arXiv subjects

Yu-Bo Liu

Publications and source records attributed to Yu-Bo Liu.

At least 19 recordsLinked to original sources

arXiv2026

Orbital-Selective Diagonal-Gap Test of Pairing in La$_3$Ni$_2$O$_7$

Recent angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) measurements on bilayer nickelate La$_3$Ni$_2$O$_7$ reveal a nearly isotropic, nodeless superconducting gap. We show that these single-particle spectra provide a symmetry-enforced test of the pairing nature. Mirror symmetry forces the hybridization between $d_{z^2}$ and $d_{x^2-y^2}$ orbitals to vanish along the Brillouin-zone (BZ) diagonal. Consequently, the gaps on the diagonal portions of the $\alpha/\beta$ and $\gamma$ Fermi pockets separately probe the intrinsic pairing strengths of the $d_{x^2-y^2}$ and $d_{z^2}$ sectors. A pairing state with pure or dominant intrinsic $d_{z^2}$-orbital pairing component produces nodes or near-nodes on the BZ diagonal of the $\alpha/\beta$ pockets and is thus challenged by ARPES and STM, whereas the pairing state with dominant intrinsic $d_{x^2-y^2}$-orbital pairing component has a nearly isotropic, nodeless gap on all pockets which well fits ARPES and STM data. The dominant $d_{x^2-y^2}$-orbital pairing is compatible with Hund's-rule-driven pairing mechanisms.

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arXiv2026

Charge-4e/6e superconductivity and chiral metal from 3D chiral superconductor

Unconventional superconductivity (SC) characterized by multi-fermion orderings has attracted substantial attention. However, previous studies have largely focused on 2D systems or 3D systems with effective 2D symmetries. Here, we investigate the vestigial phases arising from thermal fluctuations of chiral SC in 3D systems governed by the cubic $O_h$ point group. By constructing low-energy effective Hamiltonians via Ginzburg-Landau analysis and conducting Monte Carlo simulations, we systematically investigate the phase fluctuations of chiral orders within the $E_g$ and $T_{2g}/T_{1u}$ irreducible representations (IRRPs). We identify a phase diagram topology different from 2D counterparts, where the multi-phase intersection manifests as a tetracritical point rather than the triple point typically found in 2D systems. We elucidate the evolution of these phases under thermal fluctuations. Our findings reveal that for both $E_g$ and $T_{2g}/T_{1u}$ IRRPs, the primary chiral orders could melt into a chiral metallic phase across specific parameter regimes. Moreover, for the $E_g$ IRRP, phase fluctuation could also induce a charge-$4e$ phase under certain regime, while for the $T_{2g}$ and $T_{1u}$ IRRPs, it leads to a higher-order charge-$6e$ SC state. Our work paves the way for exploring exotic vestigial orders driven by non-trivial 3D crystalline symmetries.

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arXiv2026

Filling and Interlayer Superexchange Control Superconductivity in La$_3$Ni$_2$O$_7$

A central puzzle in bilayer nickelate superconductors is why pressure, epitaxial strain, oxygen stoichiometry, and chemical substitution produce systematic but apparently different changes in the superconducting T$_\text{c}$. Here we show that these trends can be organized by a two-parameter control principle based on the $d_{x^2-y^2}$-orbital filling $n_x$ and the effective interlayer antiferromagnetic superexchange $J_\perp$. Starting from a physical picture in which the nearly half-filled $d_{z^2}$ orbital supplies localized spin correlations while the nearly quarter-filled $d_{x^2-y^2}$ orbital carries superconductivity, we study an effective bilayer $t-J_\parallel-J_\perp$ model with parameters constrained by first-principles calculations. Combined slave-boson mean-field and density-matrix renormalization group calculations place realistic La$_3$Ni$_2$O$_7$ in an overdoped-cuprate-like regime where T$_\text{c}$ is governed mainly by the pairing scale. In this regime, hole doping reduces $n_x$ and suppresses T$_\text{c}$, whereas tuning routes that enhance $J_\perp$ raise T$_\text{c}$. This framework accounts for the suppression by over-oxidation and Ca/Sr substitution, the half-dome oxygen-stoichiometry response in the film, the enhancement by Nd/Sm substitution in the pressurized bulk, the right-triangle-like pressure dependence in the bulk, and the enhancement by compressive strain in the film. It also separates clean carrier doping from oxygen-vacancy tuning: clean electron doping mainly increases $n_x$, whereas oxygen vacancies weaken the apical-oxygen-mediated exchange path and introduce disorder. This leads to a falsifiable materials-design prediction: clean electron doping should enhance the pairing scale if introduced without oxygen vacancies or strong pair-breaking disorder.

