Momentum–Krylov Correspondence

• Momentum–Krylov correspondence is a framework that equates the time derivative of Krylov complexity with the proper radial momentum of an infalling probe in holographic setups.
• It employs a Krylov-space formulation via the Lanczos algorithm, bridging boundary operator dynamics with bulk geodesic motion in models like AdS₃/CFT₂ and N=4 SYM.
• This approach reveals diverse behaviors—from exponential to oscillatory growth—and highlights open challenges in finite-temperature regimes and complex internal structures.

Searching arXiv for recent and foundational papers on the momentum–Krylov correspondence and closely related holographic Krylov complexity work. Momentum–Krylov correspondence is a holographic proposal that identifies the growth rate of Krylov complexity with a bulk momentum variable, typically the proper radial momentum of an infalling massive probe, and in some formulations with the canonical momentum conjugate to wormhole length. In its most common form, for boundary time $t$ and proper radial coordinate $\rho$ or $\bar\rho$, it reads $\dot C_K(t)=-P_\rho(t)$ or $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$. The correspondence emerged in AdS$_3$/CFT$_2$ analyses and has since been developed in AdS$_5$-sliced constructions for ${\cal N}=4$ SYM, in double-scaled SYK and JT or sine–dilaton gravity, in non-relativistic Lifshitz and hyperscaling-violating geometries, and in top-down confining backgrounds, where it provides a geometric description of operator growth and, in confining cases, oscillatory complexity dynamics (Fatemiabhari et al., 24 Nov 2025, Fu et al., 26 Oct 2025, Fadafan et al., 30 Jun 2026, Fatemiabhari et al., 27 Nov 2025).

1. Krylov-space formulation of operator growth

The correspondence is built on the standard Krylov-space description of Heisenberg evolution. For an operator $\mathcal O$ and Hamiltonian $\rho$0,

$\rho$1

Applying the Lanczos algorithm to the Liouvillian $\rho$2 generates an orthonormal Krylov basis $\rho$3 satisfying

$\rho$4

with real positive Lanczos coefficients $\rho$5. Expanding the evolved operator as

$\rho$6

the amplitudes obey a one-dimensional tight-binding-like evolution, and Krylov complexity is the expectation value of the position operator on the semi-infinite chain,

$\rho$7

In many chaotic models $\rho$8 at large $\rho$9, which yields exponential growth of $\bar\rho$0, while in scale-invariant, zero-temperature, or confining settings power-law growth, saturation, or oscillatory behavior can appear (Fadafan et al., 30 Jun 2026, Fatemiabhari et al., 27 Nov 2025).

A parallel state-space construction is often called spread complexity. For a state $\bar\rho$1, one builds a Krylov basis $\bar\rho$2 by tridiagonalizing $\bar\rho$3, expands $\bar\rho$4, and defines

$\bar\rho$5

This distinction matters because several exact holographic results are first formulated for state Krylov complexity and only then reinterpreted as momentum–Krylov statements (Fatemiabhari et al., 24 Nov 2025, Fu et al., 26 Oct 2025).

2. Geometric prescription in holography

The most direct holographic statement identifies the rate of Krylov growth with proper radial momentum. In a bulk metric

$\bar\rho$6

one introduces the proper radial coordinate by

$\bar\rho$7

For a massive point particle moving on a purely radial geodesic, with radial canonical momentum $\bar\rho$8, the proper momentum is

$\bar\rho$9

The momentum–Krylov correspondence is then

$\dot C_K(t)=-P_\rho(t)$0

With $\dot C_K(t)=-P_\rho(t)$1, one obtains

$\dot C_K(t)=-P_\rho(t)$2

The minus sign reflects the choice of an infalling trajectory, for which radial momentum is inward while complexity grows outward in Krylov space (Fadafan et al., 30 Jun 2026).

In the AdS$\dot C_K(t)=-P_\rho(t)$3-sliced-by-AdS$\dot C_K(t)=-P_\rho(t)$4 construction for $\dot C_K(t)=-P_\rho(t)$5 SYM, the proper coordinate is not a single bulk radial coordinate but an effective distance along the actual trajectory,

$\dot C_K(t)=-P_\rho(t)$6

and the corresponding proper momentum is

$\dot C_K(t)=-P_\rho(t)$7

The proposed dictionary is

$\dot C_K(t)=-P_\rho(t)$8

with $\dot C_K(t)=-P_\rho(t)$9 the UV regulator of the CFT state (Fatemiabhari et al., 24 Nov 2025).

