3D Monitored Kitaev Models

• 3D Monitored Kitaev models are theoretical frameworks defined on tricoordinated lattices that host bond-directional spin interactions and Majorana fermions.
• They reveal unique spectroscopic signatures, such as flux gaps and threshold behaviors, distinguishing nodal lines, Weyl points, and Fermi surfaces.
• They also bridge layered and fully 3D systems, combining experimental probes like INS, ESR, and RIXS with thermodynamic monitoring to explore quantum phases.

Searching arXiv for recent and foundational papers on 3D Kitaev models, spectroscopy/monitoring, and multilayer stacked Kitaev systems. “3D monitored Kitaev models” does not denote a single standardized model class in the present literature. The phrase instead intersects three tightly related but conceptually distinct directions: exactly solvable and perturbed three-dimensional Kitaev spin liquids on tricoordinated lattices; monitored 3D Kitaev physics in the condensed-matter sense of being probed by dynamical observables such as inelastic neutron scattering, electron spin resonance, resonant inelastic X-ray scattering, NMR, susceptibility, and thermal probes; and, more loosely, multilayer or stacked Kitaev-honeycomb systems that route two-dimensional Kitaev physics toward realistic quasi-three-dimensional materials (Smith et al., 2016, Halász et al., 2017, Yoshitake et al., 2017, Merino et al., 2024). By contrast, the modern quantum-information meaning of “monitored dynamics”—continuous or repeated measurements producing nonunitary evolution and measurement-induced entanglement transitions—has so far been developed in the cited corpus only for two-dimensional Kitaev-type monitored circuits, not for genuine 3D lattices (Vijayvargia et al., 20 Sep 2025, Kalpada et al., 25 Nov 2025).

1. Three-dimensional Kitaev systems as the substrate of “monitoring”

Three-dimensional Kitaev models are defined on tricoordinated lattices whose nearest-neighbor bonds can be partitioned into $x$, $y$, and $z$ types, permitting the bond-directional Hamiltonian

$$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$

This structure underlies the hyperhoneycomb, hyperoctagon, hyperhexagon, stripyhoneycomb, hypernonagon, and related $(p,3)$ lattices studied in the 3D Kitaev literature (Smith et al., 2016, O'Brien et al., 2015, Mishchenko et al., 2019, Eschmann et al., 2020).

The exact solution proceeds through Majorana fractionalization. In one common notation,

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$

with static $\mathbb Z_2$ bond operators

$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$

so that the spin Hamiltonian becomes a free-Majorana hopping problem in a static gauge background (Smith et al., 2016, Smith et al., 2015, O'Brien et al., 2015). The physically conserved gauge-invariant quantities are loop or plaquette operators such as

$$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$

or, equivalently, products of bond terms around closed loops (Smith et al., 2016).

A defining three-dimensional feature is that gauge-flux excitations are generally loop-like rather than point-like. This distinction is central for thermodynamics, spectroscopy, and topological excitations. In the gapped strong-coupling limit of a 3D Kitaev model introduced on a trivalent lattice, the low-energy theory reduces to a commuting plaquette Hamiltonian on an effective diamond lattice, with excitations constrained to form closed loops in an embedded lattice; the shortest such loops are fermionic and braid nontrivially with larger loops, giving a phase $\pi$ when a fermion winds through a loop (Mandal et al., 2011). In more general 3D Kitaev models, the same loop-based gauge structure reappears thermodynamically as vison-loop proliferation and spectroscopically as localized flux-loop insertions created by spin flips (Yoshitake et al., 2017, Eschmann et al., 2020).

The low-energy Majorana sector of 3D Kitaev spin liquids is richer than in the two-dimensional honeycomb model. Symmetry analysis and exact solutions show that depending on lattice geometry and the projective implementation of time-reversal and inversion, the itinerant Majorana fermions may form Majorana Fermi surfaces, nodal lines, or Weyl points (O'Brien et al., 2015). Representative cases are the hyperoctagon lattice with a Majorana Fermi surface, the hyperhoneycomb lattice with nodal lines, and the hyperhexagon lattice with Weyl points (Smith et al., 2016).

