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The foot, the fan, and the cuprate phase diagram: Fermi-volume-changing quantum phase transitions

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that both the 'foot' and the 'fan' of cuprate strange metal behavior are governed by quantum phase transitions in a disordered metal, with the fan described by a two-dimensional Yukawa-SYK field theory whose SYK-type…

desk verdict Sachdev's perspective cleanly frames the foot/fan program around two Fermi-volume-changing transitions, but the fan's extended-boson assumption remains the load-bearing open point. read the letter →

arxiv 2501.16417 v8 pith:6XPNIDU6 submitted 2025-01-27 cond-mat.str-el hep-thquant-ph

Subir Sachdev This is my paper · ORCID
classification cond-mat.str-elhep-thquant-ph
keywords cupratesuperconductorsstrangemetalquantumcriticalfanfractionalizedFermiliquidpseudogapSYKmodelsurfacereconstructionGriffithsphase
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to explain the two anomalous metallic regimes of the hole-doped cuprate phase diagram with a single framework based on quantum phase transitions in a disordered metal. The low-temperature 'foot' of strange metal behavior is attributed to the onset of spin density wave order, where disorder localizes the critical bosons and produces an extended Griffiths-type phase. The high-temperature 'fan' is attributed to a Fermi-volume-changing transition between a Fermi liquid and a fractionalized Fermi liquid (FL*), a state whose small Fermi surface violates Luttinger's rule because part of the electronic charge is carried by a background spin liquid. The fan's critical field theory maps to a two-dimensional Yukawa-SYK model with spatially random couplings, whose SYK-type solution gives linear-in-temperature resistivity, Planckian optical scattering, a $\sim T\ln(1/T)$ entropy, and a singular thermopower, in agreement with transport and optical experiments. A sympathetic reader cares because this would unify the two strange metal regimes and identify the pseudogap metal as a thermal window above a fractionalized state that confines into $d$-wave superconductivity at lower temperatures.

What carries the argument

The load-bearing object is the two-dimensional Yukawa-SYK (2dYSYK) model: a lattice of fermions coupled through a complex Higgs boson with spatially random Yukawa couplings and a random potential. In the fan regime all fermionic and bosonic eigenmodes are extended, so the disorder self-averages and the Green's functions satisfy closed SYK-type equations whose solutions give marginal-Fermi-liquid self-energies, linear-in-$T$ resistivity, and a Planckian relaxation time. The boson propagator $D(i\Omega,q) \sim [Kq^2 + c_1|\Omega| - i c_2\Omega + m^2(T)]^{-1}$ carries the essential difference from the spin-density-wave theory: the $-ic_2\Omega$ term encodes particle-hole asymmetry and produces the singular thermopower contribution.

What would settle it

A falsifying observation would be a spatially resolved neutron or resonant inelastic x-ray scattering measurement in the foot regime of an underdoped cuprate showing that the critical spin fluctuations are spatially extended and homogeneous at the lowest energies, since the paper's theory predicts localized bosonic eigenmodes with an exponentially decaying envelope and a logarithmic growth of localization length as energy is lowered; alternatively, a Seebeck coefficient in the fan that lacks the predicted linear-$T$ 'skewed marginal Fermi liquid' enhancement would rule out the FL-FL* critical point for that doping.

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Extended reading notes

Core claim

The central claim is that the quantum critical fan of the hole-doped cuprates is controlled by a Fermi-volume-changing quantum phase transition without symmetry breaking, from a large-Fermi-surface Fermi liquid (FL) to a fractionalized Fermi liquid (FL*), and that in the presence of spatial disorder this transition is described by the two-dimensional Yukawa-SYK model. The FL* phase has a small Fermi surface that violates the Luttinger count because the missing electron states are replaced by a deconfined spin liquid, and the same pseudogap metal can also be described by a dual 'holon metal' in which the roles of spinons and holons are exchanged. The field theory carries a pronounced particle-hole asymmetry represented by the $-ic_2\Omega$ term in the Higgs boson propagator, an asymmetry absent in the symmetry-breaking theory of the foot; this term generates a singular 'skewed marginal Fermi liquid' enhancement of thermopower. Solving the self-consistent SYK equations in the regime of extended fermionic and bosonic eigenmodes reproduces linear-in-temperature resistivity, the Planckian optical scattering rate, the observed optical conductivity, and the thermal scaling of the entropy as $\sim T\ln(1/T)$ rather than the extensive zero-temperature entropy of the pure SYK model. The paper further describes how lowering temperature confines the fractionalized excitations, with Higgs condensation producing $d$-wave superconductivity, charge order, and a vortex-core structure consistent with observations.