A broader Complexity $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$0 Volume version of Momentum/Complexity correspondence relates the CV growth rate on an extremal slice $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$1 to an integrated matter momentum flux,

$\dot C(t)=-P_{\bar\rho}(t)/\epsilon$2

where $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$3 is the momentum flux projected along an infall field $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$4, and $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$5 is a geometric remainder. This relation is exact for any solution of Einstein’s equations in $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$6 dimensions and for any spherically symmetric solution in arbitrary dimensions (Barbon et al., 2020).

3. AdS realizations: from $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$7 to $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$8 SYM

The earliest concrete realizations use $\dot C(t)=-P_{\bar\rho}(t)/\epsilon$9. For states created by a primary operator with Euclidean regulator $_3$0, Caputa et al. obtained

$_3$1

for a CFT on a circle, on the infinite line, and at finite temperature, respectively. These growth rates were matched by geodesic motion in $_3$2 through the proper-momentum relation $_3$3 (Fatemiabhari et al., 24 Nov 2025).

The extension to $_3$4 SYM uses an $_3$5 geometry sliced by $_3$6. Motion confined to the $_3$7 subspace is interpreted as the Krylov complexity of the Beisert $_3$8 subsector generated by $_3$9 and $_2$0, while general motion in $_2$1 is interpreted as the Krylov complexity of a larger, effectively full $_2$2 sector. The same proper-momentum law remains the operative proposal,

$_2$3

In the hypothetical limit in which motion in the extra radial coordinate $_2$4 is frozen, the formulas reduce exactly to the $_2$5 expressions, which is the main consistency check of the construction (Fatemiabhari et al., 24 Nov 2025).

A key feature of the $_2$6 analysis is that the “complexity direction” is no longer an ordinary radial coordinate but the proper distance $_2$7 along a two-coordinate trajectory. This makes the correspondence intrinsically kinematical: the complexity rate is determined by a local proper momentum, not by a global volume or action functional (Fatemiabhari et al., 24 Nov 2025).

4. Double-scaled SYK, JT gravity, and wormhole velocity

In double-scaled SYK, the correspondence takes an exact operator-algebraic form. The relevant Krylov operator is the chord-number operator $_2$8, and the dual bulk observable is the geodesic-length operator $_2$9, with dictionary

$_5$0

For the thermal state evolved by the DSSYK transfer matrix $_5$1,

$_5$2

The growth rate is

$_5$3

where

$_5$4

is the wormhole-velocity operator in sine–dilaton gravity. This equality is exact in the double-scaled ensemble and establishes the first leg of a threefold duality: $_5$5 The last identification, with the proper momentum $_5$6 of an infalling particle in AdS$_5$7, holds semiclassically and only in the early-time or low-energy regime (Fu et al., 26 Oct 2025).

The same framework extends beyond the first moment. Higher Krylov moments $_5$8 are argued to probe connected replica wormhole contributions, the logarithmic variant probes replica saddle structure, and Krylov entropy equals the von Neumann entropy of the parent-geometry density matrix obtained after tracing out baby universes (Fu et al., 26 Oct 2025).

DSSYK also supplies one of the sharpest “momentum–Krylov” diagnostics in coherent states. For the boundary wormhole-velocity operator

$_5$9

one has

${\cal N}=4$0

The sign of ${\cal N}=4$1 in a coherent state is therefore the sign of ${\cal N}=4$2, and negative wormhole velocity is interpreted as a boundary diagnostic of firewall-like or white-hole-like structures via bulk reconstruction (Fu et al., 26 Oct 2025).

5. Non-relativistic and confining holographic backgrounds

Fadafan and Mohammadi Mozaffar extended the correspondence to Lifshitz and hyperscaling-violating geometries. In pure Lifshitz,

${\cal N}=4$3

the radial geodesic can be solved exactly,

${\cal N}=4$4

and the proper momentum is

${\cal N}=4$5

Hence

${\cal N}=4$6

so pure Lifshitz exhibits universal quadratic growth for all dynamical exponents ${\cal N}=4$7. In hyperscaling-violating geometries, the early-time behavior remains quadratic, but the late-time growth exponent changes to

${\cal N}=4$8

A special case, ${\cal N}=4$9, yields

$\mathcal O$0

which is oscillatory with a logarithmic envelope and marks a qualitative transition away from power-law growth (Fadafan et al., 30 Jun 2026).

