Superconducting Long-Range Order

• Superconducting long-range order is characterized by nonzero off-diagonal correlators that signal macroscopic phase coherence and underpin zero resistance and the Meissner effect.
• Local observables such as charge fluctuations, spin fluctuations, and entanglement entropy provide clear experimental signatures of superconducting transitions even when nonlocal measurements are challenging.
• Quantum simulators and lattice models validate that subtle changes in local diagonal quantities can robustly indicate the onset of global superconducting order in complex many-body systems.

Superconducting long-range order (LRO) refers to the macroscopic phase coherence characterizing the superconducting state, in which the off-diagonal elements of the reduced density matrix (i.e., anomalous correlators or pair-pair correlations) acquire nonzero expectation values over system-spanning distances. This phenomenon underpins the essential physical properties of superconductors, including zero resistance and the Meissner effect, and is closely related to quantum phase transitions in correlated electron systems. The microscopic and emergent aspects of superconducting LRO have been studied in a broad range of contexts, from lattice models with interactions and disorder to real materials and engineered quantum systems.

1. Emergence of Long-Range Order in Microscopic Models

Superconducting LRO is conventionally defined through the algebraic structure of the many-body ground state, specifically the presence of off-diagonal long-range order (ODLRO) in the two-particle reduced density matrix. In practice, a nonzero value

$$\lim_{|i-j|\to\infty} \langle c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger c_{j\downarrow} c_{j\uparrow} \rangle \neq 0$$

signals phase coherence across the system. While these nonlocal observables are not directly accessible in traditional experiments, recent research establishes that single-site diagonal quantities – specifically local charge and spin fluctuations, occupation probabilities, and entanglement entropy computed from the on-site reduced density matrix – show critical signatures precisely at the superconducting transition, even for ODLRO phases (Sanino et al., 14 Jul 2025).

For the one-dimensional extended Hubbard model at half-filling, the single-site reduced density matrix is

$$\rho_{(j)} = w_0|0\rangle\langle0| + w_\uparrow|\uparrow\rangle\langle\uparrow| + w_\downarrow|\downarrow\rangle\langle\downarrow| + w_2|2\rangle\langle2|$$

where $w_0,w_\uparrow,w_\downarrow,w_2$ denote the probabilities for empty, singly, and doubly occupied states. Even though ODLRO by definition involves correlators between distant sites, the superconducting transition induces sharp extrema, discontinuities, or inflection points in single-site diagonal quantities:

• The charge fluctuation $\langle\delta n\rangle = w_0 + w_2 - (w_0 - w_2)^2$
• The spin fluctuation $\langle\delta s^z\rangle = w_\uparrow/2$
• The single-site entanglement entropy $$S = -[w_0 \log_2 w_0 + w_\uparrow \log_2 w_\uparrow + w_\downarrow \log_2 w_\downarrow + w_2 \log_2 w_2]$$

The relation $$\langle\delta n\rangle \simeq 1 - 4\langle\delta s^z\rangle$$ holds in the regime of weak particle–hole symmetry breaking, aligning local fluctuations with the onset of nonlocal superconducting correlations. The phase transition from a spin-density wave (SDW) to a spin-triplet superconducting (TS) phase is consistently marked by a minimum in entanglement entropy and a pronounced feature in the charge and spin fluctuations. Notably, particle–hole symmetry breaking, which precedes the emergence of ODLRO, minimally perturbs these single-site observables, providing robust experimental markers (Sanino et al., 14 Jul 2025).

2. Symmetry Breaking, Phase Transitions, and Criticality

Superconducting transitions are distinguished by a continuous breaking of particle–hole symmetry that preconditions the onset of LRO, but this symmetry breaking only weakly affects local diagonal observables. In the studied model, as the TS phase develops, the difference $(w_0 - w_2)$ – corresponding to the asymmetry between vacant and doubly occupied sites – acts as an order parameter for particle–hole symmetry, transitioning smoothly across the critical point. However, its impact on local charge fluctuation and entanglement is quadratic and negligible over a broad parameter regime.

