Absence of Floating Phase in Superconductors with Time-reversal Symmetry Breaking on any Lattice

Abstract

Due to the interplay of multi-component order parameters (e.g., a twisted bilayer superconductor with inter-layer Josephson coupling or a frustrated ($n\ge 3$)-band superconductor), a superconductor can possess a $U(1)\times \mathbb{Z}_2$ symmetry, corresponding to the superconducting $T_c$ and time-reversal symmetry breaking transition $T_\text{TRSB}$, respectively. It was then conjectured that in this class of Hamiltonians, there exists a vast parameter regime $\mathcal{O}$ such that the system exhibits vestigial TRSB, i.e., $T_\text{TRSB} > T_c$, while at the boundary $\partial \mathcal{O}$, the system possesses a single phase transition $T_\text{TRSB}=T_c$. In this paper, we provide evidence towards this conjecture by mathematically eliminating the possibility of a floating phase, i.e., $T_\text{TRSB} < T_c$, for the strong coupling regime. More specifically, we prove that the correlation functions of $U(1)$ spins are bounded above by that of $\mathbb{Z}_2$ spins for all temperatures and lattice structures (e.g., $\mathbb{Z}^d$ for all $d$). In particular, this guarantees the existence of high-$T_c$ TRSB (and consequently topological) superconductivity in a large class of Hamiltonians. Note that the same property can also be proven for a certain parameter regime ($Δ\ge 4/5$) of the generalized XY model on any lattice structure, despite belonging to an entirely distinct class of $U(1)\times \mathbb{Z}_2$ Hamiltonians.

Figures (7)

• Figure 1: Schematic sketch. Using the example of a 2-component system can2021highmaccari2022effects with phases $\phi^\pm$, the parameter space involving $(\nu,J_2)$ describes the coupling strength $\nu$ of the current-current interaction $-\nu \nabla \phi^+ \cdot \nabla \phi^-$ and that of an inter-component $J_2 \cos 2(\phi^+ -\phi^-)$ term. See, e.g., Eq. (1) of Ref. maccari2022effects. The regime $\sO$ is where vestigial order $T_\text{TRSB} >T_c$ is expected to occur, while the boundary $\partial \sO$ denotes points in which there is only a single phase transition $T_\text{TRSB} =T_c$. In particular, it includes the critical regime $\ell \subseteq \partial \sO$, i.e., $\nu=0$, which has been studied extensively via numerics (in 2D song2022phase and in 3D bojesen2014phasemaccari2022effects). The regimes $\ell_\infty, \partial \sO_\infty, \sO_\infty$ correspond to the strong coupling limit, i.e., $J_2 \to \infty$. We emphasize that the shape of $\partial \sO$ should not be taken too seriously, but from numerics (in 2D liu2023charge and 3D maccari2022effects), it's expected that for finite $J_2>0$, there exists vestigial order for large $\nu$ and a single phase transition at small $\nu$. Similarly, the size of $\partial \sO_\infty$ should not be taken seriously, but from numerics song2022phasebojesen2014phase, it's expected that $\ell_\infty \subseteq \partial \sO_\infty$. Also write $\bar{\sO}\equiv \sO \cup \partial \sO$ and similarly $\bar{\sO}_\infty$.
• Figure 2: Phase diagram of 2D generalized XY model (Eq. \ref{['eq:general-XY']}) song2021hybridcarpenter1989phasenui2018correlationhubscher2013stiffness. We write $T_\text{ferro},T_\text{nem}$ instead of $T_\text{TRSB},T_c$ since physically, the $\dZ_2$ transition is into a ferromagnetic state instead of TRSB, while the $U(1)$ transition is into a nematic state carpenter1989phase.
• Figure 3: Cluster Representation on $\dZ^2$. (a). Some Ising spin configuration chosen from the Ising Gibbs state. (b). A constructed subgraph $\omega$ based on the Ising spin configuration in Fig. \ref{['fig:cluster-spin']}. Note that not all ordered edges are included in $\omega$, since an edge $e$ is included with some probability. However, all disordered edges are not included (based on the edge probability defined in Eq. \ref{['eq:cluster-Is-edge']}). (c). The subgraph after integrating over all possible spin configurations.
• Figure 4: Absence of Floating Phase. The $x,y$ axes denote the $\alpha, \lambda$ coupling described in Eq. \ref{['eq:U1-Z2-general']} (where we have omitted the edge subscript). The red dot denotes the critical Hamiltonian $\lambda=\alpha=0$ defined in Eq. \ref{['eq:U1-Z2']} while the diagonal lines describes $\lambda = \alpha$. Within the green region, there exists no floating phase on any lattice structure. Within the blue region, if one compares Eq. \ref{['eq:U1-Z2-general']} with the generalized XY model in Eq. \ref{['eq:general-XY']}, the SC transition $T_c$ may further split into a nematic $T_\text{nem}$ and $T_\text{ferro}$ transition and thus the order of transitions among $T_\text{TRSB},T_\text{nem},T_\text{ferro}$ is unclear.
• Figure 5: Current representation of the Ising model on $\dZ^2$. (a) A current configuration $n:E\to \dN$ contributing to the partition function $Z^\text{Is}$. (b) A current configuration contributing to the unnormalized correlation function $Z^\text{Is}\langle \tau_0 \tau_R\rangle^\text{Is}$ where the large dots denote the lattice sites $0,R$. (c). A double current representation, where the red and green lines representation distinct configurations $n_1,n_2$ contributing to the unnormalized correlation function $(Z^\text{Is}\langle \tau_0 \tau_R\rangle^\text{Is})^2$. The large dots denote the lattice sites $0,R$. (d). A double current representation, where only the trace $\hat{n}$ (compared to Fig. \ref{['fig:current-Is-double']}) is shown.

Theorems & Definitions (41)

• Theorem 1
• proof : Sketch of Proof
• Theorem 2: see Appendix \ref{['app:cluster']}
• proof : Sketch of Proof
• Theorem 3: see Appendix \ref{['app:cluster-general']}
• proof : Sketch of Proof
• Lemma 4
• proof
• Theorem 5: Cluster-Flip Invariance
• proof