2. “Monitoring” as spectroscopy: INS, ESR, and RIXS

In the 3D Kitaev literature, the most developed meaning of “monitored” is experimentally probed. The key observables are the dynamical spin structure factor for inelastic neutron scattering and ESR, and channel-resolved resonant inelastic X-ray scattering (Smith et al., 2016, Smith et al., 2015, Halász et al., 2017).

For INS and ESR, the central object is the dynamical spin structure factor

$y$0

with $y$1 (Smith et al., 2016). In an ideal Kitaev model, a local spin operator does not create a bare magnon. Instead, it flips a bond variable, inserts flux excitations, and launches Majorana matter in the altered gauge background. In 3D this leads to a local quantum quench involving a static flux-loop insertion plus dynamical Majorana matter fermions (Smith et al., 2016). As a result, the spin response is thresholded by a flux gap,

$y$2

even when the Majorana sector is gapless (Smith et al., 2016, Smith et al., 2015).

The threshold line shape diagnoses the type of 3D Majorana metal. If the Majorana density of states vanishes as

$y$3

then the structure factor above threshold behaves as

$y$4

For the hyperhoneycomb nodal-line case, $y$5 rises linearly above threshold, while for the hyperhexagon Weyl case it rises quadratically (Smith et al., 2016). By contrast, for a Majorana Fermi surface with finite $y$6, the threshold is a genuine Majorana X-ray-edge singularity

$y$7

with

$y$8

(Smith et al., 2016). This separation between divergent, linear, and quadratic threshold behavior is one of the most concrete monitored signatures of distinct 3D Kitaev spin liquids.

The hyperhoneycomb INS problem was analyzed in detail in a thermodynamic-limit exact treatment. There the response remains broad and continuum-like but exhibits a response gap even in the gapless QSL phase, because any spin flip necessarily creates flux loops (Smith et al., 2015). The same work found that the response is dominated by single-Majorana final states, accounting for $y$9 of the response at $z$0 for isotropic couplings, which is why INS acts as relatively direct spectroscopy of the Majorana density of states (Smith et al., 2015).

RIXS refines this monitored picture because it resolves distinct channels. The full intensity is written as

$z$1

with amplitudes decomposed into spin-conserving and non-spin-conserving channels (Halász et al., 2017). In the non-spin-conserving channels, the leading term reduces to the spin-polarized INS amplitude and is dominated by flux-creating processes. In the spin-conserving channel, the first nontrivial contribution creates no fluxes and probes the Majorana matter sector directly through a two-fermion continuum

$z$2

This permits direct identification of whether the 3D spin liquid hosts nodal lines, Weyl points, or Fermi surfaces (Halász et al., 2017). A universal additional signature is the suppression of the spin-conserving RIXS response near $z$3, traced to destructive interference between sublattices and interpreted as a spectroscopic manifestation of symmetry fractionalization (Halász et al., 2017).

3. Thermodynamic and dynamical monitoring of the 3D gauge sector

A second major sense of monitoring is through temperature-dependent response functions that track the emergent $z$4 gauge sector. The most complete such analysis for a true 3D Kitaev lattice is the QMC+CTQMC study of the hyperhoneycomb model (Yoshitake et al., 2017).

There the monitored observables are the magnetic susceptibility,

$z$5

the NMR relaxation rate $z$6, the dynamical spin structure factor

$z$7

and a direct gauge-flux fluctuation measure

$z$8

(Yoshitake et al., 2017). The isotropic hyperhoneycomb model shows a high-temperature crossover

$z$9

associated with itinerant matter Majoranas, and a genuine 3D finite-temperature transition

$$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$0

driven by the proliferation of looplike gauge fluxes (Yoshitake et al., 2017). Near $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$1, $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$2, $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$3, and $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$4 all develop sharp peaks, and $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$5 shows strong quasi-elastic weight above the transition and a shift of the low-energy peak to higher $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$6 below it, signaling flux-gap opening (Yoshitake et al., 2017). The comparison with the two-dimensional honeycomb model is explicit: in 2D the corresponding lower scale is only a crossover at $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$7, whereas in 3D the dynamical signatures are singular (Yoshitake et al., 2017).

