Casimir amplitudes in topological quantum phase transitions

Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.

Figure 1

3D generalization of the Su-Schrieffer-Heeger (SSH) model. In this figure, we have a staking of alternating layers named by $A$ and $B$ along the $z$ direction with hopping $t_1$ and $t_2<0$. $L$ is the size of the system.

Figure 2

Normalized finite-size contribution to the free-energy density $\varepsilon =\frac{E{a}_z^2}{t_1{L}^2}$ versus $g=\frac{t_1-t_2}{t_1}$ for various values of the length $L$ ($L$ is the size of the system along the $z$ direction) with $L=100a_z$ (squares), $L=125a_z$ (circles), $L=150a_z$ (triangles), and $L=200a_z$ (diamonds), such that $a_z$ is the lattice constant along the $z$ direction. Here $v=0.01\text{eV}\times \text{meters}$, $t_1=1\text{eV}$. The topological quantum phase transition emerges when $t_2=t_c=1$. Therefore, the trivial phase occurs for the region $g<0$, while for $g>0$ we have a topological phase. At the critical point the finite-size free-energy density assumes its maximum value.

Figure 3

Normalized Casimir pressure $p=\frac{F{a}_z^3}{t_1{L}^2}$ versus $g=\frac{t_1-t_2}{t_1}$ for various values of the length $L$ ($L$ is the size of the system along the $z$ direction) with $L=100a_z$ (squares), $L=125a_z$ (circles), $L=150a_z$ (triangles), and $L=200a_z$ (diamonds), such that $a_z$ is the lattice constant along the $z$ direction. Here, $v=0.01\text{eV}\times \text{meters}$, $t_1=1\text{eV}$ and the topological quantum phase transition emerges when $t_2=t_c=1$. Therefore, the trivial phase occurs for the region $g>0$, while for $g<0$ we have a topological phase.

Figure 4

Normalized Casimir's pressure $p=\frac{F{a}_z^3}{t_1{L}^2}$ in the topological phase ($t_2=1.025$ and $t_1=1$) versus the number os layers $n$ ($n=\frac{L}{a_z}$, $L$ is the size of the system along the $z$ direction and $a_z$ is the constant lattice this direction) for various values of the Rashba spin-orbit coupling $v$, such that, $v=0.01\text{eV}\times \text{meters}$ (squares), $v=0.02\text{eV}\times \text{meters}$(circles), and $v=0.04\text{eV}\times \text{meters}$(triangles). The pressure is negative, which indicates the attractive force between the boundaries of the finite region.

Figure 5

The region of integration for Eq. (C4) is shown in gray.