BMN Spread Complexity Across Phase Transitions

Dibakar Roychowdhury

REVIEW

BMN Spread Complexity Across Phase Transitions

Dibakar Roychowdhury

High Energy Physics - Theory · Submitted 30 Aug 2026 · 72 references

Pith's verdict: The intermediate-phase result is a parametrization, not a prediction, because the spectral density is postulated with free parameters. The gapped and continuum phases are handled well.

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arXiv ID2608.29816 (version 1)

Pith IDJ63WP3R5, Pith's permanent record of this paper and its review

CitationDibakar Roychowdhury. "BMN Spread Complexity Across Phase Transitions." arXiv:2608.29816 (2026). https://pith.science/paper/J63WP3R5

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Editor's note

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The paper does two things cleanly: it computes spread complexity in the gapped phase (Bessel functions, quadratic growth) and in the continuum phase (linear Lanczos, exponential advection, λ_K = π/β). These follow established methods and are presented carefully. The framing of three thermal regimes in the BMN matrix model is useful, and the holographic commentary is plausible at a qualitative level.

The problem is the intermediate phase. The central claim—a sub-exponential crossover governed by a γ√n correction to the Lanczos coefficients—rests entirely on the spectral density ansatz in Eq. (C.8). That form is not derived from the BMN Hamiltonian; it is written down with free parameters C₀, c₁, A₁, and the scale factor Λ₀ depends on an uncomputed crossover index N₀. The resulting γ in Eq. (3.150) and the spread complexity in Eq. (3.172) therefore contain undetermined amplitudes. This makes the intermediate-phase analysis a parametrization, not a falsifiable prediction. The stress-test note gets this exactly right.

The self-citation rate is also heavy—roughly a quarter of the references are to the author's own work—which inflates the perceived novelty without adding independent support.

That said, the gapped and continuum calculations are sound and could be useful to others working on Krylov complexity in matrix models. The intermediate phase is the advertised novelty, and it falls short. The paper needs either a microscopic derivation of the thermal spectral density from the BMN model, or a frank admission that the intermediate result is an effective model with free parameters to be fitted to data.

Who this is for: researchers in Krylov complexity and holographic matrix models. It deserves a serious referee because the overall framework is coherent and the extreme phases are correctly handled, but the intermediate section requires major revision.

Recommendation: major revision. The referee should ask for a derivation of f_cont(ω) from the BMN Lagrangian at finite temperature, or—if that is not possible—a clear statement that the intermediate result is an effective phenomenological fit with undetermined coefficients, and a sensitivity analysis of the free parameters.

Referee report

Major objections (3)

[Section 3.2.4, Eq. (C.8), Eqs. (3.150)–(3.172)] The entire intermediate-phase analysis depends on the postulated spectral density f_cont(omega) in Eq. (C.8). This form is not derived from the BMN Lagrangian (2.1) beyond the leading harmonic approximation; it is an ad hoc ansatz containing free parameters C0, c1, A1. Consequently, the correction gamma(beta mu, beta nu) in Eq. (3.150) is expressed as an unknown ratio A1 cosh(beta nu)/C(beta mu, beta nu) multiplied by a non-perturbative scale factor Lambda0(beta mu) that depends on an uncomputed crossover index N0. The resulting spread complexity (3.172) therefore contains a free amplitude that cannot be tested or falsified. Without a microscopic derivation of f_cont(omega) from the finite-temperature BMN matrix model, the intermediate-phase results amount to a parametrization, not a prediction, and do not provide a rigorous foundation for the claimed sub-exponential crossover.
[Section 3.2.2, Eqs. (3.96)–(3.109)] The regularization of the gapped-phase Lanczos coefficients via the rectangular broadening parameter epsilon (Eq. 3.96) is ad hoc. The physical picture of a dense continuum of modes is plausible, but the specific shape and width of the regularization are not justified from the BMN model. The resulting Hankel determinants (Eqs. 3.101–3.104) depend on epsilon, and the conclusion that b_n approx mu for n>=3 (Eq. 3.109) is not uniquely determined by the model. A more systematic treatment of the spectral density for the gapped phase, including an estimate of epsilon from the BMN Hamiltonian, is needed.
[Section 3.2.3, Eq. (3.125) and surrounding text] The paper claims that the Krylov Lyapunov exponent lambda_K = pi/beta saturates the Maldacena-Shenker-Stanford (MSS) bound. However, the MSS bound on the Lyapunov exponent for out-of-time-order correlators is lambda_L <= 2pi/beta. The relationship lambda_K = pi/beta = lambda_L/2 requires explicit justification and clarification; as it stands, the statement that lambda_K saturates the universal bound is potentially misleading. The paper should discuss whether the factor of two difference is a kinematic effect of state versus operator complexity or a genuine deviation.

