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SuperMatics

Science

The physics the platform searches inside.

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.

Independently tested · Lee et al., PRL 2026 ↗ · full treatment under Evidence

The Tc record line

A few degrees of critical temperature is the difference between a lab result and a power line.

Forty years of progress. Two physics families. Most of the region above the liquid-nitrogen line has barely been searched. MEL searches it, including the unconventional and high-Tc regimes where conventional AI cannot.

Nb₃Ge23 K
MgB₂39 K
YBCO92 K
BSCCO110 K
Hg-1223135 K
LaH₁₀250 K
RT293 K
LN₂ · 77 K
  • 0 K
  • 50 K
  • 100 K
  • 150 K
  • 200 K
  • 250 K
  • 300 K

MEL hunt range · above LN₂ · novel and unconventional families

Conventional · BCSUnconventional · cuprateConfirmed recordFrontier prize

  1. YBCO·92 K

    YBa₂Cu₃O₇ · 92 K

    The first cuprate to break liquid nitrogen. Workhorse of every commercial HTS magnet today.

  2. BSCCO·110 K

    Bi-2223 · 110 K

    Long-wire HTS used in fault-current limiters and motors. Higher Tc, harder fabrication.

  3. Hg-1223·135 K

    Hg-1223 · 135 K

    Ambient-pressure cuprate record. The highest Tc humanity has confirmed without extreme pressure.

  4. LaH₁₀·250 K

    LaH₁₀ · 250 K · 170 GPa

    The all-time record, but only stable inside a diamond-anvil cell. Not deployable.

  5. RT·293 K

    Room temperature · 293 K

    The reference point. No confirmed material superconducts here at ambient pressure; MRI without liquid helium waits on one that does.

Public physics values · BCS conventional ceiling · cuprate family · ambient-pressure and high-pressure records

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.

Axis
DFT-first pipelines· status quo
MEL platform· SuperMatics

Representation

Charge density · single-particle bands

✓ supported

Broken-symmetry state fields, written for high-Tc superconductors directly

✓ supported

Correlated regime

Silently fails on quantum-regime materials, the exact systems where breakthroughs happen

✗ blind spot

Built for it. The framework starts there.

✓ supported

Generative search

Brute-force over composition; no prior

✗ blind spot

Candidates are physically valid by construction. No impossible chemistry, no wasted compute.

✓ supported

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

✓ supported

Time to candidate

Weeks of simulation per material

✗ blind spot

Hundreds per day, prioritized by predicted Tc

✓ supported

Comparison framing · cuprate, nickelate, kagome families

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.

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.

Q*

ψ_CDW

Charge density wave

Periodic modulation of electron density at Q*.

Q*

M_Q (SDW)

Spin density wave

Spatially modulated magnetic moment.

Q*

Δ_PDW

Pair density wave

Cooper-pair condensate modulated at finite Q.

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.

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)

real space · n(r)
CuO₂ plane · CDW
Q*ΓXMkₓk_y
reciprocal space · ñ(q)
Brillouin zone · Fermi arcs
One observable, two projections. The same Q* sets the spacing of the charge modulation in real space and the satellite location in reciprocal space, and pinpoints which segments of the Fermi surface the order gaps out. Shown at Q* ≈ 0.31 r.l.u. (units of 2π/a), the canonical cuprate charge-order wavevector, a period of ≈ 3.2 lattice constants.

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.

F = 0Re ψF−|ψ₀|+|ψ₀|
cross-section · Im ψ = 0
ψ = 0 (max)φ⟨ψ⟩Re ψIm ψ
perspective · valley of vacua
φ parameterizes the Goldstone mode
F(ψ) = α|ψ|² + β|ψ|⁴α < 0, β > 0

Free-energy functional

F[{ψᵢ}] = ∫ d³r [ Σᵢ ( αᵢ(T)|ψᵢ|² + βᵢ|ψᵢ|⁴ + Kᵢ|∇ψᵢ|² ) + Σᵢⱼ γᵢⱼ|ψᵢ|²|ψⱼ|² ]
Each order parameter ψᵢ carries its own quadratic, quartic, and gradient terms: αᵢ(T) = aᵢ(T − Tc,ᵢ) changes sign at that order’s transition, βᵢ > 0 bounds the energy from below, and Kᵢ penalizes spatial gradients (stiffness). The biquadratic terms γᵢⱼ couple distinct orders, summed over pairs i < j: γᵢⱼ < 0 is cooperative, the orders condense together and Tc is enhanced, while γᵢⱼ > 0 is competitive. Symmetry-allowed trilinear lock-in terms appear when the ordering wavevectors are commensurate. MEL minimizes F over the symmetry-allowed fields for the material’s space group and ordering wavevector.
Demo · MEL classifier

The same model, across every family.

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.

real space · ψ(r)
Q* ≈ 0.25 r.l.u.
Q*Γ
reciprocal space · |ψ̃(q)|²
FT-STS · predicted

Classification

Bi₂Sr₂CaCu₂O₈₊δ

Cu-based · d-wave

Order parameters

Δ_SC (d-wave)ψ_CDWpseudogap

Predicted Tc

92K

Measured Tc

91K

Charge order in the pseudogap regime couples to the superconducting dome through a PDW-mediated channel.

experimentally validated
live · MEL classifier v0.4numerical model · indicative for site demo

Experimental signatures

FT-STS · |g(q,ω)|
Bragg peaks + Q* satellites
+Q*−Q*Γ

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
Phase space

Each order has an address.

