Superconductor–Ising Model: Theory & Applications

• Superconductor–Ising Model is a framework that maps superconducting phases to Ising spin variables via vortex states, Majorana modes, and frustration effects.
• The model is realized in various setups including frustrated Josephson junction arrays and quantum circuits with transverse-field Ising Hamiltonians.
• It provides insights into unconventional pairing, strong spin–orbit coupling effects, and emergent topological orders in superconducting materials.

The Superconductor–Ising Model refers to the formal and physical correspondences between superconducting systems—especially those with unconventional order, frustration, or spin–orbit coupling—and statistical Ising models, as well as the realization of Ising-type Hamiltonians in superconducting materials and circuits. The mapping is central to both quantum many-body theory, where Ising variables encode macroscopic degrees of freedom (e.g., vortex charge, Majorana modes, spin polarization), and to the engineering of quantum hardware (classical and quantum Ising machines, Josephson junction networks). The analogies extend from classical antiferromagnetic/superconducting models on doped lattices, through frustrated Josephson networks, to strong spin-orbit-coupled Ising superconductors in two dimensions.

1. Ising Models Emergent from Josephson Junction Arrays

Josephson networks with geometrical frustration realize effective Ising models by imposing topological constraints on the macroscopic phase configuration. In a vertex-sharing frustrated Kagome lattice, 0- and π-Josephson junctions generate a degeneracy in the triangular plaquette potential,

$$U(\phi_1,\phi_2) = E_J[2+\alpha - \cos\phi_1 - \cos\phi_2 - \alpha\cos(\phi_1+\phi_2)]$$

with the flux quantization constraint $\phi_1 + \phi_2 + \phi_3 = 0$ and frustration parameter $\alpha < \alpha_c = -\frac{1}{2}$. The two minima in the triangular potential correspond to vortex and antivortex states, which are mapped to Ising spin variables $\sigma = \pm 1$ (Neyenhuys et al., 2023).

An effective classical Hamiltonian for the spin variables is derived: $$H_{\mathrm{classical}} = -\sum_{i<j} J_{ij}\sigma_i\sigma_j$$ where $J_{ij}$ is highly anisotropic and long-ranged, decaying exponentially along some lattice directions and algebraically along others due to topological constraints. Quantum fluctuations generate a transverse-field term ($\propto \sum_i \sigma_i^x$), yielding a quantum Ising model with tunable couplings and entanglement properties in the ground state.

2. Superconductor–Ising Mapping: Theoretical Framework

The microscopic origins of superconductor–Ising mappings vary with context:

• Antiferromagnetic Superconductivity Models: Sewell's model (Sewell, 2015) of electrons on a cubic lattice with antiferromagnetic Ising couplings and infinite-U Hubbard repulsion is

$$H = H_{\mathrm{kin}} + H_{\mathrm{Ising}} + H_{\mathrm{rep}}$$

where $$H_{\mathrm{Ising}} = J \sum_{\langle x,y \rangle} S^z_x S^z_y$$ and $H_{\mathrm{kin}}$ enables tunneling with strict no-double-occupancy. At half-filling, the system realizes a Mott insulator with long-range antiferromagnetic order (Ising symmetry-breaking).

• Doping and Schafroth Pairing: Introducing holes, one obtains an effective Hamiltonian for $n$ holes with attractive nearest-neighbour potential $-J$ and short-range repulsion, supporting Schafroth (bosonic) pairing and therefore ODLRO and superconductivity, coexisting with background magnetic order given sufficient repulsion to prevent large clusters.

3. Quantum Circuit and Hardware Realization

Quantum electronic circuits based on superconducting islands and engineered Josephson junctions can implement transverse-field Ising Hamiltonians (Roy, 2023). Networks of Josephson parametric oscillators (JPOs) (Razmkhah et al., 2023) realize hardware spins ($\sigma_i = \pm 1$) via phase bistability, with programmable two-body couplings and LHZ-style constraints to solve optimization problems. Each JPO-based tile operates as an effective Ising node, tunable in coupling strength and robust to noise.

4. Topological Phases: Kitaev Chain and Ising Equivalence

A one-dimensional p-wave superconductor (Kitaev chain) at $\Delta = t$, $\mu = 0$ is formally equivalent to an Ising chain via Jordan–Wigner mapping (Greiter et al., 2014, Chhajed, 2020): $$H_{\mathrm{Kitaev}} = -t \sum_{j=1}^{N-1}(c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j + c_j c_{j+1} + c_{j+1}^\dagger c_j^\dagger)$$ maps to

$$H_{\mathrm{Ising}} = -t \sum_{j=1}^{N-1}\sigma^x_j \sigma^x_{j+1}$$

Spectral and eigenstate equivalence does not imply physical equivalence: Ising symmetry-breaking yields local magnetization, whereas the Kitaev chain possesses nonlocal string order and Majorana zero modes, manifesting topological rather than conventional order.

