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[Youhei Yamaji](https://orcid.org/0000-0002-4055-8792)

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[Self-Energy Spectroscopy and Artificial Neural Network](https://mdr.nims.go.jp/datasets/3431154c-3506-4179-bfca-e7ebc44aa0bd)

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Self-Energy Spectroscopy and Artificial Neural NetworkSelf-Energy Spectroscopy and Artificial Neural NetworkYouhei Yamaji+Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),Tsukuba, Ibaraki 305-0044, Japan(Received September 19, 2024; revised January 14, 2025; accepted January 14, 2025; published online February 26, 2025)The analysis of spectroscopy data has played an important role in untangling the complex dynamics of many-bodyelectrons in quantum materials and making their emergent properties understandable. Spectroscopy measurementsprovide us with the responses of the many-body electrons in materials when energy and momentum are injected. Theseresponses have been analyzed using a single-particle picture augmented by self-energy, which quantifies the deviationfrom simple free fermion excitation. While the self-energy is not directly observed in spectroscopy, it has been extractedfrom the obtained spectra by solving inverse problems. Especially, for superconductors, the analysis of self-energy is akey to understanding the origin of the superconductivity. The recent rise of machine learning has served to update theself-energy analysis of spectroscopy data and opened a new avenue for understanding the entangled nature of many-body electrons. In this article, self-energy analysis using the flexibility of neural networks is reviewed and positioned inthe research trajectory from Bardeen–Cooper–Schrieffer superconductors to copper-oxide high-temperature super-conductors.1. IntroductionIn condensed matter physics community, the rise ofmaterials informatics1) has sparked renewed interest inBayesian approaches2) and machine learning methods,3)which have emerged alongside the rapid development ofdeep learning, leading to the current rise of generativeartificial intelligence. Such interests have further expandedinto various research fields, such as computer visionincluding image processing, natural language processing,and speech processing, as Bayesian and machine learningapproaches have been developed across multiple disciplines.Even though usage of large database has attractedconsiderable attention, such as virtual screening or recom-mendation of synthesizable materials on databases consistingof large amount of experimental data and synthetic datagenerated by simulation,4) analysis of a small set of data(higher dimensional but from a single or several materials)has been actively studied in the current research trend. Whilethe former data-driven approaches, such as exploration ofstructure-property correlations, have been augmented by thedatabases and automated simulation,5) it is also interesting tountangle information encoded in the experimental observa-tion of a single material by utilizing a broader range oftheoretical or numerical tools, whose range has beenexpanded by the Bayesian approaches and machine learning.An important direction of machine-learning approaches isto find principal components, or extract a low-rank structurecrucial to structure-property correlations, from the given data,which have been, in certain instances, represented by anincomplete matrix or tensor. Irrespectively of supervisedor unsupervised learning, it is important to extract theseprincipal components, which may be called control parame-ters, reaction coordinates, or a latent space, depending on thecontext. These are not only axes, curves, or surfaces in avector space Rd but also basis functions in a Hilbert space.The extracted principal components may improve inter-pretation of data, as in the standard principal componentanalysis while the low-rank structures have been used forprediction of missing data. The automation of the researchprocesses is also augmented by boosting interpretationprocess of the given data when human involvement canbecome a bottleneck. New interpretation of old problems isexpected, as well.The machine learning and related approaches have alsoattracted attention as generators of complicated (probability)distributions, which are useful in condensed matter physics.Indeed, probability distributions generated by machine-learning approaches have been used to imitate underlyingprobability distributions of many-body configuration invariational wave function6) and self-learning Monte Carlomethods.7) The generation of complicated probability dis-tribution itself have been sophisticated under mutualinfluence between studies on neural networks and many-body physics, which results in, for example, neural canonicaltransformations.8)In the present article, we focus on spectroscopy ofquantum materials as an example of research fields wheretraditional theoretical tools to analyze spectra have beendeveloped alongside the recent development inspired bymachine-learning approaches. The spectroscopy providesus with energy-dependent materials’ responses to externalperturbation, which often contain crucial information suchas charge gap in superconductors and origin of super-conductivity.9)When we visually examine a crystalline sample, we canqualitatively speculate whether it conducts electric currentbased on its appearance, such as whether it is transparent,black, or exhibits a metallic luster. These characteristics arerelated to the material’s response to incident light, whichoften provides us with clues to understand behaviors ofelectrons confined in the crystal. (Here, surface structures ofthe materials may also influence the response more thanintrinsic properties of electrons inside.)To access quantitative information encoded in theseresponses, sophisticated technologies for spectroscopy havebeen developed alongside theoretical tools to analyzeobserved spectra. Absorption or photoluminescence has beenstudied not only with visible light but also with electro-magnetic waves beyond the visible light range. Neutrons orother particles have been also utilized. Materials may absorbenergy ħ! and momentum ħk (in the following we set ħ ¼ 1)Journal of the Physical Society of Japan 94, 031005 (2025)https://doi.org/10.7566/JPSJ.94.031005Special TopicsMachine Learning Physics031005-1 ©2025 The Physical Society of Japanmaintain attribution to the author(s) and the title of the article, journal citation, and DOI.©2025 The Author(s)This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work mustJ. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25https://orcid.org/0000-0002-4055-8792https://doi.org/10.7566/JPSJ.94.031005http://creativecommons.org/licenses/by/4.0/http://crossmark.crossref.org/dialog/?doi=10.7566%2FJPSJ.94.031005&domain=pdf&date_stamp=2025-02-26from incident particles, while they may emit electrons inresponse to the absorbed energy and momentum. Thetunneling phenomena of electrons has been also utilized forspectroscopy.Indeed, many-body electrons in crystalline solids havebeen a proving ground for spectroscopy and relatedtheoretical tools. Experimental technologies and theoreticaltools for inelastic x-ray and neutron scattering, tunnelingmeasurement, and photoemission spectroscopy have beendeveloped through the application to correlated electrons inquantum materials such as superconductors, topologicalmaterials, and so on. To interpret spectroscopy data, incooperation with the development of forward approaches,inverse approaches have been studied since the 1960s as well.While we will focus on many-body effects in spectroscopy,which will be introduced in the following, there are variousmachine-learning approaches on analysis of spectroscopydata, which are beyond the scope of the present article. Toremove human involvement or replace intuitive but biasedmodel, machine learning approaches have been introduced,which are expected to accelerate the data analysis. Therehave been studies to make automatic analysis of the hugespectroscopic data possible.10,11) To improve spectroscopydata itself, the machine learning approaches have beenintroduced. Denoising of the photoemission spectroscopydata12,13) is an example. Deep neural networks have been alsoutilized to remove structured noise in the photoemissiondata14) as well. Aside from significant influence of statisticalmechanics on the early stage of the development of themachine learning, including neural networks, the Bayesianapproaches have been utilized in condensed matter physics,even though frequentist statistics had been taught as thestandard approach in undergraduate and graduate schools. Anexample is the maximum entropy method in the analyticalcontinuation.15) The Bayesian approaches in analytic con-tinuation have