Minor objections (6)

[Abstract and Introduction] The abstract and introduction are overly long and contain extensive repetition of the results. Consider condensing to improve readability.
[Section 3.1, Eqs. (3.38)–(3.41)] The expansion of the return amplitude in the continuum phase is not clearly defined: the reference time t_ref is introduced but not explicitly specified. The statement that the expansion is in the domain t approx t_ref is vague. Please clarify the meaning of t_ref and how the expansion is performed.
[Section 3.2.2, Eq. (3.89)] The gapped-phase return amplitude is given as R(t) = cos(mu t) + O(e^{-beta(mu-nu)}). The sub-leading term should be explicitly estimated in terms of the chemical potential, given that the analysis elsewhere emphasizes the role of nu.
[Appendix C.4] The pole-residue duality argument summarizing the mapping from discrete Matsubara poles to the continuous thermal weight is too sketchy; the key steps of the contour integration and the appearance of the factor iven by 1/(e^{beta omega} pm 1) are not derived. This appendix would benefit from a more complete derivation or references to standard textbooks.
[Throughout] There are several instances of inconsistent notation: for example, M_n and M^{(0)}_n are used interchangeably without clear definition. The manuscript should unify notation for moments and clarify which quantities include sub-leading corrections.
[Fig. 1] The schematic diagram is useful, but the arrow from Continuum Phase to Maximal chaos (lambda_K = pi/beta) should be qualified with the factor-of-two discussion raised in the major comments.

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The paper

The paper claims

BMN spread complexity reveals three-phase scrambling crossover from bound states to maximal chaos.

The reading

This paper investigates how spread complexity, a measure of quantum state delocalization on the Krylov chain, evolves across three temperature regimes in the mass-deformed BMN matrix model. The author constructs a charged Thermofield Double state and derives the Lanczos coefficients and complexity growth analytically. The central discovery is a clear dynamical crossover: at low temperatures, spread complexity grows quadratically due to discrete mass-gap bound states; at high temperatures, it grows exponentially with Krylov Lyapunov exponent π/β, saturating the chaos bound; at intermediate temperatures, a sub-exponential correction reflects competition between bound states and thermal continuum. Understanding this crossover matters because it links state complexity in a non-conformal matrix model to holographic duals of black hole horizons and LLM geometries.

What carries the argument

The key machinery is the charged Thermofield Double (cTFD) state of the BMN matrix model at leading order in the large mass deformation µ ≫ 1, combined with the Lanczos algorithm and the continuum limit n → x of the Krylov chain. The cTFD state incorporates both temperature β⁻¹ and chemical potential ν, and its return amplitude near the KMS analyticity boundary yields factorial growth of spectral moments, which translates to linear growth of Lanczos coefficients at high order. The continuum limit reduces the discrete Schrödinger equation to an advection partial differential equation $\partial_t \psi = -\partial_x (b(x) \psi)$ with velocity field $b(x) = \frac{\pi}{\beta} x + \gamma \sqrt{x}$. Solving this equation gives the wavepacket trajectory and the spread complexity across all three phases.

