Working document · theory of high-temperature superconductivity

The Cuprate Problem

Forty years after Bednorz and Müller, the copper oxides still have no agreed theory. This document is not an attempt to supply one. It is an attempt to state precisely what is known, what is merely believed, and which of the remaining questions are small enough to actually answer — so that the answering can start.

Audience field theorist, not a cuprate specialist Sections 13 Questions posed 32 Status living draft

What would count as an answer

Before asking questions it is worth being explicit about what a successful answer looks like, because the field routinely talks past itself on exactly this point.

The standard against which everything gets measured is BCS, so it pays to be precise about what BCS actually achieved. It did not compute transition temperatures from first principles; Eliashberg theory and decades of electron–phonon calculation did that, later. What Bardeen, Cooper and Schrieffer supplied was something logically prior and more valuable: they identified the instability of the normal state, named the order parameter, wrote down a controlled effective description of the ordered phase, and in doing so explained a dozen unrelated-looking experiments at once. The numbers came afterwards because the concepts were right.

So there are four separable deliverables, and conflating them is the single most common source of confusion in this field:

The pairing interaction. What, in the microscopic Hamiltonian, makes two electrons want to occupy a common bound state? This is the question people mean by “the mechanism”, and it is the one closest to settled.
The normal state. What is the metal that becomes unstable? Cooper's calculation presupposes a Fermi liquid with sharp quasiparticles at the Fermi surface. Over most of the cuprate phase diagram there is no such thing. This is the part that is genuinely not understood.
The scale of the transition temperature. Given the interaction and the normal state, what fixes $T_c\simeq 100\,\mathrm{K}$ rather than $1\,\mathrm{K}$ or $1000\,\mathrm{K}$? This is a question about which of several competing energy scales is the bottleneck, and it turns out to have a rather clean partial answer.
The phase diagram. Why does superconductivity sit in a dome, flanked by an antiferromagnet on one side and a Fermi liquid on the other, with charge order, a pseudogap and a strange metal in between?

The thesis of this document

Item 1 is in much better shape than the community's rhetoric suggests. Three independent lines of work — exact algebra on the $t$–$J$ Hamiltonian, scanning-tunnelling measurements that correlate local pair density with the local charge-transfer energy, and ab initio low-energy Hamiltonians solved variationally — now point at the same thing: a local, essentially instantaneous attraction in the nearest-neighbour spin-singlet channel, generated by superexchange, that is to say by the Coulomb repulsion itself.

Items 2 and 3 are where the problem actually lives. A research programme that spends its effort hunting for a better glue is, on this reading, solving a problem that is mostly solved; one that attacks the normal state and the bottleneck on $T_c$ is aimed at the open part. That judgement is contestable and §9 names who contests it — but it is the judgement that organises the question bank in §11.

One more piece of framing. There is no small parameter here. In the cuprates $U/t\approx 8\text{--}12$, the interaction is comparable to the bandwidth, the system is two-dimensional so fluctuations are strong, and several competing ordered states lie within a few per cent of each other in energy. Weak coupling, large $N$, large dimension and semiclassical expansions all fail quantitatively, and they do not fail in a way that can be patched. The consequence for research strategy is sharp: the products that survive are bounds, symmetry statements, exact identities, controlled limits of deliberately modified models, and numerics inside a validated window. Anything that looks like an exact solution of the real problem is either a redefinition of the problem or an error. §12 returns to this as a list of traps.

The energy scales

Almost everything puzzling about the cuprates is visible in a single plot of their energy scales. Get this picture into your hands before anything else.

Below are the characteristic energies of a hole-doped cuprate on a logarithmic axis, in millielectronvolts along the top and in kelvin along the bottom. Both units are worth carrying, because the theory literature speaks in meV and the experimental literature in K, and the factor is $1\,\mathrm{meV}=11.6\,\mathrm{K}$.

Figure 1 · the scale hierarchy Values for an optimally doped single-layer cuprate. Colour marks provenance: verdigris for superconducting scales, indigo for the electronic scales that come out of the chemistry, copper for the anomalous normal-state scale $T^{*}$. The phonon band is drawn in grey because it is a spectator. The bottom rule shows aluminium for calibration.

Four things to extract from it, each of which constrains theory hard.

The superexchange is enormous. $J\approx 130\,\mathrm{meV}\approx 1500\,\mathrm{K}$ is the largest magnetic exchange of any known material class, ten to a hundred times that of a typical magnetic insulator. It is large for a specific and understandable reason: copper in the $3d^{9}$ configuration puts a single hole in a single orbital, $3d_{x^2-y^2}$, whose lobes point directly at the bridging oxygens, and the Cu–O–Cu bond angle is $180^\circ$, which maximises the virtual hopping path. If pairing derives from $J$, a mechanism with a $1500\,\mathrm{K}$ characteristic scale is available — and $T_c/J\approx 0.06$ is then not a surprisingly large number but a rather modest one.

$T_c$ is the smallest scale in the problem by two orders of magnitude. It is $1/17$ of $J$, $1/1000$ of $U$. A theory that gets $T_c$ from a near-cancellation of large quantities is structurally fragile, and almost every mean-field treatment of this problem is exactly that. Conversely, the smallness of $T_c$ relative to everything else is a hint that some other, smaller quantity is the real bottleneck. §7 identifies a candidate.

The gap-to-$T_c$ ratio is wrong for weak coupling. With an antinodal gap $\Delta_0\approx 35\,\mathrm{meV}$ and $T_c=91\,\mathrm{K}$ in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$,

$$\frac{2\Delta_0}{k_B T_c}\;\approx\;9.6 ,\qquad\text{against}\qquad \frac{2\Delta_0}{k_B T_c}\Big|_{\text{weak-coupling }d}=4.28 .$$

The gap is roughly twice too large for its $T_c$, and on the underdoped side it gets worse: the gap grows as $T_c$ falls. In BCS those two quantities are locked together by one equation. Here they are plainly controlled by different physics, which is the first and bluntest piece of evidence for the amplitude-versus-stiffness picture of §7.

The phonons are in the wrong place and of the wrong size. The oxygen stretching modes sit near $60\,\mathrm{meV}$, below $J$ and below the pseudogap scale. This does not prove phonons are irrelevant — they demonstrably renormalise spectra, appear as kinks in ARPES dispersions near $70\,\mathrm{meV}$, and may well assist — but it does mean they cannot be the organising scale, and the small oxygen-isotope exponent near optimal doping (§3, fact 11) closes the door on them as the primary actor.

A caution about “no small parameter”

It is tempting to read $J/t\approx 0.3$ as a small parameter and expand in it. Resist. The doped system has a kinetic scale $\sim t$ per carrier but a magnetic scale $\sim J$ per bond, and at doping $p$ there are $p$ carriers against $2$ bonds per site, so the two terms compete at $p\sim J/t$, which is precisely the doping range of interest. The near-degeneracy of uniform and striped states (§5) is this competition in action. Any expansion in $J/t$ silently picks a winner.

Fourteen facts a theory must fit

These are the observations that have survived thirty-plus years of repetition across many materials and many groups. Each is followed by the constraint it places on theory. Treat the list as a scoring rubric: any candidate mechanism can be graded against it in an afternoon.

First the map. The cuprate phase diagram is drawn in temperature against hole doping $p$, the number of holes added per copper atom, which is the one tuning parameter that does essentially everything.

Figure 2 · the phase diagram Schematic, with scales typical of YBa$_2$Cu$_3$O$_{6+y}$. The superconducting dome runs from $p\approx0.05$ to $0.27$, peaking near $0.16$. $T^{*}$ is a crossover, not a transition, and different probes put it at different temperatures — a fact that is itself evidence (fact 8). The copper wash is the strange-metal region, drawn as a fan emanating from $p^{*}\approx0.19$ where $T^{*}$ extrapolates to zero. Charge order is dashed because it is short-ranged and field-sensitive at zero field.

1 The parent compound is a charge-transfer insulator, not a band insulator. La$_2$CuO$_4$ has an odd number of electrons per unit cell. Band theory therefore predicts a metal; it is in fact an insulator with a $1.5\text{--}2\,\mathrm{eV}$ gap. The gap is not between copper's lower and upper Hubbard bands but between occupied oxygen $2p$ states and the upper Hubbard band, which is why it is called charge-transfer rather than Mott–Hubbard: the cheapest charge excitation moves a hole from Cu to O, costing $\Delta_{\rm CT}$, not from Cu to Cu, costing $U_d\approx9\,\mathrm{eV}$. Constraint Any starting point built on a Fermi sea of non-interacting band electrons begins on the wrong side of a metal–insulator transition. The interaction must be treated non-perturbatively from the outset, and the distinction between $U_d$ and $\Delta_{\rm CT}$ matters because it is $\Delta_{\rm CT}$ that controls $J$.
2 There are well-formed spin-$\tfrac12$ moments with an exceptionally large exchange. Néel order at $T_N\approx325\,\mathrm{K}$ in the undoped material, with $J\approx130\text{--}140\,\mathrm{meV}$ measured independently by two-magnon Raman scattering, inelastic neutron scattering of the spin-wave dispersion, and NMR. The magnon spectrum is quantitatively described by the spin-$\tfrac12$ square-lattice Heisenberg model. Constraint This is the single most successful quantitative prediction in the whole subject, and it validates the chain of reductions in §4 at zero doping. It does not validate them at finite doping, which is the whole difficulty.
3 Doped holes go onto the oxygens, then bind into local singlets. X-ray absorption shows the added holes have predominantly O $2p$ character, not Cu $3d$. Zhang and Rice argued in 1988 that such a hole binds to the existing Cu hole in a local singlet, delocalised over the four oxygens of a CuO$_4$ plaquette, and that this composite object behaves as a single spinless hole moving in the spin background. Constraint This is the load-bearing step that licenses a one-band model. It is an approximation, it has critics, and its validity is directly computable — see Q-A4.
4 A few per cent of holes destroys the magnet; five per cent makes a superconductor. Long-range Néel order is gone by $p\approx0.02\text{--}0.03$. Superconductivity appears at $p\approx0.05$ and disappears by $p\approx0.27$. Constraint Holes are extraordinarily efficient at frustrating antiferromagnetism, which is a statement about the motion of a hole in a spin background, not about magnetism alone. It also means superconductivity does not require static magnetic order — only the short-range singlet correlations that outlive it.
5 The order parameter has $d_{x^2-y^2}$ symmetry. This is proven, not inferred. Phase-sensitive experiments settled it: the corner-SQUID interferometry of Wollman and Van Harlingen, and the tricrystal ring experiments of Tsuei and Kirtley, both detect the sign change of the gap directly by counting flux quanta. Power-law rather than exponential low-temperature penetration depth, specific heat and thermal conductivity independently confirm line nodes. Constraint The gap changes sign on the Fermi surface. No interaction that is attractive and structureless in momentum can do this. The pairing interaction must be repulsive in the $s$-channel and strongly momentum-dependent — equivalently, short-ranged and sign-structured in real space. §6 turns this into a one-line criterion.
6 The pairs are small. In-plane coherence length $\xi_{ab}\approx1.5\text{--}2\,\mathrm{nm}$ against a lattice constant $a=3.8\,\mathrm{\AA}$, so $\xi/a\approx4\text{--}5$ and $k_F\xi\approx10$. In aluminium $k_F\xi\sim10^{4}$ and about $10^{6}$ pairs overlap within a coherence volume; in a cuprate the number is of order ten. Constraint The system sits in the crossover between BCS and Bose–Einstein condensation, closer to BCS but nowhere near the extreme-BCS limit in which mean-field theory is exact. Phase fluctuations of the order parameter are therefore not a small correction — they are a leading effect, which is §7.
7 The superfluid density counts the doped holes, not the electrons. Uemura's muon-spin-rotation scaling: on the underdoped side $T_c\propto n_s/m^{*}$, the zero-temperature superfluid density, and $n_s\propto p$. Optical conductivity says the same thing: the low-frequency Drude weight grows in proportion to $p$, with spectral weight transferred down from the charge-transfer scale as doping proceeds. Constraint This is the operational definition of “Mottness”. The number of low-energy degrees of freedom is $p$, not $1-p$, and it is dynamically generated: spectral weight moves across electronvolt scales as you change a chemical composition. A rigid-band picture cannot produce this, and it is also the direct cause of fact 6 — few carriers means low stiffness means short screening means small pairs.
8 A gap opens above $T_c$, and it is not the superconducting gap. Below a crossover $T^{*}$, which rises to $\sim300\,\mathrm{K}$ at low doping and falls to zero near $p^{*}\approx0.19$, the uniform spin susceptibility and the electronic specific heat are suppressed, the antinodal spectral weight near $(\pi,0)$ is gapped, and ARPES sees a disconnected Fermi arc centred on the nodal direction rather than a closed Fermi surface. Different probes give systematically different $T^{*}$. Constraint Two logically distinct readings, and distinguishing them is arguably the central open question of §8: either $T^{*}$ marks pair formation without phase coherence, in which case the pseudogap is the superconducting gap seen above $T_c$; or it marks a separate instability with its own order parameter. An open Fermi arc also violates Luttinger's theorem unless something restores it — see Q-C5.
9 Charge density wave order is everywhere in the underdoped regime. Resonant X-ray scattering finds incommensurate charge modulations with period roughly four lattice constants in every hole-doped cuprate family examined, strongest near $p\approx1/8$, onsetting near $150\,\mathrm{K}$, growing when superconductivity is suppressed by field and shrinking below $T_c$. In the lanthanum-based compounds the modulation becomes static stripes of charge and spin, and at $p=1/8$ exactly, $T_c$ collapses. Constraint Superconductivity has a near-degenerate competitor, and the competition is decided by tiny energy differences. This is both a problem — no numerical method resolves such differences reliably — and a clue: near-degeneracy between a uniform superfluid and a density-modulated state is the hallmark of frustrated phase separation, and it suggests the two states descend from a common parent (§9, framework 5).
10 At optimal doping the normal state is a strange metal, and it is strange in a very specific way. The in-plane resistivity is linear in temperature from just above $T_c$ to beyond $1000\,\mathrm{K}$ with no change of slope and no saturation. At $1000\,\mathrm{K}$ the sheet resistance per CuO$_2$ layer has reached $\approx h/e^{2}\approx26\,\mathrm{k\Omega}$, the Mott–Ioffe–Regel limit at which the mean free path equals a lattice spacing — and it keeps climbing linearly. The slope corresponds to a scattering rate $\hbar/\tau=\alpha\,k_BT$ with $\alpha\approx1$, the so-called Planckian value, reproduced across several families and, remarkably, across other materials entirely. Constraint This is the hardest single fact in the subject. A quasiparticle whose decay rate equals its own thermal energy is not a quasiparticle; a resistivity past the Mott–Ioffe–Regel limit cannot be a mean free path; and $T$-linearity over three decades of temperature cannot come from a single boson, a Debye scale or a quantum-critical fan of finite width. See Q-C1.
11 The oxygen isotope effect is nearly absent at optimal doping and large when underdoped. The exponent $\alpha=-\,d\ln T_c/d\ln M$ is $\lesssim0.1$ near $p=0.16$, against $0.5$ for a pure phonon mechanism, but rises towards and beyond $0.5$ in strongly underdoped La$_{2-x}$Sr$_x$CuO$_4$. There is also a measurable isotope effect on $T^{*}$ and on the penetration depth. Constraint Phonons do not set $T_c$ at optimal doping. They are not absent either: the doping dependence of $\alpha$ is itself a signal, most naturally read as lattice coupling to whatever sets $T_c$ on the underdoped side — which, by fact 7, is the stiffness.
12 In the underdoped regime, condensation lowers the kinetic energy. The optical conductivity sum rule relates the integrated low-frequency spectral weight to the kinetic energy of the carriers. In BCS, forming the condensate costs kinetic energy and gains more potential energy; the integrated weight decreases below $T_c$. In underdoped cuprates the measurements of Molegraaf and co-workers, and subsequent work, find the opposite sign: the low-frequency weight increases, so condensation is kinetic-energy driven. Constraint A sign, not a number — which makes it unusually valuable. Any mechanism in which a boson is exchanged to produce an attraction in the potential-energy channel predicts the BCS sign. Mechanisms in which pairing allows coherent hole motion that the unpaired state could not support — resonating valence bonds, or Mott-derived “attraction from reduced repulsion” — predict the observed sign. See Q-B4.
13 In high magnetic field the underdoped Fermi surface is small. Quantum oscillations in YBa$_2$Cu$_3$O$_{6.5}$, once superconductivity is suppressed, reveal a pocket enclosing about $2\%$ of the Brillouin zone, consistent with a carrier density of order $p$ rather than the $1+p$ of a large hole-like Fermi surface. The Hall number correspondingly changes from $p$ to $1+p$ as doping crosses $p^{*}$. Constraint Something reconstructs the Fermi surface below $p^{*}$. The conventional candidate is the charge order of fact 9 folding the zone; the exotic candidate is a fractionalised Fermi liquid, in which topological order lets the surface be small without any broken translation symmetry. These make different predictions and the distinction is decidable — Q-C5.
14 $T_c^{\max}$ varies by a factor of four across families, and the variation is systematic. $38\,\mathrm{K}$ in La$_{2-x}$Sr$_x$CuO$_4$, $93\,\mathrm{K}$ in YBa$_2$Cu$_3$O$_{7}$, $134\,\mathrm{K}$ in HgBa$_2$Ca$_2$Cu$_3$O$_{8}$ at ambient pressure and about $164\,\mathrm{K}$ under $30\,\mathrm{GPa}$. Within the families $T_c^{\max}$ rises with the number of CuO$_2$ layers per unit cell up to three and then falls; it correlates with the apical-oxygen distance; and Pavarini and co-workers showed in 2001 that it tracks a band-structure “range parameter” essentially equivalent to $|t'|/t$, the next-nearest-neighbour hopping. Constraint There is a quantitative materials trend to predict, which is a much sharper test than reproducing a dome. A theory that cannot order La, Y, Bi, Tl and Hg correctly is not yet a theory of $T_c$. See Q-D3, Q-D4.