A broader thermodynamic classification across elementary 3D tricoordinated lattices was then developed by sign-free Majorana quantum Monte Carlo (Eschmann et al., 2020). Across a broad class of bipartite lattices, the ground-state flux sector is organized by the elementary plaquette length $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$8: $$\hat{H} = -\frac{J_x}{2}\sum_{\langle jk\rangle_x}\hat{\sigma}_j^x\hat{\sigma}_k^x -\frac{J_y}{2}\sum_{\langle jk\rangle_y}\hat{\sigma}_j^y\hat{\sigma}_k^y -\frac{J_z}{2}\sum_{\langle jk\rangle_z}\hat{\sigma}_j^z\hat{\sigma}_k^z .$$9 gives 0-flux, while $(p,3)$0 gives $(p,3)$1-flux, extending Lieb-type intuition beyond the cases where Lieb’s theorem is rigorously applicable (Eschmann et al., 2020). The low-temperature gauge transition correlates strongly with the smallest vison gap $(p,3)$2 rather than simply with loop length. For example, $(p,3)$3 has $(p,3)$4 and $(p,3)$5, while $(p,3)$6 has $(p,3)$7 and $(p,3)$8 (Eschmann et al., 2020).

That classification also isolates two important exceptions. The $(p,3)$9 lattice exhibits gauge frustration, because local flux preferences cannot be simultaneously satisfied; the low-temperature average flux becomes

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$0

and the transition is strongly suppressed to $$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$1 (Eschmann et al., 2020). The odd-loop hypernonagon $$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$2 instead has $$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$3, intertwining gauge ordering with spontaneous time-reversal breaking and producing a single first-order transition at

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$4

into a crystalline $$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$5 gauge-ordered phase (Eschmann et al., 2020, Mishchenko et al., 2019).

Thermodynamic tensor-network work on the hyperhoneycomb Kitaev model reinforces the same 3D picture. In the thermodynamic limit, the specific heat shows a high-temperature crossover at

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$6

for the isotropic case and a lower gauge-ordering transition estimated at

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$7

to be compared with a quoted QMC benchmark

$$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$8

(Jahromi et al., 2020). The entropy drops from $$\hat{\sigma}_j^\alpha = i \hat b_j^\alpha \hat c_j,$$9 to $\mathbb Z_2$0 at the high-temperature crossover, then to zero below the gauge-ordering scale, expressing 3D fractionalization as a two-stage process (Jahromi et al., 2020).

4. Gauge order, topological excitations, and Majorana topology in 3D

The monitored signatures of 3D Kitaev matter are inseparable from the structure of the gauge sector and the topology of the Majorana bands. In the gapped strong-coupling phase of a 3D Kitaev model on a trivalent lattice, degenerate perturbation theory produces an effective Hamiltonian on the diamond lattice,

$\mathbb Z_2$1

with commuting loop operators $\mathbb Z_2$2 and loop excitations constrained by local, surface, and volume constraints (Mandal et al., 2011). The elementary shortest-loop excitations are fermions, and noncontractible loop and surface operators produce an $\mathbb Z_2$3-fold ground-state degeneracy on the three-torus (Mandal et al., 2011). This provides a canonical example of 3D Kitaev topological order in the low-energy gapped regime.

In gapless phases, the classification is instead in terms of Majorana semimetals. A symmetry-based analysis of elementary 3D tricoordinated Kitaev lattices shows that projective time-reversal and inversion symmetries determine whether the itinerant Majoranas realize Fermi surfaces, nodal lines, or Weyl points (O'Brien et al., 2015). Inversion may act projectively as

$\mathbb Z_2$4

and time reversal as

$\mathbb Z_2$5

so the zero-energy manifold is set by the pair $\mathbb Z_2$6 rather than by nonprojective Bloch-band symmetry alone (O'Brien et al., 2015). This explains, for example, why the hyperhoneycomb $\mathbb Z_2$7 carries nodal lines, while $\mathbb Z_2$8 can host Weyl nodes even with physical time-reversal and inversion symmetry preserved at the spin level (O'Brien et al., 2015).

