Lecture Topics and Background References
Prof. Enrico Arrigoni (10 – 14 October)
Title: "Master equation versus Keldysh Green's functions for correlated quantum systems out of equilibrium"
Plan of the lectures:
1) Master Equation, Closed vs open quantum systems. System bath and reduced density matrix
2) Quantum operations, Kraus Operators, Markovian assumption, Lindblad master equation, dissipation.
3) Solution methods for the many body case. Superfermion representation.
4) Relation with Keldysh Green's functions. When is a bath Markovian?
5) How to treat the non Markovian case. Application to correlated quantum impurities and DMFT
Required background
- Advanced quantum mechanics, second quantisation, quantum statistics (standard physics master courses), density matrix.
- Equilibrium Many body physics, Hubbard model, e.g. [1] , [2]
- Nonequilibrium (Keldysh) Green's function ( rst week of training course will be a prerequisite) [3] , [4].
Topics of the lectures
- Master Equation, Closed vs open quantum systems. System bath and reduced density matrix.
Markovian assumption, derivation of Lindblad master equation, dissipation.
For these two topics the lecture will be mainly based on Ref. [5] (cap IV) and Ref. [6] (cap. 1-3). A lot of stu can be also found also in [7], which is the standard textbook here. - Solution methods for the many body case. Superfermion representation. [6] , [8].
- Relation with Keldysh Green's functions. When is a fermionic bath Markovian ? [3] , [4] , [6] , [7].
- How to treat the non Markovian case. Application to correlated quantum impurities and DMFT [9] , [10], [11], [12], [13].
Literature
[1] J. W. Negele, H. Orland, Quantum many-particle systems , Vol. 68 of Frontiers in physics, Addison-Wesley, Redwood City, Calif., 1988.
[2] A. L. Fetter, J. D.Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971.
[3] H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors , Springer, Heidelberg, 1998.
[4] J. Rammer, H. Smith , Quantum eld-theoretical methods in transport theory of metals , Rev. Mod. Phys. 58 (1986) 323 - 359.
[5] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg , Atom-photon interactions: basic processes and applications, Wiley-VCH, Weinheim, 2004.
[6] G. Schaller, Open Quantum Systems Far from Equilibrium, Lecture notes in physics, Springer, Heidelberg, 2014.
[7] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, England, 2009.
[8] A. A. Dzhioev, D. S. Kosov, Super-fermion representation of quantum kinetic equations for the electron transport problem, J. Chem. Phys. 134 (2011) 044121.
[9] E. Arrigoni, M. Knap, W. von der Linden, Nonequilibrium Dynamical Mean Field Theory: an auxiliary Quantum Master Equation approach, Phys. Rev. Lett. 110 (2013) 086403.
[10] A. Dorda, M. Nuss, W. von der Linden, E. Arrigoni, Auxiliary master equation approach to non - equilibrium correlated impurities, Phys. Rev. B 89 (2014) 165105.
[11] A. Dorda, M. Ganahl, H. G. Evertz, W. von der Linden, E. Arrigoni, Auxiliary master equation approach within matrix product states: Spectral properties of the nonequilibrium Anderson impurity model, Phys. Rev. B 92 (2015) 125145.
[12] A. Dorda, M. E. Sorantin, W. von der Linden, E. Arrigoni, Optimized auxiliary representation of a non-Markovian environment by a Lindblad equation , arXiv:1608.04632 (2016).
[13] F. Schwarz, M. Goldstein, A. Dorda, E. Arrigoni, A. Weichselbaum, J. von Delft, Lindblad - Driven Discretized Leads for Non - Equilibrium Steady - State Transport in Quantum Impurity Models: Recovering the Continuum Limit , arXiv:1604.02050 (2016).