The same geodesic prescription has been applied to confining top-down backgrounds. In the Anabalón–Ross soliton, which has a smooth infrared end-of-space and a mass gap, radial geodesics can be solved exactly in terms of Jacobi elliptic functions, and the proper momentum oscillates because the probe falls from a UV cutoff to the IR wall, reflects, and returns. The resulting complexity also oscillates. The interpretation advanced there is that the UV cutoff together with the IR end-of-space induces a finite effective Hilbert-space truncation, so Krylov complexity does not grow indefinitely but exhibits revivals (Fatemiabhari et al., 27 Nov 2025).

A systematic study across several top-down confining duals sharpened this point into a universality claim: in all geometries with a smooth infrared end-of-space, Krylov complexity exhibits oscillatory behaviour; the oscillation frequency is controlled by the confinement scale, while the amplitude depends on both the ultraviolet cutoff and the infrared scale. Additional conserved charges modify these patterns without altering their qualitative structure (Fatemiabhari et al., 19 Feb 2026).

A complementary large-$\mathcal O$1 field-theoretic perspective arrives at a related distinction by a different route. In non-compact space, even free QFTs can show exponential Krylov growth because continuous momentum produces a continuous spectrum with an exponential tail. After compactifying space so that

$\mathcal O$2

continuous momentum no longer masks the mass spectrum. In that regime, confined phases with discrete masses yield oscillatory or bounded Krylov behaviour, whereas deconfined phases with continuous spectra yield exponential growth. In holographic examples this distinction is realized as thermal AdS or AdS soliton versus AdS black hole or black D4 geometries (Anegawa et al., 2024).

6. Scope, related correspondences, and open problems

The momentum–Krylov correspondence is most precise when it is formulated as a local-in-time statement about a proper momentum or wormhole velocity. Several nearby constructions use “moment” or phase-space data to organize Krylov dynamics but are conceptually distinct. For density-matrix operators, the relevant moment-generating object is

$\mathcal O$3

for Krylov complexity and

$\mathcal O$4

for spread complexity, which implies an early-time relation $\mathcal O$5 for general pure states and an exact all-time relation $\mathcal O$6 in two-dimensional maximally entangled cases (Caputa et al., 2024). For unitary maps with a classical limit, the quantum Krylov basis converges to the classical Krylov basis in the $\mathcal O$7 limit, making phase-space evolution in $\mathcal O$8 a direct controller of Krylov growth (Scialchi et al., 11 Mar 2026). In Floquet systems, an autocorrelation sequence $\mathcal O$9 can be converted into a sequence of Krylov angles $\rho$00, and hence into a Floquet inhomogeneous transverse-field Ising chain; failure of the reconstruction signals that the autocorrelation is not generated by unitary dynamics (Yeh et al., 2024). In a different mathematical direction, matching the moment-generating function

$\rho$01

to a truncated exponential ansatz reproduces the Rayleigh–Ritz energies and overlaps in the Krylov space generated by $\rho$02 (Amore et al., 2011).

These adjacent constructions suggest that “momentum–Krylov” can denote more than one bridge: proper radial momentum in holography, wormhole velocity in two-dimensional gravity, phase-space transport in semiclassical dynamics, or moment-generating data in recursion methods. The holographic correspondence discussed above is the one that directly identifies a geometric momentum with the time derivative of Krylov complexity (Fu et al., 26 Oct 2025, Fadafan et al., 30 Jun 2026).

Several open problems remain explicit in the current literature. Zero-temperature geodesic constructions do not reproduce the exponential finite-temperature Krylov growth often seen in field theory; the full microscopic interpretation of exponents such as $\rho$03 in terms of Lanczos coefficients is not yet worked out; the $\rho$04 construction leaves the precise characterization of the “time-dependent $\rho$05” sector open; and extensions to motion on internal spaces, finite-temperature black holes, higher-point functions, and more general operator insertions remain unfinished (Fadafan et al., 30 Jun 2026, Fatemiabhari et al., 24 Nov 2025, Fatemiabhari et al., 19 Feb 2026, Fu et al., 26 Oct 2025). A plausible implication is that the most durable content of the correspondence is not a single universal formula for complexity itself, but a robust identification of complexity growth rate with a momentum-like bulk observable whenever the dual dynamics admits an effective radial description.