Turning a First Order Quantum Phase Transition Continuous by Fluctuations: General Flow Equations and Application to $d$-Wave Pomeranchuk Instability

We derive renormalization group equations which allow us to treat order parameter fluctuations near quantum phase transitions in cases where an expansion in powers of the order parameter is not possible. As a prototypical application, we analyze the nematic transition driven by a $d$-wave Pomeranchuk instability in a two-dimensional electron system. We find that order parameter fluctuations suppress the first order character of the nematic transition obtained at low temperatures in mean-field theory, so that a continuous transition leading to quantum criticality can emerge.

Figure 1

Effective potential $U(\varphi )$ for various values of $\Lambda$ between ${\Lambda }_0=e^{-1}$ and 0, at $\mu =-0.78$ and $T=0.05$.

Figure 2

Critical temperatures versus chemical potential for ${\Lambda }_0=e^{-1}$ (larger dome with dots and crosses) and ${\Lambda }_0=1$ (smaller dome with dots). The mean-field transition line is also shown for comparison.

Figure 3

Order parameter ${\varphi }_0$ as a function of $\mu$ at $T=0$ near the left edge of the nematic dome for ${\Lambda }_0=e^{-1}$ and ${\Lambda }_0=1$.