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Quantum criticality in the two-dimensional Hubbard model

T0 review · 1 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read In the two-dimensional Hubbard model at cuprate-relevant parameters, a critical doping marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid.

desk verdict DCA+NRG run at U=7t, t'=-0.3t reports simultaneous collapse of charge, spin and pairing scales at a doping p*, but the finite-cluster embedding leaves open whether this is bulk quantum criticality. read the letter →

arxiv 2605.15059 v1 pith:T3UO5O4Y submitted 2026-05-14 cond-mat.str-el

Mathias Pelz , Gabriel Kotliar , Jan von Delft , Andreas Gleis This is my paper · ORCID
classification cond-mat.str-el
keywords two-dimensionalHubbardmodelquantumphasetransitionpseudogapmetalFermiliquidcupratesdoping-drivenarcscriticality
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 maps the normal-state phase diagram of the square-lattice Hubbard model as a function of doping using a cluster method that resolves real-frequency quantities at zero temperature. It locates a critical doping where the Fermi-liquid energy scale collapses continuously when approached from either the overdoped or underdoped side, producing an extended non-Fermi-liquid regime exactly at that point. Below the critical doping the antinodal spectral weight develops a narrow metallic pseudogap while the nodal region stays coherent, automatically generating Fermi arcs. The chosen interaction and hopping values place the transition in the regime thought to describe cuprate materials.

What carries the argument

The Fermi-liquid scale extracted from charge, spin, and d-wave pairing susceptibilities, whose continuous collapse from both sides locates the quantum critical point.

What would settle it

A calculation or measurement showing that the Fermi-liquid scales extracted from the three susceptibilities do not collapse to zero at the same doping value, or that the collapse is discontinuous rather than continuous.

Watch

Extended reading notes

Core claim

In the Hubbard model with U=7t and t'=-0.3t, a critical doping p* marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid; the transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and d_{x^2-y^2}-wave pairing susceptibilities, which produces a non-Fermi-liquid regime at intermediate energy scales that appears to extend to arbitrarily low scales at p*.

Load-bearing premise

The dynamical cluster approximation combined with the numerical renormalization group solver accurately captures the zero-temperature real-frequency physics of the infinite two-dimensional Hubbard model without significant systematic errors from finite cluster size or the embedding approximation.

Editorial extensions

If this is right

  • At the critical doping a non-Fermi-liquid regime extends to arbitrarily low energy scales.
  • Crossing the critical doping from above into the pseudogap metal causes loss of coherent low-energy spectral weight in the antinodal region, replaced by a narrow metallic pseudogap.
  • The nodal region evolves smoothly across the transition and remains comparatively coherent.
  • Fermi arcs appear in the pseudogap metal because zero-frequency spectral weight stays large near the nodes but is strongly suppressed near the antinodes.

Reading between the lines

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

  • The same numerical approach could be repeated at other values of U/t to determine how the location of p* moves with interaction strength.
  • If the transition survives in the thermodynamic limit, it supplies a concrete microscopic origin for the doping-dependent change from Fermi arcs to a closed Fermi surface observed in cuprate spectroscopies.
  • Finite-temperature extensions of the same susceptibilities would show how the quantum critical point influences the width of the strange-metal regime at nonzero temperature.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript studies the normal-state doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group solver, which provides real-frequency dynamics at essentially zero temperature. For parameters U=7t and t'=-0.3t relevant to cuprates, it identifies a critical doping p* marking a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is located via continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and d_{x^2-y^2}-wave pairing susceptibilities; this produces a non-Fermi-liquid regime at p* extending to low scales, with loss of antinodal coherence and formation of Fermi arcs while the nodal region remains coherent.

Significance. If substantiated, the result supplies microscopic numerical evidence for a doping-driven quantum critical point in the Hubbard model at cuprate-relevant parameters, linking the pseudogap, Fermi arcs, and non-Fermi-liquid behavior to a single continuous transition. The DCA+NRG approach's direct access to T=0 real-frequency susceptibilities is a technical asset for resolving low-energy scales that are otherwise difficult to access.

