Synthetic gauge geometry does not guarantee many-body locality

Key claim

An internal-state synthetic dimension can reproduce a Hofstadter hopping graph and its gauge-invariant flux exactly within the single-particle model. The same identification need not preserve locality once particles interact.

Atoms in distant internal states can occupy the same real-space Wannier orbital. Contact collisions may then couple every pair of synthetic sites in a column with comparable strength.

The kinetic graph and the interaction metric are separate pieces of a many-body Hamiltonian. A faithful simulation of the first does not supply the second.

Theme

Gauge-equivalent hopping geometry and interaction locality in synthetic dimensions.

Guiding question

When does the map

justify treating an internal state $m$ as a spatial coordinate? Which many-body conclusions require more than this map?

Setup

Take a one-dimensional optical lattice with sites $n$ and spacing $a$. Each atom has $W=2F+1$ internal states

Ordinary tunnelling moves an atom along $n$. Raman transitions move it between neighboring internal states and imprint a position-dependent phase. A representative tight-binding Hamiltonian is

with

for the Raman momentum transfer used in the simplest construction. The absence of couplings beyond $m=\pm F$ gives open boundaries in the synthetic direction.

For approximately $SU(W)$-symmetric bosonic contact interactions, projection into the lowest Wannier orbital gives

The Hamiltonian is local in the physical coordinate $n$. Its range in the coordinate $m$ will be the central issue.

Platform matters

The locality problem discussed here concerns internal-state synthetic dimensions with contact collisions. Frequency modes, momentum lattices, photonic systems, and engineered state-dependent interactions can realize different kernels.

Analysis

1. The plaquette flux is an exact graph statement

Ignore the common real hopping amplitude and assign oriented link variables

Traverse the plaquette

The ordered product is

The sign changes if the loop orientation is reversed. The signed flux in the convention above is

Under a local basis change

each link gains endpoint phases. They cancel around a closed loop, leaving $W_{◻}$ invariant.

This establishes a uniform Abelian lattice flux. Within the tight-binding Hilbert space, the synthetic system and a charged particle on the corresponding Hofstadter strip have the same one-body connectivity and link phases.

Celi and collaborators used this construction to predict Hofstadter spectra and edge modes localized near $m=\pm F$. Subsequent cold-atom experiments observed chiral dynamics in synthetic Hall ribbons.

What “exact” means

The graph equivalence holds for the modeled one-body Hamiltonian. Raman detunings, unequal ${\mathrm{\Omega }}_m$, additional atomic levels, heating, and finite strip width remain experimental corrections.

A wider internal-state manifold separates the two synthetic edges and supplies more bulk-like orbitals. It does not automatically reproduce a two-dimensional thermodynamic limit: the range of $m$, boundary couplings, and interaction scaling must also be specified.

2. Contact locality becomes all-to-all synthetic coupling

Expand the $SU(W)$-symmetric interaction:

Two atoms in different internal states $m$ and $m^{\prime }$ interact with strength $U$ whenever they occupy the same real-space site $n$. That strength is independent of

In the synthetic strip metric, the density interaction kernel is therefore

up to species-dependent scattering lengths and the same-state term. Its range along $m$ grows with $W$.

A short-range two-dimensional interaction would instead obey a decay condition such as

as the two-dimensional separation becomes large. Power-law interactions require their own decay bounds; an all-to-all column interaction lies outside the usual short-range case.

The mismatch can be stated without metaphor:

Structure	Distance it uses	Result in the internal-state construction
One-body hopping	Graph distance on $(n,m)$	Nearest-neighbor Hofstadter strip
Contact collision	Physical overlap of atomic wavefunctions	Local in $n$, generically long-range in $m$
Edge	Termination of the hopping graph	Physical absence of further coupled internal states
Many-body locality	Decay of interaction terms in a chosen metric	Not supplied by the hopping graph

The statistics of the atoms add another layer. Bosonic and fermionic particles can occupy the same hopping graph but have different on-site constraints and exchange signs. Single-particle graph equivalence does not determine those data either.

3. Why locality enters topological phase equivalence

For short-range lattice Hamiltonians, a uniformly gapped path

supports quasi-adiabatic continuation. Under the required locality and gap assumptions, one constructs a quasi-local unitary that transports the ground-state subspace along the path.

This is the basis for treating two gapped Hamiltonians as representatives of the same many-body phase. Lieb–Robinson bounds and related stability results control how rapidly the continuation spreads information.

An interpolation containing interactions of unbounded range in the synthetic metric does not satisfy the standard short-range hypothesis. A nonclosing gap along such a path can connect states that no finite-depth or quasi-local two-dimensional evolution could connect.

