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Solvers for Quantum Impurity Problems with Superconducting Baths
Second title: Tutorial for PhD students with Rok Žitko
Provider: Faculty of Science

Activity no.: 5879-21-11-31 
Enrollment deadline: 05/10/2021
PlaceNiels Bohr Institute
Date and time04.10.2021, at: 00:00 - 22.10.2021, at: 16:00
Regular seats10
ECTS credits2.50
Contact personJulie Meier    E-mail address: juliemh@nbi.ku.dk
Enrolment Handling/Course OrganiserJesper Nygård    E-mail address: nygard@nbi.ku.dk
Written languageEnglish
Teaching languageEnglish partially in English
Study boardStudy Board of Physics, Chemistry and Nanoscience
Exam requirementsPlease see the teacher for more information
Exam formContinuous assessment
Exam formAndet/Other
Exam aidsWithout aids
Grading scalePassed / Not passed
Criteria for exam assessmentPlease see the teacher for more information
Course workload
Course workload categoryHours
Preparation32.50
Lectures20.00
Tutorials10.00

Sum62.50


Content
Tentative dates for the course are 4-21 October. A schedule will be sent out to participants.

A quantum impurity system consists of a local object described by a discrete set of levels that is coupled to a continuum of itinerant states. Such systems are ubiquitous, the most well-known being magnetic dopants, quantum dots, and qubits. The corresponding Hamiltonians are deceptively simple: their eigenstates are highly non-trivial many-body states that often require advanced numerical techniques for their study. This course will introduce several approaches, especially the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG), with a focus on problems where the bath is superconducting and the continuum excitations are Bogoliubov quasiparticles. We will discuss both the theoretical underpinnings of these methods and the practical aspects of their applications, with hands-on tutorials on NRG and DMRG calculations. We will cover standard models (Kondo, Anderson), basic notions of renormalization group, Lanczos algorithm, tensor networks, BCS and Richardson models of superconductivity, proximity effect, Yu-Shiba-Rusinov and other subgap states, Josephson junctions, and a number of other key concepts.

Learning outcome
Skills
The students will learn how to apply advanced numerical techniques, in particular DMRG and NRG, to the study of impurity models in quantum many-body physics.

Use of Mathematica and C++ codes to implement NRG techniques.

Use of C++ codes and iTensor to implement DMRG calculations.

Knowledge
The students will learn:
a) the basics of Kondo and Anderson impurity models.
b) the basic theoretical foundation of numerical renormalization group
c) the basic theoretical foundation of density matrix renormalization group
d) main physical properties of several low-dimensional many-body problems (for instance the AKLT model, and several models related to superconducting systems)

Competences
The students will acquire a background to utilize numerical techniques for the analysis of strongly-correlated quantum physics problems, which may be
applied to many condensed matter systems at the focus of current research activities.

Literature
Hewson: The Kondo problem to heavy fermions, Cambridge University Press (1993)
Wilson: The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975)
Krishnamurthy, Wilkins, Wilson: Renormalization-group approach to the Anderson model of dilute magnetic alloys. I. Static properties for the symmetric case, Phys. Rev. B 21, 1003 (1980)
Krishnamurthy, Wilkins, Wilson: Renormalization-group approach to the Anderson model of dilute magnetic alloys. II. Static properties for the asymmetric case, Phys. Rev. B 21, 1044 (1980)
Satori, Shiba, Sakai, Shimizu: Numerical Renormalization Group Study of Magnetic Impurities in Superconductors, J. Phys. Soc. Japan, 61, 3239 (1992)
Bulla, Costi, Pruschke: Numerical renormalization group for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008)
Schöllwock: The density-matrix renormalization group in the age of matrix product states, Annals of Physics, 326, 96 (2011)
ITensor tutorials, http://itensor.org/docs.cgi?page=tutorials&vers=cppv3
Pavesic, Bauernfeind, Zitko: Yu-Shiba-Rusinov states in superconducting islands with finite charging energy, arxiv:2101.10168
Gull et al., Continuous-time Monte Carlo methods for quantum impurity models, Rev. Mod. Phys. 83, 349 (2011)

Target group
PhD students with a background in condensed matter or quantum many-body physics.

Lecturers
Rok Žitko
Senior research associate at the Jožef Stefan Institute, Slovenia

Remarks
For further information about the course, please contact: Michele Burrello michele.burrello@nbi.ku.dk

For administrative support re. sign-up, please contact: Julie Meier Hansen juliemh@nbi.ku.dk

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