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Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models
Speaker: Sebastiano Peotta
Work by: Murad Tovmasyan3, Sebastiano Peotta1, Sebastian Huber3, Päivi Törmä1
1. COMP Centre of Excellence and Department of Applied Physics, Aalto University School of Science, Finland
2. Institute for Theoretical Physics, ETH Zürich, Switzerland
ECT*, TrentoSuperfluid and Pairing Phenomena from
Cold Atomic Gases to Neutron Stars20th – 24th March 2017
Outline
Introduction
Lattice systems: Bloch theorem, band structure and effective mass
Superfluid weight and superfluid mass
Flat bands enhance the superconducting critical temperature
Our work Mean field approach to flat band superfluidity (briefly, Päivi’s talk 23.03)
Perturbative approach to flat band superfluidity
Perspectives
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
References to our works
SP, P. Törmä, Superfluidity in topologically nontrivial flat bands,Nature Communications 6, 8944 (2015)
Aleksi Julku, SP, Tuomas I. Vanhala, Dong-Hee Kim, P. Törmä, Geometric origin of superfluidity in the Lieb lattice flat band, Phys. Rev. Lett. 117, 045303 (2016)
Murad Tovmasyan, SP, P. Törmä, Sebastian Huber, Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models, Phys. Rev. B 94, 245149 (2016)
Long Liang, Tuomas I. Vanhala, SP, Topi Siro, Ari Harju, and P. Törmä,
Band geometry, Berry curvature and superfluid weight, Phys. Rev. B 95, 024515 (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
What is the effect of the crystalline lattice structure on superconductivity?
Outstanding question which is still unanswered for High-Tc superconductors
Bloch theorem and band structure
Bands in the continuum
Bloch theorem:Diagonalization of the single-particle Hamiltonian with periodic potential leads to the band energy dispersions and the periodic Bloch functions
band energies
Bloch functions
Definition of effective mass tensor
Periodic potential
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Wannier functions
In the basis of the Wannier functions of the lowest band a hopping Hamiltonian is obtained
Wannier functionsWannier functions
Wannier functions form a localized and translationally invariant orthonormal basis
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Simple lattices and composite lattices
Simple lattices1 orbital per unit cell
Composite lattices1 < orbitals per unit cell
Honeycomb lattice
Location of the orbitals Location of the orbitals
label the sublattices (orbitals)label the unit cells
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Effect of the lattice structure on the superconducting properties
The effective mass enters in the expression for the London penetration depth
What if effective mass is infinite (FLAT BAND)?
Naively: Infinite penetration depth implies no Meissner effect, superconductivity is lost
Not quite so, a more meaningful quantity is the superfluid weight
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Superfluid density and superfluid weight from the free energy
Definition of superfluid density and superfluid weight
Cooper pair velocity
Cooper pair momentum
Supercurrent density
It makes sense to define the superfluid weight even for a flat band (infinite mass)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Mean-field ansatz
Free energy change
Why are flat band interesting?
Answer: high density of states implies high critical temperature according to BCS theory
Flat bands and room-temperature superconductivity
Flat bandDispersive band
Partial flat band of surface states in rhombohedral graphite
Volovik, J Supercond Nov Magn (2013) 26:2887Kopnin, Heikkilä, Volovik, PRB 83, 220503(R) (2011)Khodel, V. A., & Shaginyan, V. R., JETP Lett. 51, 553-555 (1990); Phys. Rep. 294, 1-134 (1994)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Motivation: Ultracold gas experiments in optical lattices
• Haldane model: Jotzu et al., Nature 515, 237 (2014)
• Harper model: Aidelsburger et al., PRL 111, 185301 (2013), Miyake et al., PRL 111, 185302 (2013)
• Lieb lattice: S. Taie et al., Science Advances 1, e1500854 (2015)
Copper-oxide planes of High-Tc superconductors
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Main questions
A flat band is insulating for any filling in absence of interaction or disorder.
