SLIDE 1
N
Aline Ramires
Institute for Theoretical Studies - ETH - Zürich
ITP - Heidelberg University - 13th June 2017
"From a theoretical tool to the lab"
Cold Quantum Coffee
SLIDE 2
- Established in 2013;
- Interdisciplinary institute dedicated to research in mathematics,
theoretical physics and theoretical computer science;
- Currently: 9 Junior Fellows and 6 Senior Fellows,
- Support from Dr. Max Rössler and the Walter Haefner Foundation.
The Institute for Theoretical Studies
300m
ETH - Hauptgebäude
5km
ETH - ITP
SLIDE 3 Outline
- Strongly Correlated Systems
Local Moment formation and the Kondo Effect Heavy Fermions
Spin and Time-reversal: Symplectic-N Decoupling spin Hamiltonians
- Q: Just a theoretical tool?
Enlarged symmetries with ultracold atoms
SLIDE 4
The Periodic Table of Elements
SLIDE 5
- J. L. Smith and E. A. Kmetko, J. of the Less-Common Metals (1982)
Increasing localization Increasing localization Fermi Liquid Magnetism
More localized orbitals ⇒ Enhanced interactions ⇒ Strong Correlations 3d
4f 5f
FS
The Smith-Kmetko Diagram
"Electrons in the brink of localization"
SLIDE 6 Electrons in the brink of localization ⇒ Easily tunable
Cuprates
Fradkin, Nature Physics (2012)
Schematic diagram YBa2Cu3Ox
Heavy Fermions
Knebel, J. Phys. Soc. Jpn. (2011)
CeRhIn5 Ba(Fe1-xCox)As2
Fe-pnictides
Tranquada, Physics 3 (2010)
T(K) x
Strange metal Strange Metal
Examples of Strongly Correlated Systems
T(K)
Strange metal Strange Metal
Repeating theme ⇒ What can we learn from HF?
SLIDE 7 Effective models and local moment formation
Anderson Impurity Model
U+εf
HAtomic Infinite-U Anderson Model
Conduction sea Localized Orbital
Requires: At low T only the spin DOF remains. Kondo Impurity Model
AFM
- P. Coleman, Introd. to Many Body Physics (2015)
SLIDE 8 Poor-man scaling and the Kondo Effect
- J. Kondo, Prog. in Theor. Phys. 32, 1, 37 (1964)
- P. W. Anderson, J. Phys. C: Solid State Phys. 3, 2436, 2 (1970)
What if we want to keep renormalizing? Kondo Impurity Model
E ρ(E)
D - δD D
States to be removed States to be removed
Below TK: Singlet Bound State
SLIDE 9 Kondo Lattice Model
AFM QCP FL
?
- S. Doniach, Physica B (1977)
Doniach Phase Diagram
RKKY Temperature
Ruderman-Kittel-Kasuya-Yosida (1954-57)
Energy Scales in Heavy Fermions
Kondo Temperature
- J. Kondo, Prog. in Theor. Phys. 32, 1, 37 (1964)
Impurity: Singlet Bound State Lattice: HEAVY Fermi Liquid
SLIDE 10
Take-home messages I & II: The effective models to describe them usually start from a Kondo lattice model, which is written in terms of local moments and cannot be treated perturbatively. There is a class of materials called heavy fermion systems in which electrons are very strongly interacting.
SLIDE 11 Large-N Approach
No natural small energy scale:
- G. t’Hooft, Nucl Phys B 71, 461 (1973)
- E. Witten, Nucl Phys B 160, 57 (1979)
Introduce an artificial small parameter: 1/N Quantum Chromodynamics
Barions (N-body singlets)
Condensed Matter
?
Cooper Pairs Valence Bonds
- L. Balents, Nature (2010)
SLIDE 12 Symplectic-N Approach Motivation to keep time reversal
- R. Flint et. al., Nature (2008)
Time-reversal: Requirement of consistency Symplectic condition
Now we have a generalisation of spin operators which are well behaved under the time-reversal operation.
Generators:
Generalized Spin Operators
SLIDE 13
Decoupling Spin Hamiltonians
SP(N) properly accounts for Frustration and Superconductivity!
Superconductivity / Valence Bonds
SP(N) Symmetry
Hopping / Hybridization
SU(N) Symmetry
SLIDE 14 Take-home messages III & IV:
Θ
CeRhIn5
Large-N generalisations are useful for the description of strongly correlated materials. The symplectic-N approach seems to provide a more appropriate generalisation of spin operators
SLIDE 15
Q: Are these models with enlarged symmetries only theoretical tools or can they be real?
I feel like a heavy fermion!
SLIDE 16
Cold Atoms and enlarged symmetries
At ultra-low temperatures and in the low density limit, we can model interacting atoms with contact interactions. f: Hyperfine Spin (Total angular momentum of the atom) F: Total angular momentum of the PAIR of atoms which is scattering Total angular momentum conservation.
SLIDE 17
Cold Atoms and enlarged symmetries
Note that only even-F channels contribute to scattering: Taking α <-> β and using properties of the CGC: So for both Bosons (η = 1 and 2f even) and Fermions (η = -1 and 2f odd): We find: F = 0 , 2 , 4 , 6 , …
SLIDE 18 Cold Atoms and enlarged symmetries
SU(N) Symmetry
Number of particles in each flavour = nα, is a conserved quantity. Realization: Alkaline-Earth atoms in this case the interaction vertex simplifies to:
SP(N) Symmetry
“Colour magnetization” = nα-n-α is a conserved quantity. Condition: Define:
*Naturally satisfied for
- A. Ramires arXiv 1606.08709 (2017)
SLIDE 19 Cold Atoms and enlarged symmetries
- T. Maier, PhD Thesis (2015)
Dipolar character Already Condensed
1) Not strong dipole-dipole interaction 2) Stable Elements 3) Fermionic Isotopes with f > 1/2
Realizes SP(N) * Realizes SU(N) SP(10) SP(6) SP(8)
SLIDE 20
Take-home messages V & VI: It is possible to realise systems with enlarged symmetries in cold atomic systems SP(N) is a current challenge for experimentalists.
SLIDE 21 Conclusion
I feel like a heavy fermion!
- Motivated by heavy fermion systems
- Looked for appropriate theoretical tools: Symplectic-N
- Q: Are these models with enlarged symmetries real?
- Cold atoms can realise SU(N) and SP(N) symmetries
