Topology induced emergent dynamic gauge theory in an extended Kane-Mele-Hubbard model Xi Luo (Fudan) @Taipei January 5, 2015 arXiv: 1408.5730.

Topology induced emergent dynamic gauge theory in an

extended Kane-Mele-Hubbard model

Xi Luo (Fudan)

@Taipei January 5, 2015

arXiv: 1408.5730

Collaborators

• Yue Yu (Chinese Academy of Science, Fudan, and Collaborative Innovation Center of Advanced Microstructures)

• Long Liang (Chinese Academy of Science)

Outline

• Introduction• Emergence of the Proca theory• Emergent QED3

• Conclusions and discussions

Introduction

• Confinement and deconfinement phase transition of gauge fields is crucial in particle physics and condensed matter physics.

Introduction

• Eg.1 QCD, still a mystery.

– (Alford etal. Rev.Mod.Phys 2008)

Introduction

• Eg.2 spin-charge separation, superconductor, and etc.

– (high field magnet laboratory, Radboud University)

Introduction

• Haldane model (Haldane PRL 88’)– NNN coupling with a phase

– Breaking IS leads to a trivial insulator – Breaking TRS leads to a topological Chern insulator

– Intrinsic property of band structure without external magnetic field, Hall conductance, QAHE

Introduction

• Experimental realization (Jotzu etal. Nature 2014)– Ultracold fermionic atoms in a periodically

modulated optical honeycomb lattice

Introduction

• Thirring model (Fradkin & Schaposnik, PLB 94’)– Bosonization in 3 dimensions– Massive Thirring model

– Maxwell-Chern-Simons theory

– Hubbard-Stratonovich transformation

Outline

• Introduction• Emergence of the Proca theory• Emergent QED3

• Conclusions and discussions

Emergence of the Proca theory

• Kane-Mele model without Rashba interaction (Kane and Mele, PRL, 2005)

• Dirac fermion dispersion– Around Dirac points – Mass gap with– Chern number

• TI (SPT) in 2D

Emergence of the Proca theory

• With a current-current interaction

– where is the physical current:

– NN Hubbard interaction for j0 - terms

Emergence of the Proca theory

• Effective field theory (doubled Thrring model)

• After a Hubbard-Stratonovich transformation and integrate out the Fermions (mutual CS)

Emergence of the Proca theory

• Effective field theory

– As U becomes stronger, the excitation energy for the gauge field will be lower than the charge gap. Then we can integrate out the charge current.

– Define

Emergence of the Proca theory

• Effective field theory– The spin gap is lower than the charge gap means

that mA is smaller than min{Δ , t}. This requires,

– In this case, a Proca theory emerges. The efforts to determine the limit of the photon mass are going along a long time. Our results give a playground to see what happens if there is a massive photon.

Emergence of the Proca theory

• Collective excitation lies in the charge gap, the GS is still TI.

Emergence of the Proca theory

• Effective field theory

– Finite Proca mass is consistent with no gapless excitation in the bulk of a topological insulator.

– Emergence of gauge field due to topology other than mean field theory.

Emergence of the Proca theory

• Correlation function and Bragg spectroscopy (Stamper-Kurn etal. PRL 99’)

– M≠0

Emergence of the Proca theory

• In the zero mass limit (c->0, or large λ), a compact U(1) Maxwell theory emerges and the monopole condensation will induce charge confinement.

• Correlation function and Bragg spectroscopy– M=0, monopole condensation

Outline

• Introduction• Emergence of the Proca theory• Emergent QED3

• Conclusions and discussions

Emergent QED3

• Model – Emergence of a Chern-Simons term in the gauge

theory requires to break the TRS. Distinguishing the hoppings and the couplings by spins will do so.

– an extra fermion χ is put in, which may or may not have a non-trivial Chern number and serves as the matter field in the emergent QED3.

Emergent QED3

• Interaction

• Effective theory

• “Charge” carried by χ fermion

Emergent QED3

• Experimental phenomena (PHE and QAHE)– Integrate out the χ fermion

– The current response

Emergent QED3

• When there is a static spatial distribution of the densities of the spinful fermions in the bulk, the Proca equations are simplified

– The PHE and QAHE responding to the "electric" field, i.e., the fluctuation of the gradient of the spin density, can be observed either individually or combinatorially.

Conclusions and Discussions

• 1) Emergent Proca and QED3 from a weak interacting Kane-Mele model

• 2) Emergence of gauge field due to topology• 3) Confinement • 4) PHE and QAHE• 5) Rashba, non abelian • 6) Experiments for Proca theory• 7) (4+1)d generalization, second Chern number?

Thank you!