Condensed Matter Physics

F. M. Marchetti & R. Sanchez

Professors	Francesca Maria Marchetti Tel. 91 497 5590 e-mail: francesca.marchetti (at) uam.es Departamento de Fisica Teorica de la Materia Condensada 6th floor office 606 Rafael Sanchez Tel. 91 497 5806 e-mail: rafael.sanchez (at) uam.es Departamento de Fisica Teorica de la Materia Condensada 5th floor office 505	[from Alkali&Quantum Gases MIT]	[from Wikimedia Commons]

Calendar & classroom

Classes are Monday to Thursday always at 10:30

This year ALL classes are face-to-face!! classroom 01.13.AU.405 - room 405 module 13)

JAUNARY
Mon 31 Introduction class
FEBRUARY
	Tue 1 RS Lecture 1	Wed 2 RS Lecture 2	Thu 3 RS Problem 1
Mon 7 RS Lecture 3	Tue 8 RS Lecture 4	Wed 9 RS Lecture 5	Thu 10 RS Problem 2
Mon 14 RS Lecture 6	Tue 15 RS Lecture 7	Wed 16 RS Lecture 8	Thu 17 RS Problem 3
Mon 21 RS Lecture 9	Tue 22 RS Lecture 10	Wed 23 RS Lecture 11	Thu 24 RS Problem 4
Mon 28 RS Lecture 12
MARCH
	Tue 1 RS Lecture 13	Wed 2 RS Lecture 14	Thu 3 RS Problem 5
Mon 7 RS Lecture 15	Tue 8 RS Lecture 16	Wed 9 RS Lecture 17	Thu 10 RS Problem 6
Mon 14 RS Lecture 18	Tue 15 RS Lecture 19	Wed 16 RS Lecture 20	Thu 17 FMM Lecture 1
Mon 21 FMM Lecture 2	Tue 22 FMM Lecture 3	Wed 23 FMM Lecture 4	Thu 24 FMM Problem 1
Mon 28 FMM Lecture 5	Tue 29 FMM Lecture 6	Wed 30 FMM Lecture 7	Thu 31 FMM Problem 2
APRIL
Mon 4 FMM Lecture 8	Tue 5 FMM Lecture 9	Wed 6 FMM Lecture 10	Thu 8 FMM Problem 3
Mon 11	Tue 12	Wed 13	Thu 14
Mon 18	Tue 19 FMM Lecture 11	Wed 20 FMM Lecture 12	Thu 21 FMM Problem 4
Mon 25 FMM Lecture 13	Tue 26 FMM Lecture 14	Wed 27 FMM Lecture 15	Thu 28 FMM Problem 5
MAY
Mon 2	Tue 3 FMM Lecture 16	Wed 4 FMM Lecture 17	Thu 5 FMM Problem 6
Mon 9 FMM Lecture 18	Tue 10 FMM Lecture 19	Wed 11 FMM Lecture 20	Thu 12 FMM Problem 7
Mon 16 Tutorials	Tue 17 Tutorials	Wed 18 Tutorials	Thu 19 Presentations	Fri 20 Presentations
Mon 23	Tue 24	Wed 25	Thu 26
Mon 30 Exam (ordinary)	Tue 31
JUNE
		Wed 22 Exam (extraord)

Material complem.

Part I (RS):

Lecture 1
Lecture 2&3&4
Lecture 5
Lecture 6&7
Lecture 8&9&10
Lecture 11&12&13
Lecture 14&15&16
Lecture 17&18
Lecture 19
Lecture 20
Lecture 21

Problem set 1
Problem set 2
Problem set 3
Problem set 4
Problem set 5

Part II (FMM):

Lecture 1
Lecture 2&3&4
Lecture 5
Lecture 6&7
Lecture 8&9&10
Lecture 11&12&13
Lecture 14&15&16
Lecture 17&18
Lecture 19
Lecture 20
Lecture 21

Problem set 1
Problem set 2
Problem set 3
Problem set 4
Problem set 5
VOLVER ARRIBA

VOLVER ARRIBA

N.B. The supplementary materials underlined in yellow are (advanced) research articles among which you can choose one or two articles for the 15 minute presentation (+ 5 minute discussion) at the end of the course. **Presentations are an essential part of the evaluation**

Part I: Many electron systems, quantum properties
Lectures 1& 2&3 Mathematical tools: Second quantization formalism Wick's theorem
Lectures 4&5&6 Review on non-interacting electron gases Electrons in a potential. The jellium model Perturbation theory	Problem set 1
Lectures 7-11 Coulomb interactions. Mean field approximations Hartree-Fock approximation Screening	Problem set 2
Lecture 12&13&14 Tight-binding models. Graphene. Hubbard model Phase transitions	Problem set 3
A. H. Castro Neto et al., The electronic properties of graphene, Rev. Mod. Phys. 81 , 109 (2009). T. Ihn, Semiconductor nanostructures (Oxford Univ. Press, 2013).
Lectures 15 Fermi liquid theory
L. D. Landau, The theory of a Fermi liquid, Sov. Phys. JETP 3, 920 (1957).

