what we talk about when we talk about fermion...
TRANSCRIPT
Raymond CarverAnton Chekhov of American literature
➢ Explore the struggles of real life that individuals face in society
➢ There isn't enough of anything as long as we live. But at intervals a sweetness appears and, given a chance prevails
➢ If we’re lucky, writer and reader alike, we’ll finish the last line or two of a short story and then sit for a minute, quietly. … , collect ourselves, writers and readers alike, get up, “created of warm blood and nerves” as a Chekhov character puts it, and go on to the next thing: Life. Always life.
➢ Rich analytic literature, sum particular series of diagrams
➢ The ultimate desire is to obtain the exact non-FL forms of fermionic and bosonic propagators in D>1
➢ Alternative numerical approaches QMC
➢ Lattice models, large sizes and low T
➢ Numerics and Analytics would converge
Monte Carlo
Casino de Monte-Carlo, Since 1858
清 · 同治二年 (Late Qing dynasty
2nd year of emperor Tongzhi)
Monte Carlo Method
● Widely used in statistical and quantum many-body physics
● Unbiased: statistical error
●Central limit theorem
Optimization
Numerical integration
Generating probability distributions
➢ Wiki/Monte_Carlo_method
Monte Carlo Method
● Markov chain Monte Carlo is a way to do important sampling
● Distribution of converges to the Boltzmann distribution
● Observable can be measured from a Markov chain
Heisenberg model:● Quantum magnetism / Optical lattice● Phase transition and critical phenomena● Spectral properties● Quantum spin liquids…...
Quantum Monte Carlo
Hubbard model:● Metal-Insulator transition● Magnetic order● Spectral properties● Unconventional superconductivity…...
● Determinantal QMC for fermions
● World-line/SSE QMC for bosons/spins
System sizes:
Time discretization:
Computation effort scales linearly with
Parallelization: ~ 103 CPUs, ~ 106 CPU hours
Tianhe-1 Tianhe-2
Determinantal quantum Monte Carlo
Computing Resources
Tianhe-1: 5PetaFLOPS
K-computer: 10PetaFLOPS
Tianhe-2: 100PetaFLOPS
TaihuLight: 100PetaFLOPs
Tianhe-3: 1000PetaFLOPS
GigaFLOPS: 10^9
TeraFLOPS: 10^12
PetaFLOPS: 10^15
ExaFLOPS: 10^18
亿: 10^8
兆: 10^12
京: 10^16
垓: 10^20
…
恒河沙数: infty
Determinantal quantum Monte Carlo
Path-integral & Trotter-Suzuki decomposition
Free fermion (Slater) determinant
此页留白,大家看黑板。
➢ Blankenbecler et. al., Phys. Rev. D 24, 2278 (1981)➢ Hirsch, Phys. Rev. B 28, 4059(R) (1983)➢ Hirsch, Phys. Rev. B 31, 4403 (1985)
Determinantal quantum Monte Carlo
Path-integral & Trotter-Suzuki decomposition
Discrete Hubbard-Stratonovich transformation
➢ Blankenbecler et. al., Phys. Rev. D 24, 2278 (1981)➢ Hirsch, Phys. Rev. B 28, 4059(R) (1983)➢ Hirsch, Phys. Rev. B 31, 4403 (1985)
Determinantal quantum Monte Carlo
Path-integral & Trotter-Suzuki decomposition
Discrete Hubbard-Stratonovich transformation
➢ Blankenbecler et. al., Phys. Rev. D 24, 2278 (1981)➢ Hirsch, Phys. Rev. B 28, 4059(R) (1983)➢ Hirsch, Phys. Rev. B 31, 4403 (1985)➢ Assaad, Phys. Rev. B 71, 075103 (2005)➢ Assaad and Evertz, Lec. Notes. In Phys. 739 (2008)
Determinantal quantum Monte Carlo
Write Path-integral into determinant
Monte Carlo sampling in configuration space
此页留白,大家看黑板。
➢ Blankenbecler et. al., Phys. Rev. D 24, 2278 (1981)➢ Hirsch, Phys. Rev. B 28, 4059(R) (1983)➢ Hirsch, Phys. Rev. B 31, 4403 (1985)➢ Assaad, Phys. Rev. B 71, 075103 (2005)➢ Assaad and Evertz, Lec. Notes. In Phys. 739 (2008)
➢ Revealing Fermionic Quantum Criticality from New Monte Carlo TechniquesarXiv:1904.07355
Quantum Monte Carlo
● Hubbard-Stratonovich Transformation
● QMC measurements
➢ Y. D. Liao, arXiv:1901.11424
➢ PRX 7, 031052 (2017)
SAC
SACH. Shao A. Sandvik
Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., PRB 99, 245150 (2019)
Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., PRB 99, 245150 (2019)
Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., PRB 99, 245150 (2019)
Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., PRB 99, 245150 (2019)
http://ziyangmeng.iphy.ac.cn/teaching.html
Teaching materials
● World-line/SSE QMC for bosons
● Determinantal QMC for fermions
Fermions couple to critical bosonic modes● Itinerant quantum critical point● Non-Fermi-liquid● Self-Learning Monte Carlo methods● Matter fields couple to gauge fields● Alegbraic spin liquid, orthogonal metal
…...