been further developed.16)The present article is organized as follows. In Sect. 2, wereview the theoretical representation of many-body electronsystems, covering topics from many-body wave functions tothe self-energy function, an analytical function that encodesthe effects of many-body interactions, interconnected throughthe Green function formalism. Then, in Sect. 3, severalself-energy models are introduced. To analyze a categoryof Bardeen–Cooper–Schrieffer-type superconductors whereelectrons strongly interact with quantized vibration of ionsforming crystals, physicists had combined forward problemwith inverse problem to extract many-body nature of many-body electrons. To analyze high-temperature superconduc-tors, in which mutual Coulomb repulsion among electronsbecomes relevant, further self-energy models, includingneural network representations, will be introduced. Theapplications of the self-energy models to spectroscopy datais reviewed in Sect. 4. Section 5 is devoted to summary andfuture perspectives.2. Many-body Effects in SpectroscopyElectrons confined solids are typical examples of quantummany-body systems. Starting from the few electrons in atomsand small molecules,17,18) the many-body electron systemshave been targets of intensive researches.19) Magnetismand superconductivity are typical examples of phenomenaemerging in the many-body electrons. Nowadays, from thequantum-information point view, properties of the many-body electrons originating from structures of the quantumentanglement attract attention. Especially, topological many-body states of matters have been intensively studied.However, even in a simple many-body electron system,analysis of the wave function will be complicated, as follows.First of all, the many-body Schrödinger equation for themany-body electrons is hard to solve even for few electronsconfined around a nuclear and even when the quantum natureof the nuclear is ignored (Born–Oppenheimer approxima-tion). Naively speaking, we may map the many-bodySchrödinger equation to a (generalized) eigenvalue problemfor a sparse Hamiltonian matrix. For the sake of simplicity,we focus on non-relativistic systems in the following. Whenwe consider Norb single-particle orbitals and Ne electrons, thelinear dimension of the Hamiltonian matrix is ð2NorbÞ!=Ne!=ð2Norb � NeÞ!, where the factor 2 comes from the spindegrees of freedom of electrons. If you know the quantumnumbers, such as total spin=angular momentum for electronsconfined in an atom or the total momentum for electronsconfined in a crystalline solid, you can reduce the lineardimension. However, in general, the dimension of the matrixgrows exponentially when Norb or Ne increases. Thus,researchers have tried to find smart way to truncate the basisset or introduced approximations.Even when we obtain the many-body eigenstate wavefunctions, j�i, the wave functions themselves would be akind of the black box that generates a probability amplitudeas an output from given the location of the electrons as inputdata. Let us introduce one of the simplest example,interacting S ¼ 1=2 spins, or qubits, described by simpleeffective Hamiltonians, such as the Heisenberg Hamiltonian.The Heisenberg Hamiltonian effectively describes insulatingphases of hydrogen crystals in Mott’s gedanken experi-ment20) and La2CuO4.21) For example, the many-body wavefunction of the 36 spins is naively represented by a 236dimensional complex vector, which would require 1 TiB(10244 bytes) of storage. Each element of the vectorrepresents the probability amplitude for a given spinconfiguration.Then, how do we understand the properties of and extractuseful information from the wave function? In the history ofthe many-body physics, there have been seminal works thatsucceed to make the black box (at least partially) white.Typical examples of them would be the roton theory22–24)for liquid helium 4 and the Fermi liquid theory,25,26)originally for liquid helium 3 but later applied to othermany-body fermion systems, including many-body electronsin metals.These theories explain what happens in a many-bodysystem when energy and momentum are injected. The Fermiliquid theory states that a series of excited states are labeledby a particle-like excitation with energy �� measured fromthe ground state and many-body momentum k. Here, themomentum appears in the phase factor of the eigenvalue ofthe operator T̂� that translates the system with an interval � asT̂�j�i ¼ eþik��j�i.Let us introduce a concrete description of many-bodyquantum systems. As the simplest example, we first focus onan electron gas, defined by the following Hamiltonian,J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-2 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25Ĥ ¼ �X�Zd3r12me�̂y�ðrÞr2�̂�ðrÞþX�Zd3r �̂y�ðrÞvextðrÞ�̂�ðrÞþX�;�0Zd3rZd3r0�̂y�ðrÞ�̂�ðrÞ�̂y� 0 ðr 0Þ�̂� 0 ðr 0Þ2�jr � r 0j ; ð1Þwhere me is the electron mass, vext is the external electrostaticpotential, ϵ is the dielectric constant, and �̂ðrÞ is the fieldoperator given by�̂�ðrÞ ¼ V�1=2Xkeþik�rĉk�: ð2ÞHere, ĉyk� (ĉk�) is the creation (annihilation) operator for anelectron with spin σ and momentum k. We assume thatelectrons are in a cube of volume V with periodic boundaryconditions. The external potential vext is constant in theelectron gas.In many cases, we can assume that the ground state wavefunction j�0i has zero momentum or is translationallyinvariant as T̂�j�0i ¼ j�0i. Then, a simple way to constructa state with finite momentum is, for example, adding anelectron with k to the ground state as ĉyk�j�0i. Indeed, thestate has finite momentum k as T̂�ðĉyk�j�0iÞ ¼ eþik��ĉyk�j�0i.When we follow the dynamics of such a state with finitemomentum, we can introduce a Green function that describesthe propagation of a single particle or single hole asG�ðk; !Þ ¼ h�0jĉk� 1! þ i� þ � � Ĥ þ E0ĉyk�j�0iþ h�0jĉyk�1! þ i� þ � þ Ĥ � E0ĉk�j�0i ð3Þ¼Xmh�0jĉk�jmNeþ1ihmNeþ1jĉyk�j�0i! þ i� þ � � ENeþ1m þ E0þXnh�0jĉyk�jnNe�1ihnNe�1jĉk�j�0i! þ i� þ � þ ENe�1n � E0; ð4Þwhere E0 is the ground state energy that satisfies Ĥj�0i ¼E0j�0i and η is a positive broadening factor. Here, ENeþ1m isan eigenvalue corresponding to an Ne þ 1 electron eigenstatejmNeþ1i of Ĥ while ENe�1n is an eigenvalue correspondingto an Ne � 1 electron eigenstate jnNe�1i of Ĥ. From theconstruction above, the Green function is an analyticcomplex function represented by the summation of poles oforder at most one. When we take the retarded representationas in Eq. (4), these poles are located in the lower half plane.By introducing a single-particle dispersion �ðkÞ, we cantransform the Lehmann representation Eq. (4) as,G�ðk; !Þ ¼ 1! þ i� þ � � �ðkÞ � �ðk; !Þ ; ð5Þwhere �ðk; !Þ is a self-energy that encodes deviationfrom the single particle picture, and satisfies the followingrelation,�ðk; !Þ ¼ � 1�Zd!0Im�ðk; !0Þ! þ i� � !0ð6Þ¼ 1�Zd!0Pðk; !0Þ! þ i� � !0: ð7ÞHere, Pðk; !Þ is a positive definite distribution. We note thatthere is an ambiguity in the choice of the single-particledispersion; The ω independent part of the self-energy, forexample, the Hartree term, can be absorbed in �ðkÞ.By extracting the imaginary part of the Green function, weobtain the momentum and energy resolved density of states,Aðk; !Þ ¼ �ð1=�Þ ImGðk; !Þ, called a spectral weight orspectral function. As discussed in the following sections,photoemission (! ≲ 0) and inverse photoemission (! ≳ 0)spectroscopy enable us to access the spectral function.27,28)Even when the mutual Coulomb repulsion is strong, if apole at ! ¼ �ðkFÞ � � � i� has a finite residue Z defined byZ�1 ¼ 1 � @@!Re�ðkF; !Þ����!¼0; ð8Þthe Fermi liquid theory is valid. Here, we note that theresidue Z is the weight of the particle-like excitation.In the homogeneous electron gas, the bare single-particledispersion, �ðkÞ ¼ jkj2=2me, will be renormalized due to theCoulomb repulsion. By introducing the effective mass,m� ¼1 � @@!Re�ðkF; !Þ����!¼01meþ 1kF@@kRe�ðkF; !Þ����!¼0; ð9Þwe can approximate the original Green function as,G�ðk; !Þ ’ Z! þ i= ðk; !Þ � kFm�ðjkj � kFÞþ ðincoherent partÞ: ð10ÞHere, the life time of the particle-like excitation  ðk; !Þdefined as 1= ðk; !Þ ¼ �Z Im�ðk; !