Odd-loop 3D Kitaev lattices introduce a further layer of structure. On the hypernonagon lattice, the odd plaquette length forces

$\mathbb Z_2$9

so spontaneous time-reversal breaking occurs already at the level of the flux sector (Mishchenko et al., 2019). Extensive QMC and variational analysis identify at least five distinct chiral spin-liquid phases—A0F, AF, AFII, SI, and SII—all with crystalline ordering of the $$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$0 fluxes (Mishchenko et al., 2019). At the isotropic point the system enters the AFII phase at

$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$1

with a flux-loop gap

$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$2

(Mishchenko et al., 2019). Depending on flux-crystal pattern and anisotropy, the itinerant Majorana sector may be fully gapped, contain Weyl nodes with Fermi arcs, or host nodal lines with drumhead states (Mishchenko et al., 2019).

This separation between gauge-crystal order and Majorana topology is especially important. The hypernonagon study shows explicitly that flux-order boundaries and Majorana gapped/gapless boundaries do not generally coincide; a fixed flux crystal can support different Majorana topologies as couplings are varied (Mishchenko et al., 2019). A plausible implication is that any future genuine 3D measurement-driven “monitored” construction will need to track gauge-sector ordering and matter-sector entanglement or band topology as distinct, though coupled, structures.

5. Stacked and multilayer Kitaev systems as a quasi-3D route

A different but physically important route into “3D monitored Kitaev models” is through stacks of 2D Kitaev-honeycomb layers. The multilayer Kitaev model of stacked honeycomb planes coupled by interlayer Heisenberg exchange is not a fully three-dimensional bond-dependent Kitaev lattice, but it is explicitly motivated by the three-dimensional structure of layered candidate materials such as $$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$3-RuCl$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$4 and H$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$5LiIr$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$6O$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$7 (Merino et al., 2024).

The model is

$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$8

with

$$\hat u_{jk} = i \hat b_j^{\alpha_{jk}} \hat b_k^{\alpha_{jk}},$$9

and

$$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$0

Here $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$1 is the number of honeycomb layers and $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$2 is an ordinary interlayer Heisenberg exchange (Merino et al., 2024). An Abrikosov-fermion mean-field treatment plus projective symmetry analysis finds self-consistent quantum spin-liquid states built from layer-by-layer Kitaev ansätze whose bond amplitudes alternate in sign from one layer to the next (Merino et al., 2024).

The principal result is an even–odd effect in the number of layers. For small $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$3, the system is just a direct product of $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$4 single-layer Kitaev spin liquids. Beyond a first critical coupling $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$5, a new multilayer QSL appears. For even $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$6 it is gapped, because the Dirac cones of the single-layer KSL gap out over an intermediate regime $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$7. For odd $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$8 it remains gapless even at arbitrarily large $$\hat{\mathcal W}_\gamma=\prod_{l\in\gamma}\tilde K_l,\qquad \tilde K_l=-i\hat u_{jk},$$9, because one effective half-odd-integer degree of freedom remains unpaired (Merino et al., 2024). The paper calls these states hybrid QSLs, emphasizing that they are not just independent copies of the single-layer KSL (Merino et al., 2024).

For concrete cases, with ferromagnetic interlayer coupling the bilayer has $\pi$0 and $\pi$1, while the trilayer has a single transition near $\pi$2. With antiferromagnetic coupling the bilayer has $\pi$3 and $\pi$4, and the trilayer again has only one transition near $\pi$5 (Merino et al., 2024). Under a $\pi$6 magnetic field,

$\pi$7

the multilayer systems inherit a field-induced gapped chiral descendant of the Kitaev spin liquid, and antiferromagnetic interlayer exchange stabilizes this gapped chiral KSL over a broader $\pi$8-$\pi$9 window (Merino et al., 2024).

