Abstract
New two-dimensional systems such as the surfaces of topological insulators (TIs) and graphene offer the possibility of experimentally investigating situations considered exotic just a decade ago. These situations include the quantum phase transition of the chiral type in electronic systems with a relativistic spectrum. Phonon-mediated (conventional) pairing in the Dirac semimetal appearing on the surface of a TI causes a transition into a chiral superconducting state, and exciton condensation in these gapless systems has long been envisioned in the physics of narrow-band semiconductors. Starting from the microscopic Dirac Hamiltonian with local attraction or repulsion, the Bardeen-Cooper-Schrieffer type of Gaussian approximation is developed in the framework of functional integrals. It is shown that owing to an ultrarelativistic dispersion relation, there is a quantum critical point governing the zero-temperature transition to a superconducting state or the exciton condensed state. Quantum transitions having critical exponents differ greatly from conventional ones and belong to the chiral universality class. We discuss the application of these results to recent experiments in which surface superconductivity was found in TIs and estimate the feasibility of phonon pairing.
Highlights
A topological insulator (TI) is a novel state of matter in materials with strong spin–orbit interactions that creates topologically protected surface states [1,2,3]
The critical exponents in this region differ from the one called “quantum Gaussian (BCS)” in Ref. [36], and the available results were obtained using ε expansion [41, 42, 69] (ε = 4 − d, where d = 2 + 1 is the space–time dimension); 1/N, where N is the number of fermionic species on the surface [41, 42]; or functional renormalization group (RG) [70] and Monte Carlo simulations [71,72,73,74,75]
The universality class of the superconducting transition according to the classification proposed in Refs. [41, 42] is the chiral XY [symmetry of order parameter U (1)] for N = 1
Summary
A topological insulator (TI) is a novel state of matter in materials with strong spin–orbit interactions that creates topologically protected surface states [1,2,3]. Attempts to experimentally identify second-order transitions governed by the QCP in condensed matter included quantum magnets [36], superconductor–insulator transitions [43], and more recently exciton condensation in graphene [4, 5, 44] and other Dirac semimetals, including TIs. In the last two cases, the broken symmetry is often termed “chiral”. The quantum phase transitions in a Dirac semimetal due to local interactions, both attractive (superconductivity) and repulsive (exciton condensation), are studied, with an emphasis on their distinct criticality. Their expressions via a partition function [Eq (3)] are given in Appendix A
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