The conclusion is limited but important:

Nor does one quantized response invariant repair the missing hypothesis. A many-body Chern number is defined through twisted boundary conditions and can remain quantized in settings where the interaction metric is nonlocal. It does not, by itself, certify robustness against every local perturbation or establish the full pattern of long-range entanglement.

Gap plus locality

“The gap did not close” has phase-classification force only after the allowed Hamiltonian path and its locality class have been fixed.

A 2026 preprint by Geraghty, Nardin, Mazza, and Rizzi studies an extended Harper–Hofstadter model with an infinite-range interaction within each synthetic column. In their finite-size calculations, increasing that interaction connects a bosonic Laughlin-type fractional Chern regime to a momentum-space Tao–Thouless-like charge-density wave without closing the many-body gap.

The computed many-body Chern number and fitted topological entanglement entropy remain approximately unchanged, while the particle entanglement spectrum restructures and robustness to a local perturbation is lost. The authors interpret the large-interaction regime as lacking genuine locality-protected topological order.

This is evidence from a recent numerical preprint, not a general theorem. Its scope depends on the model, finite-size scaling, the definition of locality, and the diagnostics used. It nevertheless supplies a concrete counterexample to the inference that a preserved gap and Chern number alone establish equivalence as local two-dimensional phases.

4. Gauge algebra and interaction locality are independent

Synthetic links can also be matrix-valued:

For a closed path $C$, the ordered product is

Under a local non-Abelian gauge transformation based at $i_0$,

Thus the Wilson-loop matrix is gauge covariant. Its trace, determinant, and eigenvalues are gauge invariant. Experiments have realized tunable synthetic $SU(2)$ fields and measured their non-Abelian effect on chiral dynamics.

Changing scalar links into matrix-valued links enriches the one-body parallel transport. It does not alter the range of a density interaction unless the interaction engineering changes as well.

The two design questions are independent:

1. Which connection governs hopping and interference?
2. Which metric controls the decay of many-body couplings?

Schemes based on state-dependent spatial separation, Rydberg dressing, cavity mediation, or fast interaction Trotterization can reshape the second answer. Such engineering is an additional construction, not a consequence of synthetic gauge flux.

False claim to diagnose

If an interacting synthetic-dimension system remains gapped and its many-body Chern number stays quantized during a deformation, it must remain in the same two-dimensional topological phase.

The inference omits the class of allowed deformations. Standard adiabatic phase equivalence restricts the path to Hamiltonians that remain sufficiently local in the physical metric used to define the phase.

A gap and a Chern number remain valuable diagnostics. They do not show that a nonlocal path can be replaced by a quasi-local one.

What follows — and what does not

Statement	Status
$W_{◻}={\mathrm{e}}^{-\mathrm{i}\gamma }$ for the chosen loop orientation.	Exact within the tight-binding model
A local rephasing can change the plaquette flux.	False
Open internal-state boundaries give edges of the hopping graph.	Exact graph statement
Every edge mode of the ideal graph survives arbitrary experimental imperfections.	False
$SU(W)$ contact collisions are local in the physical lattice direction.	True after the stated single-band projection
The same interaction is short-range in the synthetic coordinate.	False for the all-to-all column term
Increasing $W$ makes the one-body strip wider.	True
Increasing $W$ makes fixed-strength column interactions local.	False
A uniformly gapped local path supports quasi-local continuation.	True under the standard locality and gap assumptions
A gapped nonlocal path proves equivalence as local 2D phases.	False
A many-body Chern number alone diagnoses all forms of topological order.	False
The 2026 preprint proves a universal theorem for every synthetic platform.	False; it presents model-specific numerical evidence
Non-Abelian link matrices automatically repair interaction locality.	False

Exercise

Close the synthetic direction into a ring with $W$ states. Let the density interaction at each physical site have kernel

where $d$ is understood modulo $W$.

1. Recompute the plaquette Wilson loop and show explicitly that local $U(1)$ endpoint phases cancel.

2. Evaluate

   $$\tilde{V}(k)=\sum _{d=0}^{W-1}V(d){\mathrm{e}}^{-\mathrm{i}kd},k=\frac{2\pi l}{W}.$$Copy LaTeX

3. Compare the result with a nearest-neighbor synthetic interaction

   $$V_{\mathrm{n}\mathrm{n}}(d)=U({\delta }_{d,1}+{\delta }_{d,W-1}).$$Copy LaTeX

4. Decide whether increasing $W$ at fixed $U$ produces a local interacting two-dimensional limit.

5. Explain what changes if one Kac-normalizes the all-to-all coupling as $U/W$. Does that make it short-range?

Hint 1

For the allowed momenta,

$$\sum _{d=0}^{W-1}{\mathrm{e}}^{-\mathrm{i}kd}=W{\delta }_{k,0}.$$Copy LaTeX

Hint 2

A short-range kernel has a smooth momentum dependence that reflects a finite set of real-space separations. Extensivity and locality are different requirements.