Þ, which is expandedas 1= 0 þ !2 around ! � 0. Then, we can interpret theGreen function as follows: A plane wave of jkj � kFpropagates with the renormalized kF=m� and decays intoother plane waves with a life time τ.The incoherent part shows rich structure in many-bodysystems. In the homogeneous electron gas, plasmon satellitesor sidebands29) are generated by dressed single-particleexcitations involving plasmons in addition to the renormal-ized quasiparticle excitations.30,31)When we describe crystalline solids, we may introduce adifferent basis set, instead of the plane waves in the vacuum.A typical choice is Wannier orbitals often localized aroundthe ions. It has been known that a subspace expanded by asmall subset of the Wannier orbitals will be relevant to thelow- to room-temperature properties of the solid. When oneperforms photoemission spectroscopy and, thus, extract anelectron from the solid as discussed in the following sections,one need to pay attention to the matrix element between theWannier orbitals and plane waves. The Wannier orbitals arealso relevant to scanning tunneling spectroscopy.36)In the periodic crystals, it is convenient to separate a wavenumber into a wave number in the first Brillouin zone k anda reciprocal lattice vector G as k! k þ G. Then, the fieldoperator �̂�ðrÞ is rewritten by introducing a set of theWannier orbitals, f�iRðrÞg, as,�̂�ðrÞ ¼Xw;R�wRðrÞd̂wR�; ð11ÞJ. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-3 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25where �wRðrÞ is the wth Wannier orbital in an unit labeledby the lattice R. While the choice of the Wannier orbitalsis not unique, the maximally localized Wannier orbitals37,38)have been often employed in the literature. The creation(annihilation) operator for the plane wave in the vacuum,ĉykþG� (ĉkþG�), and that for the Wannier orbital, d̂ywR� (d̂wR�)are transformed into each other through a unitary trans-formation as,ĉkþG� ¼Xw;RZd3re�iðkþGÞ�rV1=2�wRðrÞ� �d̂wR�: ð12ÞIn the series of the copper oxide superconductors, theelectronic states around the Fermi level mainly consist of theWannier orbitals consisting of Cu 3dx2�y2 atomic orbitals andO 2p� atomic orbitals39) (see Refs. 40 and 41 as examples ofthe recent studies).For ab initio single-orbital effective Hamiltonian, we mayhave a single Wannier orbital in each unit cell labeled by R.Then, we simplify the index for the creation and annihilationoperators, as ðw;RÞ ! i, where i is now a site index. Then,the ab initio single-orbital Hamiltonian is given byĤeff ¼ �Xi; j;�tijd̂yi�d̂j� þ UXin̂i"n̂i#þ 12Xi≠jVi; jðn̂i" þ n̂i#Þðn̂j" þ n̂j#Þ; ð13Þwhere n̂i� ¼ d̂yi�d̂i�.For simplicity, we rewrite the Green function by using thesingle Wannier orbital as,G�ðk; !Þ ¼ h�0jd̂k� 1! þ i� þ � � Ĥeff þ E0d̂yk�j�0iþ h�0jd̂yk�1! þ i� þ � þ Ĥeff � E0d̂k�j�0i¼ 1! þ i� þ � � �dðkÞ � �dðk; !Þ ; ð14Þwhere k is a wave number in the first Brillouin zone and�dðkÞ is the single-particle dispersion obtained by the Fouriertransformation of �tij.By restricting ourself to one-dimensional lattice, andomitting the further neighbor tij other than the nearestneighbor hoppings t (> 0) and the long-range effectiveCoulomb repulsion Vij, we show an example of the self-energy �dðk; !Þ in Fig. 1. Here, the Green function is simplycalculated by effectively constructing excited states with onemore or less electrons from the ground state j�0i, by using adynamical variational Monte Carlo method,32) and, then, theself-energy is obtained through Eq. (14).A simple spectral function from the bare band dispersion,ð�=�Þ=½ð! þ � þ 2t cos kÞ2 þ �2�, is significantly modifiedby �dðk; !Þ. There are several features found in spectralfunctions of the one-dimensional correlated electrons, whichare called the Hubbard bands, and spinon or holonbranches.33–35) While the results of the finite-size simulationof the Hubbard model are shown in Fig. 1, the accumulationof poles in self-energy at the thermodynamic limit generatesbranch-cut singularities in the spectral functions of the one-dimensional Hubbard model, as studied in the Tomonaga–Luttinger model.42–44)To describe many-body electrons in superconductingphases, we need to introduce several theoretical details,especially related to the number of electrons, as follows. Forthose not interested in these details, you may skip thefollowing seven paragraphs and go directly to the paragraphthat includes Eq. (25).The seminal article by Yang45) introduced a fundamentalapproach to examine superconductivity in a given ground-state wave function with a fixed number of electrons. Asexplained in Ref. 45, the two-particle density matrix plays animportant role and is defined by��1�2 1 2j1 j2‘1‘2¼ h�0jd̂yj1�1 d̂yj2�2d̂‘1 1 d̂‘2 2 j�0ih�0j�0i : ð15ÞBy combining the first four indices into the index J ¼ ð j1; �1;j2; �2Þ and the last four indices into L ¼ ð‘1;  1; ‘2;  2Þ, wecan treat the density matrix as matrix �JL ¼ ��1�2 1 2j1 j2‘1‘2. Whenthe largest singular value of �JL is proportional to the numberof electrons Ne, an off-diagonal long-range order sponta-neously emerges, corresponding to superconductivity. Thelargest-singular-value singular vector provides us with thespatial and internal structures of the superconducting pairs.However, it is difficult to relate the two-particle densitymatrix to the self-energy.In a standard procedure to examine spontaneous symmetrybreakings, a small symmetry breaking field is introduced. Wecan find a ground state wave function as a function of thesystem size and the small but finite symmetry breaking field.Then, there are two important limits: one is the thermody-namic limit, where 1=Ne approaches zero; the other is thelimit in which the amplitude of the symmetry-breaking fieldgoes to zero. When we take the zero symmetry breaking field 0  1 0 0.2 0.4 0.6 0.8 0  1-20-20-10-10 0 0 10 10 20 20 0  1-4 0 4 8 12-40-30-20-10 0-4 0 4 8 12-4 0 4 8 12Fig. 1. (Color online) Spectral function and self-energy for a one-dimensional Hubbard Hamiltonian. The spectral function Adðk; !Þ for U=t ¼ 8 and 32 siteswith 28 electrons is shown in the left most panel, which is obtained by dynamical variational Monte Carlo methods.32) Here, the broadening factor is set to�=t ¼ 0:125. The real and imaginary part of the retarded Green function, �dðk; !Þ, are shown in the middle and right most panels, respectively. A simplespectrum from the bare band dispersion, ð�=�Þ=½ð! þ � þ 2t cos kÞ2 þ �2�, is significantly modified by �dðk; !Þ. There are several features in correlatedelectrons, such as Hubbard bands, spinon and holon branches.33–35)J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-4 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25limit after taking the thermodynamic limit, we can judgewhether symmetry breaking spontaneously occurs.When we want to examine the superconducting order, wemay introduce the following symmetry breaking field �Jk fora spin-singlet pairing,Ĥeff  Ĥeff � �XkðJkd̂k"d̂�k# þ J �k d̂y�k#d̂yk"Þ: ð16ÞThe symmetry breaking field inevitably mixes the dynamicsof ↑-spin (↓-spin) electrons with those of ↓-spin (↑-spin)holes. Then, the number of electrons is not fixed. Thethermodynamic limit should be defined by the expectationvalue of the electron number operator, N̂ ¼Pj;� d̂yj�d̂j�. Theground-state wave function discussed so far is based on avacuum j0i and creation operators multiplied to the vacuum.When the spin-singlet superconducting pair in the low-energysubspace is examined according to the BCS theory, thevacuum j0i is often replaced byjvaci ¼Ykd̂y�k#j0i; ð17Þwhere the product of the creation operators is taken over theentire Brillouin zone. The ground state wave function isgiven in grand canonical form asjf�0i ¼Xm2ZjAmjXm02ZjAm0 j2e2mi’j�Neþ2m0 i; ð18Þwhere j�N0 i is the normalized projection of the ground-statewave function onto the N electron sector, Am is a coefficient,φ is a Uð1Þ phase, and Ne ¼ hf�0jN̂jf�0i=hf�0jf�0i is assumedto hold.By introducing the grand-canonical ground-state wavefunction, jf�0i, into the Green function in Eq. (14), we cancalculate the properties of the superconducting states.However, there are more convenient formulations of thesuperconducting states based on the self-energy. We canbegin with the Green function along the time axis,G"ðk; tÞ ¼ �i�ðtÞhf�0j½d̂k"ðtÞd̂yk" þ d̂yk"d̂k"ðtÞ�jf�0i; ð19Þwhere the Heisenberg representation of the operators isdefined as d̂k�ðtÞ ¼ eiðĤeff��N̂Þtd̂k�e�iðĤeff��N̂Þt. The Fouriertransformation of G"ðk; tÞ, of course, reproduces a represen-tation similar to Eq. (14). To take into account the mixture ofthe electron and hole dynamics, the Green function for the↓-spin particles is given by,G#ð�k; tÞ ¼ �i�ðtÞhf�0j½d̂y�k#ðtÞd̂�k# þ d̂�k#d̂y�k#ðtÞ�jf�0i:ð20ÞDue to the symmetry breaking field �Jk, G"ðk; tÞ, andG#ð�k; tÞ are not independent of each other. Furthermore,the real-time evolution of these Green functions inevitablyinvolves the following anomalous Green functions,F"#ðk; tÞ ¼ �i�ðtÞhf�0j½d̂k"ðtÞd̂�k# þ d̂�k#d̂k"ðtÞ�jf�0i; ð21ÞandF#"ðk; tÞ ¼ �i�ðtÞhf�0j½d̂y�k#ðtÞd̂yk" þ d̂yk"d̂y�k#ðtÞ�jf�0i: ð22ÞFor example, Ref. 46 examines the superconducting statewith an explicit wave function and the Green functions alongthe imaginary time axis, as summarized in the article’sAppendix.Then, the four Green functions constitute the Namburepresentation47) of the Green function,Ĝðk; tÞ ¼G"ðk; tÞ F"#ðk; tÞF"#ðk; tÞ G#ð�k; tÞ" #: ð23ÞWhile the Fourier transformation of the Green functionmatrix may provide us with the spectral (Lehmann)representation as in Eq. (14), the time evolution of the Greenfunction matrix may offer another representation.To gain insight into the Green function matrix, we canexamine the non-interacting limit. When U and Vij are set tozero in Eq. (13), the time evolution of the matrix Ĝðk; tÞ issimply solved in the ω domain asĜðk; ! þ i�Þ ¼ ! þ i� � �dðkÞ þ � ��J �k��Jk ! þ i� þ �dðkÞ � �" #�1; ð24Þwhere a symmetry, �ðkÞ ¼ �ð�kÞ, is assumed.When we follow the time evolution of Ĝðk; tÞ under the influence of the mutual electron–electron interactions, there appearmore complicated Green functions involving four or more operators, beyond the four components, G", G#, F"#, and F#".However, these complicated Green functions were shown to be deconvoluted into four Green functions and the twocomponents of the self-energy.9,48) Here, we assume that the thermodynamic limit has already been taken, and then we take the�! 0 limit afterward.To describe effects of many-body interactions in the superconducting phases, we will introduce the following two-component self-energy. Here, we assume a single orbital system and introduce the self-energy into the 2 � 2 matrixrepresentation of the Green function as,Ĝðk; 
