第５回HPC-Phys勉強会

ボルツマン機械を用いて解き明かす
高温超伝導の発現機構

Mechanism of Formation of High-Temperature Superconductivity
Revealed by Boltzmann Machine Learning
Youhei Yamaji
Department of Applied Physics, The University of Tokyoand PRESTO, JST

Collaborators:
Acknowledgement:
Prof. Teppei Yoshida (Kyoto U.)
Prof. Takeshi Kondo (ISSP)
Prof. Atsushi Fujimori (WasedaU)
Prof. Masatoshi Imada (WasedaU, Toyota PCRI)
Y. Yamaji, T. Yoshida, A. Fujimori, and M. Imada, arXiv:1903.08060.
重点課題7

An enigmatic inverse problem:
Origin of high-temperature superconductivity
1986 High-temperature superconductivity
in copper oxides
J. G. Bednorz

K. A. Müller

Life of Electrons in Crystalline Solids

E. Fermi

Coulomb©FeynmanL. D. Landau 1956

W. Pauli

Landauʼs Fermi liquid theory
strongly interacting electrons
and free electrons are
adiabatically connectedLife of Electrons in Superconductivity
1911 Zero electric resistance of Hg

H. Kamerlingh Onnes

1957 BCS theory
J. Bardeen, L. N. Cooper, J. R. Schrieﬀer

Propagation of Free Electron

For a single electron in vacuum

After Fourier transformation
Spectral weightDescription of Many-Body Electrons
Many-body Schrödinger eq.

EOM of single-particle Greenʼs function

Damascelli, Hussain, & Shen,
Rev. Mod. Phys. 75, 473 (2003)

Rigorous Relation between
Self-Energy and Spectral Weight
Normal self-energy〜Scattering rate
Anomalous self-energy〜Rate of anomalous scattering〜ω-distribution of attractive force

Spectral weightSelf-Energy in BCS Superconductors: Scale of phonon

Eliashberg eqs. for BCS SC
SC gap function: Coupling2 x DOS of phonon

A. B. Migdal, Sov. Phys. JETP 7, 996 (1958).
G. Eliashberg, Sov. Phys. JETP 11, 696 (1960).

cf.) P. Morel, P. W. Anderson,Phys. Rev. 125, 1263 (1962).

Ratio of DOS
J. R. Schrieﬀer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10, 336 (1963).
W. L. McMillan and J. M. Rowell, Phys. Rev. Lett. 14, 108 (1965).

From STS, Σnor and Σano separately obtainedSelf-Energy in Cuprate Superconductors
B. Keimer, S. A. Kivelson, M. R. Norman, & S. Uchida, Nature 518, 179 (2015).

Competing energy scales,
EF and Epair, prevent us fromseparating Σnor and Σano

Modeling self-energy:

H. Li, et al.,Nat. Commun. 9, 26 (2018).

Extension of Eliashberg theory:
J. M. Bok, et al.,Sci. Adv. 2, e1501329 (2016).

A new approach:✔Fewer assumptions✔︎More flexible representation of Σ
to reveal unexpected physics
beyond biased expectation

Photoemission Electron
Spectroscopy of Crystalline Solids
Intensity of photoelectron

Fermi-Dirac distribution

Spectral weight
Free electrons

Correlated electrons
Damascelli, Hussain, & Shen,
Rev. Mod. Phys. 75, 473 (2003)

Spectrum of Cuprate Superconductors
for T < Tc
Angle-resolved
photoemission

2T=12K

Bi2Sr2CuO6+δ

1

T. Kondo, et al.,Nat. Phys. 7, 21 (2011).
Bi2Sr2CaCu2O8+δ
Optimally doped sample(Tc=90K)

0-0.3

-0.2

-0.1

00.1

Bi2Sr2CuO6+δ

Underdetermined Non-Linear Inverse Problem

Known: Single-component spectral weight
Unknown: Two components of self-energy

For ghow
We need to separately obtain Σnor(ω) and Σano(ω)
from a single-component spectral function A(ω)
	

