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This paper claims that DMFT's self-consistent equation is the fixed point of a holographic renormalization group on the Bethe tree, and that boundary electron correlations decay with scaling dimensions set by the fixed-point Green's functio

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load-bearing objection A careful and honest reformulation of DMFT on the Bethe lattice as a tree RG; the fixed-point identification is sound, but the scaling-dimension claims rest on an unproven derivative identity and an uncontrolled finite-p factorization. the 4 major comments →

arxiv 2509.19704 v3 pith:IDVASWCJ submitted 2025-09-24 cond-mat.str-el hep-lathep-thquant-ph

Holographic Aspects of Dynamical Mean-Field Theory

Kouichi Okunishi , Akihisa Koga This is my paper
classification cond-mat.str-el hep-lathep-thquant-ph
keywords dynamical mean-field theoryBethe latticeholographic renormalization groupscaling dimensionsMott transitionboundary correlation functionsp-adic AdS/CFTHubbard model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that dynamical mean-field theory (DMFT), the standard framework for strongly correlated electrons, has a holographic structure hidden in its underlying Bethe lattice: the renormalization group that coarse grains electrons from the outer edge toward the interior has as its fixed point precisely the DMFT self-consistency equation. The authors derive this recursion and show that boundary correlation functions of electrons decay as a power of the tree distance with scaling dimensions determined by the fixed-point DMFT Green's function, mirroring a scalar field in an effective two-dimensional anti-de Sitter space. If correct, this recasts DMFT's convergence and its Mott-transition physics as statements about a bulk/boundary correspondence, and gives a systematic way to read deep-interior properties from boundary decay rates.

Core claim

On the Bethe lattice with branching number p, the branch Green's function satisfies the recursion [G_n]^{-1} = z + mu - p t^2 G_{n+1}, combined with the impurity solution G_n = F[G_n]. The central claim is that the fixed point of this recursion, Eqs. (30)-(31), is exactly the DMFT self-consistency equation for the semicircle density of states. Around the fixed point, a perturbation in G is multiplied by lambda(z) = p[tG*(z)]^2 per RG step, and the same factor controls the power-law decay x^{-2Δ_G} of boundary electron correlations and x^{-2Δ_D} with Δ_D = 2Δ_G for density correlations. The paper reports numerical DMFT results at β=100 and p=100 showing Δ_D ≈ 1 in the metallic phase and Δ_D ≈

What carries the argument

The central mechanism is a recursive partial summation of Grassmann path integrals from the outermost generation inward, which produces an effective local action whose inverse Green's function is (z + μ) - p t^2 G_next(z). Linearizing around the fixed point gives the eigenvalue λ = p[tG*]^2, and combining this with the tree's radial coordinate r_n = p^{-n} and boundary distance x = p^{n̄} yields the scaling dimensions Δ_G = -log|tG*(iω*)|/log p and Δ_D = 2Δ_G, which satisfy the AdS scalar-field relation Δ_+ + Δ_- = 1.

Load-bearing premise

The calculation assumes, as an approximation for finite branching number p, that the many-body Green's function factorizes into free-fermionic products when each RG step integrates out a generation; this is controlled only at p → ∞, where the paper itself shows the metallic-insulating scaling contrast collapses to the free value 1.

What would settle it

Compute the exact branch Green's function and boundary two-point correlator for a finite-p Bethe lattice Hubbard model (for example p=3) using a method that does not assume the free-fermionic factorization—tensor-network or quantum Monte Carlo on the tree—and check whether the boundary decay exponent matches Δ_D = -2 log|tG*(iω*)|/log p evaluated from the DMFT fixed point. A mismatch at moderate p would falsify the claim that the DMFT fixed point governs the true boundary correlations.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The DMFT iteration's convergence rate is governed by p[tG*(iω*)]^2, tying the numerical practice of iterating the self-consistency loop to a geometric eigenvalue.
  • Metallic and insulating DMFT solutions carry distinct scaling dimensions (≈1 vs. ≈1.7), so the decay of boundary correlation functions can serve as a signature of the Mott transition.
  • The relation Δ_+ + Δ_- = 1 holds in both phases, suggesting it is a property of the tree network geometry rather than of interaction strength.
  • The descendant spectrum Δ(l) reflects the quantum Matsubara structure of the Green's function, in contrast to the purely classical Bethe-lattice Ising model.
  • In the p → ∞ limit the scaling dimension collapses to the free-fermion value 1 even in the insulating phase, so the finite-p contrast is essential for the holographic description.

