Topological origin of phase transitions in the absence of ...
Interaction-driven topological phase...
Transcript of Interaction-driven topological phase...
![Page 1: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/1.jpg)
Zi Yang Meng
( 孟子杨 )
Institute of Physics, Chinese Academy of Sciences
Interaction-driven topological phase transitions
![Page 2: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/2.jpg)
IOP, CAS
K
![Page 3: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/3.jpg)
Resonance across the Pacific
Zhong-Yi LuYuan-Yao He Han-Qing Wu
Cenke XuYi-Zhuang You
![Page 4: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/4.jpg)
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
![Page 5: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/5.jpg)
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
![Page 6: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/6.jpg)
Topological phase transitions
free fermions Single-particle gap closes and reopens
![Page 7: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/7.jpg)
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)➢ …...
![Page 8: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/8.jpg)
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)➢ …...
![Page 9: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/9.jpg)
Topological phase transitions
Interacting system
➢ Hohenadler, Meng et al., PRB 85, 115132 (2012)
QSHI AFM
Single-particle gap DOES NOT close
![Page 10: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/10.jpg)
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
![Page 11: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/11.jpg)
Model and Method
![Page 12: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/12.jpg)
Determinantal quantum Monte Carlo
Path-integral & Trotter-Suzuki decomposition
Free fermion (Slater) determinant
![Page 13: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/13.jpg)
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)
![Page 14: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/14.jpg)
Determinantal quantum Monte Carlo
Write Path-integral into determinant
Monte Carlo sampling in configuration space
![Page 15: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/15.jpg)
Quantum Monte Carlo
Hubbard-Stratonovich Transformation
QMC measurements
![Page 16: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/16.jpg)
Quantum Monte Carlo
System sizes:
Time discretization:
Computation effort scales linearly with
Parallelization: ~ 103 CPUs, ~ 106 CPU hours
Tianhe-1 Tianhe-2
![Page 17: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/17.jpg)
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
![Page 18: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/18.jpg)
Phase diagram
0.2t
![Page 19: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/19.jpg)
Phase boundary
0.2t
![Page 20: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/20.jpg)
Phase boundary
![Page 21: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/21.jpg)
Phase boundary
![Page 22: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/22.jpg)
Topological phase transition
Single-particle gap DOES NOT close.
![Page 23: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/23.jpg)
Topological phase transition
Spin gap DOES close
![Page 24: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/24.jpg)
Topological phase transition
Spin Chern number change?
![Page 25: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/25.jpg)
Topological phase transition
Spin Chern number DOES NOT change
![Page 26: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/26.jpg)
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.
![Page 27: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/27.jpg)
Bosonic edge states
Open boundary measurementsBilayer Ribbon
Zig-Zag edge
PBC
➢ H.-Q. Wu et al., arXiv:1606.05822
![Page 28: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/28.jpg)
Bosonic edge states
Open boundary measurements
➢ H.-Q. Wu et al., arXiv:1606.05822
![Page 29: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/29.jpg)
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
![Page 30: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/30.jpg)
Organizers:
Zi Yang Meng ( 孟子杨 ), Institute of Physics, CAS, [email protected]
Lei Wang ( 王磊 ), Insitute of Physics, CAS, [email protected]
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
![Page 31: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/31.jpg)
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
![Page 32: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/32.jpg)
Model and Method
![Page 33: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/33.jpg)
Fermion bilinear condensate
U term AFM state
J term Inter-layer s-wave spin-singlet pairing
Order parameter:
J term Exciton condensation
Order parameter:
![Page 34: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/34.jpg)
Phase diagram
![Page 35: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/35.jpg)
Phase boundary Exciton structure factor
![Page 36: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/36.jpg)
Phase diagram
![Page 37: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/37.jpg)
Phase diagram
![Page 38: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/38.jpg)
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
![Page 39: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/39.jpg)
Basic problem
Partition function:
Observables :
Fock space :
![Page 40: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/40.jpg)
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
![Page 41: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/41.jpg)
Topological phase transition
“single-particle edge states”
single-particle edge statesgapped out
![Page 42: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/42.jpg)
Bosonic edge mode
: powerlaw correlation
: exponential correlation
➢ Y.-Z. You et al., arXiv:1510.04278
![Page 43: Interaction-driven topological phase transitionsziyangmeng.iphy.ac.cn/files/teaching/SUSTC201607.pdf · Bilayer Kane-Mele-Hubbard model and QMC method Conventional topological phase](https://reader033.fdocuments.net/reader033/viewer/2022060610/60614498f6b7b6288e4eac20/html5/thumbnails/43.jpg)
Last lecture? Hope not.