1) Can we characterize transport in a flat band with interactions in terms of the Bloch functions?
2) Is there a relation between topological properties of the flat band and bulk transport properties?
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Isolated flat band limit
Uniform pairing assumption
Number of orbitals with nonzero pairing order parameter
flat band filling
partially filled flat band
bandwidthIsolated flat-bandapproximation
band gap
flat band isolated band
SP and P. Törmä, Nature Communications 6, 8944 (2015)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
We do mean-field BCS theory in a multiband/multiorbital attractive Fermi-Hubbard model
Superfluid weight in a flat band
bandwidthIsolated flat-bandapproximation
band gap
flat band isolated band
SP and P. Törmä, Nature Communications 6, 8944 (2015)
Proportional to the coupling constantQuantumMetric of the flat band
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Important: The quantum metric can be nonzero only in a multiband Hamiltonian
Superfluid weight in a flat bandSP and P. Törmä, Nature Communications 6, 8944 (2015)
Quantum Geometric Tensor
Provost, J. P. & Vallee, G., Commun. Math. Phys. 76, 289 (1980).
Flat band Bloch functions:Bloch functions of the noninteracting lattice Hamiltonian
Quantum metric
Berry curvature
Distance bewteen neighboring quantum states
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Superfluid weight and Chern number
Complex positive semidefinite matrix
Chern number
SP and P. Törmä, Nature Communications 6, 8944 (2015)L. Liang, T. I. Vanhala, SP, T. Siro, A. Harju, and P. Törmä, Phys. Rev. B 95, 024515 (2017)
Superfluid weight in the isolated flat band limit
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Perturbative approach to flat band superfluidity
Perturbation theory cannot describe the superconducting state since this has lower symmetry than the normal state (ideal Fermi gas or Landau-Fermi liquid)
In a flat band is the opposite: the normal state (insulator) has a much larger degeneracy than the superconducting one. Interactions lift the degeneracy of the normal state
Perturbation theory can be used in this case!
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Schrieffer-Wolff transformation
We used perturbation theory in the form of Schrieffer-Wolff transformationFor a recent review: S. Bravyi, D. P. DiVincenzo, and D. Loss, Ann. Phys. 326, 2793 (2011)
0
0
SW transformation
The SW transformation is a unitary transformation that decouples a chosen subspace and its orthogonal complement
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Effective Hamiltonian from the Schrieffer-Wolff transformation
SW effective Hamiltonian describes the low energy physics in terms of the degrees of freedom of the flat band renormalized by the interaction
Projector on the flat band many-body subspace
Perturbative expansion in powers of interaction strength over band gap
1st order: Interaction term projected on the flat band many-body subspaceComplicated many-body Hamiltonian
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Time reversal symmetry
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
We assume throughout time-reversal symmetry (TRS)This facilitates pairing (Anderson’s theorem)
Lattice Wannier functions
Lattice Bloch functions
Single-particle projector on the flat band subspace
Sublattice index
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
1st order Hamiltonian
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Projected field operator
Projected number operator
Projected attractive Hubbard interaction
Projected number operators is not idempotent!
Uniform pairing condition
On top of TRS we also assume the condition (uniform pairing)
Implies the relation used already to derive the relation between quantum metric and superfluid weight with BCS theory
Subset of orbitals on which the flat band states, as well as the Cooper pair wavefunction, are nonzero
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
A consequence of the previous condition is
Positive semidefiniteoperator
Equivalent projected Hamiltonian
Projected total particle number operator
Uniform pairing!
Pair creation operator
Wannier w.f. Bloch w.f.
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Some useful commutation relations
Consequence of time reversal symmetry
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Generator of an emergent SU(2) symmetry
Exactness of BCS wavefunction at 1st order
In a flat band the BCS wave function factorizes both in real and momentum space
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
The BCS wavefunction is exact at first order and has the minimum possible energy !
