Quantum fluctuation-induced first-order breaking of time-reversal symmetry in unconventional superconductors

TL;DR

This work shows that quantum order-parameter phase fluctuations can qualitatively alter a time-reversal-symmetry-breaking transition in a two-dimensional superconductor with competing non-degenerate pairing channels. Using a self-consistent treatment in the square-lattice $t$-$J$ model with Coulomb interactions, the authors derive a phase-fluctuation corrected free energy and find that the $s+id$ region can split into a dome that terminates at a first-order boundary with the $d$ phase, narrowing the domain of $\mathcal{T}$-broken superconductivity. The resulting first-order transition is supported by a two-minima free-energy landscape and a significant suppression of phase stiffness near the dome, with implications for disorder-induced transitions and high-temperature topological superconductivity in twisted cuprate systems. The formalism provides a framework for exploring similar fluctuation-driven effects in other multi-band or disordered superconductors and suggests experimental avenues in Josephson-junction networks and related materials.

Abstract

Spontaneous time-reversal symmetry breaking in superconductors with competing non-degenerate pairing channels is an exotic quantum phase transition that could give rise to robust topological superconductivity and unusual magnetism. It is proposed mostly in two-dimensional systems and is signaled by a nonzero relative phase between the two superconducting order parameters, hence it should particularly be prone to order-parameter phase fluctuations. Nevertheless, the existing understanding of it is still at the mean-field level. Here, we illustrate the non-negligible effects of the phase fluctuations on such quantum phase transitions using the hole-doped square-lattice $t$-$J$ model as an example. We derive the phase fluctuation-corrected free energy and show that under the quantum phase fluctuations, the time-reversal asymmetric $s+id$ phase region splits off a dome featuring a first-order border with the $d$ phase, indicating the possibility of a phase separation into the time-reversal symmetric and asymmetric phases. The phase fluctuations also narrow the range of the $s+id$ phase considerably. We further discuss the implications of our findings for recent experiments on disorder-induced first-order quantum breakdown of superconductivity and promising high-temperature topological superconductivity in twisted cuprate Josephson junctions.

Figures (4)

• Figure 1: The phase diagrams without (left panel) and with (right panel) the phase fluctuations in the chemical potential--temperature plane. The range of chemical potential $\mu\in (-1.88t, -1.833t)$ corresponds to a range of hole doping $p\in (0.626, 0.640)$ for $T\rightarrow 0$ ($T=10^{-5}t$). The lines are smooth fit to the data points, with the filled circles and thick lines (open circles and thin lines) indicating the first-order (second-order) phase transition. The gray dashed rectangles specify the zoom-in region for Figs. \ref{['fig:gap']} and \ref{['fig:stiff']}. The black star in the right panel marks the state of Fig. \ref{['fig:F']}.
• Figure 2: The superconducting gaps without (left column, denoted with subscript $0$) and with (right column) the phase fluctuations as functions of the chemical potential and temperature around the $s+id$ dome.
• Figure 3: The free energy landscape as functions of the order parameters at $T=0.0005t$ and $\mu=-1.8465t$. This state is marked by the black star in Fig. \ref{['fig:PD']}, right panel.
• Figure 4: Phase stiffness without [left panel, denoted with superscript $(0)$] and with (right panel) the phase fluctuations as functions of the chemical potential and temperature around the $s+id$ dome.