Detecting topological quantum phase transitions via the c-function

We propose the c-function as a new and accurate probe to detect the location of topological quantum critical points. As a direct application, we consider a holographic model which exhibits a topological quantum phase transition between a topologically trivial insulating phase and a gapless Weyl semimetal. The quantum critical point displays a strong Lifshitz-like anisotropy in the spatial directions, and the quantum phase transition does not follow the standard Landau paradigm. The c-function robustly shows a global feature at the quantum criticality and distinguishes, with great accuracy, the two separate zero temperature phases. Taking into account the relation of the c-function with the entanglement entropy, we conjecture that our proposal is a general feature of quantum phase transitions and that it is applicable beyond the holographic framework.

Figure 1

Sketch of the TQPT considered in this work. For our choice of parameters, the transition appears at a critical value ${\overline{M}}_c\sim 0.744$ between a topologically trivial gapped state and a Weyl semimetal phase.

Figure 2

Anomalous conductivity at low temperature $\overline{T}=0.005$ as a function of the external parameter $\overline{M}$. The dashed line indicates the position of the quantum critical point ${\overline{M}}_c\sim 0.744$. The inset displays the topological band crossing which characterizes the semimetal phase, and $E_f$ is the Fermi energy of the system.

Figure 3

Parallel and transverse c-functions at $\overline{T}=0.005$ as a function of the external parameter $\overline{M}$. The dashed line indicates the position of the quantum critical point, ${\overline{M}}_c\sim 0.744$. Both functions have normalized magnitudes for presentation reasons.

Figure 4

Normalized values of the c-functions at $T=0$ as a function of $\overline{M}$. The black lines indicate the AdS value $c/\beta (d)=2$, normalized with $\beta$. The red and blue lines display the value of $c_{\perp }/{\beta }_{\perp }(d_{\perp }),c_{\parallel }/{\beta }_{\parallel }(d_{\parallel })$ at the quantum critical point where the c-function is clearly discontinuous. The normalizations are for presentation purposes since our main focus is the position and the existence of a saddle point of the c-function at the phase transition.

Figure 5

Effective dimensions $d_{\parallel ,\perp }$ and the EE derivatives $\partial S_{\parallel ,\perp }/\partial l_{\parallel ,\perp }$ as a function of the external parameter $\overline{M}$, around the QCP. The black dots indicate the position of the saddle points, while the vertical line is the location of the QCP. Notice that the effective dimensions overestimate the fixed point, while the derivatives underestimate it. The c-function saddle point takes into account both. Note that for low and large $\overline{M}$ the quantities converge. The comparison scheme used here is for constant entangling length as $\overline{M}$ varies, and the magnitude normalizations are fixed for presentation purposes.

Figure 6

Anomalous conductivity at low temperature with respect to quartic coupling. The dashed lines denote the positions of the QCP. In the small plot the dots correspond to the QCP with respect to the quartic couplings.

Figure 7

The $X$ marks denote QCP computed by the $(c_{\perp },c_{\parallel })$-functions for a varying quartic coupling. The dots are the QCP of the system, and the error bars highlight the uncertainty of their identification due to thermal effects. The error bars are estimated using the value at which the anomalous Hall conductivity approaches the benchmark value ${\sigma }_{AHE}=0.1$. The inset highlights the difference between the actual location of the QCP and the estimate of our probe. The c-functions pinpoint the QCP with great accuracy.