Building the quantum world
Lectures 16-18 Low dimensional systems: 2D electron gas Quantum wires Quantum dots Quantum Hall effect and topology. Majorana fermions.	Problem set 4
K. von Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980). M. Buttiker, Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38 , 9375 (1988). M. Buttiker, The Quantum Hall Effect in Open Conductors, Semiconductors and Semimetals 35, 191 (1992). A. Yu. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44 , 131 (2001). R. Aguado, Majorana quasiparticles in condensed matter, I l Nuovo Cimento 40 , 523 (2017). H. van Houten and C.W.J. Beenakker, Quantum point contacts, Phys. Today 49, 22 (1996). Y. Imry and R.A. Webb, Quantum Interference and the Aharonov-Bohm Effect, Scientific American 260, 56 (1989).
Wigner cristals. E. Wigner, Trans. Faraday Soc. 34, 678-685 (1938). I. Shapir et al., Science 364, 870 (2019).
Lectures 19&20 Open quantum systems. Master equations. Single-electron transport. Coulomb blockade. Qubits	Problem set 5
L. P. Kouwenhoven et al., Electron transport in quantum dots, in Mesoscopic Electron Transport, ed L. L. Sohn, G. Schoen and L. P. Kouwenhoven (Kluwer Series E vol 345) (June 1996) p 105 - 214 W. G. van der Wiel et al., Electron transport through double quantum dots, Rev. Mod. Phys. 75 , 1 (2003). R. Hanson et al., Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007). Y. V. Nazarov, Y. M.Blanter, Quantum transport: Introduction to nanoscience (Cambridge Univ. Press, 2009). P. Barthelemy and L.M.K. Vandersypen, Quantum Dot Systems: a versatile platform for quantum simulations, Ann. Phys. (Berlin) 525, 808 (2013).
Lectures 20 Quantum transport. Scattering theory. Onsager relations.
G. B. Lesovik and I. A. Sadovskyy, Phys.-Usp. 54 , 1007 (2011 ). S. Datta, Quantum transport. Atom to transistor (Cambridge Univ. Press, 2013). G. Benenti et al., Fundamental aspects of steady-state conversion of heat to work at the nanoscale, Phys. Rep. 694, 1 (2017).

Part II: Collective quantum coherence of bosons and fermions

An introduction to BEC & superfluidity

N.B. For lecture notes and assignments, see this course page on Perusall
Other material can also be found in Moodle

Lecture 0
Presentation of part 1 of the course

Historical introduction to Bose-Einstein condensation (BEC) and sperfluidity

Introductory lecture
E. A. Cornell and C. E. Wieman, "Nobel lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments", Rev. Mod. Phys. 74, 875 (2002)
E. A. Cornell, J. R. Ensher, and C. E. Wieman, "Experiments in Dilute Atomic Bose-Einstein Condensation", proceedings of the Varenna conference on Bose-Einstein condensation (1998)
A. Griffin, "Superfluidity: three people, two papers, one prize", Physics World August 2008, 27-30

Lectures 1
Second quantisation: a reminder
The ideal Bose gas

Lectures 2&3
Bose-Einstein condensation
Gas in a 3D box
Thermodynamic limit

Lecture 4
BEC in ideal Bose gases: statistical saturation of the excited states
Harmonic trapping

Problem set 1

Solutions

* Trapped ideal gases (Tc for BEC, condensate fraction)
* Phenomenology of the ultracold gases experiments
* Measurement of energy and ground-state occupation in ultracold atomic BECs

Lectures: Introduction to experiments in ultracold atomic gases; part1, part2
Science News ''Physicists create new state of matter'', Science 269, 152 (1995)
Science Perspectives ''An Intimate gathering of bosons'', K. Burnett, Science 269, 182 (1995)
F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999)
M. H. Anderson et al., Science 269, 198 (1995)
K. B. Davis et al., PRL 75, 3969 (1995)
Insight Review Article "Bose-Einstein condensation of atomic gases", Nature 416, 211 (2002)
Review article: W. Ketterle et al., "Making, probing and understanding BECs", Proceedings of the International School of Physics "Enrico Fermi", Course CXL, edited by M. Inguscio, S. Stringari and C.E. Wieman (IOS Press, Amsterdam, 1999) pp. 67-176.

