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arXiv:2406.12967 · Physical review. B./Physical review. B

Fluctuation spectrum of critical Fermi surfaces

Haoyu Guo

Published Oct 15, 2024• 9 citations
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Abstract

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 ${K}_{\mathrm{BS}}$. 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 ${K}_{\mathrm{BS}}$. 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.

DOI
10.1103/physrevb.110.155130

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arXiv:2306.03898•Physical review. B./Physical review. B•Sep 28, 2023•20 citations
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 NNN 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{K}_{\mathrm{BS}}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{K}_{\mathrm{BS}}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.
We diagnose the stability of the Migdal-Eliashberg theory for a Fermi surface coupled to a gapless boson in 2+1 dimensions. We provide a scheme for diagonalizing the Bethe-Salpeter ladder when small-angle scattering mediated by the boson plays a dominant role. We found a large number of soft modes which correspond to shape fluctuations of the Fermi surface, and these shape deformations follow a diffusion-like dynamics on the Fermi surface. Surprisingly, the odd-parity deformations of a convex Fermi surface becomes unstable near the non-Fermi liquid regime of the Ising-Nematic quantum critical point and our finding calls for revisit of the Migdal-Eliashberg framework. The implication of the Bethe-Salpeter eigenvalues in transport will be discussed in the companion paper [H.Guo,arXiv:2311.03458].
We extend the kinetic operator formalism developed in the companion paper [H.Guo,arXiv:2311.03455] to study the general eigenvalues of the fluctuation normal modes. We apply the formalism to calculate the optical conductivity of a critical Fermi surface near the Ising-Nematic quantum critical point. We find that the conductivity is the sum of multiple conduction channels including both the soft and non-soft eigenvectors of the kinetic operator, and therefore it is not appropriate to interpret the optical conductivity using extended Drude formula for momentum conserved systems. We also show that the propagation of the FS soft modes is governed by a Boltzmann equation from which hydrodynamics emerges. We calculate the viscosity and it shows clear signature of the non-Fermi liquid physics.
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.
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\ellℓ) 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\ell = 0ℓ=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.
Role of quenched randomness in metallic quantum criticality is one of the long standing problems in condensed matter physics. An aspect of the fundamental difficulties lies in the fact that such nonmagnetic disorders lead effective interactions between abundant soft modes near the Fermi surface to be drastically enhanced particularly in the diffusive regime, where the perturbative framework does not work. Here, we revisit the problem of dirty quantum critical metals in a different angle, focusing on the hydrodynamic regime instead of the diffusive regime near the non-Fermi liquid quantum critical point. More concretely, we study effects of mutual correlations between quantum critical nematic fluctuations and weak localization corrections, and show the existence of a weak-disorder non-Fermi liquid fixed point, based on the renormalization group (RG) analysis up to the two-loop order. The two-loop order RG analysis suggests that the absence of quantum coherence in two-particle composite excitations weakens the role of weak localization corrections and allows a weakly disordered non-Fermi liquid metallic state in the hydrodynamic regime of the nematic quantum critical point. Although this dirty non-Fermi liquid metallic state may not be the true infrared stable fixed point at zero temperature, expected to be characterized by the diffusive Ohmic regime, we argue that this weak-disorder non-Fermi liquid metallic fixed point would govern the region of an intermediate energy scale, described by effective hydrodynamics of nematic quantum criticality. In this respect we believe that this research would be an important step in understanding the T−T-T−linear electrical resistivity as a characteristic feature of non-Fermi liquids and the origin of unconventional superconductivity from effective hydrodynamics of quantum criticality.
We study thermalization dynamics in a fermion-phonon variant of the Sachdev-Ye-Kitaev model coupled to an external cold thermal bath of harmonic oscillators. We find that quantum critical fermions thermalize more efficiently than phonons, in sharp contrast to the behavior in the Fermi liquid regime. In addition, after a short prethermal stage, the system acquires a quasithermal distribution given by a time-dependent effective temperature, reminiscent of ``hydrodynamic'' relaxation. All physical observables relax at the same rate which scales with the final temperature through an exponent that depends universally on the low-energy spectrum of the system and the bath. Such relaxation rate is derived using a hydrodynamic approximation in full agreement with the numerical solution of a set quantum kinetic equations derived from the Keldysh formalism for nonequilibrium Green's functions. Our results hint toward further research on the applicability of the hydrodynamic picture in the description of the late time dynamics of open quantum systems despite the absence of conserved quantities in regimes dominated by conserving collisions.