How to use the list

Facts 1–5 constrain the pairing. Facts 6, 7, 11, 12 and 14 constrain the scale of $T_c$. Facts 8, 10 and 13 constrain the normal state. Notice that the three groups are almost disjoint, and that the last group is the one where no framework in §9 scores well. That asymmetry is the argument of this document compressed into a sentence.

From crystal to Hamiltonian

Four reductions take you from $10^{23}$ interacting electrons in a real crystal to a model on a square lattice with three parameters. Each step is defensible and each step throws something away. Knowing what it throws away is how you know which failures to blame on the model and which on the method.

Step 0The real material

A layered perovskite. The electronically active part is a square plane of copper atoms bridged by oxygens at $180^\circ$, with apical oxygens above and below each copper and charge-reservoir layers in between that do nothing but donate holes. Copper is nominally $\mathrm{Cu}^{2+}$, configuration $3d^{9}$: one hole, and the crystal field plus the Jahn–Teller elongation of the apical bonds puts that hole unambiguously in $3d_{x^2-y^2}$.

Why you may proceedThe remaining $3d$ orbitals and all the reservoir states lie $\gtrsim1\,\mathrm{eV}$ away and are inert at the scales of Figure 1. This step is the safest in the chain. The one live worry is the apical oxygen, which modulates $t'$ and is implicated in the materials trend of fact 14 — so it is a spectator for the mechanism but not for $T_c^{\max}$.

keep Cu $3d_{x^2-y^2}$ and O $2p_{x},2p_{y}$; discard everything else

Step 1The three-band model (Emery, 1987)

Three orbitals per unit cell, with hybridisation $t_{pd}\approx1.3\,\mathrm{eV}$, on-site repulsions $U_d\approx9\,\mathrm{eV}$ and $U_p\approx4\,\mathrm{eV}$, and the charge-transfer energy $\Delta_{\rm CT}=\varepsilon_d-\varepsilon_p\approx1.8\,\mathrm{eV}$ separating the copper and oxygen levels. Because $\Delta_{\rm CT}<U_d$ the material is a charge-transfer insulator: this model contains fact 1 rather than assuming it.

What it costsNothing essential yet, but you have paid in tractability: three orbitals with two interaction parameters is already beyond reliable numerics on useful system sizes. This is the most faithful model that anyone can still call simple, and it is where you go when you suspect the oxygens matter — for instance when asking whether the pairing scale tracks $\Delta_{\rm CT}$, which is the observable the 2022 scanning-tunnelling work of O'Mahony, Davis and co-workers measured directly.

bind each doped O hole to its Cu hole in a local singlet

Step 2The Zhang–Rice singlet (1988)

A hole added to the four oxygens of a CuO$_4$ plaquette can be combined into the symmetric ($b_{1g}$) linear combination, which hybridises maximally with the central Cu hole. Of the resulting singlet and triplet, the singlet is pushed down by roughly $8t_{pd}^{2}/\Delta_{\rm CT}\sim1\,\mathrm{eV}$ — a large splitting, and the reason the triplet can be dropped. What remains is a spinless, charge-$+e$ object that hops from plaquette to plaquette.

What it costsThe oxygen degrees of freedom as independent actors. If anything interesting happens in the triplet channel, in two-component Cu–O physics, or through $U_p$, it is now invisible. The approximation is controlled by $t_{pd}/\Delta_{\rm CT}\approx0.7$, which is not small. Its critics — Emery and Reiter historically, and various multi-component proposals since — have a real point, and the overlap of the true cluster ground state with the Zhang–Rice state is a number you can just compute: Q-A4.

one effective orbital per Cu → one band

Step 3The one-band Hubbard model

The destination of the whole chain:

$$H=-\sum_{\langle ij\rangle\sigma}t_{ij}\,c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow},$$

on a square lattice, with $t_{ij}=t\approx0.4\,\mathrm{eV}$ for nearest neighbours, $t_{ij}=t'$ for next-nearest, and $U/t\approx8\text{--}12$. The sign and magnitude of $t'$ is material-specific: $t'/t\approx-0.15$ in La$_2$CuO$_4$, $\approx-0.3$ in YBa$_2$Cu$_3$O$_7$ and $\approx-0.4$ in the Hg and Tl compounds — and it turns out to matter far more than a next-nearest-neighbour correction has any right to.

What it costsLong-range Coulomb interaction, the phonons, the disorder introduced by the dopant atoms, interlayer coupling, and the three-dimensionality that stabilises order at finite temperature. Each of these is implicated in at least one fact from §3, and the dopant disorder in particular is implicated in the strange metal (Q-C2, Q-C3).

project out double occupancy: $U\to\infty$ at fixed $J=4t^{2}/U$

Step 3′The $t$–$J$ model

A strong-coupling rewriting rather than a further reduction. Eliminating virtual double occupancy to second order gives

$$H_{tJ}=-t\!\!\sum_{\langle ij\rangle\sigma}\!\mathcal{P}\,c^{\dagger}_{i\sigma}c_{j\sigma}\,\mathcal{P}\;+\;J\!\!\sum_{\langle ij\rangle}\!\Big(\mathbf{S}_i\cdot\mathbf{S}_j-\tfrac14 n_in_j\Big),\qquad J=\frac{4t^{2}}{U},$$

where $\mathcal{P}$ projects out doubly occupied sites. The projector is the whole difficulty: it is a hard local constraint, not a term in a Hamiltonian, and it is what makes this a gauge theory in disguise (§9, framework 3). With $t=0.4\,\mathrm{eV}$ and $U=3.2\,\mathrm{eV}$ one gets $J\approx130\,\mathrm{meV}$, which is fact 2 — so the chain closes consistently at zero doping.

What it costsCharge fluctuations at energies above $J$, and with them any physics in which virtual double occupancy does real work. It also costs a three-site hopping term that is routinely dropped and occasionally matters. In exchange you get a model in which the pairing interaction is explicit, which is the subject of §6.

The payoff of keeping track

When a calculation on the $t$–$J$ model fails to reproduce something, the chain tells you the four places to look: the method, the projector, the one-band reduction, or the discarded long-range and lattice physics. Most disagreements in the literature are arguments about which of these four is responsible, conducted without saying so. Naming the suspect explicitly is cheap and it resolves a surprising number of apparent contradictions.

What we actually know about the models

There is a persistent impression that the Hubbard model is an unsolved mystery. That impression is about fifteen years out of date, and correcting it changes what there is to work on.

The rigorous results, and how little they cover

Start with what is proven, because the list is short and it is useful to know where the solid ground ends. Lieb's theorem fixes the total spin of the half-filled Hubbard ground state on a bipartite lattice. Mermin and Wagner forbid spontaneously broken continuous symmetry at any finite temperature in two dimensions, so neither antiferromagnetism nor true superconducting long-range order exists at $T>0$ in a strictly two-dimensional cuprate — what exists is Kosterlitz–Thouless quasi-order, promoted to genuine order only by the weak interlayer coupling. Nagaoka's theorem says that at $U=\infty$ with exactly one hole the ground state is a saturated ferromagnet, a beautiful result about a thermodynamically irrelevant doping. Dyson, Lieb and Simon established Néel order in the ground state of the Heisenberg antiferromagnet for spin $\ge1$; for spin $\tfrac12$ on the square lattice it is numerically beyond doubt and still not proven.

Notice what is missing. There is no proof that the two-dimensional Hubbard model superconducts, at any $U$, at any doping. Nor is one likely, nor would it be especially illuminating. The useful knowledge in this area is numerical, and the honest question is what the numerics has earned the right to assert.

The sign problem, stated precisely

Determinantal quantum Monte Carlo writes the partition function as a sum over auxiliary-field configurations of a product of two fermion determinants, one per spin. At half filling on a bipartite lattice, particle–hole symmetry makes the two determinants complex conjugates, the product is non-negative, and the method is exact up to statistics. Dope away from half filling and the symmetry is gone: configurations acquire both signs, the average sign $\langle s\rangle$ decays as $e^{-\beta N\Delta f}$ with system size $N$ and inverse temperature $\beta$, and the variance of any estimator blows up as $\langle s\rangle^{-2}$. The computational cost to reach a given temperature therefore grows exponentially. Troyer and Wiese showed that solving the sign problem in general is NP-hard, which means it will not be solved by being clever about the algorithm.

