Zi Yang Meng

( 孟子杨 )

Institute of Physics, Chinese Academy of Sciences

Interaction-driven topological phase transitions

IOP, CAS

K

Resonance across the Pacific

Zhong-Yi LuYuan-Yao He Han-Qing Wu

Cenke XuYi-Zhuang You

Content

Results

➢ phase diagram

➢ exotic interaction-driven topological phase transition

➢ emergent bosonic SPT without sign problem

Bilayer Kane-Mele-Hubbard model and QMC method

Conventional topological phase transitions

Summary ➢ H.-Q. Wu et al., Phys. Rev. B 92, 165123 (2015) ➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195163 (2016) ➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016) ➢ H.-Q. Wu et al., arXiv:1606.05822

Content

Results

➢ phase diagram

➢ exotic interaction-driven topological phase transition

➢ emergent bosonic SPT without sign problem

Bilayer Kane-Mele-Hubbard model and QMC method

Conventional topological phase transitions

Summary ➢ H.-Q. Wu et al., Phys. Rev. B 92, 165123 (2015) ➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195163 (2016) ➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016) ➢ H.-Q. Wu et al., arXiv:1606.05822

Topological phase transitions

free fermions Single-particle gap closes and reopens

Topological phase transitions

free fermions Chern number from Berry curvature

➢ Thouless, Kohmoto, Nightingale, den Nijs, PRL 49, 405 (1982)➢ Avron, Seiler, Simon, PRL 51, 51 (1983)➢ …...

Topological phase transitions

free fermions Spin Chern number from ( ) single-particle Green's function

➢ Z. Wang, X.-L. Qi, S.-C. Zhang, PRL (2010) ➢ V. Gurarie, PRB (2011) ➢ Z. Wang, S.-C. Zhang, PRX (2012)➢ T. C. Lang et al., PRB (2013)➢ H.-H Hung et al., PRB (2013)➢ H.-H Hung et al., PRB (2014)➢ Z. .Wang, S.-C. Zhang, PRX (2014)➢ H.-H. Hung, T. C. Lang, Z. Y. Meng, MPLB (2014)➢ …...

Topological phase transitions

Interacting system

➢ Hohenadler, Meng et al., PRB 85, 115132 (2012)

QSHI AFM

Single-particle gap DOES NOT close

Content

Results

➢ phase diagram

➢ exotic interaction-driven topological phase transition

➢ emergent bosonic SPT without sign problem

Bilayer Kane-Mele-Hubbard model and QMC method

Conventional topological phase transitions

Summary ➢ H.-Q. Wu et al., Phys. Rev. B 92, 165123 (2015) ➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195163 (2016) ➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016) ➢ H.-Q. Wu et al., arXiv:1606.05822

Model and Method

Determinantal quantum Monte Carlo

Path-integral & Trotter-Suzuki decomposition

Free fermion (Slater) determinant

Determinantal quantum Monte Carlo

Path-integral & Trotter-Suzuki decomposition

Discrete Hubbard-Stratonovich transformation

➢ Blankenbecler et. al., Phys. Rev. D (1981)➢ Hirsch, Phys. Rev. B (1985)➢ Assaad, Phys. Rev. B (2005)

Determinantal quantum Monte Carlo

Write Path-integral into determinant

Monte Carlo sampling in configuration space

Quantum Monte Carlo

Hubbard-Stratonovich Transformation

QMC measurements

Quantum Monte Carlo

System sizes:

Time discretization:

Computation effort scales linearly with

Parallelization: ~ 103 CPUs, ~ 106 CPU hours

Tianhe-1 Tianhe-2

Content

Results

➢ phase diagram

➢ exotic interaction-driven topological phase transition

➢ emergent bosonic SPT without sign problem

Bilayer Kane-Mele-Hubbard model and QMC method

Conventional topological phase transitions

Summary ➢ H.-Q. Wu et al., Phys. Rev. B 92, 165123 (2015) ➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195163 (2016) ➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016) ➢ H.-Q. Wu et al., arXiv:1606.05822

Phase diagram

0.2t

Phase boundary

0.2t

Phase boundary

Phase boundary

Topological phase transition

Single-particle gap DOES NOT close.

