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Flat band s uperfluidity and q uantum m etric

Flat band s uperfluidity and q uantum m etric. Speaker: Päivi Törmä Work by: Aleksi Julku 1 , Tuomas Vanhala 1 , Long Liang 1 , Topi Siro 1 , Ari Harju 1 , Sebastiano Peotta 1 , Dong- Hee Kim 2 , Päivi Törmä 1. 1. COMP Centre of Excellence and Department of Applied Physics,

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Flat band s uperfluidity and q uantum m etric

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  1. Flat band superfluidity and quantum metric Speaker: Päivi Törmä Work by:Aleksi Julku1, Tuomas Vanhala1, Long Liang1, TopiSiro1, Ari Harju1, Sebastiano Peotta1, Dong-HeeKim2, 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 PairingPhenomenafromColdAtomicGases to Neutron Stars 20th – 24th March 2017

  2. Contents • Sebastiano Peotta, PT, Superfluidity in topologically nontrivial flat bands, Nature Communications 6, 8944 (2015) • AleksiJulku, Sebastiano Peotta, Tuomas I. Vanhala, Dong-Hee Kim, PT, Geometric origin of superfluidity in the Lieb lattice flat band, • Phys. Rev. Lett. 117, 045303 (2016) • Long Liang, Tuomas I. Vanhala, Sebastiano Peotta, Topi Siro, Ari Harju, PT, Band geometry, Berry curvature and superfluid weight, • Phys. Rev. B 95, 024515 (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, 245149 (2016) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  3. Outline • Introduction (briefly) • Motivation 1: Ultracoldgasexperiments in compositelatticeswithflatbands • 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 PairingPhenomenafromColdAtomicGases to Neutron Stars ECT* Trento, 20th – 24th March 2017

  4. Reminder 1: Bloch theorem and band structure Bloch theorem: 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 PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  5. Reminder 2: Wannierfunctions 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 PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  6. Reminder 3: Simple lattices and composite lattices Composite lattices 1 < orbitals per unit cell Simple lattices 1 orbital per unit cell Honeycomb lattice Location of the orbitals Location of the orbitals label the unit cells label the sublattices (orbitals) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  7. Why are flat bandsinteresting? • 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. Ultracoldgasexperiments in opticallatticeswithcompositestructure Flatband! • Harpermodel: Aidelsburger et al., PRL 111, 185301 (2013), Miyake et al., PRL 111, 185302 (2013) • Haldanemodel: Jotzu et al., Nature 515, 237 (2014) • Lieblattice: S. Taie, et al. Science Advances 1, e1500854 (2015) Copperoxideplanes of cupratehigh-Tcsuperconductors Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  9. Flat bands and room-temperature superconductivity Flat band Dispersive band Partial flat band of surface states in rhombohedral graphite Volovik, J Supercond Nov Magn (2013) 26:2887 Kopnin, Heikkilä, Volovik, PRB 83, 220503(R) (2011) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars 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 Bandwidth W = 1 eV Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  11. The question A flat bandis 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 Our result Bloch functions 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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 band gap Isolated flat-band approximation bandwidth isolated band flat band Quantum Metric of the flat band Only the geometric contribution survives Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars 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: Blochfunctionsaredefinedup to a k-dependentphase. An invariantcanbeconstructed 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 PairingPhenomenafromColdAtomicGases to NeutronStars 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, 024515 (2017) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars 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, 024515 (2017) Thegeometriccomponent of thesuperfluidweight is associatedwithoff-diagonalmatrixelements of thecurrentoperator. Noninteracting Bloch state No geometric effect for single band systems Inter-band processes important! Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars 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, 024515 (2017) Assuming TRS and uniform pairing: In the limit of isolated but not necessarily flat band Thegeometrictermdependsonly on theproperties of theisolatedband, evenif it involves inter-bandmatrixelements of thecurrentoperator. Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  25. Comparison between flat band and parabolic band Parabolic band Flat band Linearly proportional to the coupling constant U ! Fingerprint of the geometric origin Particle density 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 c.f. A.J. Leggett, Phys. Rev. Lett. 25, 1544 (1970); Bounds on supersolids related to (dis)connectedness of the density q Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars 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, 045303 (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 PairingPhenomenafromColdAtomicGases to NeutronStars 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, 045303 (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 PairingPhenomenafromColdAtomicGases to NeutronStars 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, 045303 (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 PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  30. Kane-Meleand 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, 024515 (2017) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  31. Kane-Meleand 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, 024515 (2017) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  32. Kane-Meleand Haldane Hubbard models: BKT transition temperature In 2D the superfluid weight sets the transition temperature of the Kosterliz-Thoulesstransition. 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, 024515 (2017) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

  33. Perspectives • Superfluid properties of ultracold gases • ongoing experiments, e.g., in Stamper-Kurn’sgroup (Berkeley), Takahashi’sgroup (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, 134423 (2016) Superfluid and PairingPhenomenafromColdAtomicGases to NeutronStars ECT* Trento, 20th – 24th March 2017

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

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