This work considers the quadrupolar Pomeranchuk instability of an isotropic Fermi liquid in d = 2 dimensions. Starting from free electrons subjected to a quadrupolar interaction one can derive an effective Ginzburg-Landau theory for the nematic order parameter [1]. The crucial point of this theory is the presence of two simultanously critical modes, which have different dynamics. There is a ballistic mode with a dynamical exponent z = 2 and a Landau-damped mode with z = 3. At zero temperature, the ballistic mode is at its upper critical dimension and interactions result in logarithmic singularities in perturbation theory. In contrast, at finite temperatures the z = 3 mode dominates thermodynamics, due to the larger available phase space. Trying to analyze the zero temperature theory one has to resum the arising divergences. In general this is done by means of renormalization group (RG) techniques. But due to the presence of two different dynamical exponents there is the problem, how to rescale frequencies. In the usual RG-scheme this rescaling is done according to ω → bzω. Since the model has two different dynamical exponents, the scheme has to be adjusted in the first place. With this RG-scheme the mass correction at zero temperature can be calculated. The resulting universality class is distinct from the Isingas well as from the XY-model. At finite temperatures, there is a complex interplay between the two modes. At criticality, the length scale ξT ∼ T−1/z separates the quantum from the classical regime where each mode has effective dimensions d + z and d, respectively. The presence of two different exponents z yields an overlap regime where the ballistic mode still has its quantum character and the damped mode is already classical. It can be shown that this overlap regime dominates the system at finite temperature and that the theory cannot be reduced to an effective classical model implying a breakdown of ”dimensional reduction”. The interplay of critical modes leads to a universal temperature dependence of the correlation length. Finally, the thermodynamical properties of the system are calculated on a Gaussian level. As expected the higher z = 3 mode dominates the behavior of the specific heat and the thermal expansion. In contrast, the compressibility of the system is governed by the mode with z = 2.
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.
We devise a dimensional regularization scheme for quantum field theories with a Fermi surface to study scaling behavior of non-Fermi liquid states in a controlled approximation. Starting from two space dimensions, the co-dimension of the Fermi surface is extended to a general value while its dimension is fixed. When the Fermi surface is coupled with a critical boson centered at zero momentum, the interaction becomes marginal at a critical space dimension
dc=5/2. A deviation from the critical dimension is used as a small parameter for a systematic expansion. We apply this method to the theory where two patches of the Fermi surface are coupled with a critical boson, and show that the Ising-nematic critical point is described by a stable non-Fermi liquid state slightly below the critical dimension. Critical exponents are computed up to the two-loop order.
We analyze a two-dimensional Pomeranchuk-nematic instability, trigger by the Landau parameter
F2<0, in the presence of a small magnetic field. Using Landau Fermi-liquid theory in the isotropic phase, we analyze the collective modes near the quantum critical point ${F}_{2}=\ensuremath{-}1,{\ensuremath{\omega}}_{c}=0$ (where ${\ensuremath{\omega}}_{c}$ is the cyclotron frequency). We focus on the effects of parity symmetry breaking on the Fermi-surface deformation. We show that the linear response approximation of the Landau-Silin equation is not sufficient to study the critical regime and it is necessary to compute corrections at least of order ${\ensuremath{\omega}}_{c}^{2}$. Identifying the slowest oscillation mode in the disordered phase, we compute the phase diagram for the isotropic/nematic phase transition in terms of
F2 and ${\ensuremath{\omega}}_{c}$.