Oral check 1. Can the phase on a single link be gauge invariant?

Oral check 2. Which experimental measurement could confirm the Hofstadter hopping graph without testing interaction locality?

Solution

For the Abelian links, a gauge transformation gives

$$U_{ij}\mapsto {\mathrm{e}}^{\mathrm{i}{\chi }_i}{U}_{ij}{\mathrm{e}}^{-\mathrm{i}{\chi }_j}.$$Copy LaTeX

Multiplication around the plaquette cancels every phase with the inverse phase from the adjacent link. The result remains

$$W_{◻}={\mathrm{e}}^{-\mathrm{i}\gamma }$$Copy LaTeX

for the chosen orientation.

For the all-to-all synthetic kernel,

$$\begin{array}{rl}\tilde{V}(k)& =U\sum _{d=1}^{W-1}{\mathrm{e}}^{-\mathrm{i}kd}\\ & =U(W{\delta }_{k,0}-1).\end{array}$$Copy LaTeX

Hence

$$\boxed{{\displaystyle \tilde{V}(0)=U(W-1),\tilde{V}(k\ne 0)=-U}}.$$Copy LaTeX

The nearest-neighbor kernel instead gives

$${\tilde{V}}_{\mathrm{n}\mathrm{n}}(k)=U({\mathrm{e}}^{-\mathrm{i}k}+{\mathrm{e}}^{+\mathrm{i}k})=2U\cos k.$$Copy LaTeX

Its real-space support stays at $d=1$ as $W$ grows. The all-to-all kernel continues to couple sites separated by $O(W)$ with amplitude $U$.

At fixed $U$, the zero-momentum eigenvalue grows as $W$. This warns that a thermodynamic limit must also specify how particle number and coupling scale. Replacing $U$ by $U/W$ makes the collective interaction energy extensive in common large-$W$ limits:

$${\tilde{V}}_{\mathrm{K}\mathrm{a}\mathrm{c}}(0)=U(1-\frac{1}{W}),{\tilde{V}}_{\mathrm{K}\mathrm{a}\mathrm{c}}(k\ne 0)=-\frac{U}{W}.$$Copy LaTeX

Kac normalization weakens every pair but leaves the support all-to-all. It restores a form of extensivity; it does not create finite-range locality.

Check your understanding

Suppose spectroscopy, a wave-packet drift, and edge-resolved imaging all agree with a Hofstadter strip. The interaction is known only through a single on-site loss measurement.

Which part of the synthetic-dimensional claim has been tested? What interaction-resolved measurement or calculation would you require before assigning a local two-dimensional many-body phase?

You may also reply with “deeper,” “too easy,” “too hard,” or your derivation.

Further Reading

• Synthetic gauge fields in synthetic dimensions — the internal-state Hofstadter construction, including its chiral edges and infinite-range synthetic interaction.
• Observation of chiral edge states with neutral fermions in synthetic Hall ribbons — experimental edge dynamics in an internal-state synthetic dimension.
• Fate of a Fractional Chern Insulator under Nonlocal Interactions in Synthetic Dimensions — the 2026 numerical preprint motivating the locality-sensitive topological discussion.
• Bose-Hubbard physics in synthetic dimensions from interaction Trotterization — a proposal to engineer effective on-site interactions along an internal-state dimension.
• Quasi-adiabatic continuation of quantum states — the role of a gapped local path in many-body phase continuation.
• Topological quantum order: stability under local perturbations — locality assumptions in stability results for topological order.
• Chiral Dynamics of Ultracold Atoms under a Tunable SU(2) Synthetic Gauge Field — experimental synthetic non-Abelian link structure and chiral dynamics.

Connections and next step

• Gauge layer: link phases and Wilson loops determine the one-body lattice connection.
• Geometric layer: the hopping graph supplies an effective coordinate and its boundaries.
• Interaction layer: physical wavefunction overlap determines which synthetic sites collide.
• Locality layer: quasi-local continuation requires decay in the metric used to define the phase.
• Diagnostic layer: a Chern number can survive after robustness and entanglement-spectrum structure change along a nonlocal path.
• Engineering layer: interaction locality must be designed and verified separately from gauge flux.
• Next step: compare the particle entanglement spectrum and response to local pinning fields across the nonlocal Laughlin-to-density-wave interpolation.
• Revisit: contrast this missing locality hypothesis with the locality assumptions behind the Floquet-loop classification in the August 3 entry.