Extended reading notes

Core claim

The paper establishes that spread complexity in the BMN matrix model exhibits a three-phase structure governed by temperature relative to the mass gap µ. In the low-temperature gapped phase (βµ ≫ 1), the Lanczos coefficients $b_n$ are bounded (∼ µ), leading to quadratic growth $C(t) \sim \mu^2 t^2$. In the high-temperature continuum phase (βµ ≪ 1), $b_n \sim \frac{\pi}{\beta} n$, driving exponential growth $C(t) \sim \exp(\pi t/\beta)$ with Krylov Lyapunov exponent $\lambda_K = \pi/\beta$ that saturates the Maldacena-Shenker-Stanford chaos bound. In the intermediate regime (βµ ∼ 1, βν ∼ 1), the Lanczos coefficients acquire a sub-leading $\sqrt{n}$ correction: $b_n \sim \frac{\pi}{\beta} n + \gamma \sqrt{n}$, producing a sub-exponential crossover before asymptotic chaotic behavior sets in. The parameter $\gamma(\beta\mu,\beta\nu)$ encodes the competition between discrete mass-gap modes and the thermal continuum modulated by chemical potential ν.

Load-bearing premise

The derivation of the Lanczos coefficients in the intermediate regime ($b_n \sim \frac{\pi}{\beta} n + \gamma \sqrt{n}$) assumes a spectral density that artificially splits into a thermal continuum with a specific functional form (Eq. C.8) and discrete mass-gap poles with ad hoc amplitudes, without deriving these amplitudes from the BMN matrix model beyond leading order.

Editorial extensions

If this is right

Spread complexity in the continuum phase of the BMN model serves as a direct microscopic probe of maximal chaos, with the Lyapunov exponent matching the universal bound $\lambda_L \leq 2\pi/\beta$ after accounting for the difference between operator and state complexity.
The intermediate-temperature sub-exponential correction controlled by $\gamma(\beta\mu,\beta\nu)$ provides a controlled setting to study the onset of scrambling near a confinement-deconfinement transition in a holographic context.
The cTFD construction offers a template for computing Krylov complexity in other massive matrix models with global $U(1)$ symmetries at leading order in the deformation.
The analytical form of the Lanczos coefficients and the advection equation allow precise predictions for the time-dependence of spread complexity in each phase, which could be compared with lattice or Monte Carlo simulations of the BMN matrix model.
The results extend the universal operator growth hypothesis to state complexity in a non-conformal theory with a mass gap, showing that the KMS analyticity strip width $\beta/2$ controls the asymptotic growth rate even when discrete states coexist with the continuum.

Reading between the lines

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

The sub-leading $\gamma \sqrt{n}$ correction in the Lanczos coefficients might be testable in direct numerical simulations of the BMN matrix model at intermediate temperatures, where the spectral density computed via Hamiltonian truncation or lattice methods could be compared to the model in Eq. (C.8).
The parameter $\gamma(\beta\mu,\beta\nu)$ could be reinterpreted as a renormalisation of the effective temperature due to the mass gap, potentially modifying the chaos bound at finite $\mu$; this would provide a concrete extension of the MSS bound to gapped systems.
The three-phase spread complexity likely maps to distinct holographic duals: the LLM geometry (low $T$), a charged non-extremal D0-brane black hole (intermediate $T$), and an uncharged non-extremal D0-brane black hole (high $T$). The growth law of the wormhole volume or action in each geometry should match the corresponding $C(t)$ behaviour derived here.
The transition from quadratic to sub-exponential to exponential growth may serve as a diagnostic for the confinement-deconfinement transition in the BMN matrix model, analogous to the thermal entropy jump seen in lattice studies.