Prior Knowledge to Solvethe Underdetermined Problem

RigorousPrior knowledge about the self-energy
Causality-Krammers-Krönig relation

Structure of self-energy-ImΣano is an odd function of ω

Physically reasonable Initial guess:Σano is conﬁned in a ﬁnite range of ωFraction of electron observed:
Determined afterwards self-consistently

0initial guess
optimized
-0.5
-0.3

-0.2

-0.1

0

Flexible Representation of Σnor to Solvethe Underdetermined Problem
Rectangular function chosen as basis
0 1 22L-1

Coeﬃcients by restricted Boltzmann machinesusu
btl subtlety btl
P. Smolensky (1986)
etety-# of paramters: (# of hidden units) x (# of visible units) = 18 x 9y
< 2# of visible units = 29 = 512

Flexible Representation of Σano to Solvethe Underdetermined Problem
Mixture distribution of Boltzmann Machine

D. H. Ackley, G. E. Hinton, & T. J. Sejnowski (1985)-# of paramters: (# of visible units)2 + # of visible units = 92 + 9< 2# of visible units = 29 = 512-Optimizing Self-Energies to Solvethe Underdetermined Problem

2

By minimizing the cost function
with prior knowledge,
optimize Σnor and Σano10-0.3

-0.2

-0.1

00.1

Avoiding Overﬁtting: Cross Validation
1. Dividing data
into 2 sets2

2. Test data generated bymaximum likelihood approach2○ training data○ test data

10-0.3□ training dataー test data average

1-0.2

-0.1

00.1

0-0.3

-0.2

-0.1

00.1

Bayesian Optimization Steps to Optimize Σ
1. Optimizing Σ with training data
Initial conditions

optimized

2. Measuring cost function with test data
3. Shifting the centers of mass inΣano

Go back to 1.

Bayesian Optimization Process to Optimize Σ10-5

center of mass of BMs0Cost function
with test data-0.1

χ2 withconstrained Σano
0.6
0.4

Re

0.2

-0.2

0-0.2
-0.3

-0.3

01020304050# of optimization steps

Im-0.2-0.1

0noise inthe experiment10-6010 20 30 40 50=1.38x10-6# of optimization steps-The cost function χ2 becomes 1/3 of χ2with constrained Σano by Li et al. (2018)-Optimization is robust against noiseReproduced Spectrum of SC Cuprates
for T < Tc

T. Kondo, et al.,Nat. Phys. 7, 21 (2011).
Bi2Sr2CaCu2O8+δ
Optimally doped sample(Tc=90K)

spectral weight (eV-1)

spectral weight (eV-1)

Angle-resolved
photoemission

1.5

2

1.5

10.5

aOP90K

0-0.3

-0.2

1Bi2Sr2CuO6+δ
0.5

b-0.1

00.1

0-0.3

-0.2

-0.1

Bi2Sr2CuO6+δ
A(ω) by optimized Σ precisely reproduces Aexp(ω)

0self-energy (eV)

Self-Energy Obtained by
the Bayesian Optimization Process
0.4
0.2
0-0.2
-0.4
-0.6
-0.8
-0.3

-0.2

-0.1

00.1

Promiment peaks found in both Σnor and Σanoat the same ω (〜65 meV)

Hidden Peak Structure in Σ Revealed
norUniversal
ω-linearImΣ
0.4
due to the peak structure:
Planckiandissipation,
0.2
Possible holographic ﬂuid

0.2

self-energy (eV)self-energy (eV)

0.4

0-0.2
-0.4

Total Σ:M. R. Norman, et al.,-0.6 PRB 60, 7585 (1999)

-0.8
-0.2

-0.15

-0.1

0

Marginal FL: C. Varma et al.,PRL 63, 1996 (1990).
-0.2 T. Valla, et al.,Science 285, 2110 (1999).

-0.4

Bi2Sr2CuO6+δ

-0.6 T. Kondo, et al.,Nature 457, 296 (2009).