Why this matters for reading the literature

Every numerical claim about the doped Hubbard model at low temperature comes from a method that trades the sign problem for a different, uncontrolled approximation: a finite cluster embedded in a mean field (dynamical cluster approximation), a cylinder of finite width with a matrix-product ansatz (DMRG), a constrained random walk with a trial-wavefunction bias (auxiliary-field QMC), or a variational state. None is exact in two dimensions at low temperature. What makes the modern results credible is not any single method but the agreement of four or five methods with unrelated biases. When you read a result, the first question is how many independent methods saw it.

The three results worth internalising

At half filling, the chain of §4 works. The Hubbard model is an antiferromagnetic insulator for all $U>0$ in two dimensions at $T=0$, and at large $U$ its magnon spectrum and exchange constant match the real material. This is the quantitative success that licences everything else.

The “plain” Hubbard model does not superconduct — it makes stripes. In 2017 Zheng, Chung, Corboz, White, Zhang, Chan and co-workers pooled auxiliary-field QMC, DMRG, infinite projected entangled pair states and density-matrix embedding on the $U=8t$, $t'=0$ model at $p=1/8$. All methods agreed on a striped ground state with a filling near one hole per two sites of stripe length, and on the near-absence of superconducting order. For twenty years this had been the field's default model; the answer turned out to be charge order, not pairing.

Switch on $t'$ and superconductivity appears. In 2024 Xu, Chung, Qin, Schollwöck, White and Zhang combined DMRG with auxiliary-field QMC on the $t$–$t'$–Hubbard model and found robust $d$-wave pairing on the hole-doped side for the physical sign of $t'$, coexisting with partially filled stripes of filling between $0.6$ and $0.8$ — and weaker pairing with Néel correlations on the electron-doped side, matching the observed electron–hole asymmetry of the real phase diagram. Independently, thermal tensor-network calculations on $t$–$t'$–$J$ cylinders produce a dome-shaped pairing region with a characteristic scale up to roughly $0.1\,J$, together with a suppressed spin susceptibility and a node/antinode dichotomy above it that looks like the pseudogap.

The consequence, which is large

A one-band Hubbard model with a realistic next-nearest-neighbour hopping reproduces, without tuning: $d_{x^2-y^2}$ pairing; a dome in doping; electron–hole asymmetry; coexistence and competition with partially filled stripes; a pairing scale of order $0.1J\approx150\,\mathrm{K}$, which is the observed order of magnitude; and a node/antinode dichotomy above the pairing scale. The model is, to a good approximation, right.

This reframes the problem completely. We are not looking for the Hamiltonian. We are looking for an understanding of a Hamiltonian we can already simulate better than we can explain — and for the reason why a seemingly innocuous parameter $t'$ acts as an on/off switch for superconductivity. That second question is concrete, narrow, and almost certainly answerable. It is Q-B3 and Q-D3.

Two caveats, both important. The cylinder geometries that DMRG needs are quasi-one-dimensional, and quasi-one-dimensional systems over-reward stripe formation and can distort the competition; the widths reached ($W\le6$) are not yet two-dimensional. And all of these methods are resolving energy differences between competing states that amount to a few thousandths of $t$ per site, which is at or below the systematic error of every method involved. The qualitative conclusions are robust because many methods agree; the quantitative phase boundaries are not yet.

Why repulsion gives $d$-wave

This is the part of the problem that is understood, and it is worth understanding properly, because it is both beautiful and short. Two independent derivations arrive at the same place from opposite ends of the coupling strength.

The weak-coupling route: a sign-changing gap can exploit repulsion

Linearise the gap equation just below the instability. With a static interaction $V(\mathbf{k}-\mathbf{k}')$ in the singlet channel and quasiparticle energies $\xi_\mathbf{k}$,

$$\Delta_\mathbf{k}=-\sum_{\mathbf{k}'}V(\mathbf{k}-\mathbf{k}')\,\frac{\tanh\!\big(\xi_{\mathbf{k}'}/2T\big)}{2\xi_{\mathbf{k}'}}\,\Delta_{\mathbf{k}'}.$$

The kernel on the right is positive, so if $V>0$ everywhere — purely repulsive — a gap of constant sign is impossible: the right-hand side would be negative where $\Delta$ is positive. This is the textbook reason repulsion does not pair, and it is only true for a gap of constant sign.

Now suppose $V(\mathbf{q})$ is sharply peaked at one wave vector $\mathbf{Q}$. Near-antiferromagnetic cuprates supply exactly that, with $\mathbf{Q}=(\pi,\pi)$, because the spin susceptibility $\chi(\mathbf{q},\omega)$ inherits the ordering wave vector of the parent antiferromagnet and the effective interaction in the singlet channel is proportional to it. Then the dominant contribution to the sum comes from $\mathbf{k}'\approx\mathbf{k}+\mathbf{Q}$, and the equation can be satisfied with a positive eigenvalue provided

$$\Delta_{\mathbf{k}+\mathbf{Q}}=-\Delta_{\mathbf{k}}\qquad\text{for }\mathbf{k}\text{ on the Fermi surface.}$$

The two minus signs cancel and repulsion becomes pairing. So the question “what is the symmetry of the gap?” reduces to “what function on the Brillouin zone is odd under translation by $\mathbf{Q}$?” On the square lattice the answer is immediate:

$$\Delta_\mathbf{k}=\frac{\Delta_0}{2}\big(\cos k_x-\cos k_y\big)\;\Longrightarrow\;\Delta_{\mathbf{k}+(\pi,\pi)}=\frac{\Delta_0}{2}\big(\!-\cos k_x+\cos k_y\big)=-\Delta_\mathbf{k},$$

exactly and for every $\mathbf{k}$, not just approximately. The $d_{x^2-y^2}$ gap is the natural eigenfunction of an antiferromagnetic kernel on a square lattice. Given facts 2 and 5 of §3, the observed pairing symmetry is essentially forced.

Figure 3 · the pairing criterion, drawn Panel a: the $d_{x^2-y^2}$ gap is positive along one axis and negative along the other, vanishing on the diagonals. The overall sign is arbitrary; only the change of sign is physical. Panel b: translating an antinodal point by $\mathbf{Q}=(\pi,\pi)$ lands on the other antinode, where the gap has the opposite sign. The nesting is imperfect, and $t'$ controls how imperfect — which is one concrete reason $t'$ matters so much (§5).

The strong-coupling route: the attraction is already in the Hamiltonian

Now the same conclusion with no boson exchange at all. Define the spin-singlet pair operator on the bond $\langle ij\rangle$,

$$\Delta^{\dagger}_{ij}=\tfrac{1}{\sqrt2}\big(c^{\dagger}_{i\uparrow}c^{\dagger}_{j\downarrow}-c^{\dagger}_{i\downarrow}c^{\dagger}_{j\uparrow}\big).$$

Within the singly-occupied subspace that the projector $\mathcal{P}$ enforces, one verifies directly — check it on the two-site singlet and triplet, it takes two lines — the identity

$$\mathbf{S}_i\cdot\mathbf{S}_j-\tfrac14 n_i n_j\;=\;-\,\Delta^{\dagger}_{ij}\Delta_{ij}.$$

The singlet has $\mathbf{S}_i\cdot\mathbf{S}_j=-\tfrac34$, $n_in_j/4=\tfrac14$ and $\Delta^{\dagger}\Delta=1$, giving $-1$ on both sides; the triplet has $+\tfrac14-\tfrac14=0$ and is annihilated by $\Delta$. Substituting into the $t$–$J$ Hamiltonian of §4:

$$H_{tJ}=-t\!\!\sum_{\langle ij\rangle\sigma}\!\mathcal{P}c^{\dagger}_{i\sigma}c_{j\sigma}\mathcal{P}\;-\;J\!\!\sum_{\langle ij\rangle}\Delta^{\dagger}_{ij}\Delta_{ij}.$$

Read that equation again

The superexchange term is an attractive interaction in the nearest-neighbour spin-singlet channel, with coupling constant $J$, sitting in the bare Hamiltonian. It is instantaneous. Nothing is exchanged. No retardation, no Migdal theorem, no $\alpha^{2}F(\omega)$, no cutoff at a boson frequency. The thing that is usually called the glue is, in this language, a rewriting of the Coulomb repulsion itself: $J=4t^{2}/U$ is large because $U$ is large, in the sense that both derive from the same matrix elements.

This is Anderson's point in his 2007 paper asking whether there is a glue in the cuprates at all, and it is why the “mechanism” debate has been so persistently confusing: two camps were using a word that means different things in weak and strong coupling.

Why does this bond attraction produce $d$-wave rather than $s$-wave order? The identity alone does not decide it; both a uniform ($\langle\Delta_x\rangle=\langle\Delta_y\rangle$) and a staggered ($\langle\Delta_x\rangle=-\langle\Delta_y\rangle$) condensate of bond singlets are allowed by the algebra. The selection comes from the kinetic term and the constraint, and it is established variationally and numerically rather than by a slick argument: Gutzwiller-projected $d$-wave states have markedly lower energy than $s$-wave ones away from half filling, and every unbiased numerical method that finds pairing finds it in the $d$ channel. At half filling exactly, the $d$-wave and staggered-flux states are related by an SU(2) gauge transformation and are physically the same state — an elegant fact and a warning that mean-field labels in this problem are not always gauge-invariant.

Is the glue question decidable? Yes, and partly decided

The two routes above are not obviously the same mechanism, and whether they are is a sharp question rather than a semantic one. The way to settle it is to compute the frequency-resolved pairing vertex in the $d$-channel — which Maier, Poilblanc and Scalapino did within the dynamical cluster approximation — and ask what fraction of the pairing strength comes from frequencies below $J$. Their answer: the spectrum of the pairing interaction does look like the spin susceptibility, and most of the weight sits below roughly $2J$. So a retarded-boson language is not wrong; it is a reorganisation in which the boson is a collective mode of the same electrons that are pairing, with a characteristic scale set by $J$.

Three independent lines have since converged on the same physical picture, which is the strongest claim this document makes:

Algebra. The $t$–$J$ identity above: a bond-singlet attraction of strength $J$, derived from $U$.
Experiment. O'Mahony, Davis and co-workers combined single-electron and electron-pair (Josephson) scanning tunnelling microscopy on Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ and found that the local pair density tracks the local charge-transfer energy site by site. Since $J\propto t_{pd}^{4}/\Delta_{\rm CT}^{3}$ in the three-band model, that is a direct measurement of a superexchange origin for the pairing strength.
Ab initio. Kaneko and Imada constructed low-energy effective Hamiltonians for HgBa$_2$Ca$_2$Cu$_3$O$_8$ from first principles and solved them with a neural-network-augmented variational solver, and describe the pairing as an emergent local attraction produced by the release of double occupancy on doping — “attraction from reduced repulsion”. Their calculation reproduces the observed ambient-pressure $T_c$ and predicts a maximum near $30\,\mathrm{GPa}$.

Algebra, a local measurement and a first-principles calculation, arriving from three directions at a local singlet attraction derived from the on-site repulsion. That is about as much convergence as this subject has ever produced.

What this does not explain

Not the magnitude of $T_c$. Not the pseudogap. Not the strange metal. Not the dome. Not the materials trend. The pairing interaction is the easy half of the problem, and declaring it settled is not a declaration of victory — it is a redirection of effort towards §7 and §8.

Amplitude versus stiffness: what actually limits $T_c$

Here is the most useful single idea in the subject, and one that a field theorist can verify with a pocket calculator in five minutes.

A superconductor needs two things: a non-zero pairing amplitude $|\Delta|$, and phase coherence of that amplitude across the sample. Write the order parameter $\Psi=|\Delta|e^{i\theta}$. There are then two temperature scales, and $T_c$ is set by whichever fails first:

$T_{\rm pair}$, at which the amplitude $|\Delta|$ forms locally. This is the scale BCS computes, and it is governed by the pairing interaction — by $J$, in the picture of §6.
$T_{\theta}$, at which the phase $\theta$ orders. This is governed by the phase stiffness $\rho_s$, the energy cost of a slow twist of $\theta$, which is proportional to the superfluid density.

In a conventional superconductor $T_\theta$ is enormous — thousands of kelvin — because the superfluid density is the full electron density. Phase order is free, $T_c=T_{\rm pair}$, and BCS mean-field theory is quantitatively excellent. In the cuprates fact 7 says the superfluid density is $p$ per copper, not $1-p$. Suddenly $T_\theta$ is not enormous.

Do the arithmetic

In two dimensions the Kosterlitz–Thouless criterion relates the transition to the areal phase stiffness, $k_BT_\theta=\tfrac{\pi}{2}\rho_s$, with the stiffness of a charge-$2e$ condensate given by $\rho_s=\hbar^{2}n_s^{2D}/4m^{*}$. Put in cuprate numbers: doping $p=0.15$ holes per copper on a lattice with $a=3.8\,\mathrm{\AA}$, so $n_s^{2D}=p/a^{2}=1.0\times10^{18}\,\mathrm{m^{-2}}$, and an effective mass $m^{*}\approx4m_e$ as measured by specific heat and quantum oscillations. Then

$$\rho_s=\frac{\hbar^{2}n_s^{2D}}{4m^{*}}\approx5\,\mathrm{meV},\qquad T_\theta=\frac{\pi}{2}\frac{\rho_s}{k_B}\approx90\,\mathrm{K}.$$

That number is not a coincidence

A back-of-envelope estimate using only the doping, the lattice constant and the effective mass — with no reference whatsoever to the pairing interaction — lands on the observed $T_c$ of YBa$_2$Cu$_3$O$_7$. Meanwhile the pairing scale from §5 and §6 is of order $0.1J\approx150\,\mathrm{K}$, comfortably above it.