Þ ¼ 
 þ � � �dðkÞ � �norðk; 
Þ ��anoðk; 
Þ��anoðk; 
Þ 
 � � þ �dðkÞ þ �norðk;�
Þ�" #�1; ð25Þwhere 
 ¼ ! þ i�, and two components, �nor and �ano,represent normal and anomalous contributions of the self-energy, respectively.While ½Ĝðk; 
Þ�11 represents a normal component of thematrix form of the Green function in Eq. (25) [the Greenfunction for ↑-spin particle, G"ðk; 
Þ], ½Ĝðk; 
Þ�12 representsan anomalous component that emerges only when the systembecomes superconducting [the anomalous Green functionmixes ↑-spin particle and ↓-spin hole, F"#ðk; 
Þ]. The 11element ½Ĝðk; 
Þ�11 has the same structure as,J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-5 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25½Ĝðk; 
Þ�11 ¼1! þ i� þ � � �dðkÞ � �norðk; !Þ �Wðk; !Þ ;ð26Þwhere the contribution from the anomalous component isgive byWðk; !Þ ¼ �anoðk; !Þ2! � � þ �dðkÞ þ �norðk;�!Þ� : ð27ÞThe two components of the self-energy are rewritten as�norðk; !Þ ¼ 1�Zd!0Pðk; !0Þ! þ i� � !0; ð28Þ�anoðk; !Þ ¼ 1�Zd!0Qðk; !0Þ � Qðk;�!0Þ! þ i� � !0; ð29Þby introducing positive definite distributions, Pðk; !Þ andQðk; !Þ. As discussed in the following sections, thesedistribution functions play pivotal roles.While one-particle spectra are the focus of this article,there are other observables that can be utilized to study themany-body effects. Rather than the propagation of anelectron or a hole excitation, the propagation of multipleelectron and hole excitations are illustrated by many-particleGreen functions, which are often useful in analyzing datafrom a range of spectroscopy (see Appendix A).As a theoretical description of many-body quantumsystems,49) diagrammatic approaches have been developed,50)in parallel to the variational wave-function approaches.51–55)A formula for equilibrium free-energy given as a functionalof the one-particle Green function has been formallyconstructed by a series expansion with respect to (screened=renormalized) interactions.50) The series is terminated at afinite order56,57) or the partial summation of a category ofdiagrams58) is taken. A Monte Carlo sampling scheme basedon the diagrams has also been developed.59) While self-energy is a crucial building block in the diagrammaticapproach, vertex functions are also vital to describe many-body effects.31,60,61) The dynamical mean-field theory62–65)and related non-perturbative approaches66) have been devel-oped. There are several extensions including vertex func-tions.67) Two-particle Green functions have also beenanalyzed by extending the dynamical mean field theory.68)3. Methodology of Self-energy AnalysisTo extract the self-energy from spectroscopy data, it isnecessary to note that the measurements are conducted overa finite energy range, and that only the imaginary or real partof the Green functions is accessible in the data. When weanalyze spectra and extract information of self-energy fromthese data, we need prior knowledge and a self-energy model,irrespective of their origin, to supplement the lack of theinformation. In standard theoretical approaches, the modelhas been often derived from first-principles formulae orapproximated ones, as reviewed below.3.1 Physics-informed self-energy model3.1.1 Angle integratedLocal probes such as superconductor–normal-metal tunneljunction or scanning tunneling spectroscopy measurementscapture the momentum integrated spectra or single-particledensity of states,Nð!Þ ¼Zddk�BZAðk; !Þ; ð30Þwhere d is the dimension of the system. When we need tospecify the phase in the following, we will add a subscript toN as Nn or Ns, which correspond to the density of states in thenormal and superconducting phase, respectively.The density of states in strong-coupling Bardeen–Cooper–Schrieffer superconductors69) have been well investigatedby the Migdal–Eiashberg–Nambu theory.47,70,71) Undercertain conditions, the ratio of the density of states in thenormal and superconducting phases is given by a simplefunction,72,73)Nsð!ÞNnð!Þ ¼ Re!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � �ð!Þ2p" #; ð31Þwhere �ð!Þ is a gap function defined as�ð!Þ ¼ �anoð!ÞZð!Þ : ð32ÞHere, Zð!Þ ¼ 1 � �norð!Þ=! introduces renormalization ofthe effective mass.The satisfactory condition to validate Eq. (31) is that �ðkÞcan be approximate as�ðkÞ ’ � þ vFðjkj � kFÞ: ð33ÞWhen the condition holds, the normal state density of statebecomes constant as Nnð!Þ / 4�mekF.We have prior knowledge that the superconducting gap�ð!Þ will be formed in a small energy window incomparison with the Fermi energy EF in the BCS super-conductors. Thus, for ω significantly larger than a typicalsuperconducting gap scale �0, Nsð!Þ=Nnð!Þ will approachunity. Then, we can access the entire ω dependence of thereal part of !=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � �ð!Þ2p. By using the Kramers–Kronigrelation, the imaginary part, thus, the whole complex function!=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � �ð!Þ2pis obtained. Once the gap function �ð!Þis obtained from Nsð!Þ=Nnð!Þ and the branch cut forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � �ð!Þ2pis chosen, the normal and anomalouscomponents of the self-energy will be analyzed.The Migdal–Eliashberg–Nambu theory70,71) provides uswith simple formulae to model the two components, �ð!Þand Zð!Þ, by introducing an effective phonon density ofstates, �ð!Þ2Fð!Þ as explained in Appendix B. Then, thedistributions, Pðk; !Þ and Qðk; !Þ, are obtained in amomentum-independent manner, as follows. The distributionPð!Þ is given by the following joint density of states ofparticle–hole and phonon excitations,Pð!Þ ¼ �Z þ10d��ð�Þ2Nnð0Þ Fð�Þ½�ð! � �ÞNsð! � �Þþ �ð�! � �ÞNsð�! � �Þ� � 1�Im�ð0Þð!Þ; ð34Þwhere the contribution from the self-energy �ð0Þð!Þ fromimpurity scattering and mutual electron–electron interactions.It is complicated to represent Qð!Þ by the densities of states,�ð�Þ2Fð�Þ and Nsð!Þ. Instead, the distributionQð!Þ, which isnot necessarily positive definite, is given byQð!Þ ¼ ��Z þ10d��ð�Þ2Nnð0Þ Fð�ÞMsð! � �ÞJ. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-6 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25� �ð!c � ! þ �Þ�ð! � � � �0Þ; ð35Þwhere Msð!Þ ¼ Re½�ð!Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 ��ð!Þ2 þ i�p�.References 74 and 75 made the formalism inverted theforward approach [Eqs. (B·1)–(B·3)] to extract �ð!Þ2Fð!Þ.From the experimentally observed density of states foreV‘ � !exp � eVh, the self-energy is extracted by using theMigdal–Eliashberg–Nambu formalism, as summarized asfollows:Nsð!expÞ=Nnð!expÞ��������!extrapolationNsð!Þ=Nnð!Þ��������!Eq: ð31Þ�ð!Þ  !Eqs: ðB1Þ­ðB3ÞZð!Þ�ð!Þ2Fð!Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}self-energy modelIn Refs. 72 and 73, physically sound phonon density of statesare assumed and examined whether these models explainexperimentally observed tunneling spectroscopy data, whileRefs. 74 and 75 inverted the procedure.3.1.2 Semi-angle resolvedVekhter and Varma extended the scheme developed inRefs. 72 and 73 to partially include momentum dependenceof the self-energy.76) A simple and useful example of thetarget systems is a two-dimensional many-body electrons.When we focus on the momentum near the Fermi surface,the momentum k ¼ ðkx; kyÞ in the Cartesian coordinate isrewritten into an angle along the Fermi surface, θ, and themomentum perpendicular to and measured from the Fermisurface, �k?, as illustrated in Fig. 2.Then, one may realize that the assumption of the linearizedband dispersion Eq. (33) can be generalized as,�ðkÞ ’ � þ vFð�Þ�k?: ð36ÞThe authors of Ref. 76 found that the density of states inEq. (31) should be replaced with a partially integratedspectral function,Nð�; !