UD (Tc=23K)

-0.05

0-0.8
-0.2

-0.15

-0.1

-0.05

Peaks exactly canceled and invisible in total Σ:
Reason why it has been overlooked for 30 years

0Peak structures in Σ and SC: T. Maier, D. Poilblanc, D. Scalapino, PRL 100, 237001 (2008).

10spectral weight (eV-1)

self-energy (eV)

0-0.2

-0.4

-0.6

-0.8
-0.3

modifiedoriginal
modifiedoriginal
-0.2

-0.1

0

original spectrum
spectrum withmodified self-energy

1-0.2

-0.1

Hidden peaks in Σnor/ano indeed generate SC gap
and explain large gap with small anomalies in spectra0What Determines Tc ?: AttractiveInteraction Estimated from ImWPEAK
Eﬀective attractive interaction:

Normalization function:

self-energy (eV)

0.4
0.2
0-0.2
-0.4
-0.6
-0.8
-0.2

-0.15

-0.1

-0.05

0

What Determines Tc ?:Planckian Dissipation from ImΣPEAK

0-0.2

-0.4

ARPES of Bi2Sr2CuO6+δ
T. Kondo, et al.,Nature 457, 296 (2009).

1 (AN)
11213142 (N)Inelastic relaxation rate:cf.) Marginal FL: C. Varma et al., PRL 63, 1996 (1990).

2OP35K-0.3 -0.2 -0.1

0Planckian dissipation with universal Γ:J. Zaanen, Nature 430, 512 (2004).

-Even in SC phase, electrons form ﬂuid
-ImΣPEAK generates both high-Tc &Planckian dissipation

1.51UD23K
OP35K
OD29K
OP90K

0.5
00153045

What Determines Tc ?

SC order parameter
102.5

20 40 60 80 100

Bi2212

20.8

1.5
10.5

0.6

0-Uemura plot

Bi22010Tc

0.2 0.4 0.6 0.8 1-Homes plot

0.40σ(Tc)Tc
superﬂuid density

Bi2201 UD23K
Bi2201 OP35K
Bi2201 OD29K0.2Y. J. Uemura, et al.,PRL 62, 2317 (1989).

Bi2212 OP90K
0246Eﬀective interaction

kBT c=g(kAN)superﬂuid density

810C. C. Homes et al.,Nature 430, 539 (2004).

Dissipationx F(kAN) x Γ(kN)

Comparison with Previous Studies
Compared with J. M. Bok, et al., Sci. Adv. 2, e1501329 (2016).

The present method-obtained Σnor and Σano
separately at both Nand AN regions
for UD, OP, and UD
cuprates-revealed cancellation ofpeaks in both Σnor and Σano
Compared with H. Li, et al.,Nat. Commun. 9, 26 (2018).

NormalAnomalous

φ=20

10-5

Cost function
with test data

The present method-showed χ2 is 2/3 smaller-revealed peak structure
responsible for SC-revealed that the peak intensity
is involved in Tc determination

0.6
0.4

Re

0.2
0-0.2
-0.3

10-6
0cooa
Im

Im-0.2-0.10

χ2 withconstrained Σano
by Li et al.
1020304050# of optimization steps

Summary
Y. Yamaji, T. Yoshida, A. Fujimori, and M. Imada, arXiv:1903.08060.

-Origin of both high-Tc SC-Cancelled each other and invisible
in total ΣPossible origin of the peak structure:
Dark fermion scenario

S. Sakai, M. Civelli, M. Imada, PRL 116, 057003 (2016).
cf.) Singularities in Σ in normal state
T. D. Stanescu & G. Kotliar, PRB 74, 125110 (2006).

What determine Tc ?

self-energy (eV)

Hidden peaks in both Σnor and Σano

0.4
0.2
0-0.2
-0.4
-0.6

-Peak intensity (〜g) & dissipation (〜Γ) -0.8-0.2

kBT c= g(kAN) x F(kAN) x Γ(kN)

Spectroscopy of hidden physics-0.15-0.1-0.050