Load-bearing premise

The whole picture depends on the assumption that a solvable model with random, large-$N$ interactions faithfully represents the universal critical physics of the real quasi-two-dimensional cuprate in the fan regime, so that self-averaging and the large-flavor limit do not change the qualitative behavior.

Editorial extensions

If this is right

  • The linear-in-temperature resistivity and Planckian scattering rate observed across the fan follow from the SYK-type solution without tuning, because the relaxation time is set by temperature rather than by the interaction strength.
  • The same field theory yields a $\sim T\ln(1/T)$ entropy at criticality instead of the extensive zero-temperature entropy of the original SYK model, matching specific-heat observations.
  • The pseudogap metal is a finite-temperature manifestation of a fractionalized Fermi liquid (or its dual holon metal), so its small Fermi pockets require no static magnetic or charge order; the two dual descriptions predict different pocket areas, $p/8$ for FL* and $p/4$ for the holon metal.
  • Lowering temperature confines the spinons, and Higgs condensation from the spin-liquid background produces $d$-wave superconducting order with four nodal quasiparticles, along with charge order and a vortex-core spectral structure.
  • The low-temperature 'foot' is an extended quantum Griffiths phase of localized spin-density-wave modes, consistent with recent neutron scattering observations of critical spin fluctuations in La$_{2-x}$Sr$_x$CuO$_4$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if the FL-FL* quantum phase transition is the correct universal theory of the fan, the same mechanism should appear in heavy-fermion materials with Fermi-volume-changing transitions, where the predicted thermopower enhancement and linear-$T$ resistivity could be isolated with more controlled disorder.
  • Editorial extension: the self-averaging assumption implies a sharp spatial distinction between the foot and the fan, with localized spin-density-wave bosons at low temperature and extended bosons at higher temperature; spatially resolved probes such as scanning tunneling spectroscopy or momentum-resolved resonant inelastic x-ray scattering could map this localization crossover in a single sample.
  • Editorial extension: the particle-hole asymmetry encoded in $c_2$ should show up as an asymmetry in the optical sum rules or in the thermopower when comparing hole-doped and electron-doped cuprates, providing a testable consequence beyond the compounds analyzed in the paper.
  • Editorial extension: a quantitative prediction of the paper's framework is that the confinement crossover temperature at which the pseudogap metal gives way to $d$-wave superconductivity should track the Higgs condensation scale; comparing this scale across different cuprate families would test whether the same fractionalized state underlies all of them.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This perspective article argues that the phase diagram of hole-doped cuprates can be understood in terms of two Fermi-volume-changing quantum phase transitions: a conventional FL–SDW transition with spatial disorder, which produces the low-temperature 'foot' of strange-metal transport, and a fractionalized FL–FL* transition without symmetry breaking, whose quantum critical 'fan' is described by a two-dimensional complex Yukawa–Sachdev–Ye–Kitaev (2dYSYK) model. The paper presents the effective Lagrangians L1 and L2 in Eqs. (7) and (10), the self-consistent SYK equations (12) for the fan, and a confinement-crossover discussion based on fermion–boson duality of the underlying spin liquid. Comparisons are made with resistivity, optical conductivity, thermopower, neutron scattering, RIXS, and quantum oscillation (Yamaji) experiments, with explicit falsifiable predictions such as p/8 Fermi pockets and a localized-boson 'foot'.

Significance. The paper is a synthesis of a broad body of prior work rather than a new derivation, but it is valuable as a coherent statement of a specific framework with falsifiable consequences. Its strengths are that the assumptions are stated explicitly (e.g., the need for extended Φ eigenmodes, the unresolved FL*/holon-metal ambiguity), and that several quantitative predictions—p/8 pockets identified via the Yamaji effect, thermopower enhancement, the spinon continuum in RIXS, and the localized SDW boson foot—are compared with independent experiments. The theory is not parameter-free, but it does make predictions that could be refuted by future measurements. If the central 2dYSYK self-averaging assumption is validated, the framework would provide a unified account of the foot and fan.

