{ "schema_version": 1, "id": "PHYS-2026-08-05-01", "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-05-01/", "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-05-01.md", "metadata": { "schema_version": 1, "id": "PHYS-2026-08-05-01", "date": "2026-08-05", "updated_at": "2026-08-05", "title": "Synthetic gauge geometry does not guarantee many-body locality", "summary": "Why an internal-state ladder can realize the Hofstadter hopping graph exactly while contact interactions remain infinite-range in the synthetic metric, and what this changes in topological phase arguments.", "language": "en", "entry_kind": "daily", "status": "published", "level": "graduate-advanced", "user_difficulty": "unrated", "domains": [ "condensed-matter", "quantum-information", "mathematical-physics" ], "estimated_minutes": 60 }, "content_markdown": "\n## Key claim\n\n**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.**\n\nAtoms 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.\n\nThe 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.\n\n## Theme\n\n**Gauge-equivalent hopping geometry and interaction locality in synthetic dimensions.**\n\n## Guiding question\n\nWhen does the map\n\n$$\n\\ket{n,m}\\longleftrightarrow\\ket{x=n,y=m}\n$$\n\njustify treating an internal state $m$ as a spatial coordinate? Which many-body conclusions require more than this map?\n\n## Setup\n\nTake a one-dimensional optical lattice with sites $n$ and spacing $a$. Each atom has $W=2F+1$ internal states\n\n$$\nm=-F,-F+1,\\ldots,F.\n$$\n\nOrdinary 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\n\n$$\nH_0=\\sum_{n,m}\n\\left[\n-t\\,a_{n+1,m}^{\\dagger}a_{n,m}\n+\\Omega_m\\ee^{-\\ii\\gamma n}\na_{n,m+1}^{\\dagger}a_{n,m}\n+\\mathrm{H.c.}\n\\right],\n$$\n\nwith\n\n$$\n\\gamma=2k_Ra\n$$\n\nfor the Raman momentum transfer used in the simplest construction. The absence of couplings beyond $m=\\pm F$ gives open boundaries in the synthetic direction.\n\nFor approximately $SU(W)$-symmetric bosonic contact interactions, projection into the lowest Wannier orbital gives\n\n$$\nH_{\\rm int}\n=\\frac{U}{2}\\sum_nN_n(N_n-1),\n\\qquad\nN_n=\\sum_m n_{n,m}.\n$$\n\nThe Hamiltonian is local in the physical coordinate $n$. Its range in the coordinate $m$ will be the central issue.\n\n> [!margin: Platform matters]\n> 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.\n\n## Analysis\n\n### 1. The plaquette flux is an exact graph statement\n\nIgnore the common real hopping amplitude and assign oriented link variables\n\n$$\nU_x(n,m)=1,\n\\qquad\nU_m(n,m)=\\ee^{-\\ii\\gamma n}.\n$$\n\nTraverse the plaquette\n\n$$\n(n,m)\\to(n+1,m)\\to(n+1,m+1)\n\\to(n,m+1)\\to(n,m).\n$$\n\nThe ordered product is\n\n$$\n\\begin{aligned}\nW_\\square\n&=U_x(n,m)U_m(n+1,m)\nU_x^\\dagger(n,m+1)U_m^\\dagger(n,m)\\\\\n&=\\ee^{-\\ii\\gamma(n+1)}\\ee^{+\\ii\\gamma n}\\\\\n&=\\boxed{\\ee^{-\\ii\\gamma}}.\n\\end{aligned}\n$$\n\nThe sign changes if the loop orientation is reversed. The signed flux in the convention above is\n\n$$\n\\frac{\\Phi_\\square}{\\Phi_0}=-\\frac{\\gamma}{2\\pi}\n\\quad(\\mathrm{mod}\\ 1).\n$$\n\nUnder a local basis change\n\n$$\na_{n,m}\\mapsto\\ee^{\\ii\\chi_{n,m}}a_{n,m},\n$$\n\neach link gains endpoint phases. They cancel around a closed loop, leaving $W_\\square$ invariant.\n\nThis 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.\n\nCeli 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.\n\n> [!margin: What “exact” means]\n> The graph equivalence holds for the modeled one-body Hamiltonian. Raman detunings, unequal $\\Omega_m$, additional atomic levels, heating, and finite strip width remain experimental corrections.\n\nA 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.\n\n### 2. Contact locality becomes all-to-all synthetic coupling\n\nExpand the $SU(W)$-symmetric interaction:\n\n$$\n\\begin{aligned}\nH_{\\rm int}\n=\\frac{U}{2}\\sum_n\n\\bigg[\n&\\sum_m n_{n,m}(n_{n,m}-1)\\\\\n&+2\\sum_{m [!margin: Gap plus locality]\n> “The gap did not close” has phase-classification force only after the allowed Hamiltonian path and its locality class have been fixed.\n\nA 2026 preprint by Geraghty, Nardin, Mazza, and Rizzi studies an extended Harper–Hofstadter model with an infinite-range interaction within each synthetic column. In their finite-size calculations, increasing that interaction connects a bosonic Laughlin-type fractional Chern regime to a momentum-space Tao–Thouless-like charge-density wave without closing the many-body gap.