Topological phase transition

Spin gap DOES close

Topological phase transition

Spin Chern number change?

Topological phase transition

Spin Chern number DOES NOT change

Topological field theory description

Non-linear model with SO(4) symmetry and topological - term

Bosonic SPT with O(4) vector fluctuations

Emergent bosonic SPT

Dimer-singlet-insulator

interaction-driven topological phase transition

➢ C. Xu and A. W. W. Ludwig, PRL (2013)➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Z. You et al., Phys. Rev. B 93, 125101 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016)

Simulate emergent bosonic SPT with minus-sign-free QMC, bosonic edge mode emerging from interacting TI.

Bosonic edge states

Open boundary measurementsBilayer Ribbon

Zig-Zag edge

PBC

➢ H.-Q. Wu et al., arXiv:1606.05822

Bosonic edge states

Open boundary measurements

➢ H.-Q. Wu et al., arXiv:1606.05822

Summary

➢ Breakdown of Green's function formalism

➢ exotic interaction-driven topological phase transition

➢ emergent bosonic SPT without sign problem

Topology + interaction: fertile soil for exciting & novel physics

dissect a sparrow ( 解剖麻雀 )

➢ H.-Q. Wu et al., Phys. Rev. B 92, 165123 (2015) ➢ Y.-Y. He et al., Phys. Rev. B 93, 115150 (2016)➢ Y.-Y. He et al., Phys. Rev. B 93, 195163 (2016) ➢ Y.-Y. He et al., Phys. Rev. B 93, 195164 (2016) ➢ H.-Q. Wu et al., arXiv:1606.05822

Organizers:

Zi Yang Meng ( 孟子杨 ), Institute of Physics, CAS, ziyangmeng@iphy.ac.cn

Lei Wang ( 王磊 ), Insitute of Physics, CAS, wanglei@iphy.ac.cn

Xi Dai ( 戴希 ), Institute of Physics, CAS

Tao Xiang ( 向涛 ), Institute of Physics, CAS

International Summer School on Computational Approaches for Quantum-Many-Body Systems

2016.08.01-2016.08.21

University of Chinese Academy of Sciences, Beijing, China

Content: Quantum Monte Carlo, DMRG, TRG, (cluster) DMFT, LDA+DMFT, ...

Webpage: http://compqmb2016.csp.escience.cn

Content

Results

➢ phase diagram with spin-orbital coupling

➢ phase diagram without spin-orbital coupling

Bilayer Kane-Mele-Hubbard model and QMC method

Topological phase transitions

Outlook

Model and Method

Fermion bilinear condensate

U term AFM state

J term Inter-layer s-wave spin-singlet pairing

Order parameter:

J term Exciton condensation

Order parameter:

Phase diagram

Phase boundary Exciton structure factor

Phase diagram

Phase diagram

World-line QMC for bosons

Determinantal QMC for fermionsHubbard model: Unconventional superconductivity Metal-Insulator transition Non-Fermi-liquid Quantum spin liquids Interaction effects on topological insulators…...

Heisenberg model: Quantum magnetism (disorder) Phase transition and critical phenomena Quantum spin liquids Quantum spin ice…...

Quantum Monte Carlo

Basic problem

Partition function:

Observables :

Fock space :

Topological phase transition

How about edge states?

Strange correlation in various interacting channels

➢ Y.-Z. You et al., PRL (2014)➢ H.-Q. Wu et al., PRB 92, 165123 (2015)

: a trivial band insulator

: many-body ground state

Topological phase transition

“single-particle edge states”

single-particle edge statesgapped out

Bosonic edge mode

: powerlaw correlation

: exponential correlation

➢ Y.-Z. You et al., arXiv:1510.04278

Last lecture? Hope not.