Þ ¼Zd� k?Að�k?; �; !Þ: ð37ÞThe ratio of the partially integrated spectral function satisfiesa relation similar to Eq. (31) as,Nsð�; !ÞNnð�; !Þ ¼ Re!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � �ð�; !Þ2p" #; ð38Þwhere �ð�; !Þ is a partially momentum-dependent gapfunction.3.1.3 Angle-resolvedIrrespective of perturbative56–58) or non-perturbative,66)theoretical and numerical methodologies have been devel-oped to reveal the momentum-dependent structure of theself-energy. However, many of them are not simple andtransferable enough to be utilized to extract self-energyinformation from experimental data.Primarily focusing on the cuprates as a typical example ofcorrelated quantum materials, there are analytical representa-tions of the self-energy that capture the essence of cupratephysics. While the momentum-independent marginal-Fermi-liquid self-energy77) has been employed in the literature, therehave been several phenomenological self-energy models78–82)to reproduce the significantly momentum-dependent natureof the self-energy of cuprate superconductors, which isevident in the formation of pseudogap.The strongly momentum-dependent self-energy found inthe two-dimensional Hubbard model by a cluster extension ofthe dynamical mean field theory83) was reproduced by thedark fermion model,80) in which hybridization between twospecies of fermions generates the pseudogap.79,84) Such asingular self-energy has been found in fermion-bosoncoupled systems at quantum critical points.81)A completely different approach emerges from the AdS=CFT correspondence or holography.85,86) A black hole ina anti-de Sitter spacetime offers a description of strangemetals.87,88) The strange metal shows �k?-dependent theimaginary part of the self-energy proportional to !2�k where�k is a linear function of �k?, which is utilized in an analysisof ARPES measurements.89)3.2 Machine-learning modelEven though a physics-informed model or a decentapproximation for the self-energy has succeeded in low-critical-temperature superconductors, it has a limitation andis often biased due to the physical mechanism behind themodel. However, the accumulation of the studies on the self-energy has revealed its proper mathematical structures. Ifthere are flexible mathematical representations for P and Q,the proper mathematical structure and the prior knowledgewill be exploited to construct a self-energy model beyond thelimitation of these models.3.2.1 Artificial neural networksArtificial neural networks may serve as flexible models torepresent these distributions, P and Q. However, we note thatthere are several different structures of neural networksdesigned for different purposes. A typical category ofartificial neural networks, widely used in recent machine-or deep-learning studies, is the feed-forward neural network.While a feed-forward neural network can represent (multi-variate) distributions, as utilized in classification tasks, it isalso suitable to use a feed-forward network to process theinput vectors or matrices (pixel data).A feed-forward neural network consists of successivetransformations of a given input vector. For an Lð0Þ dimen-sional input vector xð0Þ, a new Lð1Þ dimensional vector xð1Þ isgenerated through successive linear and non-linear trans-formation as, xð1Þ‘ ¼ �ð0ÞðPm Wð0Þ‘mxð0Þm þ bð0Þ‘ Þ. Here, W ð0Þ isa Lð1Þ � Lð0Þ real matrix and bð0Þ is a Lð0Þ dimensional realvector, which are trainable parameters that define the linearFig. 2. (Color online) Fermi surface of two-dimensional electron gasesand cuprates. While the left panel shows the Fermi surface, for the electrongas, the right panel shows the typical Fermi surface of the cuprate. In the bothFermi surfaces, a new coordinate, ð�k?; �Þ, used in the semi-angle dependentanalysis, is illustrated.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-7 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25transformation of the input vector; the non-linear scalarfunction �ð0ÞðxÞ defines the non-linear transformation. We canrepeat similar transformations as xðkþ1Þ‘ ¼ �ðkÞðPm W ðkÞ‘mxðkÞm þbðkÞ‘ Þ to construct a deep neural network. To train the neuralnetwork, we need to prepare a sizable set of input and outputdata.There are several variants of feed-forward networks. Acomplex version of the feed-forward networks, where theparameters in W ðkÞ and bðkÞ are complex, is studied as anindependent subject because the complex feed-forwardneural networks show different analytical behavior. Convolu-tional neural networks, another variation of feed-forwardnetworks, involve additional structured data operations, suchas filtering and pooling, which are designed to handle images.Another category of neural networks has been developedto represent, primarily, probability distributions, inspired bythe Boltzmann weight, e�EðsÞ=Z, for a statistical descriptionof classical many-body systems. Here, EðsÞ is the energy of agiven state represented by an array of Ising spins, s, β is theinverse temperature, and Z is a normalization constant knownas the partition function. While it is simple to sample a spinconfiguration s from the Boltzmann distribution, the fullshape of the generated probability distribution is out of reach.One example in this category is the Boltzmann machine.90,91)When an Ising spin configuration consists of a bit array� ¼ f�‘g (�‘ ¼ 0; 1) and a set of hidden variables h ¼ fhmg(hm ¼  1), a Boltzmann machine is given by a marginalprobability pð�Þ through averaging a conditional probabilitypð�jhÞ ¼ e�Eðf2�‘�1g;fhmgÞ=Z over every possible configura-tions of the hidden variables. Depending on the interactionbetween the bit array � and the hidden variables h, there areseveral variations of the Boltzmann machines. While theseBoltzmann machines have been used in a broad range ofapplications,6,92) the author and the author’s collaboratorsused them to represent the ω dependence of Pðk; !Þ andQðk; !Þ at a given k.93)3.2.2 Neural-network self-energy modelAs mentioned above, the Boltzmann machine, whichoriginates from the Boltzmann weight for interacting Isingspins, returns a value associated with a bit array. To constructthe Boltzmann machines to represent P and Q, the authorsmapped ω in a finite range, ! 2 ½��=2;þ�=2�, to L-digitbinary representation as�  ð�0; �1; . . . ; �L�1Þ; ð39Þby dividing the range ½��=2;þ�=2�. Here, �i ¼modðI=2i; 2Þ for the decimal representation Ið�Þ in the range0 ≦ Ið�Þ ≦ 2L � 1 of the grid number coordinate,Ið�Þ ¼XL�1‘¼0�‘ � 2‘: ð40ÞThe binary representation introduces a hierarchical structurelike a wavelet.94,95) As shown below, the Boltzmann machineswith parameters, whose number is proportional to a polynomi-al of L, can describe 2L degrees of freedom. Similar structuresof information compression have been utilized in matrix-product, tensor-network, or tensor-train representation.96)When ω dependence is discretized as above, the simplestchoice of the basis set to describe P and Q is a rectangularfunction, �L�ð!Þ, defined as�L�ð!Þ ¼1 for x 2 ½Ið�Þ=2L; ½1 þ Ið�Þ�=2LÞ0 otherwise�: ð41ÞThe discretized description of the self-energy is shown inFig. 3. The rectangular function is also convenient tocalculate the self-energy since the Kramers–Kronig trans-formation of �L�ð!Þ is analytically available.Then, the basis set expansions of Pðk; !Þ and Qðk; !Þ, at agiven k, are obtained asPðk; !Þ ¼X�Ckð�Þ�L�! þ �=2��  ; ð42ÞQðk; !Þ � Qðk;�!Þ ¼X�Dkð�Þ��L�! þ �=2��  � �L��=2 � !��  �; ð43Þwith two functions, Ckð�Þ and Dkð�Þ. Below, the momentumindex k is omitted for simplicity, and Ckð�Þ and Dkð�Þ aredenoted as Cð�Þ and Dð�Þ, respectively.Due to the non-negativity of Cð�Þ, it is efficientlyrepresented by the restricted Boltzmann machine(RBM),91,97,98) asCð�Þ ¼ ebXfhmg¼ 1expX‘;mð2�‘ � 1ÞW‘mhm" #¼ ebYLh�1m¼02 coshX‘ð2�‘ � 1ÞW‘m" #; ð44Þwhere Lh is the number of the hidden variables, b is the bias,and the weight W‘m only connects a visible variable �‘ and a(a) (b)(c)(d)(e)010101Fig. 3. (Color online) Boltzmann machine representation of self-energy.(a) The discretized representation of Im�norðk; !Þ ¼ �Pðk; !Þ is illustratedas a combination of red rectangles. (b) The discretized mixture representationof Im�anoðk; !Þ ¼ Qðk; !Þ � Qðk;�!Þ is shown. Whereas the total mixturedistribution for Qðk; !Þ is represented by open red rectangles, eachBoltzmann-machine distribution is illustrated by filled red, cyan, and bluerectangles. Copies of Boltzmann machines, �Qðk;�!Þ, are shown in greyrectangles. (c) The wavelet-like structure of the rectangular basis set isillustrated. From the longest wave length structure governed by �0 to theshortest wave length structure controlled by �L�1, each rectangular basis(open red rectangle) is labeled by the set of bits � ¼ ð�0; �1; . . . ; �L�1Þ(�‘ ¼ 0; 1). The structure of the restricted Boltzmann machine for Cð�Þand mixture distribution consisting of Boltzmann machines for Dð�Þ isdepicted in (d) and (e), respectively. This figure is taken from Ref. 93 underthe terms of the Creative Commons Attribution 4.0 International license.