solving metallic quantum criticality in a...
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Solving metallic quantum criticality in a casino
S T O R I E S
Zi Yang Meng 孟 子 杨
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➢ 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
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Monte Carlo
Casino de Monte-Carlo, Since 1858
清 · 同治二年 (Late Qing dynasty
2nd year of emperor Tongzhi)
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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
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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
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Partition function:
Observables:
Fock space:
Monte Carlo Method
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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
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System sizes:
Time discretization:
Computation effort scales linearly with
Parallelization: ~ 103 CPUs, ~ 106 CPU hours
Tianhe-1 Tianhe-2
Determinantal quantum Monte Carlo
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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
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Determinantal quantum Monte Carlo
Path-integral & Trotter-Suzuki decomposition
Free fermion (Slater) determinant
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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)
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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)
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Determinantal quantum Monte Carlo
Write Path-integral into determinant
Monte Carlo sampling in configuration space
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Quantum Monte Carlo
● Hubbard-Stratonovich Transformation
● QMC measurements
➢ Y. D. Liao, arXiv:1901.11424
➢ PRX 7, 031052 (2017)
SAC
SACH. Shao A. Sandvik
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Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., arXiv:1808.09994
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Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., arXiv:1808.09994
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Square lattice Hubbard model
➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., arXiv:1808.09994
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Honeycomb lattice Hubbard model
➢ ZYM, Phd Thesis (2011)
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Honeycomb lattice Hubbard model
➢ ZYM, Phd Thesis (2011)
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Honeycomb lattice Hubbard model
➢ ZYM, Phd Thesis (2011)
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http://ziyangmeng.iphy.ac.cn/teaching.html
Teaching materials
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● 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
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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
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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
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FM QCP
Ce-based heavy fermion metalunpublished data from Huiqiu Yuan’s group at Zhejiang University
Strange Metal
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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)
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Model➢ PRX 7, 031101 (2017)
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Quantum Monte Carlo
➢ Self-learning Monte Carlo Method
O(N) speedup, large lattice is possible
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"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
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Machine-learning
Self-learning Monte Carlo for fermions
➢ Self-Learning DQMC PRB 96, 041119(R) (2017)
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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
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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)
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Non-Fermi liquid
➢ PRX 7, 031101 (2017)
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FM-QCP
Our model
(2+1)d Ising model
Hertz-Millis-Moriya
➢ PRX 7, 031101 (2017)
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FM-QCP
Our model
(2+1)d Ising model
Hertz-Millis-Moriya
➢ PRX 7, 031101 (2017)
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FM-QCP
Our model
(2+1)d Ising model
Hertz-Millis-Moriya
➢ PRX 7, 031101 (2017)
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L=30, beta=30(30x30x600)
Triangle lattice: AFM-QCP➢ PRB 98, 045116 (2018)
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L=30, beta=30(30x30x600)
➢ PRB 98, 045116 (2018)
Triangle lattice: AFM-QCP
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L=30, beta=30(30x30x600)
➢ PRB 98, 045116 (2018)
Triangle lattice: AFM-QCP
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Elective Momentum Ultra-Size QMC (EMUS)
➢ PRB 99, 085114 (2019)
r-space
k-space
SLAC fermions, Lang & Laeuchli➢ arXiv:1808.01230
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EMUS-QMC
● Computational complexity
● Speedup when
● Naturally integrated in SLMC
● Generic in models
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Square lattice: AFM-QCP
➢ arXiv:1808.08878
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Square lattice: AFM-QCP
Bare boson (2+1)D Ising
➢ arXiv:1808.08878
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➢ A. Abanov, A. Chubukov, J. Schmalian, Adv. in Phys., 52, 119 (2003)
Square lattice: AFM-QCP
RG calculations seem to predict ?
➢ arXiv:1808.08878
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➢ M. Metlitski, S. Sachdev, PRB 82, 075128 (2010)
Square lattice: AFM-QCP
rotation of fermi velocity ?
➢ arXiv:1808.08878
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Fermionic QCPs with QMCs● Ferromagnetic/nematic QCP
● Antiferromagnetic QCP
● Triangle lattice
● Square lattice
➢ Revealing Fermionic Quantum Criticality from New Monte Carlo TechniquesarXiv:1904.07355
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● 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
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Directly simulate U(1) gauge field couples to fermionic matter
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U1 gauge fields coupled to Dirac fermions
➢ PRX 9, 021022 (2019)
See the Chap. 6 in Xiao-Gang Wen's Book
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U1 gauge fields coupled to Dirac fermions
➢PRX 9, 021022 (2019)
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U1 gauge fields coupled to Dirac fermions
Monopole proliferation leads to confinement of gauge field
➢ Wei Wang et al., in preparation
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Z2 gauge fields coupled to Fermi surface
➢arXiv:1904.12872
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Z2 gauge fields coupled to Fermi surface
➢arXiv:1904.12872
Normal metal Orthogonal metal
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Z2 gauge fields coupled to Fermi surface
➢arXiv:1904.12872
Normal metal Orthogonal metal
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Z2 gauge fields coupled to Fermi surface
➢arXiv:1904.12872
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
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➢ PRX 7, 031101 (2017)
➢ PRB 98, 045116 (2018)
➢ arXiv: 1808.08878
➢ 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
Fermionic Quantum Criticality
Difficulty questions
Methodologies
New paradigms in quantum Matter
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What we talk aboutWhen we talk about fermion criticality
S T O R I E S
Zi Yang Meng 孟 子 杨
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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.