Where Pith is reading between the lines

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

  • Editorial inference: if the holographic dictionary holds beyond the free-fermion approximation, exact finite-p boundary correlators on a true Bethe lattice—computable with tensor networks or quantum Monte Carlo on the tree—should deviate from the DMFT prediction, with deviations growing as p decreases; a quantitative comparison would test the approximation.
  • Editorial inference: Eq. (36), δF/δG = G*^{-2}⟨c̄c⟩², is asserted without derivation and is exact only for free fermions; verifying it numerically with an interacting impurity solver would either strengthen or split the linearized-RG argument.
  • Editorial inference: the paper's own p → ∞ limit shows the Mott signature in Δ_D is a finite-p effect, suggesting the holographic description is most useful at moderate p where DMFT is still justified, rather than at p → ∞.
  • Editorial inference: the same tree-RG construction could be applied to cluster extensions of DMFT or to real-time Green's functions, where the descendant spectrum would encode genuine dynamics rather than Matsubara structure.

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

4 major / 4 minor

Summary. The manuscript proposes a 'holographic' renormalization-group (RG) formulation of DMFT on the Bethe lattice. The central construction is a recursive partial sum over descendant branches, which yields the standard Bethe-lattice DMFT self-consistency at its fixed point. The authors then associate the convergence eigenvalue of the linearized recursion with a scaling dimension in an effective AdS2 ('p-adic AdS/CFT') description, define scaling dimensions Δ_G and Δ_D = 2Δ_G from the fixed-point Green's function, and verify qualitatively with CTQMC DMFT calculations that Δ_D distinguishes metallic and insulating solutions across the Mott transition. The paper contains a free-electron appendix in which the recursion is solved exactly and the eigenvalue λ = pt²G*² is checked.

Significance. If fully established, the paper would give an appealing conceptual bridge: DMFT's self-consistency appears as the fixed point of an RG on the Bruhat-Tits tree, and correlation functions of boundary electrons are governed by the fixed-point DMFT Green's function. The paper has genuine strengths: the recursion in Sec. II is close to the standard Bethe-lattice DMFT derivation; the free-electron calculation in Appendix A is exact and provides a concrete check; and the numerical DMFT results are standard and reproducible in principle. The central mapping from the recursion to the DMFT fixed point is sound. However, the advertised finite-p results, especially the Mott-transition contrast in Fig. 4, rely on an uncontrolled free-fermionic factorization of the many-body branch Green's function, and the key functional-derivative identity in Eq. (36) is not derived. For these reasons I view the paper as a promising reformulation whose load-bearing claims still need additional support rather than as a completed derivation.

