Abstract
We investigate the order of the topological quantum phase transition in a two dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phases are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation.
Highlights
Topological states of matter have attracted a great deal of attention during the last decade
We have extended the formalism used in Refs. [19,20,21,22,28] to investigate higher-order topological insulators (HOTIs), where the bulk-boundary correspondence relates the closure of the bandgap to the zero modes that occur at the corners of the system
We have numerically calculated the spectrum of Hamiltonian (1), which was proposed in Ref. [6], to identify the discontinuities in the derivative of the grand potential and elucidate the order of the topological quantum phase transitions (TQPTs)
Summary
Topological states of matter have attracted a great deal of attention during the last decade. Critical behavior of several models and recently this method was used to devise topological heat machines [28] We extend this thermodynamical approach to describe HOTIs, i.e., to deal with systems that have both edge and corner modes. The critical exponents were extracted by analyzing the functional dependence of the closing of either the bandgap or the Wannier gap, and the order of the phase transitions was determined using the Josephson hyperscaling relation. VI, we show that a grand potential calculated from the Wilson loop spectra, which we name the Wannier grand potential, is sensitive to all phase transitions, but with critical exponents of a system with smaller dimensions, which we identify to be the edge of the original lattice
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