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arxiv: 2605.28942 · v2 · pith:UL3ZKCNZnew · submitted 2026-05-27 · ❄️ cond-mat.supr-con · hep-th

Graph-based emulation of d-dimensional curved spaces with superconducting arrays

Mehmet Dede , Guilherme Delfino , Andr\'e L.G. Mudry , Junseok Oh , Andrew P. Higginbotham , Christopher Mudry , Claudio Chamon This is my paper

Pith reviewed 2026-06-29 09:11 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con hep-th
keywords superconducting arraysJosephson junctionsholographic dualitycurved spacesgraph emulationAdS spacesdisorder effects
0
0 comments X

The pith

Superconducting wire arrays emulate d-dimensional curved spaces by mapping graphs to Josephson-coupled phases that realize scalar field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a framework where two stacked layers of superconducting wires are coupled by Josephson junctions according to the edges of a chosen graph. Each vertex receives a wire with a rigid phase, so the phases function as a scalar field living on the geometry and topology encoded by that graph. The authors apply the method to constant-negative-curvature hyperbolic spaces and show that the resulting boundary scaling exponents remain consistent with holographic expectations even when the junction strengths are strongly disordered. A dictionary is supplied that converts circuit measurements into field-theory observables, making the platform experimentally realistic.

Core claim

Two-layer superconducting wire arrays with Josephson junctions at graph edges discretize arbitrary d-dimensional geometries; the phases on the wires then furnish scalar field theories on those geometries, and holographic duality in Anti-de Sitter space stays intact under strong disorder in the couplings.

What carries the argument

The two-layer wire array with rigid-phase wires at vertices and Josephson junctions on graph edges, which directly realizes the scalar field on the emergent curved geometry.

If this is right

  • Circuit observables become direct proxies for field-theory quantities via the supplied dictionary.
  • Holographic duality in any dimension can be studied in a controlled laboratory setting.
  • Metric fluctuations induced by disordered couplings leave boundary scaling exponents unchanged.
  • The same architecture can host physics on arbitrary nontrivial graphs and simple dynamical spacetimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The platform could let experimenters scan curvature and dimension continuously by rewiring the junction network.
  • Adding tunable time-dependent couplings might allow laboratory analogs of evolving geometries.
  • Coupling the array to existing superconducting qubit readouts could give direct access to entanglement measures on the boundary.

Load-bearing premise

The superconducting phases stay rigid and the Josephson couplings dominate over parasitic effects, so the circuit really reproduces the target field theory on the graph geometry.

What would settle it

Measurement of boundary two-point functions that deviate from the expected holographic power-law scaling once Josephson coupling disorder exceeds a moderate threshold would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2605.28942 by Andr\'e L.G. Mudry, Andrew P. Higginbotham, Christopher Mudry, Claudio Chamon, Guilherme Delfino, Junseok Oh, Mehmet Dede.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Mapping between a degree of freedom [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Bipartite graph structure associated to a single [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Triangulation of a single three-dimensional tetra [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Mesh associated with a triangulation of a smooth [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Superconducting wire implementations of bipartite graphs corresponding to nontrivial topologies: a torus and a Klein [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) The square lattice Λ is defined by the set of white circles. The medial square lattice Λ [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram at nonvanishing temperature [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Bipartite graph associated to [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Extraction of the boundary scaling dimension from [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Scaling dimension ∆( [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Scaling dimension ∆( [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Heatmap of ∆( [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Wire construction of a cube with a single over [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Square array of superconducting wires coupled [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Triangular tiling [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. (a) The cube is a three-dimensional regular poly [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. (a) Hyperbolic tiling [PITH_FULL_IMAGE:figures/full_fig_p028_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Region plot for the function, [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Representations of the dihedral group D [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Representations of a Coxeter group [PITH_FULL_IMAGE:figures/full_fig_p030_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. (a) Coxeter diagram of [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. (a) Augmented Coxeter diagram of [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Iterative construction of the orbit of the generating vertex under the action of the Coxeter group [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Iterative construction of the orbit of the generating edge under the action of the Coxeter group [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. (a) Augmented Coxeter diagram of [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
read the original abstract

We introduce a framework for emulating graphs and, through them, curved spaces of arbitrary dimension, using arrays of superconducting wires. The array consists of two stacked layers of wires, horizontal and vertical, such that wires are parallel within each layer and perpendicular between layers. By discretizing a space into a graph, assigning a superconducting wire with a rigid phase to each vertex, and coupling pairs of wires through Josephson junctions according to the graph edges, arbitrary geometries and topologies can be engineered in a controlled setting. The superconducting phases then realize scalar field theories on the emergent geometry. We establish experimentally realistic conditions for implementing these architectures and develop a dictionary relating measurable circuit observables to quantities in the emulated field theory. As an application, we develop the implementation of hyperbolic (Anti-de Sitter) spaces of constant negative curvature and use them as an experimentally accessible platform to explore holographic duality in arbitrary dimensions. We investigate the effects of disorder in the Josephson couplings, which translate into metric variations in the bulk-boundary correspondence, and analyze their impact on boundary scaling exponents both analytically and numerically, finding that holographic duality remains robust even in the presence of strong disorder. Beyond holography, the framework opens a broad range of architectural possibilities, including the exploration of physics on highly nontrivial graphs and toy models of dynamical spacetimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a framework for emulating graphs and d-dimensional curved spaces (including AdS) via stacked layers of superconducting wires coupled by Josephson junctions according to graph edges, with wire phases realizing scalar field theories on the emergent geometry. It claims to derive experimentally realistic conditions, a dictionary mapping circuit observables to field-theory quantities, and to show analytically and numerically that holographic duality remains robust under strong disorder in the Josephson couplings (which induce metric variations).