\n\nThe computed many-body Chern number and fitted topological entanglement entropy remain approximately unchanged, while the particle entanglement spectrum restructures and robustness to a local perturbation is lost. The authors interpret the large-interaction regime as lacking genuine locality-protected topological order.\n\nThis is evidence from a recent numerical preprint, not a general theorem. Its scope depends on the model, finite-size scaling, the definition of locality, and the diagnostics used. It nevertheless supplies a concrete counterexample to the inference that a preserved gap and Chern number alone establish equivalence as local two-dimensional phases.\n\n### 4. Gauge algebra and interaction locality are independent\n\nSynthetic links can also be matrix-valued:\n\n$$\nU_{ij}\\in SU(2).\n$$\n\nFor a closed path $C$, the ordered product is\n\n$$\nW_C=\\mathcal P\\prod_{(ij)\\in C}U_{ij}.\n$$\n\nUnder a local non-Abelian gauge transformation based at $i_0$,\n\n$$\nW_C\\mapsto G_{i_0}W_CG_{i_0}^{-1}.\n$$\n\nThus 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.\n\nChanging 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.\n\nThe two design questions are independent:\n\n1. Which connection governs hopping and interference?\n2. Which metric controls the decay of many-body couplings?\n\nSchemes 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.\n\n## False claim to diagnose\n\n> 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.\n\nThe 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.\n\nA gap and a Chern number remain valuable diagnostics. They do not show that a nonlocal path can be replaced by a quasi-local one.\n\n## What follows — and what does not\n\n| Statement | Status |\n| --- | --- |\n| $W_\\square=\\ee^{-\\ii\\gamma}$ for the chosen loop orientation. | Exact within the tight-binding model |\n| A local rephasing can change the plaquette flux. | False |\n| Open internal-state boundaries give edges of the hopping graph. | Exact graph statement |\n| Every edge mode of the ideal graph survives arbitrary experimental imperfections. | False |\n| $SU(W)$ contact collisions are local in the physical lattice direction. | True after the stated single-band projection |\n| The same interaction is short-range in the synthetic coordinate. | False for the all-to-all column term |\n| Increasing $W$ makes the one-body strip wider. | True |\n| Increasing $W$ makes fixed-strength column interactions local. | False |\n| A uniformly gapped local path supports quasi-local continuation. | True under the standard locality and gap assumptions |\n| A gapped nonlocal path proves equivalence as local 2D phases. | False |\n| A many-body Chern number alone diagnoses all forms of topological order. | False |\n| The 2026 preprint proves a universal theorem for every synthetic platform. | False; it presents model-specific numerical evidence |\n| Non-Abelian link matrices automatically repair interaction locality. | False |\n\n## Exercise\n\nClose the synthetic direction into a ring with $W$ states. Let the density interaction at each physical site have kernel\n\n$$\nV(d)=\n\\begin{cases}\n0,&d=0,\\\\\nU,&d=1,\\ldots,W-1,\n\\end{cases}\n$$\n\nwhere $d$ is understood modulo $W$.\n\n1. Recompute the plaquette Wilson loop and show explicitly that local $U(1)$ endpoint phases cancel.\n2. Evaluate\n\n $$\n \\widetilde V(k)\n =\\sum_{d=0}^{W-1}V(d)\\ee^{-\\ii kd},\n \\qquad\n k=\\frac{2\\pi l}{W}.\n $$\n\n3. Compare the result with a nearest-neighbor synthetic interaction\n\n $$\n V_{\\rm nn}(d)=U(\\delta_{d,1}+\\delta_{d,W-1}).\n $$\n\n4. Decide whether increasing $W$ at fixed $U$ produces a local interacting two-dimensional limit.\n5. Explain what changes if one Kac-normalizes the all-to-all coupling as $U/W$. Does that make it short-range?\n\n
\nHint 1\n\nFor the allowed momenta,\n\n$$\n\\sum_{d=0}^{W-1}\\ee^{-\\ii kd}=W\\delta_{k,0}.\n$$\n\n
\n\n
\nHint 2\n\nA short-range kernel has a smooth momentum dependence that reflects a finite set of real-space separations. Extensivity and locality are different requirements.\n\n
\n\n**Oral check 1.** Can the phase on a single link be gauge invariant?\n\n**Oral check 2.** Which experimental measurement could confirm the Hofstadter hopping graph without testing interaction locality?\n\n
