Ising-nematic Quantum Critical Point Research Articles

Fermi surface geometry and optical conductivity of a two-dimensional electron gas near an Ising-nematic quantum critical point

Fermi surface geometry and optical conductivity of a two-dimensional electron gas near an Ising-nematic quantum critical point

Optical conductivity of a metal near an Ising-nematic quantum critical point

Optical conductivity of a metal near an Ising-nematic quantum critical point

Zero sound and plasmon modes for non-Fermi liquids

We derive the quantum Boltzmann equation (QBE) by using generalized Landau-interaction parameters, obtained through the nonequilibrium Green's function technique. This is a generalization of the usual QBE formalism to non-Fermi liquid (NFL) systems, which do not have well-defined quasiparticles. We apply this framework to a controlled low-energy effective field theory for the Ising-nematic quantum critical point, in order to find the collective excitations of the critical Fermi surface in the collisionless regime. We also compute the nature of the dispersion after the addition of weak Coulomb interactions. The zero angular momentum longitudinal vibrations of the Fermi surface show a linear-in-wavenumber dispersion, which corresponds to the zero sound of Landau's Fermi liquid theory. The Coulomb interaction modifies it to a plasmon mode in the long-wavelength limit, which disperses as the square-root of the wavenumber. Remarkably, our results show that the zero sound and plasmon modes exhibit the same behaviour as in a Fermi liquid, although an NFL is fundamentally different from the former.

Low-frequency Raman response near the Ising-nematic quantum critical point: A memory-matrix approach

Recent Raman scattering experiments have revealed a "quasi-elastic peak" in $\mathrm{FeSe_{1-x}S_x}$ near an Ising-nematic quantum critical point (QCP) \cite{zhang17}. Notably, the peak occurs at sub-temperature frequencies, and softens as $T^{\alpha}$ when temperature is decreased toward the QCP, with $\alpha>1$. In this work, we present a theoretical analysis of the low-frequency Raman response using a memory matrix approach. We show that such a quasi-elastic peak is associated with the relaxation of an Ising-nematic deformation of the Fermi surface. Specifically, we find that the peak frequency is proportional to $ \tau^{-1}\chi^{-1}$, where $\chi$ is the Ising-nematic thermodynamic susceptibility, and $\tau^{-1}$ is the decay rate of the nematic deformation due to an interplay between impurity scattering and electron-electron scattering mediated by critical Ising-nematic fluctuations. We argue that the critical fluctuations play a crucial role in determining the observed temperature dependence of the frequency of the quasi-elastic peak. At frequencies larger than the temperature, we find that the Raman response is proportional to $\omega^{1/3}$, consistently with earlier predictions \cite{klein18a}.

Specific Heat of a Quantum Critical Metal.

We investigate the specific heat c, near an Ising nematic quantum critical point (QCP), using sign problem-free quantum MonteCarlo simulations. Cooling towards the QCP, we find a broad regime of temperature where c/T is close to the value expected from the noninteracting band structure, even for a moderately large coupling strength. At lower temperature, we observe a rapid rise of c/T, followed by a drop to zero as the system becomes superconducting. The spin susceptibility begins to drop at roughly the same temperature where the enhancement of c/T onsets, most likely due to the opening of a gap associated with superconducting fluctuations. These findings suggest that superconductivity and non-Fermi liquid behavior (manifested in an enhancement of the effective mass) onset at comparable energy scales. We support these conclusions with an analytical perturbative calculation.

DC resistivity near a nematic quantum critical point: Effects of weak disorder and acoustic phonons

We calculate the resistivity associated with an Ising-nematic quantum critical point in the presence of disorder and acoustic phonons in the lattice model. To perform this analysis, we use the memory-matrix transport theory, which has a crucial advantage compared to other methods of not relying on the existence of well-defined quasiparticles in the low-energy effective theory. As a result, we obtain that by including an inevitable interaction between the nematic fluctuations and the elastic degrees of freedom of the lattice (parametrized by the nemato-elastic coupling κlatt), the resistivity ρ(T) of the system as a function of temperature obeys a universal scaling form described by ρ(T)∼Tln(1∕T) at high temperatures, reminiscent of the paradigmatic strange metal regime observed in many strongly correlated compounds. For a window of temperatures comparable with κlatt3∕2εF (where εF is the Fermi energy of the microscopic model), the system displays another regime in which the resistivity is consistent with a description in terms of ρ(T)∼Tα, where the effective exponent roughly satisfies the inequality 1≲α≲2. However, in the low-temperature limit (i.e., T≪κlatt3∕2εF), the properties of the quantum critical state change in an important way depending on the types of disorder present in the system: It can either recover a Fermi-liquid-like regime described by ρ(T)∼T2 or it could exhibit yet another non-Fermi liquid regime characterized by the scaling form ρ(T)−ρ0∼T2lnT (implying in the latter case that the system would display a Kondo-like upturn in the resistivity). From a broader perspective, our results emphasize the key role played by both phonon and disorder effects in the scenario of nematic quantum criticality and might be fundamental for addressing recent transport experiments in some iron-based superconductors.

