Fock-Space Flux Models: Many-Body Graph Dynamics

• Fock-space flux models are many-body frameworks where Hilbert space basis states form graph vertices and transitions carry synthetic gauge phases.
• They enable analysis of interference via closed-loop fluxes, leading to chord-diagram solvability and revealing near-CFT₁ dynamics.
• Applications span synthetic ladders and bosonic/qubit implementations that elucidate localization mechanisms and phase transitions in configuration space.

Fock-space flux models are many-body constructions in which basis states of a Hilbert space are treated as vertices of a graph or synthetic lattice, while phases, holonomies, or synthetic gauge fields are attached to transitions between those states. In this setting, flux is not confined to real-space plaquettes: it can appear as a Peierls phase on links between many-body states, as a plaquette holonomy on a Hilbert-space graph, or as a frustration of return amplitudes in configuration space. Taken together, recent work suggests that “Fock-space flux models” is not a single canonical formalism but a family of related approaches spanning random-flux models for near-$\mathrm{CFT}_1$ physics, periodically driven synthetic ladders in Fock space, and interference-based localization mechanisms on configuration graphs (Berkooz et al., 2023, Jia, 2024, Mumford, 2022).

1. Fock space as a graph with phases and loops

A common starting point is the exact rewriting of a many-body Hamiltonian as a tight-binding problem on Fock space,

$$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$

where the basis states $\{|\alpha\rangle\}$ are interpreted as graph vertices and the nonzero matrix elements $T_{\alpha\beta}$ as edges. Closely related work isolates the underlying connectivity by introducing the adjacency matrix

$$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$

so that many-body dynamics can be viewed as a single-particle walk on a graph whose nodes are Fock states and whose edges are allowed transitions (Scoquart et al., 2024, Lazarides, 30 Apr 2026).

Within this representation, flux becomes a statement about closed paths in Hilbert space. In the Parisi hypercube reinterpretation, the hypercube is treated as the Fock-space graph of a many-body Hamiltonian, and the elementary loops carry quenched-disordered flux. The paper formulates the broader microscopic signature of near-$\mathrm{AdS}_2$/near-$\mathrm{CFT}_1$ behavior as a large Fock-space graph whose loops carry uniform, statistically distributed flux that frustrates return amplitudes and produces double-scaled SYK-style chord combinatorics (Berkooz et al., 2023). A plausible implication is that “flux” in this literature refers less to a unique microscopic degree of freedom than to a gauge-invariant organization of amplitudes on closed Hilbert-space paths.

2. Random fluxes, holonomies, and near-$\mathrm{CFT}_1$ dynamics

The most explicit random-flux construction is the Parisi hypercube model. Its Hamiltonian is written as

$$H=-\frac{1}{\sqrt d}\sum_{\mu=1}^d D_\mu,\qquad D_\mu=T_\mu^+ + T_\mu^-,$$

with phase strings determined by antisymmetric quenched-disordered flux variables $F_{\mu\nu}$. The elementary plaquette holonomy is

$$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$0

where $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$1 is the tunable disorder parameter. In this formulation, the flux frustrates return amplitudes in Fock space: each closed excursion acquires a phase depending on the loop’s shape, and the trace moments of $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$2 reduce to a chord-diagram problem in which each chord crossing contributes a factor of $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$3 (Berkooz et al., 2023).

The same combinatorics extends to a family of probe observables. For operators $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$4 built from similarly fluxed elementary hops, the mixed two-point and four-point trace structures acquire weights $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$5, $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$6, and $$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$7, reproducing the double-scaled SYK diagrammatics with the identifications

$$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$8

In the infrared regime,

$$\hat H=\sum_\alpha E_\alpha |\alpha\rangle\langle \alpha| +\sum_{\alpha\neq\beta}T_{\alpha\beta}|\alpha\rangle\langle\beta|\,,$$9

the correlators become conformal and the operator dimension is

$\{|\alpha\rangle\}$0

The paper’s central claim is that $\{|\alpha\rangle\}$1-locality is too narrow a microscopic criterion: the common ingredient is instead a many-body graph with quenched, uniform, tunable frustration on its loops (Berkooz et al., 2023).