major comments (4)
  1. [Sec. II C, Eq. (36)] The identity δF[G*]/δG* = G*^{-2}(⟨c̄c⟩*)² = 1 is asserted without derivation. For an interacting impurity, the functional derivative of the branch Green's function with respect to the Weiss field contains vertex (connected four-point) corrections; the displayed identity is the free-fermion/vertexless response. This identity is load-bearing because it converts the linearized recursion into λ = pt²G*² (Eq. 37), which in turn fixes the holographic radial scaling in Eq. (74). Please provide a derivation from the impurity action, state it as an approximation with an estimate of the vertex corrections, or test it numerically for the U≠0 fixed point.
  2. [Sec. II A, after Eq. (13), and Sec. II D] The assumption that the many-body branch Green's function factorizes as in Eq. (14), stated as 'we may assume the free fermionic factorization ... for a finite p,' converts the exact partial sum (10)-(14) into the quadratic effective medium (15)-(19). The same assumption controls the operator renormalization (40)-(53), the boundary correlators (54)-(62), and the four-point vertex Γo in Eq. (84). The approximation is controlled only as p→∞, but in that limit the authors themselves find Δ_D → 1 (Eq. (75)) and note that the scaling dimension becomes insensitive to the deep interior. Thus the finite-p Mott contrast in Fig. 4 is exactly in the regime where the approximation is uncontrolled. Please quantify the leading 1/p correction by retaining at least the first connected term, or explicitly present the finite-p results as an approximate model calculation.
  3. [Sec. IV, Eq. (81) and Fig. 4] The statement that 'the scaling dimensions capture the Mott transition' is, as it stands, a re-description: Δ_D is a one-line functional of the DMFT-computed G*, so Fig. 4 inherits the already-known metal/insulator distinction from the input Green's function. This does not invalidate the holographic RG construction, but it means the result has no independent predictive content unless the boundary correlation function itself is computed directly. To make the claim load-bearing, evaluate D_N^{αα'}(τ-τ') of Eq. (59), or verify the x^{-2ΔD} law against a finite-p lattice calculation for U>0.
  4. [Sec. IV, Eq. (84)] The descendant spectrum in Fig. 5 is computed using the free-fermionic factorization Γo_{l1,l2,l3,l4} ≃ -Go(iω_{l1})Go(iω_{l3})δ_{l1,l4}δ_{l2,l3}, rather than an actual four-point vertex of the interacting impurity. The text presents the resulting spectrum shapes as evidence that boundary correlations detect the bulk phase transition, but the plotted quantity is an effective-medium construction. This is a specific instance of the uncontrolled factorization noted above and should be flagged explicitly in the paper, with the spectrum interpreted accordingly.
minor comments (4)
  1. [Sec. II B/C] The same symbol G is used for the Weiss field and for the interacting branch Green's function. This is especially confusing in Eq. (29), where the same symbol appears on both sides of F. Introducing ℑ (or ∄) for the Weiss field would materially improve readability.
  2. [Throughout] Several typos should be corrected: 'Burhat-Tits' should be 'Bruhat-Tits'; 'sitaightforwardly' in Appendix A; 'Lenarlizing' in Appendix A; 'renromalized' in Eq. (50); 'sifted' in the caption of Fig. 5; 'brach' in several places.
  3. [Sec. III, Eqs. (63)-(66)] The coordinate definitions would be clearer if the dependence of x on ̄n(α,α') were written explicitly in Eq. (65), and if the limiting sense of the metric (66) were stated: the smearing is defined only after the lattice spacing is sent to zero in the continuum limit.
  4. [Sec. III, Eq. (77)] The relation Δ+ + Δ− = 1 follows immediately from the definition Δ+ ≡ 1 - ΔD in Eq. (74). The text should state that this is a bookkeeping identity of the proposed dictionary, not an independent check of a scalar-field equation of motion.

Circularity Check

2 steps flagged

The central numerical prediction—that the scaling dimension Δ_D captures the Mott transition—reduces by construction to the DMFT fixed-point Green's function G*, and the AdS scalar-field relation Δ_+ + Δ_- = 1 is satisfied by definitional choice rather than by derivation. The tree-RG reformulation of DMFT itself is not circular.

specific steps
  1. renaming known result [Sec. IV, Eqs. (68), (72), (81), Fig. 4]
    "ΔD ≡ 2ΔG = −2 log |tG∗(iω∗)| / log p ... In order to analyze the scaling dimension in association with DMFT, it is useful to rewrite the scaling dimension (68) into ΔD = 1 − 2 log \tilde tG∗(iω∗)/log p ... For the insulating phase ... G∗(iω∗)<2. In Fig. 4, ΔD actually has a gap between the free-fermion value ΔD≃1 and about ΔD∼1.7 for Uc1<U<Uc2."

    Δ_D is a monotone function of |tG*(iω*)|—the very fixed-point DMFT Green's function whose U-dependence already defines the metallic/insulating distinction in Fig. 3 (|ImG*|→2 at ω=0 in the metal vs |G*(iω*)|<2 with ω*≠0 in the insulator). The observation that Δ_D jumps across the Mott transition is thus a relabeling of the input G* curve, not an independent prediction. No boundary correlation function is computed separately; the claimed 'scaling dimensions capture the Mott transition' is an algebraic remapping of the DMFT solution.