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.

cuprate · T vs dopinghover any region
AFMPGSCCDWSM
T
doping →
hover to inspect

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
schematic · ordering wavevectors per phase tracked in MEL spaceregion · —

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.

  1. 01

    Grounded in established physics

    Landau theory of intertwined orders, dynamical mean-field theory for correlated electrons, and decades of cuprate phenomenology[5, 6, 9].

    peer-reviewed foundation
  2. 02

    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.

    ±1 K · n = 6
  3. 03

    Core mechanism independently confirmed

    Stanford and SLAC measured the cooperative charge-order / superconductivity coupling at the heart of MEL, in Physical Review Letters.

    PRL · 2026
  4. 04

    New predictions, openly falsifiable

    Each generated candidate carries a specific measurable signature (Q*, gap, Tc) and is routed to partner labs at UIUC, UC Berkeley, and Georgia Tech for blind confirmation.

    in validation

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

y = x · perfect prediction02550751001250255075100125Predicted Tc (K) · MEL classifier outputMeasured Tc (K) · published literatureCsV₃Sb₅Nd nickelateLa₃Ni₂O₇YBCOBSCCO-2223Hg-1223

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
Sample shown from continuous back-testing against the public superconductor literature · every measured value is a published Tc · every predicted value is a held-out MEL-classifier output
CompositionFamilyPredicted TcMeasured TcStatus
CsV₃Sb₅kagome metal2.6 K2.5 Kvalidated
Nd₀.₈Sr₀.₂NiO₂infinite-layer nickelate15 K15 Kvalidated
La₃Ni₂O₇ (high-P)bilayer nickelate81 K80 Kvalidated
YBa₂Cu₃O₇₋δYBCO · bilayer cuprate92 K92 Kvalidated
Bi₂Sr₂Ca₂Cu₃O₁₀₊δBSCCO-2223 · multilayer cuprate109 K110 Kvalidated
HgBa₂Ca₂Cu₃O₈₊δHg-1223 · mercury cuprate134 K135 Kvalidated
MEL-cd-237candidate97 K—in validation
MEL-ni-184candidate92 K—in validation
Sample of MEL-classified materials · published benchmarks where available

Rung 03 · independent confirmation · Lee et al., PRL 2026

γ < 0, measured.

The picture, in one chartOrder parameter vs T
Tcψ_CDW · cooperativeΔ_SCold view (competition)T → 0T > Tc|OP|

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.

Further reading

In the literature.

01· MEL paper

2025

Short-range modulated electron lattice and d-wave superconductivity in cuprates: a phenomenological Ginzburg-Landau framework

Kim, J., Rens, D. A., Khalid, W. & Kim, H.

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

15· MEL paper

2026

A Modulated Electron Lattice (MEL) criterion for metallic superconductivity

Kim, J., Rens, D. A., Khalid, W. & Kim, H.

arXiv:2601.14500

16

2026

Cooperative phase coherence of charge order and superconductivity in cuprates

Lee et al.

Phys. Rev. Lett. · DOI 10.1103/g41t-8456

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.

  1. Paper

    arXiv permanent identifier

    MEL framework · arXiv:2512.03368

    The short-range modulated electron lattice framework for d-wave cuprate superconductivity, co-authored by the SuperMatics founding team. Published preprint; full PDF and source.

  2. Paper

    arXiv permanent identifier

    MEL metallic criterion · arXiv:2601.14500

    Companion paper extending MEL to the unified criterion for metallic superconductivity across families.

  3. Paper

    DOI 10.1103/g41t-8456 · APS

    Independent PRL measurement · Lee et al., 2026

    Stanford / SLAC resonant soft x-ray scattering measurement of the cooperative CDW-SC mechanism MEL is built around. Not our paper; that is the point.

  4. Organization

    Public company site · Korean corporate registry

    Hyunsung TNC

    The Korean materials-science company where MEL has been in development since 2006. James Kim, Ph.D. serves as CTO.

  5. Partner

    Public company site

    CAN Superconductors

    Czech industrial HTS manufacturer building MEL-generated high-Tc candidates. Public industrial customer reference.

  6. Partner

    Public company site

    Eloi Materials (EML)

    Industrial supplier of PVD sputtering targets and precursor metal powders for thin-film and bulk synthesis of MEL candidates.

  7. Counsel

    Public law-firm directory · representation confirmable on request

    Wilson Sonsini Goodrich & Rosati

    Patent strategy and corporate counsel for SuperMatics. WSGR is the dominant US firm for venture-backed deep-tech IP.

  8. Collaboration

    Public faculty page · UIUC Department of Physics

    Madhavan group · UIUC

    Active research collaboration. STM/STS and FT-STS measurement on MEL-generated candidates.

  9. Collaboration

    Public department page · Georgia Tech

    Georgia Institute of Technology

    Active research collaboration. A second STM/STS site for spectroscopy on MEL-generated candidates, alongside UIUC.

  10. Collaboration

    Public facility page · UC Berkeley

    QB3 / Berkeley Nanofabrication Center · UC Berkeley

    Active research collaboration. Synthesis and fabrication of MEL-generated candidates.

Anything missing or unverifiable, please flag to contact@supermatics.io.n = 10 artifacts · all externally linked

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.