Prof. Massimo Capone (3 – 7 October)
Title: "Towards an understanding of superconductors and correlated materials out-of-equilibrium: mean-field approaches"
Plan of the lectures:
Lecture 1: Equilibrium Methods for Strongly Correlated Electrons: the Gutzwiller approximation and Dynamical Mean-field Theory
Training Session 1: The Mott-Hubbard transition
Lecture 2: The non-equilibrium Gutzwiller approximation
Training Session 2: Quantum quench in the Hubbard model
Lecture 3: Strongly Correlated Systems in a constant Electric field: Dissipation and Dielectric Breakdown
Training Session 3: To be announced
Lecture 4: Non-equilibrium dynamics of Superconductors. BCS superconductors, s-wave, d-wave and p+ip wave
Training Session 4: Practical implementation of the dynamics of s-wave superconductors
Lecture 5: Pump and probe dynamics of High-temperature superconductors: A theorist’s perspective
Training Session 5: To be announced
Bibiography:
- Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani and D. Mihailovic, Advances in Physics 65, 58 (2016) [free pdf can be downloaded from https://arxiv.org/abs/1601.07204]
- The out-of-equilibrium Gutzwiller approximation, M. Fabrizio, Proceedings of the Hvar 2011 Workshop on 'New materials for thermoelectric applications: theory and experiment’ [free pdf can be downloaded from https://arxiv.org/abs/1204.2175]
- Non-equilibrium phenomena in strongly correlated systems. G. Mazza, Ph.D. Thesis SISSA 2015 [http://urania.sissa.it/xmlui/handle/1963/34527]
- Synchronization in the BCS Pairing Dynamics as a Critical Phenomenon, Barankov and Levity, Phys. Rev. Lett. 96, 230403 (2006)
Prof. Martin Eckstein (3 – 7 October)
Title: "Electronic structure of correlated materials out of equilibrium: non-equilibrium dynamical mean-field theory, Martin Eckstein, Max-Planck Institute for Structure and Dynamics of Matter, Hamburg, Germany"
Plan of the lectures:
Lecture 1: Keldysh formalism: Ultra-fast dynamics of correlated electrons [3,4]
a) Basics of nonequilibrium Green's functions, the Keldysh contour, real-time path integrals and perturbation theory
b) Theoretical description of pump-probe experiments
Lecture 2: Relaxation in many-body systems beyond kinetic equations
a) From nonequilibrium Green's functions to kinetic equations [3]
b) Photo-induced dynamics of systems with electron phonon coupling
c) From collisionless relaxation to thermalization: Nonthermal melting of a spin-density wave [5]
Lecture 3: Non-equilibrium dynamical mean-field theory
a) Introduction to dynamical mean-field theory (DMFT) [4,6].
b) The quantum impurity model out of equilibrium [4,7]
c) The Mott-Hubbard metal-insulator transition out of equilibrium: How fast do quasiparticles emerge? [8]
Lecture 4: Periodically driven systems [4,10]
a) The Floquet theorem, band structure of periodically driven systems
b) Effective Hamiltonians of driven systems: Floquet Schrieffer-Wolff transformation
c) The time-periodic state: Floquet Green's functions, application to driven BCS superconductors
Lecture 5: Electrons and spins out of equilibrium: Magnetic exchange interactions in non equilibrium situations [9]
Required background
Advanced quantum mechanics, second quantisation, equilibrium Many body physics, Hubbard model, e.g., Ref. [2]
References:
[1] Overview article on ultra-fast spectroscopy in correlated materials:
J. Orenstein, "Ultrafast spectroscopy of quantum materials", Physics Today 2012.
[2] Some books on many-body physics in general (second quantization, Hubbard model, Green functions, self-energy, finite temperature formalism, path integrals):
J. W. Negele and H. Orland, "Quantum Many-Particle Systems" (Addison-Wesley, 1988).
A. Altland, B. D. Simons, "Condensed Matter Field theory" (Cambridge university Press, 2010)
[3] Some books on Keldysh formalism and nonequilibrium many-body theory:
A. Kamenev, "Field theory of Non-Equilibrium systems", (Cambridge University Press, 2011)
G. Stefanucci and R. van Leeuwen, "Nonequilibrium Many-Body Theory of Quantum Systems", (Cambridge University Press, 2013)
[4] Review on Nonequilibrium Dynamical mean field theory:
H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and Ph. Werner, "Nonequilibrium dynamical mean-field theory and its applications", Rev. Mod. Phys. 86, 779 (2014).
[5] Nonthermal criticality: N. Tsuji, M. Eckstein, and Ph. Werner, "Nonthermal antiferromagnetic order and nonequilibrium criticality in the Hubbard model", Phys. Rev. Lett. 110, 136404 (2013).
[6] Introduction to Dynamical mean-field theory and correlated materials: See, e.g.,
Chapter 1 of "The LDA+DMFT approach to strongly correlated materials" E. Pavarini, E. Koch, D. Vollhardt, A. Lichtenstein (Eds.),
Forschungszentrum Jülich (2011), http://www.cond-mat.de/events/correl11/manuscripts/correl11.pdf
[7] Ch. Gramsch, K. Balzer, M. Eckstein, and M. Kollar, "Hamiltonian-based impurity solver for nonequilibrium dynamical mean-field theory", Phys. Rev. B 88, 235106 (2013).