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-8 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25hidden variable hm, as illustrated in Fig. 3(d). The advantageof the RBM is that one can analytically trace out the hiddenvariables hm, leading to the second line in the above formula.The normal component and anomalous component ofthe self-energy could be influenced by different physicalprocesses in different energy scales. While Pð!Þ could besignificant up to several eV scale, Qð!Þ would be finitewithin a smaller energy scale. To introduce prior knowledgeon the energy scale, it is convenient to represent Dð�Þ by alocalized distribution. In Ref. 93, the authors introduced alinear combination of the simple Boltzmann machine withouthidden variables [see Fig. 3(e)] asDð�Þ ¼X�w� exp"X‘ð2�‘ � 1Þb�‘þX‘;mð2�‘ � 1ÞW�‘mð2�m � 1Þ#; ð45Þwhere a linear combination of the Boltzmann machines aretaken with coefficients, fw�g. The coefficients, the biases,fb�‘g, and the weights, fW�‘mg, will be optimized.Note that a linear combination of Gaussian distributions isone of the standard procedures to approximate a smoothfunction99) and can be used as an initial guess of Dð�Þ (thedetailed procedure is given in Appendix of Ref. 93).While the restricted Boltzmann machine with enoughnumber of hidden variables can represent any positivedefinite distribution,100,101) it is not easy to initialize therestricted Boltzmann machines. In contrast, the Boltzmannmachines without the hidden variables are explicitlyinitialized to represent several distributions, includingGaussian distributions. To reconcile the drawbacks andbenefits of the Boltzmann machine, the authors of Ref. 93choose a linear combination of the Boltzmann machines forthe anomalous part.3.2.3 Training of Boltzmann-machine modelsReference 93 introduces a training procedure for these(restricted) Boltzmann machines, which consists of inner andouter loops (as later illustrated in Fig. 4). The mean squarederror between the experimental and theoretical spectrum isminimized as the cost function in the inner loop. The costfunction is defined as,�2ðkÞ ¼ 1NdXj½Aexpðk; !jÞ � Acalðk; !jÞ�2; ð46Þwhere the photoelectron intensity is measured or calculated atNd discrete energy points f!jg. Here, the experimentalspectrum is denoted by Aexpðk; !Þ while the theoreticalspectrum, obtained by the Boltzmann-machine self-energymodels, is denoted by Acalðk; !Þ, where Acalðk; !Þ is aproduct of a Fermi-Dirac distribution fFDð!Þ (convolutedwith a Lorentzian representing an experimental resolution)and the spectral function Aðk; !Þ. To minimize �2, thevariational parameters,�C ¼ ðb; fW‘mgÞT ð47Þand�D ¼ ðfw�g; fb�‘g; fW�‘mgÞT; ð48Þin Cð�Þ and Dð�Þ, are optimized by using the gradients,gC;D ¼ @�2@�C;D: ð49ÞThe increments of these parameters, ��C and ��D, are chosento be��C ¼ � �CkS�1gCkp S�1gC; ð50Þ��D ¼ � �DkgDkp gD; ð51Þwhere �C;D are the learning rates and a real number,p 2 ð0; 1Þ, controls the amplitude of the gradient. Whilethe mixture of the Boltzmann machines, Dð�Þ, will beupdated in the outer loop as well, the restricted Boltzmannmachine Cð�Þ is updated only in the inner loop. Then, thenatural gradient method is used to update �C in Ref. 93. Theincrement of �C involves the classical Fisher informationmatrix given by,S�� ¼Xj@Pð!jÞ@�C�@Pð!jÞ@�C�: ð52ÞThe outer loop then examines the test error defined as,e�2ðkÞ ¼ 1NsynNdXj;k½Asynk ðk; !jÞ � Acalðk; !jÞ�2; ð53Þwhere a set of Nsyn synthetic spectra, fAsynk ðk; !jÞg, is used toavoid overfitting.93)3.3 WorkflowAlong with the workflow for the forward approach72,73)according to the Migdal–Eliashberg–Nambu theory, theworkflows for the self-energy estimation utilizing thephysics-informed models74–76) (in Sects. 3.1.1 and 3.1.2)and the Boltzmann-machine models (in Sect. 3.2), aresummarized in Fig. 4.4. Self-energy Analysis of Experimental Data4.1 Normal stateWhile understanding strong-coupling BCS superconduc-tivity was an important subject in an early stage of the self-energy analysis, the self-energy in the normal states havebeen studied in variety of materials. The mutual Coulombrepulsion and the electron–phonon coupling will make aconsiderable impact on the quasiparticle spectra even in thenormal metallic states. For example, strong ω dependence ofRe�ð!Þ in the electron–phonon coupled systems103–105) hasbeen shown to generate a bend in electronic dispersion,which is often called a kink. Due to development of non-perturbative theoretical analyses of the self-energy, it hasbeen theoretically revealed that the Coulomb repulsion itselfgenerate the kink structure,106–108) as well. The kinkstructures have been intensively discussed in the cupratesuperconductors, where the Coulomb repulsion plays animportant role. It is highly desirable to understand thecomplicated kink anomalies found in the quasiparticledispersion of the cuprates.109)4.1.1 Surface state of Be(0001)Analysis of self-energy in a simple system is favorable tounderstand the nature of electron–phonon couplings andelectron–electron interactions. A many-body electron systemJ. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-9 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25with spherically-symmetric quadratic band dispersion willserve as such a reference. While the simplest example wouldbe a two-dimensional electron gas emerging in semiconduc-tor interfaces, the photoemission spectroscopy of theinterfaces is not simple.110) Instead, the surface states ofmetals have been studied.111–114) One of these studies wasperformed on the Be(0001) surface.115)The authors of Ref. 115 performed a forward analysis onthe ARPES spectra of the Be(0001) surface. They introducedthe self-energy �ph modeling the electron–phonon couplingsand compared the spectral weight generated by �ph and theeffects of the electron–electron interaction. Here, �ph isestimated by using the Migdal–Eliashberg formalism in thenormal state.The authors approximated �ð!Þ2Fð!Þ as�ð!Þ2Fð!Þ ¼ �!�ð!ph � !Þ=2!ph;where λ is the strength of the electron–phonon coupling and!ph is the maximum energy scale of the phonon measured byelectron-energy loss spectroscopy.116) Here, the distributionfunction is given asPð!Þ ¼ ��ð!ph � j!jÞ�!2=4!ph þ ��ðj!j � !phÞ�!ph=4;which is an example of the self-energy model for the two-dimensional electron–phonon coupled systems.The electron–phonon coupling strength λ is estimated fromthe ratio of the renormalized electron velocity v�F and barevelocity vF through 1 þ � ¼ vF=v�F. These velocities areNONOYESYESNOMcMillan-Rowell Vekhter-VarmaBoltzmann-machine model updateCOM of BMsARPESARPESTSRegressionto determineupdate1) Self-energy by MEN theorynecessary toreproduce Ref. 102(       used to deterimine       )Migdal-Eliashberg-NambuSchrieffer-Scalapino-Wilkins(a) (b) (c)(d)1) Self-energy from (R)BM2) Spectral function3) Training error4) Increments of        , YES2) Tunneling spectrum3) Increment of phonon DOSfromfromfromNO YESupdate COM of BMs1) Green and gap functions2) Renormalization functioninitial guessphonon DOSconverges?PH asymmetry,BG, or ME?training error       converges?test error        converges?parameters of             andinitial guessFig. 4. (Color online) Workflows for forward and inverted approaches. (a) The Migdal–Eliashberg–Nambu (MEN) theory and the Scrieffer–Scalapino–Wilkins scheme provide a forward approach to calculate tunneling spectra (TS), Ncals ð!Þ=Nnð0Þ, by assuming an effective boson density of states (DOS),�ð!Þ2Fð!Þ, the self-energy, �ð0Þð!Þ, induced by electron–electron interaction, and the Coulomb pseudopotential, UC (see Appendix B). (b) The McMillan–Rowell method partially inverts the forward approach to obtain the effective phonon density of states, �ð!Þ2Fð!Þ, from the experimentally observed tunnelingspectrum, N exps ð!Þ=Nnð0Þ, supplemented by �0 from the spectrum. (c) The inverted scheme is further extended in the Vekhter–Varma method, where theARPES spectra is utilized to estimate the self-energy. Reference 102 generalizes the method by incorporating the particle–hole (PH) asymmetric component ofself-energy, �3ð�;!Þ, with separating so-called backgrounds (BG), Bsð�k?; �; !Þ, and matrix elements (ME), Mð�;!Þ, from bare photoemission spectra. (d)Workflow represents a regression procedure to train the Boltzmann-machine self-energy models, proposed in Ref. 93. The parameters, �C and �D, in Cð�Þ andDð�Þ are updated by the inner and outer loops.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-10 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25estimated from the dispersion measured by the angle-resolved photoemission117) as �ðkÞ ’ v�Fðjkj � kFÞ for�!ph ≲ ! and �ðkÞ ’ vFðjkj � kFÞ for !