Topological chiral superconductivity with spontaneous vortices and supercurrent in twisted bilayer graphene

We study $d$-wave superconductivity in twisted bilayer graphene and reveal phenomena that arise due to the moir\'e superlattice. In the $d$-wave pairing, the relative motion (RM) of two electrons in a Cooper pair can have either $d+id$ or $d-id$ symmetry with opposite angular momenta. Due to the enlarged moir\'e superlattice, the center-of-mass motion (COMM) can also carry a finite angular momentum while preserving the moir\'e periodicity. By matching the total angular momentum, which has contributions from both the RM and the COMM, Cooper pairs with $d+id$ and $d-id$ RMs are intrinsically coupled in a way such that the COMM associated with one of the RMs has a spontaneous vortex-antivortex lattice configuration. Another phenomenon is that the chiral $d$-wave state carries spontaneous bulk circulating supercurrent. The chiral $d$-wave superconductors are gapped and also topological as characterized by an integer Chern number. Nematic $d$-wave superconductors, which could be stabilized, for example, by uniaxial strain, are gapless with point nodes.Comment: 10 pages, 7 figure

Nematic and chiral superconductivity induced by odd-parity fluctuations

Recent experiments indicate that superconductivity in Bi$_2$Se$_3$ intercalated with Cu, Nb or Sr is nematic with rotational symmetry breaking. Motivated by this observation, we present a model study of nematic and chiral superconductivity induced by odd-parity fluctuations. We show that odd-parity fluctuations in the two-component $E_u$ representation of $D_{3d}$ crystal point group can generate attractive interaction in both the even-parity $s$-wave and odd-parity $E_u$ pairing channels, but repulsive interaction in other odd-parity pairing channels. Coulomb repulsion can suppress $s$-wave pairing relative to $E_u$ pairing, and thus the latter can have a higher critical temperature. $E_u$ pairing has two distinct phases: a nematic phase and a chiral phase, both of which can be realized in our model. When $s$-wave and $E_u$ pairings have similar instability temperature, we find an intermediate phase in which both types of pairing coexist.Comment: 9 pages, 2 figure

Majorana Kramers pair in a nematic vortex

A time-reversal (TR) invariant topological superconductor is characterized by a Kramers pair of Majorana zero-energy modes on boundaries and in a core of a TR invariant vortex. A vortex defect that preserves TR symmetry has remained primarily of theoretical interest, since typically a magnetic field, which explicitly breaks TR, needs to be applied to create vortices in superconductors. In this work, we show that an odd-parity topological superconductor with a nematic pairing order parameter can host a nematic vortex that preserves TR symmetry and binds a Majorana Kramers pair. Such a nematic superconductor could be realized in metal-doped Bi$_2$Se$_3$, as suggested by recent experiments. We provide an analytic solution for the zero modes in a continuous nematic vortex. In lattice, crystalline anisotropy can pin the two-component order parameter along high-symmetry directions. We show that a discrete nematic vortex, which forms when three nematic domains meet, also supports a TR pair of Majorana modes. Finally, we discuss possible experiments to probe the zero modes.Comment: 9 pages, 4 figure

Quantum Geometry and Stability of Moir\'e Flatband Ferromagnetism

Several moir\'e systems created by various twisted bilayers have manifested magnetism under flatband conditions leading to enhanced interaction effects. We theoretically study stability of moir\'e flatband ferromagnetism against collective excitations, with a focus on the effects of Bloch band quantum geometry. The spin magnon spectrum is calculated using different approaches, including Bethe-Salpeter equation, single mode approximation, and an analytical theory. One of our main results is an analytical expression for the spin stiffness in terms of the Coulomb interaction potential, the Berry curvatures, and the quantum metric tensor, where the last two quantities are geometric invariants of moir\'e bands. This analytical theory shows that Berry curvatures play an important role in stiffening the spin magnons. Furthermore, we construct an effective field theory for the magnetization fluctuations, and show explicitly that skyrmion excitations bind an integer number of electrons that is proportional to the Bloch band Chern number and the skyrmion winding number.Comment: 10 pages, 4 figure

Identification of superconducting pairing symmetry in twisted bilayer graphene using in-plane magnetic field and strain

We show how the pairing symmetry of superconducting states in twisted bilayer graphene can be experimentally identified by theoretically studying effects of externally applied in-plane magnetic field and strain. In the low field regime, superconducting critical temperature $T_c$ is suppressed by in-plane magnetic field $\boldsymbol{B}_{\parallel}$ in singlet channels, but is enhanced by weak $\boldsymbol{B}_{\parallel}$ in triplet channels, providing an important distinction. The in-plane angular dependence of the critical $\boldsymbol{B}_{\parallel, c}$ has a six-fold rotational symmetry, which is broken when strain is present. We show that anisotropy in $\boldsymbol{B}_{\parallel, c}$ generated by strain can be similar for $s$- and $d$-wave channels in moir\'e superlattices. The $d$-wave state is pinned to be nematic by strain and consequently gapless, which is distinguishable from the fully gapped $s$-wave state by scanning tunneling measurements.Comment: 5+2 pages, 4 figure

