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arXiv:2608.29816

BMN Spread Complexity Across Phase Transitions

Dibakar Roychowdhury

hep-thquant-phSubmitted Aug 30, 2026• Announced Sep 1, 2026• 0 citations
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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 $β^{-1}$ and chemical potential $ν$. Focusing on the strong-coupling regime ($μ\gg 1$), 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 ($βμ\gg 1$), discrete mass-gap bound states dominate, trapping wavepacket dispersion and producing non-chaotic, oscillatory early-time growth. Conversely, in the high-temperature continuum phase ($βμ\ll 1$), thermal excitations overwhelm the mass gap, driving a transition to a continuous advection field that exhibits maximal chaotic scrambling with a Krylov Lyapunov exponent $λ_K = π/ β$ that saturates the universal bound. In the intermediate temperature regime ($βμ\sim 1, βν\sim 1$), the interplay between mass-gap bound states and the continuous thermal background induces a sub-leading correction to the Lanczos coefficients $b_n \sim \fracπβ n + γ\sqrt{n}$, 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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Latex, 1+60 pages

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We investigate the dynamical behavior and spread complexity of quantum states within the mass-deformed BMN matrix model equipped with an emergent U(1)U(1)U(1) global charge at finite temperature β−1β^{-1}β−1 and chemical potential ννν. Focusing on the strong-coupling regime (μ≫1μ\gg 1μ≫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 (βμ≫1βμ\gg 1βμ≫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 (βμ≪1βμ\ll 1βμ≪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 λK=π/βλ_K = π/ βλK​=π/β that saturates the universal bound. In the intermediate temperature regime (βμ∼1,βν∼1βμ\sim 1, βν\sim 1βμ∼1,βν∼1), the interplay between mass-gap bound states and the continuous thermal background induces a sub-leading correction to the Lanczos coefficients bn∼πβn+γnb_n \sim \fracπβ n + γ\sqrt{n}bn​∼βπ​n+γn​, 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.
We use Krylov complexity as a typical diagnostic of quantum many body dynamics in the context of BMN matrix model at large mass gap. We calculate several physical entities, for example, the moments and return amplitudes at large mass deformation. We calculate them for both spread complexity of states as well as Krylov growth of operators in the matrix model. We propose a general expression for the moments at any given order nnn. We also discuss orthogonal polynomials for spread as well as operator Krylov complexity. In the context of the Krylov operator growth we further compute the spectral function and the density of states. A careful analysis of the spectral function near the resonance reveals various IR divergences in addition to the physical collective modes. These collective modes appear to be exceptionally stable due to large mass gap. On the other hand, the UV of the theory appears to be extremely damped due to higher scattering rates. We carefully diagnose these IR divergences near resonance which reveals that these collective modes are in fact non-propagating and should be thought of as diffusion modes or relaxation modes with infinite relaxation time. We also calculate the Krylov variance and Krylov entropy for the matrix model and in particular at early time. The linear early time growth of the Krylov entropy confirms the onset of quantum chaos in the matrix model at large mass gap.
We study the behavior of spread complexity in the context of non-Hermitian many-body localization Transition (MBLT). Our analysis has shown that the singular value spread complexity is capable of distinguishing the ergodic and many-body localization (MBL) phase from the presaturation peak height for the non-hermitian models having time-reversal symmetry (TRS) and without TRS. On the other hand, the saturation value of the thermofield double (TFD) state complexity can detect the real-complex transition of the eigenvalues on increasing disorder strength. From the saturation value, we also distinguish the model with TRS and without TRS. The charge density wave complexity shows lower saturation values in the MBL phase for the model with TRS. However, the model without TRS shows a completely different behavior, which is also captivated by our analysis. So, our investigation unravels the real-complex transition in the eigenvalues, the difference between the model having TRS and without TRS, and the effect of boundary conditions for the non-hermitian models having MBL transitions, from the Krylov spread complexity perspective.
A bstract We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.
A bstract In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the Krylov subspace . The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position 〈 n 〉 defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments 〈 n p 〉 for p > 1, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.
A bstract Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within integrable systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of chaotic systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity.
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity K(t)\mathcal{K}(t)K(t) in Random Matrix Theory (RMT). In large NNN limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient {bn}\{b_n\}{bn​} saturate to constant plateau lim⁡n→∞bn=b\lim\limits_{n\rightarrow\infty}b_n=bn→∞lim​bn​=b, rendering a linear growing complexity K(t)∼t\mathcal{K}(t)\sim tK(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value bbb to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the {bn}\{b_n\}{bn​} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that bnb_nbn​ will first exhibit linear growth with nnn, whose slope saturates the famous chaos bound. After hitting the same plateau bbb, bnb_nbn​ will then remain constant. This indicates K(t)∼e2πt/β\mathcal{K}(t)\sim e^{2πt/β}K(t)∼e2πt/β before scrambling time t∗∼O(βlog⁡β)t_*\sim O(β\logβ)t∗​∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite NNN corrections.