Krylov-Restricted Thermalization

• Krylov-restricted thermalization is equilibration confined to the dynamically accessible Krylov subspace, emphasizing state-specific evolution.
• It employs the Lanczos basis and a tridiagonal Hamiltonian formalism to reveal the geometry and connectivity underlying thermalization.
• This framework underpins phenomena in constrained many-body systems, operator growth, Floquet heating, and symmetry-resolved dynamics.

Searching arXiv for papers on Krylov-restricted thermalization and related Krylov-space thermalization frameworks. Krylov-restricted thermalization denotes equilibration that is confined to the dynamically accessible Krylov subspace generated by a chosen initial state or operator, or, in fragmented systems, to a disconnected Hilbert-space fragment that coincides with such a sector. In this formulation, thermal behavior is not assessed relative to the full Hilbert space or even the full symmetry sector alone, but relative to the connected subspace actually explored by the dynamics. The concept emerged from state-space Lanczos constructions of thermalization, was formalized in constrained many-body systems with exponentially many disconnected Krylov sectors, and has since been extended to operator growth, Floquet heating, temperature-dependent Krylov flows, and symmetry-resolved dynamics (Khlebnikov et al., 2013, Moudgalya et al., 2019, Qi et al., 2024).

1. Krylov subspaces, Krylov chains, and the dynamical reduction of thermalization

For state dynamics, the Krylov subspace generated by an initial state $|\psi_0\rangle$ is

$K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$

Because

$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$

the time evolution up to a fixed accuracy and finite time is encoded in a finite-dimensional Krylov sector. In the Lanczos basis $\{|n\rangle\}$ associated with $|\psi_0\rangle$, the Hamiltonian is tridiagonal,

$$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$

so the dynamics take the form of hopping on a one-dimensional chain indexed by Krylov level $n$ (Alishahiha et al., 2024, Khlebnikov et al., 2013).

For operator dynamics, an analogous construction begins from a seed operator and the Liouvillian superoperator. In the Floquet setting, the seed is chosen as the period-averaged Hamiltonian

$$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$

and the Liouvillian is

$$\mathcal L \hat O \equiv [\hat H_{\mathrm{eff}}^{\mathrm{fold}},\hat O].$$

Gram–Schmidt orthonormalization of repeated Liouvillian action produces a Krylov basis $\{|W_n)\}$, and if

$K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$0

then the amplitudes satisfy

$K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$1

with Lanczos coefficients $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$2 playing the role of hopping amplitudes on a semi-infinite chain. The Krylov complexity

$K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$3

is the chain wavepacket’s center of mass (Qi et al., 2024).

This reduction is the formal basis of Krylov-restricted thermalization. The relevant ergodic question is transferred from the full Hilbert space to the geometry, connectivity, and asymptotic structure of the accessible Krylov chain. In this sense, Krylov space is not merely a computational truncation; it is the dynamically generated manifold on which thermalization is tested.

2. Restricted ETH and the Krylov formulation of thermal behavior

An early formulation of the idea appeared in numerical tests of ETH inside Krylov space. For hard-core bosons on a two-dimensional lattice, thermalization was observed in a Krylov subspace of dimension $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$4 even when the full Hilbert-space dimension was $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$5 to $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$6. In that setting, the Hamiltonian projected onto the Krylov subspace was diagonalized in a Ritz basis, and ETH-like smoothness of diagonal matrix elements together with suppression of off-diagonal elements was found within the projected dynamics (Khlebnikov et al., 2013).

A more explicit Krylov-space thermalization criterion was proposed as the Krylov Thermalization Hypothesis. For a typical local observable $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$7, the matrix $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$8 in the Krylov basis should exhibit an approximately tridiagonal leading structure, with all non-nearest-neighbor entries exponentially small or $K_n=\mbox{span}\{H^p|\psi_0\rangle,\; p=0\ldots n-1\}.$9-suppressed. The proposed form is

$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$0

which is a Krylov-space analogue of ETH in the energy basis. In this framework, thermalization is “restricted” not because it fails altogether, but because its leading matrix structure is controlled by the same tridiagonal Krylov chain that governs state propagation (Alishahiha et al., 2024).

This perspective also sharpens the distinction between weak and strong thermalization. The infinite-time average of Krylov complexity,

$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$1

was proposed as a probe of that distinction: larger $$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$2 correlated with stronger thermalization, whereas the variance of Lanczos coefficients captured only part of the structure and was found to be less reliable than $$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$3 (Alishahiha et al., 2024).

3. Fragmentation, constrained dynamics, and thermalization within disconnected sectors

The notion became structurally sharp in constrained models whose Hilbert spaces fracture into exponentially many disconnected Krylov sectors. In the one-dimensional pair-hopping model

$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$4

center-of-mass conservation and fractonic mobility constraints produce a decomposition

$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$5

with exponentially many dynamically disconnected subspaces. Some exponentially large sectors are integrable and map to spin-$$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$6 XX chains; others are non-integrable and show GOE-like level statistics with $$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$7 together with weak ETH restricted to the sector. This is the setting in which the phrase “Krylov-restricted thermalization” was made explicit: the appropriate question is whether a given connected Krylov sector thermalizes within itself (Moudgalya et al., 2019).

