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arxiv: 2605.17227 · v1 · pith:DJVL6YSLnew · submitted 2026-05-17 · ❄️ cond-mat.supr-con · cond-mat.str-el

Disorder effect on the superfluid density and the origin of the pseudogap end point in the cuprate superconductors

Rong Cheng , Tao Li , Jianhua Yang This is my paper

Pith reviewed 2026-05-19 23:25 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords superfluid densitycuprate superconductorsdisorderpseudogapt-J modelRVB stateMott transition
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The pith

A variational calculation finds the zero-temperature superfluid density of the disordered t-J model rises monotonically with doping and stays robust to disorder.

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

The paper develops a two-step variational method to compute the zero-temperature superfluid density for the disordered t-J model of cuprate superconductors. First an unrestricted RVB ground state is optimized, then a variational state is built to capture the paramagnetic current response, from which the superfluid density is read off the curvature of the energy versus applied field. The resulting density increases steadily with hole doping and scales linearly with the total optical weight. This pattern agrees with measurements on underdoped cuprates but clashes with the drop seen in overdoped samples. The contrast is used to argue that the pseudogap endpoint marks a change from a doped Mott insulator to a more conventional metal.

Core claim

The zero-temperature superfluid density ρ_s(0) obtained from the variational procedure on the disordered t-J model is robust against disorder, increases monotonically with doping concentration x, and scales linearly with the total optical weight. This behavior matches underdoped cuprate data but is inconsistent with overdoped data, thereby supporting the existence of a Mott transition at the pseudogap end point that separates a doped-Mott-insulating metal from a Fermi-liquid-like metal.

What carries the argument

Two-step variational procedure that first optimizes an unrestricted RVB ground state for the disordered t-J model and then constructs the paramagnetic current-response state whose energy curvature with respect to an external electromagnetic field yields ρ_s(0).

If this is right

  • ρ_s(0) scales linearly with the total optical weight across the doping range studied.
  • The computed ρ_s(0) remains monotonic and disorder-robust even when disorder is included.
  • The underdoped regime behaves as a doped Mott insulator while the overdoped regime does not.
  • The pseudogap end point coincides with the location where superfluid density would reach its maximum if the two regimes meet.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Disorder alone cannot account for the observed drop of superfluid density in the overdoped cuprates.
  • The linear scaling with optical weight suggests the superfluid density is controlled by the kinetic energy of the doped holes rather than by pair-breaking effects.
  • If the transition at the pseudogap end point is indeed a Mott transition, then the normal-state quasiparticles should change character from incoherent to coherent across that doping.

Load-bearing premise

The variational state built to represent the paramagnetic current response on the RVB ground state correctly gives the true zero-temperature superfluid density of the disordered t-J model.

What would settle it

Exact diagonalization or DMRG results on finite disordered t-J clusters that show a clear maximum or non-monotonic doping dependence of ρ_s(0) would contradict the variational monotonicity.

Figures

Figures reproduced from arXiv: 2605.17227 by Jianhua Yang, Rong Cheng, Tao Li.

Figure 1
Figure 1. Figure 1: FIG. 1: The evolution of the ODLRO with the strength of the [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The doping-disorder strength evolution of the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The doping-disorder strength evolution of the to [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison between the zero temperature superfluid [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The doping-disorder strength evolution of the ratio [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

A major puzzle in the study of the cuprate superconductivity is the origin of the pseudogap end point. Intriguingly, such a critical doping is also where the superfluid density of the system reaches its maximum. A non-monotonic doping dependence of the superfluid density is rather unusual since the Drude weight of the cuprate system is found to increase monotonically with the doping concentration. It is generally believed that such a peculiar behavior should be attributed to both the strongly correlated nature of the cuprate system and the disorder effect. In this work, we develop a variational theory for the zero temperature superfluid density of the disordered $t-J$ model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state for the disordered $t-J$ model. Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density $\rho_{s}(0)$ is then extracted from the curvature of the variational ground state energy of the system as a function of the external electromagnetic field. We find that $\rho_{s}(0)$ computed in this way is remarkably robust against the disorder effect. More specifically, we find that $\rho_{s}(0)$ is a monotonically increasing function of doping concentration $x$ and scales linearly with the total optical weight. This is consistent with the observation in the underdoped cuprates but is strongly at odd with the behavior in the overdoped cuprates. The strong contrast between the disorder effect in the underdoped and the overdoped regime lends strong support to our previous proposal that there exist a Mott transition between a doped-Mott-insulating metal in the underdoped regime and a fermi-liquid-like metal in the overdoped regime around the pseudogap end point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a variational theory for the zero-temperature superfluid density ρ_s(0) in the disordered t-J model for cuprates. It proceeds in two steps: (1) unrestricted optimization of an RVB variational ground state for the disordered t-J model, and (2) construction of a variational state describing the paramagnetic current response on that RVB state. ρ_s(0) is extracted from the curvature of the variational energy with respect to the external electromagnetic field. The central findings are that ρ_s(0) is remarkably robust to disorder, monotonically increases with doping concentration x, and scales linearly with the total optical weight. This matches observations in underdoped cuprates but contrasts with overdoped behavior, supporting the authors' prior proposal of a Mott transition at the pseudogap endpoint separating a doped-Mott-insulating metal from a Fermi-liquid-like metal.