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arXiv2025

Critical Density-Wave Vestigial Phases of Commensurate Pair Density Wave

The pair-density-wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U(1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with $n$-times expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg-Landau theory, we get the low energy effective model Hamiltonian. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For $n\le 4$, besides the charge-4e/2e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for $n\ge 5$, the restore of the translational symmetry with increasing temperature is realized through two successive Berezinskii-Kosterlitz-Thouless transitions. Such a two-step process leads into two critical vestigial phases, i.e. the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading into a power-law decaying density-density correlation even at 2D. Our work appeals for experimental verifications.

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arXiv2025

N\'eel Ordered Magnetic Phases in Bipartite Quasicrystals

Magnetism is a fundamental research area in which the recently proposed altermagnetism (AM) has become an emergent frontier. Very recently, the quasicrystal (QC) was proposed as a possible platform to realize AM. However, the existence of AM in QCs still lacks vigorous evidence. In this work, we adopt the sign-problem-free projector quantum Monte Carlo (PQMC) algorithm to investigate the magnetic phases in the half-filled Hubbard models in various 2D bipartite QCs, and always obtain N\'eel ordered states. While the N\'eel states in bipartite crystals are usually antiferromagnetism (AFM), we find it common that those in bipartite QCs can also be AM or ferromagnetism (FM). Based on symmetry analysis, combined with our comprehensive PQMC results, we propose a general criterion for determining the magnetism classes of the N\'eel states in a bipartite QC: According to whether the two sublattices are related by the inversion, the other point-group operation, or no operation about the unique symmetry center in the QC, the corresponding N\'eel state is AFM, AM or FM, respectively. For example, our results yield AM for the two $D_4$-symmetric Thue-Morse QCs and FM for the $D_5$-symmetric Penrose QC at half-filling. Our results provide a solid foundation for experimental investigations and potential applications of different classes of magnetism in QCs.

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arXiv2025

Variation Monte Carlo Study on the bilayer $t-J_{\parallel}-J_{\perp}$ model for La$_3$Ni$_2$O$_7$

The discovery of high-temperature superconductivity (HTSC) in La$_3$Ni$_2$O$_7$ has aroused significant interest in exploring the pairing mechanism. Previous studies have proposed an effective d$_{x^2-y^2}$-orbital bilayer $t-J_{\parallel}-J_{\perp}$ model, in which the electrons of the d$_{x^2-y^2}$ orbital are charge carriers, which are subject to the intralayer antiferromagnetic (AFM) superexchange $J_{\parallel}$ and the large interlayer AFM superexchange $J_{\perp}\approx 2J_{\parallel}$, with the latter transferred from the nearly half-filled and hence localized $d_{z^2}$ orbital through the strong Hund's rule coupling. Here we study this model by the variational Monte Carlo (VMC) simulation and find a dominant interlayer s-wave pairing, in which the SC order parameters have a drastic improvement compared with those of the mean field (MF) type of theories. In real materials, the Hund's coupling is finite, leading to reduced $J_{\perp}$, dictating that the MF-type theories have difficulty explaining the HTSC. However, our VMC calculations find that even for effective $J_{\perp}$ as weak as $J_{\perp}=J_{\parallel}$, the interlayer pairing is still considerably large and can be compared with the $T_c$ observed in experiments, which is very weak in MF-type theories. This result indicates the important role of the Gutzwiller projection in improving the $T_c$, which is ignored in the MF-type theories. In addition, our results show that suppressed interlayer hopping can promote interlayer pairing, which is consistent with the fact that the interlayer hopping of the d$_{x^2-y^2}$ orbital in La$_3$Ni$_2$O$_7$ is very weak. Our research offers a new perspective for understanding the pairing mechanism of bilayer nickelates and provides a reference for recent ultra-cold atom experiments in mixed-dimensional systems.