So the cuprates appear to be phase-stiffness limited, not pairing limited, at least up to optimal doping. The glue is strong enough to pair at $150\,\mathrm{K}$; what fails at $90\,\mathrm{K}$ is the ability of $0.15$ carriers per copper to maintain phase rigidity against thermal vortex unbinding. This was Emery and Kivelson's argument in 1995, and it explains the three facts that most embarrass BCS at once: Uemura scaling (fact 7), the oversized gap ratio (§2), and the existence of a pseudogap regime where pairs exist without coherence (fact 8).

Honesty about the estimate: the real system is quasi-two-dimensional, and interlayer Josephson coupling raises $T_c$ above the strict two-dimensional Kosterlitz–Thouless value, so $T_\theta$ is a characteristic scale rather than a rigorous bound; the measured $\rho_s$ is temperature-dependent and the criterion should be applied self-consistently; and the factor-of-$4$ conventions in $\rho_s$ differ between papers. Getting all of this right, with measured stiffnesses across several families, is Q-A6 and Q-D1 — and it is a genuinely useful afternoon's work, because the conclusion is load bearing for everything downstream.

Figure 4 · one hypothesis, drawn The dome as the lower envelope of two scales with opposite doping dependence. The left flank is set by stiffness and obeys Uemura scaling; the right flank is set by the weakening of pairing as magnetic correlations die; the pseudogap region is where the amplitude exists but the phase does not. Drawn as a cartoon, because it is a cartoon — it is not derived, and §9 lists frameworks that reject it. But it organises more facts per assumption than any competitor, and it makes the falsifiable prediction that the $p$-dependence of $T_c$ below optimal doping is the $p$-dependence of $\rho_s$.

Two consequences worth stating because they redirect the search for higher $T_c$.

If $T_c$ is stiffness limited, a better glue buys nothing. On the underdoped side the amplitude is already larger than it needs to be — that is exactly what the oversized $2\Delta/k_BT_c$ is telling us. Raising $T_c$ requires raising $n_s/m^{*}$: more carriers, lighter carriers, or more strongly coupled layers. The empirical rise of $T_c^{\max}$ with the number of CuO$_2$ planes up to three, and its fall beyond (fact 14), is then not a mystery about pairing but a competition between added interlayer stiffness and the increasingly unequal doping of inner and outer planes — which is a quantitative, modellable claim (Q-D4).

The right flank of the dome is a different question from the left. Overdoped cuprates have plenty of stiffness and a dying pairing interaction as antiferromagnetic correlations vanish. Treating the two sides of the dome as one phenomenon with one explanation has cost the field a great deal of confusion. They are plausibly two different bottlenecks meeting at optimal doping.

The normal state, which may be the harder problem

Cooper's calculation assumes a Fermi liquid. Over most of the cuprate phase diagram there is no Fermi liquid to assume. Two distinct anomalies live here, and they are probably not the same anomaly.

The pseudogap: two readings, and how to tell them apart

Below $T^{*}$ the antinodal states are gapped while the nodal states remain gapless, giving the Fermi arcs of fact 8. There are two families of explanation and the distinction is not rhetorical.

Reading one: the pseudogap is the pairing gap. Amplitude forms at $T^{*}=T_{\rm pair}$, phase coherence only at $T_c=T_\theta$, and the pseudogap regime is the preformed-pair regime of Figure 4. In favour: $T^{*}$ and the gap magnitude both increase as doping decreases, tracking each other rather than tracking $T_c$; Nernst and diamagnetic signals above $T_c$; the gap has the same $d$-wave form. Against: $T^{*}$ is very large — $300\,\mathrm{K}$ at low doping, which is a long way to maintain pairs without coherence — and the pseudogap has antinodal spectral features that the superconducting gap does not.

Reading two: the pseudogap is a separate instability. A distinct order or near-order, competing with superconductivity for the same antinodal states. The candidate order parameters are a bestiary: charge density wave (observed, fact 9, but with $T_{\rm CDW}<T^{*}$, so at best a descendant); nematic order, breaking the $90^\circ$ rotation symmetry; intra-unit-cell loop currents, which polarised neutron scattering reports and other probes struggle to confirm; a $d$-density wave; a pair density wave, a superconductor that itself modulates in space; and fractionalised states with topological rather than conventional order. Cataloguing these with their distinguishing signatures is Q-E3.

The cleanest discriminator is the one the field has not fully exploited: entropy. A pairing gap and a symmetry-breaking gap remove the same spectral weight but dispose of the entropy differently, and the electronic specific heat has been measured across $p^{*}$ with enough precision to integrate. That is Q-C6.

The strange metal, and why every mechanism fails a constraint

Fact 10 deserves unpacking, because its difficulty is not widely appreciated. There are four independent constraints and almost every proposal satisfies three of them.

Four constraints on any theory of $T$-linear resistivity in the cuprates, and where the standard candidates fail.
Constraint	What it rules out
Range. $\rho\propto T$ from $T_c$ to $>1000\,\mathrm{K}$, no change of slope.	Any single boson with a characteristic frequency: phonons give $\rho\propto T$ only for $T>\Theta_D$ and $T^{5}$ below. Any quantum-critical fan of finite width in $T$.
Magnitude. Sheet resistance reaches $h/e^{2}$ and keeps rising linearly.	Any Boltzmann transport picture, since the implied mean free path falls below a lattice constant. Resistivity saturation, which every conventional metal shows and this one does not.
Slope. $\hbar/\tau=\alpha k_BT$ with $\alpha\approx1$, the same in several families and in unrelated materials.	Mechanisms with a free coupling constant, which would give a material-dependent $\alpha$. The universality is the strongest hint that something scale-free is at work.
Momentum. The current must decay, so momentum must be relaxed.	Clean critical scattering off a boson at a fixed wave vector: electron–boson collisions conserve total momentum, and a translationally invariant critical theory therefore does not give a finite resistivity at all. Something must break translation invariance.

The last row is the subtle one and it is where recent progress has come from. The phenomenology has been available since 1989, when Varma, Littlewood, Schmitt-Rink, Abrahams and Ruckenstein postulated a marginal Fermi liquid: assume a susceptibility whose imaginary part is constant in frequency up to a cutoff and proportional to $\omega/T$ below $T$, and the electron self-energy becomes

$$\Sigma(\omega)\sim\lambda\Big[\omega\ln\frac{x}{\omega_c}-\,i\,\frac{\pi}{2}x\Big],\qquad x=\max(|\omega|,T),$$

which gives a quasiparticle weight vanishing logarithmically, a scattering rate linear in $T$ with a universal coefficient, and the observed optical conductivity and tunnelling spectra. It fits beautifully. It is not a theory: the susceptibility was postulated, not derived.

Deriving it has become possible. Patel, Guo, Esterlis and Sachdev showed that a two-dimensional Fermi surface coupled to a critical boson by a Yukawa coupling that is spatially random — random in the coupling, not merely a random potential — yields exactly the marginal-Fermi-liquid self-energy, $T$-linear resistivity with a Planckian coefficient, and a $T\ln(1/T)$ specific heat. The spatial randomness is what supplies momentum relaxation, resolving the fourth constraint, and the universality of $\alpha$ emerges because the answer is controlled by a fixed point rather than by the coupling strength.

Why this is the natural entry point for a field theorist

That construction is a large-$N$ disorder-averaged field theory of Sachdev–Ye–Kitaev type. The technology is replica averaging, saddle points at large $N$, and self-consistent Green's-function equations — all of which a high-energy theorist already owns. The open questions are of a kind that yields to that technology: what happens in the pairing channel of the same model; whether the disorder can be generated rather than assumed; whether the required randomness exists in real cuprates; and what the model predicts that marginal Fermi liquid phenomenology alone does not. This is Q-C2 and Q-C3, and it is the question in this document with the best ratio of tractability to importance.

Two adjacent programmes deserve mention with their limitations stated. Holographic duality produces $T$-linear resistivity and scale-invariant non-Fermi liquids readily, and has been genuinely useful as a generator of possible scaling structures and of sum rules. Its standard constructions also carry a large zero-temperature entropy that real metals do not have, and there is no derivation connecting any of them to a lattice Hamiltonian, so it should be used as a source of structural hypotheses rather than as a theory of copper oxides. The Planckian bound $\hbar/\tau\le k_BT$ is a striking empirical regularity across many materials and is not a theorem; proposed derivations all assume something substantive. Treating it as established is a mistake, and asking what it would take to prove or break it is a legitimate research question.

The Luttinger problem, which is a gift

Luttinger's theorem says the volume enclosed by the Fermi surface of a Fermi liquid is fixed by the particle density, independent of interaction strength. Underdoped cuprates appear to violate it twice over: ARPES sees open arcs rather than closed surfaces, and quantum oscillations see pockets enclosing $\sim p$ carriers where the theorem demands $1+p$ (fact 13). There are exactly two ways out, and they are experimentally distinguishable.

Conventional. Broken translation symmetry — the charge order of fact 9 — enlarges the unit cell and folds the zone, so the theorem is satisfied in the reduced zone and the small pockets are reconstruction products. Prediction: the pockets and the charge order must appear together, with matching onset and matching wave vector.
Exotic. A fractionalised Fermi liquid, in which topological order allows a small Fermi surface of conventional quasiparticles to coexist with a gapped fractionalised sector that absorbs the missing Luttinger count, with no broken symmetry at all. Prediction: small pockets without charge order, and a thermodynamic signature of the topological sector.

Because these differ in what must accompany what, the question is decidable by existing techniques, and that makes it unusually valuable: it is a place where an abstract theorem about interacting fermions makes contact with a measurement. It is Q-C5.

Seven frameworks, side by side

These are not seven competing answers to one question. They are seven choices of which physics is in control, and several of them are compatible. Reading them as rivals is the standard mistake; reading them as different bets about the bottleneck is more productive.

1 · Phonons and Eliashberg theory

The idea. The conventional mechanism: retarded attraction from lattice deformation, with $T_c$ set by the Allen–Dynes form, roughly $T_c\sim\omega_{\log}\exp[-1.04(1+\lambda)/(\lambda-\mu^{*}(1+0.62\lambda))]$, crossing over at strong coupling to $T_c\propto\sqrt{\lambda\langle\omega^{2}\rangle}$.

Verdict for cuprates: excluded as the primary mechanism. It cannot produce a sign-changing gap from an attraction that is structureless in momentum; the isotope exponent at optimal doping is an order of magnitude too small; and the parent compound is an insulator that no electron–phonon coupling explains.

But do not skip the hydride lesson

H$_3$S superconducts at $203\,\mathrm{K}$ and LaH$_{10}$ at $250\text{--}260\,\mathrm{K}$, both under pressures near $170\,\mathrm{GPa}$, and both are described quantitatively by electron–phonon Eliashberg theory with $\lambda\approx2$. The asymptotic $T_c\propto\sqrt{\lambda\langle\omega^{2}\rangle}$ has no upper bound in $\lambda$ — the real ceiling is lattice stability, since large $\lambda$ means soft phonons means a structural instability.

Two conclusions follow, and both are load bearing. “High $T_c$” and “unconventional” are independent properties, and conflating them has confused the field for decades. And the cuprate problem is not that $100\,\mathrm{K}$ is inexplicably high — phonons reach twice that — it is that cuprates reach $100\,\mathrm{K}$ at ambient pressure with the wrong mechanism, a thousand times fewer carriers, and a non-Fermi-liquid normal state. That reframing is Q-F2.

2 · Spin-fluctuation exchange

The idea. Treat the system as a Fermi liquid close to an antiferromagnetic instability. The exchange of its own spin fluctuations, with susceptibility peaked at $(\pi,\pi)$, supplies the momentum-dependent repulsion of §6 and hence $d_{x^2-y^2}$ pairing. Implemented as fluctuation-exchange approximations, the functional renormalisation group, and the weak-coupling limit of the Hubbard model. Scalapino's 2012 review is the standard account.

What it gets right. The pairing symmetry, exactly and robustly. The correlation of pairing strength with band-structure parameters, which makes it the only framework that currently makes quantitative materials predictions. And it generalises: the same machinery predicts $s^{\pm}$ pairing in the iron pnictides and pairing in the organics, and got both right in advance. That is a real predictive track record, and it is the reason this framework retains adherents despite the objection below.

What it fails. It assumes the very thing that is missing: a Fermi liquid with sharp quasiparticles. It has no account of the Mott insulator, no account of the pseudogap, no account of the strange metal, and it gets the sign of the condensation-energy change wrong (fact 12). It is best understood as the correct weak-coupling shadow of the mechanism of §6 — which raises the sharp and answerable question of how far up in $U/t$ the shadow remains faithful: Q-B2.

3 · Resonating valence bonds and the gauge theory of a doped Mott insulator

The idea. Anderson, 1987. The undoped plane is a liquid of nearest-neighbour spin singlets; doping makes those singlets mobile, and superconductivity is the phase coherence of pairs that already exist. The technical implementation is fractionalisation: write $c_{i\sigma}=b_i^{\dagger}f_{i\sigma}$ with a bosonic holon $b$ and a fermionic spinon $f$. This representation is redundant — $b_i\to b_ie^{i\phi_i}$, $f_{i\sigma}\to f_{i\sigma}e^{i\phi_i}$ leaves $c$ invariant — so the theory carries an emergent compact U(1) gauge field, enlarged to SU(2) at half filling. The constraint $b^{\dagger}b+\sum_\sigma f^{\dagger}_\sigma f_\sigma=1$ is the Gauss law. Lee, Nagaosa and Wen's 2006 review is the canonical treatment.