This stacked-lattice route is relevant to 3D Kitaev phenomenology, but the paper is explicit that it does not study monitoring in the sense of quantum measurement dynamics, monitored circuits, or measurement-induced transitions (Merino et al., 2024). Its relevance is instead that realistic 3D Kitaev materials are often structurally layered, so finite stacks already display parity-dependent gapped and gapless hybrid QSL behavior controlled by projective layer symmetries (Merino et al., 2024).

6. Relation to measured materials, perturbed Hamiltonians, and the current boundary of genuine “monitored dynamics”

Several works place 3D Kitaev monitoring into direct material context. The review of Kitaev materials identifies $y$00-Li$y$01IrO$y$02 and $y$03-Li$y$04IrO$y$05 as the first truly three-dimensional Kitaev materials, built from tricoordinated edge-sharing IrO$y$06 octahedra and expected to realize dominant bond-directional exchange (Trebst, 2017). It emphasizes that the 3D gauge sector differs from 2D because vison excitations are closed flux loops, leading to a genuine finite-temperature gauge transition around

$y$07

for the hyperhoneycomb Kitaev model, whereas the higher fractionalization crossover occurs around

$y$08

(Trebst, 2017). This establishes why thermodynamic and dynamical probes in 3D can monitor separate matter-fermion and gauge-loop scales.

Beyond the solvable limit, the nearest-neighbor 3D hyperhoneycomb Kitaev–Heisenberg model has been mapped out both semiclassically and by pseudofermion functional renormalization group. The PFFRG phase diagram contains QSL regions near both AFM and FM Kitaev limits and four ordered phases—Néel, zigzag, FM, and stripy—with isotropic QSL windows

$y$09

(Fukui et al., 2023). That work also stresses a specifically 3D finite-temperature topological transition estimate

$y$10

for the pure hyperhoneycomb Kitaev model, corresponding heuristically to $y$11 in the PFFRG cutoff language (Fukui et al., 2023). A broader tensor-network study of the 3D hyperoctagon Kitaev–Heisenberg model similarly finds narrow QSL windows around the pure Kitaev points and conventional Landau thermal transitions in ordered phases, contrasting them with the non-Landau gauge-ordering physics of the Kitaev QSL (Jahromi et al., 2020).

What remains largely absent in the cited 3D literature is the modern measurement-theoretic meaning of monitored dynamics. The two explicitly monitored Kitaev works in the corpus—on qudit generalizations of the monitored Kitaev model and on the measurement-only honeycomb Kitaev model with added single-, three-, and four-qubit checks—are entirely two-dimensional (Vijayvargia et al., 20 Sep 2025, Kalpada et al., 25 Nov 2025). They show that measurement-only Kitaev circuits can support topological area-law, critical-law, and distinct volume-law phases, and that commuting versus noncommuting added checks organize whether plaquette flux memory is preserved (Vijayvargia et al., 20 Sep 2025, Kalpada et al., 25 Nov 2025). A plausible implication is that a genuine future “3D monitored Kitaev model” in the quantum-information sense would likely have to decide whether measurements preserve 3D loop- or membrane-like gauge invariants, whether they remain quadratic in Majoranas, and whether monitored dynamics disorders the 3D gauge sector by a process analogous to vison-loop proliferation. But those questions are not answered in the present 3D corpus.

The current state of the field therefore supports a precise summary. “3D monitored Kitaev models” is well established if “monitored” means spectroscopically probed or thermodynamically diagnosed: 3D Kitaev spin liquids are monitored by INS, ESR, RIXS, NMR, susceptibility, specific heat, and thermal transport, and these observables reveal flux thresholds, vison-loop physics, gauge-ordering transitions, and the topology of Majorana Fermi surfaces, nodal lines, and Weyl nodes (Smith et al., 2016, Smith et al., 2015, Halász et al., 2017, Yoshitake et al., 2017, Eschmann et al., 2020). It is also meaningful in a broader materials sense through stacked multilayer Kitaev models that interpolate between 2D and realistic quasi-3D layered compounds (Merino et al., 2024). But as a class of measurement-induced, nonunitary 3D quantum dynamical models, it remains essentially an open direction within the cited literature.