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arXiv2025

Classification of Magnetism and Altermagnetism in Quasicrystals

Altermagnetism (AM), an unconventional magnetic phase characterized by zero net magnetism protected by symmetry(s) other than parity-time ($\mathcal{P}\mathcal{T}$) and a resulting spin-split band, has been studied exclusively in crystalline materials. Here, we extend the framework of AM to quasicrystals (QCs). We start from a comparison between the N\'{e}el state on the square lattice and that on a $D_4$-symmetric Thue-Morse QC, with both belonging to the same $d$-wave irreducible representation (IRRP) of the $D_4$ point group. Consequently, while the former is antiferromagnetism (AFM) protected by the combined $\mathcal{P}\mathcal{T}$ and translational symmetry, the lack of translational symmetry in the latter breaks the $\mathcal{P}\mathcal{T}$ symmetry, and the additional mirror or rotation symmetry protects AM. This example suggests that AM is more common in QCs than in crystals and can be easily explored through a point-group symmetry-based classification. Therefore, we classify magnetic phases in 2D $D_n$-symmetric QCs without spin-orbit coupling, by using IRRPs of $D_n$. Consequently, the identity IRRP represents ferromagnetism, the inversion-odd 1D IRRPs for twice-of-odd $n$ represent AFM, and all the remaining 1D IRRPs represent AM, protected by either mirror or rotation symmetry. We further take the Hubbard model to verify this result in various QCs with different symmetries. Our work highlights the QC as a natural platform where AM is common among magnetic phases.

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arXiv2025

Pairing without $\gamma$-Pocket in the La$_3$Ni$_2$O$_7$ Thin Film

The recent discovery of high-temperature superconductivity (HTSC) in the La$_3$Ni$_2$O$_7$ ultrathin film at ambient pressure has aroused great research interest. The $\gamma$-pocket formed by the bonding $d_{z^2}$ band, which was previously proposed to be crucial in the pairing mechanism of pressurized bulk La$_3$Ni$_2$O$_7$, is reported to be either present or absent here by different experimental groups, giving rise to the problem: what is the pairing mechanism and pairing nature without the $\gamma$-pocket? Here, we start from a band structure obtained via density-functional-theoretical calculation, which exhibits no $\gamma$-pocket. Then, equipped with electron interactions, we study the pairing nature via combined weak- and strong- coupling approaches, which provide consistent results. In the weak-coupling study, the nesting between the $\alpha$- and $\beta$- pockets leads to an $s^\pm$-wave pairing in which the gap signs on the two pockets are opposite, as provided by our random-phase-approximation based calculations. In real-space, the pairing pattern is dominated by the interlayer pairing of the $d_{x^2-y^2}$ orbital. In the strong-coupling study, as the $d_{z^2}$ orbitals are nearly half-filled and hence localized, the $d_{x^2-y^2}$ orbitals carry the HTSC. Driven by the interlayer superexchange transferred from the $d_{z^2}$ orbital through the Hund's rule coupling, the $d_{x^2-y^2}$ orbital electrons form interlayer $s$-wave pairing, as suggested by our slave-boson-mean-field study on the related two-orbital $t$-$J$ model. Projected onto the Fermi surface, this pairing just gives the $s^\pm$-wave pattern consistent with that obtained in the weak-coupling study. Our result is consistent with that obtained in recent scanning tunneling microscopy experiment.

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arXiv2025

Band Structure and Pairing Nature of La$_3$Ni$_2$O$_7$ Thin Film at Ambient Pressure

