Microscopic theory of electron quadrupling condensates

TL;DR

The paper develops a microscopic framework for four-fermion condensates, focusing on time-reversal symmetry-breaking electron quadrupling beyond BCS. It introduces a Hubbard–Stratonovich decoupling with bosonic pair fields and a generalized mean-field/sl skeleton-diagram expansion, enabling calculations of fermionic spectral functions and thermodynamics. Applying this to a three-band, density-density interaction system motivated by Ba$_{1-x}$K$_x$Fe$_2$As$_2$, it shows TRS-breaking quadrupling can emerge above the superconducting transition when intercomponent pair correlations split, and it analyzes both $2d$ and $3d$ cases to map phase diagrams. The work further provides quantitative predictions for the specific heat and density of states, revealing subtle but measurable signatures and offering a path toward diagrammatic Monte Carlo and spectral-function studies of composite orders in multi-band superconductors.

Abstract

Electron pairing at low temperatures leads to superconductivity. A fundamental question is whether more complex states - characterized by order in four-electron composite objects, termed electron quadrupling or composite order - can exist in materials, and if so, under what conditions they emerge and what properties they exhibit. These states lie beyond the scope of Bardeen-Cooper-Schrieffer theory, and a microscopic description of them remained elusive. In the first part of the paper, we provide a general microscopic framework to describe these and the other four-fermion composite states. In the second part of the paper, we derive and solve a specific fermionic model in two and three dimensions that hosts time-reversal symmetry-breaking electron quadrupling order. The fermionic microscopic theory is used to estimate the specific heat and electron density of states.

Figures (7)

• Figure 1: Cartoon of different states in the studied system. Orange (blue) colored disks denote $s + \mathrm{i} s$ ($s - \mathrm{i} s$) Cooper pairs, with arrow schematically indicating corresponding phase $[0, 2 \pi]$. Going from right (high temperature) to left (low temperature), we depict the following states: ($T^* \approx T_{BCS} < T$) normal metal with very few preformed Cooper pairs with uncorrelated phases and an equal number of $s \pm \mathrm{i} s$ pairs (here $T_{BCS}$ should be understood as a characteristic temperature scale, not a temperature of a phase transition); ($T_{BTRS} < T < T^*$) resistive metal but with stronger local ordering effects and an increased, but still equal number of preformed $s \pm \mathrm{i} s$ pairs; ($T_{SC} < T < T_{BTRS}$) BTRS electron quadrupling state where symmetry between the number of $s \pm \mathrm{i} s$ pairs is spontaneously broken -- meaning that there is a different number of $s + \mathrm{i} s$ and $s - \mathrm{i} s$ pairs, while superconducting phases are still uncorrelated on a long-range scale. This corresponds to onset of non-zero order parameter quartic in electronic fields $\langle f_{\uparrow \alpha} f_{\downarrow \alpha} f_{\downarrow \beta}^\dagger f_{\uparrow \beta}^\dagger \rangle$. ($T < T_{SC}$) BTRS superconductor -- phases become ordered, symmetry between $s \pm \mathrm{i} s$ pairs remains spontaneously broken.
• Figure 2: The plot of the rescaled number of electron pairs $L_+,\ L_-$ in the two bosonic components at local minima of the energy function Eq. (\ref{['Omega_L_-d']}) as a function of temperature $T$ in $2d$ system for $f = h = s = 0.2$ and different values of the density-density interaction $w$. Note that as the temperature is decreased at $T^* \simeq T_{BCS}$$L$'s increase greatly, which signals the pre-formation of Cooper pairs. A single minimum corresponds to a normal state while splitting into two minima to a BTRS quadrupling state. For temperatures smaller than shown on the plot, minima disappear and $L$ goes to $+ \infty$, which corresponds to the superconducting state. When the density-density interaction $w$ is small enough quadrupling state disappears. So for $w = 0.2$ we have the following phases: superconducting for $t < t_{SC} \simeq 0.783$, quadrupling for $t_{SC} < t < t_{BTRS} \simeq 0.828$ and normal for $t > t_{BTRS}$. Cooper pairs are preformed for $t < t^* \simeq 1$.
• Figure 3: The plot of critical temperatures for transitions into BTRS quadrupling and superconducting states as a function of density-density interaction strength $w$ for $f = h = s = 0.2$. Note that for sufficiently small $w$ quadrupling phase disappears in this model.
• Figure 4: The plot of the rescaled number of electron pairs $L_+,\ L_-$ in the two bosonic components at global minima of the energy function Eq. (\ref{['Omega_L_-d']}) as a function of temperature $T$ in a $3d$ system for $f = h = s = 0.5$ and $w = 0.4$. A single minimum corresponds to a normal state while splitting into two minima to a BTRS quadrupling state. Note that when $L_\pm = 1$ we get $\epsilon_\pm = 0$ and hence the system transitions to a superconducting state for lower temperatures. So in this case we have the following phases: superconducting for $t < t_{SC} \simeq 0.945742354$, quadrupling for $t_{SC} < t < t_{BTRS} \simeq 0.945742735$, and normal for $t > t_{BTRS}$. Cooper pairs are preformed ($L_\pm$ are increased) for $t < t^* \simeq 1$.
• Figure 5: The plot of the rescaled specific heat due to electron pair propagators as a function of temperature for (top)$2d$ model with $f = h = s = 0.2$ and $w = 0.2$, (bottom)$3d$ model with $f = h = s = 0.5$ and $w = 0.4$. Full-scale plots and zoomed-in part at $T_{BTRS}$ are presented. As the temperature is lowered, the specific heat is increased due to preformed Cooper pairs at $T^*$. At the onset of the quadrupling state $T_{BTRS}$, there is a jump. At least in the weak-coupling models that are considered, the feature at the quadrupling transition is quite small relative to the dominant contribution arising from preformed electron pairs.