Thermodynamic signatures of edge states in topological insulators

Topological insulators are states of matter distinguished by the presence of symmetry-protected metallic boundary states. These edge modes have been characterized in terms of transport and spectroscopic measurements, but a thermodynamic description has been lacking. The challenge arises because in conventional thermodynamics the potentials are required to scale linearly with extensive variables such as volume, which does not allow for a general treatment of boundary effects. In this paper, we overcome this challenge with Hill thermodynamics. In this extension of the thermodynamic formalism, the grand potential is split into an extensive, conventional contribution, and the subdivision potential, which is the central construct of Hill's theory. For topologically nontrivial electronic matter, the subdivision potential captures measurable contributions to the density of states and the heat capacity: it is the thermodynamic manifestation of the topological edge structure. Furthermore, the subdivision potential reveals phase transitions of the edge even when they are not manifested in the bulk, thus opening a variety of possibilities for investigating, manipulating, and characterizing topological quantum matter solely in terms of equilibrium boundary physics.

Figure 1

(a) In blue, $D^B\left(\mu \right)/W$ is shown for $M=1$; it has been shifted up by $0.1$ for greater visibility. As expected, it vanishes in the energy gap. In red, $D^b\left(\mu \right)$ for the same parameters, which is the DOS on the edge. Since the system is not in a topological phase, the DOS at the edge also vanishes in the gap. (b) In blue, $C_v^B\left(T\right)/W$ is shown as a function of $T$ for $M=1,\mu =0$. As expected, it vanishes to linear order as $T\to 0$. In red, $C_v^b\left(T\right)$ for the same parameters. Since the system is not in a topological phase, $C_v^b$ also vanishes to linear order as $T\to 0$. (c) The same as in (a), except that the blue curve has been shifted upwards by 1, and the system is in the topological phase $M=-1$. Furthermore, the line $1/\pi$ has been added in yellow to emphasize that the edge has a nonvanishing DOS in the gap. (d) The same as in (b), but in the topological phase $M=-1$. One sees that $C_v^b$ now is linear at the edge for low $T$. In the inset, the straight line with slope $\pi /3$ has been added to emphasize the linear scaling of $C_v^b$ with temperature.

Figure 2

(a) The second derivative ${\partial }^2\varphi /\partial M^2$ is shown as a function of $M$. A kink is visible at $M=0$, indicating that the bulk undergoes a third-order phase transition (see the inset, where the discontinuity in the third derivative is depicted). (b) The first derivative $\partial {\varphi }_0/\partial M$ is shown as a function of $M$. A kink is visible at $M=0$ indicating that the edge undergoes a second-order phase transition (the discontinuity in the second derivative is depicted in the inset), which can be considered the order of the topological phase transition.

Figure 3

In red, ${\partial }^2{\varphi }_0/\partial {\mathrm{\Delta }}^2$ is shown as a function of $\mathrm{\Delta }$. The smoothing of an infinite peak is visible, which indicates the presence of a third-order phase transition at the boundary of the system. In yellow, ${\partial }^2{\varphi }_e/\partial {\mathrm{\Delta }}^2+6$ is shown, using Eq. (16). The agreement between the two curves indicates that the phase transition at the edge of the model comes from the opening of a gap for the edge states.

Figure 4

(a) The relative error between ${\mathrm{\Phi }}_W$ and $\mathrm{\Phi }$, for $\mu =T=0$ and $M=-1$. We have fitted along the interval $40\le W\le 100.$ (b) Same as in (a) but the range of $W$ has been changed. (c) Same as in (a) but for $\mu =-4/3$. (d) Same as in (b) but for $\mu =-4/3$.

Figure 5

The relative error between ${\mathrm{\Phi }}_W$ and $\mathrm{\Phi }$, for $T=0,\mu =-4/3$, and $M=-1$. We have fitted along the interval $500\le W\le 550.$

Figure 6

(a) The relative error between ${\mathrm{\Phi }}_W$ and $\mathrm{\Phi }$, for $T=0,\mu =0$, and $M=0$. We have fitted along the interval $90\le W\le 100.$ It can be seen that the linear regime has not set in for this system size. (b) The same as in (a), but for $1000\le W\le 1010.$ The linear regime has set in, and the error is negligible.

Figure 7

(a) The relative error between ${\mathrm{\Phi }}_W$ and $\mathrm{\Phi }$, for $k_{B}T=1/100,\mu =-2$, and $M=1$. We have fitted along the interval $50\le W\le 100$. The errors are of the same order as in Fig. 4. (b) The same as in (a), but for $k_{B}T=1/10$. The errors are significantly smaller, since the temperature effectively smooths out the energy spectrum.