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The recursion method is a rapidly developing approach for describing the evolution of quantum many-body systems. Calculating the dynamical characteristics of such systems — transport coefficients, time correlation functions, etc. — poses a serious challenge for theory. Traditional methods based on perturbation theory are applicable only near the integrable limit of non-interacting (quasi)particles.
The recursion method offers a fundamentally different, nonperturbative approach to describing dynamics, which, on the contrary, works better the further the system is from any integrable limits. The modern development of this method is associated with significant progress in understanding the phenomenon of operator growth. Recent research has revealed statistical regularities in this growth, which, when integrated into the recursion method, have become key to constructing a nonperturbative description of quantum dynamics.
The course is taught by researchers actively working in this field and is aimed at senior undergraduate and graduate students.
Program
1. What this course is about (introductory lecture).
Quantum theory of many-body systems: an overview of problems and methods. The phenomenon of operator growth. The recursion method and the universal operator growth hypothesis. Computer-algebraic implementation of the recursion method.
2. Quantum mechanics of many-body systems (introductory lecture).
Formalism of quantum mechanics. Schrödinger and Heisenberg pictures. Operators and superoperators. Liouvillian. Tensor product of Hilbert spaces and many-body systems. Correlation functions. Quantum quench.
3. Lanczos basis in operator space.
Krylov subspace in operator space. Lanczos basis. Tridiagonal form of the Liouvillian. Heisenberg equations in the Lanczos basis. The recursion method.
4. Orthogonal polynomials: introduction.
Necessary background from the theory of orthogonal polynomials. Construction of the Lanczos basis using orthogonal polynomials.
5. Orthogonal polynomials: application.
Matrix of moments. Expressing Lanczos coefficients through moments.
6. Universal operator growth hypothesis.
The phenomenon of operator growth. Asymptotics and extrapolation of Lanczos coefficients. Physical effects of subleading contributions. Exactly solvable model Liouvillians.
7. Laplace transform of the correlation function.
Laplace transform of correlation functions and continued fractions. Asymptotics of the correlation function. Calculation of transport coefficients.
8. Application of the recursion method to quantum magnets.
Spins on lattices. Quantum Ising model. Calculation of correlation functions and transport coefficients.
9. Pseudomode expansion.
Ruelle-Pollicott quantum resonances. Expansion of correlation functions using pseudomodes. Liouvillians with dissipation. Localization in operator space. Pseudo-mode expansion within the recursion method.
10. Nakajima-Zwanzig equation in Krylov space.
Derivation of the Nakajima-Zwanzig integral equation in Krylov space. Markovian approximation. Universal random matrix approximation.
11. Quantum quench.
Quantum quench within the recursion method. Results for spin models.
12. Perspectives and open questions (concluding lecture).
Comparison with competing methods (exact diagonalization, tensor networks, Heisenberg trotterization, quantum Monte Carlo). Application to nuclear magnetic resonance. Application to quantum computer benchmarking. Promising directions for future research. Open questions.
 RSS: Forthcoming seminars
 Lecturers
Ilin Nikolay Borisovich
Lychkovskiy Oleg Valentinovich
Teretenkov Aleksandr Evgenevich
Shirokov Ilya Evgen'evich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