Duality, DQCP, SPT transitions● DQCP& Gauge and matter fields● Emergent continuous symmetry● Symmetric mass generation
…...
Quantum Monte Carlo
Dynamics bridging experiment and theory● QMC+SAC● Dynamical Signatures of fractionalizations
topological order and spin liquids…...
(Symmetry protected) Topo.
matter
Quantum spin liquids
Deconfined quantum criticality
symmetric mass generation, gauge
couples matter
Deep connections between many apparently different problems
Xiao Yan XuKai Sun
Erez Berg
Zi Hong Liu
Yang QiGao Pei PanChuang Chen
Cenke Xu Andrey Chubukov
Our stories
Subir Sachdev
➢ Revealing Fermionic Quantum Criticality from New Monte Carlo TechniquesarXiv:1904.07355
Fakher Assaad
Fermionic Quantum Criticality
● FM / AFM / Nematic fluctuations of itinerant electron systems● Non-Fermi liquid, fluctuation induced superconductivity ● Fermionic QCP
Nature 424, 524-527 (2003)
Phys. Rev. Lett. 117, 157002 (2016)
E. Berg, S. Trebst, F. Assaad,S. Gazit,Z. Y. M.…...
Hertz-Millis-Moriya, Abanov, Maslov, Chubukov,Metzner, Vojta, Schmalian, Qimiao Si,Metlitski, Sachdev,Sung-Sik Lee, Senthil,Patrick Lee, Xiao-Gang Wen…...
Nature 518, 179 (2015)
Nature 556, 43 (2018) Nature 556, 80 (2018)
Twisted double bilayer Graphene IOP, Harvard groups
Ferromagnetic fluctuations arXiv:1903.06952 arXiv:1903.08130
FM QCP
Ce-based heavy fermion metalunpublished data from Huiqiu Yuan’s group at Zhejiang University
Strange Metal
Model
➢ Abanov, Chubukov, Schmalian, Adv. in Phys. 52, 119 (2003)
➢ Metlitski, Sachdev, PRB 82, 075127 (2010)
➢ Metlitski, Sachdev, PRB 82, 075128 (2010)
➢ Sung-Sik Lee, Annu. Rev. Condens. Matter Phys 9, 227 (2018)
"Know thyself"
(Greek: γνῶθι σεαυτόν).
Thales of Miletus (c. 624 – c. 546 BC)
Self-learning Monte Carlo
●Step too small: small difference, high acceptance
●Step too large: big difference, low acceptance
●Global update: explore the low-energy configurations
Machine-learning
Self-learning Monte Carlo for fermions
➢ Self-Learning DQMC PRB 96, 041119(R) (2017)
Complexity for getting an independent configuration: Complexity for getting an independent configuration:
Fermions coupled to bosonic mode● Itinerant quantum critical point● Non-Fermi-liquid● Electron-phonon coupling
SLMC for DQMC
SLMC for DQMC
➢ Self-Learning on electron-phonon model Phys. Rev. B 98, 041102(R) (2018)
Chuang Chen Richard ScalettarGeorge Batrouni
➢ Dirac Fermions Coupled to Phonons Phys. Rev. Lett. 122, 077601 (2019)
Elective Momentum Ultra-Size QMC (EMUS)
➢ PRB 99, 085114 (2019)
r-space
k-space
SLAC fermions, Lang & Laeuchli➢ arXiv:1808.01230
EMUS-QMC
● Computational complexity
● Speedup when
● Naturally integrated in SLMC
● Generic in models
➢ A. Abanov, A. Chubukov, J. Schmalian, Adv. in Phys., 52, 119 (2003)
Square lattice: AFM-QCP
RG calculations seem to predict ?
