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We show that effective superconducting orders generally emerge at low energy in the superconducting state of graphene with conventionally defined pairing symmetry. We study a particular interesting example, the d(x2-y2)+id(xy)(') spin singlet pairing state in graphene which can be generated by electronic correlation. We find that effectively the d+id(')-wave state is a state with mixed s-wave and exotic p+ip-wave pairing orders at low energy. This remarkable property leads to distinctive superconducting gap functions and behavior of the Andreev conductance spectra through a normal/superconducting graphene structure.

In this work, we show that the quantum compass model on a square lattice can be mapped to a fermionic model with local-density interaction. We introduce a mean-field approximation where the most important fluctuations, those perpendicular to the ordering direction, are taken into account exactly. It is found that the quantum phase transition point at J(x)=J(z) marks a first-order phase transition. We also show that the mean-field result is robust against the remaining fluctuation corrections up to the second order.

We show that the d-wave ordering in particle-hole channels, dubbed d-wave checkerboard order, possesses important physics that can sufficiently explain the scanning tunneling microscopy (STM) results in cuprates. A weak d-wave checkerboard order can effectively suppress the coherence peak in the single-particle spectrum while leaving the spectrum along the nodal direction almost unaffected. Simultaneously, it generates a Fermi arc with little dispersion around the nodal points at finite temperature that is consistent with the results of angle-resolved photoemission spectroscopy (ARPES) experiments in the pseudogap phase. We also show that there is a general complementary connection between the d-wave checkerboard order and the pair-density-wave order. Suppressing superconductivity locally or globally through phase fluctuations should induce both orders in underdoped cuprates and explain the nodal-antinodal dichotomy observed in ARPES and STM experiments.

Motivated by the duality between site-centered spin and bond-centered spin in one-dimensional system, which connects two different constructions of fermions from the same set of Majorana fermions, we show that two-dimensional models with topological orders can be constructed from certain well-known models with classical orders characterized by symmetry breaking. Topology-dependent ground state degeneracy, vanishing two-point correlation functions, and unpaired Majorana fermions on boundaries emerge naturally from such construction. The approach opens a different way to construct and characterize topological orders.