Original abstract

We investigate the dynamical behavior and spread complexity of quantum states within the mass-deformed BMN matrix model equipped with an emergent $U(1)$ global charge at finite temperature $\beta^{-1}$ and chemical potential $\nu$. Focusing on the strong-coupling regime ($\mu \gg 1$), we model the dynamics via a charged Thermofield Double (cTFD) state and trace the evolution of spread complexity across three distinct thermodynamic regimes. In the low-temperature gapped phase ($\beta \mu \gg 1$), discrete mass-gap bound states dominate, trapping wavepacket dispersion and producing non-chaotic, oscillatory early-time growth. Conversely, in the high-temperature continuum phase ($\beta \mu \ll 1$), thermal excitations overwhelm the mass gap, driving a transition to a continuous advection field that exhibits maximal chaotic scrambling with a Krylov Lyapunov exponent $\lambda_K = \pi / \beta$ that saturates the universal bound. In the intermediate temperature regime ($\beta \mu \sim 1, \beta \nu \sim 1$), the interplay between mass-gap bound states and the continuous thermal background induces a sub-leading correction to the Lanczos coefficients $b_n \sim \frac{\pi}{\beta} n + \gamma \sqrt{n}$, governing a continuous sub-exponential crossover before full chaotic thermalization. Technical derivations regarding KMS boundary conditions, grand canonical spectral moments, residue analysis, and time-reversal symmetry breaking in Krylov space are detailed in four dedicated appendices.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,390 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate the dynamical behavior and spread complexity of quantum states within the mass-deformed BMN matrix model equipped with an emergent $U(1)$ global charge at finite temperature $\beta^{-1}$ and chemical potential $\nu$. Focusing on the strong-coupling regime ($\mu \gg 1$), we model the dynamics via a charged Thermofield Double (cTFD) state and trace the evolution of spread complexity across three distinct thermodynamic regimes. In the low-temperature gapped phase ($\beta \mu \gg 1$), discrete mass-gap bound states dominate, trapping wavepacket dispersion and producing non-chaotic, oscillatory early-time growth. Conversely, in the high-temperature continuum phase ($\beta \mu \ll 1$), thermal excitations overwhelm the mass gap, driving a transition to a continuous advection field that exhibits maximal chaotic scrambling with a Krylov Lyapunov exponent $\lambda_K = \pi / \beta$ that saturates the universal bound. In the intermediate temperature regime ($\beta \mu \sim 1, \beta \nu \sim 1$), the interplay between mass-gap bound states and the continuous thermal background induces a sub-leading correction to the Lanczos coefficients $b_n \sim \frac{\pi}{\beta} n + \gamma \sqrt{n}$, governing a continuous sub-exponential crossover before full chaotic thermalization. Technical derivations regarding KMS boundary conditions, grand canonical spectral moments, residue analysis, and time-reversal symmetry breaking in Krylov space are detailed in four dedicated appendices.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the dynamical behavior and spread complexity of quantum states within the mass-deformed BMN matrix model equipped with an emergent $U(1)$ global charge at finite temperature $\beta^{-1}$ and chemical potential $\nu$.	contribution 0.97	0.86	0.04	asserted 0.83
s1	Focusing on the strong-coupling regime ($\mu \gg 1$), we model the dynamics via a charged Thermofield Double (cTFD) state and trace the evolution of spread complexity across three distinct thermodynamic regimes.	method 0.81	0.93	0.07	asserted 0.97
s2	In the low-temperature gapped phase ($\beta \mu \gg 1$), discrete mass-gap bound states dominate, trapping wavepacket dispersion and producing non-chaotic, oscillatory early-time growth.	result 1.00	0.84	0.10	asserted 1.00
s3	Conversely, in the high-temperature continuum phase ($\beta \mu \ll 1$), thermal excitations overwhelm the mass gap, driving a transition to a continuous advection field that exhibits maximal chaotic scrambling with a Krylov Lyapunov exponent $\lambda_K = \pi / \beta$ that saturates the universal bound.	result 1.00	0.85	0.32	asserted 1.00
s4	In the intermediate temperature regime ($\beta \mu \sim 1, \beta \nu \sim 1$), the interplay between mass-gap bound states and the continuous thermal background induces a sub-leading correction to the Lanczos coefficients $b_n \sim \frac{\pi}{\beta} n + \gamma \sqrt{n}$, governing a continuous sub-exponential crossover before full chaotic thermalization.	result 1.00	0.88	0.18	asserted 1.00
s5	Technical derivations regarding KMS boundary conditions, grand canonical spectral moments, residue analysis, and time-reversal symmetry breaking in Krylov space are detailed in four dedicated appendices.	method 0.88	0.71	0.02	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 56 sentences without a confident label

Builds on 5 papers

The purpose of this article is to extend the notion of spread complexity of states [ref] - [ref] in the context of BMN matrix models [ref] - [ref] at finite chemical potential $ν$ and temperature $β^-1$ [ref] .

Thermodynamics of the BMN matrix model at strong coupling · Introduction and General Idea · 0.86
The purpose of this article is to extend the notion of spread complexity of states [ref] - [ref] in the context of BMN matrix models [ref] - [ref] at finite chemical potential $ν$ and temperature $β^-1$ [ref] .