What it gets right. Most of the phase diagram's topology, from very few assumptions. Spinon pairing sets $T_{\rm pair}$, holon condensation sets $T_\theta$, and $T_c=\min$ of the two delivers the dome and the pseudogap as the spinon-paired-but-not-holon-condensed region. The pseudogap acquires a candidate identity, the staggered-flux phase. Gutzwiller-projected $d$-wave wavefunctions, the variational face of the same idea, are quantitatively good. It gets the sign of fact 12 right, because pairing enables coherent hole motion. And it is the framework in which the emergent gauge structure of a constrained Hilbert space is taken seriously, which is where a field theorist has the most immediate traction.

What it fails. The gauge fluctuations are strong and uncontrolled; there is no small parameter and no systematic expansion, so mean-field phase diagrams in this language are suggestive, not derived. Different decouplings of the same Hamiltonian give different answers, and some mean-field order parameters are not gauge-invariant, so apparently different phases can be the same state. It has produced no quantitative $T_c$ and no materials trend.

4 · Preformed pairs and phase fluctuations

The idea. Emery and Kivelson, 1995, and §7 of this document. Take the pairing as given and ask what limits $T_c$; answer: the phase stiffness, which is small because the carrier density is small.

What it gets right. The arithmetic of §7 lands on $90\,\mathrm{K}$ from three measured numbers. It explains Uemura scaling, the anomalous $2\Delta/k_BT_c$, the diamagnetic and Nernst signals above $T_c$, and the existence of a pseudogap regime — and it delivers a concrete prescription for raising $T_c$.

What it fails. It is a theory of $T_c$, not of the normal state: it says nothing about the strange metal, and it takes $T_{\rm pair}$ as input. Quantitatively, the preformed-pair regime it predicts above $T_c$ is smaller in experiment than the full pseudogap region, so at low doping either $T^{*}$ is not $T_{\rm pair}$ or there is something else in there too.

5 · Quantum criticality at $p^{*}$

The idea. A quantum critical point sits beneath the dome near $p^{*}\approx0.19$, where $T^{*}$ extrapolates to zero. Its fluctuations mediate pairing, produce $T$-linear resistivity in the fan above it, and account for the effective-mass enhancement and the logarithmic specific heat observed there.

What it gets right. It predicts that anomalies should be organised around one doping, and they largely are: the Hall number changes from $p$ to $1+p$ across $p^{*}$, the specific heat peaks there, the resistivity is most nearly $T$-linear there, and recent high-resolution resonant inelastic X-ray scattering reports quantum-critical scaling in La$_{2-x}$Sr$_x$CuO$_4$.

What it fails. Nobody has identified the order parameter that goes critical — which means this is a phenomenological frame rather than a theory. $T$-linear resistivity extends far too wide in temperature and in doping for a conventional critical fan. And the critical exponents do not match the heavy-fermion quantum-critical paradigm that inspired the picture.

6 · Intertwined orders

The idea. Fradkin, Kivelson and Tranquada, 2015. Stop treating superconductivity as primary with everything else as a competitor. Charge density waves, spin density waves, nematic order, pair density waves and superconductivity are near-degenerate descendants of a single strongly correlated parent, and which one wins is decided by small perturbations. Composite and vestigial orders — a charge-$4e$ condensate, or a nematic that survives when the order generating it has melted — appear naturally.

What it gets right. It is the only framework that takes the near-degeneracy of fact 9 seriously rather than as a nuisance, and the 2017/2024 Hubbard results of §5 are exactly what it predicts: superconductivity and partially filled stripes within a hair's breadth of each other, with the winner flipped by $t'$. It also supplies a genuinely rigorous tool, Landau theory with multiple coupled order parameters, which produces falsifiable statements about what phases must exist and what must accompany what.

What it fails. It is a symmetry-and-phenomenology framework, so it does not by itself say why the parent state is what it is, why $T_c$ is $90\,\mathrm{K}$, or what the strange metal is. Its predictions are about structure rather than magnitude.

7 · Strange metal first

The idea. Invert the usual logic. The anomalous normal state is the fundamental phenomenon and superconductivity is what happens when you cool it; so solve the metal first. Concretely: marginal Fermi liquid phenomenology; Sachdev–Ye–Kitaev-type models with spatially random Yukawa couplings, which now derive that phenomenology; holographic constructions; and exactly solvable non-Fermi liquids such as the Hatsugai–Kohmoto model, whose momentum-space-local interaction produces a Mott insulator one can solve and then superconduct.

What it gets right. It is the only programme aimed squarely at facts 8, 10 and 13, which are the facts no other framework addresses. The random-Yukawa construction delivers all four strange-metal constraints of §8 at once, including the momentum-relaxation one that defeats clean critical theories. And it is solvable, which in this subject is rare enough to be worth a great deal.

What it fails. The connection to a cuprate is by analogy, not derivation. The required spatial randomness is put in by hand and its physical origin in real materials is unestablished. The Hatsugai–Kohmoto interaction is local in momentum rather than in space, which is physically backwards and almost certainly poisons some conclusions — identifying which is Q-C4. None of these models has a $d_{x^2-y^2}$ gap for the right reason, and none produces a materials trend.

The scorecard

Six capabilities, seven frameworks. Read the columns, not the rows: the instructive pattern is that the left columns are crowded and the right ones are nearly empty.

derives the framework produces it · partial gets the structure, not the magnitude, or needs an assumption · silent says nothing · wrong predicts something contradicted.
Framework	d-wave gap	Dome	Pseudogap	Strange metal	$T_c$ magnitude	Materials trend
1 Phonons	wrong	silent	silent	wrong	partial	silent
2 Spin fluctuations	derives	partial	silent	silent	partial	derives
3 RVB / gauge theory	derives	derives	derives	partial	partial	silent
4 Phase fluctuations	silent	derives	partial	silent	derives	partial
5 Quantum criticality	partial	partial	partial	partial	silent	silent
6 Intertwined orders	partial	derives	derives	silent	silent	silent
7 Strange metal first	silent	silent	partial	derives	silent	silent

What the scorecard actually shows

Two readings, both worth carrying.

No single row is close to complete, but a specific combination nearly is. Frameworks 3, 4 and 7 together cover every column except the last: RVB supplies the pairing and the fractionalised normal state, phase fluctuations supply the magnitude of $T_c$, and the random-Yukawa construction supplies the strange metal. Whether these three are compatible — whether the strange metal of framework 7 can be the normal state of framework 3 — is an unanswered structural question and, in my judgement, the most valuable one in the document. It is Q-C2 pushed to its conclusion.

The rightmost column is the field's weakest point. Exactly one framework makes quantitative materials predictions, and it is the one with the least defensible starting assumption. A theory that could order the cuprate families correctly would be taken seriously immediately, which makes Q-D3 disproportionately valuable relative to its difficulty.

Settled, probable, contested

An explicit ledger, because the field's habit of describing the whole problem as open obscures how much of it is closed, and its habit of describing consensus positions as established obscures how much rests on assumption.

Settled

These would require an extraordinary experimental reversal to overturn, and you may build on them.

The parent compounds are charge-transfer insulators with spin-$\tfrac12$ Cu moments and $J\approx130\,\mathrm{meV}$, quantitatively described by the square-lattice Heisenberg model.
The superconducting order parameter has $d_{x^2-y^2}$ symmetry, demonstrated phase-sensitively.
Pairing is not phonon-driven at optimal doping.
Charge density wave order is a generic feature of the underdoped regime and competes with superconductivity.
The normal state at optimal doping is not a Fermi liquid, and its resistivity is $T$-linear with a Planckian coefficient past the Mott–Ioffe–Regel limit.
Superfluid density scales with doping, not with total electron count.
The one-band $t$–$t'$–Hubbard model at realistic parameters has $d$-wave pairing in competition with partially filled stripes, per multiple independent numerical methods.

Probable

Supported by several independent lines of evidence, believed by most practitioners, and not proven. Build on these but flag the dependence.

The pairing interaction is superexchange — a local, essentially instantaneous attraction in the nearest-neighbour singlet channel derived from the Coulomb repulsion. three converging lines
$T_c$ below optimal doping is limited by phase stiffness rather than by pairing strength. arithmetic of §7, Uemura scaling, gap ratio
The Zhang–Rice reduction to one band is adequate for the mechanism, though not necessarily for $T_c^{\max}$ trends. widely assumed, directly testable
$t'$ controls the competition between superconductivity and stripe order, and thereby the materials trend. numerics plus the Pavarini correlation
Momentum relaxation by spatial randomness in the coupling is necessary for Planckian transport. one model, one mechanism, not yet tested in cuprates

Contested

Genuinely open, with competent people on both sides. These are where the questions of §11 concentrate.

What the pseudogap is. Preformed pairs, a distinct broken symmetry, a fractionalised state, or several of these in different doping ranges. no consensus
Whether there is a quantum critical point at $p^{*}$, and of what. no consensus
Whether the small Fermi surface requires broken translation symmetry or topological order. no consensus
Whether a retarded-boson description of the pairing is legitimate or merely a reparametrisation. terminological and physical, tangled together
Whether the strange metal causes the superconductivity or merely coexists with it. no consensus, and rarely even posed sharply
Whether a pair density wave exists and, if so, whether it is the parent of the pseudogap. active, evidence accumulating
Whether the nickelates are the same problem. open, and resolvable within a few years

A note on how to hold this ledger

The three lists are not ordered by importance — if anything, the reverse. Notice also that the “settled” list is almost entirely experimental and the “contested” list almost entirely theoretical. That is the signature of a field with excellent data and no organising theory, which is a different predicament from one with a theory awaiting data, and it calls for a different kind of work: not more measurement, but sharper questions of the sort that existing measurement can already answer.

The question bank

Thirty-two questions in six tracks. Every one is posed so that it has an answer, so that the answer is obtainable with existing tools, and so that knowing it changes what you would do next. Each carries a stable handle — Q-A1 and so on — so that we can cite them in later work without re-deriving the context.

The cost estimates assume one person with a field theorist's background and no prior cuprate expertise, working with a laptop, public numerical libraries and library access. They are deliberately conservative. Questions whose payoff is disproportionate to their cost are marked.

Track A

Calibration

6 questions

Nothing here is research. All of it is the minimum set of calculations you must have done yourself before you can read this literature critically rather than credulously. Six weeks, and afterwards the numbers in §2 and §7 are yours rather than borrowed.

Q-A1an afternoon · by hand

What exactly does $U$ do to two electrons on two sites?

Why: Every statement about superexchange, every $J=4t^{2}/U$, and the entire singlet–triplet structure that §6 rests on originates in this one exactly solvable problem. Having done it once, you never again have to trust somebody else's factor of four.
Attack: Diagonalise the two-site Hubbard model at half filling by hand. One triplet at $E=0$; the singlet sector gives $E_s=\tfrac12\big(U-\sqrt{U^{2}+16t^{2}}\big)$ and hence $J=\tfrac12\big(\sqrt{U^{2}+16t^{2}}-U\big)$. Expand in both limits. Compute the double occupancy $\langle n_\uparrow n_\downarrow\rangle$ as a function of $U/t$.
Decides: That $J$ decreases monotonically in $U$, so maximising $J$ alone would say $U=0$ — yet pairing needs the constraint that only large $U$ supplies. That tension is the whole problem in miniature, and it is why cluster calculations find the pairing strength peaking at intermediate $U/t$ of order the bandwidth, which is where real cuprates sit.

Q-A2an afternoon · 20 lines of code

Is the two-hole ground state of a four-site plaquette $d$-wave?

Why: Four sites is the smallest system in which the question can be posed at all. If the answer is yes, then $d$-wave pairing is local chemistry rather than a Fermi-surface effect, and the correct language is real-space bonds rather than momentum-space nesting. That is a different conceptual starting point, and it is the one underlying plaquette and cluster approaches.
Attack: Exact diagonalisation of the $2\times2$ Hubbard plaquette with four, three and two electrons. Classify the two-hole ground state under the point group. Compute the pair binding energy $E_b=E(N-2)+E(N)-2E(N-1)$ as a function of $U/t$ and look for a window in which it is negative. Repeat on a four-site ring and on a $4\times4$ cluster to see what is geometry and what is physics.
Decides: Whether §6's Fermi-surface argument is the explanation or a weak-coupling rationalisation of something already true on four sites. Also gives you, for free, a feel for how violently small-cluster results depend on shape — which inoculates you against a large class of published claims.

Q-A3a week · numerics

Can repulsion alone pair, and what selects the channel?