A closely related example is a correlated-hopping model with strong Hilbert space fragmentation labeled by irreducible strings. The number of fragments grows asymptotically as $$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$8, the number of frozen states grows as $$|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle = \sum_{p=0}^\infty \frac{(-i)^p}{p!} t^p H^p|\psi_0\rangle,$$9, and the largest fragment has size $\{|n\rangle\}$0 even though $\{|n\rangle\}$1 exponentially. Globally the model fails strong ETH, with level statistics close to Poisson,

$\{|n\rangle\}$2

but inside the largest fragment the story changes: local observables narrow, standard deviations decrease roughly like $\{|n\rangle\}$3, and after adding weak disorder to lift accidental degeneracies the level statistics become GOE-like,

$\{|n\rangle\}$4

The resulting notion was termed subspace-restricted ETH, identified with Krylov-restricted thermalization (Aditya et al., 2024).

The same principle has been realized experimentally in Rydberg atom arrays with blockade, facilitation, and Floquet frequency modulation. In the constrained Hamiltonian $\{|n\rangle\}$5, the Hilbert space fragments into configurational sectors such as odd, even, and mixed subspaces. The central observation is that the system thermalizes within a connected Krylov fragment, while states in different fragments do not thermalize with one another even when they are exactly degenerate in energy. Thermalization and memory therefore coexist: staggered magnetization in the active subarray decays toward its thermal value, microstate projections become nearly uniform within the allowed fragment, and single-atom von Neumann entropy saturates to $\{|n\rangle\}$6, yet the frozen or disconnected subspaces retain the initial configurational information (Zhao et al., 2024).

A precursor to this entire line of thought lies in the study of two interacting atoms in a multimode harmonic waveguide. That work did not use the modern language of Krylov space, but it showed that a well localized perturbation can generate some chaos-like signatures without producing full thermalization. Exact eigenstates retained only $\{|n\rangle\}$7 principal components even at infinite interaction strength, and relaxed observables remained far from microcanonical values. This suggests a restricted-subspace interpretation in which the localized perturbation fails to generate the broad basis mixing required for complete thermalization (Yurovsky et al., 2010).

4. Floquet heating, topological Krylov chains, and prethermal restriction

In periodically driven many-body systems, Krylov restriction acquires a topological form. For a Floquet Hamiltonian $\{|n\rangle\}$8, the one-period propagator is

$\{|n\rangle\}$9

and the usual folded Floquet Hamiltonian obtained from $|\psi_0\rangle$0 has a spectrum restricted to the Floquet Brillouin zone $|\psi_0\rangle$1. To describe thermalization through ETH, an alternative unfolded effective Hamiltonian is constructed from the zero mode of a Liouvillian defined on the Krylov chain (Qi et al., 2024).

The key invariant is a real-space winding number,

$|\psi_0\rangle$2

where $|\psi_0\rangle$3 is the chiral symmetry superoperator and $|\psi_0\rangle$4 is built from positive- and negative-energy eigenmodes of the Liouvillian. Because

$|\psi_0\rangle$5

the spectrum comes in chiral pairs. In the short-period or high-frequency regime, the Krylov chain is SSH-like and topologically nontrivial, with $|\psi_0\rangle$6 and a boundary-localized zero mode. In the trivial regime, the local gap closes, the winding number ceases to be quantized, and the zero mode delocalizes into the bulk (Qi et al., 2024).

The zero mode satisfies

$|\psi_0\rangle$7

so it is interpreted as an unfolded effective Hamiltonian. When this mode is edge-localized, it has small Krylov complexity and acts as a quasi-local conserved structure. The long-time state is then described by a canonical ensemble

$|\psi_0\rangle$8

with $|\psi_0\rangle$9 fixed by energy matching. In this regime, energy absorption is bounded and the driven system thermalizes to a finite-energy-density, finite-temperature state. When the topology is trivial, the zero mode is not protected, spreads into the bulk, and local observables approach the infinite-temperature value (Qi et al., 2024).

The thermodynamic limit modifies this picture. As $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$0, the Lanczos coefficients $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$1 approach a constant, so the Krylov chain becomes gapless overall and an exact topological edge mode is absent. Nevertheless, in the high-frequency regime the initial part of the chain remains locally SSH-like and supports a quasi-edge mode $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$2 with lifetime $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$3. Prethermalization is then interpreted as tunnelling of this quasi-edge mode through the local gap on the Krylov chain: for $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$4 the operator wavepacket stays near the boundary and the system behaves as though it had an effective conserved Hamiltonian, while for longer times the mode leaks into the bulk and eventual heating to infinite temperature resumes. This yields a systematic construction of the prethermal Hamiltonian by building the Krylov basis from $$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$5, computing the Lanczos coefficients, identifying the local gap near the boundary, extracting the localized or quasi-local zero mode, and using it as the effective Hamiltonian governing finite-time thermalization (Qi et al., 2024).

