From Interdependent Networks to Two-Interactions Physical Systems

TL;DR

This work shows that a single-layer superconducting network with two coexisting interactions—electrical connectivity and thermal dependency—can exhibit mixed-order phase transitions previously attributed to interdependent networks. Using both experiments on TISS devices and RSJJ-based simulations, the authors demonstrate abrupt, hysteretic N-S transitions accompanied by critical scaling with a universal exponent $β=1/2$, and reveal long-lived resistance plateaus with a slowdown exponent $ζ=1/2$ near criticality. The transition strength is tunable via substrate thermal conductivity, highlighting heat diffusion as the mechanism that generates effective dependency links within a single network. These findings broaden the interdependent networks framework to include two-interaction single-layer systems, offering practical routes to engineer and control phase transitions in superconducting devices and related systems.

Abstract

Recent advances have shown that introducing dependency interactions between two superconducting networks can trigger abrupt, hysteretic normal-superconductor phase transitions. In this study, we demonstrate that such behavior can also arise in a single-network superconducting system that features two distinct types of interactions: short-range electrical connectivity and long-range thermal dependency. Using experimental and simulation methods, we show that when sufficient heat is dissipated within a single-layer disordered superconducting network, the system undergoes a mixed-order phase transition marked by both a discontinuous change in resistance and critical scaling behavior. We find that the emergence and characteristics of these abrupt transitions depend critically on the thermal conductivity of the underlying substrate, establishing heat flow as the origin of the unique phase transition. Additionally, both experimental and numerical results reveal long-lived transient states and scaling dynamics near the critical point, consistent with spontaneous branching processes observed in interdependent networks theory. These findings strongly demonstrate that complex critical phenomena, such as mixed-order transitions, previously attributed to structurally interdependent systems, can also arise within single-layer physical systems when dual interactions coexist. Our results broaden the scope of the theory and experiments of phase transitions in interdependent networks and suggest new ways to design and control phase changes in physical, biological, and technological systems where two interactions are present.

Figures (6)

• Figure 1: TISS versus ISN(a) A sketch of an ISN system. On a thermally conducting and electrically insulating substrate, a network in the shape of a two-dimensional square lattice made from $\mathrm{InO}$ is fabricated. Then, a layer of $\mathrm{AlO}$, which is thermally conducting and electrically insulating, is placed on top of the network. Last, an identical $\mathrm{InO}$ network is fabricated on top of the $\mathrm{AlO}$. (b) A sketch of a TISS. A single $\mathrm{InO}$ network is fabricated on a thermally conducting and electrically insulating substrate. (c) A zoom-in side-view image of the superconducting network taken by a scanning electron microscope, showing its geometry. (d,e) Experimental resistance versus temperature measurements at different bias currents for ISN $(L=416)$ and TISS $(L=416)$. Full symbols are for cooling, and empty symbols are for heating. For the ISN sample, electric current is driven to both networks but only the RT curves of one of the networks are presented. (f) Corresponding temperature-dependent numerical simulations of a TISS $(L=100)$. Insets show corresponding extracted values of the $\beta$ exponent (Eq. \ref{['eq:beta']}). (g,h) Experimental resistance versus time measurements of the the cascading plateau at various bias currents for ISN $(L=416)$ and TISS $(L=416)$. (i) Corresponding time-dependent numerical simulations of a TISS $(L=100)$, fixed heat-bath temperature $T=2K$, as $I_c = 121.98\mu A$. Insets show corresponding extracted values of the $\zeta$ exponent (Eq. \ref{['eq:zeta']}).
• Figure 2: Current density by scanning SQUID measurement in Nb network at low and high current regimes(a) Resistance versus temperature curves for $I_b=0.1$mA (blue curve) and $I_b=1$mA (red curve). Symbols mark selected temperature values for which we plot the spatial distribution of the current flow, and the letters refer to the panels. (b)-(g) Reconstructed current density maps of a small region of the network at $I_b=0.1$mA, showing a complex structure of currents when transitioning continuously from the N state ($T=8.7$K) to the S state ($T=6.1$K). (h)-(m) Corresponding simple structural maps at $I_b=1$mA. The current distribution differs between the superconducting state (panels g, m) and the normal state (panels b, h). In the superconducting state, shielding currents dominate the behavior, concentrating the flow near the network's edge. In the normal state the current distributes evenly between the networks' segments. Numerical results of the current density distributions are shown in the Supplementary Information Figs. S1 and S2.
• Figure 3: Substrate dependence-(a) Experimental resistance versus temperature measurements for a single network ($L=200$), at different bias currents, fabricated on silicon (red symbols) and glass (blue symbols). Full symbols are for cooling cycles and empty symbols are for heating cycles. (b,c) The extraction of the critical exponent $\beta$ from the cooling curves of glass and silicon at $I_b=100\mu A$ and $I_b=170 \mu A$ respectively. As seen, both yield the same $\beta$. (d,e) Critical temperatures of the transitions from N to S (Cooling) and S to N (heating) plotted versus $I_b$ . (f) The hysteresis width is calculated for each bias current measurement. (g)-(l) Numerical simulations are in excellent agreement with the experimental results ($L=100$). The extraction of the critical exponent $\beta$ from the cooling curves of glass and silicon at $I_b=70\mu A$ and $I_b=130 \mu A$ respectively.
• Figure S1: Network current-flow map across a continuous N-S phase transition. Numerical simulations for cooling the system under a fixed current of $I = 50\,\mu\mathrm{A}$. The network size is $L = 11$. Heat spreads uniformly according to the mean-field approximation, where the temporal diffusion parameter satisfies $D t = L^2 = 121$. We used $\gamma = 1.452 \times 10^{6}\,\mathrm{W^{-1}\,K}$. (a) Network resistance as a function of temperature during cooling from $T = 3\,\mathrm{K}$ down to $T = 1\,\mathrm{K}$, showing a continuous second-order N-S transition. (b) Representative maps of the network’s current-flow-intensity configuration at five characteristic temperatures, indicated by the square markers in panel (a). The imposed current flows from the left boundary to the right, and the total resistance is measured between the corresponding injection and extraction nodes. During the phase transition (B-D), the currents spontaneously percolate through high-density current paths.
• Figure S2: Network current-flow map across an abrupt N-S phase transition. Numerical simulations for cooling the system under a fixed current of $I = 300\,\mu\mathrm{A}$. The network size is $L = 11$. Heat spreads uniformly according to the mean-field approximation, where the temporal diffusion parameter satisfies $D t = L^2 = 121$. We used $\gamma = 1.452 \times 10^{6}\,\mathrm{W^{-1}\,K}$. (a) Network resistance as a function of temperature during cooling from $T = 3\,\mathrm{K}$ down to $T = 0.1\,\mathrm{K}$, showing an abrupt N-S phase transition. (b) Representative maps of the network’s current-flow-intensity configuration at four characteristic temperatures, indicated by the square markers in panel (a). The imposed current flows from the left boundary to the right, and the total resistance is measured between the corresponding injection and extraction nodes. In contrast to the continuous second-order transition (Fig. S1B-D), no percolation structure is observed in the high-density current paths in the vicinity of the transition point (B, C).