1. Flat band superfluidity and quantum metric
Speaker: Päivi Törmä
Work by: Aleksi Julku1, Tuomas Vanhala1, Long Liang1, Topi Siro1, Ari Harju1, Sebastiano Peotta1, Dong-Hee Kim2, Päivi Törmä1
1. COMP Centre of Excellence and Department of Applied Physics,
Aalto University School of Science, Finland
2. Department of Physics and Photon Science, School of Physics and Chemistry,
Gwangju Institute of Science and Technology, Gwangju, Korea
ECT*, Trento
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
20th – 24th March 2017

2. Contents
Sebastiano Peotta, PT, Superfluidity in topologically nontrivial flat bands, Nature Communications 6, 8944 (2015)
Aleksi Julku, Sebastiano Peotta, Tuomas I. Vanhala, Dong-Hee Kim, PT, Geometric origin of superfluidity in the Lieb lattice flat band,
Phys. Rev. Lett. 117, (2016)
Long Liang, Tuomas I. Vanhala, Sebastiano Peotta, Topi Siro, Ari Harju, PT, Band geometry, Berry curvature and superfluid weight,
Phys. Rev. B 95, (2017)
Related – already covered by Sebastiano Peotta’s talk on Tuesday
Murad Tovmasyan, Sebastiano Peotta, PT, Sebastian Huber, Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models, Phys. Rev. B 94, (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

3. Outline Introduction (briefly) Our work Perspectives
Motivation 1: Ultracold gas experiments in composite lattices with flat bands
Motivation 2: Flat bands enhance the superconducting critical temperature
Our work
Main question: is there superfluid transport in a flat band?
Mean field approach
(Perturbative approach – already covered by Sebastiano Peotta’s talk)
Numerical approach: ED, DMFT, DMRG
Perspectives
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

4. Reminder 1: Bloch theorem and band structure
Diagonalization of the single-particle Hamiltonian with periodic potential leads to the band energy dispersions and the periodic Bloch functions
Periodic potential
band energies
Bloch functions
Definition of effective mass tensor
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

5. Reminder 2: Wannier functions
Wannier functions form a localized and translationally invariant orthonormal basis
Wannier functions
In the basis of the Wannier functions of the lowest band a hopping Hamiltonian is obtained
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

6. Reminder 3: Simple lattices and composite lattices
1 orbital per unit cell
Composite lattices
1 < orbitals per unit cell
Honeycomb lattice
Location of the orbitals
Location of the orbitals
label the unit cells
label the sublattices (orbitals)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

7. Why are flat bands interesting?
Answer: high density of states implies high critical temperature according to BCS theory
Flat bands can be realized in ultracold gases
and in other systems, e.g. rhombohedral graphite.

8. Ultracold gas experiments in optical lattices with composite structure
Flat band!
Harper model: Aidelsburger et al., PRL 111, (2013), Miyake et al., PRL 111, (2013)
Haldane model: Jotzu et al., Nature 515, 237 (2014)
Lieb lattice: S. Taie, et al. Science Advances 1, e (2015)
Copper oxide planes of cuprate high-Tc superconductors
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

9. Flat bands and room-temperature superconductivity
Dispersive band
Flat band
Partial flat band of surface states in rhombohedral graphite
Volovik, J Supercond Nov Magn (2013) 26:2887
Kopnin, Heikkilä, Volovik, PRB 83, (R) (2011)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

10. Flat bands and high-temperature superconductivity
attractive Hubbard model in two dimensions with the following density of states (DOS):
1) uniform DOS
2) Square lattice, 2D van Hove singularity
Compare to the flat band case:
3) flat band DOS
U = 0.1 eV
U = 0.2 eV
Uniform DOS
0.3 K
44 K
Square lattice DOS
24 K
183 K
Flat band DOS
124 K
290 K
Bandwidth W = 1 eV
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

11. The question
A flat band is insulating for any filling in absence of interaction or disorder.
Is there transport in a flat band in the presence of interactions?

12. The essential idea: multiband approach
Observation
A flat band in a single-band lattice Hamiltonian is necessarily trivial since all the hopping matrix elements are vanishing. No transport can occur.
Approach
Use a multiband/multiorbital Hamiltonian defined on a composite lattice. The total Chern number must be zero and the interaction is to a good approximation a local Hubbard interaction.
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

13. Wannier functions in simple and composite lattices
N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

14. Why total Chern number = 0?
Theorem:
Exponentially localized Wannier functions can be constructed in 2D or 3D iff the Chern number(s) is zero
Berry connection
Berry curvature
Chern number
The zero Chern number of the composite band guarantees localized Wannier functions
The flat band can have nonzero Chern number
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

15. The essential idea: multiband approach
The composite Wannier functions for a subset of bands are well-localized. One can use a simple Fermi-Hubbard tight-binding Hamiltonian
The Wannier functions relative to the flat band are not necessary localized…
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

16. Several complementary approaches
Mean-field BCS theory
Perturbative approach
Numerics (ED, DMFT, DMRG)

17. Mean-field ansatz for the superconducting state with finite ground state supercurrent
Order parameter is frozen and uniform in the simplest BCS ground state
A state with uniform supercurrent corresponds to a plane-wave modulation of the order parameter phase
Current flow in a superconductor is a ground state property!
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