major comments (2)
  1. [§4.2, Eqs. (10)–(12)] The central claim that the fan is described by the self-averaging SYK solution (12) rests on the statement that 'the solution of (10) by (12) should be valid as long as the Φ eigenmodes remain extended'. The manuscript offers no derivation or numerical evidence that the complex Higgs boson of the disordered FL–FL* transition has extended eigenmodes in the regime of interest. This is not a pedantic gap: Section 3 and Fig. 7 show that the analogous SDW bosons of L1 localize at low energies, with a non-monotonic localization length and infinite-randomness behavior at the lowest energies. The crossover between localized (foot) and extended (fan) bosonic regimes is not computed for L2, and no argument is given for why the particle-hole asymmetric complex Higgs field should evade the Harris-disorder localization that is central to the foot description. Consequently, the abstract's statement that the 2dYSYK analysis 'successfully models' the fan observations is stronger than what is demonstrated; Eqs. (12) are an ansatz whose regime of validity is not established within the manuscript.
  2. [Figs. 13 and 15; §4.2] The quantitative comparisons supporting the 'successfully models' claim are reproduced from prior work without the parameter values or fitting details. For example, the lower panels of Fig. 13 show a 2dYSYK computation of the transport relaxation time from Ref. [52], but the manuscript does not state the values of the coupling g, the spatial disorder strength g′, the random potential v, the tuning parameter s, the number of flavors, or the temperature range used in that calculation. Similarly, Fig. 15 reproduces the p/8 pocket prediction from Ref. [147], but no expression for the pocket area as a function of doping is given, so the reader cannot assess the accuracy of the claimed agreement with the Yamaji-effect observation. Since the abstract makes a quantitative modeling claim, the absence of any fit or error measure in the present paper leaves the claim unverifiable from the manuscript alone.
minor comments (5)
  1. [Section 3, after Eq. (9)] The text says 'non-pertubative quantum Monte Carlo'; 'non-pertubative' should be 'non-perturbative'.
  2. [Fig. 1 caption] The name 'Kammerlingh Onnes' should be spelled 'Kamerlingh Onnes'.
  3. [Eq. (11)] The origin and sign of the c2 term (the particle-hole asymmetry) would benefit from one sentence of explanation, as it is the key difference between L1 and L2 and is later invoked for thermopower.
  4. [Section 4.2] The phrase 'It is remarkable that ... the critical properties ... are essentially the same' is an assertion of universality; it would be helpful to state explicitly which quantities are universal and cite the specific equations in Ref. [44] that establish this.
  5. [Section 5] The confinement crossover is discussed in terms of possible Higgs condensations, but the text acknowledges that 'remains to be understood'; this limitation is stated honestly, but the abstract's phrase 'The confinement crossover ... is also discussed' could be softened to 'is addressed qualitatively'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step identified: the fan theory is an explicit SYK-type mapping ansatz whose transport and spectroscopic output is checked against independent experiments, not a reduction of the input by construction.

full rationale

The paper's chain is: (i) identify the foot with a disordered FL-SDW QPT and solve the boson-only effective theory (8)-(9) by strong-disorder RG/self-consistent methods; (ii) identify the fan with the FL-FL* Lagrangian L2 in (10); (iii) in the assumed extended-boson regime, pass to the large-N flavor-random SYK equations (12), whose solutions are then used for resistivity, optical conductivity, thermopower, entropy, and Fermi-surface pocket comparisons. Each step is a model prescription or a nontrivial solution, not an identity. The 'successful modeling' is not definitional: the SYK limit has independent content (e.g., it fails on the zero-temperature entropy and is corrected to T ln(1/T)), and the p/8 pocket area is a geometric consequence of the FL* state, matched to an independent Yamaji-effect measurement. Heavy self-citation is present (Refs. [40,42,44,50,52,55,57]), but those works are published, calculation-based, and externally benchmarked, so they are not circular in the sense of this review. The main physical weakness — that the validity of (12) requires Phi eigenmodes to remain extended, which is assumed rather than derived, while the companion SDW analysis shows localization — is a correctness/control risk, not a circularity. No equation is equivalent to another by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The paper is a perspective; it uses established concepts such as FL*, spinons, holons, and the 2dYSYK model rather than introducing new entities. The main load-bearing parameters are the couplings of the SYK model, which are tuned in the cited computational studies to match cuprate phenomenology.

free parameters (4)
  • Yukawa coupling g
    Sets the strength of fermion-boson coupling in L1 and L2; its value is chosen in the referenced 2dYSYK computations to match transport.
  • spatial disorder strength g'
    Harris disorder coupling; controls localization of bosons and the foot regime.
  • random potential v
    Elastic scattering potential; assumed weak.
  • tuning parameter s
    Distance from the quantum critical point; used to describe foot/fan regions.
assumptions (3)
  • domain assumption Existence of a FL* phase with a background spin liquid in the hole-doped cuprates (Section 4)
    The pseudogap metal is assumed to be described by a fractionalized state with small Fermi surface and a spin liquid background, enabling Fermi volume change without symmetry breaking.
  • ad hoc to paper Validity of the 2dYSYK self-averaging treatment for the fan (Section 4.2, Eq. (12))
    The SYK equations require all fermionic and bosonic eigenmodes to be extended, and random all-to-all flavor couplings. This is an ansatz not derived from a microscopic Hubbard model.
  • domain assumption Fermionic eigenmodes remain extended in the foot regime (Section 3)
    The localized SDW bosons co-exist with extended fermions; if fermions localize, transport would be different.