Recently, evidences of superconductivity (SC) with onset $T_c$ above the McMillan limit have been detected in the La$_3$Ni$_2$O$_7$ ultrathin film grown on the LaSrAlO$_4$ substrate at ambient pressure. This progress opens a new era in the field of the nickelate superconductors. Here we perform a density-functional-theory (DFT) based calculation for the band structure of this material. The obtained DFT+$U$ band structure has the feature that the bonding $d_{z^2}$ band crosses the Fermi level, forming the hole pocket $\gamma$, consistent with the angle-resolved photoemission spectrum (ARPES). Taking the low-energy Ni-$(3d_{z^2},3d_{x^2-y^2})$ orbitals placed on the tetragonal lattice structure, we construct a 2D bilayer four-band tight-binding model which well captures the main features of the DFT+$U$ band structure. Then considering the multi-orbital Hubbard interaction, we adopt the random-phase approximation (RPA) approach to investigate the pairing nature. The obtained pairing symmetry is $s^{\pm}$ or $d_{xy}$ for the hole-doping level $\delta$ below or above 0.12, induced by the different Fermi surface nesting situations. For the realistic $\delta=0.21$ measured by the ARPES, our RPA calculations obtain the next-nearest-neighbor pairing $d_{xy}$-wave SC dominated by the $d_{z^2}$ orbital, consistent with the experimental observation that the $T_c$ enhances with the shrinking of the in-plane lattice constants. This pairing state is induced by the nesting between the different patches within the $\gamma$ pocket. Our results appeal for experimental verifications.

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arXiv2024

Origin of the Diagonal Double-Stripe Spin-Density-Wave and Potential Superconductivity in Bulk La$_3$Ni$_2$O$_{7}$ at Ambient Pressure

The discovery of high-temperature superconductivity (SC) with $T_c\approx 80$ K in the pressurized La$_3$Ni$_2$O$_{7}$ has aroused great interests. Currently, due to technical difficulties, most experiments on La$_3$Ni$_2$O$_{7}$ can only be performed at ambient pressure (AP). Particularly, various experiments have revealed the presence of spin-density wave (SDW) in the unidirectional diagonal double-stripe pattern with wave vector near $(\pi/2,\pi/2)$ in La$_3$Ni$_2$O$_{7}$ at AP. In this work, we employ first-principle calculations followed by the random phase approximation (RPA)-based study to clarify the origin of this special SDW pattern and the potential SC in La$_3$Ni$_2$O$_{7}$ at AP. Starting from our density-functional-theory band structure, we construct an eight-band bilayer tight-binding model using the Ni-$3d_{z^2}$ and $3d_{x^2-y^2}$ orbitals, which is equipped with the standard multi-orbital Hubbard interaction. Our RPA calculation reveals an SDW order driven by Fermi-surface nesting with wave vector ${Q}\approx(0,\pm0.84\pi)$ in the folded Brillouin zone (BZ). From the view of the unfolded BZ, the wave vector turns to ${Q}_0\approx\pm(0.58\pi,0.58\pi)$, which is near the one detected by various experiments. Further more, this SDW exhibits an interlayer antiferromagnetic order with a unidirectional diagonal double-stripe pattern, consistent with recent soft X-ray scattering experiment. This result suggests that the origin of the SDW order in La$_3$Ni$_2$O$_{7}$ at AP can be well understood in the itinerant picture as driven by Fermi surfaces nesting. In the aspect of SC, our RPA study yields an approximate $s^\pm$-wave spin-singlet pairing with $T_c$ much lower than that under high pressure. Further more, the $T_c$ can be strongly enhanced through hole doping, leading to possible high-temperature SC at AP.

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arXiv2024

Spin-density wave and superconductivity in La$_4$Ni$_3$O$_{10}$ under ambient pressure

We investigate the spin-density wave (SDW) behavior and the potential for superconductivity (SC) in La$_4$Ni$_3$O$_{10}$ under ambient pressure using a multi-orbital random-phase approximation (RPA). Starting with a twelve-orbital tight-binding model derived from density functional theory (DFT) calculations, we explore the influence of Hubbard interactions on SDW formation. Our analysis reveals a stripe-like SDW characterized by an incommensurate wave vector, $Q\approx(\pm 0.7\pi,0)$, suggesting a possible density wave instability in agreement with recent experiments. This configuration is driven by nesting of outer-layer Ni $d_{z^2}$ orbitals and exhibits interlayer antiferromagnetic ordering between the top and bottom NiO layers, with the middle layer serving as a node. We demonstrate that the Hund's coupling $J_H$ is the primary driver of the observed SDW. While superconductivity is absent in the undoped system under ambient pressure, it becomes attainable with appropriate hole doping ($\delta=-0.4$), resulting in a SC gap structure similar to the high-pressure phase. Our study identifies the specific conditions for realizing the ambient pressure stripe density wave: $J_H>0.16U$. Additionally, when doping leads to sufficient nesting at (0,0), the system's magnetic fluctuations transition into a stable Neel-type antiferromagnetic state, analogous to the high-pressure case.