18. Superfluid weight in a multiband/multiorbital system
Definition of superfluid weight
as the free energy variation
Conventional contribution present in the single band case
Bloch functions
Our result
Geometric contribution present only in the multiband/
multiorbital case
Can be nonzero in a flat band
S. Peotta and PT, Nature Communications 6, 8944 (2015)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

19. Superfluid weight in a flat band
S. Peotta and PT, Nature Communications 6, 8944 (2015)
Uniform pairing assumption
Isolated flat-band
approximation
bandwidth
band gap
flat band
isolated band
Quantum
Metric of the flat band
Only the geometric contribution survives
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

20. Superfluid weight in a flat band
S. Peotta and PT, Nature Communications 6, 8944 (2015)
Flat band Bloch functions:
Bloch functions are defined up to a k-dependent phase. An invariant can be constructed
Quantum Geometric Tensor
Provost, J. P. & Vallee, G., Commun. Math. Phys. 76, 289 (1980).
Distance bewteen neighboring quantum states
Quantum metric
Berry curvature
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

21. We show that:
1) Superfluidity in a flat band has a geometric origin (quantum metric)
2) The flat band superfluid weight is bounded from below by the Chern number

22. Superfluid weight and Chern number
Superfluid weight in the isolated flat band limit
Complex positive semidefinite matrix
Chern number
Local form in k-space
S. Peotta and PT, Nature Communications 6, 8944 (2015)
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

23. Superfluid weight from linear response
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
The geometric component of the superfluid weight is associated with off-diagonal matrix elements of the current operator.
Noninteracting Bloch state
No geometric effect for single band systems
Inter-band processes important!
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

24. Superfluid weight from linear response
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
Assuming TRS and uniform pairing:
In the limit of isolated but not necessarily flat band
The geometric term depends only on the properties of the isolated band, even if it involves inter-band matrix elements of the current operator.
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

25. Comparison between flat band and parabolic band
Particle density
Linearly proportional to the coupling constant U ! Fingerprint of the geometric origin
Bandwidth
Physical picture: global shift of the Fermi sphere
Physical picture: delocalization of Wannier functions.
Overlapping Cooper pairs: pairing fluctuations support transport if pairs can be created and destroyed at distinct locations ky kx q
c.f. A.J. Leggett, Phys. Rev. Lett. 25, 1544 (1970);
Bounds on supersolids related to (dis)connectedness of the density
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

26. Several complementary approaches
Mean-field BCS theory
Perturbative approach
Numerics (ED, DMFT, DMRG)

27. Lieb lattice As a case study we consider the Lieb lattice geometry
A. Julku, S. Peotta, T. Vanhala, D.-H. Kim, PT, Phys. Rev. Lett. 117, (2016)
Three bands: two dispersive and one strictly flat band (with zero Chern number)
Staggered hopping coefficients open a gap between the flat band and the dispersive bands.
Isolated flat band
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

28. Lieb lattice: superfluid weight
A. Julku, S. Peotta, T. Vanhala, D.-H. Kim, PT, Phys. Rev. Lett. 117, (2016)
on the flat band depends strongly on U
on dispersive bands roughly constant
for the trivial flat bands is non-zero and large!
Total filling (1< <2 for the flat band)
Temperature is zero here
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

29. Lieb lattice: superfluid weight geometric contribution
A. Julku, S. Peotta, T. Vanhala, D.-H. Kim, PT, Phys. Rev. Lett. 117, (2016)
The large superfluid weigh and its strong dependence on the interaction within the flat band is explained by the geometric superfluid weight contribution
Half-filled flat band ( = 1.5)
Half-filled upper band ( = 2.5)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

30. Kane-Mele and Haldane models
Band structure of both Kane-Male and Haldane models.
Quasi-flat lowest band.
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

31. Kane-Mele and Haldane Hubbard models: superfluid weight, MF vs
Kane-Mele and Haldane Hubbard models: superfluid weight, MF vs. ED results
Kane-Mele Hubbard (KMH)
Haldane Hubbard (HH)
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

32. Kane-Mele and Haldane Hubbard models: BKT transition temperature
In 2D the superfluid weight sets the transition temperature of the Kosterliz-Thouless transition. For small U it is similar to the Mean-Field BCS critical temperature
L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and PT, Phys. Rev. B 95, (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

33. Perspectives Superfluid properties of ultracold gases
ongoing experiments, e.g., in Stamper-Kurn’s group (Berkeley), Takahashi’s group (Kyoto) and
ETH Zürich
Are there known materials for which the geometric contribution to the superfluid weight is sizable?
Possibly high-Tc superconductors
I. Božović, X. He, J. Wu, and A. T. Bollinger Nature 536, 309 (2016)
Design of novel superconducting materials (e.g. carbon-based)
Effects of Bloch wave functions and quantum metric in other condensed matter systems F. Piéchon, et al., Phys. Rev. B 94, (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron Stars
ECT* Trento, 20th – 24th March 2017

34. Acknowledgements QD group, Aalto University QMP Group, Aalto
Prof. P. Törmä
S. Peotta
A. Julku
Long Liang
T. Vanhala
QMP Group, Aalto
Gwangju Institute of Science and Technology, Korea
Condensed Matter Theory and Quantum Optics, ETH Zürich
A. Harju
T. Siro
Prof. S. Huber
M. Tovmasyan
Prof. D.-H. Kim