Science
The physics behind materials intelligence.
The Modulated Electron Lattice () is the physics framework behind SuperMatics. It classifies the broken-symmetry states of high- [2] and other strongly correlated systems (, , , orbital order)[6] in a representation a generative model can actually search.
This page is the short technical pass: the framework, its core free-energy functional, the figures and interactive demos that make it measurable, a benchmark table, and the published references. For the full development, see the MEL paper.
Act IThe problem
Modern materials AI is built on a method that fails exactly where the breakthroughs are. Closing that gap is the entire reason MEL exists.
01 · The bottleneck
DFT works until correlation dominates.
Every modern materials-discovery pipeline is anchored on density functional theory, and is excellent for the weakly correlated half of the periodic table. It fails, visibly and systematically, at the materials that matter most to superconductors, batteries, magnets, and thermoelectrics: [3], unconventional pairing states[4], and intertwined-order systems[6] where electron–electron interactions break the independent-particle picture.
Where the standard stack fails
- Mott insulatorsDFT predicts metallic
- Unconventional SCNo pairing mechanism
- Stripe / CDW orderWrong Q* and amplitude
- Intertwined ordersSingle-OP framing inadequate
ML models trained on data inherit the same blind spot[14]. The search space is too large to brute-force without a prior, and the prior has to come from the physics.
Why AI alone misses the materials that matter.
Standard materials AI is trained on first-principles simulations (DFT), which silently fail on exactly the materials that matter most: high-Tc superconductors and the correlated quantum systems around them. MEL is the physics prior that fixes the search.
01
Representation
Charge density · single-particle bands
✓ supported
Broken-symmetry state fields, built to describe high-Tc superconductors directly
✓ supported
02
Correlated regime
Silently fails on quantum-regime materials, the exact systems where breakthroughs happen
✗ blind spot
Built for it. The framework starts there.
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03
Generative search
Brute-force over composition; no prior
✗ blind spot
Candidates are physically valid by construction. No impossible chemistry, no wasted compute.
✓ supported
04
Experimental closure
Predictions with no measurable experimental handle. No fast way to confirm.
✗ blind spot
Every prediction comes with a specific lab measurement that confirms or refutes it
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05
Time to candidate
Weeks of simulation per material
✗ blind spot
Hundreds per day, prioritized by predicted Tc
✓ supported
Act IIThe framework
MEL turns the tangled physics of correlated electrons into a vocabulary, and then a functional, that a generative model can actually search.
02 · The vocabulary
Correlated systems share a vocabulary.
Below their transition temperatures, electrons in correlated materials organize into ordered states, most of them spatially modulated. MEL labels each one by the symmetry it breaks and the ordering wavevector at which it modulates. Four canonical primitives generate most of the interesting phenomenology.
Each channel carries its conventional order-parameter symbol: ψ for the charge density wave, M_Q for the spin density wave, Δ for the superconducting pair density wave[8], and φ for orbital order. Each labels the order-parameter strength and the symmetry its state breaks.
ψ_CDW
Charge density wave
Periodic modulation of electron density at Q*.
M_Q (SDW)
Spin density wave
Spatially modulated magnetic moment.
Δ_PDW
Pair density wave
Cooper-pair condensate modulated at finite Q.
φ_orb
Orbital order
Symmetry-broken occupation of d / f orbitals.
All four primitives can coexist and intertwine in the same material[6, 7]. This is what motivates the multi-order-parameter formalism that follows.
03 · The hinge
Q* is one observable in two spaces.
The wavevector is the single thread that ties the broken-symmetry state in real space to the Fermi-surface geometry in reciprocal space. A modulation of period λ in the density n(r) shows up as a peak at Q* = 2π/λ in its Fourier transform. That peak sits at the wavevector connecting the Fermi-surface segments the order gaps out, the nesting condition that sets where the instability forms. In the cuprates this charge order is measured directly at Q* ≈ 0.31 r.l.u.[9, 10].
Q* · the hinge between real and reciprocal space
Q* ≈ 0.31 r.l.u. (2π/a)
04 · The formalism
Generalized , by construction.
Each order parameter is a complex field ψ = |ψ| e^(iφ) living on a sombrero potential. The broken-symmetry state lies on the valley, a continuous manifold of degenerate minima parameterized by phase φ. MEL enumerates the symmetry-allowed fields for a given crystal, writes down the symmetry-allowed couplings between them[6], and assembles a full free-energy functional.
α (coefficient)
−0.42
β (coefficient)
+1.00
Tc onset
92 K
|Q*|
0.31 r.l.u.
Free-energy functional
The same model, across families.