Scattering mechanisms and electrical transport near an Ising nematic quantum critical point

Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\rho\sim T^2$ at low temperature to sublinear at high temperature. The crossover temperature scales as $q_0^3$, where $q_0$ is the minimal wavevector for umklapp scattering. Impurity scattering leads to $\rho-\rho_0\sim T^\alpha$ ($\rho_0$ being the residual resistivity), where $\alpha$ is either larger than 2 if there is only a single Fermi sheet present, or 4/3 in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low temperature behavior depends strongly on the structure of "cold spots" on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\rho\sim T^{5/3}$. At higher temperatures there is a broad crossover regime where $\rho$ either saturates or $\rho\sim T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.

Dynamical susceptibility near a long-wavelength critical point with a nonconserved order parameter

We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform ($\mathbf{Q}=0$) Ising nematic quantum critical point of $d-$wave symmetry. The nematic order parameter is not a conserved quantity, and this permits a nonzero value of the fermionic polarization in the $d-$wave channel even for vanishing momentum and finite frequency: $\Pi(\mathbf{q} = 0,\Omega_m) \neq 0$. For weak coupling between the fermions and the nematic order parameter (i.e. the coupling is small compared to the Fermi energy), we perturbatively compute $\Pi (\mathbf{q} = 0,\Omega_m) \neq 0$ over a parametrically broad range of frequencies where the fermionic self-energy $\Sigma (\omega)$ is irrelevant, and use Eliashberg theory to compute $\Pi (\mathbf{q} = 0,\Omega_m)$ in the non-Fermi liquid regime at smaller frequencies, where $\Sigma (\omega) > \omega$. We find that $\Pi(\mathbf{q}=0,\Omega)$ is a constant, plus a frequency dependent correction that goes as $|\Omega|$ at high frequencies, crossing over to $|\Omega|^{1/3}$ at lower frequencies. The $|\Omega|^{1/3}$ scaling holds also in a non-Fermi liquid regime. The non-vanishing of $\Pi (\mathbf{q}=0, \Omega)$ gives rise to additional structure in the imaginary part of the nematic susceptibility $\chi^{''} (\mathbf{q}, \Omega)$ at $\Omega > v_F q$, in marked contrast to the behavior of the susceptibility for a conserved order parameter. This additional structure may be detected in Raman scattering experiments in the $d-$wave geometry.

Shear viscosity at the Ising-nematic quantum critical point in two-dimensional metals

In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $(k_B /\hbar)\, \eta/s$, where $\eta$ is the shear viscosity and $s$ is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension $d=2$ by an expansion below $d=5/2$. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\eta$ scales in the same manner as a chiral conductivity, and the ratio $\eta/s$ diverges at low temperature ($T$) as $T^{-2/z}$, where $z$ is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.

UV/IR mixing in non-Fermi liquids: higher-loop corrections in different energy ranges

We revisit the Ising-nematic quantum critical point with an $m$-dimensional Fermi surface by applying a dimensional regularization scheme, introduced in Phys. Rev. B 92, 035141 (2015). We compute the contribution from two-loop and three-loop diagrams in the intermediate energy range controlled by a crossover scale. We find that for $m=2 $, the corrections continue to be one-loop exact for both the infrared and intermediate energy regimes.

Finite-temperature scaling close to Ising-nematic quantum critical points in two-dimensional metals

We study finite temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent $z=2$, in contrast to $z=3$ found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\Omega/\gamma(T)$ at small momenta, where $\gamma(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte-Carlo simulations at finite temperature, where results consistent with $z=2$ were found.

Ising Nematic Quantum Critical Point in a Metal: A Monte Carlo Study

The Ising nematic quantum critical point (QCP) associated with the zero temperature transition from a symmetric to a nematic {\it metal} is an exemplar of metallic quantum criticality. We have carried out a minus sign-free quantum Monte Carlo study of this QCP for a two dimensional lattice model with sizes up to $24\times 24$ sites. The system remains non-superconducting down to the lowest accessible temperatures. The results exhibit critical scaling behavior over the accessible ranges of temperature, (imaginary) time, and distance. This scaling behavior has remarkable similarities with recently measured properties of the Fe-based superconductors proximate to their putative nematic QCP.

Hyperscaling violation at the Ising-nematic quantum critical point in two-dimensional metals

Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly-coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single non-zero exponent $\theta$, so that in a spatially isotropic state in $d$ spatial dimensions, the specific heat scales with temperature as $T^{(d-\theta)/z}$, and the optical conductivity scales with frequency as $\omega^{(d-\theta-2)/z}$ for $\omega \gg T$, where $z$ is the dynamic critical exponent. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee (Phys. Rev. B {\bf 88}, 245106 (2013)). We find that hyperscaling is violated, with $\theta =1$ in $d=2$. We expect that similar results apply to Fermi surfaces coupled to gauge fields in $d=2$.

Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering

We analyze general properties of the perturbation expansion for two-dimensional quantum critical metals with singular forward scattering, such as metals at an Ising nematic quantum critical point and metals coupled to a U(1) gauge field. We derive asymptotic properties of fermion loops appearing as subdiagrams of the contributing Feynman diagrams -- for large and small momenta. Substantial cancellations are found in important scaling limits, which reduce the degree of divergence of Feynman diagrams with boson legs. Implementing these cancellations we obtain improved power-counting estimates that yield the true degree of divergence. In particular, we find that perturbative contributions to the boson self-energy are generally ultraviolet convergent for a dynamical critical exponent $z<3$, and divergent beyond three-loop order for $z \geq 3$.

Anomalous dynamical scaling from nematic and U(1) gauge field fluctuations in two-dimensional metals

We analyze the scaling theory of two-dimensional metallic electron systems in the presence of critical bosonic fluctuations with small wave vectors, which are either due to a U(1) gauge field, or generated by an Ising nematic quantum critical point. The one-loop dynamical exponent z=3 of these critical systems was shown previously to be robust up to three-loop order. We show that the cancellations preventing anomalous contributions to z at three-loop order have special reasons, such that anomalous dynamical scaling emerges at four-loop order.

Nematic fluctuations and their wave vector in two-dimensional metals

We revisit the problem of electrons on a square lattice below half filling close to an Ising-nematic quantum critical point. For Fermi surfaces with sufficiently strong antinodal nesting, the static nematic susceptibility is maximal at the antinodal nesting wave vector within a simple RPA calculation. We present a detailed analysis of the nematic susceptibility within Eliashberg theory and show that the strong interaction between Fermions in the antinodal regions shifts the maximum of the nematic susceptibility to slightly larger wave vectors. The corresponding order is akin to the incommensurate charge-density wave with d-wave form factor found recently in some underdoped cuprate materials. At sufficiently high temperatures around $T/t \sim 0.1$ nematic fluctuations are strongest at zero wave vector.

Cooper pairing in non-Fermi liquids

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles arise in a number of physical systems. Examples include: ${\it (i)}$ quantum critical points associated with the onset of order in metals; ${\it (ii)}$ spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator; ${\it (iii)}$ Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this work, we use renormalization group techniques to investigate possible instabilities of such non-Fermi-liquids in two spatial dimensions to Cooper pairing. We consider the Ising-nematic quantum critical point as an example of a phase transition in a metal, and demonstrate that the attractive interaction mediated by the order parameter fluctuations always leads to a superconducting instability. Moreover, in the regime where our calculation is controlled, superconductivity preempts the destruction of electronic quasiparticles. On the other hand, the spinon Fermi-surface and the Halperin-Lee-Read states are stable against Cooper pairing for a sufficiently weak attractive short-range interaction; however, once the strength of attraction exceeds a critical value, pairing sets in. We describe the ensuing quantum phase transition between ${\it (i)}$ $U(1)$ and $Z_2$ spin-liquid states; ${\it (ii)}$ Halperin-Lee-Read and Moore-Read states.

Transport near the Ising-nematic quantum critical point of metals in two dimensions

We consider two-dimensional metals near a Pomeranchuk instability which breaks 90$^\circ$ lattice rotation symmetry. Such metals realize strongly-coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation, and hence a non-zero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the $d$-wave channel acts as a random "field" on the Ising-nematic order. We find contributions to the resistivity with a nearly linear temperature dependence, along with more singular terms; the most singular is the random-field contribution which diverges in the limit of zero temperature.

Functional renormalization group approach to the Ising-nematic quantum critical point of two-dimensional metals

Using functional renormalization group methods, we study an effective low-energy model describing the Ising-nematic quantum critical point in two-dimensional metals. We treat both gapless fermionic and bosonic degrees of freedom on equal footing and explicitly calculate the momentum and frequency dependent effective interaction between the fermions mediated by the bosonic fluctuations. Following earlier work by S.-S. Lee for a one-patch model, Metlitski and Sachdev [Phys. Rev. B {\bf{82}}, 075127] recently found within a field-theoretical approach that certain three-loop diagrams strongly modify the one-loop results, and that the conventional 1/N expansion breaks down in this problem. We show that the singular three-loop diagrams considered by Metlitski and Sachdev are included in a rather simple truncation of the functional renormalization group flow equations for this model involving only irreducible vertices with two and three external legs. Our approximate solution of these flow equations explicitly yields the vertex corrections of this problem and allows us to calculate the anomalous dimension $\eta_{\psi}$ of the fermion field.