Why: Turns §6 from an argument into a number, and produces the $t'$ dependence that §5 identifies as the switch between superconductivity and stripes.
Attack: Build the bare susceptibility $\chi_0(\mathbf{q})$ for the $t$–$t'$ band, take $\chi=\chi_0/(1-U\chi_0)$, form the singlet pairing vertex, and solve the linearised gap equation of §6 as an eigenvalue problem on a Fermi-surface mesh. Scan $t'/t$ from $0$ to $-0.45$ and filling from $0.75$ to $1.0$. Record the leading eigenvalue $\lambda$ and the symmetry of its eigenfunction.
Decides: Whether $d$-wave dominance and the $t'$ trend of fact 14 emerge from band structure alone. If they do at weak coupling, the materials trend may not require strong-correlation physics at all — a large simplification, and a direct route into Q-D3. Benchmarks exist in the literature, so you can check yourself.

Q-A4one to two weeks

How good is the Zhang–Rice reduction, and where does it fail?

Why: This is the most load-bearing approximation in the entire field and it is almost never checked. Every one-band result inherits its validity.
Attack: Exact diagonalisation of a CuO$_4$ cluster, then a two-plaquette Cu$_2$O$_7$ cluster, in the three-band model at $\Delta_{\rm CT}=1.8\,\mathrm{eV}$, $t_{pd}=1.3\,\mathrm{eV}$, $U_d=9\,\mathrm{eV}$, $U_p=4\,\mathrm{eV}$. Compute the overlap of the exact two-hole ground state with the Zhang–Rice singlet, the singlet–triplet splitting, and the effective one-band $t$ and $J$. Then sweep $\Delta_{\rm CT}$ and watch the overlap degrade.
Decides: Whether one-band results transfer to real materials, and — sharper — whether the apical-oxygen and $\Delta_{\rm CT}$ materials trend of fact 14 is even representable in one band. If the overlap is $0.99$, the critics are wrong; if it is $0.85$ and falling with $\Delta_{\rm CT}$, a whole literature needs an error bar.

Q-A5a day · high leverage

Where in the BCS–BEC crossover does each superconductor actually sit?

Why: Determines, material by material, whether mean-field theory is a legitimate starting point or a category error. Almost nobody writes this table down, and everybody needs it.
Attack: Tabulate $k_F\xi$, $T_c/T_F$, $\Delta/E_F$ and the number of pairs within a coherence area for: Al, Nb, MgB$_2$, optimally doped and underdoped cuprates, bulk FeSe, FeSe monolayer on SrTiO$_3$, La$_3$Ni$_2$O$_7$, twisted bilayer graphene, and the unitary Fermi gas.
Decides: Positions every material on one axis. The expected pattern — $T_c/T_F\sim10^{-5}$ for elements, $\sim10^{-2}$ for the correlated families, $0.17$ at unitarity — immediately tells you which intuitions are licensed where. Feeds Q-F1 directly.

Q-A6two weeks · best value in the document

Does the phase-stiffness ceiling really account for $T_c^{\max}$?

Why: §7's central claim currently rests on a back-of-envelope that happens to land on $90\,\mathrm{K}$. Either it survives a proper treatment, or the framing of this whole document changes.
Attack: Collect measured $\rho_s(T\to0)$ from muon spin rotation and penetration depth, and $m^{*}$ from specific heat and quantum oscillations, for LSCO, YBCO, Bi2212, Tl2201 and Hg1201 across doping. Apply the Kosterlitz–Thouless criterion self-consistently with the measured $\rho_s(T)$ rather than with its zero-temperature value. Add interlayer Josephson coupling and redo. Plot $T_c$ against $T_\theta$.
Decides: If $T_c/T_\theta$ is of order unity across families and dopings, the stiffness picture is established and the search for higher $T_c$ becomes a search for stiffness rather than for glue. If not, §7 is wrong and the dome needs a different explanation. Either outcome is worth two weeks.

Track B

The pairing interaction

5 questions

If §6 is right, this track is about converting a probable claim into a settled one and finding its edges. If §6 is wrong, this is where it breaks.

Q-B1a week (reading) to a month (computing)

Is “glue” a well-defined object when the interaction is instantaneous?

Why: Resolves a thirty-year terminological war into a computable number, and determines whether Eliashberg-like language is legitimate in this problem at all.
Attack: Compute the frequency-dependent $d$-wave pairing vertex $\Gamma_d(\omega)$ on a small cluster, and the fraction of the pairing eigenvalue $\lambda_d$ accumulated below $J$, below $2J$, and above. Do it for the Hubbard model and for the $t$–$J$ model, where by construction the interaction is instantaneous, and compare. Maier, Poilblanc and Scalapino did much of this; start by reproducing their decomposition.
Decides: If most of $\lambda_d$ comes from $\omega\lesssim J$ in both models, the two languages describe one mechanism and the question was never physical. If the Hubbard vertex carries substantial weight above $2J$ that $t$–$J$ lacks, then virtual double occupancy is doing real pairing work and the $t$–$J$ reduction is losing mechanism, not just charge fluctuations.

Q-B2two weeks (reading) to a month

How far up in $U/t$ does spin-fluctuation exchange remain faithful?

Why: Framework 2 is the only one making quantitative materials predictions, and it rests on an assumption — a Fermi-liquid normal state — that is false at cuprate coupling. Whether its predictions survive anyway is decidable.
Attack: Compute $\lambda_d$ for the same model at the same parameters by two routes: RPA/fluctuation exchange, and the dynamical cluster approximation. Sweep $U/t$ from $2$ to $12$. Find where they part company, in which direction, and whether the ordering of different $t'$ survives even where the magnitudes do not.
Decides: If they agree to $U/t\approx8$, the weak-coupling materials predictions are legitimate and Q-D3 becomes tractable cheaply. If they diverge by $U/t\approx4$, then framework 2's successes in the iron pnictides do not transfer here, and its good scorecard in §9 is borrowed credit.

Q-B3a month · falsifiable prediction

What selects $d$ over extended-$s$, and what would select $s^{\pm}$?

Why: The bilayer nickelates are a live test case whose pairing symmetry is not yet established. Predicting it in advance, from the same logic that works in cuprates and pnictides, is the kind of thing theory is for.
Attack: Extend the channel map of Q-A3 to two orbitals and a bilayer geometry, using the La$_3$Ni$_2$O$_7$ band structure: the $d_{z^{2}}$ bonding/antibonding pair plus $d_{x^2-y^2}$. Determine the leading channel and how it depends on the interlayer hopping and on the filling of the $d_{z^{2}}$ band.
Decides: Whether pairing symmetry is predictable across families from one calculation. A correct advance prediction for the nickelates would be strong evidence for the generality of the momentum-dependent-repulsion logic; a wrong one would localise the cuprate mechanism to cuprates.

Q-B4a month · discriminating

Does condensation lower kinetic or potential energy, and where does the sign flip?

Why: Fact 12 is a sign, and signs are the most valuable evidence available because they do not depend on getting a magnitude right. This one discriminates directly between boson-exchange and Mott-derived pairing.
Attack: On $t$–$J$ or Hubbard ladders and cylinders with DMRG, compute $\langle H_t\rangle$ and $\langle H_J\rangle$ (or $\langle H_U\rangle$) separately in the paired and unpaired states across doping. Locate the doping at which $\Delta E_{\rm kin}$ changes sign. Compare with the optical sum-rule measurements.
Decides: If the model reproduces the experimental sign and its doping dependence, the Mott-derived picture is confirmed inside a Hamiltonian we can compute. If the model gives the BCS sign everywhere, either the model or the experiment is wrong, and finding out which is more valuable than most things in this list.

Q-B5a month

How much pairing survives when double occupancy is forbidden by hand?

Why: Isolates the role of the projector $\mathcal{P}$, which is the difference between Hubbard and $t$–$J$ and the object on which the entire gauge theory of framework 3 is built.
Attack: Compute $d$-wave pair correlations in the Hubbard model at $U/t=8$ and in $t$–$J$ at $J=4t^{2}/U$ on identical clusters with identical boundary conditions. Then interpolate: replace the hard projector by a finite energy penalty $V_{\rm dbl}$ and tune it from zero to infinity.
Decides: Whether the constraint is the source of pairing or merely convenient bookkeeping. If pairing strengthens monotonically as the constraint is enforced, the gauge-theoretic framing is physical rather than formal — which licenses a great deal of field-theoretic machinery.

Track C

The normal state

6 questions

The open part of the problem, and the part where a field theorist's training is an advantage rather than a handicap. Q-C2 is the main line of this document.

Q-C1two weeks · do this early

Which of the four constraints kills which mechanism for $T$-linear resistivity?

Why: Builds the scoring rubric before you invest months in any one mechanism. Almost certainly saves more time than it costs.
Attack: Extend the table in §8 into a full matrix. Rows: phonons, spin fluctuations at a critical point, marginal Fermi liquid phenomenology, the random-Yukawa SYK construction, holographic duality, high-temperature Hubbard Monte Carlo, phonon-assisted bad-metal transport, and Planckian-bound arguments. Columns: the four constraints of §8, plus $\omega/T$ scaling of the optical conductivity, the $\sqrt{T^{2}+B^{2}}$ magnetoresistance scaling, and the anomalous Hall angle. Fill it from the literature, citing the specific paper for each cell.
Decides: Produces a short list of survivors, and makes explicit which cells are genuinely empty rather than merely unexamined. Expect the list to be very short, which is itself the most useful possible result.

Q-C2three to six months · the main line

Why does spatially random coupling give Planckian transport when clean criticality does not?

Why: The best-posed open problem in this document, and the one whose answer would most change the picture. It is also the question whose technology — replica averaging, large-$N$ saddle points, self-consistent Green's functions — a high-energy theorist already owns.
Attack: Reproduce the Patel–Guo–Esterlis–Sachdev construction in the simplest setting that keeps the essential structure: a Fermi surface Yukawa-coupled to a critical boson with coupling $g(\mathbf{r})$ Gaussian-random about zero mean. Disorder-average, find the large-$N$ saddle, extract the self-energy, the specific heat and $\sigma(T)$. Then — this is the actual question — identify precisely which step fails when $g$ is uniform, and whether the randomness is necessary or merely sufficient.
Decides: Converts a claimed mechanism into an understood one. The natural follow-up — what happens in the pairing channel of the same model, and whether it can host a $d$-wave instability — is the bridge between frameworks 3, 4 and 7, and is the structural question §9 identifies as the most valuable in the document.

Q-C3three weeks · no calculation

Does the $T$-linear resistivity slope correlate with disorder?

Why: The falsifiable consequence of Q-C2. If spatial randomness in the coupling is essential to Planckian transport, the Planckian coefficient must depend on how much of it there is.
Attack: Meta-analysis. Collect $d\rho/dT$ per CuO$_2$ layer and a disorder proxy — residual resistivity, dopant species and concentration, cation disorder — across samples and families, deliberately spanning the intrinsically clean (YBCO, Hg1201) and the intrinsically dirty (LSCO, Bi2201). Ask whether $\alpha$ correlates with disorder or is independent of it.
Decides: A positive correlation would be strong support for Q-C2 and would make a specific prediction for future clean samples. Independence would be a serious problem for that mechanism and would point instead to something intrinsic — which is equally worth knowing, and cheaper to learn this way than by six months of calculation.

Q-C4a month · mostly analytic

Is the Hatsugai–Kohmoto model a useful caricature of a doped Mott insulator, or a misleading one?

Why: It is the only exactly solvable Mott insulator you can then superconduct. That makes it either a gift or a trap, and which it is can be determined rather than argued about. There is a live disagreement in the literature on exactly this point, so the work is publishable.
Attack: Solve it — the interaction is diagonal in momentum, so each $\mathbf{k}$ is an independent four-state problem. Add BCS pairing and locate the Cooper instability. Then audit which conclusions survive replacing the momentum-local interaction by a space-local one: check the compressibility, the $f$-sum rule, the Luttinger count, the persistence of a Mott gap at fixed filling, and $2\Delta/k_BT_c$.
Decides: Whether the model can serve as a controlled laboratory for “superconductivity out of a non-Fermi liquid”, or whether its unphysical locality has already fixed the answers. A clean audit of which results are artefacts would be a genuine service to the field.

Q-C5three weeks, then a proposal

Do the Fermi arcs and the small pockets require broken symmetry, or topological order?

Why: Makes Luttinger's theorem do experimental work. The two answers imply qualitatively different physics for the entire underdoped region.
Attack: Assemble carrier counts from quantum oscillations, the Hall number across $p^{*}$, ARPES arc lengths and the specific heat, for YBCO and Hg1201 where the data are best. Test the conventional scenario's requirements: does the pocket appear at the same doping and field as the charge order, and does its area match the charge-order-reconstructed band structure? Then design the discriminating measurement — a thermodynamic signature of a topologically ordered sector, or pockets surviving where charge order is absent.
Decides: Either the underdoped cuprate is a conventional metal with a reconstructed Fermi surface, in which case most exotic machinery is unnecessary below $p^{*}$; or it carries topological order, which is a qualitatively new statement about matter and would reorganise the field.

Q-C6a month · underused constraint

Where does the pseudogap entropy go?

Why: Entropy is conserved, the electronic specific heat has been measured across $p^{*}$ with real precision, and almost nobody uses it as a constraint on pseudogap models.
Attack: Integrate measured $C_{\rm el}/T$ from the lowest accessible temperature to above $T^{*}$, for a doping series spanning $p^{*}$, and extract the entropy removed below $T^{*}$. Then compute the same quantity for each candidate: a $d$-wave spin gap on a spinon Fermi surface; a charge-order gap on the reconstructed bands; a preformed-pair state.
Decides: These candidates remove different amounts of entropy at different rates. A candidate off by a factor of two is eliminated — and elimination, in a field with seven live hypotheses for the pseudogap, is worth more than another hypothesis.