One classifier resolves the order-parameter content and ordering wavevector for cuprates, nickelates, stripe-phase systems, and generative candidates, and exposes the coupling that sets Tc.
Classification
Bi₂Sr₂CaCu₂O₈₊δ
Cu-based · d-wave
Order parameters
Predicted Tc
92K
Measured Tc
91K
Charge order in the pseudogap regime couples to the superconducting dome through a PDW-mediated channel.
Act IIIMade measurable
A framework is only science if a lab can check it. Every order parameter maps to a probe, and every prediction to a measurement.
05 · Experimental signatures
Designed to be measured.
MEL order parameters are defined to be directly extractable from the probes that resolve them best. The Fourier transform of an topograph picks out at the satellite peaks ()[11, 12]. picks out the charge channel[9]. Inelastic neutron scattering picks out the spin channel[7]. Transport closes the loop with and the phase-boundary line.
- STM / STSreal-space modulation, gap
- FT-STSQ* magnitude and symmetry
- RXScharge-order intensity vs T
- Neutronspin order, magnon dispersion
- TransportTc, phase boundaries
The map MEL gives the platform.
A representative cuprate-class phase diagram across doping and temperature. MEL classifies each region by its dominant order parameters, giving generative models a structured target.
Sample readout
Hover the diagram to inspect a region.
Legend
- AFM· Antiferromagnetic Mott
- PG· Pseudogap
- SC· Superconducting · d-wave
- CDW· Charge density wave
- SM· Strange metal
06 · The evidence
How far should you trust this?
A theory earns confidence in steps: rest on established physics, reproduce what is already known, survive an independent test, and make new predictions a lab can refute. MEL clears each rung, and every claim below traces to a citation or a partner measurement.
- 01peer-reviewed foundation
Grounded in established physics
Landau theory of intertwined orders, dynamical mean-field theory for correlated electrons, and decades of cuprate phenomenology[5, 6, 9].
- 02±1 K · n = 6
Reproduces known superconductors
A held-out back-test: MEL-predicted Tc lands within ±1 K of measurement across the public literature, kagome metals to mercury cuprates.
- 03PRL · 2026
Core mechanism independently confirmed
Stanford and SLAC measured the cooperative charge-order / superconductivity coupling MEL is built around, in Physical Review Letters.
- 04in validation
New predictions, openly falsifiable
Each generated candidate carries a specific measurable signature (Q*, gap, Tc) and is routed to partner labs at UIUC and UC Berkeley for blind confirmation.
Rung 02 · the back-test
MEL-classified Tc predictions on known correlated materials, held out and benchmarked against published measurements, from kagome metals through the bilayer nickelate[13] to the mercury cuprates. New candidate compositions generated inside MEL space are routed through partner labs for experimental confirmation.
Back-test·Predicted vs. measured Tc
n = 6 · max residual 1.0 K
n materials
6
published Tc
Tc range
2.5 – 135 K
kagome → Hg cuprate
Max residual
±1.0 K
largest prediction error
RMS residual
0.71 K
mean |Δ| = 0.52 K
Material families · references
- CsV₃Sb₅·Kagome metal·Ortiz et al., Phys. Rev. Mater. 2019
- Nd₀.₈Sr₀.₂NiO₂·Infinite-layer nickelate·Li et al., Nature 2019
- La₃Ni₂O₇ (high-P)·Bilayer nickelate (high-P)·Sun et al., Nature 2023
- YBa₂Cu₃O₇₋δ·Y-based bilayer cuprate·Wu et al., Phys. Rev. Lett. 1987
- Bi₂Sr₂Ca₂Cu₃O₁₀₊δ·Bi-based multilayer cuprate·Maeda et al., Jpn. J. Appl. Phys. 1988
- HgBa₂Ca₂Cu₃O₈₊δ·Hg-based cuprate·Schilling et al., Nature 1993
Rung 03 · independent confirmation · Lee et al., PRL 2026
The cooperative coupling the platform sorts on.
Old view (competition): charge order suppresses Δ_SC below Tc. Confirmed view: both the charge order ψ_CDW and Δ_SC rise together with locked phases, the cooperative regime MEL describes.
MEL’s coupled functional carries a biquadratic coupling γ |ψCDW|² |ΔSC|² between the charge order and the superconducting condensate. A negative γ makes them cooperative: condensing together lowers the energy, the wavevector locks to the lattice, and is enhanced; a positive γ makes them compete. The measurement places this system on the cooperative branch, consistent with the d-symmetry (B₁g) form factor resolved for cuprate charge order[11]. SuperMatics scores candidates on the normal-state stiffness α(q) whose softening at Q* this result isolates.
