Catastrophe theory classification of Fermi surface topological transitions in two dimensions

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Catastrophe theory classification of Fermi surface topological transitions in two dimensions

Anirudh Chandrasekaran, Alex Shtyk, Joseph J. Betouras, and Claudio Chamon

Phys. Rev. Research 2, 013355 – Published 23 March 2020

Abstract

We classify all possible singularities in the electronic dispersion of two-dimensional systems that occur when the Fermi surface changes topology, using catastrophe theory. For systems with up to seven control parameters (i.e., pressure, strain, bias voltage, etc.), the theory guarantees that the singularity belongs to one of seventeen standard types known as catastrophes. We show that at each of these singularities the density of states diverges as a power law, with a universal exponent characteristic of the particular catastrophe, and we provide its universal ratio of amplitudes of the prefactors for energies above and below the singularity. We further show that crystal symmetry restricts which types of catastrophes can occur at the points of high symmetry in the Brillouin zone. For each of the seventeen wallpaper groups in two dimensions, we list which catastrophes are possible at each high-symmetry point.

Received 22 October 2019
Accepted 24 February 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013355

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bloch wave theory Density functional theory Symmetries in condensed matter Tight-binding model Wannier function methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Anirudh Chandrasekaran ¹, Alex Shtyk², Joseph J. Betouras³, and Claudio Chamon¹

¹Department of Physics, Boston University, Boston, Massachusetts 02215, USA
²Quantlab Group, Cambridge, Massachusetts, USA
³Department of Physics and Centre for the Science of Materials, Loughborough University, Loughborough LE11 3TU, United Kingdom

Classification of critical points in energy bands based on topology, scaling, and symmetry

Noah F. Q. Yuan and Liang Fu

Phys. Rev. B 101, 125120 (2020)

Article Text

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Issue

Vol. 2, Iss. 1 — March - May 2020

Subject Areas

Condensed Matter Physics

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Images

Figure 1
The lattice for the Hamiltonian treated in Sec. 2 is shown above. It is a square lattice with a two site basis of atoms colored as black ( $A$ ) and grey ( $B$ ). The $A B$ nearest neighbor (NN) hopping in the $\hat{x}$ direction has a strength of $t_{1}$ while the $A A$ and $B B$ NN hoppings in the $\hat{y}$ direction have strength $t_{2}$ . There is also complex next nearest neighbor $A A$ and $B B$ hopping $i t^{'}$ (with direction as depicted in the figure). These hopping strengths can be tuned so that the energy bands of this Hamiltonian yield the simplest of the higher-order singularities, the fold.
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Figure 2
Constant energy contours of the dispersion $f (k_{x}, k_{y}) = k_{x}^{3} - k_{y}^{2} + t k_{x}$ for three cases: (a) $t < 0$ , (b) $t = 0$ , and (c) $t > 0$ . When $t < 0$ there is an ordinary maximum and an ordinary minimum. As $t$ is reduced to 0, these merge at the origin to give a higher-order singularity. For $t > 0$ , there is no longer a critical point.
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Figure 3
The Brillouin zone of the wallpaper group p4 shown along with the high-symmetry points. The point group of p4 is generated by $π / 2$ rotation which gets carried over to $k$ -space as a symmetry that acts about the origin. By combining the $π / 2$ and $π$ rotations with reciprocal lattice translations we can identify the other high-symmetry points. At each of the high-symmetry points, only dispersions which respect the symmetry are allowed. This places a huge restriction on the higher-order singularities that are likely to occur. In the insets, we identify the catastrophes likely to occur at each of the high-symmetry points.
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Figure 4
Constant energy contours of the dispersions: (a) $f (k_{x}, k_{y}) = - k_{y}^{2} + 6 k_{x}^{2} k_{y} - 8 k_{x}^{4}$ , (b) $g (k_{x}, k_{y}) = - k_{y}^{2} + 6 k_{x}^{2} k_{y} - 9 k_{x}^{4} + k_{x}^{6}$ , and (c) $h (k_{x}, k_{y}) = - k_{y}^{2} + 6 k_{x}^{2} k_{y}$ shown within the range $- 1 < k_{x} < 1, - 1 < k_{y} < 2$ . While the energy contours of $f$ and $g$ appear to have the same topology, they are actually not related by smooth deformation. The former corresponds to two parabolas that intersect tangentially and the latter consists of two cubic curves that intersect tangentially at (0,0). The DOS diverges differently for both these dispersions although they give the same cubic truncation $h$ . This in spite of the fact that $h$ appears to produce correct topology of Fermi surface (two curves tangentially touching at an isolated higher-order critical point) and a seemingly sensible power law divergence of DOS. This is shown to be the consequence of the fact that $f$ and $g$ are nonequivalent and have different allowed order of truncation as indicated by their determinacy.
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Figure 5
It is fairly easy to determine the isometry that results from combining two isometries. We illustrate this procedure in the figure above for two important cases. In (a), we show how combining the $x$ -axis reflection $r_{x}$ with a rotation $ρ_{θ}$ simply rotates the reflection axis by $θ / 2$ . The new reflection axis is denoted by $l$ in the figure, and the reflection operation about this line is given by $ρ_{θ} r_{x}$ . A point $p$ first gets reflected about the $x$ axis and is then rotated by an angle $θ$ about the origin. The net result is the same as reflection about $l$ . In fact $l$ is the set of all points which are invariant under $ρ_{θ} r_{x}$ . In (b), we illustrate a simple geometrical construction [24] for finding the new center of rotation when a translation $t_{a}$ is combined with a rotation $ρ_{θ}$ . The new center of rotation for the composite operation $ρ_{θ} t_{a}$ is the point $p$ which is invariant under the composite operation. $ρ_{θ} t_{a}$ is then a $θ$ rotation about $p$ .
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