Number of particlesBinding energy of a Cooper pair
Effective spin Hamiltonian
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
We have found an exact ground state of a complicated quartic Hamiltonian
Approximation: retain in the projected interaction Hamiltonian only the terms that preserve the subspace defined by
all particles are paired
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Effective spin Hamiltonian
The result is a (pseudo)spin effective Hamiltonian with the same BCS ground state and same SU(2) symmetry
Wannier functions
Pseudospin operators
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Superfluid weight from the spin effective Hamiltonian
Ansatz for a state with finite supercurrent
From the spin effective Hamiltonian we obtain that the superfluid weight is related to the overlap functional for Wannier functions
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Superfluid weight from the spin effective Hamiltonian
M. Tovmasyan, S. Peotta, PT, S. Huber, Phys. Rev. B 94, 245149 (2016)
We show that the overlap functional is related to the quantum metric
Physical picture: The local Hubbard interaction produces pair-hopping terms between overlapping Wannier functions. Cooper pairs have a finite mass!
These are terms of the form
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
The mean field result is recovered and justified
Example: Creutz ladder
Two orbitals per unit cell Simple band structure: two flat bands Wannier functions are completely localized
on four sites (plaquette states)
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
M. Creutz, Phys. Rev. Lett. 83, 2636 (1999)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Drude weight of the Creutz ladder
In one dimension is more appropriate to talk about Drude weight rather than superfluid weightD. J. Scalapino, S. R. White, and S. C. Zhang, Phys. Rev. B 47, 7995 (1993)
The line is the prediction of our analytic result
Dots are obtained using Density Matrix Renormalization Group on a chain of finite length (L = 12)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Compressibility and second order effects
Using the SW transformation we can calculate the energy up to second order in the coupling U
This is important in order to obtain a finite compressibility, which is diverging at first order.
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
Our analytic result agrees with the DMRG simulations
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Drude weight and winding number
The Creutz ladder possesses a nonzero topological invariant, the winding number
We show that the quantum metric is related to the winding numberTherefore the Drude weight is bounded from below also by the winding number
M. Tovmasyan, SP, P. Törmä, S. Huber, Phys. Rev. B 94, 245149 (2016)
The example of the Creutz ladder shows that the bound is optimal
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
Important lessons
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
● One can study superconductivity and superfluidity in a flat band using perturbation theory (compare with regular band or continuum)
● The BCS wavefunction is exact at first order
● The effective spin Hamiltonian takes the form of a Heisenberg ferromagnet
● Transport in a flat band is the combined result of interaction and overlapping Wannier functions which leads to pair hopping
Perspectives
Superfluid properties of ultracold gases ongoing experiments, e.g., in Stamper-Kurn's group (Berkeley), Takahashi' group (Kyoto) and ETH Zürich
Is it possible to generalize our results to the case of
Hamiltonians that are not time-reversal invariant?
The effective spin Hamiltonian method fails for bands with nonzero Chern number. How can we generalize it?
Is it possible to use the effective Hamiltonian to calculate the critical temperature and the transport properties?
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
References to our works
SP, P. Törmä, Superfluidity in topologically nontrivial flat bands,Nature Communications 6, 8944 (2015)
Aleksi Julku, SP, Tuomas I. Vanhala, Dong-Hee Kim, P. Törmä, Geometric origin of superfluidity in the Lieb lattice flat band, Phys. Rev. Lett. 117, 045303 (2016)
Murad Tovmasyan, SP, P. Törmä, Sebastian Huber, Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models, Phys. Rev. B 94, 245149 (2016)
Long Liang, Tuomas I. Vanhala, SP, Topi Siro, Ari Harju, and P. Törmä,
Band geometry, Berry curvature and superfluid weight, Phys. Rev. B 95, 024515 (2017)
Superfluid and Pairing Phenomena from Cold Atomic Gases to Neutron StarsECT* Trento, 20th – 24th March 2017
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