Solving metallic quantum criticality in a casino

S T O R I E S

Zi Yang Meng　 　孟 子 杨

➢ 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

Partition function:

Observables:

Fock space:

Monte Carlo Method

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

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

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., arXiv:1808.09994

Square lattice Hubbard model

➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., arXiv:1808.09994

Square lattice Hubbard model

➢ C. Chen, Bachelor Thesis (2016)➢ X.-J. Han et al., arXiv:1808.09994

Honeycomb lattice Hubbard model

➢ ZYM, Phd Thesis (2011)

Honeycomb lattice Hubbard model

➢ ZYM, Phd Thesis (2011)

Honeycomb lattice Hubbard model

➢ ZYM, Phd Thesis (2011)

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)

Model➢ PRX 7, 031101 (2017)

Quantum Monte Carlo

➢ Self-learning Monte Carlo Method

O(N) speedup, large lattice is possible

"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)

Non-Fermi liquid

➢ PRX 7, 031101 (2017)

FM-QCP

Our model

(2+1)d Ising model

Hertz-Millis-Moriya

➢ PRX 7, 031101 (2017)

FM-QCP

Our model

(2+1)d Ising model

Hertz-Millis-Moriya

➢ PRX 7, 031101 (2017)

FM-QCP

Our model

(2+1)d Ising model

Hertz-Millis-Moriya

➢ PRX 7, 031101 (2017)

L=30, beta=30(30x30x600)

Triangle lattice: AFM-QCP➢ PRB 98, 045116 (2018)

L=30, beta=30(30x30x600)

➢ PRB 98, 045116 (2018)

Triangle lattice: AFM-QCP

L=30, beta=30(30x30x600)

➢ PRB 98, 045116 (2018)

Triangle lattice: AFM-QCP

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

Square lattice: AFM-QCP

➢ arXiv:1808.08878

Square lattice: AFM-QCP

Bare boson (2+1)D Ising

➢ arXiv:1808.08878

➢ 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

Directly simulate U(1) gauge field couples to fermionic matter

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

➢PRX 9, 021022 (2019)

U1 gauge fields coupled to Dirac fermions

Monopole proliferation leads to confinement of gauge field

➢ Wei Wang et al., in preparation

Z2 gauge fields coupled to Fermi surface

➢arXiv:1904.12872

Z2 gauge fields coupled to Fermi surface

➢arXiv:1904.12872

Normal metal Orthogonal metal

Z2 gauge fields coupled to Fermi surface

➢arXiv:1904.12872

Normal metal Orthogonal metal

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

➢ 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

What we talk aboutWhen we talk about fermion criticality

S T O R I E S

Zi Yang Meng　 　孟 子 杨

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.