* Deviations from Einstein's picture of an ideal saturated Bose gas

J. R. Ensher et al., PRL 77, 4984 (1996)
R. P. Smith and Z. Hadzibabic, "Effects of interactions on Bose-Einstein condensation of an atomic gas",
arXiv:1203.2063
N. Tammuz et al., "Can a Bose Gas Be Saturated?", Phys. Rev. Lett. 106, 230401 (2011)
R. P. Smith et al., "Effects of Interactions on the Critical Temperature of a Trapped Bose Gas", Phys. Rev. Lett. 106, 250403 (2011)

* Bose-Einstein Condensation in a Uniform potential

A. L. Gaunt et al., "Bose-Einstein Condensation of Atoms in a Uniform Potential", Phys. Rev. Lett. 110, 200406 (2013)
R.Lopes et al., "Quantun Deplition of a Homogeneous Bose-Einstein Condensate", Phys. Rev. Lett. 119, 190404 (2017)

Lecture 5&6
One-body density matrix & off-diagonal long range order
Order parameter
Ground state & coherent states

Lecture 7&8&9&10
The weakly-interacting Bose gas
Excitation spectrum
The Bogoliubov transformation
Sound velocity
Healing length
Condensate depletion due to interactions
ODLRO for weakly interacting Bose gases

Problem set 2

* One-body density matrix in weakly interacting Bose gases
* Phase coherence properties of weakly interacting Bose gases
* Measurements of coherence & ODLRO: from quantum optics to ultracold gases

I. Bloch et al., Nature 403, 166 (2000)
S. Ritter et al., PRL 98, 090402 (2007)

* Measurements of the excitation spectrum of a BEC in ultracold atoms

J. Steinhauer et al., PRL 88, 120407 (2002)

Lecture 11
BEC and superfluidity:
Landau criterion
Defect moving through a superfluid
GPE and inhomogeneous Bose Einstein Condensates (optional)

The discovery of superfluidity: A. Griffin, "Superfluidity: three people, two papers, one prize", Physics World (2008).
A. J. Leggett, "Superfluidity", Rev. Mod. Phys. 71, S318 (1999)

Problem set 3

* Landau criterion, Cherenkov waves and drag force in weakly interacting Bose condensates

G. E. Astrakharchik and L. P. Pitaevskii, PRA 70, 013608 (2004)
I. Carusotto et al., PRL 97, 260403 (2006)
C. Raman et al., PRL 83, 2502 (1999)
T. W. Neely et al., PRL 104, 160401 (2010)

Problem set 4

* Zero temperature: The time dependent Gross-Pitaevskii equation
* Conservation laws: continuity equation
Stationary solutions
* Landau free energy, order parameter, 2nd order phase transition
* Small amplitude oscillations: Bogoliubov-de Gennes equations: Bogoliubov spectrum of excitations

* Superfluid velocity & quantisation of circulation
* Vortex line solutions (healing length), Rotation of superfluids, Energy of a vortex solution

G.B. Hess & W.M. Fairbank, PRL 19, 216 (1968)

R.E. Packard & T.M. Sanders, Jr., PRA 6, 799 (1972)

E.J. Yarmchuck et al., PRL 43, 214 (1979)

J.R. Abo-Shaeer et al., Science 292, 476 (2001)

* Trapped condensates: Thomas-Fermi limit
* Time of flights measurements: expansion of a BEC

* Interference between two condensates

M.R. Andrews et al., Science 275, 637 (1997)

An introduction to the BCS theory for superconductivity

* Thermal properties of fermionic ultracold gases: Fermi temperature, heat capacity

B. De Marco & D. S. Jin, "Onset of Fermi degeneracy in a trapped atomic gas", Science 285, 1703 (1999)
A. G. Truscott et al., "Observation of Fermi pressure in a gas of trapped atoms", Science 291, 2570 (2001)

Lecture 12&13&14
Reminder about the ideal Fermi gas
Weakly interacting Fermi gases
The one-pair Cooper problem
Cooper pairs

Lecture 15&16
BCS theory at zero temperature
Reduced Hamiltonian
BCS ground state: pair operators

* Fock states versus coherent states (number of particles and phase as conjugate variables)

Lecture 17&18&19
Mean-field approximation
Bogoliubov transformation: quasi-particles
Variational calculation
Gap and number equations

Problem set 5

Lecture 20&21
BCS theory at finite temperature (optional)
(Elements of the) BEC-BCS crossover theory (optional)