Coupling
γ |ψ_CDW|²|Δ_SC|²
Cooperative
γ < 0
Form factor
B₁g (d-symmetry)
Descriptor
α(q) · resolved
Scope & limits
What MEL is, and what it isn’t.
An effective theory, not a first-principles solver
MEL classifies and ranks broken-symmetry order and the couplings between them. It does not compute Tc from the bare Hamiltonian; DMFT triage and experiment fill that in.
Built for correlated families
Cuprates, nickelates, kagome metals, stripe systems. For weakly-correlated materials, where DFT already works, MEL adds nothing.
Calibrated, and experiment is the arbiter
Some couplings are fixed by fits to known materials rather than derived ab initio; predicted Tc carries an uncertainty, and a partner-lab measurement is the final word.
Act IVThe record
Every external claim in one place: the literature it rests on, and the artifacts you can verify yourself.
07 · Further reading
In the literature.
01· MEL paper
2025
A Modulated Electron Lattice (MEL) criterion for metallic superconductivity
Kim, J. et al.
arXiv:2512.03368
02
1986
Possible high-Tc superconductivity in the Ba–La–Cu–O system
Bednorz, J. G. & Müller, K. A.
Z. Phys. B 64, 189
03
2006
Doping a Mott insulator: physics of high-temperature superconductivity
Lee, P. A., Nagaosa, N. & Wen, X.-G.
Rev. Mod. Phys. 78, 17
04
2017
Unconventional superconductivity
Stewart, G. R.
Adv. Phys. 66, 75
05
1996
Dynamical mean-field theory of strongly correlated fermion systems
Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J.
Rev. Mod. Phys. 68, 13
06
2015
Colloquium: Theory of intertwined orders in high temperature superconductors
Fradkin, E., Kivelson, S. A. & Tranquada, J. M.
Rev. Mod. Phys. 87, 457
07
1995
Evidence for stripe correlations of spins and holes in copper-oxide superconductors
Tranquada, J. M. et al.
Nature 375, 561
08
2020
The physics of pair-density waves
Agterberg, D. F. et al.
Annu. Rev. Condens. Matter Phys. 11, 231
09
2016
Resonant X-ray scattering studies of charge order in cuprates
Comin, R. & Damascelli, A.
Annu. Rev. Condens. Matter Phys. 7, 369
10
2012
Long-range incommensurate charge fluctuations in (Y,Nd)Ba₂Cu₃O₆₊ₓ
Ghiringhelli, G. et al.
Science 337, 821
11
2014
Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates
Fujita, K. et al.
Proc. Natl. Acad. Sci. 111, E3026
12
2002
Imaging quasiparticle interference in Bi₂Sr₂CaCu₂O₈₊δ
Hoffman, J. E. et al.
Science 297, 1148
13
2023
Signatures of superconductivity near 80 K in La₃Ni₂O₇ under high pressure
Sun, H. et al.
Nature 621, 493
14
2019
Recent advances and applications of machine learning in solid-state materials science
Schmidt, J., Marques, M. R. G., Botti, S. & Marques, M. A. L.
npj Comput. Mater. 5, 83
Open · verifiable
Every claim on this site checks out externally.
The arXiv papers, the parent organization, the industrial partner, the patent counsel, and the active research collaborations. Each one is verifiable through a public link. Diligence starts here.
- MEL criterion · arXiv:2512.03368
Paper
arXiv permanent identifier
Short-range modulated electron lattice framework for d-wave cuprate superconductivity. Published preprint; full PDF and source.
- MEL metallic criterion · arXiv:2601.14500
Paper
arXiv permanent identifier
Companion paper extending MEL to the unified criterion for metallic superconductivity across families.
- Hyunsung TNC
Organization
Public company site · Korean corporate registry
The Korean materials-science company where the MEL framework was developed over eighteen years. James Kim, Ph.D. serves as CTO.
- CAN Superconductors
Partner
Public company site
Czech industrial HTS manufacturer building MEL-generated high-Tc candidates. Public industrial customer reference.
- Wilson Sonsini Goodrich & Rosati
Counsel
Public law-firm directory · representation confirmable on request
Patent strategy and corporate counsel for SuperMatics. WSGR is the dominant US firm for venture-backed deep-tech IP.
- Madhavan group · UIUC
Collaboration
Public faculty page · UIUC Department of Physics
Active research collaboration. STM/STS and FT-STS measurement on MEL-generated candidates.
- QB3 / Berkeley Nanofabrication Center · UC Berkeley
Collaboration
Public facility page · UC Berkeley
Active research collaboration. Synthesis and fabrication of MEL-generated candidates.
Collaborate
Push the framework further with us.
We’re talking to experimental collaborators, theoretical advisors, and groups extending MEL into new application domains. Reach us at research@supermatics.io.