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arXiv2024

The magnetic $Z_2$ topological insulator on the AA-stacked bilayer graphene

The properties displayed by graphene at van Hove singularities (VHS) have caught significant attention in recent years. The emergence of exotic quantum states at these singularities prompts investigations on their evolution within the realm of multilayer stacking structures. In our research, we delve into the study of a repulsive Hubbard model focusing on the AA-stacked bilayer graphene at VHS. Within the system's ground state, each of the top and bottom layers hosts a set of spin-density waves (SDWs). These SDWs each takes on three mutually perpendicular spin polarization directions. Importantly, there is noteworthy feature that their spin polarization directions in the two layers exist as elegant embodiments of antiferromagnetic arrangement, persvading the structure with a striking pattern. Referred to in prior research as the chiral SDWs, this intralayer density wave structure confers the system the characteristics of a Chern topological insulator. However, what is particularly fascinating is the pure divergence of the bilayer structure's topological traits when compared to its monolayer counterpart. The system exhibits a profound symmetry known as $Z_2$, preserving its invariance under the combined operations of time-reversal and interlayer exchange. Consequentely, the system's ground state manifests a seemingly trivial Chern number, yet harbors a profound and intricate nontrivial $Z_2$ topological invariant. These remarkable observations align our findings with the conceptual framework of the quantum spin Hall effect.

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arXiv2024

The $s^\pm$-Wave Superconductivity in the Pressurized La$_4$Ni$_3$O$_{10}$

Recently, evidence of superconductivity (SC) has been reported in pressurized La$_4$Ni$_3$O$_{10}$. Here we study the possible pairing mechanism and pairing symmetry in this material. Through fitting the density-functional-theory band structure, we provide a six-orbital tight-binding model. In comparison with the band structure of La$_3$Ni$_2$O$_7$, the additional non-bonding $d_{z^2}$ band is importance to the pairing mechanism here. When the multi-orbital Hubbard interactions are included, our random-phase-approximation based study yields an $s^{\pm}$-wave pairing. The dominant FS nesting with nesting vector $\mathbf{Q}_1\approx (\pi,\pi)$ is between the $\gamma$-pocket contributed by the bonding $d_{z^2}$ band top and the $\alpha_1$-pocket contributed by the non-bonding $d_{z^2}$ band bottom, leading to the strongest pairing gap amplitude and opposite gap signs within the two regimes. The dominant real-space pairing is the interlayer pairing between the $d_{z^2}$ orbitals. We have also studied the doping dependence of the pairing symmetry and $T_c$.

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arXiv2024

Nematic Superconductivity and Its Critical Vestigial Phases in the Quasi-crystal

We propose a general mechanism to realize nematic superconductivity (SC) and reveal its exotic vestigial phases in the quasi-crystal (QC). Starting from a Penrose Hubbard model, our microscopic studies suggest that the Kohn-Luttinger mechanism driven SC in the QC is usually gapless due to violation of Anderson's theorem, rendering that both chiral and nematic SCs are common. The nematic SC in the QC can support novel vestigial phases driven by pairing phase fluctuations above its $T_c$. Our combined renormalization group and Monte-Carlo studies provide a phase diagram in which, besides the conventional charge-4e SC, two critical vestigial phases emerge, i.e. the quasi-nematic (Q-N) SC and Q-N metal. In the two Q-N phases, the discrete lattice rotation symmetry is counter-intuitively ``quasi-broken'' with power-law decaying orientation correlation. They separate the phase diagram into various phases connected via Berezinskii-Kosterlitz-Thouless (BKT) transitions. These remarkable critical vestigial phases, which resemble the intermediate BKT phase in the $q$-state ($q\ge 5$) clock model, are consequence of the five- (or higher-) fold anisotropy field brought about by the unique QC symmetry, which are absent in conventional crystalline materials.