* Feshbach resonances & the BEC-BCS crossover
* BEC-BCS crossover at zero temperature: T=0 variational calculation

A.J. Leggett, "Diatomic molecules and Cooper Pairs", Modern Trends in the Theory of Condesed Matter 115 13-27, (1980)

Lecture notes "Superfluidity in Ultracold Fermi gases"

M. Greiner et al., Nature 426, 537 (2003)

C. A. Regal et al., PRL 92, 040403 (2004)

Some topic for a possible presentation

W. Ketterle & M. W. Zwierlein, Making, probing and understanding ultracold Fermi gases in Ultracold Fermi Gases , Proceedings of the International School of Physics "Enrico Fermi", Course CLXIV, Varenna, 20 - 30 June 2006, Ed. M. Inguscio, W. Ketterle, and C. Salomon (IOS Press, Amsterdam) 2008

The final course presentations will be on Thursday the 19th of May 2021 AND Fridai the 20th of May 2021 (20 min. = 15 min. presentation, 5 min. discussion)

PART 1

Quasiparticles and interactions
Landau theory of the Fermi liquid
Dirac fermions in graphene
Klein tunneling in graphene
Topological insulators
Majorana fermions
Wigner cristals
Anderson model for impurities and the Kondo effect
Oscillations in scanning tunneling microscopy

Artificial quantum systems
Conductance quantization in quantum point contacts
Qubits in quantum dots
Single electron transport in quantum dots
Quantum simulators
Electron optics with quantum Hall edge channels
Persistent currents in normal metal rings
Thermoelectric effect in quantum conductors
Observation of the Aharonov-Bohm effect in metal rings
Quantum interference and resonant tunneling
Entanglement of two qubits

PART 2

General
Phase transitions, spontaneous symmetry breaking and the Goldstone mode
Josephson effect in superfluids and superconductors
The concept of phase in superfluids and superconductors: interference between two condensates and Josephson effect
Flux quantisation and vortices in superconductors and superfluids

Fermions
Feshbach resonances & the BEC-BCS crossover
Polarised Fermi gases

Bosons
Effects of interactions on Bose-Einstein condensation of an atomic gas

Deviations from Einstein's picture of an ideal saturated Bose gas

Bose-Einstein Condensation in a Uniform potential

One-body density matrix in weakly interacting Bose gases

Phenomenology of the ultracold gases experiments

Trapped ideal gases (Tc for BEC, condensate fraction)

Measurement of energy and ground-state occupation in ultracold atomic BECs

Measurements of the excitation spectrum of a BEC in ultracold atoms
Intereference between two condensates:Fock vs. coherent states
Measurements of the phase coherence properties of weakly interacting Bose gases
Measurements of coherence & ODLRO: from quantum optics to ultracold gases
The Gross-Pitaevskii equation

Superfluid velocity & quantisation of circulation: Vortex line solutions

Landau criterion, Cherenkov waves and drag force in weakly interacting Bose condensates

Time of flights measurements: expansion of a BEC

Bibliography

VOLVER ARRIBA

H. Bruus & K. Flensberg, Many-body quantum theory in condensed matter physics (Oxford Univ. Press, 2016)
P. Phillips, Advanced solid state physics (Cambridge Univ. Press, 2012)
C. Kittel, Introduction to solid state physics (Wiley, New York, 1961)
L. Pitaevskii & S. Stringari, Bose-Einstein Condensation (Clarendon Press, Oxford, 2003)

A. J. Leggett, Quantum Liquids --- Bose Condensation and Cooper Pairing in Condensed-Matter Systems (Oxford Graduate Texts Oxford, 2006)

C. J. Pethick & H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)

Bose Einstein Condensation, ed. A. Griffin, D. W. Snoke & S. Stringari (Cambridge University Press, Cambridge, 1995)
K. Huang, Introduction to Statistical Physics (CRC Press, 2001)
Ashcroft & Mermin, Solid State Physics

P. G. de Gennes, Superconductivity of Metals and Alloys, Westview Press, Oxford (1966)

M. Tinkham, Introduction to Superconductivity, Dover Publications, New York (1996)

J. R. Schrieffer, Theory of Superconductivity, Westview Press, Oxford (1964)

Lecture notes on Superconductivity from Alfredo Levy Yeyati

Home

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Condensed Matter Physics

F. M. Marchetti & R. Sanchez

Professors

Calendar & classroom

Additional material

Bibliography

Calendar & classroom

Material complem.

Part I: Many electron systems, quantum properties

Building the quantum world

Part II: Collective quantum coherence of bosons and fermions

An introduction to BEC & superfluidity

An introduction to the BCS theory for superconductivity

Bibliography