\nSolution\n\nFor the Abelian links, a gauge transformation gives\n\n$$\nU_{ij}\\mapsto\n\\ee^{\\ii\\chi_i}U_{ij}\\ee^{-\\ii\\chi_j}.\n$$\n\nMultiplication around the plaquette cancels every phase with the inverse phase from the adjacent link. The result remains\n\n$$\nW_\\square=\\ee^{-\\ii\\gamma}\n$$\n\nfor the chosen orientation.\n\nFor the all-to-all synthetic kernel,\n\n$$\n\\begin{aligned}\n\\widetilde V(k)\n&=U\\sum_{d=1}^{W-1}\\ee^{-\\ii kd}\\\\\n&=U\\left(W\\delta_{k,0}-1\\right).\n\\end{aligned}\n$$\n\nHence\n\n$$\n\\boxed{\n\\widetilde V(0)=U(W-1),\n\\qquad\n\\widetilde V(k\\neq0)=-U\n}.\n$$\n\nThe nearest-neighbor kernel instead gives\n\n$$\n\\widetilde V_{\\rm nn}(k)\n=U(\\ee^{-\\ii k}+\\ee^{+\\ii k})\n=2U\\cos k.\n$$\n\nIts real-space support stays at $d=1$ as $W$ grows. The all-to-all kernel continues to couple sites separated by $O(W)$ with amplitude $U$.\n\nAt fixed $U$, the zero-momentum eigenvalue grows as $W$. This warns that a thermodynamic limit must also specify how particle number and coupling scale. Replacing $U$ by $U/W$ makes the collective interaction energy extensive in common large-$W$ limits:\n\n$$\n\\widetilde V_{\\rm Kac}(0)=U\\left(1-\\frac1W\\right),\n\\qquad\n\\widetilde V_{\\rm Kac}(k\\neq0)=-\\frac UW.\n$$\n\nKac normalization weakens every pair but leaves the support all-to-all. It restores a form of extensivity; it does not create finite-range locality.\n\n
\n\n## Check your understanding\n\nSuppose 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.\n\nWhich 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?\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your derivation.\n\n## Further Reading\n\n- [Synthetic gauge fields in synthetic dimensions](https://doi.org/10.1103/PhysRevLett.112.043001) — the internal-state Hofstadter construction, including its chiral edges and infinite-range synthetic interaction.\n- [Observation of chiral edge states with neutral fermions in synthetic Hall ribbons](https://doi.org/10.1126/science.aaa8736) — experimental edge dynamics in an internal-state synthetic dimension.\n- [Fate of a Fractional Chern Insulator under Nonlocal Interactions in Synthetic Dimensions](https://arxiv.org/abs/2603.16724) — the 2026 numerical preprint motivating the locality-sensitive topological discussion.\n- [Bose-Hubbard physics in synthetic dimensions from interaction Trotterization](https://arxiv.org/abs/1907.10555) — a proposal to engineer effective on-site interactions along an internal-state dimension.\n- [Quasi-adiabatic continuation of quantum states](https://arxiv.org/abs/cond-mat/0503554) — the role of a gapped local path in many-body phase continuation.\n- [Topological quantum order: stability under local perturbations](https://arxiv.org/abs/1001.0344) — locality assumptions in stability results for topological order.\n- [Chiral Dynamics of Ultracold Atoms under a Tunable SU(2) Synthetic Gauge Field](https://arxiv.org/abs/2401.03612) — experimental synthetic non-Abelian link structure and chiral dynamics.\n\n## Connections and next step\n\n- Gauge layer: link phases and Wilson loops determine the one-body lattice connection.\n- Geometric layer: the hopping graph supplies an effective coordinate and its boundaries.\n- Interaction layer: physical wavefunction overlap determines which synthetic sites collide.\n- Locality layer: quasi-local continuation requires decay in the metric used to define the phase.\n- Diagnostic layer: a Chern number can survive after robustness and entanglement-spectrum structure change along a nonlocal path.\n- Engineering layer: interaction locality must be designed and verified separately from gauge flux.\n- Next step: compare the particle entanglement spectrum and response to local pinning fields across the nonlocal Laughlin-to-density-wave interpolation.\n- Revisit: contrast this missing locality hypothesis with the locality assumptions behind the Floquet-loop classification in the August 3 entry.\n", "content_sha256": "625a2d9165202dcb10346813edce38b787b6e44cf2031ec64d5c6c0e5aef1f78", "formula_count": 75, "formulas": [ { "index": 1, "display": true, "tex": "\\ket{n,m}\\longleftrightarrow\\ket{x=n,y=m}", "line": 36, "sha256": "ebe0a97b39eeb3403839c46de0e3fc2c270138a701e60c52e5065b91321a3c33" }, { "index": 2, "display": false, "tex": "m", "line": 40, "sha256": "62c66a7a5dd70c3146618063c344e531e6d4b59e379808443ce962b3abd63c5a" }, { "index": 3, 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