Track D

The scale of $T_c$

5 questions

Bounds and trends. The only products of this subject that cannot later be shown to have been wrong, and the only ones that would tell a materials chemist what to make next.

Q-D1a month · most defensible output

What is the tightest defensible upper bound on $T_c$, and what saturates it?

Why: A bound, once proved, stays proved. And the bound that binds identifies the bottleneck, which is the practical question.
Attack: Assemble and sharpen three: the two-dimensional phase-stiffness bound of §7; the $f$-sum-rule bound of Hazra, Verma and Randeria relating $T_c$ to the optical weight; and the Bose–Einstein bound $T_c\le T_c^{\rm BEC}(n_{\rm pair},m_{\rm pair})$. Apply all three to cuprates, nickelates, hydrides, FeSe monolayer and twisted bilayer graphene. Note carefully which assumptions each bound needs.
Decides: Which constraint is active in each family. If the stiffness bound binds in cuprates while lattice stability binds in hydrides, the two families have entirely different engineering problems — and conflating them, as popular accounts do, is actively misleading.

Q-D2three weeks

Does the pairing scale track $J$ across families and under pressure?

Why: Tests §6 against a trend rather than a single number, which is a much harder test to pass by accident.
Attack: Collect $J$ from two-magnon Raman scattering and neutron spin-wave fits, and $\Delta_{\rm CT}$ from X-ray absorption and from the scanning-tunnelling work of O'Mahony and co-workers, for as many families as possible. Plot $T_c^{\max}$ and the antinodal gap against both. Include the pressure dependence, where Kaneko and Imada's calculation makes a specific claim.
Decides: A clean correlation of the gap scale with $J$, coexisting with a weaker correlation of $T_c^{\max}$ with $J$, would confirm §6 and §7 simultaneously — pairing set by $J$, $T_c$ set by something else. Absence of any correlation would be a serious problem for §6.

Q-D3a month staged, three months full

Can one band structure reproduce the $T_c^{\max}$ ordering of the cuprate families?

Why: The field's weakest column (§9) and the test any theory of $T_c$ must eventually pass. A correct ordering would be taken seriously immediately.
Attack: Take material-specific $(t,t',t'',U)$ from downfolded ab initio calculations for the La, Y, Bi, Tl and Hg compounds. Compute the $d$-wave pairing eigenvalue by the method of Q-A3, and where feasible a $T_c$ estimate by dynamical cluster approximation. Combine with the stiffness estimate of Q-A6, since §7 says both factors matter. Compare the predicted ordering with measurement.
Decides: Whether the one-band paradigm has quantitative content or only qualitative content. Stageable: the weak-coupling eigenvalue ordering alone is a month's work and already informative, and can be abandoned cheaply if it fails.

Q-D4a month

Why does $T_c$ rise with the number of CuO$_2$ layers up to three, then fall?

Why: A clean, quantitative, unexplained trend that the stiffness picture ought to be able to address — and if it can, that is an independent confirmation of §7 costing one month.
Attack: Model an $n$-layer block as $n$ Josephson-coupled two-dimensional superconductors with layer-dependent doping, taking the known NMR result that inner planes are underdoped relative to outer ones. Compute the total phase stiffness and the resulting transition temperature against $n$, and compare with measured $T_c(n)$ in the Hg and Bi families.
Decides: If interlayer stiffness plus doping inhomogeneity reproduces the non-monotonicity quantitatively, §7 gains a second independent confirmation and the prescription for raising $T_c$ becomes a concrete materials-design statement.

Q-D5two weeks · do this early

What is the falsifiable content of the ab initio programme?

Why: Kaneko and Imada predict a $T_c$ maximum near $30\,\mathrm{GPa}$ in Hg1223, reaching roughly $160\,\mathrm{K}$. Predictions with numbers are rare in this subject and must be treated as the opportunity they are.
Attack: Extract the predictions precisely: the pressure of the maximum, its height, and the claimed behaviour of the effective $U$, $t$ and intersite $V$ with pressure. Separate those already tested by existing high-pressure transport data from those not yet tested. Determine what an independent method would have to reproduce to count as confirmation.
Decides: Whether the first-principles-to-$T_c$ pipeline has become predictive. If it has, the nature of this research problem changes for everybody, including the weight to place on the rest of this document — which is why it belongs in the first month, not the last.

Track E

Intertwined orders and the dome

5 questions

Where symmetry analysis does real work. Q-E2 is paper-and-pencil field theory that produces falsifiable statements, and it is the fastest route from this document to a result of your own.

Q-E1three weeks (critical reading) to three months

How nearly degenerate are the uniform and striped states, and what tips the balance?

Why: The near-degeneracy of fact 9 is the phase diagram's organising feature, and $t'$ appears to be the switch (§5). Quantifying the degeneracy also calibrates how much to believe any numerical claim in this subject.
Attack: On the $t$–$t'$–Hubbard and $t$–$t'$–$J$ models, compute energies of uniform $d$-wave, striped and coexisting states as a function of $t'/t$ at $p=1/8$, using at least two methods with different biases. Quantify the energy difference per site in units of $t$, and compare it with each method's own systematic error.
Decides: Whether the 2017/2024 pair of results is a genuine $t'$-driven transition or a methodological artefact. The by-product — an honest error bar on energy differences at this scale — is worth as much as the answer.

Q-E2three weeks · paper and pencil

What does the Landau theory of coupled superconducting, charge-density-wave and pair-density-wave order parameters predict?

Why: Pure symmetry analysis is rigorous, cheap, immune to the numerical uncertainties that plague everything else here, and squarely in a field theorist's idiom. It converts the rhetoric of “intertwined orders” into statements that can be checked against existing data.
Attack: Write the most general free energy in the uniform order parameter $\Delta_0$, the charge-density-wave amplitude $\rho_{\mathbf{Q}}$ and the pair-density-wave amplitudes $\Delta_{\pm\mathbf{P}}$, consistent with the lattice point group, time reversal and gauge invariance. Enumerate the allowed trilinear and quartic couplings. Identify the composite orders they force — the charge-$4e$ condensate $\Delta_{\mathbf{P}}\Delta_{-\mathbf{P}}$, the charge modulation at $2\mathbf{P}$ — and the vestigial phases in which a composite survives the melting of its parents. Map the phase diagram in $T$, $p$ and $B$.
Decides: Produces a list of phases that must exist if a pair density wave does, and of observations that must occur together. Several of those are already searchable in published data, so the analysis pays off immediately rather than waiting on new experiments.

Q-E3a month · synthesis

Does the pseudogap break a symmetry, and if so which?

Why: The cleanest statement of the field's central open question, and one where the obstacle is disorganisation of existing evidence rather than missing evidence.
Attack: Build the catalogue. For each candidate — nematic, intra-unit-cell loop current, $d$-density wave, pair density wave, charge density wave, preformed pairs — record the broken symmetry, the expected signature in polarised neutron scattering, second-harmonic generation, torque magnetometry, NMR, ultrasound and the Nernst effect, and whether that signature has been seen, not seen, or disputed. Then identify the single measurement that most efficiently separates the survivors.
Decides: Narrows a bestiary to a short list and yields a concrete experimental proposal. The catalogue itself is reusable and would be cited.

Q-E4a month · synthesis

Is there a pair density wave, and is it the parent of the pseudogap?

Why: A pair density wave would unify the pseudogap, the charge order and the superconductivity into one object. That is the largest conceptual simplification currently on offer.
Attack: Establish what a pair density wave predicts that a charge density wave does not: a charge modulation at $2\mathbf{P}$ as a composite, particular Josephson signatures, a specific nodal structure. Confront the scanning-tunnelling evidence in Bi2212 and the anomalous interlayer transport in the lanthanum compounds at $p=1/8$. Use Q-E2's Landau analysis to separate necessary consequences from mere consistency.
Decides: Whether “intertwined orders” has a specific microscopic realisation or remains a framing. The discipline of demanding necessary consequences rather than consistency is what makes this answerable.

Q-E5a month · numbers against numbers

Why is $T_c$ suppressed precisely at $p=1/8$?

Why: A sharp commensuration effect in a strongly correlated system, with an identified mechanism candidate and measurable energies on both sides of the comparison. Rare in this subject.
Attack: Compute the energy gained by pinning a period-four stripe to the lattice at $p=1/8$, and compare it with the superconducting condensation energy extracted from specific-heat measurements. Ask whether the observed suppression of $T_c$ is quantitatively accounted for, or off by an order of magnitude.
Decides: A quantitative success would supply framework 6 with the numbers it currently lacks, turning near-degeneracy from a qualitative story into a calculation. A quantitative failure would indicate that the $1/8$ anomaly has a different origin.

Track F

Comparative tests

5 questions

Cheap, high-leverage, and mostly literature work. These decide whether we are studying a universal phenomenon or a peculiar material — which is the most consequential fork in the whole programme.

Q-F1two weeks · reference table

Is $T_c/T_F\approx10^{-2}$ beside a correlated insulator a law or a coincidence?

Why: Either there is a universal organising principle across seven unrelated material families, or the cuprates are special. Both answers redirect everything downstream.
Attack: Extend Q-A5's table across cuprates, iron-based superconductors, heavy fermions, organic charge-transfer salts, nickelates, twisted bilayer graphene, the unitary Fermi gas and the elemental superconductors. Columns: proximity to a correlated insulator or magnetic order, pairing symmetry and whether the gap changes sign, $T_c/T_F$, $k_F\xi$, strange metal present, pseudogap present.
Decides: If $T_c/T_F$ clusters, the controlling physics is generic and the cuprate-specific details are secondary — which would argue strongly for the bound-and-stiffness framing of §7 and Q-D1. If it does not cluster, identifying what breaks the pattern is the result.

Q-F2three weeks

What do the hydrides actually prove about the ceiling of phonon pairing?

Why: The control group. Settles which part of “high $T_c$” needs exotic physics at all, and corrects the most widespread misconception about this subject.
Attack: Work through Allen–Dynes and its strong-coupling asymptotics $T_c\propto\sqrt{\lambda\langle\omega^{2}\rangle}$, then the lattice-stability constraint: for given $\lambda$ and $\omega_{\log}$, when does the structure go unstable? Assemble measured $(\lambda,\omega_{\log},T_c,P)$ for H$_3$S, LaH$_{10}$ and the other hydrides and locate them relative to that boundary.
Decides: Whether the pressure requirement is a necessary consequence of large $\lambda$ or an accident of hydrogen chemistry. That determines whether ambient-pressure phonon superconductivity above $100\,\mathrm{K}$ is worth pursuing, which is a question with consequences beyond theory.

Q-F3three weeks · revisit twice yearly

Which cuprate-derived assumption do the nickelates break?

Why: The only genuinely new high-$T_c$ family in thirty years, and therefore the best available falsification test of every framework in §9.
Attack: Tabulate what the nickelates share with and lack relative to cuprates: La$_3$Ni$_2$O$_7$ under pressure at $T_c\approx80\,\mathrm{K}$, in compressively strained films at ambient pressure with onset near $40\,\mathrm{K}$ rising to roughly $48\,\mathrm{K}$ under added pressure, and the infinite-layer Nd$_{0.8}$Sr$_{0.2}$NiO$_2$ near $15\,\mathrm{K}$. Note the absences: no charge-transfer-insulator parent, Ni nominally $d^{8}$ rather than $d^{9}$, a bilayer $d_{z^{2}}$ bonding band that may dominate, rare-earth $5d$ self-doping bands. Then, for each framework, write down what it predicts for nickelates.
Decides: Any framework requiring a $d^{9}$ charge-transfer insulator with a large $J$ is in trouble if the nickelates prove to be the same phenomenon. Conversely, if nickelates are multiband $s^{\pm}$ physics, that strengthens framework 2 and localises the cuprate mechanism. Pairs directly with Q-B3, which makes the prediction this question would test.

Q-F4two weeks

Does FeSe on SrTiO$_3$ require a phonon contribution?

Why: A case where a conventional and an unconventional mechanism demonstrably coexist — a possibility the cuprate debate has mostly suppressed in favour of a winner-takes-all framing.
Attack: Assemble the evidence for interfacial-phonon enhancement — the replica bands in ARPES, substrate isotope substitution — against the spin-fluctuation baseline of bulk FeSe, and estimate the phonon contribution to $T_c$.
Decides: If a phonon can add tens of kelvin on top of an unconventional mechanism, then “which mechanism” is the wrong question for some materials, and the doping dependence of the cuprate isotope exponent (fact 11) may need rereading as a mixed effect rather than as an exclusion.

Q-F5two weeks

Is twisted bilayer graphene the same problem, and can it be used as a knob?

Why: It has flat bands, superconductivity adjacent to correlated insulators, and Planckian transport — but no local moments of the cuprate kind, and a bandwidth that is continuously tunable, which cuprates never offer.
Attack: List shared and unshared features. Then exploit the tunability: ask which cuprate correlations — $T_c$ against stiffness, strange-metal slope against $T_c$, pseudogap against doping — can be tested as functions of a continuously varied bandwidth by varying the twist angle.
Decides: Whether moiré systems can serve as an experimental knob on cuprate questions. That would be the most useful possible outcome of this comparison, because it converts an analogy into an instrument.