� �!ph. Thetransition from v�F to vF is observed as a kink in the banddispersion.So far, the self-energy model is determined throughexperiments other than the ARPES measurement. Theauthors also introduced a pseudoparabolic electron–electronself-energy and impurity scattering as fitting parameters toreproduce the ARPES spectra of the Be(0001) surface state.While the authors of Ref. 115 utilized several independentexperimental data to obtain a decent self-energy model, theRBM representation is capable to obtain the self-energy byusing a single ω dependence of the spectrum at a givenmomentum. As shown in Fig. 5, the self-energy obtained bythe RBM is consistent with the self-energy model in Ref. 115even though there are noise in the RBM self-energy modeldue to statistical errors in the ARPES measurement.4.1.2 SrVO3Another example is a perovskite vanadium oxide SrVO3,whose spectrum has been experimentally and theoreticallystudied as a typical correlated metal.106,118–121) By assumingthat the (retarded) self-energy satisfies a particle–holesymmetry and supplementing high-energy tails of the spectralfunction, the normal-state self-energy of SrVO3 is extractedfrom the ARPES data.120)Photoemission spectroscopy mainly provides us withinformation regarding occupied states, though unoccupiedstates above the Fermi energy can be partially observed atfinite temperatures122–124) or through pump-probe measure-ments.125,126) Therefore, we need to extract the self-energyfrom a spectrum within a finite range of binding energyobserved as Aðk; !expÞ for �!c � !exp ≲ 0.By assuming the particle–hole symmetry and high-energytails of the spectrum, and the particle–hole symmetry of theself-energy as �ð�k;�!Þ ¼ ��ðk; !Þ�, the authors obtainedthe self-energy, as follows. From these assumption, theauthors constructed the spectral function Aðk; !Þ practicallyalong the entire ω axis. The retarded Green function is thenobtained through the Hilbert transformation (or the Kramers–Kronig relation)Gðk; !Þ ¼Z þ1�1Aðk; !0Þ! þ i� � !0d!0: ð54ÞThe inverse of Gðk; !Þ provides us with �ðkÞ � � þ�norðk; !Þ. The extracted self-energy is qualitatively consis-tent with the results of a numerical simulation.106)When these assumptions hold, the normal-state self-energyis reproduced through the above scheme. If the assumptionconcerning the high-energy tails is not reliable, regressionwill be more suitable for extracting the self-energy. Whenwe wish to analyze superconductors, the present schemeprovides only the total self-energy, �norð!Þ þWð!Þ. Thus, aregression scheme based on a flexible self-energy model willbe useful for analyzing the superconducting states.4.1.3 CuprateThe normal-state self-energy in a cuprate superconductorwas also analyzed by assuming fermion-boson couplingmechanism.109) Here, the dispersion along the nodal direction� � �=4 was studied.When the momentum dependence of �ðk; !Þ along thenodal direction is negligible, the spectral function is given by� 1�Im�ð!Þ½! þ � � �ðkÞ � Re�ð!Þ�2 þ Im�ð!Þ2 :If the amplitude of the self-energy is smaller than orcomparable with the bare dispersion, the maximum of thespectral function appears along ! þ � � �ðkÞ � Re�ð!Þ ¼ 0.Therefore, by assuming a bare band dispersion �ðkÞ, thereal part of the self-energy is estimated from the positions ofthe peaks of the spectra in k; ! space, where the peak of thespectrum is determined by the momentum distribution curvesmethod. By assuming the Migdal–Eliashberg formalism, thebosonic density of states was inferred from the obtained realpart of the self-energy.109) However, the momentum de-pendence of the self-energy along the nodal direction hasbeen examined in details and found to be non-negligible.89)4.2 Superconducting state4.2.1 PbThe self-energy analysis by combining the Migdal–Eliashberg–Nambu theory, Eq. (31), and the tunnelingspectra, has succeeded to demonstrate the validity of thephonon mechanism for the low-temperature BCS super-conductors, while it sacrifices many-body effects inherent inthe normal state by assuming the normal state with constantdensity of states. Even though the theoretical frameworkthoroughly depends on the phonon mechanism and thenormal-state density of states often shows anomalies,�ð!Þ2Fð!Þ behind Pð!Þ and Qð!Þ is the extracted.While the phonon density of states is approximated by twoLorentzian distributions in Ref. 72, �ð!Þ2Fð!Þ is iterativelyupdated starting from an initial guess to minimize thedifference between the tunneling spectrum given by thecurrent �ð!Þ2Fð!Þ and that obtained in the tunnelingexperiment of Pb, in Ref. 74. Finally, the authors of 0 1 2 3-1 -0.75 -0.5 -0.25  0-0.2-0.1 0 0.1-0.2 -0.1  0Be at kFRBM model(b)(a)RBM modelFig. 5. (Color online) Self-energy models for Be(0001) surface state. Theleft panel shows the ARPES spectrum at the Fermi momentum kF (red opensquares) with the spectral function generated by the RBM self-energy model(blue crosses). The right panel illustrates the self-energy models. Here, theMigdal–Eliashberg self-energy model represented by solid curves shows theimaginary part of �ph as a piecewise function of ω. While the impurityscattering term �1=  is introduced, the small pseudoparabolic term inRef. 115 is neglected. For j!j < !ph, Im�ph is proportional to !2 whileIm�ph is constant for j!j > !ph. A peak in Re�ph for j!j ≲ !ph generates akink in the quasiparticle dispersion. The RBM self-energy model (blue andred filled squares) is consistent with the Migdal–Eliashberg self-energymodel.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-11 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25Ref. 74 obtained �ð!Þ2Fð!Þ from the tunneling spectrum ofPb, as shown in the bottom panel of Fig. 6.The detailed structures in the density of states Nð!Þ of Pbby superconductor-insulator-normal tunnelings128) is alsoobserved by angle-integrated photoemission spectroscopymeasurements.129)4.2.2 Sn and othersBy using the same formalism, the phonon density of statesof β-Sn or white tin has been studied75) as a weaker couplingsystem. The extracted phonon density of states is quantita-tively consistent with the phonon dispersion observed ininelastic neutron scatterings.130,131) Other examples of similaranalyses are found in an extensive review article.132)4.2.3 CuprateWhile the total self-energy of the cuprates has been knownfor more than two decades,133) the decomposition of thenormal and anomalous components of the self-energy wasperformed within a decade. Bok et al. applied the extensionof the Migdal–Eliashberg–Nambu theory76) to the ARPESspectra around the nodal direction.102) The validity of theself-energy analysis is limited by the usage of Eq. (31),which is hardly justified around the antinodal region (� � 0in Fig. 2). Later, Li et al. introduced phenomenological self-energy models to analyze the entire Brillouin zone of thecuprate,134) where the distribution function for the anomalouscomponent, Qð!Þ, is assumed to be finite in an extendedenergy scale. To remove constraints on these self-energymodels, the Boltzmann machines were used to represent Pand Q93) as reviewed in Sect. 3.2.At a given momentum around the antinodal region, whichis a Fermi momentum at higher temperatures than Tc, theself-energy was extracted from the energy dependence of theARPES spectrum of a category of cuprates. As shown inFig. 7, the prominent peak structures inQ and Im�ano, whichgenerate a significant superconducting gap, is compensatedby the peak structures in ImW. Then, the total self-energydoes not show such prominent anomalies.The compensation of the peak structures in the self-energyis an unexpected from the phonon mechanism, which sets aconstraint on the mechanism of the high-temperature super-conductivity of the cuprates. In addition, it is revealed thatthe incoherence of the one-particle excitations, quantitativelyestimated from the self-energy, enhances the superconductingcritical temperatures,93) which is seemingly counterintuitive.5. Summary and Future PerspectivesTo understand effects of mutual interactions in quantummany-body systems, self-energy is an important physicalquantity, which is unfortunately not an observable inexperiments. The self-energy has been indeed crucial totheoretically understand the origin of the strong-couplingBardeen–Cooper–Schrieffer (BCS) superconductivity. Bysolving inverse problems formulated with a sophisticatedmany-body perturbation theory, the self-energy was extractedfrom spectroscopy data of Pb and other strong coupling BCSsuperconductors.Alongside the development of theoretical tools andspectroscopy measurements, the rise of machine learninghas inspired researchers to formulate the inverse problemfrom a broader perspective. While there are various machine-learning-inspired approaches to analysis of spectroscopydata, we focused on photoemission spectroscopy measure-ments and reviewed the self-energy analysis by using theBoltzmann machine.The self-energy analysis of superconductors is distilledto optimization of two distribution functions, Pðk; !