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arXiv:2203.08641 · Physical review. B./Physical review. B

Phonon renormalization and Pomeranchuk instability in the Holstein model

Niklas Cichutek, Max Hansen, Peter Kopietz

Published May 31, 2022• 5 citations
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Abstract

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 $d\ensuremath{\le}3$.

DOI
10.1103/physrevb.105.205148

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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 \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions d>3d>3d>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 \ensuremathϕ3{\ensuremath{\phi}}^{3}\ensuremathϕ3-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ and the adiabatic ratio \ensuremathω0/\ensuremathϵF{\ensuremath{\omega}}_{0}/{\ensuremath{\epsilon}}_{F}\ensuremathω0​/\ensuremathϵF​ have to be fine-tuned to assume critical values of order unity, where \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ is the phonon frequency and \ensuremathϵF{\ensuremath{\epsilon}}_{F}\ensuremathϵF​ is the Fermi energy. However, for dimensions d\ensuremath≤3d\ensuremath{\le}3d\ensuremath≤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≤3d\ensuremath{\le}3d\ensuremath≤3.
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.
The stability of a quantum critical point in the O(N)O(N)O(N) universality class with respect to an elastic coupling, that preserves O(N)O(N)O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension d=3−εd=3-εd=3−ε. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for 1≤N≤41\leq N \leq 41≤N≤4, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length ξξξ. For N>4N > 4N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached the sound velocity is found to vanish only asymptotically as ξ→∞ξ\to \inftyξ→∞ such that the crystal remains microscopically stable for any finite value of ξξξ. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in d=4−εd=4-εd=4−ε, where the crystal always remains microscopically stable for finite ξξξ.
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)].
We study the interplay between critical isotropic elasticity and classical Ising criticality using a functional renormalization group (FRG) approach which is implemented such that the volume is fixed during the entire renormalization group flow. For dimensions slightly smaller than four, we use a simple truncation of the FRG flow equations to recover the fixed points of the constrained Ising model: the Gaussian fixed point G, the Ising fixed point I, the renormalized Ising fixed point R, and the spherical fixed point S. We show that the fixed points R and S are both characterized by a finite anomalous dimension y_{*}<0 of strain fluctuations, implying that the energy dispersion of longitudinal acoustic phonons exhibits a nonanalytic momentum dependence proportional to k^{1-y_{*}/2} for small momentum k. We also derive and solve flow equations for the free energy at constant strain and compute stress-strain relations in the vicinity of the fixed points. As a result, we reaffirm that Ising criticality, controlled by the fixed point I, is preempted by a bulk instability. Beyond that, we find that the stress-strain relation at R and S remains linear to leading order (Hooke's law), as long as the interaction between strain and Ising fluctuations is sufficiently weak. However, the finite anomalous dimension of strain fluctuations y_{*} gives rise to nonanalytic corrections to Hooke's law.
The stability of a quantum critical point in the O(N)O(N)O(N) universality class with respect to an elastic coupling, which preserves O(N)O(N)O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension $d=3\ensuremath{-}\ensuremath{\epsilon}$. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for $1\ensuremath{\le}N\ensuremath{\le}4$, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length $\ensuremath{\xi}$. For N>4N>4N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached, the sound velocity is found to vanish only asymptotically as $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$, such that the crystal remains microscopically stable for any finite value of $\ensuremath{\xi}$. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in $d=4\ensuremath{-}\ensuremath{\epsilon}$, where the crystal always remains microscopically stable for finite $\ensuremath{\xi}$.
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.