# KAU-ITEP Winter School

Frontiers of condensed matter physics

## 23-28 December 2019

Department of Theoretical and Mathematical Physics and Bogolyubov Institute for Theoretical Physics hold traditional Winter school on Theoretical Physics from 23 to 28th of December. During the school, three lecture courses on modern quantum condensed matter physics will be given. Each course starts from the classical questions and finishes at modern problems on which people write papers ‘here and now’.

The school is aimed towards students already familiar with many-body quantum mechanics. During the first two days of school, preparatory seminars will be organized for those, who are not sure in their level of understanding of prerequisites of the main program of the school.

It is required to register for participation in school Registration

### Preliminary schedule:

1 lecture: 10:00 - 13:00

2 lecture: 14:00 - 17:00

23 December (Mon.)

1 lecture: Preparatory seminar

2 lecture: Preparatory seminar

24 December (Tue.)

1 lecture: Preparatory seminar

2 lecture: Preparatory seminar

25 December (Wed.)

1 lecture: Introduction to quantum complexity theory

2 lecture: Introduction to quantum complexity theory

26 December (Thur.)

1 lecture: Introduction to quantum complexity theory

2 lecture: Introduction to bosonization

27 December (Fr.)

1 lecture: Introduction to bosonization

2 lecture: Non-linear transport in quantum Hall effect

28 December (Sat.)

1 lecture: Non-linear transport in quantum Hall effect

All interested are cordially invited!

Will be happy to answer all your questions at This email address is being protected from spambots. You need JavaScript enabled to view it..

**Introduction to quantum complexity theory**

**Yaroslav Herasymenko, Oleksandr Gamayun (ΔITP, The Netherlands)**

**Course description:**

Many-particle quantum systems is a fascinating and complex object of study for a modern physicist. Confinement, high-temperature superconductivity and fractional quantum Hall effect – these are just some of the examples of collective quantum phenomena. Unfortunately, such systems are extremely hard to study theoretically. Analytical methods for interacting quantum systems are usually unavailable, and the demands for a computer simulation exponentially increase with the number of particles. In this course we will lay out some ideas and principles, which allow to circumvent this barrier. In particular, we will tell how some interacting systems can be solved via mapping to non-interacting fermions; the role of the entanglement entropy for the numerical methods; and which new possibilities emerge in the area with the invention of the quantum computer. A particular emphasis will be made on the connections between these three different directions.

**Prerequisites:**

● Second quantization

● The density matrix and its properties

**Plan:**

1) One-dimensional quantum systems. Analytical solutions

1.а Quantum Ising model in the transverse magnetic field

1.b Jordan-Wigner transformation, mapping the Ising model onto Majorana fermions

1.c An overview of quantum integrability in 1D

2) Entanglement entropy and the efficient simulation of interacting quantum systems

2.а Entanglement entropy, application to the ground state of the quantum Ising model

2.b The Matrix Product State ansatz, reproducing the ground state of 1D models

2.c The Area Law for entanglement entropy and its breakdowns

3) Variational quantum algorithms

3.а The basics of quantum algorithms

3.b Variational quantum algorithms, its expected advantage

3.c Adiabaticity-inspired quantum ansatz

3.d Perspectives

**Introduction to bosonization**

**Tereza Vakhtel (ΔITP, The Netherlands)**

**Course description:**

According to Landau's Fermi Liquid theory, electrons in solids behave effectively like free fermions, even when one cannot neglect the interactions. However, this theory fails to describe interacting electrons in 1-d. At the same time, there’s a variety of fermion systems in the lab and nature that are effectively 1-d. Bosonization is a technique to represent fermionic degrees of freedom in terms of bosonic degrees of freedom. In certain cases, after such a procedure, the Hamiltonian of the system becomes quadratic or much easier to treat. We will apply this tool to a 1-d system of interacting electrons and find its low energy excitations. This example is a rare case of a strongly interacting system that can be solved almost exactly.

Lecture 2 is a somewhat more advanced version of Lecture 1, aimed at students familiar with functional field integral.

**Plan:**

**Lecture 1**

**Prerequisites: **Second quantization and band structure theory

1. Quantum many-body systems in 1-d. Motivation and examples.

2. Electrons in 1-d with Coulomb interaction

3. Charge/Spin-density representation

4. Bogoluibov transformation

5. Charge and spin density waves.

**Lecture 2**

**Prerequisites: **Functional field integral for bosonic and fermionic field theories

1. Properties of the action for the 1-d electron gas

2. Jordan-Wigner transformation

3. Construction of fermionic operators from bosonic operators

4. Free bosonic action from the physics and symmetry reasonings

5. Bosonic action for the interacting system

6. Interacting electrons in 1d in the presence of disorder (if there’s any time left)

### Nonlinear Hall Acceleration

**Oles Matsyshyn (Max Planck Institute for the Physics of Complex Systems, Dresden, Germany)**

**General description:**

In this course, we will discuss some aspects of the linear and nonlinear quantum Hall effect. The nonlinear photocurrent can play a crucial role in a solar cell industry. We can expect to obtain an absolutely alternative device architecture based on the nonlinear response. With a new solar cell architecture one can try to avoid the standard current relaxation and an efficiency limit problems. A development of such technologies can radically effect a whole solar cell industry. We will describe the transport in terms of the Berry connections. I will give a brief overview of our recent work (Phys. Rev. Lett. 123.246602) and the new physical understanding of Berry curvature dipole. Also, I will show a non-trivial mapping from strongly theoretical work in a real-world industrial implementations.

**Good to know:**

- Density Matrix
- Bloch wave functions
- Tight-binding model (TBM)

**Outline:**

1) Motivation.

2) TBM, length gauge, Lioville equation for DM.

3) Position operator and Berry connections.

4) DM perturbation theory.

5) Linear and non-linear response.

6) Sum rules.

7) Final motivation.

It is required to register for participation in school Registration