effective theory and emergent su(2) symmetry in … theory and emergent su(2) symmetry in the flat...

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)


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


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


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


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 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)


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)


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


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?


Two approaches

Mean-field BCS theory

Perturbative approach

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)


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


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 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


Several complementary approaches

Mean-field BCS theory

Perturbative approach

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)


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


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



Time reversal symmetry


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


1st order Hamiltonian



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





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.


Some useful commutation relations

Consequence of time reversal symmetry


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



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


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


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




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


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


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. Creutz, Phys. Rev. Lett. 83, 2636 (1999)


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)



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.


Our analytic result agrees with the DMRG simulations


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


The example of the Creutz ladder shows that the bound is optimal


Important lessons


● 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?


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)


Acknowledgements

QD group, Aalto UniversityProf. P. Törmä

SP A. Julku Long Liang T. Vanhala

A. Harju

QMP Group, Aalto Condensed Matter Theory and Quantum Optics, ETH Zürich

Prof. S. Huber M. Tovmasyan

Gwangju Institute of Science and Technology, Korea

Prof. D.-H. KimT. Siro

effective theory and emergent su(2) symmetry in … theory and emergent su(2) symmetry in the flat...

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