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arXiv:0907.0489 · Physical Review B

Multiscale quantum criticality: Pomeranchuk instability in isotropic metals

Margit Zacharias, P. Wölfle, Markus Garst

Published Oct 12, 2009• 57 citations
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Abstract

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.

DOI
10.1103/physrevb.80.165116

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As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, d=2d=2d=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=2z=2z=2 and the other one is Landau damped with z=3z=3z=3, thus giving rise to multiple dynamical scales. We find that at temperature T=0T=0T=0, the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension d+zd+zd+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 TTT 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=3z=3z=3 mode, the critical compressibility is found to be dominated by the z=2z=2z=2 fluctuations.
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.
We consider an isotropic Fermi liquid in two dimensions near the n=2n=2n=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 ZZZ and the vertex $\ensuremath{\Gamma}$. We derive and solve renormalization-group equations for the flow of ZZZ 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 kkk has a non-Lorentzian form. Away from the QCP, we find that the flow of ZZZ is cut at the energy scale ${\ensuremath{\omega}}_{\text{FL}}\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}1}$, associated with the z=1z=1z=1 dynamics of electrons. The ${z}_{\ensuremath{\perp}}=2$ energy scale, $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}2}$, affects the flow of ZZZ 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)(n=2)(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 ZZZ 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 ${Z}_{\ensuremath{\Gamma}}$, ZZZ, 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{g}_{c,2}gc,2​ is the n=2n=2n=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{g}_{c,2}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 DDD with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.
A metallic electron system near an orientational symmetry breaking Pomeranchuk instability is characterized by a ``soft'' Fermi surface with enhanced collective fluctuations. We analyze fluctuation effects in a two-dimensional electron system on a square lattice in the vicinity of a Pomeranchuk instability with ddd-wave symmetry, using a phenomenological model which includes interactions with a small momentum transfer only. We compute the dynamical density correlations with a ddd-wave form factor for small momenta and frequencies, the dynamical effective interaction due to a fluctuation exchange, and the electron self-energy. At the quantum critical point the density correlations and the dynamical forward scattering interaction diverge with a dynamical exponent z=3z=3z=3. The singular forward scattering leads to large self-energy corrections, which destroy Fermi liquid behavior over the whole Fermi surface except near the Brillouin zone diagonal. The decay rate of single-particle excitations, which is related to the width of the peaks in the spectral function, exceeds the excitation energy in the low-energy limit. The dispersion of maxima in the spectra flattens strongly near those portions of the Fermi surface which are remote from the zone diagonal. The contribution from classical fluctuations to the self-energy spoils $(\ensuremath{\omega}∕T)$ scaling in the quantum critical regime.
We present a functional renormalization-group calculation of the properties of a quantum critical metal in d=2d=2d=2 spatial dimensions. Our theory describes a general class of Pomeranchuk instabilities with Nb{N}_{b}Nb​ flavors of boson. At small Nb{N}_{b}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{N}_{b}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)].
Quantum critical systems with multiple dynamics possess not only one but several time scales, ${\ensuremath{\tau}}_{i}\ensuremath{\sim}{\ensuremath{\xi}}^{{z}_{i}}$, which diverge with the correlation length $\ensuremath{\xi}$. We investigate how scaling predictions are modified for the simplest case of multiple dynamics characterized by two dynamical critical exponents, z>{z}_{>}z>​ and z<{z}_{<}z<​. We argue that one should distinguish the case of coupled and decoupled multiple dynamic scaling depending on whether there exists a scaling exponent which depends on both zi{z}_{i}zi​. As an example, we study generalized ${\ensuremath{\Phi}}^{4}$ theories with multiple dynamics below their upper critical dimension, d+z<<4d+{z}_{<}<4d+z<​<4. We identify under which condition coupled scaling is generated. In this case the interaction of quantum and classical fluctuations leads to an emergent dynamical exponent, ${z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}$.