Significance. If the emulation mapping and dictionary are rigorously established, the work would provide a controllable experimental platform for scalar fields on nontrivial geometries and for testing holographic duality in arbitrary dimensions, including under disorder. The graph-based discretization approach and explicit treatment of disorder effects are strengths that could enable new toy models of dynamical spacetimes.

major comments (3)
  1. [Framework description] Framework section (abstract and main text description): the central claim that assigning rigid-phase wires to vertices and Josephson junctions to edges realizes the desired discrete scalar field theory on the emergent geometry rests on an unshown reduction of the circuit Hamiltonian; no explicit derivation or error analysis is supplied showing that parasitic effects (self-inductance, shunt capacitance, quasiparticle tunneling, phase slips) remain subdominant to the tunable Josephson energies for the disorder strengths considered.
  2. [Holographic application] Holographic duality application (abstract): the statement that duality remains robust to strong disorder is load-bearing for the main result, yet the manuscript supplies neither the analytical expressions for boundary scaling exponents nor the numerical data/figures demonstrating robustness; this absence prevents assessment of the quantitative impact of metric variations.
  3. [Dictionary relating observables] Dictionary construction: the mapping from measurable circuit observables (e.g., currents, voltages) to field-theory quantities is asserted but not constructed explicitly with equations relating Josephson energies, phases, and correlation functions; without this, the claim of experimental accessibility cannot be evaluated.
minor comments (2)
  1. [Framework] Notation for the graph Laplacian and weighted edges should be defined consistently when relating the Josephson cosine potential to the discrete action.
  2. [Abstract] The abstract mentions 'arbitrary dimension' but the concrete implementation focuses on hyperbolic spaces; a brief statement on the dimensional limitation would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our framework. We address each major comment below and indicate the revisions we will implement.

read point-by-point responses

  1. Referee: [Framework description] Framework section (abstract and main text description): the central claim that assigning rigid-phase wires to vertices and Josephson junctions to edges realizes the desired discrete scalar field theory on the emergent geometry rests on an unshown reduction of the circuit Hamiltonian; no explicit derivation or error analysis is supplied showing that parasitic effects (self-inductance, shunt capacitance, quasiparticle tunneling, phase slips) remain subdominant to the tunable Josephson energies for the disorder strengths considered.

    Authors: We agree that an explicit derivation of the effective scalar-field Hamiltonian from the full circuit model, together with a quantitative error analysis for parasitic effects, would strengthen the manuscript. In the revised version we will add a dedicated subsection (or appendix) containing the step-by-step reduction of the circuit Hamiltonian and order-of-magnitude estimates demonstrating that self-inductance, shunt capacitance, quasiparticle tunneling and phase slips remain subdominant for the experimentally realistic Josephson energies and disorder strengths considered. revision: yes

  2. Referee: [Holographic application] Holographic duality application (abstract): the statement that duality remains robust to strong disorder is load-bearing for the main result, yet the manuscript supplies neither the analytical expressions for boundary scaling exponents nor the numerical data/figures demonstrating robustness; this absence prevents assessment of the quantitative impact of metric variations.

    Authors: We acknowledge that the explicit analytical expressions for the boundary scaling exponents and the supporting numerical data were not presented with sufficient clarity or prominence. In the revision we will insert the closed-form expressions for the scaling exponents, add a dedicated paragraph summarizing the analytical robustness argument, and include (or augment) numerical figures that directly compare scaling exponents across a range of disorder strengths, thereby allowing quantitative evaluation of the effect of metric variations. revision: yes

  3. Referee: [Dictionary relating observables] Dictionary construction: the mapping from measurable circuit observables (e.g., currents, voltages) to field-theory quantities is asserted but not constructed explicitly with equations relating Josephson energies, phases, and correlation functions; without this, the claim of experimental accessibility cannot be evaluated.

    Authors: We agree that the dictionary mapping must be made fully explicit. We will revise the manuscript to include a new subsection that supplies the concrete equations relating measurable quantities (Josephson energies, phase differences, currents and voltages) to the correlation functions and other observables of the emulated scalar field theory, thereby making the experimental accessibility claim directly verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; framework rests on independent JJ physics and graph discretization.

full rationale

The paper introduces an emulation framework by assigning rigid-phase wires to graph vertices and Josephson junctions to edges to realize scalar fields on emergent geometries, then applies it to AdS holography with disorder analysis. No quoted steps reduce a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain by construction. The dictionary to observables and robustness findings are presented as consequences of the circuit-to-field mapping under stated conditions, without tautological equivalence to inputs. This is the common case of a self-contained proposal grounded in standard circuit QED.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from superconductivity and discretization; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Superconducting phases remain rigid and Josephson couplings realize the desired graph edges to emulate scalar fields on the geometry.
    Invoked in the core emulation framework description.

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discussion (0)

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Forward citations

Cited by 2 Pith papers

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    Spin-wave theory on hyperbolic lattices shows Goldstone modes have vanishing local weight and a bulk gap, allowing finite-temperature magnetic order that evades the Mermin-Wagner theorem.

  2. Experimental observation of hyperbolic spacetime dynamics

    physics.optics 2026-06 unverdicted novelty 6.0

    First experimental emulation of fermionic wave packet dynamics in Lorentzian AdS2 spacetime using photonic waveguide arrays, showing mass-independent geodesic oscillations and curvature-dependent Zitterbewegung.

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