The superconducting (SC) and charge-density-wave (CDW) susceptibilities of the two dimensional Holstein model are computed using determinant quantum Monte Carlo (DQMC), and compared with results computed using the Migdal-Eliashberg (ME) approach. We access temperatures as low as 25 times less than the Fermi energy, EF , which are still above the SC transition. We find that the SC susceptibility at low T agrees quantitatively with the ME theory up to a dimensionless electronphonon coupling λ0 ≈ 0.4 but deviates dramatically for larger λ0. We find that for large λ0 and small phonon frequency ω0 EF CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.
We introduce and solve a model of interacting electrons and phonons that is a natural generalization of the Sachdev-Ye-Kitaev-model and that becomes superconducting at low temperatures. In the normal state two Non-Fermi liquid fixed points with distinct universal exponents emerge. At weak coupling superconductivity prevents the onset of low-temperature quantum criticality, reminiscent of the behavior in several heavy-electron and iron-based materials. At strong coupling, pairing of highly incoherent fermions sets in deep in the Non-Fermi liquid regime, a behavior qualitatively similar to that in underdoped cuprate superconductors. The pairing of incoherent time-reversal partners is protected by a mechanism similar to Anderson's theorem for disordered superconductors. The superconducting ground state is characterized by coherent quasiparticle excitations and higher-order bound states thereof, revealing that it is no longer an ideal gas of Cooper pairs, but a strongly coupled pair fluid. The normal-state incoherency primarily acts to suppress the weight of the superconducting coherence peak and reduce the condensation energy. Based on this we expect strong superconducting fluctuations, in particular at strong coupling.
A Wigner crystal, a regular electron lattice arising from strong correlation effects 1-6 , is one of the earliest predicted collective electronic states. This many-body state exhibits quantum and classical phase transitions 7 and has been proposed as a basis for quantum information processing applications 8, 9 . In semiconductor platforms, two-dimensional Wigner crystals have been observed under magnetic field 10-17 or moiré-based lattice potential 18-21 where the electron kinetic energy is strongly suppressed. Here, we report bilayer Wigner crystal formation without a magnetic or confinement field in atomically thin MoSe2 bilayers separated by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states formed at symmetric (1:1) and asymmetric (4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temperatures. We attribute these features to the bilayer Wigner crystals formed from two commensurate triangular electron lattices in each layer, stabilized via inter-layer interaction 22, 23 . These bilayer Wigner crystal phases are remarkably stable and undergo quantum and thermal melting transitions above a critical electron density of up to 6 ´ 10 12 cm -2 and at temperatures of ~40 K. Our results demonstrate that atomically thin semiconductors provide a promising new platform for realizing strongly correlated electronic states, probing their electronic and magnetic phase transitions, and developing novel applications in quantum electronics and optoelectronics 24-28 .Atomically thin heterostructures made of graphene and transition metal dichalcogenide (TMD) monolayers can host a variety of correlated electronic states [29][30][31][32][33] . Recent advances in materials growth and heterostructure fabrication have enabled the preparation of high-quality heterostructures with minimal disorder [34][35][36][37][38] . The large effective masses of charge carriers 39, 40 and the weak Coulomb screening in TMDs suppress the Fermi energy and enhance electron
We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method.In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.
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