3. Bosonic and qubit implementations with conserved charge

A second line of work constructs charge-conserving near-$\{|\alpha\rangle\}$2 models directly from fluxes in bosonic and qubit Fock spaces. For canonical bosons,

$\{|\alpha\rangle\}$3

with conserved number operator

$\{|\alpha\rangle\}$4

the fluxed hopping operators are

$\{|\alpha\rangle\}$5

where $\{|\alpha\rangle\}$6 are quenched random variables. These obey the magnetic translation algebra

$\{|\alpha\rangle\}$7

A representative “chain” Hamiltonian is built from strings of $\{|\alpha\rangle\}$8 raising and $\{|\alpha\rangle\}$9 lowering operators and is explicitly noted to be not $T_{\alpha\beta}$0-local in the usual sense because each $T_{\alpha\beta}$1 is already nonlocal in occupation-number space (Jia, 2024).

The exact solution again proceeds through chord diagrams. For

$T_{\alpha\beta}$2

odd moments vanish, while even moments organize into standard double-scaled SYK chord sums with crossing weight

$T_{\alpha\beta}$3

The near-$T_{\alpha\beta}$4 regime is specified by

$T_{\alpha\beta}$5

and in that limit the partition function inherits the Schwarzian/NCFT$T_{\alpha\beta}$6 thermodynamics of double-scaled SYK. Probe operators built from independently fluxed $T_{\alpha\beta}$7 have conformal dimension

$T_{\alpha\beta}$8

and four-point functions yield the maximal Lyapunov exponent

$T_{\alpha\beta}$9

The same program admits a qubit realization with

$$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$0

and both bosonic and qubit models show random-matrix-theory level statistics down to very low energies (Jia, 2024).

A central distinction from double-scaled complex SYK is that the relevant leading moments come from empty intersections while the number of operator hoppings $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$1 is fixed, not scaled to infinity. The paper argues that this avoids certain singular charge-sector limits of double-scaled complex SYK and, for the bosonic models, avoids energetic instabilities and unwanted low-temperature ordering (Jia, 2024).

4. Synthetic gauge fields in finite Fock-space ladders

Fock-space flux can also be engineered in finite-dimensional synthetic lattices rather than in large random graphs. A driven bosonic Josephson junction with a single impurity is rewritten as a two-leg ladder in Fock space, with the “lattice sites” identified as many-body Fock states and the two “legs” given by the impurity’s two states. The periodically driven microscopic Hamiltonian is

$$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$2

with piecewise-periodic drive functions $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$3, $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$4, and $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$5. For short pulse interval $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$6, the one-period Floquet operator is generated by an effective Hamiltonian

$$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$7

with

$$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$8

Here the phase $$A_{n,m} = (1-\delta_{n,m})\,\Theta\!\left(|H_{n,m}|\right),$$9 is the synthetic flux; it appears as a Peierls phase in the BEC tunneling terms and is interpreted as the analogue of a magnetic flux through a Fock-space plaquette (Mumford, 2022).

Because the boson number $\mathrm{AdS}_2$0 is fixed, the Hilbert space has dimension $\mathrm{AdS}_2$1, and the natural basis $\mathrm{AdS}_2$2 forms a two-dimensional Fock-state lattice: one synthetic dimension of length $\mathrm{AdS}_2$3 from the BEC imbalance and one of length $\mathrm{AdS}_2$4 from the impurity state. The ground band undergoes a quantum phase transition at a critical synthetic flux $\mathrm{AdS}_2$5: for $\mathrm{AdS}_2$6, the ground state is Meissner-like, while for $\mathrm{AdS}_2$7, it becomes an Abrikosov-vortex phase with two degenerate minima. The key diagnostic is the chiral current

$\mathrm{AdS}_2$8

and the phase-basis probability distribution confirms that $\mathrm{AdS}_2$9 acts as the relevant quasimomentum variable (Mumford, 2022).