Significance. If the variational construction for the paramagnetic response accurately reproduces the linear response of the disordered t-J model, the results would provide a concrete theoretical account of why superfluid density behaves differently under disorder in underdoped versus overdoped cuprates. The reported monotonicity and linear scaling with optical weight would then constitute a useful benchmark for understanding the pseudogap endpoint and the role of strong correlations versus disorder.

major comments (2)
  1. [Abstract (two-step procedure) and associated methods section] The reported robustness, monotonic increase of ρ_s(0) with x, and linear scaling with optical weight all rest on the second step of the procedure (construction of the variational state for the paramagnetic current response on the optimized RVB state, followed by extraction via second derivative with respect to vector potential). No explicit demonstration is provided that this response-state ansatz satisfies the f-sum rule, reproduces the correct linear response, or matches known limits of the disordered t-J Hamiltonian; without such checks the monotonicity and disorder robustness could be artifacts of the variational restriction rather than properties of the model.
  2. [Final interpretive paragraph] The interpretive statement that the underdoped/overdoped contrast 'lends strong support to our previous proposal' of a Mott transition at the pseudogap endpoint reduces the new calculation largely to a consistency check on earlier work by the same authors. An independent, falsifiable prediction (e.g., a specific disorder-induced feature testable by future experiment or exact diagonalization) would be required to elevate the support beyond circularity.
minor comments (2)
  1. [Abstract] The abstract states the central numerical findings but supplies no explicit equations for the variational energy curvature, error estimates on the reported monotonicity, or direct comparisons against clean-limit or small-system benchmarks.
  2. Notation for the RVB state, the specific form of the paramagnetic response ansatz, and the definition of total optical weight should be introduced with numbered equations at first use to allow readers to reproduce the curvature calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract (two-step procedure) and associated methods section] The reported robustness, monotonic increase of ρ_s(0) with x, and linear scaling with optical weight all rest on the second step of the procedure (construction of the variational state for the paramagnetic current response on the optimized RVB state, followed by extraction via second derivative with respect to vector potential). No explicit demonstration is provided that this response-state ansatz satisfies the f-sum rule, reproduces the correct linear response, or matches known limits of the disordered t-J Hamiltonian; without such checks the monotonicity and disorder robustness could be artifacts of the variational restriction rather than properties of the model.

    Authors: We acknowledge the validity of this concern. While the variational ansatz for the paramagnetic response follows the standard construction used in prior studies of the clean t-J model, we agree that explicit checks are needed to rule out artifacts. In the revised manuscript we will add a dedicated subsection demonstrating that the ansatz satisfies the f-sum rule in the clean limit, reproduces the expected linear response for weak disorder, and is consistent with known limiting cases of the disordered t-J Hamiltonian. These additions will be placed in the methods section and will include quantitative comparisons where feasible. revision: yes

  2. Referee: [Final interpretive paragraph] The interpretive statement that the underdoped/overdoped contrast 'lends strong support to our previous proposal' of a Mott transition at the pseudogap endpoint reduces the new calculation largely to a consistency check on earlier work by the same authors. An independent, falsifiable prediction (e.g., a specific disorder-induced feature testable by future experiment or exact diagonalization) would be required to elevate the support beyond circularity.

    Authors: We agree that the interpretive paragraph can be improved to better highlight the independent character of the present calculation. The new variational results on disorder robustness and monotonic doping dependence constitute a direct computation within the disordered t-J model that was not available in our earlier work. We will revise the final paragraph to emphasize these new elements and to include a concrete, falsifiable prediction: namely, that the linear scaling of ρ_s(0) with optical weight should persist under moderate disorder in the underdoped regime, a feature that can be tested by future experiments on controlled-disorder samples or by exact diagonalization on small clusters. revision: partial

Circularity Check

0 steps flagged

Minor self-citation in interpretive conclusion; core variational calculation independent

full rationale

The paper's derivation chain consists of two explicit steps applied to the disordered t-J model: (1) unrestricted variational optimization of an RVB ground state, and (2) construction of a paramagnetic current response state on that RVB background, with ρ_s(0) extracted from the second derivative of the variational energy with respect to the vector potential. The reported outcomes—monotonic increase of ρ_s(0) with doping x and linear scaling with total optical weight—are presented as direct computational results of this procedure rather than definitions or fits. The abstract's final sentence notes that the contrast with overdoped behavior 'lends strong support to our previous proposal' of a Mott transition at the pseudogap endpoint; this is an interpretive remark that does not feed back into or justify the variational steps themselves. No equation or quantity is shown to equal its input by construction, no parameter is fitted to a subset and then relabeled a prediction, and the self-reference is confined to the concluding interpretation. The calculation therefore remains self-contained against the t-J Hamiltonian and the stated variational ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the unrestricted RVB variational ansatz for the disordered t-J model and on the auxiliary variational state constructed for the paramagnetic current response; both are domain assumptions of the resonating-valence-bond approach to strongly correlated electrons.

axioms (2)
  • domain assumption The t-J model with added disorder term is an adequate microscopic description of the cuprate plane.
    Invoked in the first sentence of the methods description in the abstract.
  • domain assumption An unrestricted RVB variational wavefunction optimized on the disordered t-J Hamiltonian yields a faithful zero-temperature ground state.
    First step of the two-step procedure stated in the abstract.

pith-pipeline@v0.9.0 · 5881 in / 1507 out tokens · 27913 ms · 2026-05-19T23:25:54.130603+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We develop a variational theory for the zero temperature superfluid density of the disordered t-J model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state... Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density ρ_s(0) is then extracted from the curvature of the variational ground state energy...

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    ρ_s(0) is a monotonically increasing function of doping concentration x and scales linearly with the total optical weight.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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