5. Lanczos growth, temperature dependence, and the limits of complexity as a chaos witness

A major line of work studies thermalization by reading it off from Lanczos coefficients and Krylov complexity. In the large-$$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$6 IP matrix model, sufficiently high temperature produces a continuous, gapless spectrum together with exponential decay of the fundamental correlator. In this regime the spectral tail

$$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$7

implies

$$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$8

The interpretation offered there is that thermalization and information loss occur in the Krylov subspace associated with the chosen operator, not necessarily in the full Hilbert space of arbitrary observables. The qualifier “restricted” is therefore literal: Krylov thermalization depends on the operator-generated subspace and on the large-$$\langle n|H|m\rangle = a_n\delta_{nm} + b_{m+1}\delta_{n,m+1} + b_m\delta_{n,m-1},$$9, high-temperature regime (Iizuka et al., 2023).

The temperature dependence of Lanczos data was subsequently placed into an integrable framework. With a thermal inner product, the $n$0-dependence of Krylov matrices becomes an isospectral Lax flow, and for Hermitian initial operators the even and odd Lanczos sectors evolve as two open Toda chains subject to matching conditions. In the low-temperature gapped regime, where $n$1 for $n$2, half of the Lanczos coefficients vanish asymptotically,

$n$3

and the time-averaged Krylov complexity becomes exponentially small,

$n$4

This suppression was explicitly described as Krylov-restricted thermalization: thermal dynamics become severely constrained in Krylov space when the thermal state cannot access excitations above the gap (Angelinos et al., 26 Aug 2025).

At the same time, a central caution has become standard: exponential Krylov complexity is necessary but not sufficient for many-body chaos. The decisive counterexample is the free $n$5 CFT or Luttinger-liquid class, where $n$6 grows exponentially but the Krylov metric

$n$7

has off-diagonal elements comparable to the diagonal ones. The proposed fast-scrambler criterion therefore combines three ingredients: exponential Krylov growth, diagonal scaling

$n$8

and negligible off-diagonal $n$9 for $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$0. A common misconception is thus ruled out: fast spreading along the Krylov chain, by itself, does not establish chaotic thermalization (Chen et al., 2024).

6. Symmetry resolution, open Krylov space, and static Krylov diagnostics

More recent work has enlarged the concept from a single unresolved Krylov chain to families of sector-resolved and effectively open Krylov problems. When a conserved charge $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$1 satisfies $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$2 and an operator is symmetry invariant, the Hilbert space decomposes as $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$3, and the operator decomposes into block operators $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$4. One may then define symmetry-resolved Krylov complexities

$$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$5

At early times the full complexity equals the weighted average of the resolved ones up to $$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$6,

$$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$7

while at later times sector mixing becomes nontrivial. In special cases, however, the charge dependence disappears and one finds Krylov complexity equipartition,

$$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$8

showing that symmetry constraints can either differentiate or erase sector dependence in restricted thermalization (Caputa et al., 2 Jul 2025).

A complementary development concerns infinite closed systems, where only a finite number of Lanczos coefficients can be computed. Rather than truncating Krylov space with a reflecting hard wall, an emergent open boundary condition is imposed near the truncation point,

$$\hat O=\hat H^{(0)}=\frac{1}{T}\int_0^T dt\,\hat H(t),$$9

producing a non-Hermitian effective Liouvillian. This partitions Krylov space into a resolved system sector of relatively local operators and an unresolved environment sector of increasingly nonlocal operators. In chaotic systems with $$\mathcal L \hat O \equiv [\hat H_{\mathrm{eff}}^{\mathrm{fold}},\hat O].$$0, the resulting modes acquire negative imaginary parts and local observables decay exponentially, with the decay rate tied to the operator-growth scale. In this picture, restricted thermalization is recast as leakage of operator weight from the local Krylov sector into a nonlocal bath (Loizeau et al., 10 Mar 2025).

A further static extension is the Krylov distribution based on the resolvent-dressed state

$$\mathcal L \hat O \equiv [\hat H_{\mathrm{eff}}^{\mathrm{fold}},\hat O].$$1

Its decomposition in the Krylov basis defines

$$\mathcal L \hat O \equiv [\hat H_{\mathrm{eff}}^{\mathrm{fold}},\hat O].$$2

which measures how inverse-energy response penetrates the dynamically accessible Krylov chain. The abstract identifies three universal regimes: saturation outside the spectral support, extensive growth within continuous spectra, and sublinear or logarithmic scaling near spectral edges and quantum critical points. Although this construction was not presented as a direct thermalization order parameter, it provides a static measure of whether response remains Krylov-localized, spreads extensively, or is bottlenecked by spectral edges or criticality (Alishahiha et al., 5 Feb 2026).

Taken together, these developments define Krylov-restricted thermalization as a family of closely related statements. Thermalization may occur within a Krylov subspace but fail globally because of fragmentation; it may be protected temporarily by a topological quasi-edge mode in operator space; it may be exponentially suppressed by a spectral gap at low temperature; and it may depend sensitively on symmetry sector, operator choice, or the coupling of the resolved Krylov sector to an effective bath. The unifying feature is that the physically relevant notion of equilibration is determined by the geometry and connectivity of dynamically accessible Krylov space, rather than by the full Hilbert space alone.