anatoly Dymarsky Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506 and Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow, Russia, 143026 (Dated: April 25, 2018) (Preprint 2018)y

Funding