  2. self definitional [Sec. III, after Eq. (74), Eq. (77)]
    "From Eqs. (71) and (74), the scaling dimensions in DMFT corresponding to A and B fields can be interpreted as Δ+ ≡ 1−ΔD and Δ− ≡ ΔD, respectively. Moreover, these scaling dimensions clearly satisfy the scalar-field relation, Δ+ + Δ− = 1."

    Δ_+ is defined as 1−Δ_D, so Δ_+ + Δ_- = 1 is an algebraic identity under the chosen definitions, not a derived property of the system. The claimed consistency with the AdS_2 scalar-field formula (76) would hold for any value of Δ_D, making the relation a definitional rearrangement rather than a predictive check.

full rationale

The tree-RG derivation in Sec. II (Eqs. 2–31) is self-contained: from the path integral and the explicit free-fermionic factorization assumption, the recursion Eq. (27) and fixed point Eqs. (30)–(31) are derived and coincide with the standard Bethe-lattice DMFT self-consistency. That part is not circular. However, the paper's numerical headline and holographic dictionary reduce to the same input G*. Eq. (72)/(81) defines Δ_D as a function of |tG*(iω*)|, so Fig. 4's gap is just the known U-dependence of G* plotted on a log scale. The scalar-field relation Δ_+ + Δ_- = 1 is enforced by defining Δ_+ = 1 − Δ_D. These are re-descriptions rather than predictions. Separately, Eq. (36) asserts δF/δG* = 1 without derivation; it is a free-fermion input needed for λ = p[tG*]^2 and Eq. (74), and it is not independently justified for the interacting impurity. The finite-p free-fermionic factorization (after Eq. 13) is an explicit uncontrolled approximation; in the p→∞ limit where it is controlled, Δ_D→1 in both phases, so the finite-p Mott contrast is precisely where the approximation matters. Self-citations (e.g., Ref. [46]) supply standard constructions and are not load-bearing in a circular way. Overall, the central 'predictions' reduce by construction—partial circularity—while the underlying RG/DMFT correspondence retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The construction introduces no fitted constants: Δ_G and Δ_D are fixed functionals of the DMFT Green's function, and the only hand-chosen inputs are physical simulation parameters (p = 100, β = 100, D̃ = 1). What the central claims rest on are: the free-fermionic factorization (exact only as p → ∞), the unproven functional-derivative relation Eq. (36), the geometric coordinate system inherited from Ref. [46], and the assumed 4-point vertex factorization Eq. (84) for the descendant spectrum. No new dynamical entities are postulated; the effective AdS2 is an interpretive re-parameterization of tree depth and distance, not a dynamical gravity dual.

axioms (5)
  • ad hoc to paper Free-fermionic factorization of the many-body branch Green's function: connected many-body correlations among the p descendant branches are neglected.
    Stated after Eq. (13): "we may assume the free fermionic factorization of the many-body Green's function as an approximation for a finite p." Load-bearing for the effective medium (Eqs. 15-19), the operator RG (Eqs. 40-53), and all boundary correlator formulas (Eqs. 54-62). Exact only in the p → ∞ limit, where the paper's new Mott-transition contrast in Δ_D disappears.
  • domain assumption The recursion (27)-(28) and fixed point (30)-(31) constitute the standard Bethe-lattice DMFT self-consistency.
    The fixed-point equations are the standard d = ∞ DMFT self-consistency for the semicircle density of states, attributed by the authors to Refs. [16-19]. This is the central identification and is standard.
  • ad hoc to paper δF[G*]/δG* = G*^{-2}⟨c̄c⟩² (Eq. 36): the response of the impurity branch Green's function to the effective bath is given by the free-fermion formula, neglecting the bath-dependence of the self-energy.
    Asserted without derivation in Sec. II C; exactly true for free fermions and needed to obtain the interacting eigenvalue λ = p[tG*]² (Eq. 37). Not justified for U ≠ 0 where the Weiss-field and impurity branch Green's functions differ by the local self-energy.
  • ad hoc to paper Holographic coordinates: r_n = p^{-n}, x = p^{\bar{n}(α,α')}, metric ds² = L²/r²(dr²+dx²) with L = 1/log p (Eqs. 63-66).
    Adopted from the Bethe-lattice Ising holographic RG of Ref. [46] (same author). An interpretive coordinate system for the tree, not derived in this paper; its one-dimensional p-adic boundary fixes d = 1 in Eq. (77).
  • ad hoc to paper Γo factorization for the 4-point vertex: Γo_{l1,l2,l3,l4} ≃ -Go(iω_l1)Go(iω_l3)δ_{l1,l4}δ_{l2,l3} (Eq. 84).
    Assumed "at the level of the effective medium in DMFT" to compute the descendant spectrum (Eq. 85, Fig. 5); the paper notes that exact evaluation of the 4-body term is a hard task. This makes Fig. 5's spectrum a model-dependent diagnostic.
invented entities (1)
  • Effective AdS2 boundary geometry (Poincaré half-plane metric ds² = L²/r²(dr²+dx²) with L = 1/log p) no independent evidence
    purpose: Assign scaling dimensions and a holographic RG interpretation to the tree-depth and leaf-distance counting of the Bethe lattice.
    Geometric re-parameterization of tree counting inherited from Ref. [46]; it carries no dynamics of its own and no falsifiable handle beyond the Green's function from which it is built. In the DMFT (p → ∞) limit L → 0 and the scaling dimensions collapse to free values, as the authors state.