This model is important conceptually because the gauge field lives on links between many-body basis states, not on real-space sites. It also makes explicit that flux-driven phases in Fock space need not rely on disorder: repulsive interactions confine the wavefunction near $\mathrm{CFT}_1$0, attractive interactions broaden it, and a second interaction-driven quantum phase transition can coincide with the Meissner-vortex transition, enhancing the sensitivity to the synthetic flux (Mumford, 2022).

5. Interference, frustration, and localization in configuration space

Not all Fock-space flux models are formulated in terms of literal gauge fields. In a broader usage, the relevant structure is interference on a graph of many-body configurations. “Localized Fock Space Cages in Kinetically Constrained Models” treats the many-body Hilbert space as a graph in which each node is a bitstring basis state and each edge corresponds to a nonzero Hamiltonian matrix element. In chiral models the graph is bipartite,

$\mathrm{CFT}_1$1

and zero modes satisfy

$\mathrm{CFT}_1$2

The resulting Fock-space cages are exact localized many-body eigenstates, typically at zero energy, produced by exact cancellation of transition amplitudes on compact subgraphs. The paper emphasizes that this is localization in the graph of basis configurations rather than ordinary real-space localization (Jonay et al., 29 Apr 2025).

Related work on many-body localization shows that the scaling of the critical disorder is governed by the combined effect of diagonal and off-diagonal Fock-space correlations. Diagonal-energy correlations alter the statistics of resonant denominators, while off-diagonal hopping correlations determine whether amplitudes along different Fock-space paths interfere coherently or behave as independent random contributions. The paper’s physical interpretation is that off-diagonal correlations can destructively interfere, suppress resonant growth, and favor localization; the decisive lesson is that critical behavior cannot be inferred from diagonal correlations alone (Scoquart et al., 2024).

A complementary graph-theoretic perspective appears in East-type constrained models. There, the Hamiltonian is again treated as hopping on Fock space, but the principal question is whether localization depends on geometric locality of spin flips or on the coarser graph structure of Fock space. By randomizing the wiring inside neighboring magnetization sectors while preserving the sector ladder, the work finds a transition between delocalized and localized phases and concludes that, for East-type constrained models, the essential ingredient is the structure of the graph in Fock space rather than geometric locality of spin flips (Lazarides, 30 Apr 2026). This suggests a broader interpretation of “flux” or “flow” in Fock space: what matters can be the organization of interfering paths through Hilbert space even when no explicit magnetic phase is inserted.

6. Scope, distinctions, and recurrent misconceptions

Several distinctions recur across the literature. First, flux in Fock space is not the same as a magnetic field in real space. In the driven Josephson-junction construction, the synthetic gauge field lives on links between many-body basis states, not on real-space sites (Mumford, 2022). In the random-flux hypercube and bosonic near-$\mathrm{CFT}_1$3 models, the relevant objects are plaquette holonomies, phase strings, and chord weights on the Hilbert-space graph rather than spatial gauge fields (Berkooz et al., 2023, Jia, 2024).

Second, Fock-space flux models are not equivalent to $\mathrm{CFT}_1$4-local models. Both the hypercube construction and the bosonic flux construction state explicitly that $\mathrm{CFT}_1$5-locality is not essential: the microscopic hallmark is instead a many-body graph with quenched, uniform, tunable frustration on its loops, and the resulting chord-diagram solvability can be realized in non-$\mathrm{CFT}_1$6-local settings (Berkooz et al., 2023, Jia, 2024).

Third, localization in Fock space should not be conflated with localization in real space. Fock-space cages may be compactly supported on a small subgraph of basis states while involving superpositions of many real-space configurations, and the many-body localization literature likewise formulates delocalization as spreading over an exponentially large graph of configurations rather than simple spatial transport (Jonay et al., 29 Apr 2025, Scoquart et al., 2024).

Taken together, these works define a coherent research area centered on attaching phases, holonomies, or structured interference to the graph of many-body states. Some models use literal synthetic gauge flux, some use uniform random loop frustration, and some emphasize destructive interference on constrained Hilbert-space graphs. The common technical move is the same: many-body dynamics are recast as motion on Fock space, and flux-like structure is introduced at the level of amplitudes around closed paths.