Sequencing, and traps

Thirty-two questions is a programme, not a plan. Here is the order I would take them in, and the failure modes that have consumed other people's years.

A six-month opening

Weeks 1–2 — put the algebra in your hands. Q-A1, Q-A2, Q-A5. Two exact diagonalisations and a table. Cheap, and they convert §2, §6 and §7 from assertions into things you have verified.
Weeks 3–6 — find out what is already eliminated and what is already predicted. Q-C1 and Q-D5. Pure reading. The first tells you which strange-metal mechanisms are already dead; the second tells you whether the ab initio route has quietly become predictive, which changes the weight to put on everything else.
Weeks 7–12 — test this document's two central claims. Q-A6 for §7's stiffness ceiling, Q-A3 for §6's pairing kernel and the $t'$ trend. If Q-A6 fails, stop and rewrite §7 before going further.
Weeks 13–16 — produce something of your own, in your own idiom. Q-E2. Paper-and-pencil Landau theory of coupled superconducting, charge-density-wave and pair-density-wave order parameters. Rigorous, independent of anybody's numerics, and it yields statements that existing data can already test.
Months 5 onward — the main line. Q-C2, with Q-C3 running in parallel as its empirical check. The bet is that the normal state is the real problem and that disorder-averaged large-$N$ field theory is the tool for it.

Background threads throughout: Q-F1 and Q-F3 as reading that accumulates, since the comparative question decides whether any of this generalises, and the nickelate situation changes every few months.

The logic of that order is worth stating explicitly, because it inverts the usual instinct. The first twelve weeks contain no original research. They are calibration and triage, and they cost six weeks of effort in exchange for the ability to read any paper in this field and judge it. In a subject this large and this contested, that ability is the scarce resource. Step 4 then produces a real result from symmetry alone, which hedges against step 5 taking longer than planned — and step 5 is a genuine bet, which is what research is supposed to be.

Eight traps

Trap 1 · fitting the phase diagram

The phase diagram has perhaps six qualitative features. Most mean-field theories of it have six or more adjustable quantities. Reproducing the diagram is therefore not evidence for anything, and a paper whose central achievement is a figure resembling Figure 2 has demonstrated only that its author can count. Demand predictions of things not used in the fitting: a materials ordering, a sign, a ratio, an exponent.

Trap 2 · treating “d-wave plus a dome” as success

Every framework in §9 that addresses pairing at all produces a $d$-wave gap, because §6 shows the symmetry is essentially forced by the lattice and the magnetism. Getting $d$-wave is table stakes, not a result.

Trap 3 · assuming the normal state away

Any calculation of $T_c$ that begins from a Fermi liquid with sharp quasiparticles has assumed away the part of the problem nobody understands. This includes almost all of the spin-fluctuation literature. The results may still be useful — framework 2's predictive record is real — but they are not derivations, and Q-B2 is how you find out how far to trust them.

Trap 4 · trusting numerics outside its validated window

Determinantal Monte Carlo has an exponential sign problem away from half filling. DMRG on four- and six-leg cylinders is not two-dimensional and systematically over-rewards stripes. Cluster dynamical mean field has cluster-size artefacts that do not decrease smoothly. Variational methods compare states whose energy differences are smaller than their own bias. Every one of these is a good method; none is reliable where the competing states differ by $10^{-3}t$ per site, which is exactly where the interesting competitions live. Count the independent methods before believing a result, and treat single-method claims as hypotheses.

Trap 5 · conflating high $T_c$ with unconventional

The hydrides reach $250\,\mathrm{K}$ by entirely conventional electron–phonon pairing. Whatever is mysterious about cuprates, it is not that $100\,\mathrm{K}$ is too high for a superconductor. It is that $100\,\mathrm{K}$ is reached at ambient pressure, with a thousand times fewer carriers, out of a non-Fermi-liquid normal state, with a sign-changing gap. Keep the actual mystery in view.

Trap 6 · treating $T^{*}$ as a phase boundary

Resistivity, Knight shift, specific heat, ARPES, Nernst and ultrasound all define a $T^{*}$, and they do not agree. Some of these are crossovers in a smooth function with no transition anywhere. Before building on a value of $T^{*}$, check which probe produced it and whether it marks anything sharp.

Trap 7 · chasing an exact solution

There is no small parameter (§1). The impulse to look for the solvable model that captures it all is natural for a field theorist and it is a trap here — as Q-C4 examines in the one case where a solvable model genuinely exists. What survives is bounds, exact identities, symmetry constraints, and controlled limits of deliberately modified models, each honest about what was modified.

Trap 8 · single-sample claims

Cuprate samples differ in disorder, oxygen ordering, inhomogeneity and surface quality, and many famous anomalies have turned out to be sample-specific. The bar for a new fact in this field is reproduction across families, not across groups studying the same crystal.

What a good answer looks like

At the end of any one of these thirty-two questions, you should be able to say a sentence of the following shape: “Quantity $X$ has value $Y$, which rules out mechanism $Z$ and requires that any surviving mechanism also produce $W$.” If the answer cannot be stated in that shape, the question was not sharp enough, and the right move is to go back and sharpen it rather than to keep calculating.

What to read, and why

Grouped by what you want from it, not by chronology. Each entry says what to take away, because the failure mode with a reading list this size is reading everything and retaining nothing.

Orientation · read these two first

Keimer, Kivelson, Norman, Uchida and Zaanen, Nature 518, 179 (2015).The standard modern overview, written by five people who disagree with each other, which is why it is trustworthy. Take the phenomenology and the statement of what is established.
Proust and Taillefer, Annual Review of Condensed Matter Physics 10, 409 (2019).The transport and thermodynamic evidence assembled by the people who measured most of it. Take facts 10, 13 and the case for $p^{*}$.

The strong-correlation framework

Anderson, Science 235, 1196 (1987).Two pages that started the resonating-valence-bond programme. Read it for the style of argument, not the details.
Lee, Nagaosa and Wen, Reviews of Modern Physics 78, 17 (2006).The canonical treatment of doping a Mott insulator, including the slave-boson gauge theory. The single most useful reference for a field theorist entering this subject, because it is written in a language you already speak.
Zhang and Rice, Physical Review B 37, 3759 (1988); Emery, Physical Review Letters 58, 2794 (1987).Steps 1 and 2 of §4, in the originals. Short, and worth reading before relying on them.

The pairing interaction

Scalapino, Reviews of Modern Physics 84, 1383 (2012).“A common thread”: the momentum-dependent-repulsion mechanism across cuprates, pnictides, organics and heavy fermions. The best single account of framework 2 and of why it keeps working.
Anderson, Science 316, 1705 (2007).“Is there a glue in cuprate superconductors?” Read it together with the Maier–Poilblanc–Scalapino reply, as a worked example of a dispute that was half semantic and half real. Background for Q-B1.
O'Mahony, Ren, Chen, Davis and co-workers, PNAS 119, e2207449119 (2022).Pair density correlated with the local charge-transfer energy, measured site by site. The most direct experimental evidence for a superexchange origin of pairing.

Models and numerics · what the simulations actually show

Qin, Schäfer, Andergassen, Corboz and Gull, Annual Review of Condensed Matter Physics 13, 275 (2022).The Hubbard model from a computational perspective: what each method can and cannot do, and where they agree. Read before trusting any numerical claim, including the ones in §5.
Zheng, Chung, Corboz, White, Zhang, Chan and co-workers, Science 358, 1155 (2017).Stripes, not superconductivity, in the plain Hubbard model at $p=1/8$. Four methods agreeing.
Xu, Chung, Qin, Schollwöck, White and Zhang, Science 384, eadh7691 (2024).Superconductivity coexisting with partially filled stripes once $t'$ is switched on, with the right electron–hole asymmetry. Read as a pair with the 2017 paper; together they are the most important numerical result in the subject.

Orders, and what sets $T_c$

Emery and Kivelson, Nature 374, 434 (1995).The phase-stiffness argument of §7, in four pages. If you read one paper from this document, read this one.
Fradkin, Kivelson and Tranquada, Reviews of Modern Physics 87, 457 (2015).Intertwined orders, and the Landau-theory machinery that Q-E2 asks you to use.
Hazra, Verma and Randeria, Physical Review X 9, 031049 (2019).Upper bounds on $T_c$ from optical sum rules. The template for Q-D1, and a model of how to do rigorous work in an unrigorous field.
Pavarini, Dasgupta, Saha-Dasgupta, Jepsen and Andersen, Physical Review Letters 87, 047003 (2001).$T_c^{\max}$ correlated with a band-structure parameter across families. The origin of the $t'$ story and the starting point for Q-D3.

The normal state

Varma, Littlewood, Schmitt-Rink, Abrahams and Ruckenstein, Physical Review Letters 63, 1996 (1989).Marginal Fermi liquid. Postulated, not derived, and still the most successful phenomenology of the strange metal.
Patel, Guo, Esterlis and Sachdev, Science 381, 790 (2023).The derivation the previous entry lacked: spatially random Yukawa coupling gives marginal-Fermi-liquid behaviour, Planckian transport and $T\ln(1/T)$ specific heat. The core reference for Q-C2, and the natural entry point for a field theorist.
Hartnoll and Mackenzie, Reviews of Modern Physics 94, 041002 (2022).Planckian dissipation surveyed honestly, by an author from each side of the holographic divide. Take the empirical regularity and the absence of a theorem.
Legros and co-workers, Nature Physics 15, 142 (2019).The measurement that pins $\alpha\approx1$ across several cuprates. The data Q-C3 builds on.
Phillips, Yeo and Huang, Nature Physics 16, 1175 (2020).Superconductivity in an exactly solvable doped Mott insulator. Read it alongside its critics; that disagreement is Q-C4.

The comparison classes

Sun and co-workers, Nature 621, 493 (2023).La$_3$Ni$_2$O$_7$ at $80\,\mathrm{K}$ under pressure: the first genuinely new high-$T_c$ family in decades. Follow the thin-film and strained-film work that followed it, which is moving fast.
Drozdov and co-workers, Nature 525, 73 (2015) and 569, 528 (2019).H$_3$S and LaH$_{10}$. The control group for Q-F2, and the cure for trap 5.
Kaneko and Imada, arXiv:2606.08181 (2026).Ab initio effective Hamiltonians for Hg1223 solved with a neural-network variational method, with a predicted $T_c$ maximum near $30\,\mathrm{GPa}$. The subject of Q-D5. Recent enough that it should be read critically rather than accepted.

On citations in this document

Attributions above are given by author and journal so they can be checked, and the numbered facts of §3 name the groups responsible where a single measurement is decisive. Everything asserted here is either in that reading list or derivable from it in a few lines. Where I have made an editorial judgement rather than reported a consensus — the thesis of §1, the stiffness reading of §7, the scorecard of §9 — I have said so at the point of claiming it.

Constraint	What it rules out
Range. \(\rho\propto T\) from \(T_c\) to \(>1000\,\mathrm{K}\), no change of slope.	Any single boson with a characteristic frequency: phonons give \(\rho\propto T\) only for \(T>\Theta_D\) and \(T^{5}\) below. Any quantum-critical fan of finite width in \(T\).
Magnitude. Sheet resistance reaches \(h/e^{2}\) and keeps rising linearly.	Any Boltzmann transport picture, since the implied mean free path falls below a lattice constant. Resistivity saturation, which every conventional metal shows and this one does not.
Slope. \(\hbar/\tau=\alpha k_BT\) with \(\alpha\approx1\), the same in several families and in unrelated materials.	Mechanisms with a free coupling constant, which would give a material-dependent \(\alpha\). The universality is the strongest hint that something scale-free is at work.
Momentum. The current must decay, so momentum must be relaxed.	Clean critical scattering off a boson at a fixed wave vector: electron–boson collisions conserve total momentum, and a translationally invariant critical theory therefore does not give a finite resistivity at all. Something must break translation invariance.

What would count as an answer

The energy scales

Fourteen facts a theory must fit

From crystal to Hamiltonian

What we actually know about the models

The rigorous results, and how little they cover

The sign problem, stated precisely

The three results worth internalising

Why repulsion gives \(d\)-wave

The weak-coupling route: a sign-changing gap can exploit repulsion

The strong-coupling route: the attraction is already in the Hamiltonian

Is the glue question decidable? Yes, and partly decided

Amplitude versus stiffness: what actually limits \(T_c\)

Do the arithmetic

The normal state, which may be the harder problem

The pseudogap: two readings, and how to tell them apart

The strange metal, and why every mechanism fails a constraint

The Luttinger problem, which is a gift

Seven frameworks, side by side

1 · Phonons and Eliashberg theory

2 · Spin-fluctuation exchange

3 · Resonating valence bonds and the gauge theory of a doped Mott insulator

4 · Preformed pairs and phase fluctuations

5 · Quantum criticality at \(p^{*}\)

6 · Intertwined orders

7 · Strange metal first

The scorecard

Settled, probable, contested

Settled

Probable

Contested

The question bank

Calibration

The pairing interaction

The normal state

The scale of \(T_c\)

Intertwined orders and the dome

Comparative tests

Sequencing, and traps

Eight traps

What to read, and why

Orientation · read these two first

The strong-correlation framework

The pairing interaction

Models and numerics · what the simulations actually show

Orders, and what sets \(T_c\)

The normal state

The comparison classes