Universal Thermodynamic Signature of Self-Dual Quantum Critical Points

Self-duality is an algebraic structure of certain critical theories, which is not encoded in the scaling dimensions and critical exponents. In this work, a universal thermodynamic signature of self-dual quantum critical points (QCPs) is proposed. It is shown that the Grüneisen ratio at a self-dual QCP remains finite as $T\to 0$, which is in sharp contrast to its universal divergence at a generic QCP without self-duality, $\mathrm{\Gamma }(T,g_c)\sim T^{-1/z\nu }$. This conclusion is drawn based on the hyperscaling theory near the QCP, and has far-reaching implications for experiments and numerical simulations.

Figure 1

The link between the critical theory and critical phenomena. In the general theory of critical phenomena, the universality class is defined based on the critical exponents, which are determined by the scaling dimensions of fields (operators) under the scale (or conformal) transformation. On the other hand, self-duality is an algebraic structure of certain critical theories, which is not encoded in the scaling dimensions. As shown in this work, the self-duality implies that the derivative of the hyperscaling function vanishes at the QCP, and the Grüneisen ratio does not diverge as $T\to 0$. This provides a universal thermodynamic signature of self-dual QCPs.

Figure 2

Left: GR of the Ising domain wall model near its QCP $g_c=1$ on finite-size lattices with periodic boundary condition. The temperature $T=1/L$. Right: $\mathrm{\Gamma }(T,g_c)=1/2$ for any lattice sizes and temperature.

Figure 3

Left: GR near the 2D topological quantum phase transition described by Eq. (19). Parameters: $t_c=1.2$, $t_v=0.8$, and $\mathrm{\Delta }=0.5$. Right: at the QCP, $\mathrm{\Gamma }(T,0)$ saturates as $T\to 0$.

Figure 4

Left: GR near the 1D chemical potential-tuned metal-insulator transition described by Eq. (21). Right: at the QCP, $\mathrm{\Gamma }(T,{\mu }_c)$ diverges as $T\to 0$. The power-law fitting yields $\mathrm{\Gamma }(T,{\mu }_c)=-0.067+0.527/T$ (dashed line).