This separation of scales manifests in a direct connection: diagonal single-site observables (traditionally assumed to be sensitive only to local or diagonal order) faithfully track transitions to ODLRO phases when symmetry breaking is sufficiently weak. The critical behavior in these observables – extrema or points of inflection in the entanglement entropy, for instance – is a fingerprint of the system’s transition to superconducting LRO. This applies to quantum phase transitions of second order, which are typified by changes in quantum correlations over all spatial scales.

3. Relation Between Local Observables and Nonlocal Order

The surprising result that single-site diagonal descriptors encode clear signatures of superconducting transitions, typically associated with nonlocal order, is rooted in how the many-body wave function reorganizes in the critical regime. While off-diagonal (e.g., pair-pair) correlators grow at the transition, local fluctuations necessarily respond because the variance in local occupancy (measured by $\langle\delta n\rangle$) and the balance between empty and doubly occupied sites reflect the change in underlying quantum correlations.

Entanglement entropy, in particular, bridges the local and nonlocal characteristics of the system: for the reduced density matrix above, $S$ approximates a monotonic function of the charge fluctuation, providing a locally accessible marker for global LRO. The minimum or cusp at the superconducting transition underscores the growing importance of quantum correlations, even when the measured observable is diagonal.

4. Implications for Experiment and Quantum Simulation

The discovery that local diagonal quantities such as on-site fluctuations and single-site entanglement detect superconducting ODLRO has significant experimental ramifications. State-of-the-art quantum simulators and platforms with single-site resolution (such as cold atom optical lattices or quantum gas microscopes) can directly measure occupations and associated fluctuations. This enables the indirect yet robust identification of superconducting transitions, bypassing the challenging measurement of nonlocal correlators.

Moreover, the invariance of entanglement entropy and fluctuation minima under weak particle–hole symmetry breaking ensures that these diagonal observables are reliable for locating the critical point, even in the presence of moderate disorder or finite-size effects.

5. Theoretical Foundations and Interpretations

The ability of single-site diagonal observables to capture ODLRO is tied to the hidden symmetry structure of the underlying Hamiltonian and the correlation-induced redistribution of local degrees of freedom at the phase transition. The continuous breaking of particle–hole symmetry serves as a "precursor" transition, but the main qualitative shift in diagonal quantities happens at the superconducting critical point.

This aligns with fundamental theoretical developments showing that entanglement entropy can act as a diagnostic for quantum phase transitions and that local fluctuations can, under general conditions, encode transitions normally associated with nonlocal order. The findings challenge the view that only off-diagonal long-range two-point functions or momentum-space condensate fractions are valid signatures of superconducting coherence.

6. Summary Table – Relationship of Local Observables to ODLRO

Single-site observable	Formula	Critical behavior at superconducting transition
Charge fluctuation	$\langle\delta n\rangle = w_0 + w_2 - (w_0 - w_2)^2$	Exhibits an extremum (minimum or inflection point)
Spin fluctuation	$\langle\delta s^z\rangle = w_\uparrow/2$	Shows step-like or inverse correlated behavior
Entanglement entropy	$S = -[w_0\log_2 w_0 + \ldots]$	Displays minimum or inflection at critical point

These relationships reflect the deep and often hidden linkage between local occupation statistics and the emergence of global superconducting coherence.

7. Broader Significance and Future Research

The observation that single-site diagonal descriptors are sensitive to ODLRO represents a departure from longstanding dogma. This facilitates new methodologies for detecting and characterizing superconducting order in theoretical and experimental systems where nonlocal observables are inaccessible. A plausible implication is that similar methodologies may be extended to other paradigmatic quantum phases exhibiting ODLRO or strong entanglement, including superfluids and topological states.

Future research may focus on the universality of these signatures across a range of lattice geometries, interaction types, and spatial dimensions, as well as their robustness to disorder and temperature effects. The framework provides a fertile ground for investigating fundamental aspects of quantum criticality and coherence in many-body systems.