Multiscale quantum criticality: Pomeranchuk instability in isotropic metals

As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, $d=2$. The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations of the Fermi surface consists of two coupled modes, critical at the same point, and characterized by different dynamical exponents: one being ballistic with dynamical exponent $z=2$ and the other one is Landau damped with $z=3$, thus giving rise to multiple dynamical scales. We find that at temperature $T=0$, the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension $d+z$. Its self-interaction leads to logarithmic singularities, which we treat with the help of the renormalization group. At finite temperature, the coexistence of two different dynamical scales gives rise to a modified quantum-to-classical crossover. It extends over a parametrically large regime with intricate interactions of quantum and classical fluctuations leading to a universal $T$ dependence of the correlation length independent of the interaction amplitude. The multiple scales are also reflected in the phase diagram and in the critical thermodynamics. In particular, we find that the latter cannot be interpreted in terms of only a single dynamical exponent whereas, e.g., the critical specific heat is determined by the $z=3$ mode, the critical compressibility is found to be dominated by the $z=2$ fluctuations.

Figure 1

Effective dimensions of the longitudinal and transversal shear fluctuations as a function of momentum $q$. The respective thermal momenta, ${\xi }_T^{-1}\sim T^{1/z}$, separate two regimes where the fluctuations have either a quantum character, $q>{\xi }_T^{-1}$, with effective dimension $d+z$ or a classical character, $q<{\xi }_T^{-1}$, with effective dimension $d=2$. There is an extended quantum-to-classical crossover (gray shaded), where the quantum regime of the transversal $z_{<}=2$ mode overlaps with the classical regime of the longitudinal mode $z_{>}=3$.

Figure 2

Phase diagram for the Pomeranchuk quantum phase transition in $d=2$. The correlation length $\xi$ and the two thermal lengths, ${\xi }_T^{>}\sim T^{-1/3}$ and ${\xi }_T^{<}\sim T^{-1/2}$, distinguish three regimes: (I) a Fermi-liquid regime where $\xi$ is the smallest scale, (II) an overlap regime where $\xi$ is sandwiched between the two thermal lengths, and (III) a quantum critical regime where $\xi \sim {\xi }_T^{>}$ is the largest scale. The line labeled $T_G\left(r\right)$ represents the Ginzburg temperature.

Figure 3

Contributions to the quadrupolar polarization of the Fermi liquid. Solid lines are fermion propagators, the dashed lines are susceptibilities of shear fluctuations, and the vertex is given by the quadrupolar gradient tensor ${\widehat{Q}}_{ij}$. (a) polarization of free fermions, (b) and (c) lowest-order corrections.

Figure 4

(Color online) Visualization of a quadrupolar fluctuation (black solid line) of the $2d$ isotropic Fermi sphere (gray solid line). (a) The longitudinal mode is damped by exciting particle-hole pairs close to the Fermi surface, Eq. (15). (b) For the transversal mode the momentum $\mathbf{q}$ of the fluctuation is close to a node such that there is not sufficient phase space for Landau damping, Eq. (16).

Figure 5

One-loop corrections for the effective theory [Eq. (18)] of the shear modes.

Figure 6

(a) Fermion self-energy and (b) vertex correction of the model [Eq. (A1)]. For clarity, we dropped the frequency labeling.