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arXiv2023

The s$^\pm$-Wave Pairing and the Destructive Role of Apical-Oxygen Deficiencies in La$_3$Ni$_2$O$_7$ Under Pressure

Recently, the bilayer perovskite nickelate La$_3$Ni$_2$O$_7$ has been reported to show evidence of high-temperature superconductivity (SC) under a moderate pressure of about 14 GPa. To investigate the superconducting mechanism, pairing symmetry, and the role of apical-oxygen deficiencies in this material, we perform a random-phase-approximation based study on a bilayer model consisting of the $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals of Ni atoms in both the pristine crystal and the crystal with apical-oxygen deficiencies. Our analysis reveals an $s^{\pm}$-wave pairing symmetry driven by spin fluctuations. The crucial role of pressure lies in that it induces the emergence of the $\gamma$-pocket, which is involved in the strongest Fermi-surface nesting. We further found the emergence of local moments in the vicinity of apical-oxygen deficiencies, which significantly suppresses the $T_c$. Therefore, it is possible to significantly enhance the $T_c$ by eliminating oxygen deficiencies during the synthesis of the samples.

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arXiv2023

Unconventional superfluidity of superconductivity on Penrose lattice

We theoretically investigate the gap function, superfluid density and the transition temperature of the superconductivity (SC) on semi-periodic Penrose lattice, where an attractive Hubbard model is adopted as an example. Firstly, we clarify that the gap function, density of states and superfluid density are all positively correlate to the extended degree of single particle states around the Fermi energy. Secondly, we identify that the paramagnetic component of the superfluid density does not decay to zero in the thermodynamic limit, which is completely different from the periodic system. The difference between the diamagnetic and paramagnetic currents keeps stable with whatever scaling, which is consistent with recent experimental results that although the superfluid density is lower than that of the periodic system, the system has bulk SC. Thirdly, we find that both the superfluid density and SC transition temperature can be boosted with the increase of disorder strength, which should be general to quasicrystal but unusual to periodic systems, reflecting the interplay between the underlying geometry and disorder.

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arXiv2023

Making Chiral Topological Superconductivities from Non-topological Superconductivities Through the Twist

In this paper, we propose a general scheme to realize chiral TSCs through the "twistronics". Suppose we have a $D_n$-symmetric monolayer superconductor, which carries non-topological SC with pairing angular momentum $L=n/2$. Here we propose that we can obtain chiral TSC with the same $L$, by stacking two such monolayers with the largest twist angle $\pi/n$, forming a Moireless quasi-crystal (QC) structure, dubbed as the twist-bilayer QC (TB-QC) here. The chiral TSC in the TB-QC is driven by the interlay Josephson coupling between the pairing order parameters of the two layers. An argument based on the universal Ginzburg-Landau theory is provided to understand this proposal. One known example which fits our proposal is the $d+id$-chiral TSC in the 45$^\circ$-twisted bilayer cuprates. Here, based on the microscopic framework developed previously to treat with the electron-electron interactions in the TB-QC, we demonstrate the application of our proposal to a new example, i.e., the $f+if$-chiral TSC obtained by twisting two properly-doped honeycomb-Hubbard-model monolayers by the angle 30$^\circ$. This example is related to the newly synthesized 30$^\circ$-twisted bilayer graphene.

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arXiv2023

Charge 4e superconductivity and chiral metal in the $45^\circ$-twisted bilayer cuprates and similar bilayers

The material realization of the charge-4e/6e superconductivity (SC) is a big challenge. Here we propose realization of the charge-4e SC and chiral metal through stacking a homo-bilayer with the largest twist angle, forming the twist-bilayer quasi-crystal (TB-QC), exampled by the 45$^\circ$-twisted bilayer cuprates and 30$^\circ$-twisted bilayer graphene. When each mononlayer hosts a pairing state with the largest pairing angular momentum, previous studies yield that the second-order interlayer Josephson coupling would drive chiral topological SC (TSC) in the TB-QC. Here we propose that, above the $T_c$ of the chiral TSC, either the total- or relative- pairing phase of the two layers can be unilateral quasi-ordered or ordered, leading to the charge-4e SC or the chiral metal phase. Based on a thorough symmetry analysis to get the low-energy effective Hamiltonian, we conduct a combined renormalization-group and Monte-Carlo study and obtain the phase diagram, which includes the charge-4e SC and chiral metal phases.

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