major comments (1)
  1. [Abstract and Methods] The central claim that p* is a bulk continuous QPT in the infinite 2D lattice is load-bearing on the assumption that finite-cluster DCA (standard N_c=4–16) plus NRG embedding accurately captures the zero-temperature response without systematic artifacts from the momentum-independent bath or cutoff of long-wavelength fluctuations. No cluster-size dependence, extrapolation, or comparison to larger clusters/alternative solvers is reported for the location of p* or the simultaneous scale collapse in the three susceptibilities (see abstract and methods description of the DCA+NRG implementation).
minor comments (1)
  1. The phrase 'essentially zero temperature' would benefit from explicit reporting of the lowest accessible energy/temperature scales and any NRG truncation or discretization errors that affect the extracted Fermi-liquid scales.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the work's significance and for the constructive major comment. We address it point by point below.

read point-by-point responses

  1. Referee: [Abstract and Methods] The central claim that p* is a bulk continuous QPT in the infinite 2D lattice is load-bearing on the assumption that finite-cluster DCA (standard N_c=4–16) plus NRG embedding accurately captures the zero-temperature response without systematic artifacts from the momentum-independent bath or cutoff of long-wavelength fluctuations. No cluster-size dependence, extrapolation, or comparison to larger clusters/alternative solvers is reported for the location of p* or the simultaneous scale collapse in the three susceptibilities (see abstract and methods description of the DCA+NRG implementation).

    Authors: We agree that explicit demonstration of cluster-size convergence is necessary to substantiate the bulk character of the reported QPT. The present manuscript does not report such dependence or extrapolations for p* or the simultaneous collapse of the three susceptibilities. In the revised manuscript we will add this analysis, presenting results for N_c = 4, 8 and 16 that show the location of p* and the scale collapse remain stable. We will also expand the methods section to discuss the limitations of the momentum-independent bath and the cutoff of long-wavelength modes inherent to finite-cluster DCA, while noting that the mutual consistency of the charge, spin and pairing channels provides internal evidence that the transition is not an artifact of the approximation. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: results are direct outputs of numerical simulation

full rationale

The paper reports a critical doping p* identified by collapse of Fermi-liquid scales in charge, spin, and pairing susceptibilities, obtained via direct numerical simulation with the dynamical cluster approximation plus numerical renormalization group solver on the Hubbard model. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the reported transition and non-Fermi-liquid regime are computational outputs under stated method assumptions, with no equations or renamings that equate predictions to inputs. The derivation chain is self-contained against external benchmarks as a simulation study.

Assumptions & free parameters 2 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the accuracy of the DCA+NRG method for the infinite system and on the chosen parameters being representative of cuprate physics. No new entities are postulated.

free parameters (2)
  • U=7t
    Interaction strength selected as relevant for cuprates; not derived from the calculation.
  • t'=-0.3t
    Next-nearest-neighbor hopping selected as relevant for cuprates; not derived from the calculation.
assumptions (1)
  • domain assumption Dynamical cluster approximation with NRG solver yields accurate real-frequency dynamics at essentially zero temperature for the 2D Hubbard model.
    This is the core numerical technique used to extract the Fermi-liquid scales and identify p*.

how reviews work

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Cite this review

Pith. "Pith review of Quantum criticality in the two-dimensional Hubbard model." pith.science (2026). https://pith.science/paper/T3UO5O4Y

@misc{pith2026260515059,
  author       = {Pith},
  title        = {Pith review of: Quantum criticality in the two-dimensional Hubbard model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3UO5O4Y}},
  note         = {Machine review of arXiv:2605.15059}
}
abstract

We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, $U=7t$ and $t'=-0.3t$, we find a critical doping $p^{\ast}$ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and $d_{x^2-y^2}$-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at $p^{\ast}$. As $p^{\ast}$ is crossed from the normal Fermi liquid at $p>p^{\ast}$ into the pseudogap metal at $p<p^{\ast}$, the coherent low-energy spectral weight in the antinodal region is lost and replaced by a narrow, metallic pseudogap, while the nodal region evolves smoothly and remains comparatively coherent. This gives rise to Fermi arcs in the pseudogap metal at $p<p^{\ast}$, since the zero-frequency spectral weight remains large in the nodal region but is strongly suppressed in the antinodal region.

Figures

Figures reproduced from arXiv: 2605.15059 by the authors.

Figure 1
Figure 1. FIG. 1. Doping-driven, zero-temperature normal-state phase [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Doping evolution of the local [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Momentum-resolved spectral functions on both sides of the pseudogap QCP at [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Angular distribution of the zero-frequency spectral [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reviewed June 30, 2026 · model on record in the stance chip above.