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We develop a unified view of topological phase transitions (TPTs) in solids by revising the classical band theory with the inclusion of topology. Reevaluating the band evolution from an “atomic crystal” (a normal insulator (NI)) to a solid crystal, such as a semiconductor, we demonstrate that there exists ubiquitously an intermediate phase of topological insulator (TI), whose critical transition point displays a linear scaling between electron hopping potential and average bond length, underlined by deformation-potential theory. The validity of the scaling relation is verified in various two-dimensional (2D) lattices regardless of lattice symmetry, periodicity, and form of electron hoppings, based on a generic tight-binding model. Significantly, this linear scaling is shown to set an upper bound for the degree of structural disorder to destroy the topological order in a crystalline solid, as exemplified by formation of vacancies and thermal disorder. Our work formulates a simple framework for understanding the physical nature of TPTs with significant implications in practical applications of topological materials.

Band theory is one of the most important developments of condensed matter and material physics, which underlines the working principle of modern electronic and optoelectronic devices. It is well known that isolated atomic levels would spread to form energy bands when atoms were brought together to form a solid [

Schematic illustration of TPT in band evolution diagram. By decreasing average bond length

The study of TPT dates back to 1970s when phenomena in quantum states of matter, such as the quantum Hall effect [

It is important to recognize that conventional phase transitions, as described by Landau theory of spontaneous symmetry breaking, exhibit a universal scaling relation of criticality. Differently, TPTs, involving no symmetry breaking, are characterized by a sudden change of topological invariants with a continuously changing system parameter. Thus, one does not expect a form of universality to be associated with TPTs. Surprisingly, however, we discover a linear scaling relation between electron hopping potential and average bond length within the framework of band theory, which is commonly applicable to TPTs in different systems albeit with different slopes, i.e., without a universal scaling exponent as for conventional phase transitions. Based on a generic tight-binding (TB) model, we demonstrate this linear scaling relation to define the critical TPT point from the atomic limit to topological solid, regardless of lattice symmetry, periodicity, and form of electron hopping. We validate this linear scaling by calculating TPTs in various 2D crystalline lattices (oblique, trigonal, square, rectangle, rhombic, etc.) as well as quasicrystalline lattices. Furthermore, we demonstrate that this linear scaling sets an upper bound for the degree of disorder to destroy the topological order in a crystal by the case studies of vacancy formation and thermal disorder.

Our TB model consists of three orbitals (

We define the average bond length

As shown in Figure

Then, the energy gap is given by

Namely, the critical bond length

To validate the above hypothesis, we first systemically calculated TPTs in various 2D periodic lattices. A trigonal lattice with an

TPT in a trigonal lattice. The parameters used here are

Remarkably, we found that for more than 60 different lattices,

Linear scaling of TPT. The linear scaling relation between the critical value of average bond length

As the definition of

We again considered a trigonal lattice with random vacancies in a wide range of concentration

TPT in crystals with disorders. Atomic configuration of a trigonal lattice with (a) random vacancies at

In addition, there is a large region of parameters in

We next investigate the effect of thermal disorder in destroying the topological phase in a 2D crystal. Due to thermal fluctuation, the interatomic distance

The mean-squared displacement

Finally, we add the critical transition points (

We note that the linear scaling as discovered by TB calculations and analyses from the “atomic limit” to TI is physically related to the well-known concept of deformation potential [

Finally, it is interesting to note that for crystals one can identify whether it is a TI by symmetry analysis of band topology to determine if TPT has occurred by linking the solid crystal to the atomic crystal assumed with the same symmetry, which have provided a powerful method to discover topological crystals [

The authors declare that there is no conflict of interest regarding the publication of this article.

H. Huang and F. Liu designed the project. H. Huang performed theoretical calculation, and all authors prepared the manuscript.

This work was supported by U.S. DOE-BES (Grant No. DE-FG02-04ER46148). This research used resources of the CHPC at the University of Utah and the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

General band evolution; details of methodology; TPTs of two-dimensional crystals and quasicrystals; random vacancy; thermal fluctuation; validity of the linear scaling for different forms of electron hopping. Figure S1: schematic illustration of topological phase transition. Figure S2: the calculation of formation of band structure from discrete levels of isolated atoms by decreasing average bong length. Figure S3: five 2D Bravais lattices. Figure S4: the eight lattices based on semiregular Archimedean tilings. Figure S5: several decorated trigonal lattices. Figure S6: atomic model of the Penrose-type and the Ammann-Beenker-type quasicrystal lattices. Figure S7: the linear scaling relation in various 2D crystalline lattices using the power-law decay function (

_{2}topological order and the quantum spin Hall effect