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Dynamical mean-field theory (DMFT) is one of the most standard theoretical frameworks for addressing strongly correlated electron systems. In this study, we explore a holographic renormalization-group (RG)-like structure inherent in DMFT, which we refer to as "AdS/DMFT", by focusing on the background Bethe-lattice network behind DMFT for electrons with a semicircle density of states. We formulate an RG transformation for the branch Green's function from the outer edge to the interior of the background Bethe-lattice network, and then find that its fixed point can be interpreted as a self-consistent solution of Green's function in DMFT. Moreover, we clarify that the scaling dimensions for the branch Green's function and the boundary correlation functions of electrons at the outer edge of the Bethe-lattice network are characterized by the fixed-point Green's function, analogous to the behavior of a scalar field in an effective two-dimensional anti-de Sitter space. We also perform DMFT computations for the Bethe-lattice Hubbard model, which illustrate that the scaling dimensions capture the Mott transition in the deep interior.

Figures

Figures reproduced from arXiv: 2509.19704 by Akihisa Koga, Kouichi Okunishi.

Figure 1
Figure 1. Figure 1: The Bethe lattice with q = 3 and N = 4 with the imaginary time axis, where the index n for dotted circles represents the generation from the center node and τ denotes the imaginary time. Note that the antiperiodic boundary condition is assumed in the τ direction, where β denotes an inverse temperature. The distance between α and α ′ in the circumference direction is defined by counting the number of nodes … view at source ↗
Figure 2
Figure 2. Figure 2: The network connectivity for a parent node [( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Imaginary part of G ∗ (iω) for U = 0.0, 1.0, 2.0, and 3.0. Solid curve represents the free electron curve ImG ∗ (iω) = 2(ω ∓ √ ω2 + 1). self-consistent loop until the result converges within nu￾merical accuracy. Then, the fixed-point solution of G∗ (τ) can be obtained in the framework of DMFT. For a non-magnetic and half-filled system (µ = U/2), it is well established that strong Coulomb interactions lead … view at source ↗
Figure 4
Figure 4. Figure 4: The scaling dimension ∆D for p = 100 at β = 100. The lower branch indicates the metallic solution, while the upper branch corresponds to the insulating one. The vertical dotted lines represent Uc1 ≃ 2.36 and Uc2 ≃ 2.55. temperature. In order to analyze the scaling dimension in association with DMFT, it is useful to rewrite the scaling dimension (68) into ∆D = − 2 log(tG˜ ∗ (iω∗ )/ √ p) log p = 1 − 2 log tG… view at source ↗
Figure 5
Figure 5. Figure 5: The scaling dimension spectrum with p = 100 at β = 100 for (a) U = 1.0 and (b) 3.0. ∆(l,l′ ) D is plotted against ls ≡ l + l ′ − 2l ∗ of the horizontal axis. The origin of the vertical axis is also sifted by ∆D. In this figure we plot the contributions only form l, l′ ≥ 0, for simplicity. V. SUMMARY AND DISCUSSIONS We have reformulated dynamical mean-field theory (DMFT) for the semicircle density of states… view at source ↗

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