[8] Photo-induced Mott transition:
M. Eckstein and Ph. Werner, "Photo-induced states in a Mott insulator", Physical Review Letters 110, 126401 (2013).
Sh. Sayyad and M. Eckstein, "Slowdown of the Electronic Relaxation Close to the Mott Transition" Phys. Rev. Lett. 117, 096403 (2016).
[9] Magnetic exchnage interactions out of equilibrium:
J.H. Mentink, K. Balzer, and M. Eckstein, "Ultrafast and reversible control of the exchange interaction in Mott insulators", Nature Communications, 6, 6708 (2015).
J.H. Mentink and M. Eckstein, "Ultrafast quenching of the exchange interaction in a Mott insulator", Physical Review Letters, 113, 057201 (2014).
[10] Floquet theory of interacting systems, some review:
M. Bukov, L. d'Alessio and A. Polkovnikov, Advances in Physics, 2015, Vol. 64, No. 2, 139 (2015)
Prof. Stefan Kaiser (10 – 14 October)
Title: "Ultrafast optical control of complex quantum materials"
Plan of the lectures:
Lecture 1: Ultrafast Science & Technology
a) How do ultrafast lasers work? What is pump probe spectroscopy? [1]
b) Which experimental techniques can probe the different dynamical properties? [2-4]
Lecture 2: Photo-doping Dynamics in Correlated Electron Systems
a) How do correlations influence the quasiparticle dynamics? [2-5]
b) On what time scales photo-induced phase transitions can occur? [5,6]
Lecture 3: Non-equilibrium Dynamics of Collective Excitations in Complex Materials
a) What kind of collective excitations can be triggered by ultrafast light pulses [7-12]
b) “Higgs-spectroscopy” and or vs Amplitudon-phason-dynamics [13-15]
Lecture 4: Non-linear Phononics and Optical Control of Superconductivity in Cuprates
a) Non-thermal Optical Control of Materials and Superconductivity [16-20]
b) Coherent phonons and non-linear phonon interactions [20-22]
Lecture 5: Control of Effective Correlations and Inducing Superconductivity in Organic Quantum Materials
a) Vibrational coupling in organic quantum materials [23,24]
b) Effective control of local electronic interactions [6,25-27]
References:
[1] Laser Spectroscopy, Basic Concepts and Instrumentation, Wolfgang Demtröder, Springer (1996) or Laser Spectroscopy, Vol. 1 and Vol. 2 , Wolfgang Demtröder, Springer Verlag
[2] N.P. Armitage, Electrodynamics of Correlated Electron Systems, arXiv:0908.1126
[3] Susan L. Dexheimer, Theraherz Spectroscopy, CRC Press (2007)
[4] D.N Basov et al., Electrodynamics of Correlated Electron Materials, Rev. Mod. Phys. 83, 471 (2011)
[5] C. Gianetti et al., arxiv:1601.07204
[6] S. Iwai, Photoinduced Phase Transitions in a-,t-, and k-type ET salts: Ultrafast Melting of the Electronic Ordering, Crystals 2, 590 (2012)
[7] Charge Density Waves in Solids, George Gruner, WestView Press or The Dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988)
[8] J. Demsar et al., PRL 83, 800 (1999)
[9] M. Eichenberger et al., Nature 468, 799 (2010)
[10] T. Rohwer et al, Nature 471, 490 (2011)
[11] J.C. Petersen et al., PRL 107, 177402 (2011)
[12] H.Y. Liu et al. PRB 88, 045104 (2013)
[13] R. Matsunaga et al. PRL 111, 057002 (2013)
[14] R. Matsunaga et al. Science 345, 1145 (2014)
[15] H. Krull et al, Nat. Comm. 7, 11921 (2016)
[16] D. Fausti et al, Science 331, 189 (2011)
[17] S. Kaiser et al, PRB 89, 84516 (2014)
[18] W. Hu et al. Nat Mat. 13, 705 (2014)
[19] D. Nicoletti et al. PRB 90, 100503 (2014)
[20] R. Mankowsky et al. Nature 516, 71 (2016)
[21] M. Foerst et al, Nat. Phys. 7, 854, (2011)
[22] A. Subedi et al. PRB 89, 220301 (2014)
[23] A. Girlando J. Phys. Chem. C, 115, 19371 (2011)
[24] M.E. Kozlov et al. Synthetic Metals 86, 2177 (1997)
[25] S. Kaiser et al, Scientific reports 4, 3823 (2014)
[26] R. Singla et al. PRL 115, 187401 (2015)
[27] M. Mitrano et al. Nature 530, 461 (2016)