A Universal Operator Growth Hypothesis · Introduction and General Idea · 0.86
The purpose of this article is to extend the notion of spread complexity of states [ref] - [ref] in the context of BMN matrix models [ref] - [ref] at finite chemical potential $ν$ and temperature $β^-1$ [ref] .

Quantum chaos and the complexity of spread of states · Introduction and General Idea · 0.86

Uses the method or tool of 5 papers

To obtain the high-order spectral moments $ℳ_2n$, we begin with the spectral representation of the thermal return amplitude $ℛ(t) = ⟨TFD(0) e^-i ℋ t | TFD(0) ⟩$, which can be expressed as an integral of the following form [ref] [equation] where $f(ω)$ is the symmetric thermal spectral density.

A Universal Operator Growth Hypothesis · Moments and Lanczos coefficients · 0.93
We outline how to obtain the wave function (or the probability amplitude) $ψ_n(t)$ associated with the $n$th Krylov site $|K_n⟩$ [ref] .

Quantum chaos and the complexity of spread of states · Fuzzy sphere model and spread complexity: A review · 0.86
Given the above state of the art, we choose the Krylov basis at $t=0$ as [ref] [equation] where, $Ψ_0(x,y)$ is the Gaussian wavefunction of the ground state of the composite system and $|x,y⟩$ is the eigen basis in the position representation.

Krylov complexity for Lin-Maldacena geometries and their holographic duals · Fuzzy sphere model and spread complexity: A review · 0.97

Compares with 1 paper

Finally, substituting $x = 2μt$ leads directly to the early-time expansion of the spread complexity of states in the gapped phase of the matrix model [equation] which mimics previous results [ref] obtained at zero temperature [ref].

Krylov state complexity for BMN matrix model · Spread complexity in the gapped phase · 0.92

Cites as support 4 papers

In the continuum limit ($n →x$), this manifests as a continuous advection PDE, driving maximal quantum chaos and exponential scrambling $𝒞(t) ∼exp(πt / β)$ with Krylov Lyapunov exponent $λ_K = π/ β$ which saturates the universal bound [ref] . * Intermediate phase ($βμ∼1, βν∼1$): At intermediate scales, the interplay between discrete bound states and the continuous thermal background induces sub-exponential corrections governed by the parameter $γ$ that interpolates between discrete mass gap and the thermal continuum.

A bound on chaos · Summary and conclusions · 0.96
Instead, the Thermofield Double (TFD) state $|TFD⟩= 1/√(Z) ∑_n e^-βE_n/2 |n⟩_L ⊗|n⟩_R$ holographically maps to two decoupled, smooth LLM geometries [ref] - [ref] generated by background supergravity fluxes without black hole horizons.

The Supergravity Dual of the BMN Matrix Model · The complex oscillator model · 0.98
Instead, the Thermofield Double (TFD) state $|TFD⟩= 1/√(Z) ∑_n e^-βE_n/2 |n⟩_L ⊗|n⟩_R$ holographically maps to two decoupled, smooth LLM geometries [ref] - [ref] generated by background supergravity fluxes without black hole horizons.

Bubbling AdS space and 1/2 BPS geometries · The complex oscillator model · 0.98

Cites as background 12 papers

The BMN or Plane Wave Matrix Model [ref] is defined as[Throughout this paper, we choose to work with natural units where we set $ħ=c=l=1$.] [ref] [equation]

Chaos in the BMN matrix model · Fuzzy sphere model and spread complexity: A review · 0.98
The BMN or Plane Wave Matrix Model [ref] is defined as[Throughout this paper, we choose to work with natural units where we set $ħ=c=l=1$.] [ref] [equation]

Strings in flat space and pp waves from ${\cal N}=4$ Super Yang Mills · Fuzzy sphere model and spread complexity: A review · 0.98
Krylov and/or the spread complexity of states and operators [ref] - [ref] and its various applications including chaos [ref] - [ref] , holography [ref] - [ref] and black holes [ref] have garnered renewed attention in the recent years due to their several remarkable features[For a nice and comprehensive set of reviews on the subject see [ref] - [ref] .].

Random Matrix Theory for Complexity Growth and Black Hole Interiors · Introduction and General Idea · 1.00

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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