5. Ising Superconductivity: Spin-Orbit Coupling and Magnetic Robustness

Ising superconductors—exemplified by monolayer NbSe$_2$, MoS$_2$, and bulk noncentrosymmetric TMD polytypes (Wickramaratne et al., 2023, Roy et al., 2024, Volavka et al., 15 Jan 2025)—feature strong out-of-plane spin-orbit coupling that "locks" Cooper pair spins perpendicular to the plane. The effective Hamiltonian reads

$$H = \sum_{\mathbf{k}} \xi_{\mathbf{k}}\, \sigma_0 + \frac{1}{2}\Delta_{\mathrm{SOC}} g(\mathbf{k})\, \sigma_z - \mu_B [B_x \sigma_x + B_y \sigma_y]$$

plus $s$-wave pairing. This spin-pinning yields in-plane critical fields $H_{c2}^{\parallel}$ much larger than the Pauli limit ($H_{P}$) by suppressing paramagnetic pair-breaking via Ising SOC: $$H_{c2}^{\parallel} \sim \frac{\Delta_{\mathrm{SOC}}}{\Delta} H_P$$ Presence of strong Ising SOC and control by crystal symmetry produce unconventional pairing channels, spin-singlet/triplet mixing, and even robust diode effects when Rashba SOC is present (Bankier et al., 19 Mar 2025).

6. Superconductor–Ising Transition and Criticality

Transverse-field Ising models describe superconductor-insulator transitions on percolation-cluster networks (Bianconi, 2013), and can be realized in off-critical chains coupled to superconducting leads (Giuliano et al., 2016), which map onto two-channel Kondo problems. The critical temperature depends on the spectral structure (maximal eigenvalue) of the underlying connectivity, peaking at the percolation threshold, with quantum fluctuations and disorder further modifying phase diagrams.

7. Ising Model Analogies and Non-Conventional Superconductivity

Site-diluted 2D antiferromagnetic Ising models capture, via Monte Carlo simulation (Ni, 2011), a phase diagram analogous to unconventional superconductors: a dome-shaped $T_c(x)$ as a function of doping, pseudogap regions, and emergent quasi-crystalline order at low temperature in large systems. The mapping is qualitative, with order parameters and energy gaps extracted phenomenologically but affording insight into collective pattern formation.

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Realization	Model Hamiltonian	Distinguishing Feature
Josephson Kagome	$H_{\rm Ising}$ with	Frustration; tunable anisotropic
array (Neyenhuys et al., 2023)	long-range $J_{ij}$	long-range coupling
Antiferromagnetic	$H_{\rm kin} + H_{\rm Ising}$	AF Mott insulator, Schafroth bosonic pairing
superconductor (Sewell, 2015)		Coexistence of AF and SC
Quantum circuit (Roy, 2023, Razmkhah et al., 2023)	Transverse-field Ising	Physical qubits and hardware encoded spin models
Kitaev chain (Greiter et al., 2014, Chhajed, 2020)	$H_{\rm Kitaev} \to H_{\rm Ising}$	Topological order, Majorana zero modes
Monolayer NbSe$\mathbf_2$ (Wickramaratne et al., 2023, Roy et al., 2024)	SOC-pinned Ising SC	Robustness to in-plane field, parity-mixed pairing

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• "Long-range Ising spins models emerging from frustrated Josephson junctions arrays with topological constraints" (Neyenhuys et al., 2023)
• "Model of Antiferromagnetic Superconductivity" (Sewell, 2015)
• "Quantum Electronic Circuits for Multicritical Ising Models" (Roy, 2023)
• "A Josephson Parametric Oscillator-Based Ising Machine" (Razmkhah et al., 2023)
• "The 1D Ising model and topological order in the Kitaev chain" (Greiter et al., 2014)
• "From Ising model to Kitaev Chain -- An introduction to topological phase transitions" (Chhajed, 2020)
• "Ising superconductivity: a first-principles perspective" (Wickramaratne et al., 2023)
• "Unconventional pairing in Ising superconductors: Application to monolayer NbSe$_2$" (Roy et al., 2024)
• "Ising superconductivity in noncentrosymmetric bulk NbSe$_2$" (Volavka et al., 15 Jan 2025)
• "Superconductor-insulator transition in a network of 2d percolation clusters" (Bianconi, 2013)
• "Junction of three off-critical quantum Ising chains and two-channel Kondo effect in a superconductor" (Giuliano et al., 2016)
• "Non-conventional Superconductors and diluted Ising Model" (Ni, 2011)

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In totality, the Superconductor–Ising Model constitutes a diverse set of physical realizations—both mathematical and experimental—where effective spin degrees of freedom, classical or quantum, encode the macroscopic states of superconducting systems and are governed by variants of the Ising Hamiltonian. The interplay of frustration, strong correlations, topological constraints, spin–orbit coupling, and circuit design produces a rich landscape connecting statistical physics, quantum materials, and scalable hardware architectures.