Critical theory of Pomeranchuk transitions via high-dimensional bosonization

Abstract

We use high-dimensional bosonization to derive an effective field theory that describes the Pomeranchuck transition in isotropic two-dimensional Fermi liquids. We find that the transition is triggered by the softening of an eigenmode that leads to spontaneous Fermi surface distortion. The resultant theory in terms of this critical mode has dynamical critical exponent $z = 2$ and the upper critical dimension is $d_c = 4-z= 2$. As a result the system is at the upper critical dimension in 2D, resulting in a Gaussian fixed point with a marginally irrelevant quartic perturbation.

Figures (2)

• Figure 1: Fluctuations and instabilities of the Fermi surface. (a) Fields $\rho(\mathbf{x}, \theta)$ are local fluctuations of the Fermi surface (solid line) from the circular reference Fermi surface (dashed line) that corresponds to the isotropic ground state. (b) Pomeranchuk instability via condensation of $\rho$ in the $\ell = 2$ channel, resulting in a nematic state with an elongated Fermi surface (red solid line). $\mathbf{n}$ is a unit vector indicating the orientation of the nematic Fermi surface, which is equivalent to $-\mathbf{n}$.
• Figure 2: Schematic representations of Fermi liquid modes close to the Pomeranchuk instability. (a) After scaling out the $|\textbf{q}|^2$ factor, the Hamiltonian $M'_{\ell\ell'}$ resembles a soft oscillator $\Phi$ weakly coupled to an infinite transmission line. (b) Near criticality ($g_2\rightarrow 0$), the oscillators in the bath are coupled infinitely strongly in units of $g_2$, thus effectively forming a single block with infinite mass. (c) The spectral function $A(\omega)$ for the soft mode for different $g_2$ and (d) quality factors $Q(g_2)$ of our theory versus Hertz-type theories.