Fluctuation response of a superconductor with temporally correlated noise

We discuss how a finite noise correlation time, which can arise through coupling to engineered nonthermal environments, affects the fluctuation-driven response in a superconductor above its critical temperature. Using the phenomenological time-dependent Ginzburg–Landau model, we formulate the stochastic dynamics within the path-integral framework. Our analysis reveals that the transport response can be enhanced when the noise correlation time becomes comparable to the intrinsic relaxation time of the superconductor. The magnitude and character of this resonant-like effect depend strongly on the system’s dimensionality.

💡 Research Summary

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The paper investigates how a finite temporal correlation of external noise influences fluctuation‑driven transport in a superconductor above its critical temperature (T > T_c). Starting from the phenomenological time‑dependent Ginzburg–Landau (TDGL) equation, the authors introduce a stochastic Langevin term η(r,t) whose two‑point correlation is taken to be exponentially decaying in time, D(t‑t′)=D₀ exp(−|t‑t′|/τ)/2τ, with amplitude D₀=2T_cγ and correlation time τ. In the limit τ→0 the noise reduces to the usual white thermal noise, while τ≫1 corresponds to strongly colored (non‑Markovian) noise. The TDGL dynamics are reformulated as a Martin‑Siggia‑Rose (MSR) action and then expressed in the Keldysh basis, separating classical (ψ_cl) and quantum (ψ_q) components.

Within a Gaussian approximation of the Ginzburg–Landau free energy (valid for ε = T/T_c−1≫Gi), the retarded and advanced propagators are G_R(k,ω)=(iγ ω+ε_k)⁻¹ and G_A=G_R*, where ε_k=a₀T_c(ε+ξ²k²). The colored noise contributes a frequency‑dependent spectral density D(ω)=D₀/(1+ω²τ²). Consequently the order‑parameter correlation function becomes C(k,ω)=G_R D(ω) G_A. The key physical parameter is the ratio of the mode‑dependent relaxation time τ_k=γ/ε_k to the noise correlation time τ. Low‑k modes (τ_k≫τ) are insensitive to the noise colour and behave as in the white‑noise case; high‑k modes (τ_k≲τ) feel the finite correlation and are progressively suppressed. This selective damping leads to a non‑monotonic dependence of observable response functions on τ.

The authors compute the Aslamazov‑Larkin type contributions to three transport coefficients: electrical conductivity σ, thermal conductivity κ, and thermoelectric coefficient α. All three are expressed as momentum integrals over products of the propagators and appropriate vertices (charge vertex 2e v(k) for σ, heat vertex ω v(k) for κ, and a mixed vertex for α). The resulting formulas contain rational functions of the dimensionless ratio τ/τ_k and, for σ, also the external frequency Ω.

Key findings:

1. Electrical conductivity – At zero frequency, σ(τ) exhibits a modest peak when τ≈τ_GL≡π/(8T_cε). In one dimension the peak is pronounced (≈1 % increase) at τ*≈0.1 τ_GL; in higher dimensions the peak is broadened and less visible because many modes contribute with a spread of τ_k. For τ≪τ_GL the system behaves as if driven by white noise; for τ≫τ_GL the conductivity decays as 1/τ, reflecting the suppression of Cooper‑pair fluctuations. In the optical regime (Ωτ_GL≫1) σ(Ω,τ) is further reduced, showing no dynamical interplay between Ω and τ.

2. Thermal conductivity – κ(τ) follows a similar trend but displays a sign change in three dimensions: as τ exceeds τ_GL, κ first drops below its white‑noise value, reaches a negative peak, and then decays as −τ⁻³⁄². This reflects the stronger suppression of low‑k modes (which dominate κ) while high‑k contributions become relatively more important. In 1D and 2D the decay is positive (∝τ⁻¹/² and τ⁻² respectively).

3. Thermoelectric coefficient – α(τ) requires a small imaginary part of the relaxation constant (γ″≪γ′) to break particle‑hole symmetry. α also shows a peak at τ≈τ* (similar to σ) and decays as τ⁻² for large τ. The peak is more pronounced in higher dimensions because the heat vertex weights higher‑k modes, which are less suppressed at intermediate τ.

The paper also discusses the shift of the critical temperature δT_c caused by fluctuations. Because colored noise cuts off high‑energy modes, the usual logarithmic enhancement of δT_c is reduced; the authors give a heuristic estimate showing δT_c ∝ −Gi ln