Ferromagnetism and superconductivity in twisted double bilayer graphene

We present a theory of competing ferromagnetic and superconducting orders in twisted double bilayer graphene (TDBG). In our theory, ferromagnetism is induced by Coulomb repulsion, while superconductivity with intervalley equal-spin pairing can be mediated by electron-acoustic phonon interactions. We calculate the transition temperatures for ferromagnetism and superconductivity as a function of moir\'e band filling factor, and find that superconducting domes can appear on both the electron and hole sides of the ferromagnetic insulator at half filling. We show that the ferromagnetic insulating gap has a dome shape dependence on the layer potential difference, which provides an explanation to the experimental observation that the ferromagnetic insulator only develops over a finite range of external displacement field. We also verify the stability of the half-filled ferromagnetic insulator against two types of collective excitations, i.e., spin magnons and valley magnons.Comment: 9 pages, 6 figure

Electron-phonon and electron-electron interaction effects in twisted bilayer graphene

By comparing with recently available experimental data from several groups, we critically discuss the manifestation of continuum many body interaction effects in twisted bilayer graphene (tBLG) with small twist angles and low carrier densities, which arise naturally within the Dirac cone approximation for the non-interacting band structure. We provide two specific examples of such continuum many body theories: one involving electron-phonon interaction and one involving electron-electron interaction. In both cases, the experimental findings are only partially quantitatively consistent with rather clear-cut leading-order theoretical predictions based on well-established continuum many body theories. We provide a critical discussion, based mainly on the currently available tBLG experimental data, on possible future directions for understanding many body renormalization involving electron-phonon and electron-electron interactions in the system. One definitive conclusion based on the comparison between theory and experiment is that the leading order 1-loop perturbative renormalization group theory completely fails to account for the electron-electron interaction effects in the strong-coupling limit of flatband moir\'e tBLG system near the magic twist angle even at low doping where the Dirac cone approximation should apply. By contrast, approximate nonperturbative theoretical results based on Borel-Pad\'e resummation or $1/N$ expansion seems to work well compared with experiments, indicating rather small interaction corrections to Fermi velocity or carrier effective mass. For electron-phonon interactions, however, the leading-order continuum theory works well except when van Hove singularities in the density of states come into play.Comment: 18 pages, 7 figure

Exciton band structure of monolayer MoS2

We address the properties of excitons in monolayer MoS$_2$ from a theoretical point of view, showing that low-energy excitonic states occur both at the Brillouin zone center and at the Brillouin-zone corners, that binding energies at the Brillouin-zone center deviate strongly from the $(n-1/2)^{-2}$ pattern of the two-dimensional hydrogenic model, and that the valley-degenerate exciton doublet at the Brillouin-zone center splits at finite momentum into an upper mode with non-analytic linear dispersion and a lower mode with quadratic dispersion. Although monolayer MoS$_2$ is a direct-gap semiconductor when classified by its quasiparticle band structure, it may well be an indirect gap material when classified by its excitation spectra.Comment: 9 pages, 6 figure

Theory of phonon-mediated superconductivity in twisted bilayer graphene

We present a theory of phonon-mediated superconductivity in near magic angle twisted bilayer graphene. Using a microscopic model for phonon coupling to moir\'e band electrons, we find that phonons generate attractive interactions in both $s$ and $d$ wave pairing channels and that the attraction is strong enough to explain the experimental superconducting transition temperatures. Before including Coulomb repulsion, the $s$-wave channel is more favorable; however, on-site Coulomb repulsion can suppress $s$-wave pairing relative to $d$-wave. The pair amplitude varies spatially with the moir\'e period, and is identical in the two layers in the $s$-wave channel but phase shifted by $\pi$ in the $d$-wave channel. We discuss experiments that can distinguish the two pairing states.Comment: 5+3 pages, 4+1 figure

Phonon-induced giant linear-in-TT resistivity in magic angle twisted bilayer graphene: Ordinary strangeness and exotic superconductivity

We study the effect of electron-acoustic phonon interactions in twisted bilayer graphene on resistivity in the high-temperature transport and superconductivity in the low-temperature phase diagram. We theoretically show that twisted bilayer graphene should have an enhanced and strongly twist-angle dependent linear-in-temperature resistivity in the metallic regime with the resistivity magnitude increasing as the twist angle approaches the magic angle. The slope of the resistivity versus temperature could approach one hundred ohms per kelvin with a strong angle dependence, but with a rather weak dependence on the carrier density. This higher-temperature density-independent linear-in-$T$ resistivity crosses over to a $T^4$ dependence at a low density-dependent characteristic temperature, becoming unimportant at low temperatures. This angle-tuned resistivity enhancement arises from the huge increase in the effective electron-acoustic phonon coupling in the system due to the suppression of graphene Fermi velocity induced by the flatband condition in the moir\'e superlattice system. Our calculated temperature dependence is reminiscent of the so-called `strange metal' transport behavior except that it is arising from the ordinary electron-phonon coupling in a rather unusual parameter space due to the generic moir\'e flatband structure of twisted bilayer graphene. We also show that the same enhanced electron-acoustic phonon coupling also mediates effective attractive interactions in $s$, $p$, $d$ and $f$ pairing channels with a theoretical superconducting transition temperature on the order of $\sim$5 K near magic angle. The fact that ordinary acoustic phonons can produce exotic non-$s$-wave superconducting pairing arises from the unusual symmetries of the system.Comment: 15 pages, 8 figures. Expanded final version with a modified title as accepted for publication in PR