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.
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 consider an isotropic Fermi liquid in two dimensions near the n=2 Pomeranchuk instability in the charge channel. The order parameter is a quadrupolar stress tensor with two bosonic shear modes with polarizations longitudinal and transverse to the quadrupolar momentum tensor. Longitudinal and transverse bosonic modes are characterized by dynamical exponents ${z}_{\ensuremath{\parallel}}=3$ and ${z}_{\ensuremath{\perp}}=2$, respectively. Previous studies have found that such a system exhibits multiscale quantum criticality with two different energy scales $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}{z}_{\ensuremath{\parallel},\ensuremath{\perp}}}$, where $\ensuremath{\xi}$ is the correlation length. We study the impact of the multiple energy scales on the electron Green's function. The interaction with the critical ${z}_{\ensuremath{\parallel}}=3$ mode is known to give rise to a local self-energy that develops a non-Fermi-liquid form, $\ensuremath{\Sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2/3}$ for frequencies larger than the energy scale $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}3}$. We find that the exchange of transverse ${z}_{\ensuremath{\perp}}=2$ fluctuations leads to logarithmically singular renormalizations of the quasiparticle residue Z and the vertex $\ensuremath{\Gamma}$. We derive and solve renormalization-group equations for the flow of Z and $\ensuremath{\Gamma}$, and show that the system develops an anomalous dimension at the nematic quantum critical point (QCP). As a result, the spectral function at a fixed $\ensuremath{\omega}$ and varying k has a non-Lorentzian form. Away from the QCP, we find that the flow of Z is cut at the energy scale ${\ensuremath{\omega}}_{\text{FL}}\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}1}$, associated with the z=1 dynamics of electrons. The ${z}_{\ensuremath{\perp}}=2$ energy scale, $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}2}$, affects the flow of Z only if one includes into the theory self-interaction of transverse fluctuations.
We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic (n=2) Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ for a model case of the long-range interaction in the nematic channel. Already within the random-phase approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue Z and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, Z, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where gc,2 is the n=2 charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but gc,2 Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction D with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.
We study the problem of the damping of collective modes close to a Pomeranchuk quantum critical point in a Fermi liquid. In analogy with problems in dissipative open quantum systems, we derive the Landau damping of a Fermi liquid by integrating out a macroscopic number of degrees of freedom from a generating functional. Being a reformulation of the linearized Boltzmann equation, this approach reproduces well-known results from the theory of Fermi liquids. We also study the Bethe-Salpeter equations within the Landau theory and discuss the implications of these results on quantum phase transitions of the Pomeranchuk type and its dynamical exponent, z. We apply our results to the electronic nematic instability and find z=3 in the collisionless limit.
We present a functional renormalization-group calculation of the properties of a quantum critical metal in d=2 spatial dimensions. Our theory describes a general class of Pomeranchuk instabilities with Nb flavors of boson. At small Nb we find a family of fixed points characterized by weakly non-Fermi-liquid behavior of the conduction electrons and $z\ensuremath{\approx}2$ critical dynamics for the order-parameter fluctuations, in agreement with the scaling observed by Schattner et al. [Phys. Rev. X 6, 031028 (2016)] for the Ising-nematic transition. Contrary to recent suggestions that this represents an intermediate regime en route to the scaling limit, our calculation suggests that this behavior may persist all the way to the critical point. As the number of bosons Nb is increased, the model's fixed-point properties cross over to $z\ensuremath{\approx}1$ scaling and non-Fermi-liquid behavior similar to that obtained by Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].
The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in powers of the relevant dimensionless electron-phonon coupling ${\ensuremath{\lambda}}_{0}$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of ${\ensuremath{\lambda}}_{0}$ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions d>3 we find that the RG flow of the Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to the well-known ultraviolet stable fixed point of ${\ensuremath{\phi}}^{3}$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling ${\ensuremath{\lambda}}_{0}$ and the adiabatic ratio ${\ensuremath{\omega}}_{0}/{\ensuremath{\epsilon}}_{F}$ have to be fine-tuned to assume critical values of order unity, where ${\ensuremath{\omega}}_{0}$ is the phonon frequency and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. However, for dimensions $d\ensuremath{\le}3$ we find that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in $d\ensuremath{\le}3$.