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Cite this review

Pith. "Pith review of The foot, the fan, and the cuprate phase diagram: Fermi-volume-changing quantum phase transitions." pith.science (2026). https://pith.science/paper/6XPNIDU6

@misc{pith2026250116417,
  author       = {Pith},
  title        = {Pith review of: The foot, the fan, and the cuprate phase diagram: Fermi-volume-changing quantum phase transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XPNIDU6}},
  note         = {Machine review of arXiv:2501.16417}
}
abstract

A Fermi liquid with a 'large' Fermi surface (FL) can have a quantum phase transition to a spin density wave state (SDW) with reconstructed 'small' Fermi pockets. Both FL and SDW phases obey the Luttinger constraints on the volume enclosed by the Fermi surfaces. Critical spin fluctuations lead to spin-singlet $d$-wave pairing, as observed in the cuprates. Studies of the influence of spatial disorder on the FL-SDW quantum phase transition predict an extended quantum-critical Griffiths-type phase at low temperatures on the large Fermi surface side. These computations agree with the 'foot' of strange metal transport, and recent low temperature neutron scattering observations on La$_{2-x}$Sr$_x$CuO$_4$. However, this theory cannot explain the higher temperature pseudogap and the 'fan' of strange metal behavior of the hole-doped cuprates. Here we need to consider underlying Fermi-volume-changing quantum phase transitions without symmetry breaking. Then the small Fermi surface phase does not obey the Luttinger constraint, and the pseudogap metal is described by thermal fluctuations above a 'fractionalized Fermi liquid' (FL*) or a 'holon metal', with the descriptions related by a duality on a background spin liquid. The quantum critical fan is described using a field theory for an underlying FL-FL* quantum phase transition in the presence of spatial disorder. This field theory can be mapped to a form which can be analyzed using the methods of the Sachdev-Ye-Kitaev model. Such an analysis successfully models linear-in-temperature resistivity, optical conductivity and thermopower observations in the quantum critical fan. The confinement crossover connecting these lower and higher temperature descriptions is also discussed.

Figures

Figures reproduced from arXiv: 2501.16417 by the authors.

Figure 1
Figure 1. Jan Zaanen and the author in 2012. In the background is the Lorentz wall [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Phase diagram of the hole-doped cuprate superconductors from Keimer [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Color density plot of the resistivity exponent [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Theories of the phase diagram of Fig [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: From Scheie et al. [63]. Neutron scattering spectrum in the triangular lattice antiferromagnet KYbSe2 (top panel), compared with the Schwinger boson (SB) theory of Ghioldi et al. [64] and DMRG. We argue for a similar phenomenology for the cuprates in [PITH_FULL_IMAGE:…
Figure 6
Figure 6. Figure 6: Phase diagram of the hole-doped cuprates from Radaelli [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Adapted from Patel et al. [50]. (a) Localization length Lα of overdamped bosonic eigenmodes of Seϕ in (9) as a function of their energy eα. (b,c,d) Pictures of the corresponding bosonic eignfunctions. The ‘foot’ is described by the localized bosons. The universal theor…
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3. Dynamical spin structure factor of the [PITH_FULL_IMAGE:figures/full_fig_p016_3.png]
Figure 10
Figure 10. Figure 10: Phases of a Kondo lattice model with f-electron spins coupled to a conduction band of c electrons of density p. At large Kondo coupling, JK, the condensation of a Higgs boson Φ leads to the conventional FL which obeys the Luttinger count. The FL* phase is obtained whe…
Figure 11
Figure 11. Figure 11: Ancilla theory of a FL* phase in a single band model. After a canonical [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Summary of FL* and FL phases of Kondo lattice and one-band models. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 14
Figure 14. Figure 14: Cartoon representations of the holon metal and FL* states. The holon metal [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: From Kaul et al. [147], with added text on top. Fermi surfaces in the square lattice Brillouin zone obtained by doping an insulating antiferromagnet in the vicinity of a quantum phase transition from the Néel state to the valence bond solid (VBS). The Néel state has 2…

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Reviewed August 10, 2026 · model on record in the stance chip above.