Þ andQðk; !Þ, which generate the normal and superconductingcomponents of the self-energy through a transformation[Eqs. (28) and (29)]. Then, we can employ various descriptionof Pðk; !Þ and Qðk; !Þ. The machine learning approachesprovide us with a variety of models for complicatedprobability distributions and, thus, a flexible description ofthe self-energy. As an example of these approaches, theBoltzmann machine representation of the self-energy and itsapplication to the copper oxide high-temperature super-conductors are reviewed in the present article.Even though the machine learning approaches are flexible,there remain several crucial problems. While the Boltzmannmachines are simple and sallow neural networks, theoptimization is not straightforward. The limitation ofavailable experimental data also remains a crucial problem.While spectra in a wide energy range are available,momentum dependence is not accessible in the tunnelingspectroscopy. In contrast, the angle-resolved photoemissionspectroscopy (ARPES) provides us with momentum de-pendence of one-particle spectra while only spectra for ωsmaller than the Fermi level are available. The validation ofFig. 6. Analysis of tunneling spectrum of Pb. Curve A represents thevoltage derivative of the ratio of tunneling conductance in the super-conducting state and normal state, in units meV−1 as a function of V � 2�0,where �0 is defined in Appendix B. Curve B is Nsð!Þ=Nnð!Þ as a functionof ! ��0 while Curve C is the dimensionless phonon density of states,�ð!Þ2Fð!Þ, as a function of ω. Reprinted figure with permission fromRef. 74. © 1965 American Physical Society.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-12 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25the obtained self-energy remains an important issue as well.Though efforts have been made to validate the self-energyobtained from the spectroscopy data (as in Appendix ofRef. 93), there could be local minima in the optimizationproblems, due to the lack of information.Therefore, we need to explore further possibilities toaugment the machine-learning approaches. A possibleapproach will be utilization of other experimental data orsimulations as prior knowledge. The momentum dependencecould be extracted from interference in scanning tunnelingmicroscope135–139) although it requires theoretical tools. Tosupplement the photoemission spectra, inverse photoemis-sion would be useful, though the resolution of the inversephotoemission is lower than the photoemission so far.Combined one- and two-particle excitations observed inresonant inelastic x-ray scattering140–142) will provide us withfurther information.Acknowledgments The author thanks Teppei Yoshida, Atsushi Fujimori,and Masatoshi Imada for collaboration and ongoing discussions on spectroscopy.The author also thanks Shiro Sakai, Takeshi Kondo, Yuhki Kohsaka, and TetsuoHanaguri for insightful discussions. The numerical data shown in Fig. 1 wasobtained by an in-house extension of mVMC143) developed by MaximeCharlebois, which is detailed in Ref. 32. We acknowledge the financial supportof JSPS Kakenhi Grant No. 23H04524. This work used computational resourcesof supercomputer Fugaku provided by R-CCS through the HPCI System ResearchProject (Project ID: hp230169). This study was also supported by MEXT as aprogram for promoting researches on the supercomputer Fugaku [AI NumericalSpectroscopy for Analyzing Emergent Structures of Quantum Entanglement inCorrelated Quantum Materials (JPMXP1020230410)] and used computationalresources of supercomputer Fugaku provided by R-CCS (Project ID: hp230213).MANA is supported by World Premier International Research Center Initiative(WPI), MEXT, Japan.Appendix A: Two-particle Green FunctionsA typical example of two-particle Green functions is adynamical spin susceptibility defined as�zzðq; !Þ ¼ h�0jŜzðqÞy 1! þ i� � Ĥ þ E0ŜzðqÞj�0i;which describes spin excitations generated by operating aspin operator,ŜzðqÞ ¼ ð1=2ÞXkðĉykþq"ĉk" � ĉykþq#ĉk#Þ;to the ground state j�0i. Similarly to the one-particle Greenfunctions, the imaginary part of the dynamical susceptibilityprovides dynamical structure factor as,Szzðq; !Þ ¼ � 1�Im �zzðq; !Þ:To extract a two-particle self-energy, we need to introduce aproper single-particle dispersion as a reference. The single-particle dispersion will be given by a dispersion relation of,for example, magnons144) or triplons.145) Even in the simpleHeisenberg model on a square lattice, the self-energy isevident as deviation from the magnon dispersion obtained bythe linear spin wave theory.146)Appendix B: Migdal–Eliashberg–Nambu TheoryThe Migdal–Eliashberg–Nambu theory47,70,71) provides ussimple formulae to model the two components, �ð!Þ andZð!Þ, by introducing an effective phonon density of states,�ð!Þ2Fð!Þ as explained below. The normal and anomalouscomponents of the self-energy are given by�norð!Þ ¼ ½1 � Zð!Þ�!¼Z 1�0d!0 Re!0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!02 � �ð!0Þ2 þ i�p" #K�ð!;!0Þþ �ð0Þð!Þ; ðB:1Þ�anoð!Þ ¼ Zð!Þ�ð!Þ¼Z !c�0d!0 Re�ð!0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!02 ��ð!0Þ2 þ i�p" #½Kþð!;!0Þ� Nnð0ÞUC�; ðB:2Þ-0.8-0.6-0.4-0.2 0 0.2 0.4-0.3 -0.2 -0.1  0  0.1-0.6-0.4-0.2 0 0.2 0.4-0.2 -0.15 -0.1 -0.05  0(c)-0.8(b) 0 0.5 1 1.5 2-0.3 -0.2 -0.1  0  0.1(a)Fig. 7. (Color online) Self-energy of a cuprate represented by theBoltzmann machines.93) (a) An experimental ARPES spectrum Aexpð!Þ ofBi2Sr2CaCuO8þ� (Tc ¼ 90K) at a Fermi momentum (red open squares)127) iscompared with the spectral function given by the Boltzmann-machine self-energy shown in (b). (b) The real and imaginary parts of normal andanomalous components of the self-energy are shown. (c) The total self-energy is decomposed into the normal component and W, where the peakstructure in the normal component is compensated by the peak in ImWð!Þ.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-13 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 02/25/25where �0 ¼ Re�ð�0Þ, !c is the cutoff frequency, Nnð0ÞUCis a Coulomb pseudopotential,147,148) and η is a positivebroadening factor. Here, the kernel functions K  are definedasK ð!;!0Þ ¼Zd� �ð�Þ2Fð�Þ� 1!0 þ ! þ � þ i�  1!0 � ! þ � � i�� �; ðB:3Þwhere Fð�Þ is the phonon density of states and �ð�Þ is theeffective electron–phonon coupling.74) When we take Fð�Þ ¼�ð� ��Þ and �ð�Þ2 ¼ Nnð0Þg2el­ph, the kernel functionsrepresent the Einstein model with the Einstein phononfrequency Ω and the dressed electron–phonon couplingconstant gel­ph.Appendix C: Vekhter–Varma ExtensionHere, we briefly explain the Vekhter–Varma extension ofthe Migdal–Eliashberg–Nambu theory and the derivation ofEq. (38). First, the 11 component of the matrix representationEq. (25) is rewritten as a function of �k?, θ, and ω asG11ð�k?; �; !Þ¼ Zð�k?; �; !Þ! þ �ð�k?; �ÞZð�k?; �; !Þ2!2 � �ð�k?Þ2 � �ð�k?; �; !Þ2¼ 1Zð�k?; �; !Þ! þ �ð�k?; �; !Þ!2 � �ð�k?; �; !Þ2 � �ð�k?; �; !Þ2;ðC:1Þwhere � ¼ �=Z and � ¼ �=Z.Then, we integrate the Green function along �k? asZG11ð�k?; �; !Þ d� k? ¼ fð�; !Þ � i�Nsð�; !Þ; ðC:2Þwhere fð�; !Þ and ��Nsð�; !Þ are the real and imaginary partof the integration, respectively.When Zðk?; �; !Þ and �ðk?; �; !Þ are k? independent, theintegral in the right hand side of Eq. (C·2) is transformed asZG11ð�k?; �; !Þ d� k? ¼ZG11ð�k?; �; !Þ Zð�; !Þvð�; �; !Þ d�;ðC:3Þwhere the velocity vð�; �; !Þ is given byvð�; �; !Þ ¼ @�@�k?�����ð�k?Þ=Z¼�: ðC:4ÞThen, the integration Eq. (C·3) is transformed asZG11ð�k?; �; !Þ Zð�; !Þvð�; �; !Þ d�¼Z! þ �!2 ��ð�; !Þ2 � �2d�vð�; �; !Þ : ðC:5ÞOnly when there is a �cutoff that satisfies vð�; �; !Þ ’ vFð�Þ(vF 2 R) for j�j < j�cutoff=Zj and maxðj!2 ��2j; !2Þ �j�cutoff=Zj2, the right hand side of Eq. 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Felser,B. A. Bernevig, and N. Regnault, Science 376, eabg9094 (2022).154) Web [https://www.cryst.ehu.es/].Youhei Yamaji was born in Tokyo, Japan, in 1981.He received D.Eng. in 2010 from The University ofTokyo. He was a postdoctoral researcher at Rutgers(2010), a specially appointed research associate(2011–2014), a specially appointed lecturer (2014–2016), a specially appointed associate professor(2016–2021) at The University of Tokyo, and asenior researcher (2021–2023) at National Institutefor Materials Science. He is now a group leader atResearch Center for Materials Nanoarchitectonics,National Institute for Materials Science. He has been working on condensedmatter theory and computational physics.J. Phys. Soc. Jpn. 94, 031005 (2025) Special Topics Y. Yamaji031005-16 ©2025 The Physical Society of Japan©2025 The Author(s)J. Phys. Soc. 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