➢ arXiv:1808.08878
➢ M. Metlitski, S. Sachdev, PRB 82, 075128 (2010)
Square lattice: AFM-QCP
rotation of fermi velocity ?
➢ arXiv:1808.08878
Fermionic QCPs with QMCs● Ferromagnetic/nematic QCP
● Antiferromagnetic QCP
● Triangle lattice
● Square lattice
➢ Revealing Fermionic Quantum Criticality from New Monte Carlo TechniquesarXiv:1904.07355
● World-line/SSE QMC for bosons
● Determinantal QMC for fermions
Fermions couple to critical bosonic modes● Itinerant quantum critical point● Non-Fermi-liquid● Self-Learning Monte Carlo methods● Matter fields couple to gauge fields● Alegbraic spin liquid, orthogonal metal
…...
Duality, DQCP, SPT transitions● DQCP& Gauge and matter fields● Emergent continuous symmetry● Symmetric mass generation
…...
Quantum Monte Carlo
Dynamics bridging experiment and theory● QMC+SAC● Dynamical Signatures of fractionalizations
topological order and spin liquids…...
(Symmetry protected) Topo.
matter
Quantum spin liquids
Deconfined quantum criticality
symmetric mass generation, gauge
couples matter
Deep connections between many apparently different problems
U1 gauge fields coupled to Dirac fermions
➢ PRX 9, 021022 (2019)
See the Chap. 6 in Xiao-Gang Wen's Book
U1 gauge fields coupled to Dirac fermions
Monopole proliferation leads to confinement of gauge field
➢ Wei Wang et al., arXiv: 1906.06929
U1 gauge fields coupled to Dirac fermions
Monopoles in QED3-Gross-Neveu theory
➢ J. A. Gracey, Phys. Rev. D 98, 085012 (2018)
➢ B. Ihrig, L. Janssen, L. N. Mihaila, and M. M. Scherer, Phys. Rev. B 98, 115163 (2018)
➢ N. Zerf, P. Marquard, R. Boyack, and J. Maciejko, Phys. Rev. B 98, 165125 (2018)
➢ R. Boyack, A. Rayyan, and J. Maciejko, Phys. Rev. B 99, 195135 (2019)
1/N Aslamazov-Larkin digrams
➢ N. Zerf, R. Boyack, P. Marquard, J. A. Gracey, and J. Maciejko, arXiv:1905.03719
➢ X.-Y. Song, Y.-C. He, A. Vishwanath, and C. Wang, arXiv:1811.11182 (2018)
➢ X.-Y. Song, C. Wang, A. Vishwanath, and Y.-C. He, arXiv:1811.11186 (2018)
➢ E. Dupuis, M. Paranjape, and W. Witczak-Krempa, arXiv:1905.02750
QED3-Gross-Neveu at O(1/Nf) and O(1/Nf^2), three loops, four loops, epsilon-expansion
Z2 gauge fields coupled to Fermi surface
Normal metal Orthogonal metal
➢Chuang Chen et al., arXiv:1904.12872
Z2 gauge fields coupled to Fermi surface
Normal metal Orthogonal metal
➢Chuang Chen et al., arXiv:1904.12872
Z2 gauge fields coupled to Fermi surface
Normal metal Orthogonal metal
Continuous phase (Higgs) transition between NM and OM without symmetry breaking
Gauge-invariant string operator
NM Z2 gauge field confinedOM Z2 gauge field deconfined
➢Chuang Chen et al., arXiv:1904.12872
➢ PRX 7, 031101 (2017)
➢ PRB 98, 045116 (2018)
➢ arXiv: 1808.08878 (PNAS)
➢ PRB 95, 041101(R) (2017)
➢ PRB 96, 041119(R) (2017)
➢ PRB 98, 041102(R) (2018)
➢ PRL 122, 077601 (2019)
➢ PRB 99, 085114 (2019)
➢ PRX 9, 021022 (2019)
➢ arXiv: 1904.12872
➢ arXiv: 1906.06929
What we talk about when we talk about fermion QMC
Difficulty questions
Methodologies
New paradigms in quantum Matter