We investigate the low-energy effective theory of a Fermi surface (FS) coupled to an Ising-nematic quantum critical point in (2 + 1) spacetime dimensions with translation symmetry. We formulate the system using the large
N Yukawa-Sachdev-Ye-Kitaev model, whose saddle point is described by the Migdal-Eliashberg equations. The low-energy physics can be revealed by studying the Gaussian fluctuation spectrum around the saddle point, which is generated by the Bethe-Salpeter (BS) kernel
KBS. Based on the Ward identities, we propose an inner product on the space of two point functions, which reveals a large number of soft modes of
KBS. These soft modes parametrize deformation of the Fermi surface, and their fluctuation eigenvalues describe their decay rates. We analytically compute these eigenvalues for a circular Fermi surface, and we discover the odd-parity modes to be parametrically longer-lived than the even-parity modes, due to the kinematic constraint of fermions scattering on a convex FS. The sign of the eigenvalues signals an instability of the Ising-nematic quantum critical point at zero temperature for a convex Fermi surface. At finite temperature, the system can be stabilized by thermal fluctuations of the critical boson. We derive an effective action that describes the soft-mode dynamics, and it leads to a linearized Boltzmann equation, where the real part of the soft-mode eigenvalues can be interpreted as the collision rates. The structure of the effective action is similar to the theory of linear bosonization of a Fermi surface. As an application of the formalism, we investigate the hydrodynamic transport of non-Fermi liquid. Analyzing the Boltzmann equation, we obtain a conventional hydrodynamic transport regime and a tomographic transport regime. In both regimes, the conductance of the system in finite geometry can be a sharp indicator for the soft-mode dynamics and non-Fermi liquid physics.
The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are
O(1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.
This paper constitutes a sequel to our theoretical efforts to determine the nature of generic low-energy deformations of the Fermi surface of a quantum-critical metal, which arises at the stable non-Fermi liquid (NFL) fixed point of a quantum phase transition. The emergent critical Fermi surface, arising right at the Ising-nematic quantum critical point (QCP), is a paradigmatic example where an NFL behaviour is induced by the strong interactions of the fermionic degrees of freedom with those of the bosonic order parameter. It is an artifact of the bosonic modes becoming massless at the QCP, thus undergoing Landau-damping at the level of one-loop self-energy. We resort to the well-tested formalism of the quantum Boltzmann equations (QBEs) for identifying the excitations. While in our earlier works, we have focused on the collisionless regime by neglecting the collision integral and assuming the bosons to be in equilibrium, here we embark on a full analysis. In particular, we take into account the bosonic part of the QBEs as well, which, however, turn out to have no effect on the solutions. Decomposing the master equation into angular-momentum (
ℓ) channels, the emergent modes are of two types: Fermi-surface deformations with discrete spectra and particle-hole excitations forming a continuous band. The long-lived zero-sound mode, which corresponds to
ℓ=0, is found to be robust against damping effects. Intriguingly, we have an infinite family of discrete modes corresponding to higher-order harmonics of the net deformation.
We consider the problem of fermions interacting with gapless long-wavelength collective bosonic modes. The theory describes, among other cases, a ferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering. We construct a controllable expansion at the QCP in two steps: we first create a non-Fermi-liquid ``zero-order'' Eliashberg-type theory, and then demonstrate that the residual interaction effects are small. We prove that this approach is justified under two conditions: the interaction should be smaller than the fermionic bandwidth, and either the band mass
mB should be much smaller than
m=kF∕vF, or the number of fermionic flavors
N should be large. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashberg theory itself includes a set of singular renormalizations which can be understood as a consequence of an effective long-range dynamic interaction between quasiparticles, generated by the Landau damping term. These singular renormalizations give rise to a negative nonanalytic
q3∕2 correction to the static spin susceptibility, and destroy a ferromagnetic QCP. We demonstrate that this effect can be understood in the framework of the ${\ensuremath{\phi}}^{4}$ theory of quantum criticality. We also show that the nonanalytic
q3∕2 correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the
q3∕2 contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.
Interactions in Fermi systems can generate a Pomeranchuk instability leading to orientational symmetry breaking, that is, nematic order. In a metallic system close to such an instability the Fermi surface is easily deformed by anisotropic perturbations and exhibits enhanced collective fluctuations. We analyze fluctuation effects in the quantum critical regime near a d-wave Pomeranchuk instability in two dimensions. Density correlations with a d-wave form factor and the dynamical forward scattering interaction diverge near the instability. The singular forward scattering leads to large self-energy contributions, which destroy Fermi liquid behavior over the whole Fermi surface except at “cold spots” on 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 the spectra flattens strongly near those portions of the Fermi surface which are remote from the zone diagonal. The decay rate for DC transport is linear in temperature except at the cold spots. We discuss the possible relevance of d-wave Fermi surface fluctuations for the “strange metal” behavior observed in the normal phase of cuprates.
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.